
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00025733/00001
Material Information
 Title:
 Energy transfer and dissociation in hyperthermal atomdiatom collisions
 Creator:
 Beard, Lyntis Halland, 1951
 Publication Date:
 1979
 Language:
 English
 Physical Description:
 viii, 222 leaves : ill. ; 28 cm.
Subjects
 Subjects / Keywords:
 Approximation ( jstor )
Atomic interactions ( jstor ) Atoms ( jstor ) Boundary conditions ( jstor ) Coordinate systems ( jstor ) Electronics ( jstor ) Energy ( jstor ) Momentum ( jstor ) Potential energy ( jstor ) Wave functions ( jstor ) Chemistry thesis Ph. D Collisions (Nuclear physics) ( lcsh ) Dissertations, Academic  Chemistry  UF Energy transfer ( lcsh )
 Genre:
 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Thesis:
 ThesisUniversity of Florida.
 Bibliography:
 Bibliography: leaves 213221.
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Lyntis H. Beard, Jr.
Record Information
 Source Institution:
 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 023005591 ( ALEPH )
05530164 ( OCLC )

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ENLru"'iY TP;ikSF'R AD JISSOCIAT ION IN HYPERTHERMAL
ATOMDIATOM COLL ISIONS
By
LYNTIS H. EARD, JR.
A DISSERTATION PRESENTED TO THF GRADUATE COUNCIL OF
THE~ UNIVERSITY OF LRD
IN PARTIAL FULFILLMENT OF THE RQIREMENTS FOR THE
DEGREE OF DOCTOR OF PHIJLOSOPHY
UNIVERSITY OF FLORI.DA i9,,3
To my Parents
ACKNOWLEDG EVENTS
I would like to express my appreciation and gratitude to my advisor, Professor David A. Micha, for suggesting the study of the problems addressed in this dissertation, and for many helpful discussions. His dedication, support and encouragement have been invaluable.
I would like to thank all the faculty members of the Quantum Theory Project, all of whom have contributed to my development as a student and scientist. in particular, I thank Pmofessor PerOlov Lwdin for providing me the opportunity to attend the summer school in Sweden and Norway.
I would also like to express my thanks in general to all the
members of the Quantum Theory Project. In particiiar, I would like to thank: Dr. Henry Kurtz, Dr. Nelson H. F. Beeb, Dr. Jack Smith and Mr. Larry Relyea, who were the source of sage computational advice; Dr. Michael Hehenberger, who pointed out the work of Shampine and GorHon; and Dr. John Bellum, who provided useful encouragement and discussion in collision theory to a somewhat bewildered but eager chemist. My special thanks go to Dr. Zeki Kuruoglu who was the sounding board for, many ideas and whose comments have helped shape much of the work presented here.
I am grateful to Miss Brenda Foye for the nic job she has done in typing the manuscript.
iii
M1y deepest appreciation goes to my wife, Adriana, for her patience, understanding and constant encouragement during the task at hand.
iv
TABLE OF CONTENTS
Pace
ACKNOWLEDGEMENTS ..... .. .. ....................... iii
ABSTRACT ........ .......................... vii
CHAPTER
INTRODUCTION ...........................
1. Theoretical Treatment of Nuclear and Electronic
Motions ..... .... .................... 2
2. Adiabatic Collision Processes .... ......... 6
3. Nonadiabatic Collision Processes ... ....... 7
4. ManyBody Theory and the Treatment of Nuclear
and Electronic Motions .... .. ............ 11
5. Analysis of Potential Energy Surface Information .... .. ..................... .... 17
6. Proposed Application of the Faddeev Formalism 26
7. Plan of Dissertation .. ........... . 30
II THE TWOBODY tMATRIX .... ................ .... 32
1. General TwoBody Scattering Theory ...... ... 34
2. Boundary Conditions for the OffShell Wave
Function ..... ................... .... 36
3. The VariablePhase and Amplitude Method . 38
4. Computational Aspects of the VPA Equations 42
5. The Comparison Potential Method .......... .... 45
6. The Bateman Method .... ... .............. 50
III NUMERICAL RESULTS FOR THE TWOBODY tMATRIX ....... 54
1. Properties of the TwoBody tMatrix ... ...... 54
2. Numerical Calculations Using the YPA
Equations ... ................... ..57
3. Numerical Results Obtained Using the Bateman
Method .... ... .. .................... 96
IV THREEBODY PROBLEM .... .... ................. 101
1. Problems with the LippmannSchwinger Equation 102
2. Multichannel Transition Operators ... ....... 103
3. The MultipleCollision Expansion .... ....... 111
V THE SINGLE COLLISION APPROXIMATION .. ......... ... 114
1. Description of Channel States ........... .... 115
2. Inelastic Scattering ... ............. .... 117
3. Dissociative Scattering ...... ............ 125
v
TABLE OF CONTENTS (Continued)
VI COLLINEAR SCATTERING. ................... 128
1. Formulation of the Collinear
Scattering Problem .... ............. ... 129
2. TwoBody tMatrix for the OneDimensional
Scattering Problem .... ............. ... 133
3. Results for Inelastic Scattering ... ...... 136
4. Dissociative Collinear Scattering ... ...... 146
VII THREEDIMENSIONAL INELASTIC SCATTERING RESULTS .... 163
1. Practical Implementation of the
Peaking Approximation ... ............ ... 163
2. Numerical Results .... ............. ... 168
3. Discussion ...... .. ................. 186
APPENDICES
I HARD CORE TWOBODY tMATRIX .... ............. ... 196
II INTEGRALS USED IN "COMPARISON POTENTIAL" METHOD .... 200
III JWKB STARTING PROCEDURE ...... .............. ... 204
IV COORDINATES FOR THE THREEBODY PROBLEM.. .......... ... 206
V ANALYTICAL QUANTITIES USED IN THE COLLINEAR
SCATTERING MODEL ..... .. ................... ..210
BIBLIOGRAPHY ..... .... .. ......................... 213
BIOGRAPHICAL SKETCH.. ..... ................. ...... ..222
\,i
Abstract of Dissertation Presented to the Graduate Council of the University cf Floorida in Partid] Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
ENERGY TRANSFER AND DISSOCIATION IN HYPERTHERMAL ATOMDIATOM COLLISIONS
BY
Lyntis H. Beard, Jr.
March 1979
Chairman: David A. Micha
Major Department: Chemistry
Energy transfer and dissociation processes in hyperthermal atomdiatom collisions are investigated using approximations based on the multiple collision expansion of the Faddeev equations. Within this manybody approach, the collision process is treated as a sequence of atomatom encounters. This allows one to obtain information on a threeatom system using only twoatom transition operator matrix elements.
A new numerical procedure, the "variablephase arnd amplitude"
method, is developed to compute the required offenergyshell twobody tmatrix elements for an arbitrary radial diatomic potential. An alternate approach based on the comparison potential method is also investigated, leading to an improved computational algorithm. Whereas these methods are based on a partial wave decomposition of the twobody toperator, the need for nonpartial wave techniques is pointed out at high energies and large momentum transfers. In this connection the Bateman method is investigated.
In the present study, the singlecollision approximation is considered for both collinear and threedimensional motions. In the collinear studies, it is shown that multiple collision effects are important in the case of
vii
inelastic scattering near threshOld, and that for the case of dissocitive scattering they become crucial. A study is also made of the validity of the single collision peaking approximation, and it is shown that this approximation breaks down when soft twoatom potentials are involved. in order to compare to experiment, threedimensional results are presented for the scattering of Li + with N 2 and CO at hyperthermal energies.
viii
CHAPTER I
INTRODUCTION
With the advent of new experimental techniques, the modernday
chemist is now more than ever able to discern the specificity and selectivity shown by che reactants and products in a chemical system. The goal of a chemist is to properly interpret and understand in a general way the events that occur in a chemical reaction. The task at hand, however, is quite formidable because of the complexity of the particles involved. This complexity implies that the investigation of the simplest chemical systems is necessary to lay the foundation of a deeper and more conceptual understanding of chemical phenomena.
In this dissertation we will be concerned with the investigation of energy transfer processes in atomdiatom collisions at hyperthermal energies. In particular, we shall be concerned with processes of the type
A + BC A + BC
+B t AC
+ C + AB
+A + B + C ,
where A, B and C represent the atoms in the atomdiatom system. Theoretically, atomdiatom systems are of interest because they exhibit the entire spectrum of possible modes of energy transfer, i.e., they allow for vibrational and rotational energy transfer, as well as electronic and translational energy transfer. Therefore, we see that the elucidation
1
of the dynamics of atomdiatom collisions would provide an impetus to the eventual understanding oil 1iore complicated systems. Our goal in this chapter will be to clarify and discuss the treatment of electronic structure in atomdiatom scattering processes. The theoretical aspects
of the scattering problem will be the subject of the remaining chapters in this dissertation.
1. Theoretical Treatment of Nuclear and Electronic Notions
A formal treatment of an atomdiatom system requires that one consider both the nuclear and electronic degrees of freedom of the system. In other words, one would have to solve a manyparticle problem. From a practical standpoint, such a problem would not be tractable unless approximations were introduced. In this section therefore, we will sketch various approaches that can be taken to treat the atomdiatom system as an effective threebody problem.
The first approach we shall consider is that of introducing approximations which allow for the separation of nuclear and electronic motions. We begin then by considering the Schrddinger equation (HE)T1 = 0(1)
where E is the energy of the system, T' is the wave function and the Hamiltonian H can be written as (in atomic units)
2 1 2 1 N 2
H yu V(X V +Vou +Hs(1)
a=1 ~i cu1+s
in the centerofmass system (Fr62). The Greek indices in Eq. (1.2) refer
to the nuclei and the Latin indices refer to the N electrons in the atom
3
diatom system. We note that the Coulomb interactions of the various particles involved are contained in Vcoul and the spinorbit interactions are contained in HsO. The electronic and nuclear mass polarization terms have been neglected. The reduced mass w is defined by the relati on
l m M
where m is the mass of nucleus a and
3
M = Z mB (1.4)
=I
is the total mass of the nuclei. In order to separate nuclear and electronic degrees of freedom, the wave function T is generally expanded in terms of a set of electronic functions {p1(r ; ri)} which depend on the electronic coordinates {ri} and only parametrically on the nuclear coordinates {r. Thus one has
(r ;Li: 1(r.; ni)Xl~r (1.5)
where Xl(r.) can be considered as the nuclear wave function describing the motion of the nuclei on the potential energy surface which characterizes the l'th electronic state [(Fr62), (Hi67)]. Equations %1.5) and (1.1) lead to the following set of coupled equations:
[T +1'1I 4 UII E]XI E [TI + TII, + UII1, (1.6)
where we have introduced the definitions
T 1 2 1 2
.. 2 Z VB (1.7)
4
N
H = Vc (1.8)
el 2 1= 1 coul
S(el) (SO) (1.9)
11, = i' i (1. )
(el ) =l 1> (1.10)
U i 1 IHe 1, '
u (SO) = d (1.12)
ill' = I@ "s I >
11' 2 1
8=1 2T ll
and
2 (1.15)
These equations are well known in the literature [(Fr62), (Hi67), (Tu76a)1 and in general form the basis of most treatments of atomic and molecular scattering problems. Our notation and discussion follow that of Tully
(Tu76a).The diagonal matrix elements U ) are the effective potentia(1.13)
d(a) ,>(1.14)
11<11
and
ill' 6
These equations are well known in the literature [(Fr62), (Hi67), (Tu76a)] and in general form the basis of most treatments of atomic and molecular scattering problems. Our notation and discussion follow that of Tully
(Tu76a)]. The diagonal matrix elements Ul11(r a) are the effective potential energy surfaces for the atomdiatom system and the nondiagonal matrix elements T 1 and U1 are the coupling elements responsible for
the promotion of electronic transitions. The diagonal term 11 is a nonadiabatic correction to U11. The diagonal term Ti can be shown to be exactly zero if the electronic functions {6,} are taken to be real (Hi67).
5
Tne neglect of all the momentum dependent coupling terms T and r' (all and 1') is known as the BornOppenheimer approximation [(Hi67), (Tu76a)l and leads to a new set of coupled equations [T + U E]X1 E UIIXI, (1.16)
a ii 1 '=1 li
in which electronic transitionsmay occur only through the nondiagonal potential terms U,1.
The effectiveness of Eq. (1.6) in properly describing collision
processes involving electronic transitions depends crucially on the choice of the functions {01}. For example, if { i} is chosen to belong to the adiabatic representation, i.e., that representation which diagonalizes the Hamiltonian
H' = Hel + H (1.17)
one obtains the set of uncoupled equations [T + V1 E]XI = 0, (1.18)
where
Vl1(r a U li(r a (1.19)
In this representation, transitions between the various electronic states are not allowed, unless one considers the momentum dependent coupling terms. The term adiabatic representation is also used in the literature for that representation in which the electronic Hamiltonian Hel is diagonal [(Hi67), (1u76a)l. There are, however, an infinite number of nonadiabatic representations. One set of representations encountered in the literature are the so called diabatic representations [(0m71), (Sm69)]. These are representations in which the electronic functions
6
approach well defined atomic states asymptotically (0n71). There is no unique recipe for constructing such representations. In general they are chosen such that the momentumdependent coupling terms T ll are small (Sm69). One should note however, as Tully (Tu76a) points cut, that diabatic representations are useful in atomatom scattering because in those situations avoided crossings in the adiabatic curves are localized to a point. In other situations where there are more nuclear degrees of freedom, as in atomdiatom scattering, one is dealing with avoided surface crossings at which large variations in the nonadiabatic couplings can occur. This indicates that one might not be able to find a diabatic representation which is suitable in a global sense.
2. Adiabatic Collision Processes
Because of the complexity of atomic and molecular collisions, most work in the literature has focused on the theory of adiabatic collision processes. The theoretical investigation of such processes is, however, by no means trivial. For example, in the case of inelastic atomdiatom scattering, Eq. (1.18) would generally involve solving a large set of coupled differential equations. If a large number of open channels for vibrational and rotational excitation exist, an accurate and economical solution to Eq. (1.18) may be difficult to obtain. In order to solve
the problem for such collision processes, further approximations have been sought which reduce the dimensionality of such coupled differential equations. A review of these techniques is given by Micha (Mi74), Lester (Le76) and Rabitz (Ra76). In the case of reactive scattering the situation is even more complicated. For an atomdiatom system, there are four arrangement channels [A + BC, B + AC, C + AB and A + B + C] and as
7
a consequence free motion in the asymptotic region is governed by different channel Hamiltonians. This aspect of reactive scattering introduces complications, because the boundary conditions become difficult to apply and in addition no unique and simple set of coordinates is suited for propagating the solution in all regions of configuration space. We note that in each rearrangement channel, a natural choice of coordinates would be that of the Jacobi coordinates i and ri, where i describes the relative motion of the "free" atom with respect to the center of mass of the diatom and ri the relative motion of the diatom. The label 'i" is a channel index (see Appendix IV). In order to overcome the difficulty with the choice of coordinates, Marcus (Ma66) introduced the concept of natural collision coordinates, e.g., a set of coordinates which goes smoothly from one set of Jacobi coordinates to the other. However, these coordinates complicate the structure of the kinetic energy operator and may in some instances become multivalued according to how they are constructed [(Ma66), (Wi77)]. A clarification of many of the problems of reactive scattering may be obtained by considering the structure and manybody aspects of the wave function [(Mi72b), (Re77)]. A general review of reactive scattering may be found in the articles by George and Ross (Ge7l), Kouri (Ko73) and Micha (Mi75a).
3. Nonadiabatic Collision Processes
Nonadiabatic collision processes have not been studied in as much
detail as adiabatic processes for two general reason. One reason is the added complexity of dealing with a coupled set of equations [Equations (1.6) and (1.16)]; the second reason is the lack of information on the potential energy surfaces and on the various couplings involved. It has
been found, ho':ever. that for some problems it is possible to overcome the first of these obstacles by treating some or all of the nuclear degrees of freedom either classically or semiclassically. As was pointed out earlier, in some instances nonadiabatic effects may be localized to certain regions of the potential energy surfaces. This, for example, would allow one to use classical trajectory methods in regions where motion along the adiabatic potential surface is allowed. In such an approach the nonadiabatic regions serve only to determine branch points at which a jump to another surface can occur. This approach, which is known as the trajectory surface hopping model (TSH), was proposed by Tully and Preston (Tu71) to study the systems [H+,D2] (Tu71), [H,D+] (Pr73) and [Ar+,H2] (Ch74). In order to determine whether a trajectory remains in the same potential energy surface or branches to another surface, the probability for hopping must be computed. A discussion of how this probability is obtained is given in the above references. We shall only mention one method; that of the well known LandauZenerStUckelbert (LZS) approximation [see for example Bransden (Br70)]. Strickly this approximation is valid only for the onedimensional case. Here, however, one applies the LZS result to a cut perpendicular to the nonadiabatic seam (Tu76a) in the potential energy surfaces. Since trajectories may encounter various branch points, we see that in many physical applications the TSH method could become computationally unmanageable.
Another approach, which is useful in treating nonadiabatic processes such as
A + BC(n1,j) A+ + BC(ni,ji) (1.20)
A + BC (n,j!) (1.21)
9
which involve charge transfer and vibrationalrotational excitation, is that given by the multiplecurve crossing model. In order to define this model, let us introduce the Jacobi coordinates 1 and r1 which are appropriate in describing the arrangement channel specified by Eq. (1.21). Furthermore, let us expand the nuclear wave function x1 1,rl) in terms of a set of vibrationrotation functions {ulV( 1;rl)} as follows,
Xl ,1 1,) =E ulV( 1r;l)ly 1), (1.22)
V
where v labels the vibrationrotation states and wl( 1) acts as a nuclear wave function for the relative motion of atom A with respect to the center of mass of the diatom BC. Note that the {u } depend only
parametrically on !. If we now use Eq. (1.6) expressed in Jacobi coordinates and Eq. (1.22), we obtain a set of coupled differential equations given by
E E" w
[E +r + Wv1v E]wl (RI) =
E E' E" l ) (1.23)
E [ + W l, ] I, ,(1.23)
11
v'I
where
E 1 2 (1.24)
2M1 RI
mA(mB+m C
ME : (1.25)
1 mAm Bm C
E' i IR R
TI1' 1M
1 1 1
Ell 2 6 ,> (1.27)
IU+ T I'y I" ,~ > (1.28)
WI 1( ) IV iu ll, 1I, I+u ill
I 1 12 (1.29)
2m 1 rl
M mB mC (1.30)
(mB+mC)
TII' m> r' (1.31)
I" I> (1.32)
and the quantities mA, mB and mC are the masses of the atoms composing the system. Equation (1.23) is the basis for treating an inelastic atomdiatom scattering problem. The diagonal matrix elements WIlIV(AI) correspond to effective potential energy curves, each of which is associated to a particular electronicvibrationalrotational state (Tu76). These potential energy curves then serve as a basis for treating charge transfer and inelastic scattering within the multiplecurve crossing model [(Ba69b), (FI71), (Gi75), (Tu76a)]. It is obvious, however, that due to the enormous number of these curves, various other approximations must be introduced. Generally one averages over the rotational states and assumes that nonadiabatic effects are confined to the avoidedcrossing regions where the LZS approximation can be used. A discussion of the approximations and implementation of this model can be found in the
work of Fisher et al. [(Fi7L) (Fi72), (Ba69h)], Gislason and Sachs (Gi75), and in the review article by fully (Tul6a). We note that Eq. (1.23) is not suitable for treating rearrangement collision processes. However, using the manybody formalism of the Faddeev equations Yuan and Micha (Yu76) have implemented multiple curve crossing information to study rearrangement scattering of the atomdiatom system [K,IBr].
4. ManyBody Theory and the Treatment of Nuclear and Electronic Motions
All the methods discussed in the previous sections involved the
tCreatment of electronic and nuclear motions essentially as two separate problems. This approach has resulted in the introduction of the concept of a potential energy surface, ime., that quantity which characterizes the electronic state of the system and serves as the effective potential governing nuclear motion. An alternative to this picture would be the simultaneous treatment of nuclear and electronic motions. We will briefly sketch how this can be accomplished by the use of manybody theories. This will be done for three reasons. First, it will serve tocontrast the more traditional methods of scattering theory based on Eq. (1.6). Secondly, it will show how one can use known twobody data as input to the solution of an inherent manybody problem. Lastly, it will serve as a reference with which to compare the approximate treatment of atomdiatom collisions given in later chapters of the present work.
The manybody formalisms that will be discussed are based on the
treatment of the threebody problem pioneered by Faddeev (Fa6l). Earlier it was pointed out that for a threeparticle system, there are four arrangement channels. This implies that free ioftion in each asymptotic
i12
region is governed by a different channel Hamiltonian. in order to clarify this, consider the Hamiltonian for three interacting atoms A, B and C (in laboratory frame),
H = Z K L4 HI + E VJK (1.33)
a I J
KWJ
where K is the kinetic energy of nucleus a = a, b, or c (associated with A, B and C respectively); HI is the electronic Hamiltonian of atom I, and VJK is the Coulomb interaction among the charges of atoms J and K (Mi77b). The channel Hamiltonian for a given rearrangement channel i[l for (A+BC), 2 for (B+AC), 3 for (C+AB)] is defined by the relation
H. = lim H (1.34)
R
and as can be seen using Eq. (1.33)
Hi = Z K + E HI + V (1.35)
In the breakup arrangement channel one can analogously define the free particle Hamiltonian
H lir H = E Kc + Z HI (1.36)
A i a I
*1
4.
ri
which can be used in Eq. (1.35) to obtain Hi = H+ Vi (1.37)
and
3
H = H + Z Vi (1.38)
i=1
where Vii s equal to VJK. The Faddeev formalism, then, is merely a way
of decomposing the total wave function of the system into channel components which can be propagated by their respective channel Hamiltonians. The result is given by the relation (Fa6l)
(E 1)Tj) = Vj 3 (I 6jk) (k) (j = 1, 2, 3) (1.39)
where the total wave function T has been decomposed into the channel components 1' ) i.e.,
3 0)
T: = E T (1.40)
j=1
A detailed derivation of the Faddeev equations will be given in Chapter IV of the present work. In the following discussion, our concern will be the use of Eq. (1.39) solely as the basis for the simultaneous treatment of nuclear and electronic motions. Micha (Mi77d) has shown that this can easily be accomplished within the framework of the method of diatomicsinmolecules (DIM) (E163). We begin by considering the total wave function for the system, keeping in mind that for structured particles it must have the form
= CAX (1.41)
where C is a normalization constant, J is the N electron antisymmetrizer and X is an unsymmetrized total wave function for both nuclear and electronic motions. Since H and .A commute, one can cast the Schridinger equation into the form
E I (H.A +. H)]X =0 (1.42)
14
or in the Faddeev form
{EA '[Hj,} ]+}X) 2 [VA I+ E 16jk)X (1.43)
k=1
where
[H;,) ]+ Hjd + J H1 (1.44)
Equation (1.39) may now be replaced by Eq. (1.43), which is more suitable for introducing DIM approximations. A symmetrized form of the Hamiltonian such as
HA = r.A ,H]+ (1.45)
has been introduced in order to avoid nonhermitian matrices (Tu76a) which occur whenever the X(j) are expanded in terms of a set of channel electronic states !j)> and used in Eq. (1.43). The channel electronic states i (J)> correspond to a collection of MO) primitive channel wave functions constructed in a step by step procedure from a set of atomic wave functions "Al' EBm and Cm' as is usually done in the DIM method [(E163), (Mi77d)]. When these channel functions are constructed, however, the arrangement structure of the channel must be considered. Therefore, one begins by constructing a collection of M0 functions { Al Bn Cm>}. Each of the atomic functions EI1 are eigenfunctions S2 and SlZ, and are composed of nonsymmetrized combinations of products of atomic orbitals centered on the nucleus of atom I. This set of functions we designate IO>. A set of MO) unsymmetrized or primitive channel wave functions [ (J)> can be obtained frm If_ > by first combining the products EImEKn to form eigenfunctionsm of S and SjZ, and
3 jZ
then combining the products of these and to form eiqenfunctions of
S2 and SZ (Mi77d). This leads to the relations > (1.46)
< 0r> = (1.47)
and
<() 1(J)> = 0) (1.48)
where and I are unit matrices and Cj) defines a channel coupling matrix between the breakup channel (channel 0) and the arrangement channel i. The unsymmetrized or primitive channel wave functions Xkj) can now be expressed as
X= ) (1.49)
where (j) denotes a M(j) x 1 column of scattering functions. In obtaining Eq. (1.49), one has assumed the approximate completeness relation
A
0 )>< i)i I (1.50)
where 1. is the unit operator in the space of many electron states in channel j (Mi77d). Using equations (1.43) and (1.49) the end result of the DIMFaddeev equations is given by
{A(JE E K 1 (j), EE + V.]+} X(j)=
a 2 I
1 I)
[A), Vj+ E (1 6 jk) C(jk) X(k) (1.51)
k j
where
A(j) <(J) (j)> (1.52)
16
c(Jk) = (1.53)
= I EK (1.54)
EI is a diagonal matrix containing atomic energies and EK is a matrix containing diatomic energies. The energies E are associated with the unbound atom in channel j and the EIK energies of the corresponding diatomic in the same channel. Equation (1.51) is a result of the BornOppenheimertype approximation
_K 15i)> W K (1.55)
and the DIM approximations
< (J) I._, HjIE(j)> A (J) Ej (1.56)
and
<(J)IA H J)> = A(j) IK (1.57)
The uanitis c(jk)
The quantities CJ in Eq. (1.51) act as recoupling coefficients among the various channels, allowing for inelastic, rearrangement and dissociative processes to occur. Furthermore, as long as the proper atomic and diatomic energies are included, Eq. (1.51) allows for electronic excitation. A detailed application of this formalism to the [H + H2] system is given in the literature [(Mi72b), (Mi77d)]. In these references it was shown that one can construct a suitable expression of the ground state potential energy surface of the system in terms of twobody interactions provided one uses spindependent potentials.
Atomdiatom collisions involving charge transfer cannot be treated properly using Eq. (1.51) because this equation assumes a unique parti
17
tion of the electrons. There is, however, an alternative approach based on the coupledchannel manybody formalism [(Ko75), (To74), (Ra77)], which is suitable for treating charge transfer. In this approach one simply considers various partitions of the Hamiltonian consistent with the atomic and molecular fragments involved in the scattering process of interest. A set of coupled equations may then be constructed that are analogous to the Faddeev equations (Mi76b).
5. Analysis of Potential Energy Surface Information
In the present work, we shall be concerned with ionmolecule reactions of the type (Li+, CO) and (Li+, N2). Since CO and N2 have identical masses and very similar molecular properties, these systems are very interesting for comparison purposes. Such a study is also attractive because a comparison between experiment and theory is now possible in light of recent molecular beam experiments [(B676), (Ea78)].
Our investigation of these systems will be based on an analysis of the various interaction potentials involved and upon the potential energy surface data available in the literature [(St75), (St76), (Th78a)]. This analysis though qualitative in nature, will serve as a guide for the development of an impulsive model which will be specified in the next section.
The only available potential energy surface information on the
(Li+, CO) and (Li+, N2) systems corresponds to a restricted number of geometries for the ground state of the system. Because of the relatively large number of electrons in these systems, accurate boundstate calculations are very expensive. This is further complicated by the large number of points required to adequately characterize the potential
energy surface. The available calculations [(St75), (St76), (Th78a)] were performed within the SCF approximation and corrected to some degree for correlation effects. SCF calculations alone cannot sufficiently describe the potential energy surface because such an approximation fails to properly describe the dissociation of closed shell systems such as N2 and CO into open shell atoms (St76). SCF methods in general fail to properly characterize the large distance multipole type interactions between atomic and molecular fragments. Staemmler [(St75), (St76)] and Thomas et al. (Th78a) corrected the SCF results by making use of CI (configuration interaction) methods.
Since the information obtained in the above calculations is restricted to the ground states of the (Li+, N2) and (Li+, CO) systems, the possibility of nearby excited states must be ascertained by considering the possible electronic states of the various atom and diatom fracments of these systems (Ma75). For example, Fig. (1i) considers some of the electronic states of the (Li N2) system when Li+ and N. are infinitely separated. The resulting potential energy curves were constructed using the information given by Bond et al. (Bo65). We note that the ground state of N2( I+ state) correlates with the two nitrogen atoms in the 4S state. Using the WignerWitmer rules (He5O), one can predict that the other states that correlate with the [N( 4S) + N( 4S)] asymptotic state are the 3E 5E+ and 7+ states. In Fig. (11) the u g u
possibility of charge transfer (Li, N2) and of electronic excitation of nitrogen (Li + N2) is also considered. Asymptotically we see that the ground state of N(2g) correlates with [(N(4S) + N+(3P)]. Of more interest is the excited electronic state which asymptotically leads to [N(4S) + N(2D)], where the 2 state corresponds to the first excited
Figure (11) Potential energy curves for the diatoms
in the reactant and product regions of the (Li+, N2) system [(Bo65), (Ma75)].
The Li+(Li) atom and the N2(N2) molecule
are at infinite separation.
20
12  II
l2 / 7+[N( 4S)+N(23 2+ ) 1.11 D);Li(S)
F2 +
71+
rA
2 E[( )+( N2i( s
21
state of nitrogen. Note that in cons tructi nr~ the curves in Fig. (11)5 the energies of Li( )and L(2S) must be included. The possibility of
reactive channels is ruled out by the closed shell nature of the (Li+, N2) system. Furthermore, the electronic excitation of Li+ [Li+( S) + N 2( Z 9)] is not considered since it would lie much higher in energy than those states already given in Fig. (11). It is apparent
then from Fig. (11), that there is a possibility of electronic excitation, since the 3+state of N~ lies only 6.1 eV (Bo65) above that of the ground 1Z+state. Charge transfer on the other hand would seem
g
unlikely.
As the Li + is brought near N2, the relative position of the various electronic states would change. If the spacing between the ground arid
excitated states grows closer, nonadiabatic effects will be important for the given system. Figure (12) illustrates what happens as the Li + atom approaches the N 2 molecule. This figure shows three planes. Plane III corresponds to the same cut of the potential energy surface given in Fig. (11) in which R (the distance between Li + and the centerofmass of N2) is infinite. Planes I and II are perpendicular to plane III and intersect that plane at r equal to zero and at r equal to the equilibrium internuclear separation (r e) of N 2. It is shown then that
the ground state potential energy of the system is decreased as Li+ approaches and increases rapidly as R goes to zero. The reason for this initial decrease of the potential as the Li + approaches N 2 is due to attractive polarization forces. This will be explained later when the long range interactions of the system are discussed. On the other hand
at small values of R, strong repulsion forces exist because of the closed shell interactions. A similar qualitative behavior is expected for the
Figure (12) Various cuts through the potential energy surface
of an atomdiatom system are illustrated. The
parameter R measures the distance of the projectile
atom to the centerofmass of the target diatom; whereas, r is the internuclear separation of the
target diatom. Planes i and II are perpendicular to plane III at r equal zero and re, respectively.
23
R=CO r=o re R=O
r, rill
De
24
excited states. It should be mentioned, however, that the reaction
+1 1+ + 1 E+
Li +S) + N2(g) E Li+S) + N2(3u1
would be spinforbidden (Geil). This conclusion is only valid if one can neglect effects such as spinorbit interaction. A discussion of the applicability of symmetry rules to collisions is given in the work by George and Ross (Ge7l) and that by Shuler (Sh53). From an experimental viewpoint, electronic excitation has not been observed at the energies [4 to 8 eV in the centerofmass system] and scattering angles studied
in the present work (B676). A similar analysis could be done for (Li +, CO); however, this system is isoelectronic with (Li+, N2) and is expected to behave similarly.
We have not considered changes of the equilibrium internuclear
separation of N2 as Li+ approaches. In order to do so, we now discuss the potential energy surface in more detail. The long range interaction between a charged atom and a diatom with a nondegenerate electronic state is dominated by electrostatic and inductive terms (Bu67), i.e.
V(R, r, 0) = VCOUL + VIND (1.58)
where
V qp + R c p
COUL R2 3 P2(coso) R4 P3(rso)
+ R5P4(coso) + . (159
+ qH p(1.59)
V 1 q2 1 [ + 2 (a,, P,2(coso) + "
IND 2 R 4 31
(1.60)
25
(a 1I + 20(L ) (1.61)
where q corresponds to the charge of the atom, the molecular parameters p, Q, Q and H correspond to the dipole, quadrupole, octapole and hexadecapole moments of the diatom, and the parameters a and a.1 correspond respectively to the parallel and perpendicular components of the diatomic polarizability relative to the internuclear axis of the molecule. The functions Pl(coso) are Legendre polynomials and 0 is the angle between the vector from the centerofmass of the diatom to the projectile atom and the internuclear axis of the diatom. Both the N2 and CO molecules have very similar values for the various parameters used in equations (1.59) and (1.60), with the exception that N2 does not have a permanent dipole moment. This difference, however, is quite important since the chargedipole interaction determines the R_2 term in Eq. (1.59), and as such dominates the long range portion of the (Li+, GO) potential. It is precisely these parameters [p, Q, etc.] which are not well specified in SCF calculations [(St75), (St76), (Th78a)]. The value of the dipole moment of CO in fact has been the subject of wide controvercy (Ne64). Experimentally this value is small and negative [pCO = 0.049 a.u.] (St76) and implies the polarity of the CO molecule is given by C0. More recent calculations (Ch76) have now shown agreement with the experimental results. Unlike the situation for N2, in a collinear reaction the potential between Li + and CO is not symmetric. If one examines the behavior of the equilibrium internuclear separation of CO as a function of R,one would find that it shows much more deviation from its value at R equal to infinity than that for N2. In the scattering experiments on these systems (B676), it is found that (Li+, CO) shows more
26
vibrational excitation than the (Li +, N2) systems. Various investigators [(B676), (St76), (Mi78b)] have attributed this observed result to the chargedipole interaction in the (Li+, CO) system. The relative insensitivity of re as a function of R for the (Li+, N2) system is shown in Fig. (13).
At small values of R the interaction of Li + with N2 and CO is
highly repulsive and can be easily characterized by an exponential potential [St76), (Th78a)]. Thomas et al. (Th78a) use the potential
V(R, r, 0) = Vl(r) + Z v,(r, R)PX(cosE) (1.62)
A
where Vl(r) is the potential for the free diatom and v,(r, R) is determined by using various parametrized functions [sums over exponentials and Rn terms] to yield the best curve fit to the potential. Equation (1.62) behaves asymptotically as Eq. (1.58). Staeimler (St76) points out that the Legendre expansion of the potential in Eq. (1.62) loses its significance for small R[R 3.5 a.u.] and may in fact not converge.
6. Proposed Application of the Faddeev Formalism
Our study of the (Li+, CO) and (Li+, N2) systems will be based on the many body formalism of the Faddeev equations. Within this approach we shall consider an iteration of the Faddeev equations (in transition operator form) which yields a miltiplecollision expansion (Mi75). This allows us to describe an atomdiatom collision process as one which takes place as a sequence of twobody encounters. Since the collision energies are large and the scattering angles are beyond the rainbow angle, it is expected that the scattering cross section will be largely
Figure (13) Contour plots of the ground state potential
energy surface for the (Li+, N2) system as a function of the NN internuclear separation (St75). Each curve corresponds to a fixed
value of R, the distance between Li+ and
the centerofmass of the N2 molecule.
115.5 ,..
R=3
E (a.u.)
116.0
R=4 =5
116.3
18 2.0 22 24
rN2 (a.u.)
29
characterized by the repulsive forces of the system. Thus, an impulsive model based on the truncation of the multiplecollision expansion would be expected to yield useful information on these systems. An advantage to this approach, as will be shown in Chapter V, is that one is able to obtain statetostate information directly from the initial and final momentum distributions of the target and product diatom, and from the various twobody potentials involved. Due to the small vibrationalrotational spacings of CO and N2, other methods such as the coupledchannel method, based on Eq. (1.23) would be computationally prohibitive (Th78b).
We shall assume in our implementation of the Faddeev equations that electronic motions have already been eliminated and that only information from the ground adiabatic potential energy surface of the system is needed. As mentioned previously, the scattering process at hyperthermal energies probes the short range part of the potential. Therefore, we shall develop a many body theory based on the decomposition of the adiabatic potential into the form (Mi79) V( ) = Vsr + Vlr (1.63)
where
3
Vsr = Vi(ri) (1.64)
sr i=1 1
and
V1r E v )(R, r)Y (R, r) (1.65)
The Vlr term denotes the long range potential and is analogous to the Legendre expansion used in Eq. (1.62). Here, however, Y (R, r) is a
30
general function depending only on the relative orientation of the atom, and diatom. The V sr term, on the other hand, corresponds to the short range potential and is characterized by a sum of manybody terms. In Eq. (1.64), Vsr was assui.1ed to be a sum of radial twobody potentials; one potential for each twoatom fragment in the atomdiatoni system. This breakdown of the potential circumvents the difficulties, mentioned in the last section, in representing the short range behavior of V by Eq. (1.62). Having introduced Eq. (1.63), we propose an impulse model in which only the short range part of the potential is kept. The resulting problem is then ideally suited for the manybody formalism of the Faddeev equations.
We notethat, at low collision energies, one expects that the
approach of Li to CO or N 2 would heavily influence the orientation of the diatom. This would be particularly true for a molecule with a d ipole moment like CO. The net result in our impulsive model would be to overestimate rotational excitation. At the high energies considered in this study, however, this effect is expected to be minimal.
7. Plan of Dissertation
The present investigation of atomdiatom collisions will be given in three major parts. The first part presents an analysis of the twobody problem and discusses various computational schemes for the determination of offshell matrix elements of the twobody transition operator (Chapters II and III). In the next part (Chapters IV and V), the formalism of the threebody problem is presented and the various approximations obtained from the multiple collision expansion are discussed. Finally, in the third part (Chapters VI and VII), computational results
31
based on the singlecollision approximation are presented for both collinear and threedimensional atomdiatom scattering processes.
CHAPTER II
THE TWOBODY tMATRIX
Scattering processes in few or manybody systems require the introduction of twobody transition operators, whose matrix elements must be known for arbitrary initial and final relative momenta of the two bodies and for arbitrary energies, i.e., they must be known in general "offtheenergyshell." In other words, the operator quantities of interest are those describing the interaction of two particles embedded in a many particle system, and as a consequence energy is not conserved locally, thus explaining the term "offshell". Although these twobody transition operators stand for basic physical concepts, their properties and values are little known for molecular systems, where the bodies are atoms or ions.
The computation of twobody tmatrix elements has long been a problem of intense interest in the field of nuclear physics. Consequently, there exists an extensive literature devoted to this subject as it pertains to potentials describing nucleonnucleon interactions. A very nice and extensive review in this regard is the work of Srivastava and Sprung (SrlS). In the chemical literature, work in this area has been sparse. However, there have now appeared various papers which deal with the computation of tmatrix elements for potentials of chemical interest, for example: van Leeuwen and Reiner (vL6l) proposed a numerical procedure based on the comparison potential approach, which has now been implemented in the work of Brumer and Shapiro (Brl5); Kuruoglu and 32
'3
Micha (Ku78) have developed a variational approach leading to a separable approximation to the tmatrix; and along more approximate methods, Korsch and Mdhlenkamp (Ko77) have developed a semiclassical approach based on the JWKB approximation which is applicable to repulsive potentials.
The principal aim of this chapter will be to present a new computational method for obtaining matrix elements of twoatom transition operators between arbitrary momentum states, of high accuracy, and applicable to any radial diatomic potential. The approach we have taken is based on equations presented some time ago (vL6l), which reduce the calculation of transition operator matrix elements to integration of inhomogeneous radial differential equations followed by numerical quadratures. Our procedure is based on the wellknown variablephase approach [(Ca63), (Ca67)], but extended to the present inhomogeneous equations. This extension shows that, unlike the situation for standard radial scattering equations, the variable phase is coupled at every distance with a variable amplitude. To emphasize this point we refer here to a "VariablePhase and Amplitude" approach. Another feature of our approach is the way in which one can treat propagation through classically forbidden regions. Along with the development of the VPA method we shall present some suggested improvements to the comparison potential method.
All the methods we have mentioned above make use of a partial wave expansion to construct the twobody tmatrix elements. When we discuss the singlecollision approximation in Chapjter V, it will become apparent that it would be desirable to have a procedure for the direct computation of these matrix elements. This will emen become crucial
34
when one deals with physical systems at energies for which a large number of partial waves are required and the computational expense becomes important. Consequently, in light of this problem, we shall also investigate a nonpartial wave approach that has been recently developed by Belyaev and coworkers (Be74), which is based on the Bateman method [(Ba22), (Ka58)] for solving integral equations.
1. General TwoBody Scattering Theory
The general computation of offshell matrix elements of the twobody toperator for simple nonsingular potentials can in general be solved without much difficulty. This may be done by direct consideration of the LippmannSchwinger equation [(Si7l), (Sr68)] t(z) = V + VGo(z)t(z), (2.1)
where
G (z) = (zH)1 (2.2)
In these relations Ho is the kinetic energy operator for relative motion, z is a complex energy parameter (z = E+is) and V is the interaction potential between two particles. V will always be assumed local and spherically symmetric. From equation (2.1) it is apparent that the problem must be reformulated for singular potentials, those which contain a hard core or have a singularity stronger that ]/r2 at the origin (Ta72). This can be accomplished by introducing the wave operator W(z) via
t(z) = VW(z), (2.3)
where
W(z) = 1 + Go(z)VW(z), (2.4)
35
and by relating the matrix elements of the toperator to those of the wave operator in a mixed coordinate momentum representation (vL61). Multiplying (2.4) by G 1(z) on the left, one finds that W(z) also satisfies
(z H V)W(z) = (z Ho). (2.5)
In a mixed coordinatemomentum representation one then obtains the inhomogeneous Schridinger equation below (1 = 1):
[z + (2m)1V2 V(r)]<_irw(z)> = [z (2m)1q2]< r4>, (2.6)
where m, r and q are respectively, the reduced mass, the relative coordinate and the momentum of the two particles being considered, and lw(z)> is the offshell state (since q2/2m t E) defined as
q
le(z)> = W(z)li> (2.7)
q
In the nuclear physics literature, (2.6) is known as the BetheGoldstone equation (Sr68). A reduced radial equation may now be obtained by introducing into equation (2.6) the following partial wave decompositions:
<44> = (21 )3/2 eIq'r
3/2 1
= (2) 3/2 i (21 + 1)P1l(qr)j1(qr)/(qr) (2.8)
1
and
= (27)3/2 E il(21 + 1)P (q'r)w1(q,kE;r)/(qr), (2.9)
1 E
where jl(qr) is the regular RiccatiBessel function and kE = 12mE1I/2 The resulting reduced radial equation is
36
d2... 1(1q2+1j
S +a v(r)w (qk ;r) = (5 q2 (qr), (2.10)
dr 2 E E r2 1 E kEE
where aE is +1 if E is positive or 1 otherwise, and v(r) = 2mV(r). The matrix elements of the toperator can now be obtained from those of the wave operator by using equation (2.3). Thus one has the relation
t(4', ; z) = f V(r) dr, (2.11)
which may also be decomposed into partial waves with the result
t(q q; z) = (47)1 Z (21 + 1 (q'q (q', q; E), (2.12)
1 1
where
t (q', q; E) : 2 1 1 J(q'r)V(r)w (q, kE; r) dr. (2.13)
1 7Tqq 1 E'
This definition of the partial wave matrix elements tl(q', q; E) follows directly from equations (2.8), (2.9) and (2.12).
2. Boundary Conditions For The OffShell Wave Function
In order to obtain the offshell wave function w (q, k E; r), a
solution to equation (2.10) must be found subject to two boundary conditions (vL61):
Wl(q, kE; r) r 0 (2.14)
E r o
and
wl(q, kE; r) jl(qr) + Clh 1)(kEr), (2.15)
r 0
where C is a constant and h(+) is a RiccatiHankel function (Ca67).
1 1
Thus, w1(q, kE; r) is a complex wave function which asymptotically
37
behaves as the sum of a component from a plane wave and that of an outgoing spherical wave, which, unlike onshell scattering expressions, depends on different momenta (Ta72). For potentials containing a hard core (2.14) must be replaced by wl(q, kE; r) 0 (2.16)
r r C
where r is the hard core radius. Once the offshell wave function is
c
obtained, the partial wave matrix elements t (q', q; E) follow directly from (2.13). However, if the potential contains a hard core, care must be taken in applying Eq. (2.13). If one integrates equation (2.10) as follows,
rc+E 2 rc++
f [ j + EE wl] dr f {Iv(r) + +( + 1) +
rc rE rCE2 2ak 2 ^
+ (E k~ q )jl(qr)} dr, (2.17)
it is apparent that the product w 1(q, kE; r)v(r) does not vanish inside the core region even though w 1(q. kE; r) does. In fact, for a potential containing a hard core (La68),
v(r)w (q, kE; r) 6(r rc) d l(q, kE; rc+) +
1 Ecdr Iq kE; rc)
(q2 aEkE) Jl (qr) (2.18)
for r< r where
 C
d d
Tr wl(q, kE; rc+) : lim jwl(q, kE; rc + c) (2.19)
C 0+
A derivation for the case of a oure hard core potential is given in Appendix I.
3. The VariablePhase And Amplitude Method
The aim of the VPA method is to take advantage of the known
oscillatory nature of the offshell wave function, 1(q, kE; r). This is done by considering a solution to Eq. (2.10) which has the form below:
U)l(q, kE; r) = l(qr) + al (q, kE; r) cos [61(q, kE; r)]Ul(k r
a1(q, kE; r) sin [fl(q, kE; r)]Vl(kEr), (2.20)
where ul(kEr) and l(kEr) are RiccatiBessel functions (jl(kEr) and nIl(kEr)) or RiccatiHankel functions (kl(kEr) and 1l(kEr)) (Ca67). The choice is determined by the nature of the region through which one is propagating, i.e., whether the region is classically allowed or not. Using equations (2.10) and (2.20), one may obtain a set of coupled equations for the amplitude, ol(q, kE; r), and the phase, 6 1(q, kE; r). These equations will now be derived using an integral equations approach (Ro67). The functions ul(kEr) and Vl(kEr) used in expression (2.20) satisfy the RiccatiBessel differential equations (Ab65) d_ + lE I(I + 1) 0
dr2 Zl r2 Z1 (2.21)
Taking the regular and irregular functions u1 and vl to be normalized such that their Wronskian, W[Ul, vl], is unity, the Green's function associated with (2.21) may be written as gl(r, r') = EU(k r ) v (kE> (2.22)
39
where r< is the lesser of r and r', and r, is the greater of r and r' (Ro67). At this point, the discussion will be restricted to positive energies, since this would not involve a loss of generality. It is readily seen that equation (2.21) is just the corresponding differential equation to (2.10) in the absence of a potential. Thus, wIl(q, kE; r) may be written as follows:
00
l(q, kE; r) = N'Jl(kEr) + I dr'gl(r, r')[(k2 q2) Jl(qr') +
v(r')w1(q, kE; r')] (2.23)
where N' is a constant. Since jl(qr) satisfies equation (2.21), the first term in the above integral, (k2 q2) f' dr'gl(r, r,)3I(qr'))
may be evaluated exactly. This may be accomplished using the relation (A2.2 ) given in Appendix II. Introducing the quantities A1(q, kE; r) and BI(q, kE; r) defined by
AI(q, kE; r) = N + k I E dr'v(r')wl(q, kE; r') nl(kEr') (2.24)
r
and
fr^
BI(q, kE; r) = kE f0 dr'v(r')wl(q, kE; r") Jl(kEr'), (2.25) where N = N' + 1, and using equation (2.22), one obtains
wl(q, kE; r) = jl(qr) + AI(q, kE; r) Jl (kEe) +
Bl(q, kE; r) nI (kjr)) (2.26)
40
The functions A1 and B behave as constants asymptotically, as can be seen from the boundary conditions (2.14) and (2.15). These constants may in turn be related to an offshell phase shift, 1(q, kE), and amplitude, c1(q, kE), in the same manner as is done for the onshell case [(Ca63), (Ca67)]. Therefore one has
Al(q, kE; r) a (q, kE) cos [61(q, kE)] (2.27)
and
B1(q, kE; r) 1 (q, kE) sin [61(q, kE)] (2.28)
r oo
The variable phase assumption consists of extending the definition of the phase and amplitude to all values of r (Ca63); thus,
A,(q, kE; r) = aI(q, kE; r) cos [Sl(q, kE; r)] (2.29)
and
Bl(q, kE; r) = al(q, kE; r) sin [5l(q, kE; r)] (2.30)
This assumption is of course valid, since 61(q, kE; rM) and aI (q, kE; rM) are just the resulting phase shift and amplitude for a potential with a cut off at r = rM (Ca67). In fact the justification of this approach is best understood by considering a finite boundary value problem and extending the boundary to infinity (Ze64). The desired set of coupled first order differential equations, the VPA equations, may now be derived using (2.29) and (2.30) for A1 and B1. To accomplish this, first consider the derivatives of A and B as defined by equation (2.24) and (2.25). From these relations it is apparent that
^ dk r d r
JlkEr) dr Al(q' RE; r) nl(kEr) dr Bl1(q, kE; r) = 0 ; (2.31)
41
therefore, the derivative of the offshell wave function has no terms involving the derivatives of A1 and Bl, i.e.,
d d d
d wml(q, kE; r) = dr ~1 (qr) + A1(q, kE; r) dr 1(kEr)
Bl(q, kE; r) Ln (kEr) (2.32)
Making use of (2.29) and (2.30), equations (2.26) and (2.32) may be written as,
W J1(qr) + a, cos (61) il (kEr)
sin (61) n1 (kEr) (2.33)
and
d d d
drl r l1(qr) + a, cos (6 )r J1(kEr)
d ^
~ sin (61) d nl(kEr) (2.34)
Taking the derivative of (2.33) and comparing it with (2.34) one obtains,
AA
dsin (61) j (kEr) + cos (61) nI (kEr) d (2.35)
pal al ^^Icos (61) J1 (kEr) sin (61) n1 (kEr) dr 1
In a similar manner taking the derivative of (2.34) and comparing it to the inhomogeneous Schridinger equation (2.10), one finally obtains, with the aid of (2.35), the "variablephase and amplitude" equations
d ^r ~rwjc2(.6
d 61(q, kE; r) = kE v(r)[ (qr)]/ (2.36)
and
d l(qr kE; r) = kE1 v(r)w l[sin(61 Ol(kEr) + cos(61)nl(kEr)].
(2.37)
42
These equations are analogous to those for the onshell case, where q = k E (Ca63). However, for the onshell case the phase and amplitude equations are uncoupled.
The boundary condition satisfied by the VPA equations at the
origin follows directly from equations (2.14) and (2.33). From these
expressions it is apparent that
sin [61(q, k E; r)] 0 (2.38)
r +o
and that the value of cx1(q, k E; r) for the same limit is arbitrary, and becomes fixed only after the asymptotic boundary condition (2.15) is applied. On the other hand, if a hard core is present
3 (qr C) (2.39)
a~ ~ ~ c q E;r
q k; r r cos(61)jl(kErc) sin(61)nl(k Erc)
and 61(q, k E; r c) is arbitrary.
4._ Computational Aspects of the VPA Equations
The procedure used in calculating w 1(q, k E; r) consists of propagating two solutions to the VPA equations (2.36) and (2.37) such that they satisfy the initial boundary condition, and constructing linear combinations of these solutions so as to satisfy the asymptotic boundary condition (2.15). It is possible to construct two linearly independent solutions that satisfy the initial boundary condition due to the inhomogeneous character of equation (2.10). This procedure is analogous to that found in reference (Br75); however, instead of a piecewise breakdown of the potential and the propagation of local solutions as given there, numerical integration is involved here.
43
An important consideration is the singular nature of the VPA equations near the origin. Equations'(2.36) and (2.37) contain the potential as a multiplicative factor. This problem can be overcome by assuming hard core boundary conditions or by using a simple step potential within the first classical turning point [(Be60), (Br75)I1. The second of these procedures was chosen in this work. The length of the first step, r1, and the value of the assumed step potential over this first step, VV. were chosen such that making any further optimal changes in these parameters would not lead to a significant change in the value of the calculated matrix elements, wl(q, kE ; r) and tl(q' q; E). The guess work involved here can be eliminated by using a JWKB approximation to the derivative of the imaginary part of the wave function near the first classical turning point. The approximations involved are given in Appendix III. From expression (2.10) it is apparent that Im[w I(q, kE; r)] satisfies the usual Schrbdinger equation and boundary condition at the origin. It therefore follows that within the first classical turning point Im[w,] is decaying exponentially, and is zero at the origin. All that is necessary then, from a computational point of view, is to start the integration in a region where the magnitude of the derivative of In1[wiI is smaller than some given value. In our calculations a value of 1010 was found to be adequate. Another important consideration in propagating the V~PA equations is the cost involved in evaluating a large number of Bessel functions. Because of thqis, the coupled equations were integrated numerically using an Adams numerical integration routine developed by Shampine and Gordon, which.is particularly suited for problems for which derivative evaluations are expensive. These codes and a discussion of their efficiency and accuracy are found in reference (Sh75).
44
The propagation of two independent solutions to the inhomogeneous
differential equation (2.10) further presents two problems. One is that
of the growth rate of the amplitude in a nonclassical region and the second is that of the propagation through classically allowed regions at negative energies. The first problem was solved as given in reference (Br75). The idea was simply to use the propagated wave functions to
construct two new linearly independent solutions to Eq. (2.10), which would have much smaller amplitudes. Therefore we shall restrict our attention to the second problem. If the energy is negative, the modified RiccatiHankel functions k1and i1are the natural basis to use for uand v 1 in expression (2.20),. the VPA form for the offshell wave function w 1* However, if the potential is attractive and the energy is such that one must integrate through a classically allowed region, the wave function in this region would be oscillatory in nature. The RiccatiHankel functions on the other hand have a nonexponential growing and decaying character; thus, propagation of the offshell wave function under these circumstances is very difficult. The solution then
is simply to replace k 1 and i I by the RiccatiBessel functions jIand n l. This in turn requires a modification of the VPA equations (2.36) and (2.37). However, the modification is minor and does not lead to any significant increase in computational effort. Let us go back then to the
original inhomogeneous Schrbdinger equation, Eq. (2.10), as it applies to negative energies, and rearrange it into the form
k2 1(1+ 1) ~ w 22 W (k2 2)jlq) (.0
E r2 v)]1=kEw E1
Treating the left hand side of this equation as an inhomogeneous term, we can adopt the same procedure that was used in section 113 to derive the desired set of modified VPA equations
d 61(q, kE; r) kEI {v(r)[w wlJl(qr)] (2.41)
+ 2k2 Iwl Jl(qr)]}/x2
and
d kE; r) = kEI {v(r)wI + 2kE [I jl(qr)]} x
x {sin(61)Jl(kEr) + cos(61)nl(kEr)}. (2.42)
A similar set of modified equations could be used for positive energies when a nonclassical region is encountered there. This would require RiccatiHankel functions. This, however, was not found to be necessary in the calculations done in this work. Stabilization was only a problem at negative energies.
5. The Comparison Potential Method
Another possible approach in solving the inhomogeneous Schrddinger equation (2.10) for the offshell wave function w (q, kE; r), is to make use of the comparison potential (CP) method. As was mentioned in the introduction to this chapter, this approach has already been presented in the literature [(vL61), (Br75)]. We shall,however, sketch an outline of the method and present some improvements to the computational algorithm for w1" These changes are based on the work of R. Gordon [(Go69), (Go7l), (Ro76)], who has investigated extensively the use of
46
local approximations to the potential in order to simplify scattering calculations. The outcome of this pursuit will then be helpful in suggesting some improvements to the VPA method, or for that matter any method which implements a numerical solution for the offshell wave function w1
The basic idea behind the CP method is to approximate the potential on a given interval in such a way that one can express the solution to the resulting differential equation in terms of known functions which present no problem computationally. To meet these requirements the potential has often been treated locally as a step potential or as a linearly varying potential (Go7l). The centrifugal term, 1(1 +1/r may also be included in the potential, and the total effective potential may be approximated (Go7l). In the present work, however, only the potential was approximated, taking it to be a collection of steppotentials. Thus one has
[d1 c (1+ 1) (= k2 'j~r
dr 2 1 (Ek 1 ,q), (.3
and
a K? C k2 v (.4
i I E E i2.4
where a. plays a role analogous to aYE and v.i is thie local approximation to v(r) For r i < r < ri+i. The advantage of thts particular choice lies in the fact that one can use Bessel functions as 1the local basis from which to construct the solution to (2.43); and since the centrifugal term is treated exactly, the convergence toward tAe asymptotic boundary condition (2.15) is accelerated in the case of tht higher partial waves (An76). Locally then, on the ith interval
47
i u+r) i + i
W' (q, kE; r) = A u1 (K r) B1 v (Kir) + jl (qr), (2.45)
where
k2 q2
C1 2 2 (2.46)
2~ q
Gi~i q
The offshell wave function is then obtained from two propagated solutions in a form analogous to that given for the VPA approach. The propagation here, however, is accomplished by matching each wave function and its derivative across every interval. This can be done by either propagating the coefficients Ai and Bi as was done in the reference
1 1
(Br75), or by propagating the wave function and its derivative directly as we propose to do. To accomplish this, we shall modify the procedure given by Rosenthal and Gordon (Ro76) for the case of the usual radial Schrddinger equation. From expression (2.45) and the corresponding expression for the derivative of the wave function on the given ith interval, it is easy to show that
Ai = W[Wl(ri) C1 J (qri)' vl(Kir il/Wi (2.47)
and
i' = W [ wI ) i A
B1 1(ri Cl j, (q, ri)' ul (Kiri)]/Wi (2.48)
where W. is a constant equal to the value of the Wronkian of u1 and vl, the two local basis functions. The i superscript on w has been removed and is not necessary because of the continuity requirement imposed on the wave function and its derivative. Using (2.47) and (2.48) in the expression for w1l Eq. (2.45), and the analogous expression
d i d_ U. V l +)
~'W1 kE r) 1d dr 'I~Kr)
+ id (r (2.49)
C1 [s j1 Iq)
for its derivative; the desired result
rij)= fi[w(ri) C Ci(qr)] + C j (qri+1) (2.50)
is obtained, where we have introduced the matrix definitions
w~)E d (2.51)
L r 1
and
A ij, (qr)1
d A) E d) (2.52)
and the coefficient matrix Cwhose components are defined by the relations
C1= url dr vAi vrl 1d A~i]IW (2.53)
C12 = {v(r i+1) u (ri) u (ri4.1) v (ri)}/W1 (2.54)
21A i. dr dr dr
and
u2 r r1.) r i+1
40,
in order to simplify the notation, we have suppressed 'Yr from the argument of u 1 and vl, as well as the "1" subscript. Equation (2.49) is appealing in the sense that it provides a simple algorithm for propagating w, and its derivative from ri to ri+i, but it is even more appealing in the sense that one need not store the coefficients Al, B I and C' for all the intervals needed in the calculation. Only one problem remains to be dealt with. In the Ci' method the tmatrix is calculated replacing (2.13) by the expression ri +1,
t (q',1 q; E) = 2 14 i f j (q'r)w'(q, kE; r) (2.57)
1T iqq rj1
If one had propagated the coefficients Ai, B1 and C' on each interval one would have a set of integrals over products of two Riccatifunctions
as is indicated by Eq. (2.45), the expression for wand Eq. (2.57), the expression for t 1, These integrals are well known and are easy to
generate (Me6l). We, however, have been able to derive expressions for the integrals needed in Eq. (2.57) directly in terms of the offshell wave function wand its derivative. A derivation of these quantities is given in Appendix II.
One of the major criticisms of the VPA approach and other numerical procedures for obtaining w,, is that t1q, q; E) must be obtained through numerical quadratures. In practice this is problematic when q and q' are larger, because of the oscillatory nature of the functions involved. Generally, when one propagates wl numerically, it is determined at smaller step sizes than are involved in the CP approach; thus, one could take advantage of the integral expressions mentioned above by implementing a breakdown of the potential into steppotentials only in
50
the evaluation of t,(q', q; E). This would then remove any problems in this regard.
6. The Bateman Method
Very little has been done in the literature with regard to nonpartial wave techniques in the evaluation of scattering amplitudes or twobody tmatrix elements. Only within the Eikonal approximation [(G159), (Su69), (Ch73)] and approximations based on the Born expansion [(Ho68), (Ra72)] has this not been the case. The attempts that have been made to solve the LippmannSchwinger equation for either the full wave function or the tmatrix have involved numerical quadratures. For example, Walters (Wa7l) has considered the LippmannSchwinger equation for the full wave function and converted it to matrix form by use of a two dimensional numerical quadrature for the case of a screened Coulomb potential. The amplitudes obtained from this procedure however, though in good agreement with exact results, showed poor convergence when the number of quadrature points was increased. Another approach that has been taken is that of Rosenthal and Kouri (Ro73), where the LippmannSchwinger equation for the scattering amplitude was reduced from its twodimensional form to a onedimensional integral equation, which was then solved numerically. Using the work of Walters (Wa7l) for comparison, these authors obtained results within 5 to 10 per cent agreement. In contrast to these methods, the approach we shall mow pursue does not involve numerical quadratures, at least in the case of certain model potentials. This procedure, known as the Batemam method, was originally implemented by Akhmadkhodzhaev, Belyaev and Wrzerfonko (Ak70) in the computation of partial wave tmatrix elements. key have, however,
extended this work to the calculation of the full tmatrix elements for the case of a Gaussian potential (Be74a). with significant success. Lim and Gianini (Li78) have gone further and applied it to the case of a Yukawa potential, obtaining encouraging results.
Formally the Bateman method is equivalent to using the Schwinger variational principle [(Li78), (Ad75)]. We shall thus present the method in this framework. Let us begin then by considering the general identity (Le69)
[A(z)] = At(z) + At(z*)t At(z*)t At(z)1 At(z), (2.58)
where A is an operator that is Lconjugation invariant (A(z) = A(z*)t) and At is its trial value. Similar operator identities have been used as a basis for studying upper and lower bounds in the theory of bound state calculations (L665). Taking A(z) to be equal to t(z) and using equation (2.3), which relates the toperator to the wave operator W(z) in the variational expression (2.58),one obtains
[t(z)] = Wt(z) + Wt(z*)tV Wt(z*)t V VG o(z)V Wt(z), (2.59) where the trial operator Wt(z) can in general be chosen as Wt(z) = If>
Here If> represents a set of n basis functions {fFi,f2,f3,...Ifn} whose expansion coefficients {C1,C2,...,Cn} are determined variationally through expression (2.59). In the usual variational procedures (5172), (Ad74), (Pa74), (Ku78), the set of functions {fi i:s usually chosen to be square integrable; here however, {f} will be chosen to be a collection of plane wave states, i.e., the set denoted by {k} or {klk2,...,kn}.
52
Thus letting i f > be equal to > and making use of equations (2.59) and (2.60), one can obtain by taking the proper variations (Go6O) the expression
t( 1', q; z) = q( )C(q, Z) ,(2.61)
where C satisfies the relation
J(z)C(q, z) = 1(4) (2.62)
and where the components of the matrices I and J are defined as li() : ivl > (2.63)
and
Jij(z) = (2.64)
Another equivalent expression to (2.61) would be
t(', 4; z) = I(')J(z)lI(q) (2.65)
but we have chosen to use (2.61) and (2.62), in order to solve a simultaneous set of linear equations instead of the matrix inversion implied by this last expression.
The integrals needed in this procedure, as shown by equations
(2.63) and (2.64), are those required for the second Born approximation [(Da51), (Le56), (Ho68)]. It is readily found however that integrals of the type are not trivial to evaluate, and in general must be done numerically (Ho68). For the particular potentials we shall be interested in, the Morse and exponential potentials, one can obtain the desired integrals from various references [(Da51), (Le56), (Li78)].
53
In conclusion we should point out the weaknesses of the Cateman method. From Eq. (2.65) it is apparent that we have obtained a separable approximation to the tmatrix. Furthermore, using Eq. (2.1) it is easy to show that this could be viewed as a direct consequence of the separable approximation
V vlk>
to the potential [(Ad74), (Ad75)]. This is an important point, since Osborn [(Os73a), (Os73b)] has shown mathematically the nonconvergence of tmatrix elements based on separable approximations to local potentials. This problem exists irrespective of the approach used in constructing the given separable approximation. Fortunately, the difficulties arise for large values of the momenta and q in Eq. (2.63), values at which the tmatrix elements are small and have no significant effect in threebody calculations (S173). Finally, there remains the problem of the inapplicability of the Bateman approach to singular potentials, a question open to further investigation.
CHAPTER III
NUMERICAL RESULTS FOR THE TWOBODY tMATRIX
Having developed the formalism of the VariablePhase and Amplitude method in the previous chapter, we will now apply it to various model potentials of physical interest. This will serve to illustrate various known analytical properties of the twobody tmatrix and to gauge the reliability of the computational procedure.
We will concentrate on calculations for the lowest g (attractive) and 3u(repulsive) potentials of H2 because of the role these play in the collision dynamics of H + H2, and because their appearance in dynamical studies can be justified within a diatomicsinmolecules treatment (Mi77b). A comparison of some of our results will be given with others obtained within a variational procedure, developed to solve the threeatom problem at low (thermal) energies (Ku78).
A study will also be made on the reliability of the Bateman method in the computation of tmatrix elements for potentials of interest in atomic and molecular scattering theory.
1. Properties of the TwoBody tMatrix
A brief summary of the various properties of the offshell tmatrix elements, tl(q', q; E), will now be given. This will prove valuable in determining the reliability of any given computational procedure used to obtain tI.
The most important properties satisfied by tl(q', q; E) are those
54
of symmetry,
tl(q', q; E) = tl(q, q'; E) (3.1)
and of offshell unitarity,
Im[tl(q', q; E)] = 7mkEtl* (kE, q'; E)tl(kE, q; E) (3.2)
The symmetry relation is valid at all energies, but the unitarity relation is valid only for positive energies. These relations are well established in the literature and will not be derived (Wa67). It suffices to say that in this case they are a direct consequence of assuming a real local potential.
If one considers the differential equation satisfied bywI(q, kE; r), Eq. (2.10), it is apparent that wl reduces to the onshell wave function when q = kE. Similarly, the offshell tmatrix elements are defined such that there is a smooth transition to the onshell behavior as the momenta involved approach the onshell quantities. This is exemplified by the relations (Wa67)
wl(q, kE; r) j (qr) 'rmqtl(kE, q; E)h1 (kEr) (3.3)
r c
and
Im[tl(q, kE; E)]
tan[61l(kE)] = Re[tl(q, kE; E)] (3.4)
which correspond to well known onshell relations for q = kE.
Another important aspect of the tmatrix elements is their behavior as a function of momenta and energy. In this respect, there are various limits of interest. If the energy E and initial momentum q are fixed, then
lim tl(q', q; E)= 0 (3.5)
q _. 0
and therefore because of syirnetry
lim t(q', q; E) = 0 (3.6)
q + 00
These relations are apparent if one considers the oscillatory nature of wl(q, kE; r) and j1(q'r) in expression (2.13) as q and q' approach infinity. A clearer understanding of relations (3.5) and (3.6) could be obtained from a semiclassical analysis of the twobody tmatrix as a function of energy and momenta. Such an analysis has been carried out by Korsch and Mbhlenkamp [(Ko76), (Ko77a)] for the case of purely repulsive potentials. These authors have classified the various regions in the qq' momentum plane in which t, corresponds to classically allowed or forbidden processes. Classically forbidden processes would, for example, be associated with complexvalued classical paths or trajectories.
As a function of energy, the tmatrix must also show continuity as the energy goes from positive to negative, and consequently
lim tl(q', q; E) = lim Re[tl(q', q; E)] (3.7)
E 0 E 0+
and
lim Im[ti(q', q; E)] = 0 (3.8)
E * 0+
For negative energies, the tmatrix elements are real and have poles at energies corresponding to the bound states of thz system. This can be established by considering the expansion of t(' ; z) in terms of the eigenfunctions of the full Hamiltonian of the system (Ne66). The result is as follows:
t(q', q; z) = (nl ')(z q) nm nm
nlm (z enl)
+ t(',q; E +)
+ f dq"[(Eq, Cq ic) + (z k)1
x t(q', i"' Eq11+) t*(q", q; qII) (3.9)
where gnlm(q) is a bound state function in the momentum representation having the eigenvalue of enl The quantities, eq, are eigenvalues to the scattering states of the system.
2. Numerical Calculations Using the VPA Equations
We will now present various results for the H2 system for the
lowest 1 g+ and 3Eu potentials [(Mi77c), (0178)]. Some of these results have already been reported in the literature [(Be78), (Ku78), (Ko77a)]. A Morse potential
V(r) = D[l ea(rro)]2 D (3.10)
was used to represent the 1E + interaction potential, and a Hulthdn potential
V(r) = Aer/a[l er/a1 (3.11)
was used for the repulsive 3 Zu interaction potentials. The parameters 0o1 0
used were D = 4.786 eV, a = 2.1123 A ro = 0.7411 A, A = 20.11 eV and
00
= 0.4984 A (Mi77c). This particular study will prove interesting, since it serves to draw a contrast between the analytic properties of t1 and w1 obtained from an attractive potential, and those obtained from a strictly repulsive potential.
58
in figures (3la) through (31c), we have plotted various components of the positive energy offshell wave function w0 for the case of a Morse potential. If one recalls the VPA expression for w given by Eq. (2.26), it is clear that one can consider w1(q, kE; r) to have two components: jl(qr) and Xl(q, kE; r), where X, is defined by X1(q, k; r)= Wl(q, kE; r) jl(qr) (3.12)
and is often called the wave defect (Sr68). The term jl(qr) in Eq. (3.12) is present in equations (2.26) and (3.12) because of the inhomogeneous character of the differential equation obeyed by wl [see Eq. (2.10)]. Thus, we see from Eq. (3.12) that for positive energies
Re[wl] = jl(qr) + Re[xl] (3.13)
and
Im[1l] = Im[Xl] (3.14)
Equation (3.13) implies that the real part of w is highly influenced by the driving term jl(qr). This can be seen in figures (31a) and (31c), which show how the function [2/w]I/2 Re[wo(K, KE; R)/Kj oscillates about the driving term [2/]1/2[jo(KR)!K]. The data used to obtain these figureswere ''= 0.23 a.u. and E = 0.01 a.u. If one were to increase the value of either K or E, the offshell wave function would become much more oscillatory. Similar results were also obtained for the Hulth~n potential. We note, however, that an attractive potential pulls the nodes of the offshell wave function in, whereas a repulsive potential pushes them out. For negative energies, the offshell wave function becomes real. An illustration of w, at negative energies
(a 4 M:
C:)
4
LLJ
co CJ
Nd
Ce C:L: e
C) c):14
0) E<'r) = C)
CNJ Cj
4) CQ M
4)
a
zi a)
P
u to UJO
60
r
u
L
Q
61
is given by figures (32), (33) and (34) for the Morse potential case. Since the tmatrix elements have poles at the bound states of the system [see Fig. (35)], we see from the relation connecting tl and wl [Eq. (2.13)] that the amplitude of w! must grow as the energy approaches a bound state eigenvalue. In Fig. (32), [2/] 1/2[(j0/K] is plotted versus R for K = 4.475 a.u. and an energy value of E =
0.1446 a.u. The large magnitude of the amplitude displayed by this function, when compared to the result at E = 0.153 a.u. given in Fig. (33), is a clear indication that the energy is quite near to an eigenvalue of the potential. A similar effect can be seen at higher partial waves, as is shown in Fig. (34) where [2/T]1/2[w5(K, KE; R)/K] is plotted versus R for K = 4.5 a.u. and E = 0.153 a.u. Note that the effect of the driving term, jI(KR), is always present in all of the figures (3.2) through (3.4). Within the interaction region of the potential, it is the wave defect X, which dominates; whereas the driving term dominates in the asymptotic region. For the Hulth~n potential, the offshell wave function does not display the large changes in magnitude mentioned above for the Morse potential, since the Hulth6n potential used in this study does not support bound states.
Various studies of the partial wave tmatrix elements as a function of energy and momenta were carried out for both the Morse and Hulth~n potentials. We have already shown the effect of the bound states on the offshell wave function wl. Their effect on the tmatrix elements is also quite important. In Table (31), we have tabulated the Morse to(q, q; E) matrix elements for q = 4.0 a.u. as a function of energy. Column I gives the VPA results and column II the corresponding results obtained by a variational method developed by Kui'glu and Micha (Ku78).
C)
0 Uj CL 43 '0
0
cr a)
C
4)
LLJ
CXL) 4) r 0 (0+) > 0
V) (U
tA
c 0
a)
Cj AI
cm
.r
LL
63 ..........................
0,4
C6
(A CY)
4) LLJ cl)
> Cl)
0)4)
Q) C:
4)
(0.0 a)
4)
4) 'C 0 (ts c a0
U a) cu (A
S
o
Z 41
ro
4 U) C) r
4)
CQ
C14 CL) r.0
CY)
LL.
65
LO
cr_
cn L2
C U1
0
14
LL
67 ............ ... 0
co
D
0 L ";f
0
0
0 0 0 0
4
04)
a
S 4J =3 0 4) CL
u = Q) S (A 4) So
0 CD
0 LLI
4j
C)
L.) 4)
0
Lf)
I
ce)
LL
Lo cu
0
X
70
Table (31) The t0(,q; E) matrix elements are
given for q = 4.0 a.u. Column I
gives the VPA calculations and column II the results obtained from a variational procedure (Ku78). (Morse
Potential)
Energy (a.u.) to(q, q; E) x 102
I iI
0.18 2.5971 2.5798
0.17 4.4008 4.3841
0.16 2.0273 2.0069
0.15 8.8552 8.7964
0.14 9.9843 10.0183
0.13 2.0411 2.0680
0.12 1.5197 1.5486
0.11 1.2378 1.2106
0.10 1.0856 1.1159
71
We note that in general the agreement is quite good. A more detailed picture about the structure of the Morse t0 matrix elements at negative energies can be seen from the results given in Table (32) and Fig. (35), each of which describes the energy range corresponding to the first four poles. The figure shows that the tmatrix elements change sign as one goes across a pole, in agreement with equation (3.9). For the Hulthdn potential, the tmatrix elements show a very smooth and plain structure as can be seen in Table (33), where to(q, q; E) is given versus energy for q = 4.5 a.u. The agreement with the variational results is again quite good. At positive energies, we have shown in Fig. (36) the behavior of the Morse to(q, q; E) matrix elements near E = 0 a.u. [q = 4.0 a.u.]. We see the continuity one expects; i.e.,both equation (3.7) and (3.8) are satisfied. In general, the tmatrix elements of both the Morse and Hulthdn potentials show a much more oscillatory structure at positive energies. Figures (3.7a) and (3.7b) illustrate this for the diagonal (K' = K) Hulth~n to(K', K; E) matrix elements. The large oscillations seen in these figures do not extend indefinitely. Korsch and M6hlenkamp (Ko77a) have extended the present calculations to higher energies and have shown that the amplitude of the tmatrix eventually decays to zero. In the comparison of their semiclassical results with the present quantum mechanical ones, they obtained good agreement. As a function of momenta, the tmatrix elements show a much greater oscillatory nature. This can be seen in Fig. (38) where the Morse to(K, K; E) matrix elements are plotted versus K for E = 0.1378 a.u., and in Fig. (39) where the Morse to(K'; K; E) matrix elements are plotted versus K' for E = 0.153 a.u. and K = 4.5 a.u. Both figures (3.8) and (3.9) show how the tmatrix goes to zero asymptotically in
72
Table (32) Selected values of t (Q, q; E) for
q = 4.475 a.u. are presented about
the first four poles. (Mlorse
Potential)
Energy (a.u.) t(K,K:E) Location of Pole
0.165078 2.6856
0.165076 5.4287
0. 1650741
0.165074 174.7388
0.165072 5.0667
0.144800 1.2866
0.144600 3.5240
0.144486
0. 144400 4.6930
0. 144200 1. 4018
0.125600 0.0290
0.125400 0.0936
0. 125268
0.125200 0.2220
0.125000 0.0656
0.107800 0.0011
0.107600 0.0026
0. 121421
0.107400 0.0243
0. 107200 0.0020
73
Table (33) Selected values of t 0(q, q; E)
for q = 4.0 a.u. are given.
Column I gives the VPA calculations and column II the results obtained from a variational procedure (NuOW. SAuWh~ Potential)
Energy (a.u.) t 0(q, q; E) x 103
I II
0.18 6.4807 6.4592
0.17 6.2683 6.2477
0.16 6.0525 6.0300
0.15 5.8277 5.8057
0.14 5.5946 5.5742
0.13 5.3574 5.3348
0.12 5.1079 5.0868
0.11 4.8526 4.8295
W
o
Z
4
1
C 11
LLJ
C)
4) 4
o
Cn
75
X104 104 (cl.U.)
Re t,(%KKE) (a.u.) I m Q KKE7.)X
4j tr
0
0
L0
LO LO
70L X
LL, :5z
C)
4)
(Ii
ce
4
0 (A
4)
E
x S
E
4)
0
xv
LL
77
LU
0
d 7s %.oo
0
0
co 0
0 L (3!)]'M) 1
LLI
C)
4)
E
4
0
(A
4) rla)
E
x
4) m
E
vo
:he 4)
C
43 0
0 4J
W S
I
Ln
U
79
0
C:
0 IT
0
C13,
Lo
0
0
co
COL (:t /I' ,I),I )
A3
4) 4) c c a) (1) E 4)
x (n
S
S o
1j
E
Ile
LLJ
to a C) 04)
0 I= 4
co
A
cn 4
LL
OA
0
CL CC) 0 (Y) T
o
r LO
w
0
0
Figure (39) tn(K', K; E) for E = 0.153 a.u.
aid K = 4.5 a.u. (Morse Potential)
,03
0.09 1 1
0.06
0.03 1
0.00
0.03 1
0.061
0.0 5.0 10.0 15.0 20.0
K' (a.u.)
K and K' [see equations (3.5) and (3.6)]. Comparable behavior was observed at positive energies as well, irrespective of the attractive or repulsive nature of the potential. Table (34) shows a momentum study for the Hulthdn potential to(q', q; E) matrix for E = 0.15 a.u. and the 1 values 0, 5 and 10. These results indicate that the diagonal q' = q tmatrix elements are consistently larger in magnitude than the offdiagonal q' t q tmatrix elements. Column I in this table gives the VPA results and Column II the corresponding variational results (Ku78). As a function of the partial wave parameter 1, the tmatrix generally shows an oscillatory structure for low partial waves and a smoothly decaying structure for high partial waves. Figures (310a) and (310b), nicely illustrate this effect for both the real and imaginary components of the Hulth~n tl(K', K; E) matrix elements (K' = K = 4.0 a.u. and E = 0.01 a.u.). Using the same data, we see in figures (311a) and (311b) plots of the real and imaginary coqmnents of the total tmatrix [t(K', K; E)] versus momentum transfer P = IK' Kj. These plots imply that the tmatrix elements become smiiler as the momentum transfer increases. In this particular case, thirtyfive partial waves were required for the total tmatrix to converge As the energy and momenta arguments become larger, this number will generally increase. Partial wave convergence will be discussed furthEr in Chapter VII, where the twobody tmatrix elements are used in the threebody atomdiatom scattering problem.
The tmatrix results we have reported here vare all obtained from Eq. (2.13) using a numerical quadrature, which in this case was a combination Simpson's and Newton's 3/8 rule (Hi56). In order to test the reliability of this procedure, we have used Eq. *(3.2) to obtain phase
85
Table (34) Selected values of t1 (q', q; E) are
given for E = 0.15 a.u. and the partial wave numbers I = 0, 5 and 10.
Column I gives the VPA calculations
ant column II the results obtained from
a variational procedure (Ku78). (Hulth~n
Potential)
1 I q t I(q', q; E) x 103
I II
0 3.0 1.0 0.7899 0.7894
2.0 4.0665 4.0693
3.0 10.187 10.179
4.0 2.1074 2.1160
5.0 0.2154 0.24765 3.0 1.0 0.0207 0.0207
2.0 1.1745 1.1755
3.0 5.3773 5.3709
4.0 1.0412 1.0434
5.0 0.0513 0.054710 3.0 1.0 0.0005 0.0009
2.0 0.1174 0.1186
3.0 1,4905 1.4857
4.0 0.2644 0.2668
5.0 0.0093 0.0125_
S. Q)
(1) 4) > 0
LLI w
.r
4J
4
4 C) 0 .
C)
4) Oil
C)
Cn
87
Lo
co
N
10 Ln Go TN
:3 4J (A 0) 43 > 0
LLI W
9
4)
41
E
4 CD 0 .
C)
4)
0 11
CL LLI
C)
S
U
LO
(Y)
00 0 AW N
N
10
00 0) Lo
C5 b
43
4)
AW 4=1
11 :3
4
S
Gi
> C)
C
LLJ
.. LLJ
t
t
4) 4
C)
4 .
0 Rlt
4) 11
0
91
00
(Yi CY) LO C\j
C\j
4)
4J
>
ui
t * LLJ
43
E
P4 4 C) 0 .
lqr 4J
0 11
CL
4
cu S
LL

Full Text 
where
(6.27)
For a symmetric hard core potential it is easy to show that
a)(q, kE; x) = (2tr)"1/2 [eiqx eiqxc eikErc eikEr] (6.28)
and
 (q kE)(q' + kÂ£) e lAqrC] ,
(6.29)
where Aq = (q1 q). Although equations (6.28) and (6.29) were obtained
from the equations outlined in this section, one would arrive at
identical results by considering a potential of the form V(x)=A0(rcr)
and taking the limit as A approaches infinity. For more complicated
potentials, such as an exponential potential including a hard core, one
must generally go to a numerical solution for co^ in Eq. (6.22). The
procedure we have chosen is completely analogous to the comparison
potential method outlined in Chapter II and therefore will not be
repeated.
3. Results for Inelastic Scattering
The quantity of interest in coll inear scattering is the transition
probability P... for going from an initial state i to a final state f.
In the case of inelastic scattering we have the well known result
[(Th68), (Ek71)]
Figure (33) [2/tt]1/2 [wq(K, Kj:; R)/K] at an energy value that lies
between the first and second bound states. (E = 0.153 a.u.
and K = 4.475 a.u., Morse Potential)
4.52
2.47
0.42
12
i
j
f
24
T
T
_J 1 I
36 48 60
vO
o
5
Tne neglect of all the momentum dependent coupling terms tjj, and ,
(all 1 and 1) is known as the BornOppenheimer approximation [(Hi67),
(Tu76a)]and leads to a new set of coupled equations
[Ta + Uii E]xl = ^ ,E=1 U11 ,X1 ^1,16)
in which electronic transitions may occur only through the nondiagonal
potential terms U,],.
The effectiveness of Eq. (1.6) in properly describing collision
processes involving electronic transitions depends crucially on the choice
of the functions {<Â¡>^}. For example, if is chosen to belong to the
adiabatic representation, i.e., that representation which diagonalizes
the Hamiltonian
H' = Hel + H
SO
one obtains the set of uncoupled equations
(1.17)
where
[t +
a
 E]x1 = 0 ,
V,(r ) = U,,(r ) .
1 a 1 i or
(1.18)
(1.19)
In this representacin, transitions between the various electronic states
are not allowed, unless one considers the momentum dependent coupling
terms. The term adiabatic representation is also used in the literature
for that representation in which the electronic Hamiltonian H^ is diag
onal [(Hi67), (Tu76a)]. There are, however, an infinite number of non
adiabatic representations. One set of representations encountered in
the literature are the so called diabatic representations [(0m71),
(Sm69)]. These are representations in which the electronic functions
38
A derivation for the case of a pure hard core potential is given in
Appendix I.
3. The VariablePhase And Amplitude Method
The aim of the VPA method is to take advantage of the known
oscillatory nature of the offshell wave function, co^(q, kÂ£; r). This
is done by considering a solution to Eq. (2.10) which has the form
below:
o,(q, kÂ£; r) = j,(qr) + a.,(q, kÂ£; r) cos [S^q, kÂ£; r)]u1 (kÂ£r)
 a,(q, kÂ£; r) sin [6, (q, kÂ£; rjjv^^r), (2.20)
A A A
where u^(kÂ£r) and v (k^r) are RiccatiBessel functions (Ji(kÂ£r) and
A A A
n1(kÂ£r)) or RiccatiHankel functions (k^(kÂ£r) and i^(kÂ£r)) (Ca67). The
choice is determined by the nature of the region through which one is
propagating, i.e., whether the region is classically allowed or not.
Using equations (2.10) and (2.20), one may obtain a set of coupled
equations for the amplitude, a,(q, kÂ£; r), and the phase, 5^(q, k^; r).
These equations will now be derived using an integral equations approach
(Ro67). The functions u, (kÂ£r) and v^(kÂ£r) used in expression (2.20)
satisfy the RiccatiBessel differential equations (Ab65)
z, + [E  ^ +2^] z, = 0 (2.21)
dr 1 / 1
Taking the regular and irregular functions uj and v, to be normalized
such that their Wronskian, W[u, v,], is unity, the Green's function
associated with (2.21) may be written as
g,(r. r') = (kEr>) ,
(2.22)
194
as a comparison to more approximate procedures.
The collinear studies in Chapter VI clearly indicate that the valid
ity of the peaking approximation may be limited to cases where the po
tential is very repulsive and hard. It was not possible to pursue this
matter in the case of threedimensional motion, because of the complex
ities arising in the angular momentum analysis. This question must be
answered, however, if one is to properly judge the validity of the single
collision approximation. We pointed out in Section V.2 that the eval
uation of the single collision transition amplitudes would entail the
introduction of partial wave expansions for the various quantities
involved [see Eq. (5.22)]. At high energies this would not be practical,
because of the large number of partial waves required to obtain conver
gence. In order to circumvent these difficulties, one must either intro
duce decoupling approximations, or make use of multidimensional quadra
ture techniques. Given the simplicity of the single collision term
[see Eq. (5.24)], the latter possibility would seem thebest alternative,
at least in the case of repulsive potentials. If the potentials support
bound states, the presence of poles at negative energies would severely
complicate the use of multidimensional quadratures.
Finally, we note that the results on collision induced dissociation
reported in Section VI.4 for collinear scattering indicate that it is
essential to include certain double collision terms if reliable results
are to be obtained. Essentially, it was demonstrated that the interac
tion between the two atoms of the dissociating diatomic must not be
neglected. Recall that the role played by this Â¡potential in the case
of inelastic scattering was merely one of determining the initial and
final momentum distributions of the target diatom. This is clearly not
203
where m, = m + m Here we note that the definition for r. is given
3y 3 Y 1
by Eq. (A4.1) can be associated with the mixed index set (iyB) which
is equal to (1CB), (2AB) or (3BA). One can further show, using equations
(A4.1) and (A4.2), that the various channel relative coordinates satisfy
the relations
r fi (ij)
may J 0
(A4.3)
and
^ = (ij)
maymBy
> m
rj m
Â£ ft.
By
(A4.4)
where (ij) is equal to +1 if i and j are in cyclic order or 1 otherwise.
M is the total mass of the system. In each case the transformation
Jacobian is equal to unity.
At this point, one can now introduce the momenta p.. and P.Â¡, con
jugate to f.. and ft respectively. These relative momenta can be defined
in terms of the p as follows,
a
V
^3 > ^y *
"W ^ mBY Pb
(A4.5)
and
p = ft k ft ^Ln
*1 M Pa M PB M Py
(A4.6)
V
Qualitatively then describes the relative motion of atom a with
respect to the pair (By) having the relative momentum p^, in the given
channel i. We can further introduce the channel reduced masses m. and
M. defined by the relations,
(A4.7)
58
In figures (3la) through (3lc), we have plotted various components
of the positive energy offshell wave function w for the case of a Morse
potential. If one recalls the VPA expression for w given by Eq. (2.26),
it is clear that one can consider wj(q, k^; r) to have two components:
A
j(qr) and xj (^ kjrl r), where xj is defined by
A
Xj(q* kÂ£; r) = o)^ (q, kÂ£; r) j^qr) (3.12)
A
and is often called the wave defect (Sr68). The term jj(qr) in Eq.
(3.12) is present in equations (2.26) and (3.12) because of the inhomo
geneous character of the differential equation obeyed by ojj [see Eq.
(2.10)]. Thus, we see from Eq. (3.12) that for positive energies
Re [oj1 ] = jj(qr) + Refx,1 (3.13)
and
Imfco, ] = ImCxj] (3.14)
Equation (3.13) implies that the real part of wj is highly influenced
A
by the driving term jj(qr). This can be seen in figures (3la) and
(3lc), which show how the function [2/tt]^2 Re[ujQ(K, K^; R)/K] oscil
lates about the driving term [2/tt] 1//2[(KR)/K]. The data used to
obtain these figures were K= 0.23 a.u. and E = 0.01 a.u. If one were
to increase the value of either K or E, the offshell wave function
would become much more oscillatory. Similar results were also obtained
for the Hulthn potential. We note, however, that an attractive poten
tial pulls the nodes of the offshell wave function in, whereas a repul
sive potential pushes them out. For negative energies, the offshell
wave function becomes real. An illustration of coj at negative energies
29
characterized by the repulsive forces of the system. Thus, an impulsive
model based on the truncation of the multiplecollision expansion would
be expected to yield useful information on these systems. An advantage
to this approach, as will be shown in Chapter V, is that one is able to
obtain statetostate information directly from the initial and final
momentum distributions of the target and product diatom, and from the
various twobody potentials involved. Due to the small vibrational
rotational spacings of CO and M2, other methods such as the coupled
channel method, based on Eq. (1.23) would be computationally prohibitive
(Th78b).
We shall assume in our implementation of the Faddeev equations that
electronic motions have already been eliminated and that only information
from the ground adiabatic potential energy surface of the system is
needed. As mentioned previously, the scattering process at hyperthermal
energies probes the short range part of the potential. Therefore, we
shall develop a many body theory based on the decomposition of the adia
batic potential into the form (Ki79)
V(H r) = Vsr + Vlr (1.63)
where
3
V, = 2 V.(r.) (1.64)
sr i=i 1 1
and
Vlr = l v?(R, r)Yx(R, r) (1.65)
X
The Vjr term denotes the long range potential and is analogous to the
A A
Legendre expansion used in Eq. (1.62). Here, however, Y^(R, r) is a
3. Numerical Results Obtained Using the Eatemar. Method
From the results given in the last section, it should be clear
that a method which gives the total tmatrix t(q, q; E) without a
partial wave expansion would be highly desirable. Such a method would
hopefully avoid the problems of computing a highly oscillatory function
for many partial waves. To this end, we now present some results
obtained using the Bateman method as outlined in Section (26).
Computationally all that is needed to obtain t(q', q; E) using the
Bateman method is the evaluation of the integrals of the type
and along with a procedure for solving a set of complex
simultaneous equations [see Eq. (2.61)]. For the Yukawa and exponential
potentials, analytic expressions for the above integrals may be found in
the references given in Section (26) [(Da51), (Le56), (Li77)]. The
integrals for any other potential which is a linear combination of
exponentials may be obtained using the relation (Br70)
+0O
V(r) = / dA
oo
p(A) eXr .
(3.15)
where p(A) is some linear combination of delta functions. For the Morse
potential, one has
p(A) = D[e2ar <5(A 2a) 2ear 6(A a)] (3.16)
The set of complex linear equations given by Eq. (2.61) was solved
using the IMSL library program LEQTIC (Ai75). Siince the input matrix
elements have quite complicated analytic expressions, their
accuracy was verified both analytically and numerically. In all cases
the analytic and numerical results agreed to ten or more decimal digits.
In the present calculations, we have found tthat the Bateman method
186
in the quantal calculations, we have plotted the radial formfactor
fQj 00j [A=Li+, E=C and C=0] as a function of jf in Fig. (76). This
figure shows a rather pronounced oscillatory behavior as a function of
jf, but with a rapidly decaying tail for high jfvalues. This decaying
tail leads to the rapid decrease in the rotational transition probability
for high jvalues.
An attempt was made to extend the present results by using the
exponential potentials given in the introduction to this section.
However, the energies involved are quite high, making it necessary to
use more than one hundred partial waves in the evaluation of the neces
sary tmatrix elements. Because of this, the partial wave tmatrix
elements involved are very small and the errors accrued at the high
partial waves did not allow convergence in the full tmatrix. As was
pointed out in Chapter III, it is these conditions which are least favor
able for numerical procedures such as the VPA method.
Generating the various differential cross section plots shown in
this section required a rather large number of tmatrix evaluations.
It is the evaluation of these functions which can be quite costly,
when numerical methods are used. Because of this, plots of the real
Pk
and imaginary components of the 2 were made as a function of the final
rotational quantum number. These results are given in figures (77a)
and (77b), Reft^] x 10 and Im[t[^] x 10 are plotted versus j^.
3. Discussion
At hyperthermal energies, the study of atomdiatom scattering
processes using coupled channel methods becomes prohibitively expensive
and impractical. It was shown in Chapter IV that iterating the AGS
Figure (73) Plot of differential cross section (arbitrary units)
versus final rotational quantum number for the (Li+, CO)
system. [ (nÂ¡, j )(0,0), nf=0, E1=4.23eV and 0p, p =49.2]'
156
and
M0lC> = <40llVIV
 <^qIT2 + "^3 + + 1^>0> (6.49)
in other words they differ by the presence of the double collision terms
(s,c) (s)
TjG0T2 and T1^oT3 In t*ie c111'near case Moi differs from Mqj by
the term T^G^. Physically, this means that it is important to con
sider the interaction between the two atoms of the dissociating diatomic.
(s)
The rather drastic difference between the results obtained from Mq^
(s c)
and Mqj in the coll inear case may on the other hand not extend to
the threedimensional results, since the "dimensionality bias" may play
an important role in exaggerating the importance of the double collision
terms present in Eq. (6.38). Nevertheless, this question must be addressed
in the study of the full threedimensional CID studies. Another aspect
of the present calculations must be emphasized. The formalism we are
using is based on the Faddeev equations and differs from that used by
Eckelt, Korsch and Philipp (Ek74), who used the formalism of Bianchi
and Fabella (Bi64). In this latter approach, these authors have intro
duced a further approximation leading to what is known as the impulse
approximation in the literature [(Mc70), (Ro67)]. Within the present
(s c P k)
approach, this approximation would amount to replacing by
M
(s,c,I)
01
*Pk, * .
c3 P3f q3i
Eq.) / dQ3 qj' >,
(6.50)
where t^ (p*
r
3f q3i; h
q3
*2
2M3
) in Eq. (6.35) has been replaced by the
halfonshell tmatrix element t^ (P3f> q3l; Eq *). The peaking argu
merits p^, q^ and are determined by the procedure given in Section
Table (35) Columns I and III are phase shift values
obtained from halfonshell quantitites,
and columns II and IV are the correspond
ing onshell quantities. All phase shifts
are Mod (it). (Morse Potential)
Â¡(E =
0.01 a.u.)
6, (E =
0.04 a.u.)
I
II
III
IV
0.8655
0.8647
1.2732
1.2757
0.8463
0.8471
0.1843
0.1819
0.4431
0.4421
1.5293
1.5270
1.5482
1.5494
0.3781
0.3805
0.5379
0.5373
0.7473
0.7446
0.3403
0.3390
1.3752
1.3780
1.0825
1.0815
0.4598
0.4627
1.4480
1.4494
0.3532
0.3524
0.9677
0.9685
1.0698
1.0702
0.6154
0.6157
1.4366
1.4477
0.3781
0.3896
0.8966
0.9154
10
183
The renormalized results were obtained by scaling the entire spectrum
so that the largest peak has the value one. We note that the differences
in all cases are minor. One of the reasons for these unexpected small
differences is that the magnitudes of the momenta that occur in the
present calculations are all similar. We caution, however, that for
softer potentials this may not be the case. As was pointed out in
Chapter m. the tmatrix may be a very oscillatory function of its
momentum arguments.
It was argued in Section V.2 that the momentum overlap volume
give by Eq. (5.36) could provide a rough measure of the probability
amplitudes. From Eq. (7.12), it is clear that two such overlap volumes
are involved. Each single collision term leads one to consider two
overlapping spheres as defined by the functions and
present in Eq. (7.4). Each of these functions is normalized, implying
that a meaniful comparison to the single collision peaking calculations
can be obtained only by normalizing the square root of the volume of
the sphere determined by . From the (Li+, CO) data used to
generate the energy loss spectra given in figures (.71) through (73),
we obtained the renormalized overlap volumes plotted in Fig. (75)
versus j^. From the three curves plotted in this figure, it is apparent
that the peaks are shifting to higher final jvaDues as the scattering
angle increases. This is the same conclusion that was drawn from
figures (71) through (73). There is also a general similarity of
the shapes of these curves to those found in the quantal results.
However, the momentumoverlap model predicts a greater amount of rota
tional excitation. The interference features are also not contained in
this simpler mode. To give an idea of the behavior of the formfactors
147
Table (63)
Transition probabilities
using the parameters m =
A = 10 and = 5.928 in
for a soft exponential potential,
1/13, a = 0.1287, rc = 2.0,
reduced units.
p10
P20
P30
P40
Single Collision Approximation
0.893
1.299
1.317
Peaking Approximation
2.174
4.063
2.81
0.878
3.107
(Be78) L. H. Beard and D. A. Micha, Chem. Pys. Lett. 53, 329 (1978).
(Bi64) L. Bianchi and L. Fabella, Nuovo Cimento 1823 (1964).
(Bi73) R. F. Bishop, Phys. Rev. C7, 479 (1973).
(Bo65) J. W. Bond, K. M. Watson and J. A. Welch, Jr., Atomic Theory of
Gas Dynamics, AddisonWesley, Reading, Massachusetts (1965),
Chapter 4.
(Bc74) A. Bogan, Phys. Rev. A9, 1230 (1974).
(Bd76) R. Bdttner, U. Ross and J. P. Toennies, J. Chem. Phys. 65, 733
(1976).
(Br30) H. C. Brinkman and H. A. Kramers, Proc. Acad. Sci. Amsterdam
33, 973 (1930).
(Br70) B. H. Bransden, Atomic Collision Theory, W. A. Benjamin, Inc.,
New York (1970), Chapter 1.
(Br75) P. Brumer and M. Shapiro, J. Chem. Phys. 63, 427 (1975).
(Bu67) A. D. Buckingham, Adv. Chem. Phys. 12[, 107 (1967).
(Ca63) F. Calogero, Nuovo Cimento 27_, 261 (1963).
(Ca67) F, Calogero, Variable Phase Approach to Potential Scattering,
Academic Press, New York (1967).
(Ch71) J. C. Y. Chen and C. J. Joachin, Physica ET3, 333 (1971).
(Ch73) J. C. Y. Chen, L. Itambro and A. L. Sinfailam, Phys. Rev. A7,
2003 (1973).
(Ch74) S. Chapman and R. K. Preston, J. Chem. Phys. 60, 650 (1974).
(Ch76) C. Chackerian, J. Chem. Phys. Â£5, 4228 (1976).
(C170) W. L. Clinton, C. M. Cosgrove and G. A. Henderson, Phys. Rev.
A6, 2357 (1970).
(Co68) J. P. Coleman, J. Phys. B 1, 567 (1968).
(Cu75) J. T. Cushing, Applied Analytical Mathematics for Physical
Scientists, John Wiley and Sons, New York (1975), Chapter 2.
(Da51) R. H. Dalitz, Proc. Roy. Soc. A206, 509 (1951).
, .0
(Da74) G. Dahlquist, A. Bjorck and N. Anderson, Numerical Methods,
Prentice Hall, Englewood Cliffs, N. J. (1974).
(Ea78) W. Eastes, U. Ross and J. P. Toennies, J. Chem. Phys. (to be
published).
9.9 0.5 8.9 18.3 277 371
AG
56
and therefore because of symmetry
lim tj (q, q; E) = 0 (3.6)
q 1 y oo
These relations are apparent if one considers the oscillatory nature of
A
toj (q, kÂ£; r) and j^ (q' r) in expression (2.13) as q and q' approach
infinity. A clearer understanding of relations (3.5) and (3.6) could
be obtained from a semiclassical analysis of the twobody tmatrix as
a function of energy and momenta. Such an analysis has been carried
out by Korsch and Mohlenkamp [(Ko76), (Ko77a)j for the case of purely
repulsive potentials. These authors have classified the various regions
in the qq' momentum plane in which t corresponds to classically allowed
or forbidden processes. Classically forbidden processes would, for
example, be associated with complexvalued classical paths or trajec
tories.
As a function of energy, the tmatrix must also show continuity as
the energy goes from positive to negative, and consequently
lim t, (q1, q; E) = lim Re[t,(q', q; E)J (3.7)
E  0 E 0+
and
lim Im[t(q', q; E)] = 0 (3.8)
E + 0+
For negative energies, the tmatrix elements are real and have poles at
energies corresponding to the bound states of the system. This can be
established by considering the expansion of t(q' q; z) in terms of the
eigenfunctions of the full Hamiltonian of the system (Ne66). The result
is as follows:
129
by either of these two factors will then determine the validity of 1D
scattering calculations. Because of these limitations in collinear
studies, we will pursue this study only as a test leading to an evalua
tion of the single collision approximation, the peaking approximation
and the importance of multiple collision terms.
1. Formulation of the Collinear Scattering Problem
For the sake of specificity, we will select channel 1 as our
initial arrangement channel, i.e. A + (B + C). At present, a harmonic
oscillator model will be used to represent the diatomic (BC). Later
in Section VI.4 where dissociation is considered, a Morse oscillator
will be introduced. The interaction potential, VÂ£, between the projec
tile atom and the ncnstruck atom will be neglected.
In order to simplify our analysis of the 1D scattering problem
and compare to other work in the literature, we shall adopt the follow
ing scaled Jacobi coordinates:
Xj = [(ir^k)1/2/?!]1/2 x:
and
Xj =[ oyc) 1/2/fi] 1/2 .
where
xx = xc xB
and
Xj = XA (VB + mCXC)/niBC
(6.1)
(6.2)
(6.3)
(6.4)
are the Jacobi coordinates defined in Appendix IV as they apply to 1D
scattering. In equations (6.1) and (6.2) we have introduced the diatomic
reduced mass = (rrigm^/rrig^) and the harmonic oscillator force constant k.
Figure (3
1)
(a) [2/tt]}/2 Re[>0(K, KÂ£; R)/K]
(b) [2/tt] 1/2 Im[ojq(K, KE; R)/K]
(c) [2/tt]1/2 [J0(KR)/K] (a.u.)
Calculations correspond to E = 0
Kj: = 4.26 a.u. and K = 0.23 a.u.
Potential)
(a.u.)
(a.u.)
.01 a.u.,
(Morse
50
the evaluation of t(q', q; E). This would then remove any problems in
this regard.
6. The Bateman Method
Very little has been done in the literature with regard to non
parti al wave techniques in the evaluation of scattering amplitudes or
twobody tmatrix elements. Only within the Eikonal approximation
[(6159), (Su69), (Ch73)] and approximations based on the Born expansion
[(Ho68), (Ra72)] has this not been the case. The attempts that have
been made to solve the LippmannSchwinger equation for either the full
wave function or the tmatrix have involved numerical quadratures. For
example, Walters (Wa71) has considered the LippmannSchwinger equation
for the full wave function and converted it to matrix form by use of a
two dimensional numerical quadrature for the case of a screened Coulomb
potential. The amplitudes obtained from this procedure however, though
in good agreement with exact results, showed poor convergence when the
number of quadrature points was increased. Another approach that has
been taken is that of Rosenthal and Kouri (Ro73), where the Lippmann
Schwinger equation for the scattering amplitude was reduced from its
twodimensional form to a onedimensional integral equation, which was
then solved numerically. Using the work of Walters (Wa71) for comparison,
these authors obtained results within 5 to 10 per cent agreement. In
contrast to these methods, the approach we shall new pursue does not
involve numerical quadratures, at least in the case of certain model
potentials. This procedure, known as the Bateman method, was originally
implemented by Akhmadkhodzhaev, Belyaev and Wrzecionko (Ak70) in the
computation of partial wave tmatrix elements. Ttey have, however,
In order to simplify the notation, we have suppressed V.." from the
argument of uj and vj, as well as the "1" subscript. Equation (2.49)
is appealing in the sense that it provides a simple algorithm for propa
gating cjj and its derivative from r^ to r^+1, but it is even more
appealing in the sense that one need not store the coefficients A^j,
and Cj for all the intervals needed in the calculation. Only one
problem remains to be dealt with. In the CP method the tmatrix is
calculated replacing (2.13) by the expression
(q', q; E)
_ 2 1
ir q'q
E
i
ri+lA
V.Â¡ / j(q'r)i^(q, kÂ£; r)
(2.57)
If one had propagated the coefficients A^j, and C, on each interval
one would have a set of integrals over products of two Riccatifunctions
as is indicated by Eq. (2.45), the expression for cjp and Eq. (2.57),
the expression for t.,. These integrals are well known and are easy to
generate (Me61). We, however, have been able to derive expressions for
the integrals needed in Eq. (2.57) directly in terms of the offshell
wave function and its derivative. A derivation of these quantities
is given in Appendix II.
One of the major criticisms of the VPA approach and other numerical
procedures for obtaining is that tj (q', q; E) must be obtained
through numerical quadratures. In practice this is problematic when
q and q' are larger, because of the oscillatory nature of the functions
involved. Generally, when one propagates w numerically, it is deter
mined at smaller step sizes than are involved in the CP approach; thus,
one could take advantage of the integral expressions mentioned above by
implementing a breakdown of the potential into steppotentials only in
of symmetry,
tj (q', q; E) = tj (q, q'; E) ,
(3.1)
and of offshell unitarity,
Im[tÂ¡ (q , q; E)] = irmk^t* (kÂ£, q'; E)tÂ¡(kE, q; E) (3.2)
The symmetry relation is valid at all energies, but the unitarity rela
tion is valid only for positive energies. These relations are well
established in the literature and will not be derived (Wa67). It
suffices to say that in this case they are a direct consequence of
assuming a real local potential.
If one considers the differential equation satisfied byw,(q, k^.; r),
Eq. (2.10), it is apparent that reduces to the onshell wave function
when q = k^. Similarly, the offshell tmatrix elements are defined
such that there is a smooth transition to the onshell behavior as the
momenta involved approach the onshell quantities. This is exemplified
by the relations (Wa67)
and
~ A(+)
w,(q, kÂ£; r) ~ j(qr) 7Tmqt1(kE, q; E)h, (kÂ£r)
r ^ oo
tan[6j (kÂ£)] =
Im[t (q, kÂ£; E)]
Re[t, (q, kE; E)]
(3.3)
(3.4)
which correspond to well known onshell relations for q = kE.
Another important aspect of the tmatrix elements is their behavior
as a function of momenta and energy. In this respect, there are various
limits of interest. If the energy E and initial momentum q are fixed,
then
lim t. (q', q; E) = 0
q 1 * oo
(3.5)
13
where V. is equal to V,.,. The Faddeev formalism, then, is merely a way
i j i\
of decomposing the total wave function of the system into channel com
ponents which can be propagated by their respective channel Hamiltonians.
The result is given by the relation (Fa61)
(E Hj)Y
3 /i,\
2 (1 
k=l J
(j = 1, 2, 3)
(1.39)
where the total wave function T has been decomposed into the channel com
ponents ^, i.e.,
'Â¥ =
3
E
j=l
(1.40)
A detailed derivation of the Faddeev equations will be given in
Chapter IV of the present work. In the following discussion, our con
cern will be the use of Eq. (1.39) solely as the basis for the simul
taneous treatment of nuclear and electronic motions. Micha (Mi77d) has
shown that this can easily be accomplished within the framework of the
method of diatomicsinmolecules (DIM) (El63). We begin by considering
the total wave function for the system, keeping in mind that for struc
tured particles it must have the form
Â¥ = Cy\X (1.41)
where C is a normalization constant, is the N electron ar.tisymmetrizer
and X is an unsymmetrized total wave function for both nuclear and elec
tronic motions. Since H and J\ commute, one can cast the Schrodinger
equation into the form
[Ej4  (H/ +/ H)]X = 0 ,
(1.42)
99
Table (37) Basis {k} used in the Bateman calcu
lations^given in Table (36). The
vector k(l) was chosen to lie along
the direction of the initial momentum
vector 4 = all other lie on the
(q'q) momentum pl_ane. is the angle
between Â£(n) and k(l).
n
6[k(n),k(l)] (radius)
tc(n)
1
0
1.816
2
0.785
0.605
3
1.571
1.816
4
2.356
0.605
5
3.1416
1.816
6
0.1963
1.816
7
0.5890
0.454
8
1.3744
1.816
9
2.1598
0.454
10
2.945
1.816
11
0.295
1.816
12
0.393
2.421
13
1.178
1.816
14
1.963
2.421
15
2.749
1.816
110
choose a different partition of the various operators involved, allow
ing one to construct equations which put more emphasis on selective
channels, which have a greater physical consequence (Hahn and Watson (Ha71),
Kouri and Levin (Ko75) and Toboman (T074)).
We have now specified all the multichannel transition operators
for the given atomdiatom system. One striking aspect of the resulting
relations, as given by Eq. (4.35), is the similarity between the expres
sions for the various transition operators. In particular if one com
pares the breakup transition operator, T to the remaining operators,
Tjj (j = 1 to 3), one discovers the relation (Ko77b)
Toi (1+ TjVTji Vo'1 !436>
or more concisely
(4.37)
The second term on the right hand side of Eq. (4.36), ^Gg*, will not
contribute to the onshell scattering amplitudes, i.e.,
Mqi = <4>0(Z)T01(Z)$1(Z)> (4.38a)
= <i(Z)> (4.38b)
where
4*jC = Wj$0 (4.39)
In this last expression, is a wave function describing the inter
J
acting pair in channel j as a continuum state. The superscript "c" has
been introduced to distinguish 4>.c from $.Â¡ which describes the inter
acting pair in channel j as a bound state. What this means physically
Figure (71) Plot of differential cross section (arbitrary units)
versus final rotational quantum number for the (Li+, CO)
system. [(ni,ji) = (0,0), nf=0, E^^SeV and ej, Â£ =37.1]
140
been renormalized according to the relation
Vn'Pn'n 'JTn'n1
(6.32)
as was suggested in reference (Cl70). If the computation of the tran
sition probabilities were exact, then the expression in the denominator
of Eq. (6.32) would be unity, because of unitarity. This procedure
brings the single collision results into closer agreement with the
exact results; however, it does break down near the threshold energies
for vibrational excitation because of the presence of singularities.
The exact reason for the presence of these singularities in the single
collision approximation is not clear. Their presence has been noted by
other authors [(Cl70), (Ek71)]. By having considered the behaviour of
Mri in near thresho1ci it was ascertained that the singu
larity could not arise there but from the 1/P^' term multiplying
IM^112 1n This would indicate that the single collision
approximation to is not good enough, and multiple collision terms
are needed. On physical grounds this is quite reasonable, since near
threshold the incoming projectile barely has enough energy to excite
the diatomic; and thus, the likelihood of multiple collisions becomes
quite high. The importance of these sequential collisions or chattering
in collinear scattering has been verified in classical trajectory cal
culations [(Se69), (Ke72)]. Analogous results to those discussed above
for Pjq are given for P^q in figures (63a) through (63d). We see
that the comparison of the single collision results with the exact
results is more favorable in this case. In Table (61) we list exact,
single collision and peaking results for P^g versus using m = 0.125
and m = 0.5. These results show that for larger values of m, the single
Figure (74) Plot of differential cross section (arbitrary units)
versus final rotational quantum number for the (Li+, N?)
system.[(ni,ji)=(0,0), nf=0, Ei=4.23eV and eÂ£, + =37.1T
pri
Figure (310a) Plot of Re[t(K, K; E)] versus 1. (K = 4.0 a.u. and
E = 0.01 a.u., Hulthn Potential)
9
which involve charge transfer and vibrationalrotational excitation, is
that given by the multiplecurve crossing model. In order to define this
model, let us introduce the Jacobi coordinates ^ and r^ which are
appropriate in describing the arrangement channel specified by Eq. (1.21).
Furthermore, let us expand the nuclear wave function Xi ^1 ^ in terms
of a set of vibrationrotation functions (u, (^^;r^)} as follows,
x, (,.?,)*Â£ u, (R Â¡.wos.), (1.22)
V
where v labels the vibrationrotation states and wv(ftj) acts as a nu
clear wave function for the relative motion of atom A with respect to
the center of mass of the diatom BC. Note that the {u^} depend only
parametrically on R.,. If we now use Eq. (1.6) expressed in Jacobi coor
dinates and Eq. (1.22), we obtain a set of coupled differential equations
given by
+ + Hlvlv E>lv'V '
rj,t + Wlvl'v,1l,vl(il)> 0.23)
V
where
t = 
JV2
2M1 R1
H, =
mA+mB+mC
(1.24)
(1.25)
= Mj <(h ^rJ^I
E'
T11*
(1.26)
Table (31) The tn(q, q; E) matrix elements are
giveri for q = 4.0 a.u. Column I
gives the VPA calculations and column
II the results obtained from a varia
tional procedure (Ku78). (Morse
Potential)
Energy (a.u.) tQ(q, q; E) x 102
I
II
0.18
2.5971
2.5798
0.17
4.4003
4.3841
0.16
2.0273
2.0069
0.15
8.8552
8.7964
0.14
9.9843
10.0183
0.13
2.0411
2.0680
0.12
1.5197
1.5486
0.11
1.2378
1.2106
0.10
1.0856
1.1159
7T
6
OJ
R (a.u.)
cr>
en
104
where H0 is the freemotion Hamiltonian for the three atom system, and
introduce the assumption
3
V = Z V. (4.3)
j=l J
i.e., that of pairwise additive potentials. Each Vj is an atomatom
potential corresponding to the asymptotic interaction in channel j, and
may be specified using the procedure outlined in Chapter I. Asymptoti
cally in each channel j, the system can be described by the channel
Hamiltonian
Hj = H0 + vj (44)
Therefore, che total Hamiltonian for the system can be written as
H = H, + (4.5)
J
where
V(j) = V V. = l 6ik Vk (4.6)
J k=1 *
is the channel interaction potential and 6, is equal to (1 6).
J K j k
Having made this connection it is apparent that iky considering various
partitions to the total Hamiltonian in terms of channel quantities,
such as H. and V^'^, one should be able to construct the various opera
J
tors needed in the reformulation of Eq. (4.1). Important in this regard
are various twobody operators, such as the channel resolvents and two
body transition operators, which we will now introduce.
The resolvent relations that are needed follow directly from equa
tions (4.2) through (4.6), using the identities '(11068)
To my Parents
103
where T(Z) is the general threebody Toperatcr, GQ(Z) is the free
particle propagator, Z is an energy parameter (E + ie) and V is the
potential for the threebody system. The problem with the boundary
conditions now manifests itself in the form of the "uniqueness" question
with regard to the solutions to Eq. (4.1) [(Sc74a), (Re77)]. Since a
threebody system allows for the existence of twobody bound states at
a continuum of energies within the range of physical interest, one must
provide additional constraints which select those solutions satisfying
the proper physical boundary conditions. Furthermore, because of the
presence of these subsystem bound states, one finds that even after the
centerofmass motion has been factored out, the kernel contains
unfactorable delta functions (Re77). Each of these delta functions
expresses the conservation of momentum of the spectator atom in the
presence of the interacting atomatom pair. A very nice discussion of
all these problems is given in the references [(Kr71), (Sc74a), (Si71),
(Re77), (Wa67)].
2. The Multichannel Transition Operators
It is clear from the discussion in the previous section, that the
LippmannSchwinger equation, Eq. (4.1), must be reformulated by speci
fically taking into account the existence of the various arrangement
channels. This can be done in a straightforward manner by considering
the structure of the potential characterizing the threebody system, and
by introducing various operators describing the interaction of two
particles in the presence of a third [(Fa61), (Si7/1), (Sc74a)].
Let us begin by considering the total Hamiltinian,
H = H0 + V,
(4.2)
21
state of nitrogen. Note that in constructing the curves in Fig. (11),
the energies of Li ( S) and Li( S) must be included. The possibility of
reactive channels is ruled out by the closed shell nature of the
(Li+, N^) system. Furthermore, the electronic excitation of Li'
. O I f
[Li ( S) + N2( Eg}] is not considered since it would lie much higher in
energy than those states already given in Fig. (11). It is apparent
then from Fig. (11), that there is a possibility of electronic excita
tion, since the Eu state of lies only 6.1 eV (Bo65) above that of
the ground Eg state. Charge transfer on the other hand would seem
uniikely.
As the Li+ is brought near N2, the relative position of the various
electronic states would change. If the spacing between the ground arid
excitated states grows closer, nonadiabatic effects will be important
for the given system. Figure (12) illustrates what happens as the Li +
atom approaches the molecule. This figure shows three planes.
Plane III corresponds to the same cut of the potential energy surface
given in Fig. (11) in which R (the distance between Li+ and the center
ofmass of N2) is infinite. Planes I and II are perpendicular to plane
III and intersect that plane at r equal to zero and at r equal to the
equilibrium internuclear separation (rg) of N^. It is shown then that
the ground state potential energy of the system is decreased as Li+
approaches and increases rapidly as R goes to zero. The reason for this
initial decrease of the potential as the Li+ approaches N2 is due to
attractive polarization forces. This will be explained later when the
long range interactions of the system are discussed. On the other hand
at small values of R, strong repulsion forces exist because of the closed
shell interactions. A similar qualitative behavior is expected for the
221
(Ta65) K. Takayanagi, Adv. At. Mol. Phys. 1. 149 (1965).
(Ta72) J. R. Taylor, Scattering Theory, John Wiley and Sons, Inc.
New York (1972).
(Th68) E. Thiele and J. Weare, J. Chem. Phys. 48, 2364 (1968).
(Th78a) L. D. Thomas, W. P. Kraemer and G. H. F. Diercksen, Chem.
Phys. 30, 33 (1978).
(Th78b) L. D. Thomas, W. P. Kraemer, G. H. F. Diercksen and P. McGuire,
Chem. Phys. 27, 237 (1978).
(To74) W. Tobocman, Phys. Rev. C9, 2466 (1974).
(Tu71) J. C. Tully and R. K. Preston, J. Chem. Phys. 55_t 562 (1971).
(Tu76a) J. C. Tully, in Dynamics of Molecular Collisions, Part B,
Ed. William H. Miller, Plenum Press, New York (1976).
(Tu76b) J. C. Tully and C. M. Truesdale, J. Chem. Phys. Â£5, 1002 (1976).
(vL61) J. M. J. van Leeuwen and A. S. Reiner, Physica 27, 99 (1961).
(Wa57) K. M. Watson, Phys. Rev. 105, 1388 (1957).
(Wa67) K. M. Watson, J. Nuttall, Topics in Several Particle Dynamics,
Holden Day, San Francisco (1967).
(Wa71) H. R. J. Walters, J. Phys. B 4, 437 (1971).
(Wi77) N. W. Witriol, J. D. Stettler, M. A. Ratner, J. R. Sabin and
S. B. Trickey, J. Chem. Phys. 66, 1141 (1977).
(Wo75) G. Wolken, Jr., J. Chem. Phys. 63^, 528 (1975).
(Yu76a) J. M. Yuan and D. A. Micha, J. Chem. Phys. 54, 1032 (1976).
(Yu76b) J. M. Yuan and D. A. Micha, J. Chem. Phys. 65, 4876 (1976).
(Ze64) Ch. Zemach, Nuovo Cimento 33, 4219 (1964).
26
vibrational excitation than the (Li N^) systems. Various investi
gators [(Bo76), (St76), (Mi78b)] have attributed this observed result
to the chargedipole interaction in the (Li+, CO) system. The relative
insensitivity of rg as a function of R for the (Li Ng) system is
shown in Fig. (13).
At small values of R the interaction of Li+ with and CO is
highly repulsive and can be easily characterized by an exponential poten
tial [St76), (Th78a)]. Thomas et al. (Th78a) use the potential
V(R, r, 0)
 Vj(r)
vx(r,
R)P^(cosO)
(1.62)
where V^(r) is the potential for the free diatom and v^(r, R) is deter
mined by using various parametrized functions [sums over exponentials
and R"n terms] to yield the best curve fit to the potential. Equation
(1.62) behaves asymptotically as Eq. (1.58). Staemmler (St76) points
out that the Legendre expansion of the potential in Eq. (1.62) loses
its significance for small R[R < 3.5 a.u.] and may in fact not converge.
6. Proposed Application of the Faddeev Formalism
Our study of the (Li+, CO) and (Li+, Ng) systems will be based on
the many body formalism of the Faddeev equations. Within this approach
we shall consider an iteration of the Faddeev equations (in transition
operator form) which yields a miltiplecollision expansion (Mi75).
This allows us to describe an atomdiatom collision process as one
which takes place as a sequence of twobody encounters. Since the colli
sion energies are large and the scattering angles are beyond the rainbow
angle, it is expected that the scattering cross section will be largely
Figure (39)
tn(K', K; E) for E = 0.153 a.u.
and K = 4.5 a.u. (Morse Potential)
Figure (31 la) Plot of Re[t(K', K; E)] versus AP = Â£' lÂ£.
(K = 4.0 a.u. and E = 0.01 a.u., Hulthn Potential)
17
tion of the electrons. There is, however, an alternative approach based
on the coupledchannel manybody formalism [(Ko75), (To74), (Ra77)],
which is suitable for treating charge transfer. In this approach one
simply considers various partitions of the Hamiltonian consistent with
the atomic and molecular fragments involved in the scattering process
of interest. A set of coupled equations may then be constructed that
are analogous to the Faddeev equations (Mi76b).
5. Analysis of Potential Energy Surface Information
In the present work, we shall be concerned with ionmolecule reac
tions of the type (Li+, CO) and (Li+, N?). Since CO and N2 have ident
ical masses and very similar molecular properties, these systems are
very interesting for comparison purposes. Such a study is also attrac
tive because a comparison between experiment and theory is now possible
in light of recent molecular beam experiments [(Bo76), (Ea78)].
Our investigation of these systems will be based on an analysis of
the various interaction potentials involved and upon the potential energy
surface data available in the literature [(St75), (St76), (Th78a)].
This analysis though qualitative in nature, will serve as a guide for
the development of an impulsive model which will be specified in the
next section.
The only available potential energy surface Information on the
(Li+, CO) and (Li+, N2) systems corresponds to a restricted number of
geometries for the ground state of the system. Because of the relatively
large number of electrons in these systems, accurate boundstate calcula
tions are very expensive. This is further complicated by the large
number of points required to adequately characterize the potential
CHAPTER IV
THE THREEBODY PROBLEM
In Chapter I we discussed the feasibility of treating an atom
diatom scattering process as an effective threebody problem in such
a way that one could make use of twobody interaction data. It was
pointed out there that one could construct a proper representation of
the interaction potential for a reactive triatomic system in terms of
twobody quantities, only if one introduced a set of spindependent
atomic pair interactions. Since our present interest lies in high
energy atomdiatom collisions, we proposed the use of a simplified
impulsive model in which the total interaction potential would be
assumed a sum of spin independent twobody potentials. Having already
addressed in Chapter I the merits and limitations to such a representa
tion of the potential, we turn now to the solution of the scattering
problem.
The formalism we shall adopt in treating the scattering of an atom
diatom system will be based on that given by Faddeev (Fa61), who
developed the first rigorous mathematical treatment of the threebody
problem. We shall not attempt to address the more mathematical aspects
of the problem. Instead, we shall restrict ourselves to a discussion
of difficulties encountered when one tries to apply the LippmannSchwinger
equation to the threebody problem and to a formal derivation of the
various transition operators of physical interest. The resulting
expressions for these transition operators will then be used to construct
101
44
The propagation of two independent solutions to the inhomogeneous
differential equation (2.10) further presents two problems. One is that
of the growth rate of the amplitude in a nonclassical region and the
second is that of the propagation through classically allowed regions
at negative energies. The first problem was solved as given in reference
(Br75). The idea was simply to use the propagated wave functions to
construct two new linearly independent solutions to Eq. (2.10), which
would have much smaller amplitudes. Therefore we shall restrict our
attention to the second problem. If the energy is negative, the modified
/s /\
RiccatiHankel functions and ij are the natural basis to use for
/v /v
uj and v^ in expression (2.20),. the VPA form for the offshell wave
function o>i. However, if the potential is attractive and the energy is
such that one must integrate through a classically allowed region, the
wave function in this region would be oscillatory in nature. The
RiccatiHankel functions on the other hand have a nonexponential grow
ing and decaying character; thus, propagation of the offshell wave
function under these circumstances is very difficult. The solution then
A A A A
is simply to replace kj and ij by the RiccatiBessel functions and n^.
This in turn requires a modification of the VPA equations (2.36) and
(2.37). However, the modification is minor and does not lead to any
significant increase in computational effort. Let us go back then to the
original inhomogeneous Schrodinger equation, Eq. (2.10), as it applies
to negative energies, and rearrange it into the form
K1 + 1)
v (r) ]co] = 2k^
 (k* + q2)j1(qr).
(2.40)
169
V(r) = Aebr (7.23)
and obtained the parameters A = 1908.51 eV and b = 5.949 A for (Li+C),
A = 5237.46 eV and b = 4.8578 A"1 for (Li+0), and the parameters
A = 3079.77 eV and b = 5.2582 for (Li+N). Using such a simple
potential does not properly describe the potential energy surface at
low energies, where there is an attractive component. However, the
experiments we shall compare to are at high energies (4 to 8eV) and the
scattering angles involved are beyond the rainbow angle (Bo76). Because
the above parameters lead to very repulsive potentials, we will simply
use a hard core interaction potential for each atomatom pair. The
data we have used for each of the two systems are given in Table (71).
We shall restrict the present investigation to the study of rota
tional excitation from the ground vibrational level. This, however, is
by no means a limitation of the present approach. The experiments to
which we shall compare our calculations are time of flight studies on
the scattering of Li+ ions from ground state and CO molecules (Bo76).
In particular, we shall compare our calculated differential cross sec
tions [in arbitrary units] as a function of final rotational quantum
number jf versus those obtained from experiment Typical results ob
tained for the (Li\ CO) system are shown in figures (71), (72) and
(73). These figures correspond to a center of mass collision energy
of 4.23 eV and center of mass scattering angles fi 37.1, 43.2 and
49.2, respectively. Initially, the target diatom is assumed to be
in its ground vibrational and rotational level. We note that as the
scattering angle is increased the overall envelcge of the plotted dis
tributions is broadened. The same effect v/as noted in the experimental
181
results (column II) to those obtained theoretically (column I). The
center of mass collision energy and scattering angle are 4.28 eV and
10 [8 in the laboratory frame]. The rainbow angle is at 5.5 in the
laboratory frame. The theoretical probabilities were obtained from
the relation
P = [
Ld fi
(n\ j 'm, j; 0
P'lPr
m,
da
dft
(n1, j'<n, j; 0
pipi)1
1
(7.24)
As noted in the results mentioned earlier iri this section, the peak
in the theoretical probabilities falls at a higher value of jf than
the peak found in the experimental results [see Table (72)]. Because
of this and the general simplicity of the present model, one cannot at
this time state whether the maxima are due to effects coming from the
potential or from dynamical effects not contained in Eq. (7.12). One
should also note that the scattering angle is not much larger than the
rainbow angle, and therefore the experimental data quoted in Table (72)
are not the most favorable to our model.
The results we have been discussing have all been obtained using
the single collision peaking approximation as it arises in the Faddeev
formalism. The use of this formalism has led to expressions involving
offshell tmatrix elements. On the other hand, the impulse approxima
tion [see Section VI.4 ] leads to halfonshell tmatrix elements. For
the present case of a hard core potential, we have done calculations
using offshell, halfonshell and onshell tmatrix elements to compute
the energy loss spectrum given in Fig. (71). The results are given in
Table (73). Column I corresponds to the offshell results, column II
to the halfonshell results and column III to the onshell results.
CHAPTER III
NUMERICAL RESULTS FOR THE TWOBODY tMATRIX
Having developed the formalism of the VariablePhase and Amplitude
method in the previous chapter, we will now apply it to various model
potentials of physical interest. This will serve to illustrate various
known analytical properties of the twobody tmatrix and to gauge the
reliability of the computational procedure.
We will concentrate on calculations for the lowest (attractive)
and (repulsive) potentials of HÂ£ because of the role these play in
the collision dynamics of H + H^, and because their appearance in
dynamical studies can be justified within a diatomicsinmolecules treat
ment (Mi77b). A comparison of some of our results will be given with
others obtained within a variational procedure, developed to solve the
threeatom problem at low (thermal) energies (Ku78).
A study will also be made on the reliability of the Bateman method
in the computation of tmatrix elements for potentials of interest in
atomic and molecular scattering theory.
1. Properties of the TwoBody tMatrix
A brief summary of the various properties of the offshell tmatrix
elements, tj(q', q; E), will now be given. This will prove valuable in
determining the reliability of any given computational procedure used to
obtain tj.
The most important properties satisfied by tj (q1, q; E) are those
54
I certify that I have read tnis study and that in my opinion
it conforms to acceptable standards of scholarly presentation ana
is fully adequate, in scope and quality, as a dissertation for the
degree of Doctor of Philosophy.
Cr" t 1, / C f '.<(< 'f' O' '''
David ATlTicha, Chairman T~
Professor of Chemistry and Physics
I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and
is fully adequate, in scope and quality, as a dissertation, for the
degree of Doctor of Philosophy.
M
Jl]
X
L
N. Yngve
Professor
Ohrn
! of Chemistry and Physics
I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and
is fully adequate, in scope and quality, as a dissertation for the
degree of Doctor of Philosophy.
Jcm^K. Sabin
Professor of Physics and Chemistry
I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and
is fully adequate, in scope and quality, as a dissertation for the
degree of Doctor of Philosophy.
Thomas L. Bailey
Professor of Physics
I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and
is fully adequate, in scope and quality, as a dissertation for the
degree of Doctor of Philosophy.
1 i' \ P \ L ft
Charles P. Luehr
Professor of Mathematics
47
w] (q, kÂ£; r) =
A1
A1
u] (^r) B v. (k.t) + Cj
0 1
(qr),
(2.45)
where
aEkl q2
2 2
aiKi q
(2.46)
The offshell wave function is then obtained from two propagated solu
tions in a form analogous to that given for the VPA approach. The prop
agation here, however, is accomplished by matching each wave function
and its derivative across every interval. This can be done by either
propagating the coefficients A.j and Bj as v/as done in the reference
(Br75), or by propagating the wave function and its derivative directly
as we propose to do. To accomplish this, we shall modify the procedure
given by Rosenthal and Gordon (Ro76) for the case of the usual radial
Sc'nrodinger equation. From expression (2.45) and the corresponding
expression for the derivative of the wave function on the given i^
interval, it is easy to show that
A] = Wtujr.) Cj j1 (qr.), v1(icir.)]/Wi (2.47)
and
B] = (ri) Cj jj (q, r.), u] (k.^)]/!^ (2.48)
A A
where W.. is a constant equal to the value of the Wronkian of uj and v^,
the two local basis functions. The i superscript on wj has been removed
and is not necessary because of the continuity requirement imposed on
the wave function and its derivative. Using (2.47) and (2.48) in the
expression for Eq. (2.45), and the analogous expression
9.65 x 12
7.72x10'2
, 1 r
H + D2
m = 0.2
n,= 0 E= 20.0
H
T
5.79 x10'2
P
3.86 x 1CT'
1.93 x 10'2
ol _
 1.00 3.40
780
T
1220
16.60
21.00
CD
109
T U)
i
1
o
T1
T1
r
Ti
fH
61
T \2)
A1
=
0
+ Go
T2
0
T2
T (2)
1
f
OJ
i
I
o
1
T3
CO
1
0
T^3)
(4.31)
At this point we can now introduce the formal halfonshell definition
(4.32)
for the transition operator T.. which describes processes originating
in channel 1 and ending in channel i (i = 0, 1, 2 or 3). Using the
definition of V^', Eq. (4.32) may be written as
TilU)*! = I
J = 1
Vo(Z)
1
Tj (j)
(4.33)
which leads to the relation
LA = ~l + 2 6. .T.G T. $
U 1 U 0 1 j=1 lj J o jl l
(4.34)
This implies the set of coupled equations
3
+ E
j=l
(4.35)
which are known in the literature as the Alt, Grassberger and Sandhas
equations, or simply, as the AGS equations (A167). These operators are
not unique however, since one could always add a quantity to each T.
J 1
which does not contribute to the transition amplitudes in the onshell
limit (Lovelace [(Lo64a), (Lo64b)]). Another possibility would be to
single collision approximation is not valid in the same range of
energies that were studied using a hard core potential, i.e. for
projectile energies less than 10 Tiu units. Typical results obtained
are given in Table (63), where the parameters used were m = 1/13,
a = 0.1287, rc = 2.0, A = 10.0 and = 5.918 Tico units. Note that for
the hard core results shown in figures (6.1) and (6.2) the transition
probabilities obtained from the single collision approximation were
quite reasonable for projectile energies one quanta beyond threshold.
This is not the case for the transition probabilities quoted in Table
(6.3), unitarity is violated even at several quanta above threshold.
Another interesting point is the discrepancy between the peaking and
nonpeaking results quoted in Table (63), indicating that the peaking
approximation should not be used for such soft potentials. Similar
cautions on the validity of the peaking approximation have also been
noted elsewhere in the literature (Co68). The reason for the poor
quality of the single collision results in this last case might lie in
the fact that for very soft potentials, the distance over which the
potential falls to zero is large and therefore the concept of a local
ized collision becomes blurred and the assumption of an impulsive
encounter would not be valid [(Se66), (Se69)].
4. Dissociative Collinear Calculations
In the previous section a harmonic oscillator was used to model
the behaviour of the diatom in an atomdiatom collision. When this
potential is replaced by a more realistic one such as the Morse poten
tial, one can allow for the possibility of collision induced dissocia
tion (CID), i.e. those processes where
51
extended this work to the calculation of the full tmatrix elements for
the case of a Gaussian potential (Be74a). with significant success. Lim
and Gianini (Li78) have gone further and applied it to the case of a
Yukawa potential, obtaining encouraging results.
Formally the Bateman method is equivalent to using the Schwinger
variational principle [(Li78), (Ad75)]. We shall thus present the
method in this framework. Let us begin then by considering the general
identity (Le69)
[A(z)J = At(z) + At(z*)+ At(z*r At(z)1 At(z), (2.58)
where A is an operator that is Lconjugation invariant (A(z) = A(z*)r)
and At is its trial value. Similar operator identities have been used
as a basis for studying upper and lower bounds in the theory of bound
state calculations (Lo65) Taking A(z) to be equal to t(z) and using
equation (2.3), which relates the toperator to the wave operator W(z)
in the variational expression (2.58), one obtains
[t(z)J = Wt(z) + Wt(z*)+V Wt(z*)+ V VGq(z)V Wt(z), (2.59)
where the trial operator W^.(z) can in general be chosen as
Wt(z) = f>
Here f> represents a set of n basis functions >fn} >
whose expansion coefficients (Cj,C2,...,Cn> are determined variationally
through expression (2.59). In the usual variatisrnal procedures (SI72),
(Ad74), (Pa74), (Ku78), the set of functions {f} iis usually chosen to
be square integrable; here however, {Â£} will be chosen to be a collec
tion of plane wave states, i.e., the set denoted by {k} or {kj,^,... ,kn}.
VI.1. Furthermore, Eckelt and Korsch (Ek71) also used the tmatrix
for the potential
157
V
3
0 x > xc
(6.51)
instead of the symmetric hard core tmatrix, Eq. (6.29). The resulting
tmatrix from Eq. (6.51) leads to an unsymmetric tmatrix. In order to
make the results obey the principle of detailed balance, the above
(6.52)
This is not necessary within the Faddeev framework, since each term in
the multiple collision expansion automatically satisfies time reversi
bility (Mi72b). We have used expression (6.52) as it applies to the
unsymmetrical hard core and were able to reproduce all the results
quoted by Eckelt and Korsch (Ek73). This provided a check for the com
puter programs that were developed to do the present study. In compar
ing the calculations obtained from f/(s,c,Pk) ancj ^ose from M(s,c,I)
using equations (6.51) and (6.52), we found that the latter yields
results that were nearly unitary, whereas the singlecollision results
were always lower in magnitude but with the same qualitative behaviour.
The sole reason for this was that different tmatrix expressions were
used in the two calculations. Hence, we feel that this indicates that
one should be wary in attaching too much significance to how well
results obtained with approximations such as (6.34), (6.35) and (6.50)
obey the uni tarity relation Eq. (6.42). One interesting feature of
these calculations as seen by comparing figures (65), (66) and (67),
APPENDIX II
INTEGRALS USED IN "COMPARISON POTENTIAL" METHOD
In Chapter II we saw the need for integrals of the product of two
RicattiBessel functions, and in particular, for the comparison poten
tial method, integrals such as those found in Eq. (2.57). Consider now
A A A /v
the integral of the product uV, where u^ and satisfy the Ricatti
Bessel differential equation (2.21), i.e.
and
7? ui + Iq? 1Xij7Ui i = 0
r
dr
'l + [q2 ^ \ = 0
(A2.la)
(A2.lb)
It is well known (Me61) that the desired integral
A A
r=r.
2  W[u,, v,]
I dr u v = L_
ri M ? q)
r=r,
if E^ f E^; on the otherhand if E = E^ (Fa71)
(A2.2)
l Vl
dr =
A I A
ijr
. {q2 lilii)) ; ,rs
(A2.3)
200
219
(0p28) J. R. Oppenheimer, Phys. Rev. 31^, 349 (1928).
(0s73a) T. A. Osborn, J. Math. Phys. L4, 373 (1973).
(0s73b) T. A. Osborn, J. Math. Phys. J4, 1485 (1973).
(Pa35) L. Pauling and E. B. Wilson, Introduction to Quantum Mechanics,
McGrawHill, New York (1935), p. 270.
(Pa74) G. L. Payne and J. D. Perez, Phys. Rev. CIO, 1584 (1974).
(Ph76) V. Philipp, H. J. Korsch and P. Eckelt, J. PHvs. B 9_, 345
(1976).
(Pr71) R. K. Preston and J. C. Tuly, J. Chem. Phys. 54, 5297 (1971).
(Pr73) R. K. Preston and R. J. Cross, Jr., J. Chem. Phys. 59_, 3616
(1973).
(Ra69) D. Rapp and T. Kassal, Chem. Rev. 69, 61 (1969).
(Ra72) H. Rabitz, Phys. Rev. A5, 620 (1972).
(Ra76) H. Rabitz, in Dynamics of Molecular Collisions, Part A,
Ed. William H. Miller, Plenum Press. New York (1976).
(Ra77) S. Rabitz and H. Rabitz, J. Chem. Phys. 67, 2964 (1977).
(Re77) E. F. Redish, University of Maryland, Department of Physics
Report TR # 77060 (1977).
(Ro67) L. S. Rodberg and R. M. Thaler, Introduction to the Quantum
Theory of Scattering, Academic Press, New York (1967).
(Ro73) C. M. Rosenthal and D. J. Kouri, Mol. Phys. 26, 549 (1973).
(Ro76) A. Rosenthal and R. G. Gordon, J. Chem. Phys. 64, 1621 (1976).
(Sc73) J. Schottler and J. P. Toennies, Chem. Phys. 2, 137 (1973).
(Sc74a) E. W. Schmid and H. Ziegelmann, The Quantum Mechanical Three
Body Problem, Pergamon Press, Oxford (1974).
(Sc74b) J. Schottler and J. P. Toennies, Chem. Phys. 4, 24 (1974).
(Sc75a) R. Schinke and J. P. Toennies, J. Chem. Phys. 62^, 4871 (1975).
(Sc75b) J. Schottler and J. P. Toennies, Chem. Phys. 1J3, 87 (9175).
(Sc75c) K. Schulten and R. Gordon, J. Math. Phys. 16, 1961 (1975).
TABLE OF CONTENTS
Pace
ACKNOWLEDGEMENTS 1
ABSTRACT vii
CHAPTER
IINTRODUCTION ..... 1
1. Theoretical Treatment of Nuclear and Electronic
Motions 2
2. Adiabatic Collision Processes 6
3. Nonadiabatic Collision Processes 7
4. ManyBody Theory and the Treatment of Nuclear
and Electronic Motions 11
5. Analysis of Potential Energy Surface Informa
tion 17
6. Proposed Application of the Faddeev Formalism 26
7. Plan of Dissertation 30
IITHE TWOBODY tMATRIX 32
1. General TwoBody Scattering Theory 34
2. Boundary Conditions for the OffShell Wave
Function 36
3. The VariablePhase and Amplitude Method ... 38
4. Computational Aspects of the VPA Equations . 42
5. The Comparison Potential Method 45
6. The Bateman Method 50
IIINUMERICAL RESULTS FOR THE TWOBODY tMATRIX 54
1. Properties of the IwoBody tMatrix 54
2. Numerical Calculations Using the fPA
Equations 57
3. Numerical Results Obtained Using the Bateman
Method 96
IVTHREEBODY PROBLEM 101
1. Problems with the LippmannSchwinger Equation 102
2. Multichannel Transition Operators 103
3. The MultipleCoilision Expansion Ill
VTHE SINGLE COLLISION APPROXIMATION 114
1. Description of Channel States 115
2. Inelastic Scattering 117
3. Dissociative Scattering 125
v
100
results were generally good, provided one again concentrates on small
momentum transfers. We do not report any offshell studies here,
although, a similar quality in the results was obtained. When we
extended the Bateman method to potentials applicable to chemical
problems, e.g. exponential potentials that are very steep and localized
in coordinate space, the results obtained were totally inadequate. The
sensitivity to the choice of (1) became very large. In light of these
findings, no further investigations were carried out. A possible
explanation of this negative outcome might lie in the fact that for a
hard potential, the interaction region is highly localized leading to
a corresponding delocalized function in momentum space. Thus, for such
potentials, one would expect the number of basis function in the set
{k} to be very high.
Figure (66) Plot of the differential energy transfer probability
(Ae) versus Ac as obtained from M^>c,Pk). The stick
spectra corresponds to inelastic scattering and the
continuous curve denotes transitions to the continuum.
T
T
200
0
1
2
R (au.)
CHAPTER VII
THREEDIMENSIONAL INELASTIC SCATTERING RESULTS
In this chapter, we will apply the single collision peaking
approximation to inelastic atomdiatom scattering processes for the
case of full threedimensional motion. Specifically, we will investi
gate the systems (Li + CO) and (Li+, N2), and discuss various numerical
results in light of recent experiments [(Bo76), (Ea78)]. This investi
gation will serve to complement other numerical results found in the
literature [(Bo74), (Ek74), (Ph76)], obtained using the impulse approx
imation [see Section V.4 ].
Within the present approach, we will approximate the inelastic
scattering amplitude by [see Section V.2 ]
M(s,Pk) Pk (2) Pk p(3)
Ml'l l2 Fl'l + t3 Fl'l
(7.1)
where the target formfactors F^j and f? are defined by Eq. (5.27).
The twobody tmatrix elements t2^ and tgrK are offtheenergy shell,
and their arguments are chosen by a procedure analogous to that used in
the collinear case in Section VI.6 One of the goals of this chapter
will be to illustrate how the target formfactors play a major role in
characterizing the energy loss spectra in atomdiatom scattering
processes.
Pk
1. Practical Implementation of the Peaking Approximation
The differential cross section for inelastic scattering processes
163
83
0.0
Figure (37a) Diagonal (K' = K) matrix elements of Re[tn(K', K; E)].
(Huthn Potential)
O'
61
is given by figures (32), (33) and (34) for the Morse potential
case. Since the tmatrix elements have poles at the bound states of
the system [see Fig. (35)], we see from the relation connecting t]
and u), [Eq. (2.13)] that the amplitude of w, must grow as the energy
1/2
approaches a bound state eigenvalue. In Fig. (32), [2/tt] 7 [cog/K]
is plotted versus R for K = 4.475 a.u. and an energy value of E =
0.1446 a.u. The large magnitude of the amplitude displayed by this
function, when compared to the result at E = 0.153 a.u. given in Fig.
(33), is a clear indication that the energy is quite near to an eigen
value of the potential. A similar effect can be seen at higher partial
waves, as is shown in Fig. (34) where [2/tt] [u^(K, K^; R)/K] is
plotted versus R for K = 4.5 a.u. and E = 0.153 a.u. Note that the
effect of the driving term, j^(KR), is always present in all of the
figures (3.2) through (3.4). Within the interaction region of the
potential, it is the wave defect xj which dominates; whereas the driving
term dominates in the asymptotic region. For the Hulthn potential, the
offshell wave function does not display the large changes in magnitude
mentioned above for the Morse potential, since the Hulthn potential
used in this study does not support bound states.
Various studies of the partial wave tmatrix elements as a function
of energy and momenta were carried out for both the Morse and Hulthn
potentials. We have already shown the effect of the bound states on
the offshell wave function co Their effect on the tmatrix elements
is also quite important. In Table (31), we have tabulated the Morse
tg(q, q; E) matrix elements for q = 4.0 a.u. as a function of energy.
Column I gives the VPA results and column II the correspondng results
obtained by a variational method developed by Kurooglu and Micha (Ku78).
Ove) cOLX(a;r>l)) aa
ACKNOWLEDGEMENTS
I would like to express rny appreciation and gratitude to my
advisor, Professor David A. Micha, for suggesting the study of the
problems addressed in this dissertation, and for many helpful dis
cussions. His dedication, support and encouragement have been
invaluable.
I would like to thank all the faculty members of the Quantum
Theory Project, all of whom have contributed to ny development as a
student and scientist. In particular, I thank Professor PerOlov
Lbwdin for providing me the opportunity to attend the summer school
in Sweden and Norway.
I would also like to express my thanks in general to all the
members of the Quantum Theory Project. In particular, I would like
to thank: Dr. Henry Kurtz, Dr. Nelson H. F. Beebe, Dr. Jack Smith
and Mr. Larry Relyea, who were the source of sage computational advice
Dr. Michael Hehenberger, who pointed out the 'work of Shampine and
Gordon; and Dr. John Bellum, who provided useful encouragement and
discussion in collision theory to a somewhat bewiTitered but eager
chemist. My special thanks go to Dr. Zeki KuruogTu who was the sound
ing board for many ideas and whose comments have helped shape much of
the work presented here.
I am grateful to Miss Brenda Foye for the nica job she has done
in typing the manuscript.
14
or in the Faddeev form
(1.43)
where
(1.44)
Equation (1.39) may now be replaced by Eq. (1.43), which is more suitable
for introducing DIM approximations. A symmetrized form of the Hamil
tonian such as
i M = \[J\ ,H] +
(1.45)
has been introduced in order to avoid nonhermitian matrices (Tu7ca) which
occur whenever the are expanded in terms of a set of channel elec
tronic states and used in Eq. (1.43). The channel electronic
states j_^> correspond to a collection of primitive channel wave
functions constructed in a step by step procedure from a set of atomic
wave functions ^and Â£Cm, as is usually done in the DIM method
[(El63), (Mi77d)]. When these channel functions are constructed, how
ever, the arrangement structure of the channel must be considered.
Therefore, one begins by constructing a collection of Mq functions
2
(I^Al^Bn^Cm^' "ac*1 t*ie a^om1c functions are eigenfunctions Sj
and and are composed of nonsymmetrized combinations of products of
atomic orbitals centered on the nucleus of atom I. This set of func
tions we designate j^>. A set of M^ unsymmetrized or primitive
channel wave functions can be obtained from Â£^> by first com
bining the products to form eigenfunctions of Sj and S^ and
15 13 11 9 7 5 3 1 1 3 5 7 9 11 13 15
E X107 (a.u.)
<_n
Re t0(K,K;E) X104 (a.u.) Im t0(K,K;E)X104 (a.u.)
Figure (77b) A plot of Im[t? ] x 10^ versus final rotational quantum
number for the(Li+, 0) interaction in the (Li+, CO)
system. [(ni,^)=(0,0), nf=0, Ej=4.23eV and ej,p^=37.1]
125
The integral in Eq. (5.36) is a simple exercise in elementary geometry
and is thus very easy to evaluate (Se70). In Chapter VII we will see
that the overlap volume, m^, can in many instances predict the relative
shape of the energy loss spectrum for inelastic scattering. However,
from Eq. (5.24) or Eq. (5.26), it is apparent that one should really
worry about the relative phases and magnitude of < and
< q11^i> 1'n the overlap region. A simple extension of the present model
would be to use the semiclassical momentum distributions inside Eq.
(5.36), as recently proposed (Mi77). This would still not address the
problem of the phases, but it would lead to a quantity which may be
used to analyze the possible mechanisms in an atomdiatom collision.
Such a model is attractive, since it would provide a simple, inexpensive
way for the theoretician or experimentalist to gain insight about energy
transfer and reactive processes in atomdiatom collisions. In any case,
the analysis that led to Eq. (5.36) may be used to determine the region
(s) (s Pk)
of integration, if one were to evaluate M^(3) or (3) by a
multidimensional quadrature scheme.
3. Dissociative Scattering
Following the same procedure used in the previous section one can
show that for dissociative scattering Eq. (5.4) leads to
t3 (p3f q3i; Ep_ ^
where
3f
p3f
2m o
(5.39)
Table (32) Selected values of tQ(q, q; E) for
q = 4.475 a.u. are presented about
the first four poles. (Horse
Potential)
Energy (a.u.) t(K,K:E) Location of Pole
0.165078
2.6856
0.165076
5.4287
0.1650741
0.165074
174.7388
0.165072
5.0667
0.144800
1.2866
0.144600
3.5240
0.144486
0.144400
4.6930
0.144200
1.4018
0.125600
0.0290
0.125400
0.0936
0.125268
0.125200
0.2220
0.125000
0.0656
0.107800
0.0011
0.107600
0.0026
0.UP421
0.107400
0.0243
0.107200
0.0020
CHAPTER II
THE TWOBODY tMATRIX
Scattering processes in few or manybody systems require the
introduction of twobody transition operators, whose matrix elements
must be known for arbitrary initial and final relative momenta of the
two bodies and for arbitrary energies, i.e., they must be known in
general "offtheenergyshellIn other words, the operator quantities
of interest are those describing the interaction of two particles
embedded in a many particle system, and as a consequence energy is not
conserved locally, thus explaining the term "offshell". Although these
twobody transition operators stand for basic physical concepts, their
properties and values are little known for molecular systems, where the
bodies are atoms or ions.
The computation of twobody tmatrix elements has long been a
problem of intense interest in the field of nuclear physics. Conse
quently, there exists an extensive literature devoted to this subject
as it pertains to potentials describing nucleonnucleon interactions.
A very nice and extensive review in this regard is the work of Srivastava
and Sprung (Sr75). In the chemical literature, work in this area has
been sparse. However, there have now appeared various papers which deal
with the computation of tmatrix elements for potentials of chemical
interest, for example: van Leeuwen and Reiner (vL61) proposed a numer
ical procedure based on the comparison potential approach, which has now
been implemented in the work of Brumer and Shapiro (Br75); Kuruoglu and
32
en
o
107
The FaddeevWatson equations do contain information on other
channel processes; however, it will be necessary to derive a set of
transition operators more suitable for rearrangement processes. In
order to accomplish this, we will begin by considering a finalchannel
decomposition for the total scattering wave function [(Wa67), (Mi72b)].
Channel 1 will be singled out as the initial channel, for the sake of
specificity. Accordingly, the total wave function will be labeled by
Â¥1(+), and it will be associated to an initial wave function ^ through
the relation (Wa67)
Â¥1(+) = W(Z)$ (4.18)
where W(Z) is the wave operator defined as
W(Z) = ie G(Z) (4.19)
It will be assumed throughout, that these are halfonshell relations,
i.e., E is equivalent to the total energy of the system, as described
initially by Using equations (4.1), (4.7) and (4.9) one can show
that
G(Z) = Go(Z) + G0(Z)T(Z)G0(Z), (4.20)
which along with Eq. (4.19) implies that
W(Z) = WQ(Z) + G0(Z)T(Z)W0(Z) (4.21)
where WQ(Z) = ie G0(Z). The finalchannel decomposition for W now
follows immediately from Eq. (4.16) and Eq. (4.21), and is
w(z) = w0(z) + z y(j)(z),
j
(4.22)
25
a
+ 2a >
(1.61)
where q corresponds to the charge of the atom, the molecular parameters
p, Q, ft and H correspond to the dipole, quadrupole, octapole and hexa
decapole moments of the diatom, and the parameters cu and correspond
respectively to the parallel and perpendicular components of the diatomic
polarizability relative to the internuclear axis of the molecule. The
functions P^(cos0) are Legendre polynomials arid 0 is the angle between
the vector R from the centerofmass of the diatom to the projectile
atom and the internuclear axis of the diatom. Beth the N~ and CO mole
cules have very similar values for the various parameters used in equa
tions (1.59) and (1.60), with the exception that Ng does not have a
permanent dipole moment. This difference, however, is quite important
since the chargedipole interaction determines the R term in Eq. (1.59),
and as such dominates the long range portion of the (Li+, CO) potential.
It is precisely these parameters [p, Q, etc.] which are not well speci
fied in SCF calculations [(St75), (St76), (Th78a)]. The value of the
dipole moment of CO in fact has been the subject of wide controvercy
(Ne64). Experimentally this value is small and negative [p^q = 0.049
a.u.] (St76) and implies the polarity of the CO molecule is given by
C0+. More recent calculations (Ch76) have now shown agreement with the
experimental results. Unlike the situation for N^, in a col linear reac
tion the potential between Li+ and CO is not symmetric. If one examines
the behavior of the equilibrium internuclear separation of CO as a func
tion of R,one would find that it shows much more deviation from its
value at R equal to infinity than that for N^. In the scattering exper
iments on these systems (B576), it is found that (Li+, CO) shows more
71
We note that in general the agreement is quite good. A more detailed
picture about the structure of the Morse tg matrix elements at negative
energies can be seen from the results given in Table (32) and fig. (35),
each of which describes the energy range corresponding to the first four
poles. The figure shows that the tmatrix elements change sign as one
goes across a pole, in agreement with equation (3.9). For the Hulthn
potential, the tmatrix elements show a very smooth and plain structure
as can be seen in Table (33), where tg(q, q; E) is given versus energy
for q = 4.5 a.u. The agreement with the variational results is again
quite good. At positive energies, we have shown in Fig. (36) the
behavior of the Morse tg(q, q; E) matrix elements near E = 0 a.u.
[q = 4.0 a.u.]. We see the continuity one expects; i.e.,both equation
(3.7) and (3.8) are satisfied. In general, the tmatrix elements of
both the Morse and Hulthn potentials show a much more oscillatory
structure at positive energies. Figures (3.7a) and (3.7b) illustrate
this for the diagonal (K1 = K) Hulthn tg(K', K; E) matrix elements.
The large oscillations seen in these figures do not extend indefinitely.
Korsch and Mohlenkamp (Ko77a) have extended the present calculations to
higher energies and have shown that the amplitude of the tmatrix even
tually decays to zero. In the comparison of their semiclassical results
with the present quantum mechanical ones, they obtained good agreement.
As a function of momenta, the tmatrix elements show a much greater
oscillatory nature. This can be seen in Fig. (38) where the Morse
tg(K, K; E) matrix elements are plotted versus K for E = 0.1378 a.u.,
and in Fig. (39) where the Morse tg(K'; K; E) matrix elements are
plotted versus K' for E = 0.153 a.u. and K = 4.5 a.u. Both figures
(3.8) and (3.9) show how the tmatrix goes to zero asymptotically in
f 5r'V"v5> f **
***'*'* y?
>.
i
I
CO
CO
116
Equation (5.8) can be proved using the relations in Appendix IV. This
equivalence amongst the various channel coordinates can be stated
formally by the relations
= 6[Rj Rj (^ ) ] 6 [ rÂ¡ ^Â¡(R^)] (5.10)
and
< PjPjjPjPj^ = 6[Pj PjiPi^OlStPj Pj (Pi, Pi) ] (5.11)
where a quantity such as Rj(R.Â¡,r.Â¡) denotes that value of Rj obtained
from its relation to R^ and r., as given by Eq. (A4.4) in Appendix IV.
Alternately, one could derive these last two equations by making use of
equations (5.5) through (5.8) and the identity operators (Re77)
1 = / dRi dr  R^ r.Â¡>< Ri ri  (5.12)
and
1 = / d^ d^  ^ pjx Pi Pi  (5.13)
The free motion in the remaining channels can be described by
(Mi 75)
= Xj(Pi) 4>i (ni0Â¡mj) (i = 1, 2 or 3), (5.14)
where x.j(P.j) is a relative motion state and is a vibrotor state
characterized by the quantum numbers n., j. and m^.. In the coordinate
ii Ji
space representation one has
= ,
(5.15)
120
(Ek71), (Ek74)]. Within this approximation one factors the tmatrix,
t3, out of the integral sign in Eq. (5.24) yielding the expression
M
r;Pk)(3) t^k /d53<*1'h1'xl1l*1>
(5.26)
where tPk denotes that value of t, obtained for some suitably chosen
value of Q3, say QPk. The criteria for selecting QPk will be discussed
later in Section VI.I. The assumption one has introduced here is that
t3 is a smooth and slowly varying function of Q3 within the range of
integration determined by < 1 q^' > and < q^<}>j>, which on the other
hand, are highly oscillatory functions (Ek71). Physically, the peaking
approximation implies that we have factored the transition amplitude,
(s) Pk
Mj,' (3), into a term t3 which contains information on the dynamics of
the collision process, and a term
/ dQ3 < 4^ >< q114>1>
(5.27)
called a target "form factor," which contains information characterizing
the size and shape of the target, along with information on the initial
and final momentum distributions of the particles involved, i.e. the
restoring forces of the diatomic target. We see then that the role of
the potential for the target diatomic, in this case V3, is one of deter
mining the initial and final momentum distributions [(Ek71), (Mi75b)].
Consider now the validity of the single collision approximation.
We know that in general multiple collision expansions are applicable
only at high energies [(Ch71), (Ch73), (Mi75b)]. From a mathematical
point of view, the convergence properties of expansions such as (5.1)
35
and by relating the matrix elements of the toperator to those of the
wave operator in a mixed coordinate momentum representation (vL61).
Multiplying (2.4) by G^(z) on the left, one finds that W(z) also
satisfies
(z Hq V)W(z) = (z H0). (2.5)
In a mixed coordinatemomentum representation one then obtains the
inhomogeneous Schrodinger equation below (f) = 1):
[z + (2m)_1V2 V(r)] = [z (2m)1q2], (2.6)
where m, r and q are respectively, the reduced mass, the relative
coordinate and the momentum of the two particles being considered, and
 is the offshell state (since q2/2m f E) defined as
a^(z)> = W(z)q> (2.7)
In the nuclear physics literature, (2.6) is known as the BetheGoldstone
equation (Sr68). A reduced radial equation may now be obtained by
introducing into equation (2.6) the following partial wave decompositions
= (2tt)"3/2 e^r
= (2tt) 3/2 E i1 (21 + 1)P1 (q*r)j1 (qr)/(qr) (2.8)
and
= (2tt)3/^ Z i1 (21 + l)P1(q?)a>1(q,kE;r)/(qr), (2.9)
where j^qr) is the regular RiccatiBessel function and kÂ£ = 12mE1//2.
The resulting reduced radial equation is
Figure (3
2) [2/tt]1/2 to)0(K, Ke; R)/K] at E = 0.1446 a.u., an
energy value near the second pole. (K = 4.475 a.u,
Morse Potential)
115
(5.4) as well as the factors which determine their validity.
1. Description of Channel States
For each arrangement channel i, i = 1, 2 or 3, we shall introduce
the set of Jacobi coordinates ft and r, along with the corresponding
conjugate momenta and p.. The momenta and p^ describe, respec
tively, the relative momentum and the diatomic relative momentum in
the given channel i. A definition of all these coordinates in terms
ordinary laboratory coordinates for the three atom system is given in
Appendix IV.
Working with the given relative coordinates in the center of mass
system, we can describe the free threeparticle motion in channel 0 by
the plane wave
= < R, Pi>< T, IPi> (55)
where + ^
< RiPi> = (2tt)3/2 e1 Pl Rl (5.6)
and
(2,)3/2 e1 Pi
4
(5.7)
Note that any one of the channel coordinates can be used to describe the
system, since
* Rj + Pj
j = 1,2,3), (5.8)
and thus
< RjrjlPJpj> = < RirilpiPi>
(5.9)
My deepest appreciation goes to my wife, Adriana, for her patience,
understanding and constant encouragement during the task at hand.
iv
APPENDIX I
HARD CORE TWOBODY tMATRIX
We will now derive a closed analytical expression for the two
body tmatrix for a hard core potential using the method of Laugh!in
and Scott (La68), i.e. the tmatrix elements for a potential of the
form
V = X 6 (rc r) (Al.l)
in the limit of X going to infinity. The step function 6 is defined
by Eq. (5.37). The offshell wave function co^ can be obtained immedi
ately from the appropriate radial differential equation, Eq. (2.10),
and the physical boundary conditions, equations (2.15) and (2.16). The
result is
Vqrc)
i(q> kr; r) = j (qr)
1 1 1 C(l)
Vc)
hi(1)(kErc).
(A1.2)
The tmatrix can now be obtained from the definition of t^(q1, q; E)
given by equation (2.13); however, as was pointed out in Section II.2 ,
one must make use of Eq. (2.18). From this we obtain
Vq' E) = kE; rc+>
(q2 kE2) / J1(q'r)j1(qr)dr]. (A1.3)
196
148
A + (BC) A + B + C .
The presence of three unbound particles in the exit channel is what
makes CID very difficult to study from a theoretical point of view.
As was pointed out in Chapter I, because of the nature of the exit
channel, coupled channel procedures lead to a continuous infinite set
of coupled integrodifferential equations [(Wo75), (Kn77)], and would
thus be very difficult to solve. The approach we shall take, suggested
in Section V.3 with the single collision approximation, offers a
much more tractable procedure. Specifically, the approximations we
shall use are
01 = V^f ^3i EP3f> liln> <6'34>
and
M0lCPk) = ^ / dQ3 <$1C'q1'> (6.35)
the one dimensional analogs of equations (5.33) and (5.44). Clearly
the amount of work necessary within this latter approach is comparable
in magnitude to the amount that was spent on the inelastic problem, and
this is certainly not true of the coupled channel procedure [(Kn77),
(Mi75), (Sh77), (Wo75)].
Using a Morse potential,
V^x^ = D[ 1 e"axl]2, (6.36)
for the target diatomic does not affect the validity of the transforma
tions used in Section VI.I to obtain Eq. (6.6), the Schrbdinger equation
for a particle of mass m hitting an oscillator of unit mass vibrating
about an equilibrium position [(Se66), (Sc75a), ((Ek72)]. Energies are
11
work of Fisher et al. [(Fi71), (Fi72), (Ba69b)], Gislason and Sachs
(Gi75), and in the review article by Tully (Tu76a). We note that Eq.
(1.23) is not suitable for treating rearrangement collision processes.
However, using the manybody formalism of the Faddeev equations Yuan and
Micha (Yu76) have implemented multiple curve crossing information to
study rearrangement scattering of the atomdiatom system [K,IBr].
4. ManyBody Theory and the Treatment of Nuclear and Electronic Motions
All the methods discussed in the previous sections involved the
treatment of electronic and nuclear motions essentially as two separate
problems. This approach has resulted in the introduction of the concept
of a potential energy surface, i.e., that quantity which characterizes
the electronic state of the system and serves as the effective potential
governing nuclear motion. An alternative to this picture would be the
simultaneous treatment of nuclear and electronic motions. We will
briefly sketch how this can be accomplished by the use of manybody
theories. This will be done for three reasons. First, it will serve
to contrast the more traditional methods of scattering theory based on
Eq. (1.6). Secondly, it will show how one can use known twobody data
as input to the solution of an inherent manybody problem. Lastly, it
will serve as a reference with which to compare the approximate treat
ment of atomdiatom collisions given in later chapters of the present
work.
The manybody formalisms that will be discussed are based on the
treatment of the threebody problem pioneered by Faddeev (Fa61). Earlier
it was pointed out that for a threeparticle system, there are four
arrangement channels. This implies that free motion in each asymptotic
ENERGY
IN
TRANSFER AND DISSOCIATION
ATOMDIATOM COLLISIONS
HYPERTHERMAL
By
LYNTIS H. BEARD, JR.
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1979
134
Introducing the definitions
0 if [sgn(q)sgn(x)] = 1
<
tt if [sgn (q) sgn (x) ] = 1
r e x
1^(0) = [cos(e)]1
and
f^qr) = <
one can shov; that
. 1 i
e1clx = I i1
1=0
cos(qr) if 1 =0
sin(qr) if 1 = 1
nj [cos(eqx)] f] (qr)
and
(6.16)
(6.17)
(6.18)
(6.19)
(6.20)
1/2 ^ 1
(jo(q, k ; x) e = (2tt) S i q,[cos(6qx)]w, (q,k ;r);
t H 1=0 1 1 L
where
(6.21)
2
[ + (aEkE2 v(r))] ^(q, kÂ£; r) = (a^2 q2)f](qr) (6.22)
and v(r) = 2mV(r). The function takes the place of the Legendre
A
polynomial and f^ that of the RiccatiBessel function found in
the partial wave expansion for the threedimensional offshell wave
function, (z)>. However, the expansions used in equations (6.19)
and (6.20) are not partial wave expansions; they are an artifact of the
way we have chosen to solve for the offshell wave function. Alternative
formulations of onedimensional Schrodinger equations in terms of
parity expansions or in terms of phase shifts could also be pursued
Figure (12) Various cuts through the potential energy surface
of an atomdiatom system are illustrated. The
parameter R measures the distance of the projectile
atom to the centerofmass of the target diatom;
whereas, r is the internuclear separation of the
target diatom. Planes I and II are perpendicular
to plane III at r equal zero and r respectively.
73
Table (33) Selected values of t(q, q; E)
for q = 4.0 a.u. areugiven.
Column I gives the VPA calcula
tions and column II the results
obtained from a variational pro
cedure (Ku78). (Hulthn Poten
tial )
Energy (a.u.)
Vq>
q; E) x 103
I
II
0.18
6.4807
6.4592
0.17
6.2683
6.2477
0.16
6.0525
6.0300
0.15
5.8277
5.8057
0.14
5.5946
5.5742
0.13
5.3574
5.3348
0.12
5.1079
5.0868
0.11
4.8526
4.8295
jf
o
10
20
25
29
30
35
40
45
50
60
(73) A comparison of differential cross section data obtained using offshell
(column I), halfonshell (column II) and onshell (column III) tmatrix elements.
Renormalized results have been obtained by setting the maximum peak in energy loss
spectra equal to one. The data used is for the (Li + CO) system as given in the
caption for Fig. (71).
do / n
dfl (ni~0
.ji =0 > nf=0,jf)
I
II
III
Renormalized
Renormalized
Renormalized
2.27( 4)
2.43( 3)
2.12( 4)
2.38( 3)
2.12( 4)
2.60( 3)
5.87( 4)
6.31( 3)
4.10( 4)
4.60( 3)
4.97( 4)
6.10( 3)
2.40( 2)
2.57( 1)
2.29( 2)
2.57( 1)
2.20( 2)
2.70( 1)
2.09( 2)
2.24( 1)
1.99( 2)
2.23( 1)
1.88( 2)
2.31( 1)
9.31( 2)
1.00
8.91( 2)
1.00
8.15( 2)
1.00
7.69( 2)
8.26( 1)
7.38( 2)
8.29( 1)
5.36( 2)
6.58( 1)
9.55( 3)
1.03( 1)
8.81( 3)
9.89( 2)
4.70( 3)
5.77( 2)
5.26( 2)
5.65( 1)
4.71( 2)
5.29( 1)
3.30( 2)
4.05( 1)
5.66( 3)
6.08( 2)
5.05( 3)
5.67( 2)
3.29( 3)
4.03( 2)
3.84( 5)
4.13( 4)
3.43( 5)
3.85( 4)
2.03( 5)
2.49( 4)
1.09(11)
1.17(10)
9.94(12)
1.12(10)
4.7 9 (12)
5.88(11)
CO
o 12 24 36 48 60
j,
I
rx>
This dissertation was submitted to the Graduate Faculty of
the Department of Chemistry in the College of Liberal Arts and
Sciences and to the Graduate Council, and was accepted as partial
fulfillment of the requirements for the degree of Doctor of
Philosophy.
March 1979
Dean, Graduate School
126
m n m
3i
= P +
li
AC
m.
nlAB mBC
If
m
AB
If
(5.40)
P3f = q3. APj (5.41)
and
?i = plf ir~ Afi <542>
i ir mbc
Equations (5.40) through (5.42) follow from the relations in Appendix
IV. One does not have an integral in Eq. (5.38) as was the case for
(s)
*^111(3) since the final state $ is characterized by a product of
plane waves as given by Eq. (5.5). Furthermore, Eq. (5.38) is simpler
than Eq. (5.24), since it contains only the halfonshell tmatrix.
This simplification however, does not help as much as one would think.
Experimentally, Eq. (5.38) would imply the need for a coincidence study,
one in which two of the exit particles are detected simultaneously
[(Sc74a), (Sh77)]. This is something that has not been done so far
[(Sc73), (Sc74b), (Sc75)], and thus one must average ]M^^ over p^
to obtain a proper comparison with experiment (Sc73). Because of energy
conservation, only an average over the orientation p would be needed.
We will address only the simpler problem of collinear scattering, and
will postpone further discussion on this matter to the next chapter.
In section IV.2 we presented an alternate expression for Mqj, Eq.
(4.38b). Within the single collision approximation this relation leads
to
(5.43)
Figure (7
6) A plot of versus final rotational quantum number
for the (Li+, CO) system. [(nj,^)=(0,0), nf=0, Ej=4.23eV
and =37.1]
11
168
for the present case in which only lower vibrational levels are con
sidered.
Finally, one should note that the evaluation of the target form
factor or the differential cross section requires the evaluation of 3j
symbols to very high j values, when a large amount of rotational excita
tion is involved. For the systems we shall study in the following
section, j values up to sixty will be considered. Often the straight
forward use of recursion relations found in standard texts [(Ed74),
(Mi61)] will lead to computational problems. To avoid this difficulty,
we have adopted the algorithms given by Schulten and Gordon (Sc75c)
which are stable to very high j values (j>100).
2. Numerical Results
The results of various calculations on the systems (Li+, CO) and
(Li+, N2) will now be presented. Within the present manybody approach,
these systems are convenient because the mass ratios are small and
repulsive forces are dominant in the collision energies of interest.
Using equations (5.33) and (5.34), one obtains the mass ratios 0.15
and 0.27 for the (Li+, CO) system, and the mass ratio 0.2 for the
(Li+, Ng) system. With these mass ratios, the single collision approx
imation should be applicable, since the ratios are less than 0.5 [see
Section VI.3 ]. The potential energy hypersurfaces have been recently
calculated by Staemmler [(St75), (St76)]. Using the information found
in these references, Micha et al. (Mi78b) have extracted twobody poten
tial data corresponding to the collinear configuration of each of the
above systems. For the interaction between the projectile and each of
the target atoms, they used a potential of the form
12
region is governed by a different channel Hamiltonian. In order to
clarify this, consider the Hamiltonian for three interacting atoms A, B
and C (in laboratory frame),
H = Z K + Z H, + Z V
a r I i
(1.33)
K/J
where is the kinetic energy of nucleus a = a, b, or c (associated with
A, B and C respectively); Hj is the electronic Hamiltonian of atom I,
and Vj^ is the Coulomb interaction among the charges of atoms J and K
(Mi77b). The channel Hamiltonian for a given rearrangement channel
i[l for (A+BC), 2 for (B+AC), 3 for (C+AB)] is defined by the relation
H. = 1im H (1.34)
R.**
and as can be seen using Eq. (1.33)
(1.35)
H. = E K + E Ht + V, .
^ a y I JK
In the breakup arrangement channel one can analogously define the free
particle Hamiltonian
(1.36)
which can be used in Eq. (1.35) to obtain
Hi Ho + Vi
(1.37)
and
3
H = H + E V.
o i
(1.38)
10
E"
(1.27)
T
I
(1.29)
(1.30)
(1.31)
(1.32)
and the quantities m^ and mc are the masses of the atoms composing
the system. Equation (1.23) is the basis for treating an inelastic atom
diatom scattering problem. The diagonal matrix elements Wlvlv^l
correspond to effective potential energy curves, each of which is asso
ciated to a particular electronicvibrationalrotational state (Tu76).
These potential energy curves then serve as a basis for treating charge
transfer and inelastic scattering within the multiplecurve crossing
model [(Bac9b), (Fi71), (Gi75), (Tu76a)]. It is obvious, however, that
due to the enormous number of these curves, various other approximations
must be introduced. Generally one averages over the rotational states
and assumes that nonadiabatic effects are confined to the avoidedcross
ing regions where the LZS approximation can be used. A discussion of
the approximations and implementation of this model can be found in the
203
for
/ rv, ( 377 u )dr
(q2q22)1i[r(^u1)(^;i)
 V ( 4 u ) (a2 HL+J1 ]r: v 1
V dr V vqi r2 1 vr
1
 2qx2 / u,v dr}
rl
(A2.ll)
qj f Equation (A2.ll) may be proven using the identity
[K 4: vj v.][ d2
dr "1' V1JL dr2
U] + (qf 1(1 1))u13> = 0, (A2.12)
which follows from equations (A2.ia) and (A2.1b).
193
equations led to a multiple collision expansion, which in effect un
coupled the various transition amplitudes. Because of this, approxima
tions based on the multiple collision expansion allow for the direct
computation of the transition probabilities of interest without the
need for introducing expansions in terms of intermediate states. For
problems such as collision induced dissociation, we saw in Chapter VI
that enormous simplifications are obtained using such a manybody
approach. We conclude then that approximations such as the ones we
have pursued in this dissertation warrant further investigation as a
viable and very practical approach to the study of atomdiatom colli
sion at high energies.
From a practical standpoint, the calculations given in the last
section for inelastic scattering demonstrate that useful qualitative
information is obtainable with the single collision peaking approxima
tion. However, it is clear that if more reliable quantitative informa
tion is desired, one must input more realistic twobody tmatrix data.
In light of the difficulties encountered in obtaining converged tmatrix
elements at high energies and large momenta, it is clear that an improved
algorithm for the high partial waves must be sought. One possibility in
this regard would be the implementation of semiclassical procedures.
On the other hand, perhaps the most practical solution would be to
pursue nonpartial wave techniques. Even though the Bateman method used
in Chapter HI did not work for singular potentials, one could modify this
procedure using a twopotential formula (Ro67). This would entail
separating the hard repulsive core from the tail of the potential, and
using the Bateman method for the softer component. The need for an
accurate numerical procedure still remains, however, since it will serve
220
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113
By convention, P. and p^. will be used to denote the momentum charac
terizing an initial or final state $.j, and the variables and q.. will
be reserved for the intermediate momenta (Mi75b). Using the definition
of T3, Eq. (4.11), we see that
where t^ is a twobody tmatrix describing the interaction between
atoms A and B. is the channel reduced mass defined by Eq. (A4.8)
in Appendix IV. Note that the energy argument of t^ corresponds to
the total energy minus the kinetic energy of the spectator atom. From
Eq. (5.22) one can also ascertain the fundamental difference between
the threebody quantity and the twobody quantity t^ On the energy
plane the pole structure of t^ is replaced by branch points in the
structure of (Lo64a). Matrix elements such as < q31
evaluated via equations (5.11) through (5.19), yielding the result
< ^3 <^3^1> ~ ^^3 ^3^1 ^3^< ^1^1 ^3^I<+>i> *
(5.23)
(s)
Thus, the expression for M (3), Eq. (5.21), leads to
) (5.24)
where
(5.25b)
(5.25a)
(5.25c)
119
*1 V53^l' > V % A?1 (525d)
and APj = P^' P^ is the momentum transfer. As can be seen from Eq.
(5.24), M^j (3) has a very simple structure. All that is needed in
its evaluation is the product of the initial and final momentum distri
bution amplitudes, along with the twobody tmatrix. This simplicity,
however, is quite deceiving. The integrand in Eq. (5.24) actually has
a multicenter nature, as can be seen from equations (5.25a) through
(5.25d). In essence the problem we are facing is the familiar one deal
ing with the angular momentum analysis of an atomdiatom scattering
process. The usual approach taken in solving problems of this nature
involves the introduction of various partial wave expansions for each
of the quantities involved [(Ah65), (Ba69a), (El69), (Ha70), (0m64)].
However, this does not seem practical for high energies, which require
a large number of partial waves. Another possibility, would be that of
using a threedimensional quadrature scheme. This is particularly
attractive, since the integrand of Eq. (5.24) contains the product of
* .
the momentum wave functions < j' jq^ > and < q^{, which would limit
f
the range of integration over to some overlap region between the two
amplitudes (Mi75b). There is one major problem though, and it is the
idea of a threedimensional quadrature itself. Multidimensional
quadratures are generally somewhat unreliable [(D.a74), (St67)]. This
however, is an open question requiring further investigation. Finally,
there remains the possibility of introducing approximations into the
evaluation of (3). In this regard, we will now consider what is
known in the literature as the "peaking" approximation [(Ro67), (Mc70),
40
The functions A, and B behave as constants asymptotically, as can be
1 1
seen from the boundary conditions (2.14) and (2.15). These constants
may in turn be related to an offshell phase shift, 6^(q, k ), and
amplitude, a^(q, k^), in the same manner as is done for the onshell
case [(Ca63), (Ca67)j. Therefore one has
A,(q, k ; r) ~ a (q, kp) cos [6,(q, kJ] (2.27)
t p > co  L
and
B,(q, kE; r) ~ a (q, kÂ£) sin [6,(q, kÂ£)] (2.28)
P > CO
The variable phase assumption consists of extending the definition of
the phase and amplitude to all values of r (Ca63); thus,
A'j (q, kÂ£; r) = ^(q, kÂ£; r) cos [fi^q, kÂ£; r)] (2.29)
and
Bj(q, kE; r) = aj (q, kE; r) sin [6^q, kÂ£; r)] (2.30)
This assumption is of course valid, since 6j (q, kÂ£; r^) and a^(q, kÂ£; rM)
are just the resulting phase shift and amplitude for a potential with
a cut off at r = r^ (Ca67). In fact the justification of this approach
is best understood by considering a finite boundary value problem and
extending the boundary to infinity (Ze64). The desired set of coupled
first order differential equations, the VPA equations, may now be
derived using (2.29) and (2.30) for Aj and To accomplish this,
first consider the derivatives of A^ and B^ as defined by equation
(2.24) and (2.25). From these relations it is apparent that
VkEr) Mq kE; r) i(kEr) Vq V r) = o ;
(2.31)
117
where
< RiIxi(Pi)> = < RiPi> = x (Pi.R)
and
< ri(J)i> =
Similarly, in the momentum representation one obtains
< Pj' Pi' I V = < Pi' Pi>< Pi' l4>i>
or
< Pj' Pj $i> = <5("Pj1 Pi )4i (ni, j'i, mj ; pil ) ,
where
5i (ni, ji; p ) = / dr. < l^x r.  1>
is a diatomic momentum distribution amplitude.
(5.16)
(5.17
(5.18)
(5.19)
(5.20)
2. Inelastic Scattering
The singlecollision inelastic scattering amplitude, was
given by Eq. (5.3). We will denote each term in this expression,
'< I "P j I ^ Mjij(j); and for the sake of specificity, we will
study the j = 3 term. Working in the momentum representation, we see
that
,(s)
(3) = / dQ3 dq3" dQ3 dq3 < ^ Q3 q^ >< Q3 q^ T3IQ3 q3> x
x1> .
(5.21)
based on the singlecollision approximation are presented for both
coll inear and threedimensional atomdiatom scattering processes.
15
then combining the products of these and Â£n to
\J i
form eigenfunctions of
2
S and Sz (Mi77d). This leads to the relations
5(J)> lig> 4^ '
(1.46)
(1.47)
and
<^(j)r(J)> = r(j)
(1.48)
where _[g anc* 1'^ are unit matrices and C^ defines a channel coupling
matrix between the breakup channel (channel 0) and the arrangement
channel i. The unsymmetrized or primitive channel wave functions
can now be expressed as
x(j) = r(j)>x(j) f (1.49)
where denotes a M^' x 1 column of scattering functions. In ob
taining Eq. (1.49), one has assumed the approximate completeness rela
tion
ll(j)>
V
A
where 1. is the unit operator in the space of many electron states in
sJ
channel j (Mi77d). Using equations (1.43) and (1.49) the end result of
the DIMFaddeev equations is given by
{A(j)E Z K Jr[A(j), Z E, + V.]+} x(j) =
a I J
i[A(j), V ] Z (1 6.J C(jk) X(k) (1.51)
J k JK
A(J) = .
where
(1.52)
Figure (63) Plot of ?2q versus the kinetic energy of the projectile
in units of hw
(a) Exact Results
(b) Single Collision Results
(c) Results From Peaking Approximation
(d) Renormalized Single Collision Results
(m = 0.125, V} = Harmonic Oscillator Potential,
= Hard Core Interaction).
1.0
T
T
0.75
050
025
0
0
12
24
T
36
48
60
T
S.65 x 102
1 1
H + D2
7.72 x102
m = 0.2
nÂ¡=0,E' = 20.0
579 x 102
P
1
3.86 x10'2
1.93 x10"2
0
j L
1.00 3.40
780
T
12.20
1560
21.00
vJT
CO
211
ap (nn'+li)
F,,, = A r(s ,+ s + i  ) ttt
JL 1 n n n n a (2s ,+1) ,
x 3F2 (n, V + sn + if n' n + 2sn + *
A .AP
 n + l; 1) ,
a
(A5.5)
where
Vn
(z).
Vn (4D)iP/a f
cl
{a(2sn+l)n[n!r(2sn)]_1}1/2 ,
2D n 1/2 ,
r(z+n)
"W
(A5.6)
(A5.7)
(A5.8)
(A5.9)
^2 1S a hypergeometric function (Ab65) defined by the relation
/ v n (_n)k(a)k(b)k zk
3F2(n, a, b; c, d; z) (c)k(d)k FT (A5.10)
and the quantities a and D are the Morse potential parameters defined by
Eq. (6.36).
The form factor F^ for dissociative scattering in the case of a
Morse oscillator can be derived following the analysis given by Strachan
(St35). Using the notation given by Eckelt and Korsch (Ek73), one obtains
r.ii(s) r(snn^)r(l2s)r(n+iif)
01 E',n r(l+ns+sn)r(lssniy)
41
therefore, the derivative of the offshell wave function has no terms
involving the derivatives of A and Bj, i.e.,
wj (q, kE; r) = J^qr) + Aj(q, kÂ£; r) ^ J1(kÂ£r)
 B^q, kE; r) ^ n1(kÂ£r) (2.32)
Making use of (2.29) and (2.30), equations (2.26) and (2.32) may be
written as,
and
w1 = jj(qr) + a1 cos (6^ jj (kÂ£r)
A
 a1 sin (6j) n (kÂ£r)
Br 1 = clr h + al cos ^61 ^ BF J*1 (kEr)
(2.33)
 a sin (6,) d n,(kEr) .
(2.34)
Taking the derivative of (2.33) and comparing it with (2.34) one obtains,
A A
sin (6j) j (kpr) + cos (6j) nj (kpr) d (2.35)
dF i = al
cos (6j) jj (kEr) sin (6j) nj (kÂ£r) dr 1
In a similar manner taking the derivative of (2.34) and comparing it to
the inhomogeneous Schrodinger equation (2.10), one finally obtains, with
the aid of (2.35), the "variablephase and amplitude" equations
df 6j (q, kÂ£; r) = k"1 v(r)[wj j^qrM/c^ (2.36)
and
HFai(q, kE; r) = v(r)w1 [sin(6^) (kÂ£r) + cos^ )n} (kÂ£r)].
(2.37)
84
K and K' [see equations (3.5) and (3.6)]. Comparable behavior was
observed at positive energies as well, irrespective of the attractive
or repulsive nature of the potential. Table (34) shows a momentum
study for the Hulthn potential tg(q', q; E) matrix for E = 0.15 a.u.
and the 1 values 0, 5 and 10. These results indicate that the diagonal
q1 = q tmatrix elements are consistently larger in magnitude than the
offdiagonal q! / q tmatrix elements. Column I in this table gives
the VPA results and Column II the corresponding variational results
(Ku78). As a function of the partial wave parameter 1, the tmatrix
generally shows an oscillatory structure for low partial waves and a
smoothly decaying structure for high partial waves. Figures (310a)
and (310b), nicely illustrate this effect for both the real and imagi
nary components of the Hulthn t(K', K; E) matrix elements (K' = K = 4.0
a.u. and E = 0.01 a.u.). Using the same data, #e see in figures (3lla)
and (3llb) plots of the real and imaginary confonents of the total
tmatrix [t(K1, K; E)] versus momentum transfer AP = K' KÂ¡. These
plots imply that the tmatrix elements become snail er as the momentum
transfer increases. In this particular case, thirtyfive partial waves
were required for the total tmatrix to converge As the energy and
momenta arguments become larger, this number will) generally increase.
Parcial wave convergence will be discussed further in Chapter VII,
where the twobody tmatrix elements are used in the threebody atom
diatom scattering problem.
The tmatrix results we have reported here were all obtained from
Eq. (2.13) using a numerical quadrature, which in this case was a com
bination Simpson's and Newton's 3/8 rule (Hi56). In order to test the
reliability of this procedure, we have used Eq. '(33.2) to obtain phase
Figure (35) Calculated pole structure of
tg(K, K; E). (Morse Potential)
34
when one deals with physical systems at energies for which a large number
of partial waves are required and the computational expense becomes im
portant. Consequently, in light of this problem, we shall also investi
gate a nonpartial wave approach that has been recently developed by
Belyaev and coworkers (Be74), which is based on the Bateman method
[(Ba22), (Ka58)] for solving integral equations.
1. General TwoBody Scattering Theory
The general computation of offshell matrix elements of the twobody
toperator for simple nonsingular potentials can in general be solved
without much difficulty. This may be done by direct consideration of
the LippmannSchwinger equation [(Si71), (Sr68)]
t(z) = V + VG0(z)t(z), (2.1)
where
G0(z) = (zH0)'1. (2.2)
In these relations HQ is the kinetic energy operator for relative motion,
z is a complex energy parameter (z = E+ie) and Â¥ is the interaction
potential between two particles. V will always be assumed local and
spherically symmetric. From equation (2.1) it is apparent that the
problem must be reformulated for singular potentials, those which contain
a hard core or have a singularity stronger that fyr^ at the origin
(Ta72). This can be accomplished by introducing the wave operator W(z)
via
t(z) = VW(z), (2.3)
where
W(z) = 1 + G0(z)VW(z),
(2.4)
1.0
T
075
Li* CO
0.50
0.25
0
0
T
1
36
P*
/
6.0 8.0 100
(a.u.)
as
cr,
Ill
is that one can describe the breakup process as one would a rearrange
ment process, but the interacting pair in the exit channel must be
assumed to lie in a continuum state.
3. The MultipleCollision Expansion
In practical applications, the exact treatment of multichannel
scattering processes via the coupled AGS equations for is generally
not possible. This is particularly true at high energies, where the
number of accessible states in each given arrangement channel becomes
quite large. Fortunately, the form of the AGS equations suggests that
one might be able to avoid these difficulties by considering their
iterated form. We will now discuss the implications of this approach
as it pertains to atomdiatom collisions.
If one now iterates Eq. (4.35), one obtains the series
Til = ^ilV1 + ^ ^i/jlTj +
J
+ 16. .6..6 .T.G T, + ...
jk ij jk kl 3 o k
(4.40)
The physical meaning of this series becomes clear if one keeps in mind
the nature of the operators involved. Consider for example the first
term in Eq. (4.40). From the discussion in the previous section it is
obvious that this term does not contribute to either inelastic or break
up processes. Therefore, it plays a role only in the case of rearrange
ment scattering. Since this term does not contain any interaction terms,
it is often called the "spectator stripping" term [(Mi72b), (Mi72c)].
The second term in the series contains only the operators T.Â¡, which are
twobody transition operators in a threebody space. Hence, these terms
94
shifts from the halfonshell tmatrix elements. Typical results are
shown in Table (35), where Morse potential phase shifts are tabulated
for the energies E = 0.01 and E = 0.04 a.u. Columns I and III correspond
to phase shifts obtained using Eq. (3.2), whereas columns II and IV list
analogous results obtained from the numerical integration of the onshell
Schrodinger equation. A De Vogelaere integration scheme (Le68) was used
for the onshell calculations using the procedure developed by Bernstein
(Be60). Comparably good results were obtained for various other poten
tial data. Other factors that demonstrate accuracy of the present method
are symmetry and unitarity [see equations (3.1) and (3.2)]. Generally,
agreement to three or four decimal places was obtained. An exception
to this occurred at high values of momenta where the tmatrix elements
are quite small in magnitude. This indicates that the mesh size, over
which the offshell wave function is determined and over which the quad
rature in Eq. (2.13) is carried out, may be too large. Given the extreme
oscillatory nature of the functions involved, a smaller step size would
be desirable. However, a compromise on the step size is necessary to
prevent the accumulation of error from being too large.
It is in this respect that the comparison potential method offers
an advantage, since the quadratures involved in that procedure are done
analytically. We found though, that the VPA procedure showed more
stability in the propagation of the offshell wave function through non
classical regions. This, however, warrants further investigation, since
it is a function of the efficiency and sophistication of the computer
codes used.
Figure (6
Plot of Pjq versus the kinetic energy of the projectile
in units of hto
(a) Exact Results
(b) Single Collision Results
(c) Results From Peaking Approximation
(d) Renormalized Single Collision Results
(m = 0.125, Vj = Harmonic Oscillator Potential,
Vg = Hard Core Interaction).
1.0
075
050
0.25 
0
O 371
o 43.2
x 49.2
0
X
X
10
e
20
30
it
40 50 60
CO
O'"
APPENDIX V
ANALYTICAL QUANTITIES USED IN THE COLLI NEAR SCATTERING MODEL
The computation of the peaking transition probabilities for vibra
tional excitation and collision induced dissociation reported in Chapter
VI require the evaluation of the form factors:
A
Fri = (A5.1)
and
A
Fq1 = (A5.2)
For the case of inelastic scattering, F^,^ may be found in the liter
ature for both the harmonic oscillator and the Morse potentials [(Sh60),
(Ra69), (En69), (Ek71), (Ek72)]. In the case of a harmonic oscillator
potential [(Sh60), (Ek71)] one can easily show that
11
= (1)
n+"' (n'!n!)1/2 e'V2
(no>
n'+n
*Snx n ~k
x kfQ k!(n'k)!(nk)! (A5*3)
where
nQ = (AP)2/2 (A5.4)
and kmx is the smaller of n and n'. The corresponding relation for the
Morse oscillator [(En69), (Ek72)] is given by
210
18
energy surface. The available calculations [(St75), (St76), (Th78a)]
were performed within the SCF approximation and corrected to some degree
for correlation effects. SCF calculations alone cannot sufficiently
describe the potential energy surface because such an approximation
fails to properly describe the dissociation of closed shell systems such
as N? and CO into open shell atoms (St76). SCF methods in general fail
to properly characterize the large distance multipole type interactions
between atomic and molecular fragments. Staemmler [(St75), (St76)] and
Thomas et al. (Th78a) corrected the SCF results by making use of Cl
(configuration interaction) methods.
Since the information obtained in the above calculations is re
stricted to the ground states of the (Li+, N^) and (Li+, CO) systems,
the possibility of nearby excited states must be ascertained by consider
ing the possible electronic states of the various atom and diatom frag
ments of these systems (Ma75). For example, Fig. (1i) considers some
of the electronic states of the (Li+, N^) system when Li+ and N9 are
infinitely separated. The resulting potential energy curves were con
structed using the information given by Bond et al. (Bo65). We note
that the ground state of N2(*Â£* state) correlates with the two nitrogen
atoms in the state. Using the WignerWitmer rules (He50), one can
predict that the other states that correlate with the [N(S) + N(^S)]
asymptotic state are the ^E*, ^E+ and 'E^ states. In Fig. (11) the
possibility of charge transfer (Li, N*) and of electronic excitation of
j 'fc
nitrogen (Li N^) is also considered. Asymptotically we see that the
ground state of Np^E*) correlates with [(N() + N+(3P)]. Of more
interest is the excited electronic state which asymptotically leads to
4 2 2
[N( S) + N( D)], where the D state corresponds to the first excited
52
Thus letting  Â£ > be equal to t > and making use of equations (2.59)
and (2.60), one can obtain by taking the proper variations (Go50) the
expression
t(q', q; z) = Â£+(q')Â£(q, z) (2.61)
where C_ satisfies the relation
J(z)C(q, z) = I(q) (2.62)
and where the components of the matrices Â£ and Â£ are defined as
Ij (q) = ^iVq> (2.63)
and
Jij(z) = <^V VGo(z)Vkj> (2.64)
Another equivalent expression to (2.61) would be
t(q', ; z) = (2.65)
but we have chosen to use (2.61) and (2.62), in order to solve a
simultaneous set of linear equations instead of the matrix inversion
implied by this last expression.
The integrals needed in this procedure, as shown by equations
(2.63) and (2.64), are those required for the second Born approximation
[(Da51), (Le56), (Ho68)]. It is readily found however that integrals
of the type are not trivial to evaluate, and in general
must be done numerically (Ho68). For the particular potentials we shall
be interested in, the Morse and exponential potentials, one can obtain
the desired integrals from various references [(Da51), (Le56), (Li78)].
and (5.2) can only be ascertained by detailed consideration of the
potentials involved. In practice, this turns out to be quite difficult;
and so, one must resort to computational trial and error. One can,
however, put forth various rules based on physical considerations which
give some indication of the validity of the multiple collision expansion
and in particular that of the single collision approximation. We note
that equations (5.1) and (5.2) are expansions in terms of the quantities
T. and Gq. Because of the definition of G0, (Z H0)_1, it is apparent
that these expansions should indeed exhibit better convergence properties
at higher energies. One cannot be definitive however, since the operators
{Tj} are present in each term. In general these operators would not be
expected to cause difficulty for most potentials of chemical interest
[(Ch71), (Mi75b)]. Since T.Â¡ contains V.Â¡ to all orders, it remains well
behaved even for singular potentials. As to the validity of the single
collision approximation, one can see from a classical mechanical picture
that such a collision process must be impulsive. In other words the
projectile must strike and leave the interaction region fast enough so
that the restoring forces of the target diatomic do not lead to sequen
tial collisions. This latter assertion implies that for the impulse
approximation to be valid, the projectile energy must be larger than
the internal excitation energy of the target. Dmsidering the expres
sions for T^, T21 and TQ1 obtained from Eq. (4.35), it follows that
T, i = + T~ G T, G T.. + G T G TQ1 +
11 2 2 o 1 o 11 2 o 3 o 31
T3 + T3 Go T1 Go T11 + T3 Go T2 So T21 '
(5.28)
Using the relation
165
and summed over the final magnetic quantum numbers.
Introducing the vibrator states
= u^rj) Y^irj) (7.6)
and the expansion (Me61)
y y
lYAPlrl = 4tt E iA Y*y (APj) YAu(?1)jx(YAP1r1) (7.7)
Ay
into Eq. (7.4), one can show that (Ph76)
f!?} = 4tt E iX C(.r!. f(;?, .. Y* (AP ) ,
1 1 X\i J m'jm n'j'njA Ay 1
(7.8)
where we have defined the radial formfactor
f^l, .. = / drir.2 u*,.,(r1)u (r,) j, (Isp API r,)
n'j'njA J 1 1 n'j' V nj 1 AMmBC 1
(7.9)
and the coefficient
r(Ay)
j'm'jm
(7.10)
The integral in Eq. (7.10) is well known (Ed74) and leads to the relation
cXv)
j V jm
(l)m' (47r)1/2[(2j *+l) (2X+1) (2j+l)]1/2 I
j' A j
0 0 0
(7.11)
(2)
The above expression for F [Eq. (7.8)] and the corresponding equation
(3) 1 1
for FJ, lead to considerable simplification of Eq. (7.5). The alge
braic manipulations are quite straightforward, although tedious. Since
they are given in the literature (Ph76), we will simply state the result
105
(A B)l = Al + Al B(A B)'1
(4.7)
and
(A B)'1 = A"1 + (A B)"1 BA'1
(4.8)
The expressions obtained through Eq. (4.7) would be
G(Z) = Gj(Z) + Gj(Z)v(j)g(Z)
(4.9)
and
Gj(Z) = Gq(Z) + G0(Z)VjGj(Z),
(4.10)
where G(Z) = (Z H)"1 and Gj(Z) = (Z Hj)1. These equations remain
valid for j = 0 if one defines Vq = 0, so that v(0) = V. The twobody
transition operators can be defined as
(4.11)
Tj(Z) = Vj + G0(Z)VjTj(Z),
analogous to the definition given for t(z) given in chapter II. At
embedded in a threebody space; i.e., they characterize the behavior of
two interacting particles in the presence of a third spectator particle.
This should be clear from the structure of GQ, which is the free propa
gator for the entire threebody system.
Let us return now to the LippmannSchwinger equation, and use
Eq. (4.2) to express the threebody Toperators as
T(Z) = l t(J)(Z) ,
j=l
where we have introduced the definition
(4.12)
t(J')(Z) = Vj + VjG0(Z)T(Z) .
(4.13)
Figure (11) Potential energy curves for the diatoms
in the reactant and product regions of
the (Li+, N2) system [(Bo65),,(Ma75)].
The Li+(Li) atom and the N2(^2) molecule
are at infinite separation.
Figure (51) Kinematical construction for in
elastic scattering.
CO
CO
ENERGY
IN
TRANSFER AND DISSOCIATION
ATOMDIATOM COLLISIONS
HYPERTHERMAL
By
LYNTIS H. BEARD, JR.
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1979
To my Parents
ACKNOWLEDGEMENTS
I would like to express rny appreciation and gratitude to my
advisor, Professor David A. Micha, for suggesting the study of the
problems addressed in this dissertation, and for many helpful dis
cussions. His dedication, support and encouragement have been
invaluable.
I would like to thank all the faculty members of the Quantum
Theory Project, all of whom have contributed to ny development as a
student and scientist. In particular, I thank Professor PerOlov
Lbwdin for providing me the opportunity to attend the summer school
in Sweden and Norway.
I would also like to express my thanks in general to all the
members of the Quantum Theory Project. In particular, I would like
to thank: Dr. Henry Kurtz, Dr. Nelson H. F. Beebe, Dr. Jack Smith
and Mr. Larry Relyea, who were the source of sage computational advice
Dr. Michael Hehenberger, who pointed out the 'work of Shampine and
Gordon; and Dr. John Bellum, who provided useful encouragement and
discussion in collision theory to a somewhat bewiTitered but eager
chemist. My special thanks go to Dr. Zeki KuruogTu who was the sound
ing board for many ideas and whose comments have helped shape much of
the work presented here.
I am grateful to Miss Brenda Foye for the nica job she has done
in typing the manuscript.
My deepest appreciation goes to my wife, Adriana, for her patience,
understanding and constant encouragement during the task at hand.
iv
TABLE OF CONTENTS
Pace
ACKNOWLEDGEMENTS 1
ABSTRACT vii
CHAPTER
IINTRODUCTION ..... 1
1. Theoretical Treatment of Nuclear and Electronic
Motions 2
2. Adiabatic Collision Processes 6
3. Nonadiabatic Collision Processes 7
4. ManyBody Theory and the Treatment of Nuclear
and Electronic Motions 11
5. Analysis of Potential Energy Surface Informa
tion 17
6. Proposed Application of the Faddeev Formalism 26
7. Plan of Dissertation 30
IITHE TWOBODY tMATRIX 32
1. General TwoBody Scattering Theory 34
2. Boundary Conditions for the OffShell Wave
Function 36
3. The VariablePhase and Amplitude Method ... 38
4. Computational Aspects of the VPA Equations . 42
5. The Comparison Potential Method 45
6. The Bateman Method 50
IIINUMERICAL RESULTS FOR THE TWOBODY tMATRIX 54
1. Properties of the IwoBody tMatrix 54
2. Numerical Calculations Using the fPA
Equations 57
3. Numerical Results Obtained Using the Bateman
Method 96
IVTHREEBODY PROBLEM 101
1. Problems with the LippmannSchwinger Equation 102
2. Multichannel Transition Operators 103
3. The MultipleCoilision Expansion Ill
VTHE SINGLE COLLISION APPROXIMATION 114
1. Description of Channel States 115
2. Inelastic Scattering 117
3. Dissociative Scattering 125
v
TABLE OF CONTENTS (Continued)
Paqe
VI COLL INEAR SCATTERING 128
1. Formulation of the Coll inear
Scattering Problem 129
2. TwoBody tMatrix for the OneDimensional
Scattering Problem 133
3. Results for Inelastic Scattering 136
4. Dissociative Coll inear Scattering 146
VII THREEDIMENSIONAL INELASTIC SCATTERING RESULTS .... 163
1. Practical Implementation of the
Peaking Approximation 163
2. Numerical Results 168
3. Discussion 186
APPENDICES
I HARD CORE TWOBODY tMATRIX 196
II INTEGRALS USED IN "COMPARISON POTENTIAL" METHOD .... 200
III JWKB STARTING PROCEDURE 204
IV COORDINATES FOR THE THREEBODY PROBLEM 206
VANALYTICAL QUANTITIES USED IN THE COLLINEAR
SCATTERING MODEL 210
BIBLIOGRAPHY 213
BIOGRAPHICAL SKETCH 222
VI
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
ENERGY TRANSFER AND DISSOCIATION IN HYPERTHERMAL
ATOMDIATOM COLLISIONS
By
Lyntis H. Beard, Or.
March 1979
Chairman: David A. Micha
Major Department: Chemistry
Energy transfer and dissociation processes in hyperthermal atom
diatom collisions are investigated using approximations based on the
multiple collision expansion of the Faddeev equations. Within this many
body approach, the collision process is treated as a sequence of atom
atom encounters. This allows one to obtain information on a threeatom
system using only twoatom transition operator matrix elements.
A new numerical procedure, the "variablephase and amplitude"
method, is developed to compute the required offenergyshel1 twobody
tmatrix elements for an arbitrary radial diatomic potential. An alter
nate approach based on the comparison potential method is also investi
gated, leading to an improved computational algorithm. Whereas these
methods are based on a partial wave decomposition of the twobody topera
tor, the need for nonpartial wave techniques is pointed out at high
energies and large momentum transfers. In this connection the Bateman
method is investigated.
In the present study, the single collision approximation is considered
for both collinear and threedimensional motions. In the collinear studies,
it is shown that multiple collision effects are important in the case of
vii
inelastic scattering near threshold, and that for the case of dissocia
tive scattering they become crucial. A study is also made of the valid
ity of the single collision peaking approximation, and it is shown that
this approximation breaks down when soft twoatom potentials are involved.
In order to compare to experiment, threedimensional results are presented
for the scattering of Li+ with N~ and CO at hyperthermal energies.
viii
CHAPTER
T
INTRODUCTION
With the advent of new experimental techniques, the modernday
chemist is now more than ever able to discern the specificity and selec
tivity shown by che reactants and products in a chemical system. The
goal of a chemist is to properly interpret and understand in a general
way the events that occur in a chemical reaction. The task at nand,
however, is quite formidable because of the complexity of the particles
involved. This complexity implies that the investigation of the simples
chemical systems is necessary to lay the foundation of a deeper and more
conceptual understanding of chemical phenomena.
In this dissertation we will be concerned with the investigation
of energy transfer processes in atomdiatom collisions at hyperthermal
energies. In particular, we shall be concerned with processes of the
type
A + BC A + BC
 B + AC
 C + AB
* A + B + C ,
where A, B and C represent the atoms in the atomdiatom system. Theoret
ically, atomdiatom systems are of interest because they exhibit the en
tire spectrum of possible modes of energy transfer, i.e., they allow for
vibrational and rotational energy transfer, as well as electronic and
translational energy transfer. Therefore, we see that the elucidation
1
2
of the dynamics of atomdiatom collisions would provide an impetus to
the eventual understanding of more complicated systems. Our goal in
this chapter will be to clarify and discuss the treatment of electronic
structure in atomdiatom scattering processes. The theoretical aspects
of the scattering problem will be the subject of the remaining chapters
in this dissertation.
1. Theoretical Treatment of Nuclear and Electronic Motions
A formal treatment of an atomdiatom system requires that one con
sider both the nuclear and electronic degrees of freedom of the system.
In other words, one would have to solve a manyparticle problem. From
a practical standpoint, such a problem would not be tractable unless
approximations were introduced. In this section therefore, we will
sketch various approaches that can be taken to treat the atomdiatom
system as an effective threebody problem.
The first approach we shall consider is that of introducing approx
imations which allow for the separation of nuclear and electronic motions.
We begin then by considering the Schrodinger equation
(HE)T = 0 (1.1)
where E is the energy of the system, 'F is the wave function and the Hamil
tonian H can be written as (in atomic units)
a
12
j L vf + V
1 = 1 1
coul
+ H
SO
(1.1)
in the centerofmass system (Fr62). The Greek indices in Eq. (1.2) refer
to the nuclei and the Latin indices refer to the N electrons in the atom
3
diatom system. We note that the Coulomb interactions of the various
particles involved are contained in VCQU and the spinorbit interactions
are contained in H^q. The electronic and nuclear mass polarization
terms have been neglected. The reduced mass is defined by the rela
tion
where m is the mass of nucleus a and
a
M
3
l
3=1
m.
0.4)
is the total mass of the nuclei. In order to separate nuclear and elec
tronic degrees of freedom, the wave function T is generally expanded in
terms of a set of electronic functions {<Â¡>. (r ; r.)} which depend on the
electronic coordinates {r.} and only parametrically on the nuclear coor
dinates {r }. Thus one has
'F(r ; r.) = E (fu (r ; r,. )xi(r )
_a i ^ rl'a i Ara/
(1.5)
where xÂ¡ Oy) can be considered as the nuclear wave function describing
the motion of the nuclei on the potential energy surface which charac
terizes the Tth electronic state [(Fr62), (Hi67)]. Equations {1.5) and
(1.1) lead to the following set of coupled equations:
[VTiV uirEJx, = vyir +Tir + uivlxv
(1.6)
where we have introduced the definitions
T . I j J_ v2
01 2 6=1 "6 6
(1.7)
4
Hel 7 . 7i + Vcoul
I = 1
(1.8)
U =u(eD + u(SO)
uit un1 un
(1.9)
ui i ^ <i  Â¡
(1.10)
Uj 11 <(f>i I I >>
(1.11)
T
IT
2
= Z
6=1
1
2u3
T
IT
2
Z
6=1
(1.12)
(1.13)
4t = ^iiVi^ (1*14)
and
1T = d15)
These equations are well known in the literature [(Fr62), (Hi67), (Tu76a)],
and in general form the basis of most treatments of atomic and molecular
scattering problems. Our notation and discussion follow that of Tully
(Tu76a)].The diagonal matrix elements Ujj(r^) are the effective potential
energy surfaces for the atomdiatom system and the nondiagonal matrix
elements and U^, are the coupling elements responsible for
the promotion of electronic transitions. The diagonal term is a non
adiabatic correction to Ujj. The diagonal term rj can be shown to be
exactly zero if the electronic functions {<}>^} are taken to be real (Hi67).
5
Tne neglect of all the momentum dependent coupling terms tjj, and ,
(all 1 and 1) is known as the BornOppenheimer approximation [(Hi67),
(Tu76a)]and leads to a new set of coupled equations
[Ta + Uii E]xl = ^ ,E=1 U11 ,X1 ^1,16)
in which electronic transitions may occur only through the nondiagonal
potential terms U,],.
The effectiveness of Eq. (1.6) in properly describing collision
processes involving electronic transitions depends crucially on the choice
of the functions {<Â¡>^}. For example, if is chosen to belong to the
adiabatic representation, i.e., that representation which diagonalizes
the Hamiltonian
H' = Hel + H
SO
one obtains the set of uncoupled equations
(1.17)
where
[t +
a
 E]x1 = 0 ,
V,(r ) = U,,(r ) .
1 a 1 i or
(1.18)
(1.19)
In this representacin, transitions between the various electronic states
are not allowed, unless one considers the momentum dependent coupling
terms. The term adiabatic representation is also used in the literature
for that representation in which the electronic Hamiltonian H^ is diag
onal [(Hi67), (Tu76a)]. There are, however, an infinite number of non
adiabatic representations. One set of representations encountered in
the literature are the so called diabatic representations [(0m71),
(Sm69)]. These are representations in which the electronic functions
6
approach well defined atomic states asymptotically (0m71). There is no
unique recipe for constructing such representations. In general they
are chosen such that the momentumdependent coupling terms xjj, are
small (Sm69). One should note however, as Tully (Tu76a)points out, that
diabatic representations are useful in atomatom scattering because in
those situations avoided crossings in the adiabatic curves are localized
to a point. In other situations where there are more nuclear degrees
of freedom, as in atomdiatom scattering, one is dealing with avoided
surface crossings at which large variations in the nonadiabatic couplings
can occur. This indicates that one might not be able to find a diabatic
representation which is suitable in a global sense.
2. Adiabatic Collision Processes
Because of the complexity of atomic and molecular collisions, most
work in the literature has focused on the theory of adiabatic collision
processes. The theoretical investigation of such processes is, however,
by no means trivial. For example, in the case of inelastic atomdiatom
scattering, Eq. (1.18) would generally involve solving a large set of
coupled differential equations. If a large number of open channels for
vibrational and rotational excitation exist, an accurate and economical
solution to Eq. (1.18) may be difficult to obtain. In order to solve
the problem for such collision processes, further approximations have
been sought which reduce the dimensionality of such coupled differential
equations. A review of these techniques is given by Micha (Mi74), Lester
(Le76) and Rabitz (Ra76). In the case of reactive scattering the situa
tion is even more complicated. For an atomdiatom system, there are
four arrangement channels [A + BC, B + AC, C + AB and A + B + C] and as
a consequence Tree motion in the asymptotic region is governed by differ
ent channel Hamiltonians. This aspect of reactive scattering introduces
complications, because the boundary conditions become difficult to apply
and in addition no unique and simple set of coordinates is suited for
propagating the solution in all regions of configuration space. We note
that in each rearrangement channel, a natural choice of coordinates would
be that of the Jacobi coordinates and r., where F^. describes the rela
tive motion of the "free" atom with respect to the center of mass of the
diatom and r. the relative motion of the diatom. The label "i" is a
channel index (see Appendix IV). In order to overcome the difficulty
with the choice of coordinates, Marcus (Ma66) introduced the concept of
natural collision coordinates, e.g., a set of coordinates which goes
smoothly from one set of Jacobi coordinates to the other. However,
these coordinates complicate the structure of the kinetic energy opera
tor and may in some instances become multivalued according to how they
are constructed [(Ma66), (Wi77)]. A clarification of many of the problems
of reactive scattering may be obtained by considering the structure and
manybody aspects of the wave function [(Mi72b), (Re77)]. A general
review of reactive scattering may be found in the articles by George and
Ross (Ge71), Kouri (Ko73) and Micha (Mi75a).
3. Nonadiabatic Collision Processes
Nonadiabatic collision processes have not been studied in as much
detail as adiabatic processes for two general reason. One reason is the
added complexity of dealing with a coupled set of equations [Equations
(1.6) and (1.16)]; the second reason is the lack of information on the
potential energy surfaces and on the various couplings involved. It has
8
been found, however, that for some problems it is possible to overcome
the first of these obstacles by treating some or all of the nuclear de
grees of freedom either classically or semiclassically. As was pointed
out earlier, in some instances nonadiabatic effects may be localized to
certain regions of the potential energy surfaces. This, for example,
would allow one to use classical trajectory methods in regions where
motion along the adiabatic potential surface is allowed. In such an
approach the nonadiabatic regions serve only to determine branch points
at which a jump to another surface can occur. This approach, which is
known as the trajectory surface hopping model (TSH), was proposed by
Tully and Preston (Tu71) to study the systems [H+,] (Tu71), [H,]
(Pr73) and [Ar+,] (Ch74). In order to determine whether a trajectory
remains in the same potential energy surface or branches to another sur
face, the probability for hopping must be computed. A discussion of
how this probability is obtained is given in the above references. We
shall only mention one method; that of the well known LandauZener
Stckelbert (LZS) approximation [see for example Bransden (Br70)J.
Strickly this approximation is valid only for the onedimensional case.
Here, however, one applies the LZS result to a cut perpendicular to the
nonadiabatic seam (Tu76a) in the potential energy surfaces. Since trajec
tories may encounter various branch points, we see that in many physical
applications the TSH method could become computationally unmanageable.
Another approach, which is useful in treating nonadiabatic processes
such as
A + BC(n1,j) A+ + BC"(ni,ji) (1.20)
A + BC (nj.jp
(1.21)
9
which involve charge transfer and vibrationalrotational excitation, is
that given by the multiplecurve crossing model. In order to define this
model, let us introduce the Jacobi coordinates ^ and r^ which are
appropriate in describing the arrangement channel specified by Eq. (1.21).
Furthermore, let us expand the nuclear wave function Xi ^1 ^ in terms
of a set of vibrationrotation functions (u, (^^;r^)} as follows,
x, (,.?,)*Â£ u, (R Â¡.wos.), (1.22)
V
where v labels the vibrationrotation states and wv(ftj) acts as a nu
clear wave function for the relative motion of atom A with respect to
the center of mass of the diatom BC. Note that the {u^} depend only
parametrically on R.,. If we now use Eq. (1.6) expressed in Jacobi coor
dinates and Eq. (1.22), we obtain a set of coupled differential equations
given by
+ + Hlvlv E>lv'V '
rj,t + Wlvl'v,1l,vl(il)> 0.23)
V
where
t = 
JV2
2M1 R1
H, =
mA+mB+mC
(1.24)
(1.25)
= Mj <(h ^rJ^I
E'
T11*
(1.26)
10
E"
(1.27)
T
I
(1.29)
(1.30)
(1.31)
(1.32)
and the quantities m^ and mc are the masses of the atoms composing
the system. Equation (1.23) is the basis for treating an inelastic atom
diatom scattering problem. The diagonal matrix elements Wlvlv^l
correspond to effective potential energy curves, each of which is asso
ciated to a particular electronicvibrationalrotational state (Tu76).
These potential energy curves then serve as a basis for treating charge
transfer and inelastic scattering within the multiplecurve crossing
model [(Bac9b), (Fi71), (Gi75), (Tu76a)]. It is obvious, however, that
due to the enormous number of these curves, various other approximations
must be introduced. Generally one averages over the rotational states
and assumes that nonadiabatic effects are confined to the avoidedcross
ing regions where the LZS approximation can be used. A discussion of
the approximations and implementation of this model can be found in the
11
work of Fisher et al. [(Fi71), (Fi72), (Ba69b)], Gislason and Sachs
(Gi75), and in the review article by Tully (Tu76a). We note that Eq.
(1.23) is not suitable for treating rearrangement collision processes.
However, using the manybody formalism of the Faddeev equations Yuan and
Micha (Yu76) have implemented multiple curve crossing information to
study rearrangement scattering of the atomdiatom system [K,IBr].
4. ManyBody Theory and the Treatment of Nuclear and Electronic Motions
All the methods discussed in the previous sections involved the
treatment of electronic and nuclear motions essentially as two separate
problems. This approach has resulted in the introduction of the concept
of a potential energy surface, i.e., that quantity which characterizes
the electronic state of the system and serves as the effective potential
governing nuclear motion. An alternative to this picture would be the
simultaneous treatment of nuclear and electronic motions. We will
briefly sketch how this can be accomplished by the use of manybody
theories. This will be done for three reasons. First, it will serve
to contrast the more traditional methods of scattering theory based on
Eq. (1.6). Secondly, it will show how one can use known twobody data
as input to the solution of an inherent manybody problem. Lastly, it
will serve as a reference with which to compare the approximate treat
ment of atomdiatom collisions given in later chapters of the present
work.
The manybody formalisms that will be discussed are based on the
treatment of the threebody problem pioneered by Faddeev (Fa61). Earlier
it was pointed out that for a threeparticle system, there are four
arrangement channels. This implies that free motion in each asymptotic
12
region is governed by a different channel Hamiltonian. In order to
clarify this, consider the Hamiltonian for three interacting atoms A, B
and C (in laboratory frame),
H = Z K + Z H, + Z V
a r I i
(1.33)
K/J
where is the kinetic energy of nucleus a = a, b, or c (associated with
A, B and C respectively); Hj is the electronic Hamiltonian of atom I,
and Vj^ is the Coulomb interaction among the charges of atoms J and K
(Mi77b). The channel Hamiltonian for a given rearrangement channel
i[l for (A+BC), 2 for (B+AC), 3 for (C+AB)] is defined by the relation
H. = 1im H (1.34)
R.**
and as can be seen using Eq. (1.33)
(1.35)
H. = E K + E Ht + V, .
^ a y I JK
In the breakup arrangement channel one can analogously define the free
particle Hamiltonian
(1.36)
which can be used in Eq. (1.35) to obtain
Hi Ho + Vi
(1.37)
and
3
H = H + E V.
o i
(1.38)
13
where V. is equal to V,.,. The Faddeev formalism, then, is merely a way
i j i\
of decomposing the total wave function of the system into channel com
ponents which can be propagated by their respective channel Hamiltonians.
The result is given by the relation (Fa61)
(E Hj)Y
3 /i,\
2 (1 
k=l J
(j = 1, 2, 3)
(1.39)
where the total wave function T has been decomposed into the channel com
ponents ^, i.e.,
'Â¥ =
3
E
j=l
(1.40)
A detailed derivation of the Faddeev equations will be given in
Chapter IV of the present work. In the following discussion, our con
cern will be the use of Eq. (1.39) solely as the basis for the simul
taneous treatment of nuclear and electronic motions. Micha (Mi77d) has
shown that this can easily be accomplished within the framework of the
method of diatomicsinmolecules (DIM) (El63). We begin by considering
the total wave function for the system, keeping in mind that for struc
tured particles it must have the form
Â¥ = Cy\X (1.41)
where C is a normalization constant, is the N electron ar.tisymmetrizer
and X is an unsymmetrized total wave function for both nuclear and elec
tronic motions. Since H and J\ commute, one can cast the Schrodinger
equation into the form
[Ej4  (H/ +/ H)]X = 0 ,
(1.42)
14
or in the Faddeev form
(1.43)
where
(1.44)
Equation (1.39) may now be replaced by Eq. (1.43), which is more suitable
for introducing DIM approximations. A symmetrized form of the Hamil
tonian such as
i M = \[J\ ,H] +
(1.45)
has been introduced in order to avoid nonhermitian matrices (Tu7ca) which
occur whenever the are expanded in terms of a set of channel elec
tronic states and used in Eq. (1.43). The channel electronic
states j_^> correspond to a collection of primitive channel wave
functions constructed in a step by step procedure from a set of atomic
wave functions ^and Â£Cm, as is usually done in the DIM method
[(El63), (Mi77d)]. When these channel functions are constructed, how
ever, the arrangement structure of the channel must be considered.
Therefore, one begins by constructing a collection of Mq functions
2
(I^Al^Bn^Cm^' "ac*1 t*ie a^om1c functions are eigenfunctions Sj
and and are composed of nonsymmetrized combinations of products of
atomic orbitals centered on the nucleus of atom I. This set of func
tions we designate j^>. A set of M^ unsymmetrized or primitive
channel wave functions can be obtained from Â£^> by first com
bining the products to form eigenfunctions of Sj and S^ and
15
then combining the products of these and Â£n to
\J i
form eigenfunctions of
2
S and Sz (Mi77d). This leads to the relations
5(J)> lig> 4^ '
(1.46)
(1.47)
and
<^(j)r(J)> = r(j)
(1.48)
where _[g anc* 1'^ are unit matrices and C^ defines a channel coupling
matrix between the breakup channel (channel 0) and the arrangement
channel i. The unsymmetrized or primitive channel wave functions
can now be expressed as
x(j) = r(j)>x(j) f (1.49)
where denotes a M^' x 1 column of scattering functions. In ob
taining Eq. (1.49), one has assumed the approximate completeness rela
tion
ll(j)>
V
A
where 1. is the unit operator in the space of many electron states in
sJ
channel j (Mi77d). Using equations (1.43) and (1.49) the end result of
the DIMFaddeev equations is given by
{A(j)E Z K Jr[A(j), Z E, + V.]+} x(j) =
a I J
i[A(j), V ] Z (1 6.J C(jk) X(k) (1.51)
J k JK
A(J) = .
where
(1.52)
16
C(jk) = <5(J)5(k)> ,
(1.53)
(1.54)
is a diagonal matrix containing atomic energies and ^JK 1S a matrix
containing diatomic energies. The energies Ej are associated with the
unbound atom in channel j and the E_IK energies of the corresponding dia
tomic in the same channel. Equation (1.51) is a result of the Born
Oppenheimertype approximation
A(j') Ka (1.55)
and the DIM approximations
Ej (1.56)
and
A(j) Iik (157)
f i k 1
The quantities 1 in Eq. (1.51) act as recoupling coefficients among
the various channels, allowing for inelastic, rearrangement and dissoci
ative processes to occur. Furthermore, as long as the proper atomic
and diatomic energies are included, Eq. (1.51) allows for electronic ex
citation. A detailed application of this formalism to the [K + W^\ sys
tem is given in the literature [(Mi72b), (Mi77d)]. In these references
it was shown that one can construct a suitable expression of the ground
state potential energy surface of the system in terms of twobody inter
actions provided one uses spindependent potentials.
Atomdiatom collisions involving charge transfer cannot be treated
properly using Eq. (1.51) because this equation assumes a unique parti
17
tion of the electrons. There is, however, an alternative approach based
on the coupledchannel manybody formalism [(Ko75), (To74), (Ra77)],
which is suitable for treating charge transfer. In this approach one
simply considers various partitions of the Hamiltonian consistent with
the atomic and molecular fragments involved in the scattering process
of interest. A set of coupled equations may then be constructed that
are analogous to the Faddeev equations (Mi76b).
5. Analysis of Potential Energy Surface Information
In the present work, we shall be concerned with ionmolecule reac
tions of the type (Li+, CO) and (Li+, N?). Since CO and N2 have ident
ical masses and very similar molecular properties, these systems are
very interesting for comparison purposes. Such a study is also attrac
tive because a comparison between experiment and theory is now possible
in light of recent molecular beam experiments [(Bo76), (Ea78)].
Our investigation of these systems will be based on an analysis of
the various interaction potentials involved and upon the potential energy
surface data available in the literature [(St75), (St76), (Th78a)].
This analysis though qualitative in nature, will serve as a guide for
the development of an impulsive model which will be specified in the
next section.
The only available potential energy surface Information on the
(Li+, CO) and (Li+, N2) systems corresponds to a restricted number of
geometries for the ground state of the system. Because of the relatively
large number of electrons in these systems, accurate boundstate calcula
tions are very expensive. This is further complicated by the large
number of points required to adequately characterize the potential
18
energy surface. The available calculations [(St75), (St76), (Th78a)]
were performed within the SCF approximation and corrected to some degree
for correlation effects. SCF calculations alone cannot sufficiently
describe the potential energy surface because such an approximation
fails to properly describe the dissociation of closed shell systems such
as N? and CO into open shell atoms (St76). SCF methods in general fail
to properly characterize the large distance multipole type interactions
between atomic and molecular fragments. Staemmler [(St75), (St76)] and
Thomas et al. (Th78a) corrected the SCF results by making use of Cl
(configuration interaction) methods.
Since the information obtained in the above calculations is re
stricted to the ground states of the (Li+, N^) and (Li+, CO) systems,
the possibility of nearby excited states must be ascertained by consider
ing the possible electronic states of the various atom and diatom frag
ments of these systems (Ma75). For example, Fig. (1i) considers some
of the electronic states of the (Li+, N^) system when Li+ and N9 are
infinitely separated. The resulting potential energy curves were con
structed using the information given by Bond et al. (Bo65). We note
that the ground state of N2(*Â£* state) correlates with the two nitrogen
atoms in the state. Using the WignerWitmer rules (He50), one can
predict that the other states that correlate with the [N(S) + N(^S)]
asymptotic state are the ^E*, ^E+ and 'E^ states. In Fig. (11) the
possibility of charge transfer (Li, N*) and of electronic excitation of
j 'fc
nitrogen (Li N^) is also considered. Asymptotically we see that the
ground state of Np^E*) correlates with [(N() + N+(3P)]. Of more
interest is the excited electronic state which asymptotically leads to
4 2 2
[N( S) + N( D)], where the D state corresponds to the first excited
Figure (11) Potential energy curves for the diatoms
in the reactant and product regions of
the (Li+, N2) system [(Bo65),,(Ma75)].
The Li+(Li) atom and the N2(^2) molecule
are at infinite separation.
E (eV)
i
00
o
oo
ro
O
21
state of nitrogen. Note that in constructing the curves in Fig. (11),
the energies of Li ( S) and Li( S) must be included. The possibility of
reactive channels is ruled out by the closed shell nature of the
(Li+, N^) system. Furthermore, the electronic excitation of Li'
. O I f
[Li ( S) + N2( Eg}] is not considered since it would lie much higher in
energy than those states already given in Fig. (11). It is apparent
then from Fig. (11), that there is a possibility of electronic excita
tion, since the Eu state of lies only 6.1 eV (Bo65) above that of
the ground Eg state. Charge transfer on the other hand would seem
uniikely.
As the Li+ is brought near N2, the relative position of the various
electronic states would change. If the spacing between the ground arid
excitated states grows closer, nonadiabatic effects will be important
for the given system. Figure (12) illustrates what happens as the Li +
atom approaches the molecule. This figure shows three planes.
Plane III corresponds to the same cut of the potential energy surface
given in Fig. (11) in which R (the distance between Li+ and the center
ofmass of N2) is infinite. Planes I and II are perpendicular to plane
III and intersect that plane at r equal to zero and at r equal to the
equilibrium internuclear separation (rg) of N^. It is shown then that
the ground state potential energy of the system is decreased as Li+
approaches and increases rapidly as R goes to zero. The reason for this
initial decrease of the potential as the Li+ approaches N2 is due to
attractive polarization forces. This will be explained later when the
long range interactions of the system are discussed. On the other hand
at small values of R, strong repulsion forces exist because of the closed
shell interactions. A similar qualitative behavior is expected for the
Figure (12) Various cuts through the potential energy surface
of an atomdiatom system are illustrated. The
parameter R measures the distance of the projectile
atom to the centerofmass of the target diatom;
whereas, r is the internuclear separation of the
target diatom. Planes I and II are perpendicular
to plane III at r equal zero and r respectively.
24
excited states. It should be mentioned, however, that the reaction
Li + (1S) + N2(!Eg) v Li + (1S) + M3!*)
would be spinforbidden (Ge71). This conclusion is only valid if one
can neglect effects such as spinorbit interaction. A discussion of
the applicability of symmetry rules to collisions is given in the work
by George and Ross (Ge71) and that by Shuler (Sh53). From an experimental
viewpoint, electronic excitation has not been observed at the energies
[4 to 8 eV in the centerofmass system] and scattering angles studied
in the present work (Bo76). A similar analysis could be done for
(Li+, CO); however, this system is isoelectronic with (Li + fO and is
expected to behave similarly.
We have not considered changes of the equilibrium internuclear
separation of N2 as Li+ approaches. In order to do so, we now discuss
the potential energy surface in more detail. The long range interaction
between a charged atom and a diatom with a nondegenerate electronic
state is dominated by electrostatic and inductive terms (Bu67), i.e.
(1.58)
where
VC0UL jjf + jjf P2('COse'i f jjf P3(0)
(1.59)
 a. ))McosG) + ...],
eL
(1.60)
25
a
+ 2a >
(1.61)
where q corresponds to the charge of the atom, the molecular parameters
p, Q, ft and H correspond to the dipole, quadrupole, octapole and hexa
decapole moments of the diatom, and the parameters cu and correspond
respectively to the parallel and perpendicular components of the diatomic
polarizability relative to the internuclear axis of the molecule. The
functions P^(cos0) are Legendre polynomials arid 0 is the angle between
the vector R from the centerofmass of the diatom to the projectile
atom and the internuclear axis of the diatom. Beth the N~ and CO mole
cules have very similar values for the various parameters used in equa
tions (1.59) and (1.60), with the exception that Ng does not have a
permanent dipole moment. This difference, however, is quite important
since the chargedipole interaction determines the R term in Eq. (1.59),
and as such dominates the long range portion of the (Li+, CO) potential.
It is precisely these parameters [p, Q, etc.] which are not well speci
fied in SCF calculations [(St75), (St76), (Th78a)]. The value of the
dipole moment of CO in fact has been the subject of wide controvercy
(Ne64). Experimentally this value is small and negative [p^q = 0.049
a.u.] (St76) and implies the polarity of the CO molecule is given by
C0+. More recent calculations (Ch76) have now shown agreement with the
experimental results. Unlike the situation for N^, in a col linear reac
tion the potential between Li+ and CO is not symmetric. If one examines
the behavior of the equilibrium internuclear separation of CO as a func
tion of R,one would find that it shows much more deviation from its
value at R equal to infinity than that for N^. In the scattering exper
iments on these systems (B576), it is found that (Li+, CO) shows more
26
vibrational excitation than the (Li N^) systems. Various investi
gators [(Bo76), (St76), (Mi78b)] have attributed this observed result
to the chargedipole interaction in the (Li+, CO) system. The relative
insensitivity of rg as a function of R for the (Li Ng) system is
shown in Fig. (13).
At small values of R the interaction of Li+ with and CO is
highly repulsive and can be easily characterized by an exponential poten
tial [St76), (Th78a)]. Thomas et al. (Th78a) use the potential
V(R, r, 0)
 Vj(r)
vx(r,
R)P^(cosO)
(1.62)
where V^(r) is the potential for the free diatom and v^(r, R) is deter
mined by using various parametrized functions [sums over exponentials
and R"n terms] to yield the best curve fit to the potential. Equation
(1.62) behaves asymptotically as Eq. (1.58). Staemmler (St76) points
out that the Legendre expansion of the potential in Eq. (1.62) loses
its significance for small R[R < 3.5 a.u.] and may in fact not converge.
6. Proposed Application of the Faddeev Formalism
Our study of the (Li+, CO) and (Li+, Ng) systems will be based on
the many body formalism of the Faddeev equations. Within this approach
we shall consider an iteration of the Faddeev equations (in transition
operator form) which yields a miltiplecollision expansion (Mi75).
This allows us to describe an atomdiatom collision process as one
which takes place as a sequence of twobody encounters. Since the colli
sion energies are large and the scattering angles are beyond the rainbow
angle, it is expected that the scattering cross section will be largely
Figure (13) Contour plots of the ground state potential
energy surface for the (Li+, N?) system as a
function of the NN internuclear separation
(St75). Each curve corresponds to a fixed
value of R, the distance between Li+ and
the ceriterofmass of the N2 molecule.
28
(a.u.
18
20
22
24
29
characterized by the repulsive forces of the system. Thus, an impulsive
model based on the truncation of the multiplecollision expansion would
be expected to yield useful information on these systems. An advantage
to this approach, as will be shown in Chapter V, is that one is able to
obtain statetostate information directly from the initial and final
momentum distributions of the target and product diatom, and from the
various twobody potentials involved. Due to the small vibrational
rotational spacings of CO and M2, other methods such as the coupled
channel method, based on Eq. (1.23) would be computationally prohibitive
(Th78b).
We shall assume in our implementation of the Faddeev equations that
electronic motions have already been eliminated and that only information
from the ground adiabatic potential energy surface of the system is
needed. As mentioned previously, the scattering process at hyperthermal
energies probes the short range part of the potential. Therefore, we
shall develop a many body theory based on the decomposition of the adia
batic potential into the form (Ki79)
V(H r) = Vsr + Vlr (1.63)
where
3
V, = 2 V.(r.) (1.64)
sr i=i 1 1
and
Vlr = l v?(R, r)Yx(R, r) (1.65)
X
The Vjr term denotes the long range potential and is analogous to the
A A
Legendre expansion used in Eq. (1.62). Here, however, Y^(R, r) is a
30
general function depending only on the relative orientation of the atom
and diatom. The V$r term, on the other hand, corresponds to the short
range potential and is characterized by a sum of manybody terms. In
Eq. (1.64), V was assumed to be a sum of radial twobody potentials;
one potential for each twoatom fragment in the atomdiatom system.
This breakdown of the potential circumvents the difficulties, mentioned
in the last section, in representing the short range behavior of V by
Eq. (1.62). Having introduced Eq. (1.63), we propose an impulse model
in which only the short range part of the potential is kept. The re
sulting problem is then ideally suited for the manybody formalism of
the Faddeev equations.
We note that, at low collision energies, one expects that the
approach of Li+ to CO or would heavily influence the orientation of
the diatom. This would be particularly true for a molecule with a
dipole moment like CO. The net result in our impulsive model would be
to overestimate rotational excitation. At the high energies considered
in this study, however, this effect is expected to be minimal.
7. Plan of Dissertation
The present investigation of atomdiatom collisions will be given
in three major parts. The first part presents an analysis of the two
body problem and discusses various computational schemes for the deter
mination of cffshell matrix elements of the twobody transition opera
tor (Chapters II and III). In the next part (Chapters IV and V), the
formalism of the threebody problem is presented and the various approx
imations obtained from the multiple collision expansion are discussed.
Finally, in the third part (Chapters VI and VII), computational results
based on the singlecollision approximation are presented for both
coll inear and threedimensional atomdiatom scattering processes.
CHAPTER II
THE TWOBODY tMATRIX
Scattering processes in few or manybody systems require the
introduction of twobody transition operators, whose matrix elements
must be known for arbitrary initial and final relative momenta of the
two bodies and for arbitrary energies, i.e., they must be known in
general "offtheenergyshellIn other words, the operator quantities
of interest are those describing the interaction of two particles
embedded in a many particle system, and as a consequence energy is not
conserved locally, thus explaining the term "offshell". Although these
twobody transition operators stand for basic physical concepts, their
properties and values are little known for molecular systems, where the
bodies are atoms or ions.
The computation of twobody tmatrix elements has long been a
problem of intense interest in the field of nuclear physics. Conse
quently, there exists an extensive literature devoted to this subject
as it pertains to potentials describing nucleonnucleon interactions.
A very nice and extensive review in this regard is the work of Srivastava
and Sprung (Sr75). In the chemical literature, work in this area has
been sparse. However, there have now appeared various papers which deal
with the computation of tmatrix elements for potentials of chemical
interest, for example: van Leeuwen and Reiner (vL61) proposed a numer
ical procedure based on the comparison potential approach, which has now
been implemented in the work of Brumer and Shapiro (Br75); Kuruoglu and
32
33
Micha (Ku78) have developed a variational approach leading to a
separable approximation to the tmatrix; and along more approximate
methods, Korsch and Mohlenkamp (Ko77) have developed a semi classical
approach based on the JWKB approximation which is applicable to repulsive
potentials.
The principal aim of this chapter will be to present a new computa
tional method for obtaining matrix elements of twoatom transition oper
ators between arbitrary momentum states, of high accuracy, and applicable
to any radial diatomic potential. The approach we have taken is based
on equations presented some time ago (vL61), which reduce the calculation
of transition operator matrix elements to integration of inhomogeneous
radial differential equations followed by numerical quadratures. Our
procedure is based on the wellknown variablephase approach [(Ca63),
(Ca67)], but extended to the present inhomogeneous equations. This
extension shows that, unlike the situation for standard radial scatter
ing equations, the variable phase is coupled at every distance with a
variable amplitude. To emphasize this point we refer here to a
"VariablePhase and Amplitude" approach. Another feature of our
approach is the 'way in which one can treat propagation through classi
cally forbidden regions. Along with the development of the VPA method
we shall present some suggested improvements to the comparison potential
method.
All the methods we have mentioned above make use of a partial
wave expansion to construct the twobody tmatrix elements. When we
discuss the singlecollision approximation in Chapter V, it will become
apparent that it would be desirable to have a procedure for the direct
computation of these matrix elements. This will even become crucial
34
when one deals with physical systems at energies for which a large number
of partial waves are required and the computational expense becomes im
portant. Consequently, in light of this problem, we shall also investi
gate a nonpartial wave approach that has been recently developed by
Belyaev and coworkers (Be74), which is based on the Bateman method
[(Ba22), (Ka58)] for solving integral equations.
1. General TwoBody Scattering Theory
The general computation of offshell matrix elements of the twobody
toperator for simple nonsingular potentials can in general be solved
without much difficulty. This may be done by direct consideration of
the LippmannSchwinger equation [(Si71), (Sr68)]
t(z) = V + VG0(z)t(z), (2.1)
where
G0(z) = (zH0)'1. (2.2)
In these relations HQ is the kinetic energy operator for relative motion,
z is a complex energy parameter (z = E+ie) and Â¥ is the interaction
potential between two particles. V will always be assumed local and
spherically symmetric. From equation (2.1) it is apparent that the
problem must be reformulated for singular potentials, those which contain
a hard core or have a singularity stronger that fyr^ at the origin
(Ta72). This can be accomplished by introducing the wave operator W(z)
via
t(z) = VW(z), (2.3)
where
W(z) = 1 + G0(z)VW(z),
(2.4)
35
and by relating the matrix elements of the toperator to those of the
wave operator in a mixed coordinate momentum representation (vL61).
Multiplying (2.4) by G^(z) on the left, one finds that W(z) also
satisfies
(z Hq V)W(z) = (z H0). (2.5)
In a mixed coordinatemomentum representation one then obtains the
inhomogeneous Schrodinger equation below (f) = 1):
[z + (2m)_1V2 V(r)] = [z (2m)1q2], (2.6)
where m, r and q are respectively, the reduced mass, the relative
coordinate and the momentum of the two particles being considered, and
 is the offshell state (since q2/2m f E) defined as
a^(z)> = W(z)q> (2.7)
In the nuclear physics literature, (2.6) is known as the BetheGoldstone
equation (Sr68). A reduced radial equation may now be obtained by
introducing into equation (2.6) the following partial wave decompositions
= (2tt)"3/2 e^r
= (2tt) 3/2 E i1 (21 + 1)P1 (q*r)j1 (qr)/(qr) (2.8)
and
= (2tt)3/^ Z i1 (21 + l)P1(q?)a>1(q,kE;r)/(qr), (2.9)
where j^qr) is the regular RiccatiBessel function and kÂ£ = 12mE1//2.
The resulting reduced radial equation is
36
[2 + ok2  v(r)Jfa) (q,k ;r) = {o k2 q)j (qr), (2.10)
where is +1 if E is positive or 1 otherwise, and v(r) = 2mV(r).
The matrix elements of the toperator can now be obtained from those of
the wave operator by using equation (2.3). Thus one has the relation
t(q\ q; z) = / V(r) df, (2.11)
which may also be decomposed into partial waves with the result
t(q\ q; z) = (tt)"1 E (21 + 1)P, (q1 q)t, (q, q; E), (2.12)
where
tj (q1 > q; E) =  / j] (q'r)V(r)w (q, kÂ£; r) dr. (2.13)
This definition of the partial wave matrix elements tÂ¡(q, q; E) follows
directly from equations (2.8), (2.9) and (2.12).
2. Boundary Conditions For The OffShell Wave Function
In order to obtain the offshell wave function w^(q, k^; r), a
solution to equation (2.10) must be found subject to two boundary condi
tions (vL61):
^(q, kE; r) r ; o 0 (2.14)
and
(q, r) ~ j, (qr) + C,h{+^(k r), (2.15)
where is a constant and hj+) is a RiccatiHankel function (Ca67).
Thus, w^(q, k^; r) is a complex wave function which asymptotically
37
behaves as the sum of a component from a plane wave and that of an out
going spherical wave, which, unlike onshell scattering expressions,
depends on different momenta (Ta72). For potentials containing a hard
core (2.14) must be replaced by
w,(q, kp; r) ~ 0 (2.16)
L r > r
c
where r is the hard core radius. Once the offshell wave function is
c
obtained, the partial wave matrix elements t (q1, q; E) follow directly
from (2.13). However, if the potential contains a hard core, care must
be taken in applying Eq. (2.13). If one integrates equation (2.10) as
follows,
rc+e ? rc+e m ,\
/ [wV + 0Â£kE 03j3 dr = / {[v(r) + J ] ^ +
rce rce r2
+ (oÂ£kj q2)j(qr)} dr, (2.17)
it is apparent that the product to^(q, k^.; r)v(r) does not vanish inside
the core region even though w^(q, k^.; r) does. In fact, for a potential
containing a hard core (La68),
v(r)w1 (q, kE; r) = 6(r rÂ£) ^^(q, kÂ£; rQ+) +
(q2 aÂ£k2) ^ (qr) (2.18)
for r < r where
c
!im wj (q, rQ + e) .
e  o+
,(q, kE; rc+) =
(2.19)
38
A derivation for the case of a pure hard core potential is given in
Appendix I.
3. The VariablePhase And Amplitude Method
The aim of the VPA method is to take advantage of the known
oscillatory nature of the offshell wave function, co^(q, kÂ£; r). This
is done by considering a solution to Eq. (2.10) which has the form
below:
o,(q, kÂ£; r) = j,(qr) + a.,(q, kÂ£; r) cos [S^q, kÂ£; r)]u1 (kÂ£r)
 a,(q, kÂ£; r) sin [6, (q, kÂ£; rjjv^^r), (2.20)
A A A
where u^(kÂ£r) and v (k^r) are RiccatiBessel functions (Ji(kÂ£r) and
A A A
n1(kÂ£r)) or RiccatiHankel functions (k^(kÂ£r) and i^(kÂ£r)) (Ca67). The
choice is determined by the nature of the region through which one is
propagating, i.e., whether the region is classically allowed or not.
Using equations (2.10) and (2.20), one may obtain a set of coupled
equations for the amplitude, a,(q, kÂ£; r), and the phase, 5^(q, k^; r).
These equations will now be derived using an integral equations approach
(Ro67). The functions u, (kÂ£r) and v^(kÂ£r) used in expression (2.20)
satisfy the RiccatiBessel differential equations (Ab65)
z, + [E  ^ +2^] z, = 0 (2.21)
dr 1 / 1
Taking the regular and irregular functions uj and v, to be normalized
such that their Wronskian, W[u, v,], is unity, the Green's function
associated with (2.21) may be written as
g,(r. r') = (kEr>) ,
(2.22)
39
where r< is the lesser of r and r', and r> is the greater of r and r'
(Ro67). At this point, the discussion will be restricted to positive
energies, since this would not involve a loss of generality. It is
readily seen that equation (2.21) is just the corresponding differential
equation to (2.10) in the absence of a potential. Thus, oj^(q, kÂ£; r)
may be written as follows:
00
u(q, kÂ£; r) = N'(kÂ£r) + / dr'g^r, r') [ (k q2) jj (qr') +
v(r )aj] (q, kE*, r1)] (2.23)
A
where N' is a constant. Since j^(qr) satisfies equation (2.21), the
first term in the above integral,
(kE q2) C dr'9i r)Vqr') *
may be evaluated exactly. This may be accomplished using the relation
(A2.2 ) given in Appendix II. Introducing the quantities A^(q, kÂ£; r)
and B^(q, kÂ£; r) defined by
Aj (q, kE; r) = N + k^1 / dr'v(r' )uj (q, kÂ£; r') n1(kÂ£r') (2.24)
and
B(q. kÂ£; r) = k"1 /[ dr'v(r' )w, (q, kÂ£; r') J, (kÂ£r1), (2.25)
where N = N' + 1, and using equation (2.22), one obtains
l(q* kE; r) = (qr) + A^ (q, kE; r) j] (k^)! +
 Bj (q, kE; r) n] (k^r)) .
(2.26)
40
The functions A, and B behave as constants asymptotically, as can be
1 1
seen from the boundary conditions (2.14) and (2.15). These constants
may in turn be related to an offshell phase shift, 6^(q, k ), and
amplitude, a^(q, k^), in the same manner as is done for the onshell
case [(Ca63), (Ca67)j. Therefore one has
A,(q, k ; r) ~ a (q, kp) cos [6,(q, kJ] (2.27)
t p > co  L
and
B,(q, kE; r) ~ a (q, kÂ£) sin [6,(q, kÂ£)] (2.28)
P > CO
The variable phase assumption consists of extending the definition of
the phase and amplitude to all values of r (Ca63); thus,
A'j (q, kÂ£; r) = ^(q, kÂ£; r) cos [fi^q, kÂ£; r)] (2.29)
and
Bj(q, kE; r) = aj (q, kE; r) sin [6^q, kÂ£; r)] (2.30)
This assumption is of course valid, since 6j (q, kÂ£; r^) and a^(q, kÂ£; rM)
are just the resulting phase shift and amplitude for a potential with
a cut off at r = r^ (Ca67). In fact the justification of this approach
is best understood by considering a finite boundary value problem and
extending the boundary to infinity (Ze64). The desired set of coupled
first order differential equations, the VPA equations, may now be
derived using (2.29) and (2.30) for Aj and To accomplish this,
first consider the derivatives of A^ and B^ as defined by equation
(2.24) and (2.25). From these relations it is apparent that
VkEr) Mq kE; r) i(kEr) Vq V r) = o ;
(2.31)
41
therefore, the derivative of the offshell wave function has no terms
involving the derivatives of A and Bj, i.e.,
wj (q, kE; r) = J^qr) + Aj(q, kÂ£; r) ^ J1(kÂ£r)
 B^q, kE; r) ^ n1(kÂ£r) (2.32)
Making use of (2.29) and (2.30), equations (2.26) and (2.32) may be
written as,
and
w1 = jj(qr) + a1 cos (6^ jj (kÂ£r)
A
 a1 sin (6j) n (kÂ£r)
Br 1 = clr h + al cos ^61 ^ BF J*1 (kEr)
(2.33)
 a sin (6,) d n,(kEr) .
(2.34)
Taking the derivative of (2.33) and comparing it with (2.34) one obtains,
A A
sin (6j) j (kpr) + cos (6j) nj (kpr) d (2.35)
dF i = al
cos (6j) jj (kEr) sin (6j) nj (kÂ£r) dr 1
In a similar manner taking the derivative of (2.34) and comparing it to
the inhomogeneous Schrodinger equation (2.10), one finally obtains, with
the aid of (2.35), the "variablephase and amplitude" equations
df 6j (q, kÂ£; r) = k"1 v(r)[wj j^qrM/c^ (2.36)
and
HFai(q, kE; r) = v(r)w1 [sin(6^) (kÂ£r) + cos^ )n} (kÂ£r)].
(2.37)
42
These equations are analogous to those for the onshell case, where
q = kE (Ca63). However, for the onshell case the phase and amplitude
equations are uncoupled.
The boundary condition satisfied by the VPA equations at the
origin follows directly from equations (2.14) and (2.33). From these
expressions it is apparent that
sin [6,(q, kE; r)] ~ 0 (2.38)
r o
and that the value of aj(q, kÂ£; r) for the same limit is arbitrary, and
becomes fixed only after the asymptotic boundary condition (2.15) is
applied. On the other hand, if a hard core is present
 i,(qr )
cti(q, kp, ) *" ^
1 L r * rc cos(61)j1(kErc) sin(61 )n, (kÂ£rc)
(2.39)
and <5(q, kE; r ) is arbitrary.
4. Computational Aspects of the VPA Equations
The procedure used in calculating ^(q, kE; r) consists of propa
gating two solutions to the VPA equations (2.36) and (2.37) such that
they satisfy the initial boundary condition, and constructing linear
combinations of these solutions so as to satisfy the asymptotic boundary
condition (2.15). It is possible to construct two linearly independent
solutions that satisfy the initial boundary condition due to the
inhomogeneous character of equation (2.10). This procedure is analogous
to that found in reference (Br75); however, instead of a piecewise
breakdown of the potential and the propagation of local solutions as
given there, numerical integration is involved here.
43
An important consideration is the singular nature of the VPA
equations near the origin. Equations (2.36) and (2.37) contain the
potential as a multiplicative factor. This problem can be overcome by
assuming hard core boundary conditions or by using a simple step poten
tial within the first classical turning point [(Be60), (Br75)]. The
second of these procedures was chosen in this work. The length of the
first step, r^, and the value of the assumed step potential over this
first step, Vj, were chosen such that making any further optimal changes
in these parameters would not lead to a significant change in the value
of the calculated matrix elements, ^(q, k^; r) and tj (q1, q; E). The
guess work involved here can be eliminated by using a JWKB approximation
to the derivative of the imaginary part of the wave function near the
first classical turning point. The approximations involved are given in
Appendix III. From expression (2.10) it is apparent that Im[w^(q, kr; r)]
satisfies the usual Schrodinger equation and boundary condition at the
origin. It therefore follows that within the first classical turning
point Im[u)] is decaying exponentially, and is zero at the origin. Ail
that is necessary then, from a computational point of view, is to start
the integration in a region where the magnitude of the derivative of
Im[coj] is smaller than some given value. In our calculations a value of
10 was found to be adequate. Another important consideration in
propagating the VPA equations is the cost involved, in evaluating a
large number of Bessel functions. Because of this, the coupled equations
were integrated numerically using an Adams numerical integration routine
developed by Shampine and Gordon, which*is particularly suited for
problems for which derivative evaluations are expensive. These codes
and a discussion of their efficiency and accuracy are found in reference
(Sh75).
44
The propagation of two independent solutions to the inhomogeneous
differential equation (2.10) further presents two problems. One is that
of the growth rate of the amplitude in a nonclassical region and the
second is that of the propagation through classically allowed regions
at negative energies. The first problem was solved as given in reference
(Br75). The idea was simply to use the propagated wave functions to
construct two new linearly independent solutions to Eq. (2.10), which
would have much smaller amplitudes. Therefore we shall restrict our
attention to the second problem. If the energy is negative, the modified
/s /\
RiccatiHankel functions and ij are the natural basis to use for
/v /v
uj and v^ in expression (2.20),. the VPA form for the offshell wave
function o>i. However, if the potential is attractive and the energy is
such that one must integrate through a classically allowed region, the
wave function in this region would be oscillatory in nature. The
RiccatiHankel functions on the other hand have a nonexponential grow
ing and decaying character; thus, propagation of the offshell wave
function under these circumstances is very difficult. The solution then
A A A A
is simply to replace kj and ij by the RiccatiBessel functions and n^.
This in turn requires a modification of the VPA equations (2.36) and
(2.37). However, the modification is minor and does not lead to any
significant increase in computational effort. Let us go back then to the
original inhomogeneous Schrodinger equation, Eq. (2.10), as it applies
to negative energies, and rearrange it into the form
K1 + 1)
v (r) ]co] = 2k^
 (k* + q2)j1(qr).
(2.40)
45
Treating the left hand side of this equation as an inhomogeneous term,
we can adopt the same procedure that was used in section 113 to
derive the desired set of modified VPA equations
^ <5, (q, kE; r) = k^1 {v(r)[w^ ooj jj (qr)] (2.41)
+ 2kE [u) J1 (qr)]}/a^
and
^^(q, kÂ£; r) = k"1 Mr)^ + 2kE j(qr)]} x
x {sin(fi.)j.(kEr) + cos(61 )n1 (kÂ£r)}. (2.42)
A similar set of modified equations could be used for positive energies
when a nonclassical region is encountered there. This would require
RiccatiHankel functions. This, however, was not found to be necessary
in the calculations done in this work. Stabilization was only a problem
at negative energies.
5. The Comparison Potential Method
Another possible approach in solving the inhomogeneous Schrodinger
equation (2.10) for the offshell wave function w^(q, k^; r), is to make
use of the comparison potential (CP) method. As was mentioned in the
introduction to this chapter, this approach has already been presented
in the literature [(vL61), (Br75)]. We shall,however, sketch an outline
of the method and present some improvements to the computational
algorithm for w,. These changes are based on the work of R. Gordon
[(Go69), (Go71), (Ro76)], who has investigated extensively the use of
46
local approximations to the potential in order to simplify scattering
calculations. The outcome of this pursuit will then be helpful in
suggesting some improvements to the VPA method, or for that matter any
method which implements a numerical solution for the offshell wave
function 1.
The basic idea behind the CP method is to approximate the potential
on a given interval in such a way that one can express the solution to
the resulting differential equation in terms of known functions which
present no problem computationally. To meet these requirements the
potential has often been treated locally as a step potential or as a
O
linearly varying potential (Go71). The centrifugal term, 1(1 + l)/r ,
may also be included in the potential, and the total effective potential
may be approximated (Go71). In the present work, however, only the
potential was approximated, taking it to be a collection of step
potentials. Thus one has
aiKi
HI + 1)
]uj
(aEk^ q2) (qr) ^
(2.43)
and
Vi = aEkE vi (2.44)
where ck plays a role analogous to and v^ is the local approximation
to v(r) for r. Â£ r < r.j+1. The advantage of this particular choice lies
in the fact that one can use Bessel functions as the local basis from
which to construct the solution to (2.43); and since the centrifugal
term is treated exactly, the convergence toward tike asymptotic boundary
condition (2.15) is accelerated in the case of the higher partial waves
(An76). Locally then, on the V interval
47
w] (q, kÂ£; r) =
A1
A1
u] (^r) B v. (k.t) + Cj
0 1
(qr),
(2.45)
where
aEkl q2
2 2
aiKi q
(2.46)
The offshell wave function is then obtained from two propagated solu
tions in a form analogous to that given for the VPA approach. The prop
agation here, however, is accomplished by matching each wave function
and its derivative across every interval. This can be done by either
propagating the coefficients A.j and Bj as v/as done in the reference
(Br75), or by propagating the wave function and its derivative directly
as we propose to do. To accomplish this, we shall modify the procedure
given by Rosenthal and Gordon (Ro76) for the case of the usual radial
Sc'nrodinger equation. From expression (2.45) and the corresponding
expression for the derivative of the wave function on the given i^
interval, it is easy to show that
A] = Wtujr.) Cj j1 (qr.), v1(icir.)]/Wi (2.47)
and
B] = (ri) Cj jj (q, r.), u] (k.^)]/!^ (2.48)
A A
where W.. is a constant equal to the value of the Wronkian of uj and v^,
the two local basis functions. The i superscript on wj has been removed
and is not necessary because of the continuity requirement imposed on
the wave function and its derivative. Using (2.47) and (2.48) in the
expression for Eq. (2.45), and the analogous expression
48
d i
dr
)
.J *
i i rU
(q. kE; r) = Aj[^ru1 (^r)] + b] [3 v] (^r)]
+ cÂ¡ & h ^ ]
(2.49)
for its derivative; the desired result
A
u(ri+1) = Ci[u(ri) C]j^(qri)3 + Cj (qri+1)
(2.50)
is obtained, where we have introduced the matrix definitions
w (r)
and
i(r) =
(O
jWi
dr 1
jj (qr)
d
dr J1
Ji (qr)
(2.51)
(2.52)
and the coefficient matrix C_., whose components are defined by the
relations
C11 = {u1 [d7>}/Wj
(2.53)
Ci2 iv(r,..,.,) u (r.) u ( ) v (r,)}/W
i+1
i+r
i' i
(2.54)
"21
 lafWr^jlt^rpn/H, (2.55)
and
'22 E {[3Fv(ri+l)I u(ri> '3Fi
(2.56)
In order to simplify the notation, we have suppressed V.." from the
argument of uj and vj, as well as the "1" subscript. Equation (2.49)
is appealing in the sense that it provides a simple algorithm for propa
gating cjj and its derivative from r^ to r^+1, but it is even more
appealing in the sense that one need not store the coefficients A^j,
and Cj for all the intervals needed in the calculation. Only one
problem remains to be dealt with. In the CP method the tmatrix is
calculated replacing (2.13) by the expression
(q', q; E)
_ 2 1
ir q'q
E
i
ri+lA
V.Â¡ / j(q'r)i^(q, kÂ£; r)
(2.57)
If one had propagated the coefficients A^j, and C, on each interval
one would have a set of integrals over products of two Riccatifunctions
as is indicated by Eq. (2.45), the expression for cjp and Eq. (2.57),
the expression for t.,. These integrals are well known and are easy to
generate (Me61). We, however, have been able to derive expressions for
the integrals needed in Eq. (2.57) directly in terms of the offshell
wave function and its derivative. A derivation of these quantities
is given in Appendix II.
One of the major criticisms of the VPA approach and other numerical
procedures for obtaining is that tj (q', q; E) must be obtained
through numerical quadratures. In practice this is problematic when
q and q' are larger, because of the oscillatory nature of the functions
involved. Generally, when one propagates w numerically, it is deter
mined at smaller step sizes than are involved in the CP approach; thus,
one could take advantage of the integral expressions mentioned above by
implementing a breakdown of the potential into steppotentials only in
50
the evaluation of t(q', q; E). This would then remove any problems in
this regard.
6. The Bateman Method
Very little has been done in the literature with regard to non
parti al wave techniques in the evaluation of scattering amplitudes or
twobody tmatrix elements. Only within the Eikonal approximation
[(6159), (Su69), (Ch73)] and approximations based on the Born expansion
[(Ho68), (Ra72)] has this not been the case. The attempts that have
been made to solve the LippmannSchwinger equation for either the full
wave function or the tmatrix have involved numerical quadratures. For
example, Walters (Wa71) has considered the LippmannSchwinger equation
for the full wave function and converted it to matrix form by use of a
two dimensional numerical quadrature for the case of a screened Coulomb
potential. The amplitudes obtained from this procedure however, though
in good agreement with exact results, showed poor convergence when the
number of quadrature points was increased. Another approach that has
been taken is that of Rosenthal and Kouri (Ro73), where the Lippmann
Schwinger equation for the scattering amplitude was reduced from its
twodimensional form to a onedimensional integral equation, which was
then solved numerically. Using the work of Walters (Wa71) for comparison,
these authors obtained results within 5 to 10 per cent agreement. In
contrast to these methods, the approach we shall new pursue does not
involve numerical quadratures, at least in the case of certain model
potentials. This procedure, known as the Bateman method, was originally
implemented by Akhmadkhodzhaev, Belyaev and Wrzecionko (Ak70) in the
computation of partial wave tmatrix elements. Ttey have, however,
51
extended this work to the calculation of the full tmatrix elements for
the case of a Gaussian potential (Be74a). with significant success. Lim
and Gianini (Li78) have gone further and applied it to the case of a
Yukawa potential, obtaining encouraging results.
Formally the Bateman method is equivalent to using the Schwinger
variational principle [(Li78), (Ad75)]. We shall thus present the
method in this framework. Let us begin then by considering the general
identity (Le69)
[A(z)J = At(z) + At(z*)+ At(z*r At(z)1 At(z), (2.58)
where A is an operator that is Lconjugation invariant (A(z) = A(z*)r)
and At is its trial value. Similar operator identities have been used
as a basis for studying upper and lower bounds in the theory of bound
state calculations (Lo65) Taking A(z) to be equal to t(z) and using
equation (2.3), which relates the toperator to the wave operator W(z)
in the variational expression (2.58), one obtains
[t(z)J = Wt(z) + Wt(z*)+V Wt(z*)+ V VGq(z)V Wt(z), (2.59)
where the trial operator W^.(z) can in general be chosen as
Wt(z) = f>
Here f> represents a set of n basis functions >fn} >
whose expansion coefficients (Cj,C2,...,Cn> are determined variationally
through expression (2.59). In the usual variatisrnal procedures (SI72),
(Ad74), (Pa74), (Ku78), the set of functions {f} iis usually chosen to
be square integrable; here however, {Â£} will be chosen to be a collec
tion of plane wave states, i.e., the set denoted by {k} or {kj,^,... ,kn}.
52
Thus letting  Â£ > be equal to t > and making use of equations (2.59)
and (2.60), one can obtain by taking the proper variations (Go50) the
expression
t(q', q; z) = Â£+(q')Â£(q, z) (2.61)
where C_ satisfies the relation
J(z)C(q, z) = I(q) (2.62)
and where the components of the matrices Â£ and Â£ are defined as
Ij (q) = ^iVq> (2.63)
and
Jij(z) = <^V VGo(z)Vkj> (2.64)
Another equivalent expression to (2.61) would be
t(q', ; z) = (2.65)
but we have chosen to use (2.61) and (2.62), in order to solve a
simultaneous set of linear equations instead of the matrix inversion
implied by this last expression.
The integrals needed in this procedure, as shown by equations
(2.63) and (2.64), are those required for the second Born approximation
[(Da51), (Le56), (Ho68)]. It is readily found however that integrals
of the type are not trivial to evaluate, and in general
must be done numerically (Ho68). For the particular potentials we shall
be interested in, the Morse and exponential potentials, one can obtain
the desired integrals from various references [(Da51), (Le56), (Li78)].
53
In conclusion we should point out the weaknesses of the Bateman
method. From Eq. (2.55) it is apparent that we have obtained a
separable approximation to the tmatrix. Furthermore, using Eq. (2.1)
it is easy to show that this could be viewed as a direct consequence
of the separable approximation
V = V>1 <Â£V (2.66)
to the potential [(Ad74), (Ad75)]. This is an important point, since
Osborn [(0s73a), (0s73b)] has shown mathematically the nonccnvergence
of tmatrix elements based on separable approximations to local poten
tials. This problem exists irrespective of the approach used in con
structing the given separable approximation. Fortunately, the diffi
culties arise for large values of the momenta q1 and q in Eq. (2.63),
values at which the tmatrix elements are small and have no significant
effect in threebody calculations (SI73). Finally, there remains the
problem of the inapplicability of the Bateman approach to singular
potentials, a question open to further investigation.
CHAPTER III
NUMERICAL RESULTS FOR THE TWOBODY tMATRIX
Having developed the formalism of the VariablePhase and Amplitude
method in the previous chapter, we will now apply it to various model
potentials of physical interest. This will serve to illustrate various
known analytical properties of the twobody tmatrix and to gauge the
reliability of the computational procedure.
We will concentrate on calculations for the lowest (attractive)
and (repulsive) potentials of HÂ£ because of the role these play in
the collision dynamics of H + H^, and because their appearance in
dynamical studies can be justified within a diatomicsinmolecules treat
ment (Mi77b). A comparison of some of our results will be given with
others obtained within a variational procedure, developed to solve the
threeatom problem at low (thermal) energies (Ku78).
A study will also be made on the reliability of the Bateman method
in the computation of tmatrix elements for potentials of interest in
atomic and molecular scattering theory.
1. Properties of the TwoBody tMatrix
A brief summary of the various properties of the offshell tmatrix
elements, tj(q', q; E), will now be given. This will prove valuable in
determining the reliability of any given computational procedure used to
obtain tj.
The most important properties satisfied by tj (q1, q; E) are those
54
of symmetry,
tj (q', q; E) = tj (q, q'; E) ,
(3.1)
and of offshell unitarity,
Im[tÂ¡ (q , q; E)] = irmk^t* (kÂ£, q'; E)tÂ¡(kE, q; E) (3.2)
The symmetry relation is valid at all energies, but the unitarity rela
tion is valid only for positive energies. These relations are well
established in the literature and will not be derived (Wa67). It
suffices to say that in this case they are a direct consequence of
assuming a real local potential.
If one considers the differential equation satisfied byw,(q, k^.; r),
Eq. (2.10), it is apparent that reduces to the onshell wave function
when q = k^. Similarly, the offshell tmatrix elements are defined
such that there is a smooth transition to the onshell behavior as the
momenta involved approach the onshell quantities. This is exemplified
by the relations (Wa67)
and
~ A(+)
w,(q, kÂ£; r) ~ j(qr) 7Tmqt1(kE, q; E)h, (kÂ£r)
r ^ oo
tan[6j (kÂ£)] =
Im[t (q, kÂ£; E)]
Re[t, (q, kE; E)]
(3.3)
(3.4)
which correspond to well known onshell relations for q = kE.
Another important aspect of the tmatrix elements is their behavior
as a function of momenta and energy. In this respect, there are various
limits of interest. If the energy E and initial momentum q are fixed,
then
lim t. (q', q; E) = 0
q 1 * oo
(3.5)
56
and therefore because of symmetry
lim tj (q, q; E) = 0 (3.6)
q 1 y oo
These relations are apparent if one considers the oscillatory nature of
A
toj (q, kÂ£; r) and j^ (q' r) in expression (2.13) as q and q' approach
infinity. A clearer understanding of relations (3.5) and (3.6) could
be obtained from a semiclassical analysis of the twobody tmatrix as
a function of energy and momenta. Such an analysis has been carried
out by Korsch and Mohlenkamp [(Ko76), (Ko77a)j for the case of purely
repulsive potentials. These authors have classified the various regions
in the qq' momentum plane in which t corresponds to classically allowed
or forbidden processes. Classically forbidden processes would, for
example, be associated with complexvalued classical paths or trajec
tories.
As a function of energy, the tmatrix must also show continuity as
the energy goes from positive to negative, and consequently
lim t, (q1, q; E) = lim Re[t,(q', q; E)J (3.7)
E  0 E 0+
and
lim Im[t(q', q; E)] = 0 (3.8)
E + 0+
For negative energies, the tmatrix elements are real and have poles at
energies corresponding to the bound states of the system. This can be
established by considering the expansion of t(q' q; z) in terms of the
eigenfunctions of the full Hamiltonian of the system (Ne66). The result
is as follows:
57
t(q q;
z) = I
nlm
^enl ; q1 {'Z ~ Â£q)
(z Enl)
Jnlm
(q')
nl
n
(q)
+ t(q, q; eq+)
+ j dq!l [ (eqi. cq is)'1 + (z Â£k)"1]
x t(q' q"; eqn+) t*(q", q; cqn) (3.9)
where ^nm(q) is a bound state function in the momentum representation
having the eigenvalue of enÂ¡. The quantities, eq, are eigenvalues to
the scattering states of the system.
2. Numerical Calculations Using the VPA Equations
We will now present various results for the system for the
lowest *Lg+ and ^Lu potentials [(Mi77c), (0178)]. Some of these results
have already been reported in the literature [(Be78), (Ku78), (Ko77a)].
A Morse potential
V(r) = D[1 e_a(r"ro)]2 D (3.10)
was used to represent the ^Ig+ interaction potential, and a Hultnn
potential
V(r) = Ae'r/a[l e"r/a]_1 (3.11)
3
was used for the repulsive I interaction potentials. The parameters
0_i o
used were D = 4.786 eV, a = 2.1123 A 1, rQ = 0.7411 A, A = 20.11 eV and
o
a = 0.4984 A (Mi77c). This particular study will! prove interesting,
since it serves to draw a contrast between the analytic properties of
t and toj obtained from an attractive potential, and those obtained from
a strictly repulsive potential.
58
In figures (3la) through (3lc), we have plotted various components
of the positive energy offshell wave function w for the case of a Morse
potential. If one recalls the VPA expression for w given by Eq. (2.26),
it is clear that one can consider wj(q, k^; r) to have two components:
A
j(qr) and xj (^ kjrl r), where xj is defined by
A
Xj(q* kÂ£; r) = o)^ (q, kÂ£; r) j^qr) (3.12)
A
and is often called the wave defect (Sr68). The term jj(qr) in Eq.
(3.12) is present in equations (2.26) and (3.12) because of the inhomo
geneous character of the differential equation obeyed by ojj [see Eq.
(2.10)]. Thus, we see from Eq. (3.12) that for positive energies
Re [oj1 ] = jj(qr) + Refx,1 (3.13)
and
Imfco, ] = ImCxj] (3.14)
Equation (3.13) implies that the real part of wj is highly influenced
A
by the driving term jj(qr). This can be seen in figures (3la) and
(3lc), which show how the function [2/tt]^2 Re[ujQ(K, K^; R)/K] oscil
lates about the driving term [2/tt] 1//2[(KR)/K]. The data used to
obtain these figures were K= 0.23 a.u. and E = 0.01 a.u. If one were
to increase the value of either K or E, the offshell wave function
would become much more oscillatory. Similar results were also obtained
for the Hulthn potential. We note, however, that an attractive poten
tial pulls the nodes of the offshell wave function in, whereas a repul
sive potential pushes them out. For negative energies, the offshell
wave function becomes real. An illustration of coj at negative energies
Figure (3
1)
(a) [2/tt]}/2 Re[>0(K, KÂ£; R)/K]
(b) [2/tt] 1/2 Im[ojq(K, KE; R)/K]
(c) [2/tt]1/2 [J0(KR)/K] (a.u.)
Calculations correspond to E = 0
Kj: = 4.26 a.u. and K = 0.23 a.u.
Potential)
(a.u.)
(a.u.)
.01 a.u.,
(Morse
en
o
61
is given by figures (32), (33) and (34) for the Morse potential
case. Since the tmatrix elements have poles at the bound states of
the system [see Fig. (35)], we see from the relation connecting t]
and u), [Eq. (2.13)] that the amplitude of w, must grow as the energy
1/2
approaches a bound state eigenvalue. In Fig. (32), [2/tt] 7 [cog/K]
is plotted versus R for K = 4.475 a.u. and an energy value of E =
0.1446 a.u. The large magnitude of the amplitude displayed by this
function, when compared to the result at E = 0.153 a.u. given in Fig.
(33), is a clear indication that the energy is quite near to an eigen
value of the potential. A similar effect can be seen at higher partial
waves, as is shown in Fig. (34) where [2/tt] [u^(K, K^; R)/K] is
plotted versus R for K = 4.5 a.u. and E = 0.153 a.u. Note that the
effect of the driving term, j^(KR), is always present in all of the
figures (3.2) through (3.4). Within the interaction region of the
potential, it is the wave defect xj which dominates; whereas the driving
term dominates in the asymptotic region. For the Hulthn potential, the
offshell wave function does not display the large changes in magnitude
mentioned above for the Morse potential, since the Hulthn potential
used in this study does not support bound states.
Various studies of the partial wave tmatrix elements as a function
of energy and momenta were carried out for both the Morse and Hulthn
potentials. We have already shown the effect of the bound states on
the offshell wave function co Their effect on the tmatrix elements
is also quite important. In Table (31), we have tabulated the Morse
tg(q, q; E) matrix elements for q = 4.0 a.u. as a function of energy.
Column I gives the VPA results and column II the correspondng results
obtained by a variational method developed by Kurooglu and Micha (Ku78).
Figure (3
2) [2/tt]1/2 to)0(K, Ke; R)/K] at E = 0.1446 a.u., an
energy value near the second pole. (K = 4.475 a.u,
Morse Potential)
T
T
200
0
1
2
R (au.)
Figure (33) [2/tt]1/2 [wq(K, Kj:; R)/K] at an energy value that lies
between the first and second bound states. (E = 0.153 a.u.
and K = 4.475 a.u., Morse Potential)
7T
6
OJ
R (a.u.)
cr>
en
Figure (34) 12/tt]1/2 [w5(K, Ke; R)/K] for K = 4.5 a.u. and E = 0.153 a.u.
(Morse Potential)
/
6.0 8.0 100
(a.u.)
as
cr,
Figure (35) Calculated pole structure of
tg(K, K; E). (Morse Potential)
O'
Table (31) The tn(q, q; E) matrix elements are
giveri for q = 4.0 a.u. Column I
gives the VPA calculations and column
II the results obtained from a varia
tional procedure (Ku78). (Morse
Potential)
Energy (a.u.) tQ(q, q; E) x 102
I
II
0.18
2.5971
2.5798
0.17
4.4003
4.3841
0.16
2.0273
2.0069
0.15
8.8552
8.7964
0.14
9.9843
10.0183
0.13
2.0411
2.0680
0.12
1.5197
1.5486
0.11
1.2378
1.2106
0.10
1.0856
1.1159
71
We note that in general the agreement is quite good. A more detailed
picture about the structure of the Morse tg matrix elements at negative
energies can be seen from the results given in Table (32) and fig. (35),
each of which describes the energy range corresponding to the first four
poles. The figure shows that the tmatrix elements change sign as one
goes across a pole, in agreement with equation (3.9). For the Hulthn
potential, the tmatrix elements show a very smooth and plain structure
as can be seen in Table (33), where tg(q, q; E) is given versus energy
for q = 4.5 a.u. The agreement with the variational results is again
quite good. At positive energies, we have shown in Fig. (36) the
behavior of the Morse tg(q, q; E) matrix elements near E = 0 a.u.
[q = 4.0 a.u.]. We see the continuity one expects; i.e.,both equation
(3.7) and (3.8) are satisfied. In general, the tmatrix elements of
both the Morse and Hulthn potentials show a much more oscillatory
structure at positive energies. Figures (3.7a) and (3.7b) illustrate
this for the diagonal (K1 = K) Hulthn tg(K', K; E) matrix elements.
The large oscillations seen in these figures do not extend indefinitely.
Korsch and Mohlenkamp (Ko77a) have extended the present calculations to
higher energies and have shown that the amplitude of the tmatrix even
tually decays to zero. In the comparison of their semiclassical results
with the present quantum mechanical ones, they obtained good agreement.
As a function of momenta, the tmatrix elements show a much greater
oscillatory nature. This can be seen in Fig. (38) where the Morse
tg(K, K; E) matrix elements are plotted versus K for E = 0.1378 a.u.,
and in Fig. (39) where the Morse tg(K'; K; E) matrix elements are
plotted versus K' for E = 0.153 a.u. and K = 4.5 a.u. Both figures
(3.8) and (3.9) show how the tmatrix goes to zero asymptotically in
Table (32) Selected values of tQ(q, q; E) for
q = 4.475 a.u. are presented about
the first four poles. (Horse
Potential)
Energy (a.u.) t(K,K:E) Location of Pole
0.165078
2.6856
0.165076
5.4287
0.1650741
0.165074
174.7388
0.165072
5.0667
0.144800
1.2866
0.144600
3.5240
0.144486
0.144400
4.6930
0.144200
1.4018
0.125600
0.0290
0.125400
0.0936
0.125268
0.125200
0.2220
0.125000
0.0656
0.107800
0.0011
0.107600
0.0026
0.UP421
0.107400
0.0243
0.107200
0.0020
73
Table (33) Selected values of t(q, q; E)
for q = 4.0 a.u. areugiven.
Column I gives the VPA calcula
tions and column II the results
obtained from a variational pro
cedure (Ku78). (Hulthn Poten
tial )
Energy (a.u.)
Vq>
q; E) x 103
I
II
0.18
6.4807
6.4592
0.17
6.2683
6.2477
0.16
6.0525
6.0300
0.15
5.8277
5.8057
0.14
5.5946
5.5742
0.13
5.3574
5.3348
0.12
5.1079
5.0868
0.11
4.8526
4.8295
Figure (36) Behavior of tg around E = 0 a.u. (Morse Potential)
15 13 11 9 7 5 3 1 1 3 5 7 9 11 13 15
E X107 (a.u.)
<_n
Re t0(K,K;E) X104 (a.u.) Im t0(K,K;E)X104 (a.u.)
Figure (37a) Diagonal (K' = K) matrix elements of Re[tn(K', K; E)].
(Huthn Potential)
Ove) cOLX(a;r>l)) aa
Figure (37b) Diagonal (K', K) matrix elements of Im[tQ(K', K; E)].
(Hulthn Potential)
Â¡m t0(K,K;E)X103 (a.u.)
2
 2
 4
 6
 8
10 
K= 4.5 (a.u.)
H*H (Hulthen Potential)
_1_
J I I L
0.00
0.04
0.0 8
Energy (a.u.)
0.1 2
0.16
Figure (38) Diagonal (K = K) matrix elements
for to(K', K; E). (Morse Potential)
HH (Morse Potential)
E = 0.1378 (a.u.)
i
8
K (a.u.)
12
i
Figure (39)
tn(K', K; E) for E = 0.153 a.u.
and K = 4.5 a.u. (Morse Potential)
83
0.0
84
K and K' [see equations (3.5) and (3.6)]. Comparable behavior was
observed at positive energies as well, irrespective of the attractive
or repulsive nature of the potential. Table (34) shows a momentum
study for the Hulthn potential tg(q', q; E) matrix for E = 0.15 a.u.
and the 1 values 0, 5 and 10. These results indicate that the diagonal
q1 = q tmatrix elements are consistently larger in magnitude than the
offdiagonal q! / q tmatrix elements. Column I in this table gives
the VPA results and Column II the corresponding variational results
(Ku78). As a function of the partial wave parameter 1, the tmatrix
generally shows an oscillatory structure for low partial waves and a
smoothly decaying structure for high partial waves. Figures (310a)
and (310b), nicely illustrate this effect for both the real and imagi
nary components of the Hulthn t(K', K; E) matrix elements (K' = K = 4.0
a.u. and E = 0.01 a.u.). Using the same data, #e see in figures (3lla)
and (3llb) plots of the real and imaginary confonents of the total
tmatrix [t(K1, K; E)] versus momentum transfer AP = K' KÂ¡. These
plots imply that the tmatrix elements become snail er as the momentum
transfer increases. In this particular case, thirtyfive partial waves
were required for the total tmatrix to converge As the energy and
momenta arguments become larger, this number will) generally increase.
Parcial wave convergence will be discussed further in Chapter VII,
where the twobody tmatrix elements are used in the threebody atom
diatom scattering problem.
The tmatrix results we have reported here were all obtained from
Eq. (2.13) using a numerical quadrature, which in this case was a com
bination Simpson's and Newton's 3/8 rule (Hi56). In order to test the
reliability of this procedure, we have used Eq. '(33.2) to obtain phase
85
c.
Table (34)
Selected values of tj (q', q; E) are
given for E = 0.15 a.u. and the par
tial wave numbers 1 = 0, 5 and 10.
Column I gives the VPA calculations
ant column II the results obtained from
a variational procedure (Ku78). (Hulthn
Potential)
1 q'
q
t, (q1 q; E)
x 103
I
II
0 3.0
1.0
0.7899
0.7894
2.0
4.0665
4.0693
3.0
10.187
10.179
4.0
2.1074
2.1160
5.0
0.2154
0.2476
5 3.0
1.0
0.0207
0.0207
2.0
1.1745
1.1755
3.0
5.3773
5.3709
4.0
1.0412
1.0434
5.0
0.0513
0.0547
10 3.0
1.0
0.0005
0.0009
2.0
0.1174
0.1186
3.0
1.4905
1.4857
4.0
0.2644
0.2668
5.0
0.0093
0.0125
Figure (310a) Plot of Re[t(K, K; E)] versus 1. (K = 4.0 a.u. and
E = 0.01 a.u., Hulthn Potential)
1.1
0.5
0.2
0.8
1.4
2.1
O 7 14 21 28 35
L
CD
Figure (310b) Plot of Im[ti(K, K; E)] versus 1. (K = 4.0 a.u. and
E = 0.01 a.u., Hulthn Potential)
0.2
0.3
0.8
 1.4
1.9
2.5
0 7 14 21 28 35
L
CO
Figure (31 la) Plot of Re[t(K', K; E)] versus AP = Â£' lÂ£.
(K = 4.0 a.u. and E = 0.01 a.u., Hulthn Potential)
Figure (3llb)
Plot of Im[t(K', Â£; E)] versus AP = K' 
(K= 4.0 a.u. and E = 0.01 a.u., Hulthn P
r+ 7^4
1.9
0.3 
1.4
3.0
4.7
6.3
/ \
/ \
\
r /
i
0
1.6
\
3.2
4.8
AP (a.u.)
Uj
94
shifts from the halfonshell tmatrix elements. Typical results are
shown in Table (35), where Morse potential phase shifts are tabulated
for the energies E = 0.01 and E = 0.04 a.u. Columns I and III correspond
to phase shifts obtained using Eq. (3.2), whereas columns II and IV list
analogous results obtained from the numerical integration of the onshell
Schrodinger equation. A De Vogelaere integration scheme (Le68) was used
for the onshell calculations using the procedure developed by Bernstein
(Be60). Comparably good results were obtained for various other poten
tial data. Other factors that demonstrate accuracy of the present method
are symmetry and unitarity [see equations (3.1) and (3.2)]. Generally,
agreement to three or four decimal places was obtained. An exception
to this occurred at high values of momenta where the tmatrix elements
are quite small in magnitude. This indicates that the mesh size, over
which the offshell wave function is determined and over which the quad
rature in Eq. (2.13) is carried out, may be too large. Given the extreme
oscillatory nature of the functions involved, a smaller step size would
be desirable. However, a compromise on the step size is necessary to
prevent the accumulation of error from being too large.
It is in this respect that the comparison potential method offers
an advantage, since the quadratures involved in that procedure are done
analytically. We found though, that the VPA procedure showed more
stability in the propagation of the offshell wave function through non
classical regions. This, however, warrants further investigation, since
it is a function of the efficiency and sophistication of the computer
codes used.
Table (35) Columns I and III are phase shift values
obtained from halfonshell quantitites,
and columns II and IV are the correspond
ing onshell quantities. All phase shifts
are Mod (it). (Morse Potential)
Â¡(E =
0.01 a.u.)
6, (E =
0.04 a.u.)
I
II
III
IV
0.8655
0.8647
1.2732
1.2757
0.8463
0.8471
0.1843
0.1819
0.4431
0.4421
1.5293
1.5270
1.5482
1.5494
0.3781
0.3805
0.5379
0.5373
0.7473
0.7446
0.3403
0.3390
1.3752
1.3780
1.0825
1.0815
0.4598
0.4627
1.4480
1.4494
0.3532
0.3524
0.9677
0.9685
1.0698
1.0702
0.6154
0.6157
1.4366
1.4477
0.3781
0.3896
0.8966
0.9154
10
3. Numerical Results Obtained Using the Eatemar. Method
From the results given in the last section, it should be clear
that a method which gives the total tmatrix t(q, q; E) without a
partial wave expansion would be highly desirable. Such a method would
hopefully avoid the problems of computing a highly oscillatory function
for many partial waves. To this end, we now present some results
obtained using the Bateman method as outlined in Section (26).
Computationally all that is needed to obtain t(q', q; E) using the
Bateman method is the evaluation of the integrals of the type
and along with a procedure for solving a set of complex
simultaneous equations [see Eq. (2.61)]. For the Yukawa and exponential
potentials, analytic expressions for the above integrals may be found in
the references given in Section (26) [(Da51), (Le56), (Li77)]. The
integrals for any other potential which is a linear combination of
exponentials may be obtained using the relation (Br70)
+0O
V(r) = / dA
oo
p(A) eXr .
(3.15)
where p(A) is some linear combination of delta functions. For the Morse
potential, one has
p(A) = D[e2ar <5(A 2a) 2ear 6(A a)] (3.16)
The set of complex linear equations given by Eq. (2.61) was solved
using the IMSL library program LEQTIC (Ai75). Siince the input matrix
elements have quite complicated analytic expressions, their
accuracy was verified both analytically and numerically. In all cases
the analytic and numerical results agreed to ten or more decimal digits.
In the present calculations, we have found tthat the Bateman method
97
worked quite well for soft potentials such as the Yukawa potential
studied by Walters (Wa71), and Rosenthal and Kouri (Ro73) [V(r) =
1.1825 r"1 e_r in units such that h = m = aQ (unit of length) = 1].
This can be seen in Table (36), where we have tabulated the onshell
tmatrix t(q', q; E) for q1 = q = kE = 1.816 a^1 as a function of the
angle Qqiq between q* and q. Comparing the Bateman results to the VPA
results, we see that the Bateman results are much better for small
values of the momentum transfer q' q. This was found to be true
y
in all the studies that were made. The set of plane wave states {k}
that was used in the above calculations is given in Table (37). Because
of the rotational invariance of the potentials involved, it is only
necessary to choose the set of vectors {k, k^, ..., kn) to lie on the
(q1 q) plane. If one specifies q as the reference vector, it is then
only necessary to specify and kp. The results given in Table (36)
were quite insensitive to any variation in the choice of {k}. Better
agreement was obtained, however, if the set included both q1 and q as
members of the set. A comparison with the calculations given by Walters
(Wa71), and Rosenthal and Kouri (Ro73) is also given. The agreement
between these results and those obtained from the VPA method is better
than that obtained from the Bateman method. The Bateman method, however,
is much easier to apply in the present case than either of these other
methods. Another observation that can be made from Table (36) is that
the Bateman method yields better agreement for the imaginary component
of t than for the real component. The reason for this difference is
not obvious and remains an open question.
We did extend the present study to soft exponentials with parameters
and units comparable to those shown above for the Yukawa case. The
Table (36) Comparison of onshell tmatrix elements for the attractive Yukawa potential
V = 1.1825 r"ler given by Walters (Wa71) [Units such that h = m = aQ = 1
are used]. The parameters E = 1.649 and q1 = q = kE = 1.816 a01 are used.
eq1 q
(degrees)
t(q1, q;
E) x 102
VPA
Bateman
Walters (Wa7l)
Rosenthal and Kouri (Ro73)
0
5.529il.872
5.778il.817
5.484il.877
5.623il.786
10
4.986il.841
5.212il.791
20
3.811il.755
4.024il.708
30
2.664il.632
2.900il.556
40
l.819il.490
2.051il.367
60
0.818il.208
l.016il.147
90
0.201i0.880
0.384i0.866
0.187i0.897
0.229i0.856
120
0.215i0.679
0.212i0.672
150
0.1040.574
0.091i0.581
180
0.125i0541
0.083i0.558
0.139iO.552
0.12710.512
CO
99
Table (37) Basis {k} used in the Bateman calcu
lations^given in Table (36). The
vector k(l) was chosen to lie along
the direction of the initial momentum
vector 4 = all other lie on the
(q'q) momentum pl_ane. is the angle
between Â£(n) and k(l).
n
6[k(n),k(l)] (radius)
tc(n)
1
0
1.816
2
0.785
0.605
3
1.571
1.816
4
2.356
0.605
5
3.1416
1.816
6
0.1963
1.816
7
0.5890
0.454
8
1.3744
1.816
9
2.1598
0.454
10
2.945
1.816
11
0.295
1.816
12
0.393
2.421
13
1.178
1.816
14
1.963
2.421
15
2.749
1.816
100
results were generally good, provided one again concentrates on small
momentum transfers. We do not report any offshell studies here,
although, a similar quality in the results was obtained. When we
extended the Bateman method to potentials applicable to chemical
problems, e.g. exponential potentials that are very steep and localized
in coordinate space, the results obtained were totally inadequate. The
sensitivity to the choice of (1) became very large. In light of these
findings, no further investigations were carried out. A possible
explanation of this negative outcome might lie in the fact that for a
hard potential, the interaction region is highly localized leading to
a corresponding delocalized function in momentum space. Thus, for such
potentials, one would expect the number of basis function in the set
{k} to be very high.
CHAPTER IV
THE THREEBODY PROBLEM
In Chapter I we discussed the feasibility of treating an atom
diatom scattering process as an effective threebody problem in such
a way that one could make use of twobody interaction data. It was
pointed out there that one could construct a proper representation of
the interaction potential for a reactive triatomic system in terms of
twobody quantities, only if one introduced a set of spindependent
atomic pair interactions. Since our present interest lies in high
energy atomdiatom collisions, we proposed the use of a simplified
impulsive model in which the total interaction potential would be
assumed a sum of spin independent twobody potentials. Having already
addressed in Chapter I the merits and limitations to such a representa
tion of the potential, we turn now to the solution of the scattering
problem.
The formalism we shall adopt in treating the scattering of an atom
diatom system will be based on that given by Faddeev (Fa61), who
developed the first rigorous mathematical treatment of the threebody
problem. We shall not attempt to address the more mathematical aspects
of the problem. Instead, we shall restrict ourselves to a discussion
of difficulties encountered when one tries to apply the LippmannSchwinger
equation to the threebody problem and to a formal derivation of the
various transition operators of physical interest. The resulting
expressions for these transition operators will then be used to construct
101
102
a multiple collision expansion, which, as was pointed out in Chapter I,
can be used to describe the scattering of an atom and a diatom as a
sequence of atomatom encounters [(Ch71),(Mi75b)].
1. Problems with the LippmannSchwinger Equation
The main difficulty with the threebody problem lies in the
existence of various arrangement channels. By this we mean, that given
three particles (atoms) A, B and C, one can have any of four possible
arrangement channels, i.e.,
channel (1) corresponding to A + (B, C),
channel (2) corresponding to B + (A, C),
channel (3) corresponding to C + (A, B),
and channel (0) corresponding to A + B + C,
where an interacting pair is denoted by enclosing the given particles
in parentheses. This implies that for reactive scattering there are
difficulties in implementing the boundary conditions, at least in the
usual coordinate space formulations of the scattering problem [(Ma66),
(Li71), (Mi72a)], and for dissociative scattering there are even problems
with regard to the proper formulation of the boundary conditions (Me76).
For rearrangement scattering the problems begin with the choice of
coordinates, since the coordinates that are best suited for describing
the motion of the threebody system in the entrance channel are not the
most appropriate ones for describing the motion in the exit channel
(Ma66). Going further, let us consider an integral equation for the
threebody problem analogous to the LippmannSchwinger equation for the
twobody transition operator, i.e., the equation
T(Z) = V + VG0(Z)T(Z)
(4.1)
103
where T(Z) is the general threebody Toperatcr, GQ(Z) is the free
particle propagator, Z is an energy parameter (E + ie) and V is the
potential for the threebody system. The problem with the boundary
conditions now manifests itself in the form of the "uniqueness" question
with regard to the solutions to Eq. (4.1) [(Sc74a), (Re77)]. Since a
threebody system allows for the existence of twobody bound states at
a continuum of energies within the range of physical interest, one must
provide additional constraints which select those solutions satisfying
the proper physical boundary conditions. Furthermore, because of the
presence of these subsystem bound states, one finds that even after the
centerofmass motion has been factored out, the kernel contains
unfactorable delta functions (Re77). Each of these delta functions
expresses the conservation of momentum of the spectator atom in the
presence of the interacting atomatom pair. A very nice discussion of
all these problems is given in the references [(Kr71), (Sc74a), (Si71),
(Re77), (Wa67)].
2. The Multichannel Transition Operators
It is clear from the discussion in the previous section, that the
LippmannSchwinger equation, Eq. (4.1), must be reformulated by speci
fically taking into account the existence of the various arrangement
channels. This can be done in a straightforward manner by considering
the structure of the potential characterizing the threebody system, and
by introducing various operators describing the interaction of two
particles in the presence of a third [(Fa61), (Si7/1), (Sc74a)].
Let us begin by considering the total Hamiltinian,
H = H0 + V,
(4.2)
104
where H0 is the freemotion Hamiltonian for the three atom system, and
introduce the assumption
3
V = Z V. (4.3)
j=l J
i.e., that of pairwise additive potentials. Each Vj is an atomatom
potential corresponding to the asymptotic interaction in channel j, and
may be specified using the procedure outlined in Chapter I. Asymptoti
cally in each channel j, the system can be described by the channel
Hamiltonian
Hj = H0 + vj (44)
Therefore, che total Hamiltonian for the system can be written as
H = H, + (4.5)
J
where
V(j) = V V. = l 6ik Vk (4.6)
J k=1 *
is the channel interaction potential and 6, is equal to (1 6).
J K j k
Having made this connection it is apparent that iky considering various
partitions to the total Hamiltonian in terms of channel quantities,
such as H. and V^'^, one should be able to construct the various opera
J
tors needed in the reformulation of Eq. (4.1). Important in this regard
are various twobody operators, such as the channel resolvents and two
body transition operators, which we will now introduce.
The resolvent relations that are needed follow directly from equa
tions (4.2) through (4.6), using the identities '(11068)
105
(A B)l = Al + Al B(A B)'1
(4.7)
and
(A B)'1 = A"1 + (A B)"1 BA'1
(4.8)
The expressions obtained through Eq. (4.7) would be
G(Z) = Gj(Z) + Gj(Z)v(j)g(Z)
(4.9)
and
Gj(Z) = Gq(Z) + G0(Z)VjGj(Z),
(4.10)
where G(Z) = (Z H)"1 and Gj(Z) = (Z Hj)1. These equations remain
valid for j = 0 if one defines Vq = 0, so that v(0) = V. The twobody
transition operators can be defined as
(4.11)
Tj(Z) = Vj + G0(Z)VjTj(Z),
analogous to the definition given for t(z) given in chapter II. At
embedded in a threebody space; i.e., they characterize the behavior of
two interacting particles in the presence of a third spectator particle.
This should be clear from the structure of GQ, which is the free propa
gator for the entire threebody system.
Let us return now to the LippmannSchwinger equation, and use
Eq. (4.2) to express the threebody Toperators as
T(Z) = l t(J)(Z) ,
j=l
where we have introduced the definition
(4.12)
t(J')(Z) = Vj + VjG0(Z)T(Z) .
(4.13)
106
This is a final channel decomposition in which each term corresponds to
a collision event where the last step is an interaction through the
potential V, [(Wa67), (Mi72b)]. It should be apparent by comparing the
J
expressions for and T, that there is a close relation between the
J
two. Indeed, if t(j) is expressed as
t(j)(Z) = T. (Z)[1 + x(j)(Z)] (4.14)
J
where x^) is a distortion term, one is immediately led to the relation
3
x(j)(Z) = Z 6..G (Z)T(i)(Z) (4.15)
1 = 1 J 0
Finally, substituting Eq. (4.15) in Eq. (4.14), one obtains
T(J)(Z) = T.(Z) + T.(Z)G (Z) Z 6 (4.16)
J J 0 i=l J'
a set of three coupled equations. In matrix form the result is
T<1>
T1
o Tl Tl
TU)
t(2)
=
T2
+
t2 o t2
Go
l(2)
j(3)

_T3.
T3 T3 0
.
j(3)
Equations analogous to these were first derived by Watson (Wa57),
though Faddeev (Fa61) was the first to present a rigorous analysis of
their mathematical properties; as a consequence, they are known as the
FaddeevWatson equations. One important property of these equations is
that they become connected upon one iteration and therefore contain no
unfavorable delta functions. From a practical viewpoint, these equa
tions are the most appropriate expressions to use in describing scatter
ing processes in which both the entrance and exit channels correspond
to the breakup channel.
107
The FaddeevWatson equations do contain information on other
channel processes; however, it will be necessary to derive a set of
transition operators more suitable for rearrangement processes. In
order to accomplish this, we will begin by considering a finalchannel
decomposition for the total scattering wave function [(Wa67), (Mi72b)].
Channel 1 will be singled out as the initial channel, for the sake of
specificity. Accordingly, the total wave function will be labeled by
Â¥1(+), and it will be associated to an initial wave function ^ through
the relation (Wa67)
Â¥1(+) = W(Z)$ (4.18)
where W(Z) is the wave operator defined as
W(Z) = ie G(Z) (4.19)
It will be assumed throughout, that these are halfonshell relations,
i.e., E is equivalent to the total energy of the system, as described
initially by Using equations (4.1), (4.7) and (4.9) one can show
that
G(Z) = Go(Z) + G0(Z)T(Z)G0(Z), (4.20)
which along with Eq. (4.19) implies that
W(Z) = WQ(Z) + G0(Z)T(Z)W0(Z) (4.21)
where WQ(Z) = ie G0(Z). The finalchannel decomposition for W now
follows immediately from Eq. (4.16) and Eq. (4.21), and is
w(z) = w0(z) + z y(j)(z),
j
(4.22)
108
where
y(J)(Z) = G0(Z)T(J)(Z)W0(Z) (4.23)
Introducing the FaddeevWatson equations for into Eq. (4.23) yields
y(J)(Z) = Yj(Z) + Z 5J.iG0(Z)Tj(Z)Y(i)(Z), (4.24)
having defined Y(Z) as
J
Yj(Z) = G0(Z)Tj(Z)W0(Z) = Wj(Z) W0(Z). (4.25)
Applying W(Z), as given by Eq. (4.22) to yields the desired final
channel decomposition for the wave function,
Y1(+) = Z (4.26)
j
where
T (J) = y(J') $ (4.27)
Here we have made use of Eq. (4.25) and the halfonshell relations
W0(Z)^i = ie G0(Z)$1 = 0 (4.28)
and
WjtZjij = ie Gj(Z)*1 = (4.29)
The Faddeev equations for the total wave function,
Tl(j) = 6ji$i + z I..G0(2)T.(1)T^) (4.30)
are now easily obtained from equations (4.24) and (4.27). The corre
sponding matrix form of these equations is
109
T U)
i
1
o
T1
T1
r
Ti
fH
61
T \2)
A1
=
0
+ Go
T2
0
T2
T (2)
1
f
OJ
i
I
o
1
T3
CO
1
0
T^3)
(4.31)
At this point we can now introduce the formal halfonshell definition
(4.32)
for the transition operator T.. which describes processes originating
in channel 1 and ending in channel i (i = 0, 1, 2 or 3). Using the
definition of V^', Eq. (4.32) may be written as
TilU)*! = I
J = 1
Vo(Z)
1
Tj (j)
(4.33)
which leads to the relation
LA = ~l + 2 6. .T.G T. $
U 1 U 0 1 j=1 lj J o jl l
(4.34)
This implies the set of coupled equations
3
+ E
j=l
(4.35)
which are known in the literature as the Alt, Grassberger and Sandhas
equations, or simply, as the AGS equations (A167). These operators are
not unique however, since one could always add a quantity to each T.
J 1
which does not contribute to the transition amplitudes in the onshell
limit (Lovelace [(Lo64a), (Lo64b)]). Another possibility would be to
110
choose a different partition of the various operators involved, allow
ing one to construct equations which put more emphasis on selective
channels, which have a greater physical consequence (Hahn and Watson (Ha71),
Kouri and Levin (Ko75) and Toboman (T074)).
We have now specified all the multichannel transition operators
for the given atomdiatom system. One striking aspect of the resulting
relations, as given by Eq. (4.35), is the similarity between the expres
sions for the various transition operators. In particular if one com
pares the breakup transition operator, T to the remaining operators,
Tjj (j = 1 to 3), one discovers the relation (Ko77b)
Toi (1+ TjVTji Vo'1 !436>
or more concisely
(4.37)
The second term on the right hand side of Eq. (4.36), ^Gg*, will not
contribute to the onshell scattering amplitudes, i.e.,
Mqi = <4>0(Z)T01(Z)$1(Z)> (4.38a)
= <i(Z)> (4.38b)
where
4*jC = Wj$0 (4.39)
In this last expression, is a wave function describing the inter
J
acting pair in channel j as a continuum state. The superscript "c" has
been introduced to distinguish 4>.c from $.Â¡ which describes the inter
acting pair in channel j as a bound state. What this means physically
Ill
is that one can describe the breakup process as one would a rearrange
ment process, but the interacting pair in the exit channel must be
assumed to lie in a continuum state.
3. The MultipleCollision Expansion
In practical applications, the exact treatment of multichannel
scattering processes via the coupled AGS equations for is generally
not possible. This is particularly true at high energies, where the
number of accessible states in each given arrangement channel becomes
quite large. Fortunately, the form of the AGS equations suggests that
one might be able to avoid these difficulties by considering their
iterated form. We will now discuss the implications of this approach
as it pertains to atomdiatom collisions.
If one now iterates Eq. (4.35), one obtains the series
Til = ^ilV1 + ^ ^i/jlTj +
J
+ 16. .6..6 .T.G T, + ...
jk ij jk kl 3 o k
(4.40)
The physical meaning of this series becomes clear if one keeps in mind
the nature of the operators involved. Consider for example the first
term in Eq. (4.40). From the discussion in the previous section it is
obvious that this term does not contribute to either inelastic or break
up processes. Therefore, it plays a role only in the case of rearrange
ment scattering. Since this term does not contain any interaction terms,
it is often called the "spectator stripping" term [(Mi72b), (Mi72c)].
The second term in the series contains only the operators T.Â¡, which are
twobody transition operators in a threebody space. Hence, these terms
112
describe processes in which the incoming atom interacts with only one
of the target atoms in the diatomic, while the third atom acts as a
spectator. The higher order terms in Eq. (4.40) would then correspond
to double and higher order encounters. The role of the {V in a"
these terms is quite important, for they exclude the possibility of
terms leading to disconnected diagrams and rule out terms which do not
contribute to the given process on physical grounds (Ch71). Because of
this conceptual picture, this series, Eq. (4.40), is known as a multiple
collision expansion [(Ch71), (Mi75b)].
Each term in the multiplecollision expansion has now been asso
ciated with a direct mechanism for an atomdiatom scattering event.
What one hopes is that only a limited number of these terms give a signi
ficant contribution to the scattering amplitude. The success of results
obtained by truncating the expansion would then be indicative of the
types of mechanisms involved in a given physical system. For reactive
scattering, there already exists ample evidence to corroborate the
existence of a stripping mechanism in certain atomdiatom reactions,
e.g., Minturn, Datz and Becker (Mi67) have shown that a classical
spectator stripping model properly describes the location of peaks in
the product distribution of various alkaliatomhalogenmolecule reactions
[Cs + Br^, K + (I^ or Br^)]. Within the context of the multiple colli
sion expansion,similar success was obtained by Yuan and Micha [(Yu76a),
(Yu76b)], who used the spectator stripping model to study the systems
[Ar+, (H^, D2, HD)] and [K + [l^, B^, IBr)]. It should be pointed out
that the expression for the transition amplitudes,
Mnr = <*ilG0'i(z,lii> .
1,
(4.41)
obtained within the stripping model is equivalent to the Qppenheimer
(0p28), and Brinkman, Kramers approximation (Br30),
113
i 1 il
> V $i>
i j i
(4.42)
which was introduced in the study of electron capture processes in
atomic physics (Mc70). The extension of these studies to include
multiplecollision effects has generally been pursued only within the
framework of classical mechanics [(Ba64), (Su68), (Ge69), (Ma74),
(Ma76)], but the success of these models has been impressive consider
ing their simplicity. We defer the study of single and higer order
terms to the next chapter, where a more detailed analysis will be
given.
CHAPTER V
THE SINGLE COLLISION APPROXIMATION
In this chapter we will investigate the single collision
approximation as it applies to inelastic and dissociative atom
diatom collisions. As can be seen from Eq. (4.35), the multiple
collision expansion for the transition operators of interest is
T11 = T2 + T3 + T2GoT3 + T3GoT2 +
(5.1)
for inelastic scattering, and
T01 V1 + T2 + T3 + T2GoT3 + T3GoT2 +
+ T.G T, + T.G T, +
1 o 2 1 o 3
(5.2)
for dissociative scattering. The single collision approximation to
the scattering amplitudes would then be
Ml'i = <*1'IT2 + T3IV (53)
and
M) .
01
= < <*o'T2 + T
IV
(5.4)
where the higher order terms in equations (5.1) and (5.2) have been
neglected. Note that the term GQ in Eq. (5.2) does not contribute to
the breakup amplitude in the onshell limit. In the following sections
we will discuss the practical implementation of equations (5.3) and
114
115
(5.4) as well as the factors which determine their validity.
1. Description of Channel States
For each arrangement channel i, i = 1, 2 or 3, we shall introduce
the set of Jacobi coordinates ft and r, along with the corresponding
conjugate momenta and p.. The momenta and p^ describe, respec
tively, the relative momentum and the diatomic relative momentum in
the given channel i. A definition of all these coordinates in terms
ordinary laboratory coordinates for the three atom system is given in
Appendix IV.
Working with the given relative coordinates in the center of mass
system, we can describe the free threeparticle motion in channel 0 by
the plane wave
= < R, Pi>< T, IPi> (55)
where + ^
< RiPi> = (2tt)3/2 e1 Pl Rl (5.6)
and
(2,)3/2 e1 Pi
4
(5.7)
Note that any one of the channel coordinates can be used to describe the
system, since
* Rj + Pj
j = 1,2,3), (5.8)
and thus
< RjrjlPJpj> = < RirilpiPi>
(5.9)
116
Equation (5.8) can be proved using the relations in Appendix IV. This
equivalence amongst the various channel coordinates can be stated
formally by the relations
= 6[Rj Rj (^ ) ] 6 [ rÂ¡ ^Â¡(R^)] (5.10)
and
< PjPjjPjPj^ = 6[Pj PjiPi^OlStPj Pj (Pi, Pi) ] (5.11)
where a quantity such as Rj(R.Â¡,r.Â¡) denotes that value of Rj obtained
from its relation to R^ and r., as given by Eq. (A4.4) in Appendix IV.
Alternately, one could derive these last two equations by making use of
equations (5.5) through (5.8) and the identity operators (Re77)
1 = / dRi dr  R^ r.Â¡>< Ri ri  (5.12)
and
1 = / d^ d^  ^ pjx Pi Pi  (5.13)
The free motion in the remaining channels can be described by
(Mi 75)
= Xj(Pi) 4>i (ni0Â¡mj) (i = 1, 2 or 3), (5.14)
where x.j(P.j) is a relative motion state and is a vibrotor state
characterized by the quantum numbers n., j. and m^.. In the coordinate
ii Ji
space representation one has
= ,
(5.15)
117
where
< RiIxi(Pi)> = < RiPi> = x (Pi.R)
and
< ri(J)i> =
Similarly, in the momentum representation one obtains
< Pj' Pi' I V = < Pi' Pi>< Pi' l4>i>
or
< Pj' Pj $i> = <5("Pj1 Pi )4i (ni, j'i, mj ; pil ) ,
where
5i (ni, ji; p ) = / dr. < l^x r.  1>
is a diatomic momentum distribution amplitude.
(5.16)
(5.17
(5.18)
(5.19)
(5.20)
2. Inelastic Scattering
The singlecollision inelastic scattering amplitude, was
given by Eq. (5.3). We will denote each term in this expression,
'< I "P j I ^ Mjij(j); and for the sake of specificity, we will
study the j = 3 term. Working in the momentum representation, we see
that
,(s)
(3) = / dQ3 dq3" dQ3 dq3 < ^ Q3 q^ >< Q3 q^ T3IQ3 q3> x
x1> .
(5.21)
113
By convention, P. and p^. will be used to denote the momentum charac
terizing an initial or final state $.j, and the variables and q.. will
be reserved for the intermediate momenta (Mi75b). Using the definition
of T3, Eq. (4.11), we see that
where t^ is a twobody tmatrix describing the interaction between
atoms A and B. is the channel reduced mass defined by Eq. (A4.8)
in Appendix IV. Note that the energy argument of t^ corresponds to
the total energy minus the kinetic energy of the spectator atom. From
Eq. (5.22) one can also ascertain the fundamental difference between
the threebody quantity and the twobody quantity t^ On the energy
plane the pole structure of t^ is replaced by branch points in the
structure of (Lo64a). Matrix elements such as < q31
evaluated via equations (5.11) through (5.19), yielding the result
< ^3 <^3^1> ~ ^^3 ^3^1 ^3^< ^1^1 ^3^I<+>i> *
(5.23)
(s)
Thus, the expression for M (3), Eq. (5.21), leads to
) (5.24)
where
(5.25b)
(5.25a)
(5.25c)
119
*1 V53^l' > V % A?1 (525d)
and APj = P^' P^ is the momentum transfer. As can be seen from Eq.
(5.24), M^j (3) has a very simple structure. All that is needed in
its evaluation is the product of the initial and final momentum distri
bution amplitudes, along with the twobody tmatrix. This simplicity,
however, is quite deceiving. The integrand in Eq. (5.24) actually has
a multicenter nature, as can be seen from equations (5.25a) through
(5.25d). In essence the problem we are facing is the familiar one deal
ing with the angular momentum analysis of an atomdiatom scattering
process. The usual approach taken in solving problems of this nature
involves the introduction of various partial wave expansions for each
of the quantities involved [(Ah65), (Ba69a), (El69), (Ha70), (0m64)].
However, this does not seem practical for high energies, which require
a large number of partial waves. Another possibility, would be that of
using a threedimensional quadrature scheme. This is particularly
attractive, since the integrand of Eq. (5.24) contains the product of
* .
the momentum wave functions < j' jq^ > and < q^{, which would limit
f
the range of integration over to some overlap region between the two
amplitudes (Mi75b). There is one major problem though, and it is the
idea of a threedimensional quadrature itself. Multidimensional
quadratures are generally somewhat unreliable [(D.a74), (St67)]. This
however, is an open question requiring further investigation. Finally,
there remains the possibility of introducing approximations into the
evaluation of (3). In this regard, we will now consider what is
known in the literature as the "peaking" approximation [(Ro67), (Mc70),
120
(Ek71), (Ek74)]. Within this approximation one factors the tmatrix,
t3, out of the integral sign in Eq. (5.24) yielding the expression
M
r;Pk)(3) t^k /d53<*1'h1'xl1l*1>
(5.26)
where tPk denotes that value of t, obtained for some suitably chosen
value of Q3, say QPk. The criteria for selecting QPk will be discussed
later in Section VI.I. The assumption one has introduced here is that
t3 is a smooth and slowly varying function of Q3 within the range of
integration determined by < 1 q^' > and < q^<}>j>, which on the other
hand, are highly oscillatory functions (Ek71). Physically, the peaking
approximation implies that we have factored the transition amplitude,
(s) Pk
Mj,' (3), into a term t3 which contains information on the dynamics of
the collision process, and a term
/ dQ3 < 4^ >< q114>1>
(5.27)
called a target "form factor," which contains information characterizing
the size and shape of the target, along with information on the initial
and final momentum distributions of the particles involved, i.e. the
restoring forces of the diatomic target. We see then that the role of
the potential for the target diatomic, in this case V3, is one of deter
mining the initial and final momentum distributions [(Ek71), (Mi75b)].
Consider now the validity of the single collision approximation.
We know that in general multiple collision expansions are applicable
only at high energies [(Ch71), (Ch73), (Mi75b)]. From a mathematical
point of view, the convergence properties of expansions such as (5.1)
and (5.2) can only be ascertained by detailed consideration of the
potentials involved. In practice, this turns out to be quite difficult;
and so, one must resort to computational trial and error. One can,
however, put forth various rules based on physical considerations which
give some indication of the validity of the multiple collision expansion
and in particular that of the single collision approximation. We note
that equations (5.1) and (5.2) are expansions in terms of the quantities
T. and Gq. Because of the definition of G0, (Z H0)_1, it is apparent
that these expansions should indeed exhibit better convergence properties
at higher energies. One cannot be definitive however, since the operators
{Tj} are present in each term. In general these operators would not be
expected to cause difficulty for most potentials of chemical interest
[(Ch71), (Mi75b)]. Since T.Â¡ contains V.Â¡ to all orders, it remains well
behaved even for singular potentials. As to the validity of the single
collision approximation, one can see from a classical mechanical picture
that such a collision process must be impulsive. In other words the
projectile must strike and leave the interaction region fast enough so
that the restoring forces of the target diatomic do not lead to sequen
tial collisions. This latter assertion implies that for the impulse
approximation to be valid, the projectile energy must be larger than
the internal excitation energy of the target. Dmsidering the expres
sions for T^, T21 and TQ1 obtained from Eq. (4.35), it follows that
T, i = + T~ G T, G T.. + G T G TQ1 +
11 2 2 o 1 o 11 2 o 3 o 31
T3 + T3 Go T1 Go T11 + T3 Go T2 So T21 '
(5.28)
Using the relation
122
Gi Go = Go Ti Go (529)
which follows from the definition of and T., and equations (4.7)
and (4.8), we see that Eq. (5.28) is equivalent to
T11 T2 + VG1 Go>Tll + VG3 Go>T31 +
T3 + "^"3(^1 ~ G0)Tn + "^3(^2 Go^21 (5.30)
a formal expression that implies the weaker the various potentials ,
the smaller the contribution from the corresponding multiple collision
terms. Other important considerations regarding multiple collisions
relate to the size and geometry of the target involved [(Meol), (Mi78)].
If dg and d^ are the diameters of the atoms in the diatomic; and Rg^,
the average value of r^; then, multiple collision terms are less impor
tant when
Pu Rbc 1 (5.31)
and
(dB + dc)/2 < RgC (5.32)
(Mi78). The relative masses of the atoms in the atomdiatomic system
are also very important. The likelihood of multiple collisions for
inelastic scattering diminishes if
mA mB
mc M
< 1
and
(5.33)
mA mC
mB M
< 1
(5.34)
123
[(Sg66), (Se69), (Ek71), (Ek74)], A further discussion of the impor
tance of the above mass factors will be taken up in chapter VI for the
case of col linear scattering.
We will now consider some kinematical rules which follow from
Eq. (5.24), i.e. the definition of (3). It has been pointed out
(Mi75) that the functions < ^2' I^ 1* > anc* < are eac^ 1ca^ized
within a given spherical region; implying that the degree of overlap is
(s)
a measure of the size of the transition amplitude (3). A good
quantitative estimate of the volume of each sphere can be obtained by
taking q I and q 'I equal to their classical values; and thus
3 1 l'max I31 1 max M
I3il
< ^1 Lax
[2m1W1ni] 1/2
(5.35a)
and
iV
< !q 11
'H1 'max
= [2m W nf] 1/2
11
(5.35b)
where W ni and W^nf are initial and final classical vibrational energy
of the diatomic BC. Based on this analysis a kinematical construction
can be made as illustrated in Fig. (51). A concise statement of the
overlap volume may be written as (Fe71)
V3) /dQ3 etlSj' llnaxq1']8[q1lmaxq1l] (5.36)
where q^' and q^ are given by equations (5.25c) and (5.25d). The func
tion 9(x) is known as the Heaviside step function and is defined as
(Cu75)
e(x)
1 x > 0
(
0 x < 0
(5.37)
Figure (51) Kinematical construction for in
elastic scattering.
125
The integral in Eq. (5.36) is a simple exercise in elementary geometry
and is thus very easy to evaluate (Se70). In Chapter VII we will see
that the overlap volume, m^, can in many instances predict the relative
shape of the energy loss spectrum for inelastic scattering. However,
from Eq. (5.24) or Eq. (5.26), it is apparent that one should really
worry about the relative phases and magnitude of < and
< q11^i> 1'n the overlap region. A simple extension of the present model
would be to use the semiclassical momentum distributions inside Eq.
(5.36), as recently proposed (Mi77). This would still not address the
problem of the phases, but it would lead to a quantity which may be
used to analyze the possible mechanisms in an atomdiatom collision.
Such a model is attractive, since it would provide a simple, inexpensive
way for the theoretician or experimentalist to gain insight about energy
transfer and reactive processes in atomdiatom collisions. In any case,
the analysis that led to Eq. (5.36) may be used to determine the region
(s) (s Pk)
of integration, if one were to evaluate M^(3) or (3) by a
multidimensional quadrature scheme.
3. Dissociative Scattering
Following the same procedure used in the previous section one can
show that for dissociative scattering Eq. (5.4) leads to
t3 (p3f q3i; Ep_ ^
where
3f
p3f
2m o
(5.39)
126
m n m
3i
= P +
li
AC
m.
nlAB mBC
If
m
AB
If
(5.40)
P3f = q3. APj (5.41)
and
?i = plf ir~ Afi <542>
i ir mbc
Equations (5.40) through (5.42) follow from the relations in Appendix
IV. One does not have an integral in Eq. (5.38) as was the case for
(s)
*^111(3) since the final state $ is characterized by a product of
plane waves as given by Eq. (5.5). Furthermore, Eq. (5.38) is simpler
than Eq. (5.24), since it contains only the halfonshell tmatrix.
This simplification however, does not help as much as one would think.
Experimentally, Eq. (5.38) would imply the need for a coincidence study,
one in which two of the exit particles are detected simultaneously
[(Sc74a), (Sh77)]. This is something that has not been done so far
[(Sc73), (Sc74b), (Sc75)], and thus one must average ]M^^ over p^
to obtain a proper comparison with experiment (Sc73). Because of energy
conservation, only an average over the orientation p would be needed.
We will address only the simpler problem of collinear scattering, and
will postpone further discussion on this matter to the next chapter.
In section IV.2 we presented an alternate expression for Mqj, Eq.
(4.38b). Within the single collision approximation this relation leads
to
(5.43)
127
The latter expression would lead to, e.g.,
H<0;C>(3> / d3 < *3 (V vE iK'v <544)
an expression analogous to Eq. (5.24), where however, <4>^,lq^> is a
continuum momentum wave function for the diatomic BC in channel 1.
The consequence of Eq. (5.43) will also be explored in the next chapter
in connection with coll inear scattering.
CHAPTER VI
COLLINEAR SCATTERING
In the previous chapter we proposed the application of the single
collision approximation to the study of inelastic and dissociative
atomdiatom scattering processes. We will now consider this approxima
tion for the case of coll inear scattering, which offers a simple and
inexpensive check on the validity and limitations of such an approach.
There is also the advantage of having an extensive literature from
which one can ascertain the relative merit of the present manybody
treatment when compared to other quantum mechanical, semiclassical or
classical treatments of the collinear scattering problem. An extensive
survey of collinear scattering may be found in the reviews by Shin (Sh76),
Rapp and Kassal (Ra69), and Takayanagi [(Ta63), (Ta65)].
From a physical standpoint it is hoped that a study of collinear
scattering will provide useful information about translationalvibra
tional energy transfer in atomdiatom collisions. The collinear config
uration is in general the most effective one for vibrational excitation
[(Sh76), (Ra69)]. However, one must be careful in generalizing the
results obtained from such a model. Collinear (1D) and full three
dimensional (3D) dynamical calculations differ in two very important
aspects (Be74b): the amount of phasespace available in 1D collisions
is less than that for 3D collisions, i.e. 1D scattering calculations
suffer from a "dimensionalitybias"; secondly, steric factors play a
more important role in 3Dcol 1isions. The importance of the role played
128
129
by either of these two factors will then determine the validity of 1D
scattering calculations. Because of these limitations in collinear
studies, we will pursue this study only as a test leading to an evalua
tion of the single collision approximation, the peaking approximation
and the importance of multiple collision terms.
1. Formulation of the Collinear Scattering Problem
For the sake of specificity, we will select channel 1 as our
initial arrangement channel, i.e. A + (B + C). At present, a harmonic
oscillator model will be used to represent the diatomic (BC). Later
in Section VI.4 where dissociation is considered, a Morse oscillator
will be introduced. The interaction potential, VÂ£, between the projec
tile atom and the ncnstruck atom will be neglected.
In order to simplify our analysis of the 1D scattering problem
and compare to other work in the literature, we shall adopt the follow
ing scaled Jacobi coordinates:
Xj = [(ir^k)1/2/?!]1/2 x:
and
Xj =[ oyc) 1/2/fi] 1/2 .
where
xx = xc xB
and
Xj = XA (VB + mCXC)/niBC
(6.1)
(6.2)
(6.3)
(6.4)
are the Jacobi coordinates defined in Appendix IV as they apply to 1D
scattering. In equations (6.1) and (6.2) we have introduced the diatomic
reduced mass = (rrigm^/rrig^) and the harmonic oscillator force constant k.
130
The particular form of equations (6.1) and (6.2) allow us to work in
energy units of hw,
w = [k/m^1^2
(6.5)
The resulting Schrodinger equation for the threebody system in the
centerofmass frame is (Ra69)
where
(6.7)
(6.8)
and Xj) is the interaction between the projectile A and the
struck atom B. M is the total mass of the system (m^ + mB + mc).
Note that and are not the same as before. They correspond to a
modification of the original potentials consistent with the new set of
coordinates and units. A pictorial representation of the coordinates
used is given in figures (6la) and (6lb). From Eq. (6.6) and Fig.
(6lb) it should be clear that we have transformed our former three
body problem into one where an oscillator of unit mass, vibrating about
an equilibrium position, is struck by a particle having an effective mass
m; or equivalently, into a problem where a harmomdic oscillator having a
particle of mass and one of unit mass, is struck by a particle of mass
m. We note, as pointed out by Secrest and Johnson (Se66), that the mass
parameter m corresponds to many different atomdiiatom problems and is
thus useful in studying the importance of mass effects in atomdiatom
collisions.
X1
i
Figure (61) Coordinates used in coll inear
scattering.
(a) Jacobi coordinates
(b) Scaled Jacobi coordinates
132
Using the physical picture given above, we can now state the
single collision approximation for inelastic scattering as
M'l = M' (3) = (69)
where we have used Eq. (5.3) and have dropped the term Mp (2), since
\Â¡2 is neglected in the given model. In the momentum representation,
Eq. (6.9) leads to
Mri = f <% *3 (q3 q3; E (610)
the analog of Eq. (5.24) obtained in the threedimensional case. The
various channel momentum coordinates present in Eq. (6.10) can be
obtained from Appendix IV, if one considers a three particle system
with mass m^ = m, mg = 1 and m^ = as illustrated in Fig. (6lb).
In channel 1, P^ and p, are the conjugate momenta to and Xp
respectively. The result for the coordinates q^, q^ q^ and q^ in
Eq. (6.10) is analogous to that given by equations (5.25a) through
(5.25d), but with the masses as given above. The peaking approximation
to Eq. (6.10) is
M
(s,Pk)
1 1
1APlxl[n>
(6.11)
A
where is the relative position operator for the BC pair, n and n'
are the initial and final vibrational quantum numbers characterizing
2 and p respectively. The twobody tmatrix t^9^ is factored out
from Eq. (6.10) at some value of Q^*. Once this choice is made q^'* and
q^ are obtained from equations (5.25a) through (5.25d). Noting that the
133
Gaussian envelopes of the harmonic oscillator functions and
in Eq. (6.10) peak when q^ and q^ equal zero, we have chosen
Q3* as (Ek74)
Q3* \ [Qjtqj1 0) + Q3(q3 0)1 (6.12)
which is equivalent to
Q3* = \ (P{ + Pj), (6.13)
as can be seen from equations (5.25c) and (5.25d). The choice of Q^*
is in general quite arbitrary as long as it remains within the overlap
region of the two momentum amplitudes. We have found M^, to be
quite insensitive to the choice of Q^*.
2. TwoBody tMatrix for the OneDimensional Scattering Problem
In the present treatment of coll inear scattering we shall always
maintain a close parallel to the 3D collision problem. Because of
this, we will deal only with symmetric potentials, i.e. those such that
V(x) = V(x). Furthermore, we will assume the presence of a hard core
in order to take into account the physical impenetrability of the
particles involved.
The analogue to Eq. (2.6), the differential equation satisfied
by the offshell wave function, is
[ + z = [z (2m)*1 q2] (6.14)
dx2 q
where
= (2tt)1/2 eiqx .
(6.15)
134
Introducing the definitions
0 if [sgn(q)sgn(x)] = 1
<
tt if [sgn (q) sgn (x) ] = 1
r e x
1^(0) = [cos(e)]1
and
f^qr) = <
one can shov; that
. 1 i
e1clx = I i1
1=0
cos(qr) if 1 =0
sin(qr) if 1 = 1
nj [cos(eqx)] f] (qr)
and
(6.16)
(6.17)
(6.18)
(6.19)
(6.20)
1/2 ^ 1
(jo(q, k ; x) e = (2tt) S i q,[cos(6qx)]w, (q,k ;r);
t H 1=0 1 1 L
where
(6.21)
2
[ + (aEkE2 v(r))] ^(q, kÂ£; r) = (a^2 q2)f](qr) (6.22)
and v(r) = 2mV(r). The function takes the place of the Legendre
A
polynomial and f^ that of the RiccatiBessel function found in
the partial wave expansion for the threedimensional offshell wave
function, (z)>. However, the expansions used in equations (6.19)
and (6.20) are not partial wave expansions; they are an artifact of the
way we have chosen to solve for the offshell wave function. Alternative
formulations of onedimensional Schrodinger equations in terms of
parity expansions or in terms of phase shifts could also be pursued
135
[(Li73), (Eb65), (Ka61)]. We have chosen the above formulation because
of its close parallel to the treatment found in Chapter II. In using
this approach however, one must give close consideration to the boundary
conditions placed on wj. If the potential does net contain an. infinitely
repulsive hard core, transmission from one side of the potential barrier
to the other must be taken into consideration. Since we have assumed
the presence of a hard core, must satisfy the
w, ~ 0 (6.23)
1 r^c
boundary condition at the hard core position rc and the asymptotic
boundary condition
f (qr) + cst eikr if E > 0
W1 <
r +
f^(qr) + cst eKr if E < 0 (6.24)
N
Note that when the asymptotic boundary of the full offshell wave
function o)(q, k^; x) is considered, equation (6.21) takes properly into
account the direction of the incoming v/ave.
The tmatrix may also be expanded in a form analogous to the
partial wave expansion of the threedimensional tmatrix. Making use
of the definition of t,
+O0
t(q1 q; E) = / dx V(x) (6.25)
00 '
and equations (6.15), (6.20) and (6.21) one obtains
t(q\ q; E) = I ^(cosG^) t^q', q; E) ,
(6.26)
where
(6.27)
For a symmetric hard core potential it is easy to show that
a)(q, kE; x) = (2tr)"1/2 [eiqx eiqxc eikErc eikEr] (6.28)
and
 (q kE)(q' + kÂ£) e lAqrC] ,
(6.29)
where Aq = (q1 q). Although equations (6.28) and (6.29) were obtained
from the equations outlined in this section, one would arrive at
identical results by considering a potential of the form V(x)=A0(rcr)
and taking the limit as A approaches infinity. For more complicated
potentials, such as an exponential potential including a hard core, one
must generally go to a numerical solution for co^ in Eq. (6.22). The
procedure we have chosen is completely analogous to the comparison
potential method outlined in Chapter II and therefore will not be
repeated.
3. Results for Inelastic Scattering
The quantity of interest in coll inear scattering is the transition
probability P... for going from an initial state i to a final state f.
In the case of inelastic scattering we have the well known result
[(Th68), (Ek71)]
137
P 1 = "D2 ,
n'n P1P 1 1 1 11' 1 1
1 1
(6.30)
which is approximated by
2 2
3 4tt m
n'n = P1'P1
M(^I2
'lT
(6.31)
within the single collision approximation.
We will now consider various results obtained for a collinear
problem where the interaction between the projectile atom and its colli
sion partner is a hard core. These results are analogous to those found
in references (Ek71), (Cl70), and (Se66); our objective, however, is to
show the degree of discrepancy of the full single collision approxima
tion, and the peaking approximation within the framework of the multiple
collision expansion of the Faddeev equations. In figures (62a) through
(62c) we have plotted the transition probability P,n versus the kinetic
energy Ej of the projectile (Ej
 Pli
2M
10
) for the case m = 0.125. The
1
exact results obtained by using the coupled channels method of Shuler
and Zwanzig (Sh60) are given in Fig. (62a). The single collision non
peaking results, or those obtained from Eq. (6.10), are given in Fig.
(62b); and the peaking results, or those obtained from Eq. (6.11) are
given in Fig. (62c). A summary of the various integrals and analytical
quantities needed in these calculations is given in Appendix V. We note
that the nonpeaking results are closer to the exact results than the
peaking ones; however, the discrepancy is not large, and the greater
computational simplicity of the latter approximation makes it particu
larly attractive. Figure (62d) gives nonpeaking results that have
Figure (6
Plot of Pjq versus the kinetic energy of the projectile
in units of hto
(a) Exact Results
(b) Single Collision Results
(c) Results From Peaking Approximation
(d) Renormalized Single Collision Results
(m = 0.125, Vj = Harmonic Oscillator Potential,
Vg = Hard Core Interaction).
f 5r'V"v5> f **
***'*'* y?
>.
i
I
CO
CO
140
been renormalized according to the relation
Vn'Pn'n 'JTn'n1
(6.32)
as was suggested in reference (Cl70). If the computation of the tran
sition probabilities were exact, then the expression in the denominator
of Eq. (6.32) would be unity, because of unitarity. This procedure
brings the single collision results into closer agreement with the
exact results; however, it does break down near the threshold energies
for vibrational excitation because of the presence of singularities.
The exact reason for the presence of these singularities in the single
collision approximation is not clear. Their presence has been noted by
other authors [(Cl70), (Ek71)]. By having considered the behaviour of
Mri in near thresho1ci it was ascertained that the singu
larity could not arise there but from the 1/P^' term multiplying
IM^112 1n This would indicate that the single collision
approximation to is not good enough, and multiple collision terms
are needed. On physical grounds this is quite reasonable, since near
threshold the incoming projectile barely has enough energy to excite
the diatomic; and thus, the likelihood of multiple collisions becomes
quite high. The importance of these sequential collisions or chattering
in collinear scattering has been verified in classical trajectory cal
culations [(Se69), (Ke72)]. Analogous results to those discussed above
for Pjq are given for P^q in figures (63a) through (63d). We see
that the comparison of the single collision results with the exact
results is more favorable in this case. In Table (61) we list exact,
single collision and peaking results for P^g versus using m = 0.125
and m = 0.5. These results show that for larger values of m, the single
Figure (63) Plot of ?2q versus the kinetic energy of the projectile
in units of hw
(a) Exact Results
(b) Single Collision Results
(c) Results From Peaking Approximation
(d) Renormalized Single Collision Results
(m = 0.125, V} = Harmonic Oscillator Potential,
= Hard Core Interaction).
E? (Kinetic Energy)
14?
Table (61) Comparison of exact and approximate results for P
10
Ej
(Units of hw)
P10(m = 0.125)
PjQ(m 0.5)
Peaking
Single
Collision
Exact
Peaking
Single
Col 1ision
Exact
1.001
18.60
9.21
0.004
65.162
38.955
0.010
1.2
0.856
0.614
0.201
1.845
1.455
0.435
1.5
0.757
0.609
0.349
0.989
0.848
0.715
1.8
0.720
0.611
0.478
0.586
0.521
0.912
2.0
0.698
0.606
0.554
0.417
0.377
0.987
2.2
0.676
0.597
0.514
0.297
0.271
0.674
2.5
O.f 40
0.577
0.499
0.178
0.165
0.382
2.8
0.603
0.550
0.464
0.106
0.099
0.121
3.0
0.577
0.530
0.434
0.075
0.071
0.007
3.2
0.550
0.509
0.439
0.053
0.050
0.001
3.5
0.511
0.476
0.424
0.031
0.030
0.049
3.8
0.471
0.442
0.410
0.018
0.017
0.127
4.0
0.446
0.420
0.400
0.013
0.012
0.126
4.2
0.421
0.398
0.378
0.009
0.009
0.112
4.5
0.385
0.365
0.351
0.005
0.005
0.089
S*
CO
144
collision and peaking approximations break down. Secrest (Se69) has
shown that as m becomes larger multiple collisions become very important;
and thus, one would expect the single collision approximation to yield
incorrect results.
Other calculations using a potential of the type
00 if iXI C rc
x > r
(6.33)
c
were also carried out. In order to verify the reliability of the com
parison potential procedure used in evaluating the tmatrix, calcula
tions were performed using large values of A and a in Eq. (6.33) and
a comparison was made with the corresponding hard core results. Table
(62) gives one such study for P^g using the peaking approximation.
The parameters used in this set of calculations were m = 0.5, A = 20.0
and rc = 2.0, where the units are those compatible with the scaled
Jacobi coordinates given by equations (6.1) and (6.2). We note that
the exponential potential results approach the hard core values as a
increases in magnitude. Another important test of the single collision
approximation is its applicability to softer potentials. In the litera
ture there exists an extensive study of col linear scattering using a
soft exponential potential [(Sc75a), (C170), (An69), (Se66)]. Unfor
tunately the potential used is not that given in (6.33), but one that
is exponential throughout space, i.e. VgiXg) = &Â£'aX3 for all values of
Xg. Therefore, it is not at all clear that one should compare the
results using Eq. (6.33) with those reported in the literature. The
results obtained with Eq. (6.33) using small a d indicate that the
145
Table (62) Convergence study showing how the transition probability
P^q for an exponential potential approaches that for a
hard core potential as a is increased in magnitude
(m = 0.5, A = 20.0 and rc = 2.0 in reduced units).
(Units of hu>)
(Hard Core)
(a=4.0)
(cel.5)
(ce0.5)
1.1
2.580
2.745
4.007
16.661
1.3
1.458
1.515
2.046
7.136
1.5
0.989
1.015
1.299
3.661
1.8
0.586
0.594
0.703
1.385
2.0
0.417
0.420
0.475
0.739
2.3
0.250
0.251
0.265
0.288
2.5
0.178
0.177
0.180
0.154
single collision approximation is not valid in the same range of
energies that were studied using a hard core potential, i.e. for
projectile energies less than 10 Tiu units. Typical results obtained
are given in Table (63), where the parameters used were m = 1/13,
a = 0.1287, rc = 2.0, A = 10.0 and = 5.918 Tico units. Note that for
the hard core results shown in figures (6.1) and (6.2) the transition
probabilities obtained from the single collision approximation were
quite reasonable for projectile energies one quanta beyond threshold.
This is not the case for the transition probabilities quoted in Table
(6.3), unitarity is violated even at several quanta above threshold.
Another interesting point is the discrepancy between the peaking and
nonpeaking results quoted in Table (63), indicating that the peaking
approximation should not be used for such soft potentials. Similar
cautions on the validity of the peaking approximation have also been
noted elsewhere in the literature (Co68). The reason for the poor
quality of the single collision results in this last case might lie in
the fact that for very soft potentials, the distance over which the
potential falls to zero is large and therefore the concept of a local
ized collision becomes blurred and the assumption of an impulsive
encounter would not be valid [(Se66), (Se69)].
4. Dissociative Collinear Calculations
In the previous section a harmonic oscillator was used to model
the behaviour of the diatom in an atomdiatom collision. When this
potential is replaced by a more realistic one such as the Morse poten
tial, one can allow for the possibility of collision induced dissocia
tion (CID), i.e. those processes where
147
Table (63)
Transition probabilities
using the parameters m =
A = 10 and = 5.928 in
for a soft exponential potential,
1/13, a = 0.1287, rc = 2.0,
reduced units.
p10
P20
P30
P40
Single Collision Approximation
0.893
1.299
1.317
Peaking Approximation
2.174
4.063
2.81
0.878
3.107
148
A + (BC) A + B + C .
The presence of three unbound particles in the exit channel is what
makes CID very difficult to study from a theoretical point of view.
As was pointed out in Chapter I, because of the nature of the exit
channel, coupled channel procedures lead to a continuous infinite set
of coupled integrodifferential equations [(Wo75), (Kn77)], and would
thus be very difficult to solve. The approach we shall take, suggested
in Section V.3 with the single collision approximation, offers a
much more tractable procedure. Specifically, the approximations we
shall use are
01 = V^f ^3i EP3f> liln> <6'34>
and
M0lCPk) = ^ / dQ3 <$1C'q1'> (6.35)
the one dimensional analogs of equations (5.33) and (5.44). Clearly
the amount of work necessary within this latter approach is comparable
in magnitude to the amount that was spent on the inelastic problem, and
this is certainly not true of the coupled channel procedure [(Kn77),
(Mi75), (Sh77), (Wo75)].
Using a Morse potential,
V^x^ = D[ 1 e"axl]2, (6.36)
for the target diatomic does not affect the validity of the transforma
tions used in Section VI.I to obtain Eq. (6.6), the Schrbdinger equation
for a particle of mass m hitting an oscillator of unit mass vibrating
about an equilibrium position [(Se66), (Sc75a), ((Ek72)]. Energies are
149
still measured in units of tiu, but oj in this case is
a) = a^D/r^)172, (6.37)
implying that the potential parameters a and D are related by
a = (2D)'1/2 (6.38)
The Morse vibrational and continuum functions and jy q^'>
present in equations (6.34) and (6.35) are well known analytically
[(Ek73), (En69), (FI71), (St35)] and there is no problem in their
evaluation. The same is true for the form factor
F01 = / dQ3 y lq1,> (6.39)
present in the expression for f,i(s,c,Pk) (^k73), and the analogous one
for inelastic scattering (Ek72). A summary of all these analytical
expressions may be found in Appendix V.
The construction of the transition probabilities from must be
handled with care. The relation
Pn(AEl) [cPj' Cj' TuP1n>2 (6.40)
analogous to the expression for P , Eq. (6.30), is actually a differ
ential energy transfer probability. In Eq. (6.40) <Â£^1 characterizes
the continuum state, Jy , of the diatomic, and Ae = e^' en. For
inelastic scattering the differential energy transfer probability would
be (Ek72)
P (Ae) e P ( ^t f1
nv n n v dn '
(6.41)
150
where Ac = e t en The unitarity relation when dissociation is
allowed would be (Le69)
N
Z
n'=0
P ,
n n
Pn(Ae)dAe = 1 ,
(6.42)
where the integral is over the energy range allowed by energy conserva
tion, which in this case is from the threshold of dissociation
(Dn = D en) for the Morse oscillator to the kinetic energy, avail
able from the projectile. If Eq. (6.34) is used to obtain Pn(Ae), one
also has the problem that the exit channel is characterized by
where
= (Pj Px) (6.43)
whereas
<Â£j' e> = 6(e e) (6.44)
The transformation between the two normalizations is given by the rela
tion
6(ei e) = [6(p Pj) + 6(Pl' + P1)]p(p1) (6.45)
where
3Â£i i i Pi i
p(pl) = (6.46)
1 oPj m^
is the density of states (Le69). This implies that
pn(as) = vri l^ll + lToilpin>l2 <647>
where one finds that the particles of the diatomic are being scattered
151
to the left and right. An analogous result is obtained from Eq. (6.40),
since the Morse continuum function has two components which
asymptotically correspond to plane waves moving to the left and right.
This result is a direct consequence of the boundary conditions and the
requirement that be a bounded function (St35).
(s) (s,c,Pk)
The calculations using the expressions for Mq^; and M^ have
led to rather unexpected results. When the differential energy transfer
probability from the ground state was plotted versus Ac for the system
H + D2 (m = 0.2, n = 0), Mq^ led to a discontinuity at the threshold
( c n p 1/ ^
for dissociation, whereas did not. This may be seen by com
paring figures (64) and (65). The interaction potential was taken
to be a hard core and the value 12.28 in units of hw was used for the
Morse parameter D in Eq. (6.38). Eckelt and Korsch (Ek73) have reported
a similar study on this system, using a theory which leads to an expres
f s c P k)
sion analogous to that for M^ but with loss of time reversal
(c\
invariance. On the other hand the plane wave approximation M^',
Eq. (6.34), has been applied only to the threedimensional scattering
problem (Sh77), but not to the collinear problem. Because of the
discrepancy obtained in the present study between these two approxima
tions, the difference between them must be clearly understood. We note
that it is important to investigate that region of the energy loss
spectrum encompassing high vibrational excitation and transitions to
the continuum. This was not done in the threedimensional study men
tioned above. If one returns to the formal expressions [equations (5.1)
and (5.2)3 used in deriving (6.34) and (6.35), we see that they differ
as follows:
M01^ <$o!T2 + 13^l>
(6.48)
Figure (64) Plot of the differential energy transfer probability
(s')
Pq(Ae) versus Ac as obtained from The stick
spectra correspond to inelastic scattering and the
continuous curve denotes transitions to the continuum.
The continuous part of the spectrum is scaled by a
factor of 1/2.
T
S.65 x 102
1 1
H + D2
7.72 x102
m = 0.2
nÂ¡=0,E' = 20.0
579 x 102
P
1
3.86 x10'2
1.93 x10"2
0
j L
1.00 3.40
780
T
12.20
1560
21.00
vJT
CO
Figure (65) Plot of the differential energy transfer probability
Pq(Ae) versus Ae as obtained from jhe stick
spectra corresponds to inelastic scattering and the
continuous curve denotes transitions to the continuum.
9.65 x 12
7.72x10'2
, 1 r
H + D2
m = 0.2
n,= 0 E= 20.0
H
T
5.79 x10'2
P
3.86 x 1CT'
1.93 x 10'2
ol _
 1.00 3.40
780
T
1220
16.60
21.00
CD
156
and
M0lC> = <40llVIV
 <^qIT2 + "^3 + + 1^>0> (6.49)
in other words they differ by the presence of the double collision terms
(s,c) (s)
TjG0T2 and T1^oT3 In t*ie c111'near case Moi differs from Mqj by
the term T^G^. Physically, this means that it is important to con
sider the interaction between the two atoms of the dissociating diatomic.
(s)
The rather drastic difference between the results obtained from Mq^
(s c)
and Mqj in the coll inear case may on the other hand not extend to
the threedimensional results, since the "dimensionality bias" may play
an important role in exaggerating the importance of the double collision
terms present in Eq. (6.38). Nevertheless, this question must be addressed
in the study of the full threedimensional CID studies. Another aspect
of the present calculations must be emphasized. The formalism we are
using is based on the Faddeev equations and differs from that used by
Eckelt, Korsch and Philipp (Ek74), who used the formalism of Bianchi
and Fabella (Bi64). In this latter approach, these authors have intro
duced a further approximation leading to what is known as the impulse
approximation in the literature [(Mc70), (Ro67)]. Within the present
(s c P k)
approach, this approximation would amount to replacing by
M
(s,c,I)
01
*Pk, * .
c3 P3f q3i
Eq.) / dQ3 qj' >,
(6.50)
where t^ (p*
r
3f q3i; h
q3
*2
2M3
) in Eq. (6.35) has been replaced by the
halfonshell tmatrix element t^ (P3f> q3l; Eq *). The peaking argu
merits p^, q^ and are determined by the procedure given in Section
VI.1. Furthermore, Eckelt and Korsch (Ek71) also used the tmatrix
for the potential
157
V
3
0 x > xc
(6.51)
instead of the symmetric hard core tmatrix, Eq. (6.29). The resulting
tmatrix from Eq. (6.51) leads to an unsymmetric tmatrix. In order to
make the results obey the principle of detailed balance, the above
(6.52)
This is not necessary within the Faddeev framework, since each term in
the multiple collision expansion automatically satisfies time reversi
bility (Mi72b). We have used expression (6.52) as it applies to the
unsymmetrical hard core and were able to reproduce all the results
quoted by Eckelt and Korsch (Ek73). This provided a check for the com
puter programs that were developed to do the present study. In compar
ing the calculations obtained from f/(s,c,Pk) ancj ^ose from M(s,c,I)
using equations (6.51) and (6.52), we found that the latter yields
results that were nearly unitary, whereas the singlecollision results
were always lower in magnitude but with the same qualitative behaviour.
The sole reason for this was that different tmatrix expressions were
used in the two calculations. Hence, we feel that this indicates that
one should be wary in attaching too much significance to how well
results obtained with approximations such as (6.34), (6.35) and (6.50)
obey the uni tarity relation Eq. (6.42). One interesting feature of
these calculations as seen by comparing figures (65), (66) and (67),
Figure (66) Plot of the differential energy transfer probability
(Ae) versus Ac as obtained from M^>c,Pk). The stick
spectra corresponds to inelastic scattering and the
continuous curve denotes transitions to the continuum.
3 3 9 'i 5 21
A
LT1
JD
Figure (67) Plot of the differential energy transfer probability
Pjq(Ac) versus Ac as obtained from m(s,c,Pk). y^e
spectra corresponds to inelastic scattering and the con
tinuous curve denotes transitions to the continuum.
9.9 0.5 8.9 18.3 277 371
AG
162
which give plots of P^, P,, and P.q versus Ae. is that the envelope
always reflects the initial momentum distribution of the target diatomic.
This supports the idea discussed in Section V.2 of the information
obtainable from either the simple overlap volume weighed by the semi
classical momentum distribution, or from the form factor FQ1 present
in the expression for M^' Figure (66) is also interesting in
that it is an example of how the present approach can be used to study
scattering of excited targets, something which is quite difficult to
do with other coupled channel procedures.
CHAPTER VII
THREEDIMENSIONAL INELASTIC SCATTERING RESULTS
In this chapter, we will apply the single collision peaking
approximation to inelastic atomdiatom scattering processes for the
case of full threedimensional motion. Specifically, we will investi
gate the systems (Li + CO) and (Li+, N2), and discuss various numerical
results in light of recent experiments [(Bo76), (Ea78)]. This investi
gation will serve to complement other numerical results found in the
literature [(Bo74), (Ek74), (Ph76)], obtained using the impulse approx
imation [see Section V.4 ].
Within the present approach, we will approximate the inelastic
scattering amplitude by [see Section V.2 ]
M(s,Pk) Pk (2) Pk p(3)
Ml'l l2 Fl'l + t3 Fl'l
(7.1)
where the target formfactors F^j and f? are defined by Eq. (5.27).
The twobody tmatrix elements t2^ and tgrK are offtheenergy shell,
and their arguments are chosen by a procedure analogous to that used in
the collinear case in Section VI.6 One of the goals of this chapter
will be to illustrate how the target formfactors play a major role in
characterizing the energy loss spectra in atomdiatom scattering
processes.
Pk
1. Practical Implementation of the Peaking Approximation
The differential cross section for inelastic scattering processes
163
164
is formally related to the scattering amplitude by the relation (Le69)
da
dft
(l'l)
(2zL\4m 2 J
[ h > nl P,
1' 11
(7.2)
Here, we continue to work with the Jacobi coordinates introduced in
Chapter V. Since only inelastic processes will be considered, we have
chosen channel 1 as the entrance and exit arrangement channel. Intro
ducing the peaking approximation as given by Eg. (7.1), we see that
4 P,'
m.
(1^1) = r*L) m2 11 itPk F(2) (_Â£_ \ + tPk f(3) 1B_ + ,,2
dnu 1; {b> Pj 1*2 P1' 1 ^mBC ^rl; + t3 Fri v mBC APi)! ,
(7.3)
where
.mC
F'l (^BC AFl) =
APl* rÂ¡ I niJ imj,>
(7.4)
r
and r^ is the position operator for the relative coordinate of the
diatom (BC). Note that the initial and final states of the target
diatom are characterized by the quantum numbers {n^, jp m^}, with a
prime denoting the final exit state. To simplify our notation, we will
drop the "1" subscript in the above labels and the "j" subscript in the
magnetic quantum number itk.
In the experiments we shall discuss in the next section, the mag
netic quantum numbers are not analyzed. Therefore, the differential
cross section expression given by Eq. (7.3) will be replaced by
^ (n1 j'n,j;0+iÂ£ ) = (^)4
PlPl
^ Pj(2j+1) m'm
tPk f'(2)I + tPk f(3)i2
2 1"1 3 l'l1 (75)
where we have averaged over the initial magnetic quantum numbers and
165
and summed over the final magnetic quantum numbers.
Introducing the vibrator states
= u^rj) Y^irj) (7.6)
and the expansion (Me61)
y y
lYAPlrl = 4tt E iA Y*y (APj) YAu(?1)jx(YAP1r1) (7.7)
Ay
into Eq. (7.4), one can show that (Ph76)
f!?} = 4tt E iX C(.r!. f(;?, .. Y* (AP ) ,
1 1 X\i J m'jm n'j'njA Ay 1
(7.8)
where we have defined the radial formfactor
f^l, .. = / drir.2 u*,.,(r1)u (r,) j, (Isp API r,)
n'j'njA J 1 1 n'j' V nj 1 AMmBC 1
(7.9)
and the coefficient
r(Ay)
j'm'jm
(7.10)
The integral in Eq. (7.10) is well known (Ed74) and leads to the relation
cXv)
j V jm
(l)m' (47r)1/2[(2j *+l) (2X+1) (2j+l)]1/2 I
j' A j
0 0 0
(7.11)
(2)
The above expression for F [Eq. (7.8)] and the corresponding equation
(3) 1 1
for FJ, lead to considerable simplification of Eq. (7.5). The alge
braic manipulations are quite straightforward, although tedious. Since
they are given in the literature (Ph76), we will simply state the result
166
x
(
Because of the symmetry properties of the 3j coefficients in Eq. (7.11),
the sum in Eq. (7.12) is restricted to those values of A for which the
relations
(7.13)
j' j  Â£ A Â£ j + j
and
(7.14)
j' + j + A = even
are satisfied. For homonuclear diatoms, the (1)J+J'phase factor in
Eq. (7.12) implies that odd Aj transitions are symmetry forbidden
[(Ek74), (Bo76)].
The differential cross section can now be obtained by the evaluation
of the radial form factors defined by Eq. (7.9) and of the offshell
twobody tmatrix elements. In the present investigation, we will
assume the radial functions unj to be harmonic oscillator wave function
with modified frequency arguments allowing for rotational distortion
[(Pa35), (Yu76a)]. Adopting the treatment given by Pauling and Wilson
(Pa35), the resulting expression for the radial wave function is
(7.15)
1/2 e[C/xnj]2
Hn(4/xnj), (7.16)
where
(7.19)
? 6j(j+1)Bgh 1/2
a) = [< +
cm,r 2
1 e
and
nj
re[l +
j(j+l)Beh
3j(j+l)Beh +^cm1re2(ue)2
]
(7.20)
The function Hn in Eq. (7.16) is the nth Hermite polynomial. Anharmonic
effects could be incorporated into Eq. (7.18) by replacing we by an
effective vibrational constant (Yu76a)
o)n = we 03exe(n + 1/2) + toeye(n + 1/2)2 .
(7.21)
The corresponding vibrationalrotational eigenenergies are
(he)"1 Wnj = )n(n + 1/2) + Bej(j + 1) De[j(j + l)]2 (7.22)
where the equilibrium rotational constant Be, the centrifugal distortion
De and are all in units of cm"1 (Le70).
It is clear that the presence of the Gaussian term in Eq. (7.16)
limits the range of integration used in evaluating the radial form
factors. Therefore, the numerical evaluation of Eq. (7.9) is rather
straightforward and no complications occur. We have used a 32point
Gauss integration procedure as suggested by Philipp and coworkers (Ph76).
The use of a higher order quadrature led to no significant improvement
168
for the present case in which only lower vibrational levels are con
sidered.
Finally, one should note that the evaluation of the target form
factor or the differential cross section requires the evaluation of 3j
symbols to very high j values, when a large amount of rotational excita
tion is involved. For the systems we shall study in the following
section, j values up to sixty will be considered. Often the straight
forward use of recursion relations found in standard texts [(Ed74),
(Mi61)] will lead to computational problems. To avoid this difficulty,
we have adopted the algorithms given by Schulten and Gordon (Sc75c)
which are stable to very high j values (j>100).
2. Numerical Results
The results of various calculations on the systems (Li+, CO) and
(Li+, N2) will now be presented. Within the present manybody approach,
these systems are convenient because the mass ratios are small and
repulsive forces are dominant in the collision energies of interest.
Using equations (5.33) and (5.34), one obtains the mass ratios 0.15
and 0.27 for the (Li+, CO) system, and the mass ratio 0.2 for the
(Li+, Ng) system. With these mass ratios, the single collision approx
imation should be applicable, since the ratios are less than 0.5 [see
Section VI.3 ]. The potential energy hypersurfaces have been recently
calculated by Staemmler [(St75), (St76)]. Using the information found
in these references, Micha et al. (Mi78b) have extracted twobody poten
tial data corresponding to the collinear configuration of each of the
above systems. For the interaction between the projectile and each of
the target atoms, they used a potential of the form
169
V(r) = Aebr (7.23)
and obtained the parameters A = 1908.51 eV and b = 5.949 A for (Li+C),
A = 5237.46 eV and b = 4.8578 A"1 for (Li+0), and the parameters
A = 3079.77 eV and b = 5.2582 for (Li+N). Using such a simple
potential does not properly describe the potential energy surface at
low energies, where there is an attractive component. However, the
experiments we shall compare to are at high energies (4 to 8eV) and the
scattering angles involved are beyond the rainbow angle (Bo76). Because
the above parameters lead to very repulsive potentials, we will simply
use a hard core interaction potential for each atomatom pair. The
data we have used for each of the two systems are given in Table (71).
We shall restrict the present investigation to the study of rota
tional excitation from the ground vibrational level. This, however, is
by no means a limitation of the present approach. The experiments to
which we shall compare our calculations are time of flight studies on
the scattering of Li+ ions from ground state and CO molecules (Bo76).
In particular, we shall compare our calculated differential cross sec
tions [in arbitrary units] as a function of final rotational quantum
number jf versus those obtained from experiment Typical results ob
tained for the (Li\ CO) system are shown in figures (71), (72) and
(73). These figures correspond to a center of mass collision energy
of 4.23 eV and center of mass scattering angles fi 37.1, 43.2 and
49.2, respectively. Initially, the target diatom is assumed to be
in its ground vibrational and rotational level. We note that as the
scattering angle is increased the overall envelcge of the plotted dis
tributions is broadened. The same effect v/as noted in the experimental
170
Table (71) Parameters used in peaking calculations
[(He50), (Mi 78b)]
Spectroscopic
C0[X1Z+]
N2[XlSg]
Constants
we (cm1)
2170.21
2359.62
Be (cm1)
1.9313
2.010
re (X)
1.1281 A
1.094 X
0
AtomAtom
Pair
re(A)
(Li+ 
C)
1.17
(Li+ 
0)
1.05
(Li+ 
N)
1.05
Figure (71) Plot of differential cross section (arbitrary units)
versus final rotational quantum number for the (Li+, CO)
system. [(ni,ji) = (0,0), nf=0, E^^SeV and ej, Â£ =37.1]
o 12 24 36 48 60
j,
I
rx>
Figure (7
2) Plot of differential cross section (arbitrary units)
versus final rotational quantum number for the (Li+, CO)
system. [ (nj ,j.Â¡ )=(0,0), nf=0, E]=4.23eV and 0p. p =43.2]
1.0
T
075
Li* CO
0.50
0.25
0
0
T
1
36
P*
Figure (73) Plot of differential cross section (arbitrary units)
versus final rotational quantum number for the (Li+, CO)
system. [ (nÂ¡, j )(0,0), nf=0, E1=4.23eV and 0p, p =49.2]'
1.0
Li4, CO
075 
050
025 
0 
0
II
i
12
24
60
^4
CT
177
results (B676). In each of these calculations, the peak in the distri
bution envelope fell at a jfvalue that was higher than the one observed
experimentally. In Fig. (71) the peak occurs at jf = 29, whereas the
peak in the corresponding experimental curve falls at jf = 20. A
similar result was obtained by Micha et al. (Mi78b) in their spacetime
correlation function approach. In their studies, they also assume the
single collision approximation. As a possible reason for this discrep
ancy between theory and experiment, they suggested that the lockin
effect of neglected longrange anisotropy would lead to an apparent
increase in the moment of inertia of the target diatomic. Upon treat
ing the moment of inertia as a parameter and increasing its value, they
were able to obtain agreement with experiment. In order to compare the
above results to those obtained using a homonuclear target, we have
plotted in Fig. (74) the differential cross section versus final rota
tional quantum number of the (Li+, ^) system, using the same energy
and scattering angle data used to obtain Fig. (71). We note that the
structure obtained for the (Li+, I^) system is awch simpler than that
for the (Li+, CO) system. Part of this simplicity is due to the fact
that NÂ£ is a homonuclear diatomic and odd Aj transitions are symmetry
forbidden. The interference pattern in the (Li*,, N2) results is also
less pronounced than that in the (Li + CO) resulits.
The experimental data to which we have compared the above results
(Bo76), do not indicate an interference structure. This, however, is
due to lack of resolution. Other experiments on (Li+, CO) have recently
been done which show various maxima in the energy loss spectra for rota
tional excitation (Ea78). In Table (72), we conpare the probabilities
of rotational excitation obtained from the analysis of experimental
Figure (74) Plot of differential cross section (arbitrary units)
versus final rotational quantum number for the (Li+, N?)
system.[(ni,ji)=(0,0), nf=0, Ei=4.23eV and eÂ£, + =37.1T
pri
1.0
T
T
0.75
050
025
0
0
12
24
T
36
48
60
180
Table (72) Comparison of probabilities of
rotational excitation for the
(Li+, CO) system (n.. ,j.Â¡ ) = (0,0),
n^=0, E^4.28 eV and 6^p^=10
Column I gives theoretical results
and Column II the ones obtained
from experiment (Ea78).
P
jf
I
II
0
0.018
0.058
1
0.011
0.100
2
0.002
0.100
3
0.027
0.040
4
0.045
0.0
5
0.009
0.0
6
0.228
0.0
7
0.143
0.0
8
0.082
0.0
9
0.231
0.0
10
0.056
0.0
11
0.102
0.006
12
0.028
0.059
13
0.015
0.096
14
0.004
0.074
181
results (column II) to those obtained theoretically (column I). The
center of mass collision energy and scattering angle are 4.28 eV and
10 [8 in the laboratory frame]. The rainbow angle is at 5.5 in the
laboratory frame. The theoretical probabilities were obtained from
the relation
P = [
Ld fi
(n\ j 'm, j; 0
P'lPr
m,
da
dft
(n1, j'<n, j; 0
pipi)1
1
(7.24)
As noted in the results mentioned earlier iri this section, the peak
in the theoretical probabilities falls at a higher value of jf than
the peak found in the experimental results [see Table (72)]. Because
of this and the general simplicity of the present model, one cannot at
this time state whether the maxima are due to effects coming from the
potential or from dynamical effects not contained in Eq. (7.12). One
should also note that the scattering angle is not much larger than the
rainbow angle, and therefore the experimental data quoted in Table (72)
are not the most favorable to our model.
The results we have been discussing have all been obtained using
the single collision peaking approximation as it arises in the Faddeev
formalism. The use of this formalism has led to expressions involving
offshell tmatrix elements. On the other hand, the impulse approxima
tion [see Section VI.4 ] leads to halfonshell tmatrix elements. For
the present case of a hard core potential, we have done calculations
using offshell, halfonshell and onshell tmatrix elements to compute
the energy loss spectrum given in Fig. (71). The results are given in
Table (73). Column I corresponds to the offshell results, column II
to the halfonshell results and column III to the onshell results.
jf
o
10
20
25
29
30
35
40
45
50
60
(73) A comparison of differential cross section data obtained using offshell
(column I), halfonshell (column II) and onshell (column III) tmatrix elements.
Renormalized results have been obtained by setting the maximum peak in energy loss
spectra equal to one. The data used is for the (Li + CO) system as given in the
caption for Fig. (71).
do / n
dfl (ni~0
.ji =0 > nf=0,jf)
I
II
III
Renormalized
Renormalized
Renormalized
2.27( 4)
2.43( 3)
2.12( 4)
2.38( 3)
2.12( 4)
2.60( 3)
5.87( 4)
6.31( 3)
4.10( 4)
4.60( 3)
4.97( 4)
6.10( 3)
2.40( 2)
2.57( 1)
2.29( 2)
2.57( 1)
2.20( 2)
2.70( 1)
2.09( 2)
2.24( 1)
1.99( 2)
2.23( 1)
1.88( 2)
2.31( 1)
9.31( 2)
1.00
8.91( 2)
1.00
8.15( 2)
1.00
7.69( 2)
8.26( 1)
7.38( 2)
8.29( 1)
5.36( 2)
6.58( 1)
9.55( 3)
1.03( 1)
8.81( 3)
9.89( 2)
4.70( 3)
5.77( 2)
5.26( 2)
5.65( 1)
4.71( 2)
5.29( 1)
3.30( 2)
4.05( 1)
5.66( 3)
6.08( 2)
5.05( 3)
5.67( 2)
3.29( 3)
4.03( 2)
3.84( 5)
4.13( 4)
3.43( 5)
3.85( 4)
2.03( 5)
2.49( 4)
1.09(11)
1.17(10)
9.94(12)
1.12(10)
4.7 9 (12)
5.88(11)
CO
183
The renormalized results were obtained by scaling the entire spectrum
so that the largest peak has the value one. We note that the differences
in all cases are minor. One of the reasons for these unexpected small
differences is that the magnitudes of the momenta that occur in the
present calculations are all similar. We caution, however, that for
softer potentials this may not be the case. As was pointed out in
Chapter m. the tmatrix may be a very oscillatory function of its
momentum arguments.
It was argued in Section V.2 that the momentum overlap volume
give by Eq. (5.36) could provide a rough measure of the probability
amplitudes. From Eq. (7.12), it is clear that two such overlap volumes
are involved. Each single collision term leads one to consider two
overlapping spheres as defined by the functions and
present in Eq. (7.4). Each of these functions is normalized, implying
that a meaniful comparison to the single collision peaking calculations
can be obtained only by normalizing the square root of the volume of
the sphere determined by . From the (Li+, CO) data used to
generate the energy loss spectra given in figures (.71) through (73),
we obtained the renormalized overlap volumes plotted in Fig. (75)
versus j^. From the three curves plotted in this figure, it is apparent
that the peaks are shifting to higher final jvaDues as the scattering
angle increases. This is the same conclusion that was drawn from
figures (71) through (73). There is also a general similarity of
the shapes of these curves to those found in the quantal results.
However, the momentumoverlap model predicts a greater amount of rota
tional excitation. The interference features are also not contained in
this simpler mode. To give an idea of the behavior of the formfactors
Figure (75) A plot of momentumoverlap volumes versus final rotational
quantum number for the (Li + CO) system. [ (n,,j, )e(0,0),
nf=0, E^^eV and Q^^i=37Io]
1.0
075
050
0.25 
0
O 371
o 43.2
x 49.2
0
X
X
10
e
20
30
it
40 50 60
CO
O'"
186
in the quantal calculations, we have plotted the radial formfactor
fQj 00j [A=Li+, E=C and C=0] as a function of jf in Fig. (76). This
figure shows a rather pronounced oscillatory behavior as a function of
jf, but with a rapidly decaying tail for high jfvalues. This decaying
tail leads to the rapid decrease in the rotational transition probability
for high jvalues.
An attempt was made to extend the present results by using the
exponential potentials given in the introduction to this section.
However, the energies involved are quite high, making it necessary to
use more than one hundred partial waves in the evaluation of the neces
sary tmatrix elements. Because of this, the partial wave tmatrix
elements involved are very small and the errors accrued at the high
partial waves did not allow convergence in the full tmatrix. As was
pointed out in Chapter III, it is these conditions which are least favor
able for numerical procedures such as the VPA method.
Generating the various differential cross section plots shown in
this section required a rather large number of tmatrix evaluations.
It is the evaluation of these functions which can be quite costly,
when numerical methods are used. Because of this, plots of the real
Pk
and imaginary components of the 2 were made as a function of the final
rotational quantum number. These results are given in figures (77a)
and (77b), Reft^] x 10 and Im[t[^] x 10 are plotted versus j^.
3. Discussion
At hyperthermal energies, the study of atomdiatom scattering
processes using coupled channel methods becomes prohibitively expensive
and impractical. It was shown in Chapter IV that iterating the AGS
Figure (7
6) A plot of versus final rotational quantum number
for the (Li+, CO) system. [(nj,^)=(0,0), nf=0, Ej=4.23eV
and =37.1]
11
CO
CO
Figure (77a) A plot of Re[k] x 106 versus final rotational quantum
number for the (Li+, 0) interaction in the (Li+, CO)
system. [(nÂ¡, j ) = (0,0), nf=0, Ej=4.23eV and 6p,p =37.1]
4.52
2.47
0.42
12
i
j
f
24
T
T
_J 1 I
36 48 60
vO
o
Figure (77b) A plot of Im[t? ] x 10^ versus final rotational quantum
number for the(Li+, 0) interaction in the (Li+, CO)
system. [(ni,^)=(0,0), nf=0, Ej=4.23eV and ej,p^=37.1]
3.87
1.50
0.88
3.25 ^
24
jf
0
12
T
6
o
36
48 60
o
ro
193
equations led to a multiple collision expansion, which in effect un
coupled the various transition amplitudes. Because of this, approxima
tions based on the multiple collision expansion allow for the direct
computation of the transition probabilities of interest without the
need for introducing expansions in terms of intermediate states. For
problems such as collision induced dissociation, we saw in Chapter VI
that enormous simplifications are obtained using such a manybody
approach. We conclude then that approximations such as the ones we
have pursued in this dissertation warrant further investigation as a
viable and very practical approach to the study of atomdiatom colli
sion at high energies.
From a practical standpoint, the calculations given in the last
section for inelastic scattering demonstrate that useful qualitative
information is obtainable with the single collision peaking approxima
tion. However, it is clear that if more reliable quantitative informa
tion is desired, one must input more realistic twobody tmatrix data.
In light of the difficulties encountered in obtaining converged tmatrix
elements at high energies and large momenta, it is clear that an improved
algorithm for the high partial waves must be sought. One possibility in
this regard would be the implementation of semiclassical procedures.
On the other hand, perhaps the most practical solution would be to
pursue nonpartial wave techniques. Even though the Bateman method used
in Chapter HI did not work for singular potentials, one could modify this
procedure using a twopotential formula (Ro67). This would entail
separating the hard repulsive core from the tail of the potential, and
using the Bateman method for the softer component. The need for an
accurate numerical procedure still remains, however, since it will serve
194
as a comparison to more approximate procedures.
The collinear studies in Chapter VI clearly indicate that the valid
ity of the peaking approximation may be limited to cases where the po
tential is very repulsive and hard. It was not possible to pursue this
matter in the case of threedimensional motion, because of the complex
ities arising in the angular momentum analysis. This question must be
answered, however, if one is to properly judge the validity of the single
collision approximation. We pointed out in Section V.2 that the eval
uation of the single collision transition amplitudes would entail the
introduction of partial wave expansions for the various quantities
involved [see Eq. (5.22)]. At high energies this would not be practical,
because of the large number of partial waves required to obtain conver
gence. In order to circumvent these difficulties, one must either intro
duce decoupling approximations, or make use of multidimensional quadra
ture techniques. Given the simplicity of the single collision term
[see Eq. (5.24)], the latter possibility would seem thebest alternative,
at least in the case of repulsive potentials. If the potentials support
bound states, the presence of poles at negative energies would severely
complicate the use of multidimensional quadratures.
Finally, we note that the results on collision induced dissociation
reported in Section VI.4 for collinear scattering indicate that it is
essential to include certain double collision terms if reliable results
are to be obtained. Essentially, it was demonstrated that the interac
tion between the two atoms of the dissociating diatomic must not be
neglected. Recall that the role played by this Â¡potential in the case
of inelastic scattering was merely one of determining the initial and
final momentum distributions of the target diatom. This is clearly not
195
sufficient when dissociation is involved. He have shown in Section IV.2
that adding the extra double collision terms [T^GqT^ and TjGqT^ in the
expansion for Tq^] is equivalent to using a continuum state to describe
the dissociating target diatom. In the single collision approximation,
a plane wave was used for this purpose. Given the disparity between
the results obtained from these approximations, we feel that it would be
very important to extend the present coll inear study to the case of
threedimensional motion.
APPENDIX I
HARD CORE TWOBODY tMATRIX
We will now derive a closed analytical expression for the two
body tmatrix for a hard core potential using the method of Laugh!in
and Scott (La68), i.e. the tmatrix elements for a potential of the
form
V = X 6 (rc r) (Al.l)
in the limit of X going to infinity. The step function 6 is defined
by Eq. (5.37). The offshell wave function co^ can be obtained immedi
ately from the appropriate radial differential equation, Eq. (2.10),
and the physical boundary conditions, equations (2.15) and (2.16). The
result is
Vqrc)
i(q> kr; r) = j (qr)
1 1 1 C(l)
Vc)
hi(1)(kErc).
(A1.2)
The tmatrix can now be obtained from the definition of t^(q1, q; E)
given by equation (2.13); however, as was pointed out in Section II.2 ,
one must make use of Eq. (2.18). From this we obtain
Vq' E) = kE; rc+>
(q2 kE2) / J1(q'r)j1(qr)dr]. (A1.3)
196
The integral in Eq. (A1.3) can be solved analytically using relation
(A2.2) in Appendix II. With a little algebra one can also show that
rr2 q2kE2 d
W q E> T XW Vqrc> drT jl(qV +
q q ^
+ j1(q,rc>rj1(qrc) +
 (q'rc) J1 (qi"c) In h1TkErc) (A1.4)
where the and hj^ are Bessel functions related to the Ricatti
/v ^ (1)
Bessel functions jj and h^v ; through the relations (Ab65)
(z) = z j1 (z) (A1.5)
and
h(1)(z) = z h(1)(z) (A1.6)
Equation (A1.4) has been reduced to the form given by van Leeuv/en and
Reiner (vL61), though these authors used a derivation different from
the one given above. It should be pointed out that there exist many
errors in the literature (Bi73) where people have neglected the first
term in equation (A1.3). The reason for this error is the neglect of
Eq. (2.18) in the above procedure when one applies Eq. (2.13).
It is interesting to note that in obtaining the total tmatrix by
summing over partial waves, the first and second term in Eq. (A1.4) lead
to the simple closed expression
198
t(5, J. E) . (E SLi ,,2
m 27TAq
r 2
rc
mm lm
L Y*m(^,)Ylm^)jl(q'rc)j1(<,rc> Lln h<1)(kErc>> (A1'7>
where
Aq q* q (A1.8)
In obtaining Eq. (A1.7), we have made use of Eq. (2.12) and the identi
ties (vL61)
(2w)3 / dr _ e1q'7c = lilil! *5 > (A1.9)
drc 2,r2 Aq
i Â£ Ytm (q,) Ylm(q) J1(q'rc> 5^ jl(qrc> (A1'10)
In order to clarify the meaning of the first term in expression (A1.7),
it is instructive to consider the expression Eq. (A1.2), sum it over
partial waves to obtain u>(q', q; E) and use this expression to obtain
the total tmatrix. Doing this one obtains
q; E> (fsrE) rc:
J1(Aq rc)
2u^ Aq
+
(Al.ll)
If one were to approximate w(q, E; r ) by a plane wave, one would get
199
t(q', q; E) 
2 JiUqrc)
r
c 2
2tt Aq
(A1.12)
Thus, we see that such an expression is a planewave or Born approxima
tion to the hard core tmatrix. Other approximations based on addition
formulas such as Eq. (A1.10) could be pursued. This for example has
been done by Bethe, Brandow and Petschek (Be63) in applications of the
BruecknerGoldstone theory of nuclear matter; however, approximations
analogous to those used by these authors were not found to be satis
factory in the present case of atomdiatom collisions at hyperthermal
energies.
APPENDIX II
INTEGRALS USED IN "COMPARISON POTENTIAL" METHOD
In Chapter II we saw the need for integrals of the product of two
RicattiBessel functions, and in particular, for the comparison poten
tial method, integrals such as those found in Eq. (2.57). Consider now
A A A /v
the integral of the product uV, where u^ and satisfy the Ricatti
Bessel differential equation (2.21), i.e.
and
7? ui + Iq? 1Xij7Ui i = 0
r
dr
'l + [q2 ^ \ = 0
(A2.la)
(A2.lb)
It is well known (Me61) that the desired integral
A A
r=r.
2  W[u,, v,]
I dr u v = L_
ri M ? q)
r=r,
if E^ f E^; on the otherhand if E = E^ (Fa71)
(A2.2)
l Vl
dr =
A I A
ijr
. {q2 lilii)) ; ,rs
(A2.3)
200
201
The derivation of equations (A2.2) and (A2.3) is straight forward and
is usually given in standard texts on quantum mechanics (Me61) or in
the theory of special functions [(Fa71), (Gr66)]. If we now replace
A
one of the RicattiBessel function, say v^, in the left hand side of
Eq. (A2.2) by where satisfies Eq. (2.43) in the interval
[r^, ] one can actually follow a derivation analogous to that used
to obtain expression (A2.2), and the result is
r2
A
/ u,aJ dr
r.
A
w[ooq1 U] ]
2 2
[Vi V
'? (q2o k 2) r2 .
+ / j,(qr)u dr.
(q2 ) rl
rl
(A2.4)
In order to clarify this, we will now derive Eq. (A2.4). The starting
point is the identity
[
dr
2 1
+ (q lilil
)u 1 +
 U11 l1 + (aiKi2 +2 ^ ^ (aEkE2 q2)Ji = 0
(A2.5)
^ 1
which is obviously true since u^ and coj satisfy Eq. (A2.1a) and Eq.
(2.43) respectively. Rearranging and using the identity
d ur i id ~ d i
dr W[U u^ ^ ? U1 ~ ui ~2 W1
dr dr 1
(A2.6)
one obtains the desired relation, Eq. (A2.4). An expression analogous
to Eq. (A2.3) is also obtainable for the case in which q^ = a^ic^ in
Eq. (A2.4). Here one would use the identities
202
ru,[ wi1 + (q2 11^ (aFkF2 q2)ji(qr)n = 0
1 dr2 1 1 r2 1 E L 1
3F 3F11 )( BFl) 
/ d i w d A n / d ~ w d
r( uÂ¡1 )(  u, + r( u )( y (o
dr 1 dr2 I dr 1 dr 1
rÂ¡^ *
)!
dr
and
(Ir t^i2 1(1 r2 ) rwl1 V = 2qi Vi1'
(q2 1^1 p ^ ) [ru1 ( cd i ) + rw 1 ( u1) u^1 ] .
r
These relations can easily be verified using equations (A2.1a)
The final answer is
r2
r i *
J to u,dr
rl
2q
Vdrt apa.,1 )( ^G,) u,1 ( ^ u,)
 (qj2 1(1 V ) nolGj] 2 +
r2 rl
+ (q2 aEkÂ£2) / (qr)( p u] )dr}
(A2.7)
(A2.8)
(A2.9)
and (2.43).
(A2.10)
Finally, the integral on the right hand side of Eq. (A2.10) can be
obtained from the relation
203
for
/ rv, ( 377 u )dr
(q2q22)1i[r(^u1)(^;i)
 V ( 4 u ) (a2 HL+J1 ]r: v 1
V dr V vqi r2 1 vr
1
 2qx2 / u,v dr}
rl
(A2.ll)
qj f Equation (A2.ll) may be proven using the identity
[K 4: vj v.][ d2
dr "1' V1JL dr2
U] + (qf 1(1 1))u13> = 0, (A2.12)
which follows from equations (A2.ia) and (A2.1b).
APPENDIX III
JWKB STARTING PROCEDURE
In propagating the numerical solution to the radial Schrodinger
equation
ip(r) + [k2 U(r) K1 t U]4,(r) = 0, (A3.1)
d rc r2
one generally starts at some rQ within the first turning point (Be60)
such that ^ is small. Trial calculations are then performed to test
the sensitivity of the final wave function to the starting point. If
the potential is very singular near the origin, this sensitivity is
indeed small (Be60); however, if the potential is not singular or if
the singularity is not very strong, this may not be the case. The
latter observation was found to be true for some of the calculations
reported in Chapter III, e.g. the offshell wave function studies done
for the Morse and Hulthn potentials. In order to remove this arbi
trariness we have adopted a JWKB procedure proposed by Bellum (Be75a).
It is well known that the JW^B solution to Eq. (A3.1) near the
first turning point, rt> has the form [(Mo58), (La37)]
*(r) [*$] [A J1/3 (w(r)) + B J_1/3 (w(r))] (A3.2)
where A and B are constants specified by the boundary conditions,
q(r) = [k2 U(r) l1 + ^/2)2 ]1/2 (A3.3)
r2
204
205
w(r) = / q(r')
rx
dr'
(A3.4)
and Jv is a order Bessel function. Using the asymptotic form of
Eq. (A3.2) one finds that (Mo58)
i>(r) \ tq(r)31/2 e~ !w(r^ 1
and
(A3.5)
di^(r) _1
dr 2
[q(r)]
1/2
1 dlnq(r)] dw(r)1 w(r)
l2 dr dr J e
(A3.6)
for r
to the effective potential,
Jef
= U(r) + 1 k2 (r rt)
(A3.7)
leading to the relation
cMr) I a \ 11/2 pilcl(r)3 (rt r)/(uef k2)}
dr 2 qin 6 J (A3.8)
for r < r^. Equation (A3.8) can now be used as a good estimate to the
magnitude of to the left of the first turning point. In the
computations reported in Chapter III, a value of ^lll) less than 10~10
was found to be suitable.
APPENDIX IV
COORDINATES FOR THE THREEBODY PROBLEM
Consider three distinguishable particles (atoms) A, B and C. We
shall associate with each atom a, a equal to A, B or C, the atomic mass
nr,, the position coordinate r and the momentum coordinate p In each
a r a ra
arrangement channel i, i equal to 1, 2 or 3, of the three particle
system one is concerned with a free atom a and an interacting pair
(By); thus, it proves convenient to work with relative Jacobi coordinates.
These relative position coordinates shall be defined according to
Fig. (A41), where e.g., R^ is used to denote the position of particle
A relative to the center of mass of the (BC) pair with the relative
position coordinate r^. In order to give a general description of these
coordinates, it is useful to introduce the labels (ijk) and (aBy), which
indicate a cyclic permutation of the channel indices (123) and the par
ticle indices (ABC), respectively. Furthermore, we shall associate the
index i with the index a, j with B and k with y. The reason for this
association becomes clear if one considers the fact, e.g., that in
channel 1 particle A is free and particles B and C are bound. The
desired Jacobi coordinates can then be defined as follows,
(A4.1)
and
mBrB + W
mBy
(A4.2)
206
207
C
Figure (A41) Internal and relative position
vectors r.Â¡ and ft, i = 1, 2, 3,
respectively for channels 1, 2
and 3.
203
where m, = m + m Here we note that the definition for r. is given
3y 3 Y 1
by Eq. (A4.1) can be associated with the mixed index set (iyB) which
is equal to (1CB), (2AB) or (3BA). One can further show, using equations
(A4.1) and (A4.2), that the various channel relative coordinates satisfy
the relations
r fi (ij)
may J 0
(A4.3)
and
^ = (ij)
maymBy
> m
rj m
Â£ ft.
By
(A4.4)
where (ij) is equal to +1 if i and j are in cyclic order or 1 otherwise.
M is the total mass of the system. In each case the transformation
Jacobian is equal to unity.
At this point, one can now introduce the momenta p.. and P.Â¡, con
jugate to f.. and ft respectively. These relative momenta can be defined
in terms of the p as follows,
a
V
^3 > ^y *
"W ^ mBY Pb
(A4.5)
and
p = ft k ft ^Ln
*1 M Pa M PB M Py
(A4.6)
V
Qualitatively then describes the relative motion of atom a with
respect to the pair (By) having the relative momentum p^, in the given
channel i. We can further introduce the channel reduced masses m. and
M. defined by the relations,
(A4.7)
209
and
y By
(A4.8)
to cast equations (A4.5) and (A4.6) into the form
* = A %
(A4.9)
and
? .A A + V
i i ma m3y
(A4.10)
The relation between the various channel momenta can now be written
simply as
mft rtuM
_e_ n. (ij) L p.
maym8y J
i % j
(A4.ll)
and
!5P. + (ij) p.
%Y J J
P.
1
(A4.12)
APPENDIX V
ANALYTICAL QUANTITIES USED IN THE COLLI NEAR SCATTERING MODEL
The computation of the peaking transition probabilities for vibra
tional excitation and collision induced dissociation reported in Chapter
VI require the evaluation of the form factors:
A
Fri = (A5.1)
and
A
Fq1 = (A5.2)
For the case of inelastic scattering, F^,^ may be found in the liter
ature for both the harmonic oscillator and the Morse potentials [(Sh60),
(Ra69), (En69), (Ek71), (Ek72)]. In the case of a harmonic oscillator
potential [(Sh60), (Ek71)] one can easily show that
11
= (1)
n+"' (n'!n!)1/2 e'V2
(no>
n'+n
*Snx n ~k
x kfQ k!(n'k)!(nk)! (A5*3)
where
nQ = (AP)2/2 (A5.4)
and kmx is the smaller of n and n'. The corresponding relation for the
Morse oscillator [(En69), (Ek72)] is given by
210
211
ap (nn'+li)
F,,, = A r(s ,+ s + i  ) ttt
JL 1 n n n n a (2s ,+1) ,
x 3F2 (n, V + sn + if n' n + 2sn + *
A .AP
 n + l; 1) ,
a
(A5.5)
where
Vn
(z).
Vn (4D)iP/a f
cl
{a(2sn+l)n[n!r(2sn)]_1}1/2 ,
2D n 1/2 ,
r(z+n)
"W
(A5.6)
(A5.7)
(A5.8)
(A5.9)
^2 1S a hypergeometric function (Ab65) defined by the relation
/ v n (_n)k(a)k(b)k zk
3F2(n, a, b; c, d; z) (c)k(d)k FT (A5.10)
and the quantities a and D are the Morse potential parameters defined by
Eq. (6.36).
The form factor F^ for dissociative scattering in the case of a
Morse oscillator can be derived following the analysis given by Strachan
(St35). Using the notation given by Eckelt and Korsch (Ek73), one obtains
r.ii(s) r(snn^)r(l2s)r(n+iif)
01 E',n r(l+ns+sn)r(lssniy)
212
AP ap ap
X ,F9 (n,s+s +i^, s+s +i; l+2s ,n+i%; 1)
it na na n a
+ [s S] ,
(A5.ll)
where
A = Ae'An(4D)1AP/a,
e n
(A5.12)
V =(sA')1/4,
(A5.13)
s = i(2Â£')1/2/a
(A5.14)
<5 (s) = arg[r(2s)/r(JgsD)] = 6(s)
(A5.15)
and [s s] indicates that a term identical to the first one must be
included with s replaced by s. We note that the above expression for
Fqi requires the evaluation of complex gamma functions. These quantities
may be evaluated using the Stirling's formula (Fr74)
lnT(z) = j ln(2ir) + (z h) In (z) z
1
1260z5
(A5.16)
This relation is valid for the condition
iarg(z)  < ir (A5.17)
and will yield ten digit accuracy if z[_>15. The recursion relation
T(z + 1) = zr(z) (A5.18)
can be used along with Eq. (A5.16) for z<15.
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H. Miller, Plenum Press, New York (1976), p. 131.
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Oxford (1971).
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BIOGRAPHICAL SKETCH
Lyntis Hall and Beard, Jr. was born on September 13, 1951, in
Tegucigalpa, Honduras, Central America. He attended grammar school
in San Pedro Sul a, Honduras. In 1964 he came to the United States to
pursue his secondary education, receiving his high school degree from
Tylertown High School (Tylertown, Mississippi). From 1968 to 1972 he
attended Mississippi State University, where he received a B.S. degree
in chemistry. From September 1972 until the present he has been a
graduate student in the Quantum Theory Project and the Department of
Chemistry at the University of Florida. He attended both the Winter
Institute (Gainesville, 1972 and 1973) and the Summer Institute (Uppsala,
1974) in quantum chemistry and physics organized by the Quantum Theory
Project of the University of Florida and the Quantum Theory Group of
the University of Uppsala in Sweden.
222
I certify that I have read tnis study and that in my opinion
it conforms to acceptable standards of scholarly presentation ana
is fully adequate, in scope and quality, as a dissertation for the
degree of Doctor of Philosophy.
Cr" t 1, / C f '.<(< 'f' O' '''
David ATlTicha, Chairman T~
Professor of Chemistry and Physics
I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and
is fully adequate, in scope and quality, as a dissertation, for the
degree of Doctor of Philosophy.
M
Jl]
X
L
N. Yngve
Professor
Ohrn
! of Chemistry and Physics
I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and
is fully adequate, in scope and quality, as a dissertation for the
degree of Doctor of Philosophy.
Jcm^K. Sabin
Professor of Physics and Chemistry
I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and
is fully adequate, in scope and quality, as a dissertation for the
degree of Doctor of Philosophy.
Thomas L. Bailey
Professor of Physics
I certify that I have read this study and that in my opinion
it conforms to acceptable standards of scholarly presentation and
is fully adequate, in scope and quality, as a dissertation for the
degree of Doctor of Philosophy.
1 i' \ P \ L ft
Charles P. Luehr
Professor of Mathematics
This dissertation was submitted to the Graduate Faculty of
the Department of Chemistry in the College of Liberal Arts and
Sciences and to the Graduate Council, and was accepted as partial
fulfillment of the requirements for the degree of Doctor of
Philosophy.
March 1979
Dean, Graduate School
0.2
0.3
0.8
 1.4
1.9
2.5
0 7 14 21 28 35
L
CO
Figure (67) Plot of the differential energy transfer probability
Pjq(Ac) versus Ac as obtained from m(s,c,Pk). y^e
spectra corresponds to inelastic scattering and the con
tinuous curve denotes transitions to the continuum.
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FILES
Figure (77a) A plot of Re[k] x 106 versus final rotational quantum
number for the (Li+, 0) interaction in the (Li+, CO)
system. [(nÂ¡, j ) = (0,0), nf=0, Ej=4.23eV and 6p,p =37.1]
205
w(r) = / q(r')
rx
dr'
(A3.4)
and Jv is a order Bessel function. Using the asymptotic form of
Eq. (A3.2) one finds that (Mo58)
i>(r) \ tq(r)31/2 e~ !w(r^ 1
and
(A3.5)
di^(r) _1
dr 2
[q(r)]
1/2
1 dlnq(r)] dw(r)1 w(r)
l2 dr dr J e
(A3.6)
for r
to the effective potential,
Jef
= U(r) + 1 k2 (r rt)
(A3.7)
leading to the relation
cMr) I a \ 11/2 pilcl(r)3 (rt r)/(uef k2)}
dr 2 qin 6 J (A3.8)
for r < r^. Equation (A3.8) can now be used as a good estimate to the
magnitude of to the left of the first turning point. In the
computations reported in Chapter III, a value of ^lll) less than 10~10
was found to be suitable.
8
been found, however, that for some problems it is possible to overcome
the first of these obstacles by treating some or all of the nuclear de
grees of freedom either classically or semiclassically. As was pointed
out earlier, in some instances nonadiabatic effects may be localized to
certain regions of the potential energy surfaces. This, for example,
would allow one to use classical trajectory methods in regions where
motion along the adiabatic potential surface is allowed. In such an
approach the nonadiabatic regions serve only to determine branch points
at which a jump to another surface can occur. This approach, which is
known as the trajectory surface hopping model (TSH), was proposed by
Tully and Preston (Tu71) to study the systems [H+,] (Tu71), [H,]
(Pr73) and [Ar+,] (Ch74). In order to determine whether a trajectory
remains in the same potential energy surface or branches to another sur
face, the probability for hopping must be computed. A discussion of
how this probability is obtained is given in the above references. We
shall only mention one method; that of the well known LandauZener
Stckelbert (LZS) approximation [see for example Bransden (Br70)J.
Strickly this approximation is valid only for the onedimensional case.
Here, however, one applies the LZS result to a cut perpendicular to the
nonadiabatic seam (Tu76a) in the potential energy surfaces. Since trajec
tories may encounter various branch points, we see that in many physical
applications the TSH method could become computationally unmanageable.
Another approach, which is useful in treating nonadiabatic processes
such as
A + BC(n1,j) A+ + BC"(ni,ji) (1.20)
A + BC (nj.jp
(1.21)
122
Gi Go = Go Ti Go (529)
which follows from the definition of and T., and equations (4.7)
and (4.8), we see that Eq. (5.28) is equivalent to
T11 T2 + VG1 Go>Tll + VG3 Go>T31 +
T3 + "^"3(^1 ~ G0)Tn + "^3(^2 Go^21 (5.30)
a formal expression that implies the weaker the various potentials ,
the smaller the contribution from the corresponding multiple collision
terms. Other important considerations regarding multiple collisions
relate to the size and geometry of the target involved [(Meol), (Mi78)].
If dg and d^ are the diameters of the atoms in the diatomic; and Rg^,
the average value of r^; then, multiple collision terms are less impor
tant when
Pu Rbc 1 (5.31)
and
(dB + dc)/2 < RgC (5.32)
(Mi78). The relative masses of the atoms in the atomdiatomic system
are also very important. The likelihood of multiple collisions for
inelastic scattering diminishes if
mA mB
mc M
< 1
and
(5.33)
mA mC
mB M
< 1
(5.34)
3 3 9 'i 5 21
A
LT1
JD
123
[(Sg66), (Se69), (Ek71), (Ek74)], A further discussion of the impor
tance of the above mass factors will be taken up in chapter VI for the
case of col linear scattering.
We will now consider some kinematical rules which follow from
Eq. (5.24), i.e. the definition of (3). It has been pointed out
(Mi75) that the functions < ^2' I^ 1* > anc* < are eac^ 1ca^ized
within a given spherical region; implying that the degree of overlap is
(s)
a measure of the size of the transition amplitude (3). A good
quantitative estimate of the volume of each sphere can be obtained by
taking q I and q 'I equal to their classical values; and thus
3 1 l'max I31 1 max M
I3il
< ^1 Lax
[2m1W1ni] 1/2
(5.35a)
and
iV
< !q 11
'H1 'max
= [2m W nf] 1/2
11
(5.35b)
where W ni and W^nf are initial and final classical vibrational energy
of the diatomic BC. Based on this analysis a kinematical construction
can be made as illustrated in Fig. (51). A concise statement of the
overlap volume may be written as (Fe71)
V3) /dQ3 etlSj' llnaxq1']8[q1lmaxq1l] (5.36)
where q^' and q^ are given by equations (5.25c) and (5.25d). The func
tion 9(x) is known as the Heaviside step function and is defined as
(Cu75)
e(x)
1 x > 0
(
0 x < 0
(5.37)
195
sufficient when dissociation is involved. He have shown in Section IV.2
that adding the extra double collision terms [T^GqT^ and TjGqT^ in the
expansion for Tq^] is equivalent to using a continuum state to describe
the dissociating target diatom. In the single collision approximation,
a plane wave was used for this purpose. Given the disparity between
the results obtained from these approximations, we feel that it would be
very important to extend the present coll inear study to the case of
threedimensional motion.
BIOGRAPHICAL SKETCH
Lyntis Hall and Beard, Jr. was born on September 13, 1951, in
Tegucigalpa, Honduras, Central America. He attended grammar school
in San Pedro Sul a, Honduras. In 1964 he came to the United States to
pursue his secondary education, receiving his high school degree from
Tylertown High School (Tylertown, Mississippi). From 1968 to 1972 he
attended Mississippi State University, where he received a B.S. degree
in chemistry. From September 1972 until the present he has been a
graduate student in the Quantum Theory Project and the Department of
Chemistry at the University of Florida. He attended both the Winter
Institute (Gainesville, 1972 and 1973) and the Summer Institute (Uppsala,
1974) in quantum chemistry and physics organized by the Quantum Theory
Project of the University of Florida and the Quantum Theory Group of
the University of Uppsala in Sweden.
222
166
x
(
Because of the symmetry properties of the 3j coefficients in Eq. (7.11),
the sum in Eq. (7.12) is restricted to those values of A for which the
relations
(7.13)
j' j  Â£ A Â£ j + j
and
(7.14)
j' + j + A = even
are satisfied. For homonuclear diatoms, the (1)J+J'phase factor in
Eq. (7.12) implies that odd Aj transitions are symmetry forbidden
[(Ek74), (Bo76)].
The differential cross section can now be obtained by the evaluation
of the radial form factors defined by Eq. (7.9) and of the offshell
twobody tmatrix elements. In the present investigation, we will
assume the radial functions unj to be harmonic oscillator wave function
with modified frequency arguments allowing for rotational distortion
[(Pa35), (Yu76a)]. Adopting the treatment given by Pauling and Wilson
(Pa35), the resulting expression for the radial wave function is
(7.15)
1/2 e[C/xnj]2
Hn(4/xnj), (7.16)
where
1.1
0.5
0.2
0.8
1.4
2.1
O 7 14 21 28 35
L
CD
24
excited states. It should be mentioned, however, that the reaction
Li + (1S) + N2(!Eg) v Li + (1S) + M3!*)
would be spinforbidden (Ge71). This conclusion is only valid if one
can neglect effects such as spinorbit interaction. A discussion of
the applicability of symmetry rules to collisions is given in the work
by George and Ross (Ge71) and that by Shuler (Sh53). From an experimental
viewpoint, electronic excitation has not been observed at the energies
[4 to 8 eV in the centerofmass system] and scattering angles studied
in the present work (Bo76). A similar analysis could be done for
(Li+, CO); however, this system is isoelectronic with (Li + fO and is
expected to behave similarly.
We have not considered changes of the equilibrium internuclear
separation of N2 as Li+ approaches. In order to do so, we now discuss
the potential energy surface in more detail. The long range interaction
between a charged atom and a diatom with a nondegenerate electronic
state is dominated by electrostatic and inductive terms (Bu67), i.e.
(1.58)
where
VC0UL jjf + jjf P2('COse'i f jjf P3(0)
(1.59)
 a. ))McosG) + ...],
eL
(1.60)
1.9
0.3 
1.4
3.0
4.7
6.3
/ \
/ \
\
r /
i
0
1.6
\
3.2
4.8
AP (a.u.)
Uj
132
Using the physical picture given above, we can now state the
single collision approximation for inelastic scattering as
M'l = M' (3) = (69)
where we have used Eq. (5.3) and have dropped the term Mp (2), since
\Â¡2 is neglected in the given model. In the momentum representation,
Eq. (6.9) leads to
Mri = f <% *3 (q3 q3; E (610)
the analog of Eq. (5.24) obtained in the threedimensional case. The
various channel momentum coordinates present in Eq. (6.10) can be
obtained from Appendix IV, if one considers a three particle system
with mass m^ = m, mg = 1 and m^ = as illustrated in Fig. (6lb).
In channel 1, P^ and p, are the conjugate momenta to and Xp
respectively. The result for the coordinates q^, q^ q^ and q^ in
Eq. (6.10) is analogous to that given by equations (5.25a) through
(5.25d), but with the masses as given above. The peaking approximation
to Eq. (6.10) is
M
(s,Pk)
1 1
1APlxl[n>
(6.11)
A
where is the relative position operator for the BC pair, n and n'
are the initial and final vibrational quantum numbers characterizing
2 and p respectively. The twobody tmatrix t^9^ is factored out
from Eq. (6.10) at some value of Q^*. Once this choice is made q^'* and
q^ are obtained from equations (5.25a) through (5.25d). Noting that the
3
diatom system. We note that the Coulomb interactions of the various
particles involved are contained in VCQU and the spinorbit interactions
are contained in H^q. The electronic and nuclear mass polarization
terms have been neglected. The reduced mass is defined by the rela
tion
where m is the mass of nucleus a and
a
M
3
l
3=1
m.
0.4)
is the total mass of the nuclei. In order to separate nuclear and elec
tronic degrees of freedom, the wave function T is generally expanded in
terms of a set of electronic functions {<Â¡>. (r ; r.)} which depend on the
electronic coordinates {r.} and only parametrically on the nuclear coor
dinates {r }. Thus one has
'F(r ; r.) = E (fu (r ; r,. )xi(r )
_a i ^ rl'a i Ara/
(1.5)
where xÂ¡ Oy) can be considered as the nuclear wave function describing
the motion of the nuclei on the potential energy surface which charac
terizes the Tth electronic state [(Fr62), (Hi67)]. Equations {1.5) and
(1.1) lead to the following set of coupled equations:
[VTiV uirEJx, = vyir +Tir + uivlxv
(1.6)
where we have introduced the definitions
T . I j J_ v2
01 2 6=1 "6 6
(1.7)
E? (Kinetic Energy)
14?
45
Treating the left hand side of this equation as an inhomogeneous term,
we can adopt the same procedure that was used in section 113 to
derive the desired set of modified VPA equations
^ <5, (q, kE; r) = k^1 {v(r)[w^ ooj jj (qr)] (2.41)
+ 2kE [u) J1 (qr)]}/a^
and
^^(q, kÂ£; r) = k"1 Mr)^ + 2kE j(qr)]} x
x {sin(fi.)j.(kEr) + cos(61 )n1 (kÂ£r)}. (2.42)
A similar set of modified equations could be used for positive energies
when a nonclassical region is encountered there. This would require
RiccatiHankel functions. This, however, was not found to be necessary
in the calculations done in this work. Stabilization was only a problem
at negative energies.
5. The Comparison Potential Method
Another possible approach in solving the inhomogeneous Schrodinger
equation (2.10) for the offshell wave function w^(q, k^; r), is to make
use of the comparison potential (CP) method. As was mentioned in the
introduction to this chapter, this approach has already been presented
in the literature [(vL61), (Br75)]. We shall,however, sketch an outline
of the method and present some improvements to the computational
algorithm for w,. These changes are based on the work of R. Gordon
[(Go69), (Go71), (Ro76)], who has investigated extensively the use of
177
results (B676). In each of these calculations, the peak in the distri
bution envelope fell at a jfvalue that was higher than the one observed
experimentally. In Fig. (71) the peak occurs at jf = 29, whereas the
peak in the corresponding experimental curve falls at jf = 20. A
similar result was obtained by Micha et al. (Mi78b) in their spacetime
correlation function approach. In their studies, they also assume the
single collision approximation. As a possible reason for this discrep
ancy between theory and experiment, they suggested that the lockin
effect of neglected longrange anisotropy would lead to an apparent
increase in the moment of inertia of the target diatomic. Upon treat
ing the moment of inertia as a parameter and increasing its value, they
were able to obtain agreement with experiment. In order to compare the
above results to those obtained using a homonuclear target, we have
plotted in Fig. (74) the differential cross section versus final rota
tional quantum number of the (Li+, ^) system, using the same energy
and scattering angle data used to obtain Fig. (71). We note that the
structure obtained for the (Li+, I^) system is awch simpler than that
for the (Li+, CO) system. Part of this simplicity is due to the fact
that NÂ£ is a homonuclear diatomic and odd Aj transitions are symmetry
forbidden. The interference pattern in the (Li*,, N2) results is also
less pronounced than that in the (Li + CO) resulits.
The experimental data to which we have compared the above results
(Bo76), do not indicate an interference structure. This, however, is
due to lack of resolution. Other experiments on (Li+, CO) have recently
been done which show various maxima in the energy loss spectra for rota
tional excitation (Ea78). In Table (72), we conpare the probabilities
of rotational excitation obtained from the analysis of experimental
Figure (38) Diagonal (K = K) matrix elements
for to(K', K; E). (Morse Potential)
145
Table (62) Convergence study showing how the transition probability
P^q for an exponential potential approaches that for a
hard core potential as a is increased in magnitude
(m = 0.5, A = 20.0 and rc = 2.0 in reduced units).
(Units of hu>)
(Hard Core)
(a=4.0)
(cel.5)
(ce0.5)
1.1
2.580
2.745
4.007
16.661
1.3
1.458
1.515
2.046
7.136
1.5
0.989
1.015
1.299
3.661
1.8
0.586
0.594
0.703
1.385
2.0
0.417
0.420
0.475
0.739
2.3
0.250
0.251
0.265
0.288
2.5
0.178
0.177
0.180
0.154
216
(Go69) R. G. Gordon, J. Chem. Phys. 51, 14 (1969).
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(Gr56) A. Gray and G. 8. Mathews, A Treatise on Bessel Functions,
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(Ha70) E. P. Harper, Y. E. Kim and A. Tubis, Phys. Rev. C2, 877 (1970).
(Ha71) Y. Hahn and K. M. Watson, Phys. Rev. A5, 1718 (1971).
(He50) G. Herzberg, Molecular Spectra and Molecular Structure, Van
Nostrand Reinhold, New York (1950), Vol. 1.
(Hi56) F. B. Hildebrand, Introduction to Numerical Analysis, McGraw
Hill, New York (1956), pp. 7176.
(Hi67) J. 0. Hirschfelder and W. J. Meath, Adv. Chem. Phys. 12_, 3
(1967).
(Ho68) A. R. Holt and B. L. Moiseiwitsch, Adv. At. Mol. Phys. Â£, 143
(1968).
(Ka58) L. V. Kantarovich and V. I. Krylov, Approximate Methods of
Higher Analysis, Interscience, New York T1958), Chap. II.
(Ka61) A. H. Kahn, Amer. J. Phys. 29, 77 (1961).
(Ka76) A. Kafri. Y. Shimoni, R. D. Levine and S. Alexander, Chem.
Phys. 13, 323 (1976).
(Ke58) E. C. Kemble, The Fundamental Principles of Quantum Mechanics
with Elementary Applications, Dover, New York (1958), Chap. III.
(Ke72) G. M. Kendall, J. Chem. Phys. 58, 3523 (1972).
(Kn77) E. W. Knapp and D. J. Diestler, J. Chem. Phys. 67, 4969 (1977).
(Ko73) D. J. Kouri, in Energy, Structure and Reactivity, eds. D. W.
Smith and W. B. McRae, New York, Wiley (1973).
(Ko75) D. J. Kouri and F. S. Levin, Nuc. Phys. A250, 127 (1975).
(Ko76) H. J. Korsch, Phys. Rev. A14, 1645 (1976).
(Ko77a) H. J. Korsch and R. Mohlenkamp, J. Phys. B 10, 3451 (1977).
(Ko77b) K. L. Kowalski, Phys. Rev. C15, 42 (1977).
(Kr71) H. Kruger, Max Planck Institute fr Stromungsforschung,
Gottingen, Report No. 110, 1971 (unpublished).
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TABLE OF CONTENTS (Continued)
Paqe
VI COLL INEAR SCATTERING 128
1. Formulation of the Coll inear
Scattering Problem 129
2. TwoBody tMatrix for the OneDimensional
Scattering Problem 133
3. Results for Inelastic Scattering 136
4. Dissociative Coll inear Scattering 146
VII THREEDIMENSIONAL INELASTIC SCATTERING RESULTS .... 163
1. Practical Implementation of the
Peaking Approximation 163
2. Numerical Results 168
3. Discussion 186
APPENDICES
I HARD CORE TWOBODY tMATRIX 196
II INTEGRALS USED IN "COMPARISON POTENTIAL" METHOD .... 200
III JWKB STARTING PROCEDURE 204
IV COORDINATES FOR THE THREEBODY PROBLEM 206
VANALYTICAL QUANTITIES USED IN THE COLLINEAR
SCATTERING MODEL 210
BIBLIOGRAPHY 213
BIOGRAPHICAL SKETCH 222
VI
Figure (36) Behavior of tg around E = 0 a.u. (Morse Potential)
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(A167) E. 0. Alt, P. Grassberger and W. Sandhas, Nuc. Phys. B2, 167
(1967).
(An76) P. Andersen, Chem. Phys. Lett. 44, 317 (1976).
(Ba22) H. Bateman, Proc. Roy. Soc. A100, 441 (1922).
(Ba64) D. R. Bates, C. J. Cook and F. J. Smith, Proc. Phys. Soc.
(London) 83, 49 (1964).
(Ba69a) R. Balian and E. Brezin, Nuovo Cimento 61B, 403 (1969).
(Ba69b) E. Bauer, E. R. Fisher and F. R. Gilmore, J. Chem. Phys. 51,
4173 (1969). ~
(Ba75) G. G. BalintKurti, Adv. Chem. Phys. 30, 137 (1975).
(Be60) R. B. Bernstein, J. Chem. Phys. 3^3, 795 (1960).
(Be63) H. A. Bethe, B. H. Brandow and A. G. Petschek, Phys. Rev. 129,
225 (1963).
(Be74a) V. B. Belyaev, E. Wrzecionko and B. F. Irgaziev, Sov. J.
Nucl. Phys. 20, 664 (1974).
(Be74b) R. B. Bernstein and R. D. Levine, Chem. Phys. Lett. 29, 314
(1974).
(Be75) J. C. Bellum, unpublished notes.
213
106
This is a final channel decomposition in which each term corresponds to
a collision event where the last step is an interaction through the
potential V, [(Wa67), (Mi72b)]. It should be apparent by comparing the
J
expressions for and T, that there is a close relation between the
J
two. Indeed, if t(j) is expressed as
t(j)(Z) = T. (Z)[1 + x(j)(Z)] (4.14)
J
where x^) is a distortion term, one is immediately led to the relation
3
x(j)(Z) = Z 6..G (Z)T(i)(Z) (4.15)
1 = 1 J 0
Finally, substituting Eq. (4.15) in Eq. (4.14), one obtains
T(J)(Z) = T.(Z) + T.(Z)G (Z) Z 6 (4.16)
J J 0 i=l J'
a set of three coupled equations. In matrix form the result is
T<1>
T1
o Tl Tl
TU)
t(2)
=
T2
+
t2 o t2
Go
l(2)
j(3)

_T3.
T3 T3 0
.
j(3)
Equations analogous to these were first derived by Watson (Wa57),
though Faddeev (Fa61) was the first to present a rigorous analysis of
their mathematical properties; as a consequence, they are known as the
FaddeevWatson equations. One important property of these equations is
that they become connected upon one iteration and therefore contain no
unfavorable delta functions. From a practical viewpoint, these equa
tions are the most appropriate expressions to use in describing scatter
ing processes in which both the entrance and exit channels correspond
to the breakup channel.
207
C
Figure (A41) Internal and relative position
vectors r.Â¡ and ft, i = 1, 2, 3,
respectively for channels 1, 2
and 3.
Figure (75) A plot of momentumoverlap volumes versus final rotational
quantum number for the (Li + CO) system. [ (n,,j, )e(0,0),
nf=0, E^^eV and Q^^i=37Io]
48
d i
dr
)
.J *
i i rU
(q. kE; r) = Aj[^ru1 (^r)] + b] [3 v] (^r)]
+ cÂ¡ & h ^ ]
(2.49)
for its derivative; the desired result
A
u(ri+1) = Ci[u(ri) C]j^(qri)3 + Cj (qri+1)
(2.50)
is obtained, where we have introduced the matrix definitions
w (r)
and
i(r) =
(O
jWi
dr 1
jj (qr)
d
dr J1
Ji (qr)
(2.51)
(2.52)
and the coefficient matrix C_., whose components are defined by the
relations
C11 = {u1 [d7>}/Wj
(2.53)
Ci2 iv(r,..,.,) u (r.) u ( ) v (r,)}/W
i+1
i+r
i' i
(2.54)
"21
 lafWr^jlt^rpn/H, (2.55)
and
'22 E {[3Fv(ri+l)I u(ri> '3Fi
(2.56)
97
worked quite well for soft potentials such as the Yukawa potential
studied by Walters (Wa71), and Rosenthal and Kouri (Ro73) [V(r) =
1.1825 r"1 e_r in units such that h = m = aQ (unit of length) = 1].
This can be seen in Table (36), where we have tabulated the onshell
tmatrix t(q', q; E) for q1 = q = kE = 1.816 a^1 as a function of the
angle Qqiq between q* and q. Comparing the Bateman results to the VPA
results, we see that the Bateman results are much better for small
values of the momentum transfer q' q. This was found to be true
y
in all the studies that were made. The set of plane wave states {k}
that was used in the above calculations is given in Table (37). Because
of the rotational invariance of the potentials involved, it is only
necessary to choose the set of vectors {k, k^, ..., kn) to lie on the
(q1 q) plane. If one specifies q as the reference vector, it is then
only necessary to specify and kp. The results given in Table (36)
were quite insensitive to any variation in the choice of {k}. Better
agreement was obtained, however, if the set included both q1 and q as
members of the set. A comparison with the calculations given by Walters
(Wa71), and Rosenthal and Kouri (Ro73) is also given. The agreement
between these results and those obtained from the VPA method is better
than that obtained from the Bateman method. The Bateman method, however,
is much easier to apply in the present case than either of these other
methods. Another observation that can be made from Table (36) is that
the Bateman method yields better agreement for the imaginary component
of t than for the real component. The reason for this difference is
not obvious and remains an open question.
We did extend the present study to soft exponentials with parameters
and units comparable to those shown above for the Yukawa case. The
108
where
y(J)(Z) = G0(Z)T(J)(Z)W0(Z) (4.23)
Introducing the FaddeevWatson equations for into Eq. (4.23) yields
y(J)(Z) = Yj(Z) + Z 5J.iG0(Z)Tj(Z)Y(i)(Z), (4.24)
having defined Y(Z) as
J
Yj(Z) = G0(Z)Tj(Z)W0(Z) = Wj(Z) W0(Z). (4.25)
Applying W(Z), as given by Eq. (4.22) to yields the desired final
channel decomposition for the wave function,
Y1(+) = Z (4.26)
j
where
T (J) = y(J') $ (4.27)
Here we have made use of Eq. (4.25) and the halfonshell relations
W0(Z)^i = ie G0(Z)$1 = 0 (4.28)
and
WjtZjij = ie Gj(Z)*1 = (4.29)
The Faddeev equations for the total wave function,
Tl(j) = 6ji$i + z I..G0(2)T.(1)T^) (4.30)
are now easily obtained from equations (4.24) and (4.27). The corre
sponding matrix form of these equations is
133
Gaussian envelopes of the harmonic oscillator functions and
in Eq. (6.10) peak when q^ and q^ equal zero, we have chosen
Q3* as (Ek74)
Q3* \ [Qjtqj1 0) + Q3(q3 0)1 (6.12)
which is equivalent to
Q3* = \ (P{ + Pj), (6.13)
as can be seen from equations (5.25c) and (5.25d). The choice of Q^*
is in general quite arbitrary as long as it remains within the overlap
region of the two momentum amplitudes. We have found M^, to be
quite insensitive to the choice of Q^*.
2. TwoBody tMatrix for the OneDimensional Scattering Problem
In the present treatment of coll inear scattering we shall always
maintain a close parallel to the 3D collision problem. Because of
this, we will deal only with symmetric potentials, i.e. those such that
V(x) = V(x). Furthermore, we will assume the presence of a hard core
in order to take into account the physical impenetrability of the
particles involved.
The analogue to Eq. (2.6), the differential equation satisfied
by the offshell wave function, is
[ + z = [z (2m)*1 q2] (6.14)
dx2 q
where
= (2tt)1/2 eiqx .
(6.15)
APPENDIX III
JWKB STARTING PROCEDURE
In propagating the numerical solution to the radial Schrodinger
equation
ip(r) + [k2 U(r) K1 t U]4,(r) = 0, (A3.1)
d rc r2
one generally starts at some rQ within the first turning point (Be60)
such that ^ is small. Trial calculations are then performed to test
the sensitivity of the final wave function to the starting point. If
the potential is very singular near the origin, this sensitivity is
indeed small (Be60); however, if the potential is not singular or if
the singularity is not very strong, this may not be the case. The
latter observation was found to be true for some of the calculations
reported in Chapter III, e.g. the offshell wave function studies done
for the Morse and Hulthn potentials. In order to remove this arbi
trariness we have adopted a JWKB procedure proposed by Bellum (Be75a).
It is well known that the JW^B solution to Eq. (A3.1) near the
first turning point, rt> has the form [(Mo58), (La37)]
*(r) [*$] [A J1/3 (w(r)) + B J_1/3 (w(r))] (A3.2)
where A and B are constants specified by the boundary conditions,
q(r) = [k2 U(r) l1 + ^/2)2 ]1/2 (A3.3)
r2
204
164
is formally related to the scattering amplitude by the relation (Le69)
da
dft
(l'l)
(2zL\4m 2 J
[ h > nl P,
1' 11
(7.2)
Here, we continue to work with the Jacobi coordinates introduced in
Chapter V. Since only inelastic processes will be considered, we have
chosen channel 1 as the entrance and exit arrangement channel. Intro
ducing the peaking approximation as given by Eg. (7.1), we see that
4 P,'
m.
(1^1) = r*L) m2 11 itPk F(2) (_Â£_ \ + tPk f(3) 1B_ + ,,2
dnu 1; {b> Pj 1*2 P1' 1 ^mBC ^rl; + t3 Fri v mBC APi)! ,
(7.3)
where
.mC
F'l (^BC AFl) =
APl* rÂ¡ I niJ imj,>
(7.4)
r
and r^ is the position operator for the relative coordinate of the
diatom (BC). Note that the initial and final states of the target
diatom are characterized by the quantum numbers {n^, jp m^}, with a
prime denoting the final exit state. To simplify our notation, we will
drop the "1" subscript in the above labels and the "j" subscript in the
magnetic quantum number itk.
In the experiments we shall discuss in the next section, the mag
netic quantum numbers are not analyzed. Therefore, the differential
cross section expression given by Eq. (7.3) will be replaced by
^ (n1 j'n,j;0+iÂ£ ) = (^)4
PlPl
^ Pj(2j+1) m'm
tPk f'(2)I + tPk f(3)i2
2 1"1 3 l'l1 (75)
where we have averaged over the initial magnetic quantum numbers and
28
(a.u.
18
20
22
24
CHAPTER
T
INTRODUCTION
With the advent of new experimental techniques, the modernday
chemist is now more than ever able to discern the specificity and selec
tivity shown by che reactants and products in a chemical system. The
goal of a chemist is to properly interpret and understand in a general
way the events that occur in a chemical reaction. The task at nand,
however, is quite formidable because of the complexity of the particles
involved. This complexity implies that the investigation of the simples
chemical systems is necessary to lay the foundation of a deeper and more
conceptual understanding of chemical phenomena.
In this dissertation we will be concerned with the investigation
of energy transfer processes in atomdiatom collisions at hyperthermal
energies. In particular, we shall be concerned with processes of the
type
A + BC A + BC
 B + AC
 C + AB
* A + B + C ,
where A, B and C represent the atoms in the atomdiatom system. Theoret
ically, atomdiatom systems are of interest because they exhibit the en
tire spectrum of possible modes of energy transfer, i.e., they allow for
vibrational and rotational energy transfer, as well as electronic and
translational energy transfer. Therefore, we see that the elucidation
1
Figure (65) Plot of the differential energy transfer probability
Pq(Ae) versus Ae as obtained from jhe stick
spectra corresponds to inelastic scattering and the
continuous curve denotes transitions to the continuum.
CHAPTER VI
COLLINEAR SCATTERING
In the previous chapter we proposed the application of the single
collision approximation to the study of inelastic and dissociative
atomdiatom scattering processes. We will now consider this approxima
tion for the case of coll inear scattering, which offers a simple and
inexpensive check on the validity and limitations of such an approach.
There is also the advantage of having an extensive literature from
which one can ascertain the relative merit of the present manybody
treatment when compared to other quantum mechanical, semiclassical or
classical treatments of the collinear scattering problem. An extensive
survey of collinear scattering may be found in the reviews by Shin (Sh76),
Rapp and Kassal (Ra69), and Takayanagi [(Ta63), (Ta65)].
From a physical standpoint it is hoped that a study of collinear
scattering will provide useful information about translationalvibra
tional energy transfer in atomdiatom collisions. The collinear config
uration is in general the most effective one for vibrational excitation
[(Sh76), (Ra69)]. However, one must be careful in generalizing the
results obtained from such a model. Collinear (1D) and full three
dimensional (3D) dynamical calculations differ in two very important
aspects (Be74b): the amount of phasespace available in 1D collisions
is less than that for 3D collisions, i.e. 1D scattering calculations
suffer from a "dimensionalitybias"; secondly, steric factors play a
more important role in 3Dcol 1isions. The importance of the role played
128
16
C(jk) = <5(J)5(k)> ,
(1.53)
(1.54)
is a diagonal matrix containing atomic energies and ^JK 1S a matrix
containing diatomic energies. The energies Ej are associated with the
unbound atom in channel j and the E_IK energies of the corresponding dia
tomic in the same channel. Equation (1.51) is a result of the Born
Oppenheimertype approximation
A(j') Ka (1.55)
and the DIM approximations
Ej (1.56)
and
A(j) Iik (157)
f i k 1
The quantities 1 in Eq. (1.51) act as recoupling coefficients among
the various channels, allowing for inelastic, rearrangement and dissoci
ative processes to occur. Furthermore, as long as the proper atomic
and diatomic energies are included, Eq. (1.51) allows for electronic ex
citation. A detailed application of this formalism to the [K + W^\ sys
tem is given in the literature [(Mi72b), (Mi77d)]. In these references
it was shown that one can construct a suitable expression of the ground
state potential energy surface of the system in terms of twobody inter
actions provided one uses spindependent potentials.
Atomdiatom collisions involving charge transfer cannot be treated
properly using Eq. (1.51) because this equation assumes a unique parti
APPENDIX IV
COORDINATES FOR THE THREEBODY PROBLEM
Consider three distinguishable particles (atoms) A, B and C. We
shall associate with each atom a, a equal to A, B or C, the atomic mass
nr,, the position coordinate r and the momentum coordinate p In each
a r a ra
arrangement channel i, i equal to 1, 2 or 3, of the three particle
system one is concerned with a free atom a and an interacting pair
(By); thus, it proves convenient to work with relative Jacobi coordinates.
These relative position coordinates shall be defined according to
Fig. (A41), where e.g., R^ is used to denote the position of particle
A relative to the center of mass of the (BC) pair with the relative
position coordinate r^. In order to give a general description of these
coordinates, it is useful to introduce the labels (ijk) and (aBy), which
indicate a cyclic permutation of the channel indices (123) and the par
ticle indices (ABC), respectively. Furthermore, we shall associate the
index i with the index a, j with B and k with y. The reason for this
association becomes clear if one considers the fact, e.g., that in
channel 1 particle A is free and particles B and C are bound. The
desired Jacobi coordinates can then be defined as follows,
(A4.1)
and
mBrB + W
mBy
(A4.2)
206
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
ENERGY TRANSFER AND DISSOCIATION IN HYPERTHERMAL
ATOMDIATOM COLLISIONS
By
Lyntis H. Beard, Or.
March 1979
Chairman: David A. Micha
Major Department: Chemistry
Energy transfer and dissociation processes in hyperthermal atom
diatom collisions are investigated using approximations based on the
multiple collision expansion of the Faddeev equations. Within this many
body approach, the collision process is treated as a sequence of atom
atom encounters. This allows one to obtain information on a threeatom
system using only twoatom transition operator matrix elements.
A new numerical procedure, the "variablephase and amplitude"
method, is developed to compute the required offenergyshel1 twobody
tmatrix elements for an arbitrary radial diatomic potential. An alter
nate approach based on the comparison potential method is also investi
gated, leading to an improved computational algorithm. Whereas these
methods are based on a partial wave decomposition of the twobody topera
tor, the need for nonpartial wave techniques is pointed out at high
energies and large momentum transfers. In this connection the Bateman
method is investigated.
In the present study, the single collision approximation is considered
for both collinear and threedimensional motions. In the collinear studies,
it is shown that multiple collision effects are important in the case of
vii
1.0
Li4, CO
075 
050
025 
0 
0
II
i
12
24
60
^4
CT
30
general function depending only on the relative orientation of the atom
and diatom. The V$r term, on the other hand, corresponds to the short
range potential and is characterized by a sum of manybody terms. In
Eq. (1.64), V was assumed to be a sum of radial twobody potentials;
one potential for each twoatom fragment in the atomdiatom system.
This breakdown of the potential circumvents the difficulties, mentioned
in the last section, in representing the short range behavior of V by
Eq. (1.62). Having introduced Eq. (1.63), we propose an impulse model
in which only the short range part of the potential is kept. The re
sulting problem is then ideally suited for the manybody formalism of
the Faddeev equations.
We note that, at low collision energies, one expects that the
approach of Li+ to CO or would heavily influence the orientation of
the diatom. This would be particularly true for a molecule with a
dipole moment like CO. The net result in our impulsive model would be
to overestimate rotational excitation. At the high energies considered
in this study, however, this effect is expected to be minimal.
7. Plan of Dissertation
The present investigation of atomdiatom collisions will be given
in three major parts. The first part presents an analysis of the two
body problem and discusses various computational schemes for the deter
mination of cffshell matrix elements of the twobody transition opera
tor (Chapters II and III). In the next part (Chapters IV and V), the
formalism of the threebody problem is presented and the various approx
imations obtained from the multiple collision expansion are discussed.
Finally, in the third part (Chapters VI and VII), computational results
Figure (37b) Diagonal (K', K) matrix elements of Im[tQ(K', K; E)].
(Hulthn Potential)
144
collision and peaking approximations break down. Secrest (Se69) has
shown that as m becomes larger multiple collisions become very important;
and thus, one would expect the single collision approximation to yield
incorrect results.
Other calculations using a potential of the type
00 if iXI C rc
x > r
(6.33)
c
were also carried out. In order to verify the reliability of the com
parison potential procedure used in evaluating the tmatrix, calcula
tions were performed using large values of A and a in Eq. (6.33) and
a comparison was made with the corresponding hard core results. Table
(62) gives one such study for P^g using the peaking approximation.
The parameters used in this set of calculations were m = 0.5, A = 20.0
and rc = 2.0, where the units are those compatible with the scaled
Jacobi coordinates given by equations (6.1) and (6.2). We note that
the exponential potential results approach the hard core values as a
increases in magnitude. Another important test of the single collision
approximation is its applicability to softer potentials. In the litera
ture there exists an extensive study of col linear scattering using a
soft exponential potential [(Sc75a), (C170), (An69), (Se66)]. Unfor
tunately the potential used is not that given in (6.33), but one that
is exponential throughout space, i.e. VgiXg) = &Â£'aX3 for all values of
Xg. Therefore, it is not at all clear that one should compare the
results using Eq. (6.33) with those reported in the literature. The
results obtained with Eq. (6.33) using small a d indicate that the
Â¡m t0(K,K;E)X103 (a.u.)
2
 2
 4
 6
 8
10 
K= 4.5 (a.u.)
H*H (Hulthen Potential)
_1_
J I I L
0.00
0.04
0.0 8
Energy (a.u.)
0.1 2
0.16
137
P 1 = "D2 ,
n'n P1P 1 1 1 11' 1 1
1 1
(6.30)
which is approximated by
2 2
3 4tt m
n'n = P1'P1
M(^I2
'lT
(6.31)
within the single collision approximation.
We will now consider various results obtained for a collinear
problem where the interaction between the projectile atom and its colli
sion partner is a hard core. These results are analogous to those found
in references (Ek71), (Cl70), and (Se66); our objective, however, is to
show the degree of discrepancy of the full single collision approxima
tion, and the peaking approximation within the framework of the multiple
collision expansion of the Faddeev equations. In figures (62a) through
(62c) we have plotted the transition probability P,n versus the kinetic
energy Ej of the projectile (Ej
 Pli
2M
10
) for the case m = 0.125. The
1
exact results obtained by using the coupled channels method of Shuler
and Zwanzig (Sh60) are given in Fig. (62a). The single collision non
peaking results, or those obtained from Eq. (6.10), are given in Fig.
(62b); and the peaking results, or those obtained from Eq. (6.11) are
given in Fig. (62c). A summary of the various integrals and analytical
quantities needed in these calculations is given in Appendix V. We note
that the nonpeaking results are closer to the exact results than the
peaking ones; however, the discrepancy is not large, and the greater
computational simplicity of the latter approximation makes it particu
larly attractive. Figure (62d) gives nonpeaking results that have
135
[(Li73), (Eb65), (Ka61)]. We have chosen the above formulation because
of its close parallel to the treatment found in Chapter II. In using
this approach however, one must give close consideration to the boundary
conditions placed on wj. If the potential does net contain an. infinitely
repulsive hard core, transmission from one side of the potential barrier
to the other must be taken into consideration. Since we have assumed
the presence of a hard core, must satisfy the
w, ~ 0 (6.23)
1 r^c
boundary condition at the hard core position rc and the asymptotic
boundary condition
f (qr) + cst eikr if E > 0
W1 <
r +
f^(qr) + cst eKr if E < 0 (6.24)
N
Note that when the asymptotic boundary of the full offshell wave
function o)(q, k^; x) is considered, equation (6.21) takes properly into
account the direction of the incoming v/ave.
The tmatrix may also be expanded in a form analogous to the
partial wave expansion of the threedimensional tmatrix. Making use
of the definition of t,
+O0
t(q1 q; E) = / dx V(x) (6.25)
00 '
and equations (6.15), (6.20) and (6.21) one obtains
t(q\ q; E) = I ^(cosG^) t^q', q; E) ,
(6.26)
102
a multiple collision expansion, which, as was pointed out in Chapter I,
can be used to describe the scattering of an atom and a diatom as a
sequence of atomatom encounters [(Ch71),(Mi75b)].
1. Problems with the LippmannSchwinger Equation
The main difficulty with the threebody problem lies in the
existence of various arrangement channels. By this we mean, that given
three particles (atoms) A, B and C, one can have any of four possible
arrangement channels, i.e.,
channel (1) corresponding to A + (B, C),
channel (2) corresponding to B + (A, C),
channel (3) corresponding to C + (A, B),
and channel (0) corresponding to A + B + C,
where an interacting pair is denoted by enclosing the given particles
in parentheses. This implies that for reactive scattering there are
difficulties in implementing the boundary conditions, at least in the
usual coordinate space formulations of the scattering problem [(Ma66),
(Li71), (Mi72a)], and for dissociative scattering there are even problems
with regard to the proper formulation of the boundary conditions (Me76).
For rearrangement scattering the problems begin with the choice of
coordinates, since the coordinates that are best suited for describing
the motion of the threebody system in the entrance channel are not the
most appropriate ones for describing the motion in the exit channel
(Ma66). Going further, let us consider an integral equation for the
threebody problem analogous to the LippmannSchwinger equation for the
twobody transition operator, i.e., the equation
T(Z) = V + VG0(Z)T(Z)
(4.1)
The integral in Eq. (A1.3) can be solved analytically using relation
(A2.2) in Appendix II. With a little algebra one can also show that
rr2 q2kE2 d
W q E> T XW Vqrc> drT jl(qV +
q q ^
+ j1(q,rc>rj1(qrc) +
 (q'rc) J1 (qi"c) In h1TkErc) (A1.4)
where the and hj^ are Bessel functions related to the Ricatti
/v ^ (1)
Bessel functions jj and h^v ; through the relations (Ab65)
(z) = z j1 (z) (A1.5)
and
h(1)(z) = z h(1)(z) (A1.6)
Equation (A1.4) has been reduced to the form given by van Leeuv/en and
Reiner (vL61), though these authors used a derivation different from
the one given above. It should be pointed out that there exist many
errors in the literature (Bi73) where people have neglected the first
term in equation (A1.3). The reason for this error is the neglect of
Eq. (2.18) in the above procedure when one applies Eq. (2.13).
It is interesting to note that in obtaining the total tmatrix by
summing over partial waves, the first and second term in Eq. (A1.4) lead
to the simple closed expression
201
The derivation of equations (A2.2) and (A2.3) is straight forward and
is usually given in standard texts on quantum mechanics (Me61) or in
the theory of special functions [(Fa71), (Gr66)]. If we now replace
A
one of the RicattiBessel function, say v^, in the left hand side of
Eq. (A2.2) by where satisfies Eq. (2.43) in the interval
[r^, ] one can actually follow a derivation analogous to that used
to obtain expression (A2.2), and the result is
r2
A
/ u,aJ dr
r.
A
w[ooq1 U] ]
2 2
[Vi V
'? (q2o k 2) r2 .
+ / j,(qr)u dr.
(q2 ) rl
rl
(A2.4)
In order to clarify this, we will now derive Eq. (A2.4). The starting
point is the identity
[
dr
2 1
+ (q lilil
)u 1 +
 U11 l1 + (aiKi2 +2 ^ ^ (aEkE2 q2)Ji = 0
(A2.5)
^ 1
which is obviously true since u^ and coj satisfy Eq. (A2.1a) and Eq.
(2.43) respectively. Rearranging and using the identity
d ur i id ~ d i
dr W[U u^ ^ ? U1 ~ ui ~2 W1
dr dr 1
(A2.6)
one obtains the desired relation, Eq. (A2.4). An expression analogous
to Eq. (A2.3) is also obtainable for the case in which q^ = a^ic^ in
Eq. (A2.4). Here one would use the identities
149
still measured in units of tiu, but oj in this case is
a) = a^D/r^)172, (6.37)
implying that the potential parameters a and D are related by
a = (2D)'1/2 (6.38)
The Morse vibrational and continuum functions and jy q^'>
present in equations (6.34) and (6.35) are well known analytically
[(Ek73), (En69), (FI71), (St35)] and there is no problem in their
evaluation. The same is true for the form factor
F01 = / dQ3 y lq1,> (6.39)
present in the expression for f,i(s,c,Pk) (^k73), and the analogous one
for inelastic scattering (Ek72). A summary of all these analytical
expressions may be found in Appendix V.
The construction of the transition probabilities from must be
handled with care. The relation
Pn(AEl) [cPj' Cj' TuP1n>2 (6.40)
analogous to the expression for P , Eq. (6.30), is actually a differ
ential energy transfer probability. In Eq. (6.40) <Â£^1 characterizes
the continuum state, Jy , of the diatomic, and Ae = e^' en. For
inelastic scattering the differential energy transfer probability would
be (Ek72)
P (Ae) e P ( ^t f1
nv n n v dn '
(6.41)
209
and
y By
(A4.8)
to cast equations (A4.5) and (A4.6) into the form
* = A %
(A4.9)
and
? .A A + V
i i ma m3y
(A4.10)
The relation between the various channel momenta can now be written
simply as
mft rtuM
_e_ n. (ij) L p.
maym8y J
i % j
(A4.ll)
and
!5P. + (ij) p.
%Y J J
P.
1
(A4.12)
151
to the left and right. An analogous result is obtained from Eq. (6.40),
since the Morse continuum function has two components which
asymptotically correspond to plane waves moving to the left and right.
This result is a direct consequence of the boundary conditions and the
requirement that be a bounded function (St35).
(s) (s,c,Pk)
The calculations using the expressions for Mq^; and M^ have
led to rather unexpected results. When the differential energy transfer
probability from the ground state was plotted versus Ac for the system
H + D2 (m = 0.2, n = 0), Mq^ led to a discontinuity at the threshold
( c n p 1/ ^
for dissociation, whereas did not. This may be seen by com
paring figures (64) and (65). The interaction potential was taken
to be a hard core and the value 12.28 in units of hw was used for the
Morse parameter D in Eq. (6.38). Eckelt and Korsch (Ek73) have reported
a similar study on this system, using a theory which leads to an expres
f s c P k)
sion analogous to that for M^ but with loss of time reversal
(c\
invariance. On the other hand the plane wave approximation M^',
Eq. (6.34), has been applied only to the threedimensional scattering
problem (Sh77), but not to the collinear problem. Because of the
discrepancy obtained in the present study between these two approxima
tions, the difference between them must be clearly understood. We note
that it is important to investigate that region of the energy loss
spectrum encompassing high vibrational excitation and transitions to
the continuum. This was not done in the threedimensional study men
tioned above. If one returns to the formal expressions [equations (5.1)
and (5.2)3 used in deriving (6.34) and (6.35), we see that they differ
as follows:
M01^ <$o!T2 + 13^l>
(6.48)
33
Micha (Ku78) have developed a variational approach leading to a
separable approximation to the tmatrix; and along more approximate
methods, Korsch and Mohlenkamp (Ko77) have developed a semi classical
approach based on the JWKB approximation which is applicable to repulsive
potentials.
The principal aim of this chapter will be to present a new computa
tional method for obtaining matrix elements of twoatom transition oper
ators between arbitrary momentum states, of high accuracy, and applicable
to any radial diatomic potential. The approach we have taken is based
on equations presented some time ago (vL61), which reduce the calculation
of transition operator matrix elements to integration of inhomogeneous
radial differential equations followed by numerical quadratures. Our
procedure is based on the wellknown variablephase approach [(Ca63),
(Ca67)], but extended to the present inhomogeneous equations. This
extension shows that, unlike the situation for standard radial scatter
ing equations, the variable phase is coupled at every distance with a
variable amplitude. To emphasize this point we refer here to a
"VariablePhase and Amplitude" approach. Another feature of our
approach is the 'way in which one can treat propagation through classi
cally forbidden regions. Along with the development of the VPA method
we shall present some suggested improvements to the comparison potential
method.
All the methods we have mentioned above make use of a partial
wave expansion to construct the twobody tmatrix elements. When we
discuss the singlecollision approximation in Chapter V, it will become
apparent that it would be desirable to have a procedure for the direct
computation of these matrix elements. This will even become crucial
46
local approximations to the potential in order to simplify scattering
calculations. The outcome of this pursuit will then be helpful in
suggesting some improvements to the VPA method, or for that matter any
method which implements a numerical solution for the offshell wave
function 1.
The basic idea behind the CP method is to approximate the potential
on a given interval in such a way that one can express the solution to
the resulting differential equation in terms of known functions which
present no problem computationally. To meet these requirements the
potential has often been treated locally as a step potential or as a
O
linearly varying potential (Go71). The centrifugal term, 1(1 + l)/r ,
may also be included in the potential, and the total effective potential
may be approximated (Go71). In the present work, however, only the
potential was approximated, taking it to be a collection of step
potentials. Thus one has
aiKi
HI + 1)
]uj
(aEk^ q2) (qr) ^
(2.43)
and
Vi = aEkE vi (2.44)
where ck plays a role analogous to and v^ is the local approximation
to v(r) for r. Â£ r < r.j+1. The advantage of this particular choice lies
in the fact that one can use Bessel functions as the local basis from
which to construct the solution to (2.43); and since the centrifugal
term is treated exactly, the convergence toward tike asymptotic boundary
condition (2.15) is accelerated in the case of the higher partial waves
(An76). Locally then, on the V interval
180
Table (72) Comparison of probabilities of
rotational excitation for the
(Li+, CO) system (n.. ,j.Â¡ ) = (0,0),
n^=0, E^4.28 eV and 6^p^=10
Column I gives theoretical results
and Column II the ones obtained
from experiment (Ea78).
P
jf
I
II
0
0.018
0.058
1
0.011
0.100
2
0.002
0.100
3
0.027
0.040
4
0.045
0.0
5
0.009
0.0
6
0.228
0.0
7
0.143
0.0
8
0.082
0.0
9
0.231
0.0
10
0.056
0.0
11
0.102
0.006
12
0.028
0.059
13
0.015
0.096
14
0.004
0.074
53
In conclusion we should point out the weaknesses of the Bateman
method. From Eq. (2.55) it is apparent that we have obtained a
separable approximation to the tmatrix. Furthermore, using Eq. (2.1)
it is easy to show that this could be viewed as a direct consequence
of the separable approximation
V = V>1 <Â£V (2.66)
to the potential [(Ad74), (Ad75)]. This is an important point, since
Osborn [(0s73a), (0s73b)] has shown mathematically the nonccnvergence
of tmatrix elements based on separable approximations to local poten
tials. This problem exists irrespective of the approach used in con
structing the given separable approximation. Fortunately, the diffi
culties arise for large values of the momenta q1 and q in Eq. (2.63),
values at which the tmatrix elements are small and have no significant
effect in threebody calculations (SI73). Finally, there remains the
problem of the inapplicability of the Bateman approach to singular
potentials, a question open to further investigation.
36
[2 + ok2  v(r)Jfa) (q,k ;r) = {o k2 q)j (qr), (2.10)
where is +1 if E is positive or 1 otherwise, and v(r) = 2mV(r).
The matrix elements of the toperator can now be obtained from those of
the wave operator by using equation (2.3). Thus one has the relation
t(q\ q; z) = / V(r) df, (2.11)
which may also be decomposed into partial waves with the result
t(q\ q; z) = (tt)"1 E (21 + 1)P, (q1 q)t, (q, q; E), (2.12)
where
tj (q1 > q; E) =  / j] (q'r)V(r)w (q, kÂ£; r) dr. (2.13)
This definition of the partial wave matrix elements tÂ¡(q, q; E) follows
directly from equations (2.8), (2.9) and (2.12).
2. Boundary Conditions For The OffShell Wave Function
In order to obtain the offshell wave function w^(q, k^; r), a
solution to equation (2.10) must be found subject to two boundary condi
tions (vL61):
^(q, kE; r) r ; o 0 (2.14)
and
(q, r) ~ j, (qr) + C,h{+^(k r), (2.15)
where is a constant and hj+) is a RiccatiHankel function (Ca67).
Thus, w^(q, k^; r) is a complex wave function which asymptotically
E (eV)
i
00
o
oo
ro
O
202
ru,[ wi1 + (q2 11^ (aFkF2 q2)ji(qr)n = 0
1 dr2 1 1 r2 1 E L 1
3F 3F11 )( BFl) 
/ d i w d A n / d ~ w d
r( uÂ¡1 )(  u, + r( u )( y (o
dr 1 dr2 I dr 1 dr 1
rÂ¡^ *
)!
dr
and
(Ir t^i2 1(1 r2 ) rwl1 V = 2qi Vi1'
(q2 1^1 p ^ ) [ru1 ( cd i ) + rw 1 ( u1) u^1 ] .
r
These relations can easily be verified using equations (A2.1a)
The final answer is
r2
r i *
J to u,dr
rl
2q
Vdrt apa.,1 )( ^G,) u,1 ( ^ u,)
 (qj2 1(1 V ) nolGj] 2 +
r2 rl
+ (q2 aEkÂ£2) / (qr)( p u] )dr}
(A2.7)
(A2.8)
(A2.9)
and (2.43).
(A2.10)
Finally, the integral on the right hand side of Eq. (A2.10) can be
obtained from the relation
3.87
1.50
0.88
3.25 ^
24
jf
0
12
T
6
o
36
48 60
o
ro
37
behaves as the sum of a component from a plane wave and that of an out
going spherical wave, which, unlike onshell scattering expressions,
depends on different momenta (Ta72). For potentials containing a hard
core (2.14) must be replaced by
w,(q, kp; r) ~ 0 (2.16)
L r > r
c
where r is the hard core radius. Once the offshell wave function is
c
obtained, the partial wave matrix elements t (q1, q; E) follow directly
from (2.13). However, if the potential contains a hard core, care must
be taken in applying Eq. (2.13). If one integrates equation (2.10) as
follows,
rc+e ? rc+e m ,\
/ [wV + 0Â£kE 03j3 dr = / {[v(r) + J ] ^ +
rce rce r2
+ (oÂ£kj q2)j(qr)} dr, (2.17)
it is apparent that the product to^(q, k^.; r)v(r) does not vanish inside
the core region even though w^(q, k^.; r) does. In fact, for a potential
containing a hard core (La68),
v(r)w1 (q, kE; r) = 6(r rÂ£) ^^(q, kÂ£; rQ+) +
(q2 aÂ£k2) ^ (qr) (2.18)
for r < r where
c
!im wj (q, rQ + e) .
e  o+
,(q, kE; rc+) =
(2.19)
217
(La37) R. E. Langer, Phys. Rev. 51, 659 (1937).
(La58) R. Laugnlin and B. L. Scott, Phys. Rev. 171, 1196 (1968).
(Le56) R. R. Lewis, Jr., Phys. Rev. 102, 537 (1956).
(Le58) W. A. Lester, Jr. J. Comp. Phys. 3, 322 (1968).
(Le69) R. D. Levine, Quantum Mechanics of Molecular Rate Processes,
Oxford University Press, London ~(969X> Chap. 3.
(Le70) I. N. Levine, Quantum Chemistry, Allyn and Bacon, Boston
(1970), Vol. II, Chap. 3.
(Le76) William A. Lester, Jr., in Dynamics of Molecular Collisions,
Part A, Ed. William H. Miller, Plenum Press (1976).
(Li71) J. C. Light, Adv. Chem. Phys. 19^, 1 (1971).
(Li73) H. J. Lipkin, Quantum Mechanics: New Approaches to Selected
Topics, NorthHblland, Amsterdam (1973), Chap. 8.
(Li78) T. K. Lim and J. Giannini, Phys. Rev. A18, 521 (1978).
(Lo64a) C. Lovelace, in Strong Interactions and High Energy Physics,
ed. R. G. Moorhouse, Oliver and Boyd, London (1964).
(Lo64b) C. Lovelace, Phys. Rev. 135, B1225 (1964).
(Lo65) PerOlov Lowdin, J. Chem. Phys. 43, 3175 (1965).
(Ld68) PerOlov Lowdin, Phys. Rev. 139, A357 (1968).
(MaoG) R. A. Marcus, J. Chem. Phys. 45, 4493(1966).
(Ma74) B. H. Mahan, in Interactions Between Ions and Molecules,
ed. P, Ausloos, Plenum, New York (1974).
(Ma75) B. H. Mahan, Accts. Chem. Res. 8, 55 (1975).
(Ma76) B. H. Mahan, W. E. W. Ruska and J. S. Winn, J. Chem. Phys. 65,
3888 (1976). ~
(Mc70) M. R. C. McDowell and J. P. Coleman, Introduction to the
Theory of IonAtom Collisions, NorthHoli and, Amsterdam (1970).
(Me61) A. Messiah, Quantum Mechanics, NorthHoli and, Amsterdam (1961),
Vols. I and II.
(Me76) S. P. Merkuriev, C. Gignoux and A. Laverne, Ann. Phys. (N. Y.)
99, 30 (1976).
112
describe processes in which the incoming atom interacts with only one
of the target atoms in the diatomic, while the third atom acts as a
spectator. The higher order terms in Eq. (4.40) would then correspond
to double and higher order encounters. The role of the {V in a"
these terms is quite important, for they exclude the possibility of
terms leading to disconnected diagrams and rule out terms which do not
contribute to the given process on physical grounds (Ch71). Because of
this conceptual picture, this series, Eq. (4.40), is known as a multiple
collision expansion [(Ch71), (Mi75b)].
Each term in the multiplecollision expansion has now been asso
ciated with a direct mechanism for an atomdiatom scattering event.
What one hopes is that only a limited number of these terms give a signi
ficant contribution to the scattering amplitude. The success of results
obtained by truncating the expansion would then be indicative of the
types of mechanisms involved in a given physical system. For reactive
scattering, there already exists ample evidence to corroborate the
existence of a stripping mechanism in certain atomdiatom reactions,
e.g., Minturn, Datz and Becker (Mi67) have shown that a classical
spectator stripping model properly describes the location of peaks in
the product distribution of various alkaliatomhalogenmolecule reactions
[Cs + Br^, K + (I^ or Br^)]. Within the context of the multiple colli
sion expansion,similar success was obtained by Yuan and Micha [(Yu76a),
(Yu76b)], who used the spectator stripping model to study the systems
[Ar+, (H^, D2, HD)] and [K + [l^, B^, IBr)]. It should be pointed out
that the expression for the transition amplitudes,
Mnr = <*ilG0'i(z,lii> .
1,
(4.41)
43
An important consideration is the singular nature of the VPA
equations near the origin. Equations (2.36) and (2.37) contain the
potential as a multiplicative factor. This problem can be overcome by
assuming hard core boundary conditions or by using a simple step poten
tial within the first classical turning point [(Be60), (Br75)]. The
second of these procedures was chosen in this work. The length of the
first step, r^, and the value of the assumed step potential over this
first step, Vj, were chosen such that making any further optimal changes
in these parameters would not lead to a significant change in the value
of the calculated matrix elements, ^(q, k^; r) and tj (q1, q; E). The
guess work involved here can be eliminated by using a JWKB approximation
to the derivative of the imaginary part of the wave function near the
first classical turning point. The approximations involved are given in
Appendix III. From expression (2.10) it is apparent that Im[w^(q, kr; r)]
satisfies the usual Schrodinger equation and boundary condition at the
origin. It therefore follows that within the first classical turning
point Im[u)] is decaying exponentially, and is zero at the origin. Ail
that is necessary then, from a computational point of view, is to start
the integration in a region where the magnitude of the derivative of
Im[coj] is smaller than some given value. In our calculations a value of
10 was found to be adequate. Another important consideration in
propagating the VPA equations is the cost involved, in evaluating a
large number of Bessel functions. Because of this, the coupled equations
were integrated numerically using an Adams numerical integration routine
developed by Shampine and Gordon, which*is particularly suited for
problems for which derivative evaluations are expensive. These codes
and a discussion of their efficiency and accuracy are found in reference
(Sh75).
57
t(q q;
z) = I
nlm
^enl ; q1 {'Z ~ Â£q)
(z Enl)
Jnlm
(q')
nl
n
(q)
+ t(q, q; eq+)
+ j dq!l [ (eqi. cq is)'1 + (z Â£k)"1]
x t(q' q"; eqn+) t*(q", q; cqn) (3.9)
where ^nm(q) is a bound state function in the momentum representation
having the eigenvalue of enÂ¡. The quantities, eq, are eigenvalues to
the scattering states of the system.
2. Numerical Calculations Using the VPA Equations
We will now present various results for the system for the
lowest *Lg+ and ^Lu potentials [(Mi77c), (0178)]. Some of these results
have already been reported in the literature [(Be78), (Ku78), (Ko77a)].
A Morse potential
V(r) = D[1 e_a(r"ro)]2 D (3.10)
was used to represent the ^Ig+ interaction potential, and a Hultnn
potential
V(r) = Ae'r/a[l e"r/a]_1 (3.11)
3
was used for the repulsive I interaction potentials. The parameters
0_i o
used were D = 4.786 eV, a = 2.1123 A 1, rQ = 0.7411 A, A = 20.11 eV and
o
a = 0.4984 A (Mi77c). This particular study will! prove interesting,
since it serves to draw a contrast between the analytic properties of
t and toj obtained from an attractive potential, and those obtained from
a strictly repulsive potential.
39
where r< is the lesser of r and r', and r> is the greater of r and r'
(Ro67). At this point, the discussion will be restricted to positive
energies, since this would not involve a loss of generality. It is
readily seen that equation (2.21) is just the corresponding differential
equation to (2.10) in the absence of a potential. Thus, oj^(q, kÂ£; r)
may be written as follows:
00
u(q, kÂ£; r) = N'(kÂ£r) + / dr'g^r, r') [ (k q2) jj (qr') +
v(r )aj] (q, kE*, r1)] (2.23)
A
where N' is a constant. Since j^(qr) satisfies equation (2.21), the
first term in the above integral,
(kE q2) C dr'9i r)Vqr') *
may be evaluated exactly. This may be accomplished using the relation
(A2.2 ) given in Appendix II. Introducing the quantities A^(q, kÂ£; r)
and B^(q, kÂ£; r) defined by
Aj (q, kE; r) = N + k^1 / dr'v(r' )uj (q, kÂ£; r') n1(kÂ£r') (2.24)
and
B(q. kÂ£; r) = k"1 /[ dr'v(r' )w, (q, kÂ£; r') J, (kÂ£r1), (2.25)
where N = N' + 1, and using equation (2.22), one obtains
l(q* kE; r) = (qr) + A^ (q, kE; r) j] (k^)! +
 Bj (q, kE; r) n] (k^r)) .
(2.26)
2
of the dynamics of atomdiatom collisions would provide an impetus to
the eventual understanding of more complicated systems. Our goal in
this chapter will be to clarify and discuss the treatment of electronic
structure in atomdiatom scattering processes. The theoretical aspects
of the scattering problem will be the subject of the remaining chapters
in this dissertation.
1. Theoretical Treatment of Nuclear and Electronic Motions
A formal treatment of an atomdiatom system requires that one con
sider both the nuclear and electronic degrees of freedom of the system.
In other words, one would have to solve a manyparticle problem. From
a practical standpoint, such a problem would not be tractable unless
approximations were introduced. In this section therefore, we will
sketch various approaches that can be taken to treat the atomdiatom
system as an effective threebody problem.
The first approach we shall consider is that of introducing approx
imations which allow for the separation of nuclear and electronic motions.
We begin then by considering the Schrodinger equation
(HE)T = 0 (1.1)
where E is the energy of the system, 'F is the wave function and the Hamil
tonian H can be written as (in atomic units)
a
12
j L vf + V
1 = 1 1
coul
+ H
SO
(1.1)
in the centerofmass system (Fr62). The Greek indices in Eq. (1.2) refer
to the nuclei and the Latin indices refer to the N electrons in the atom
Table (61) Comparison of exact and approximate results for P
10
Ej
(Units of hw)
P10(m = 0.125)
PjQ(m 0.5)
Peaking
Single
Collision
Exact
Peaking
Single
Col 1ision
Exact
1.001
18.60
9.21
0.004
65.162
38.955
0.010
1.2
0.856
0.614
0.201
1.845
1.455
0.435
1.5
0.757
0.609
0.349
0.989
0.848
0.715
1.8
0.720
0.611
0.478
0.586
0.521
0.912
2.0
0.698
0.606
0.554
0.417
0.377
0.987
2.2
0.676
0.597
0.514
0.297
0.271
0.674
2.5
O.f 40
0.577
0.499
0.178
0.165
0.382
2.8
0.603
0.550
0.464
0.106
0.099
0.121
3.0
0.577
0.530
0.434
0.075
0.071
0.007
3.2
0.550
0.509
0.439
0.053
0.050
0.001
3.5
0.511
0.476
0.424
0.031
0.030
0.049
3.8
0.471
0.442
0.410
0.018
0.017
0.127
4.0
0.446
0.420
0.400
0.013
0.012
0.126
4.2
0.421
0.398
0.378
0.009
0.009
0.112
4.5
0.385
0.365
0.351
0.005
0.005
0.089
S*
CO
4
Hel 7 . 7i + Vcoul
I = 1
(1.8)
U =u(eD + u(SO)
uit un1 un
(1.9)
ui i ^ <i  Â¡
(1.10)
Uj 11 <(f>i I I >>
(1.11)
T
IT
2
= Z
6=1
1
2u3
T
IT
2
Z
6=1
(1.12)
(1.13)
4t = ^iiVi^ (1*14)
and
1T = d15)
These equations are well known in the literature [(Fr62), (Hi67), (Tu76a)],
and in general form the basis of most treatments of atomic and molecular
scattering problems. Our notation and discussion follow that of Tully
(Tu76a)].The diagonal matrix elements Ujj(r^) are the effective potential
energy surfaces for the atomdiatom system and the nondiagonal matrix
elements and U^, are the coupling elements responsible for
the promotion of electronic transitions. The diagonal term is a non
adiabatic correction to Ujj. The diagonal term rj can be shown to be
exactly zero if the electronic functions {<}>^} are taken to be real (Hi67).
6
approach well defined atomic states asymptotically (0m71). There is no
unique recipe for constructing such representations. In general they
are chosen such that the momentumdependent coupling terms xjj, are
small (Sm69). One should note however, as Tully (Tu76a)points out, that
diabatic representations are useful in atomatom scattering because in
those situations avoided crossings in the adiabatic curves are localized
to a point. In other situations where there are more nuclear degrees
of freedom, as in atomdiatom scattering, one is dealing with avoided
surface crossings at which large variations in the nonadiabatic couplings
can occur. This indicates that one might not be able to find a diabatic
representation which is suitable in a global sense.
2. Adiabatic Collision Processes
Because of the complexity of atomic and molecular collisions, most
work in the literature has focused on the theory of adiabatic collision
processes. The theoretical investigation of such processes is, however,
by no means trivial. For example, in the case of inelastic atomdiatom
scattering, Eq. (1.18) would generally involve solving a large set of
coupled differential equations. If a large number of open channels for
vibrational and rotational excitation exist, an accurate and economical
solution to Eq. (1.18) may be difficult to obtain. In order to solve
the problem for such collision processes, further approximations have
been sought which reduce the dimensionality of such coupled differential
equations. A review of these techniques is given by Micha (Mi74), Lester
(Le76) and Rabitz (Ra76). In the case of reactive scattering the situa
tion is even more complicated. For an atomdiatom system, there are
four arrangement channels [A + BC, B + AC, C + AB and A + B + C] and as
199
t(q', q; E) 
2 JiUqrc)
r
c 2
2tt Aq
(A1.12)
Thus, we see that such an expression is a planewave or Born approxima
tion to the hard core tmatrix. Other approximations based on addition
formulas such as Eq. (A1.10) could be pursued. This for example has
been done by Bethe, Brandow and Petschek (Be63) in applications of the
BruecknerGoldstone theory of nuclear matter; however, approximations
analogous to those used by these authors were not found to be satis
factory in the present case of atomdiatom collisions at hyperthermal
energies.
150
where Ac = e t en The unitarity relation when dissociation is
allowed would be (Le69)
N
Z
n'=0
P ,
n n
Pn(Ae)dAe = 1 ,
(6.42)
where the integral is over the energy range allowed by energy conserva
tion, which in this case is from the threshold of dissociation
(Dn = D en) for the Morse oscillator to the kinetic energy, avail
able from the projectile. If Eq. (6.34) is used to obtain Pn(Ae), one
also has the problem that the exit channel is characterized by
where
= (Pj Px) (6.43)
whereas
<Â£j' e> = 6(e e) (6.44)
The transformation between the two normalizations is given by the rela
tion
6(ei e) = [6(p Pj) + 6(Pl' + P1)]p(p1) (6.45)
where
3Â£i i i Pi i
p(pl) = (6.46)
1 oPj m^
is the density of states (Le69). This implies that
pn(as) = vri l^ll + lToilpin>l2 <647>
where one finds that the particles of the diatomic are being scattered
Figure (310b) Plot of Im[ti(K, K; E)] versus 1. (K = 4.0 a.u. and
E = 0.01 a.u., Hulthn Potential)
Figure (64) Plot of the differential energy transfer probability
(s')
Pq(Ae) versus Ac as obtained from The stick
spectra correspond to inelastic scattering and the
continuous curve denotes transitions to the continuum.
The continuous part of the spectrum is scaled by a
factor of 1/2.
inelastic scattering near threshold, and that for the case of dissocia
tive scattering they become crucial. A study is also made of the valid
ity of the single collision peaking approximation, and it is shown that
this approximation breaks down when soft twoatom potentials are involved.
In order to compare to experiment, threedimensional results are presented
for the scattering of Li+ with N~ and CO at hyperthermal energies.
viii
85
c.
Table (34)
Selected values of tj (q', q; E) are
given for E = 0.15 a.u. and the par
tial wave numbers 1 = 0, 5 and 10.
Column I gives the VPA calculations
ant column II the results obtained from
a variational procedure (Ku78). (Hulthn
Potential)
1 q'
q
t, (q1 q; E)
x 103
I
II
0 3.0
1.0
0.7899
0.7894
2.0
4.0665
4.0693
3.0
10.187
10.179
4.0
2.1074
2.1160
5.0
0.2154
0.2476
5 3.0
1.0
0.0207
0.0207
2.0
1.1745
1.1755
3.0
5.3773
5.3709
4.0
1.0412
1.0434
5.0
0.0513
0.0547
10 3.0
1.0
0.0005
0.0009
2.0
0.1174
0.1186
3.0
1.4905
1.4857
4.0
0.2644
0.2668
5.0
0.0093
0.0125
42
These equations are analogous to those for the onshell case, where
q = kE (Ca63). However, for the onshell case the phase and amplitude
equations are uncoupled.
The boundary condition satisfied by the VPA equations at the
origin follows directly from equations (2.14) and (2.33). From these
expressions it is apparent that
sin [6,(q, kE; r)] ~ 0 (2.38)
r o
and that the value of aj(q, kÂ£; r) for the same limit is arbitrary, and
becomes fixed only after the asymptotic boundary condition (2.15) is
applied. On the other hand, if a hard core is present
 i,(qr )
cti(q, kp, ) *" ^
1 L r * rc cos(61)j1(kErc) sin(61 )n, (kÂ£rc)
(2.39)
and <5(q, kE; r ) is arbitrary.
4. Computational Aspects of the VPA Equations
The procedure used in calculating ^(q, kE; r) consists of propa
gating two solutions to the VPA equations (2.36) and (2.37) such that
they satisfy the initial boundary condition, and constructing linear
combinations of these solutions so as to satisfy the asymptotic boundary
condition (2.15). It is possible to construct two linearly independent
solutions that satisfy the initial boundary condition due to the
inhomogeneous character of equation (2.10). This procedure is analogous
to that found in reference (Br75); however, instead of a piecewise
breakdown of the potential and the propagation of local solutions as
given there, numerical integration is involved here.
162
which give plots of P^, P,, and P.q versus Ae. is that the envelope
always reflects the initial momentum distribution of the target diatomic.
This supports the idea discussed in Section V.2 of the information
obtainable from either the simple overlap volume weighed by the semi
classical momentum distribution, or from the form factor FQ1 present
in the expression for M^' Figure (66) is also interesting in
that it is an example of how the present approach can be used to study
scattering of excited targets, something which is quite difficult to
do with other coupled channel procedures.
CHAPTER V
THE SINGLE COLLISION APPROXIMATION
In this chapter we will investigate the single collision
approximation as it applies to inelastic and dissociative atom
diatom collisions. As can be seen from Eq. (4.35), the multiple
collision expansion for the transition operators of interest is
T11 = T2 + T3 + T2GoT3 + T3GoT2 +
(5.1)
for inelastic scattering, and
T01 V1 + T2 + T3 + T2GoT3 + T3GoT2 +
+ T.G T, + T.G T, +
1 o 2 1 o 3
(5.2)
for dissociative scattering. The single collision approximation to
the scattering amplitudes would then be
Ml'i = <*1'IT2 + T3IV (53)
and
M) .
01
= < <*o'T2 + T
IV
(5.4)
where the higher order terms in equations (5.1) and (5.2) have been
neglected. Note that the term GQ in Eq. (5.2) does not contribute to
the breakup amplitude in the onshell limit. In the following sections
we will discuss the practical implementation of equations (5.3) and
114
Table (36) Comparison of onshell tmatrix elements for the attractive Yukawa potential
V = 1.1825 r"ler given by Walters (Wa71) [Units such that h = m = aQ = 1
are used]. The parameters E = 1.649 and q1 = q = kE = 1.816 a01 are used.
eq1 q
(degrees)
t(q1, q;
E) x 102
VPA
Bateman
Walters (Wa7l)
Rosenthal and Kouri (Ro73)
0
5.529il.872
5.778il.817
5.484il.877
5.623il.786
10
4.986il.841
5.212il.791
20
3.811il.755
4.024il.708
30
2.664il.632
2.900il.556
40
l.819il.490
2.051il.367
60
0.818il.208
l.016il.147
90
0.201i0.880
0.384i0.866
0.187i0.897
0.229i0.856
120
0.215i0.679
0.212i0.672
150
0.1040.574
0.091i0.581
180
0.125i0541
0.083i0.558
0.139iO.552
0.12710.512
CO
130
The particular form of equations (6.1) and (6.2) allow us to work in
energy units of hw,
w = [k/m^1^2
(6.5)
The resulting Schrodinger equation for the threebody system in the
centerofmass frame is (Ra69)
where
(6.7)
(6.8)
and Xj) is the interaction between the projectile A and the
struck atom B. M is the total mass of the system (m^ + mB + mc).
Note that and are not the same as before. They correspond to a
modification of the original potentials consistent with the new set of
coordinates and units. A pictorial representation of the coordinates
used is given in figures (6la) and (6lb). From Eq. (6.6) and Fig.
(6lb) it should be clear that we have transformed our former three
body problem into one where an oscillator of unit mass, vibrating about
an equilibrium position, is struck by a particle having an effective mass
m; or equivalently, into a problem where a harmomdic oscillator having a
particle of mass and one of unit mass, is struck by a particle of mass
m. We note, as pointed out by Secrest and Johnson (Se66), that the mass
parameter m corresponds to many different atomdiiatom problems and is
thus useful in studying the importance of mass effects in atomdiatom
collisions.
obtained within the stripping model is equivalent to the Qppenheimer
(0p28), and Brinkman, Kramers approximation (Br30),
113
i 1 il
> V $i>
i j i
(4.42)
which was introduced in the study of electron capture processes in
atomic physics (Mc70). The extension of these studies to include
multiplecollision effects has generally been pursued only within the
framework of classical mechanics [(Ba64), (Su68), (Ge69), (Ma74),
(Ma76)], but the success of these models has been impressive consider
ing their simplicity. We defer the study of single and higer order
terms to the next chapter, where a more detailed analysis will be
given.
Figure (13) Contour plots of the ground state potential
energy surface for the (Li+, N?) system as a
function of the NN internuclear separation
(St75). Each curve corresponds to a fixed
value of R, the distance between Li+ and
the ceriterofmass of the N2 molecule.
(7.19)
? 6j(j+1)Bgh 1/2
a) = [< +
cm,r 2
1 e
and
nj
re[l +
j(j+l)Beh
3j(j+l)Beh +^cm1re2(ue)2
]
(7.20)
The function Hn in Eq. (7.16) is the nth Hermite polynomial. Anharmonic
effects could be incorporated into Eq. (7.18) by replacing we by an
effective vibrational constant (Yu76a)
o)n = we 03exe(n + 1/2) + toeye(n + 1/2)2 .
(7.21)
The corresponding vibrationalrotational eigenenergies are
(he)"1 Wnj = )n(n + 1/2) + Bej(j + 1) De[j(j + l)]2 (7.22)
where the equilibrium rotational constant Be, the centrifugal distortion
De and are all in units of cm"1 (Le70).
It is clear that the presence of the Gaussian term in Eq. (7.16)
limits the range of integration used in evaluating the radial form
factors. Therefore, the numerical evaluation of Eq. (7.9) is rather
straightforward and no complications occur. We have used a 32point
Gauss integration procedure as suggested by Philipp and coworkers (Ph76).
The use of a higher order quadrature led to no significant improvement
215
(Eb65) J. H. Eberly, Amer. J. Physics 33, 771 (1955).
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University Press, Princeton (1974), Chapter 4.
(Ek71) P. Eckelt and H. J. Korsch, Chem. Phys. Lett. _U, 313 (1971).
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(Ek74) P. Eckelt, H. J. Korsch and V. Philipp, J. Phys. B 7_, 1649
(1974).
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Cimento 64A, 16 (1969).
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to Gamma, Beta, Legendre and Bessel Functions, Dover Publications,
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Systems, McGrawHill Book Co., New York (1*971), p. 28.
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W. E. Brittin et a 1., Interscience, New York (1959), Vol. 1.
(Go60) G. Goertzel and N. Traili, Some Mathematical Methods of Physics,
McGrawHill, New York (1960), Chap. 15.
Figure (3llb)
Plot of Im[t(K', Â£; E)] versus AP = K' 
(K= 4.0 a.u. and E = 0.01 a.u., Hulthn P
r+ 7^4
2
*1 O
iu
M167) R. E. Minturn, S. Datz and R, L. Becker, J. Chem. Phys. 44,
1149 (1966).
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(Mi74) D. A. Micha, Adv. Quantum Chem. 8, 231 (1974).
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J. L. Calais et al., Plenum, New York (1976).
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HH (Morse Potential)
E = 0.1378 (a.u.)
i
8
K (a.u.)
12
i
X1
i
Figure (61) Coordinates used in coll inear
scattering.
(a) Jacobi coordinates
(b) Scaled Jacobi coordinates
212
AP ap ap
X ,F9 (n,s+s +i^, s+s +i; l+2s ,n+i%; 1)
it na na n a
+ [s S] ,
(A5.ll)
where
A = Ae'An(4D)1AP/a,
e n
(A5.12)
V =(sA')1/4,
(A5.13)
s = i(2Â£')1/2/a
(A5.14)
<5 (s) = arg[r(2s)/r(JgsD)] = 6(s)
(A5.15)
and [s s] indicates that a term identical to the first one must be
included with s replaced by s. We note that the above expression for
Fqi requires the evaluation of complex gamma functions. These quantities
may be evaluated using the Stirling's formula (Fr74)
lnT(z) = j ln(2ir) + (z h) In (z) z
1
1260z5
(A5.16)
This relation is valid for the condition
iarg(z)  < ir (A5.17)
and will yield ten digit accuracy if z[_>15. The recursion relation
T(z + 1) = zr(z) (A5.18)
can be used along with Eq. (A5.16) for z<15.
170
Table (71) Parameters used in peaking calculations
[(He50), (Mi 78b)]
Spectroscopic
C0[X1Z+]
N2[XlSg]
Constants
we (cm1)
2170.21
2359.62
Be (cm1)
1.9313
2.010
re (X)
1.1281 A
1.094 X
0
AtomAtom
Pair
re(A)
(Li+ 
C)
1.17
(Li+ 
0)
1.05
(Li+ 
N)
1.05
a consequence Tree motion in the asymptotic region is governed by differ
ent channel Hamiltonians. This aspect of reactive scattering introduces
complications, because the boundary conditions become difficult to apply
and in addition no unique and simple set of coordinates is suited for
propagating the solution in all regions of configuration space. We note
that in each rearrangement channel, a natural choice of coordinates would
be that of the Jacobi coordinates and r., where F^. describes the rela
tive motion of the "free" atom with respect to the center of mass of the
diatom and r. the relative motion of the diatom. The label "i" is a
channel index (see Appendix IV). In order to overcome the difficulty
with the choice of coordinates, Marcus (Ma66) introduced the concept of
natural collision coordinates, e.g., a set of coordinates which goes
smoothly from one set of Jacobi coordinates to the other. However,
these coordinates complicate the structure of the kinetic energy opera
tor and may in some instances become multivalued according to how they
are constructed [(Ma66), (Wi77)]. A clarification of many of the problems
of reactive scattering may be obtained by considering the structure and
manybody aspects of the wave function [(Mi72b), (Re77)]. A general
review of reactive scattering may be found in the articles by George and
Ross (Ge71), Kouri (Ko73) and Micha (Mi75a).
3. Nonadiabatic Collision Processes
Nonadiabatic collision processes have not been studied in as much
detail as adiabatic processes for two general reason. One reason is the
added complexity of dealing with a coupled set of equations [Equations
(1.6) and (1.16)]; the second reason is the lack of information on the
potential energy surfaces and on the various couplings involved. It has
Figure (7
2) Plot of differential cross section (arbitrary units)
versus final rotational quantum number for the (Li+, CO)
system. [ (nj ,j.Â¡ )=(0,0), nf=0, E]=4.23eV and 0p. p =43.2]
Figure (34) 12/tt]1/2 [w5(K, Ke; R)/K] for K = 4.5 a.u. and E = 0.153 a.u.
(Morse Potential)
127
The latter expression would lead to, e.g.,
H<0;C>(3> / d3 < *3 (V vE iK'v <544)
an expression analogous to Eq. (5.24), where however, <4>^,lq^> is a
continuum momentum wave function for the diatomic BC in channel 1.
The consequence of Eq. (5.43) will also be explored in the next chapter
in connection with coll inear scattering.
198
t(5, J. E) . (E SLi ,,2
m 27TAq
r 2
rc
mm lm
L Y*m(^,)Ylm^)jl(q'rc)j1(<,rc> Lln h<1)(kErc>> (A1'7>
where
Aq q* q (A1.8)
In obtaining Eq. (A1.7), we have made use of Eq. (2.12) and the identi
ties (vL61)
(2w)3 / dr _ e1q'7c = lilil! *5 > (A1.9)
drc 2,r2 Aq
i Â£ Ytm (q,) Ylm(q) J1(q'rc> 5^ jl(qrc> (A1'10)
In order to clarify the meaning of the first term in expression (A1.7),
it is instructive to consider the expression Eq. (A1.2), sum it over
partial waves to obtain u>(q', q; E) and use this expression to obtain
the total tmatrix. Doing this one obtains
q; E> (fsrE) rc:
J1(Aq rc)
2u^ Aq
+
(Al.ll)
If one were to approximate w(q, E; r ) by a plane wave, one would get

