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 Permanent Link:
 https://ufdc.ufl.edu/UF00097498/00001
Material Information
 Title:
 Electronic and dynamical aspects of diatomic systems /
 Added title page title:
 Diatomic systems, Electronic and dynamical aspects of
 Creator:
 Bellum, John Curtis, 1945
 Publication Date:
 1976
 Copyright Date:
 1976
 Language:
 English
 Physical Description:
 xiii, 218 leaves : ill., diagrs., graphs ; 28 cm.
Subjects
 Subjects / Keywords:
 Angular momentum ( jstor )
Atoms ( jstor ) Electronics ( jstor ) Electrons ( jstor ) Energy ( jstor ) Ionization ( jstor ) Mathematical variables ( jstor ) Memory interference ( jstor ) Orbitals ( jstor ) Wave functions ( jstor ) Collisions (Nuclear physics) ( lcsh ) Dissertations, Academic  Physics  UF Physics thesis Ph. D
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 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Thesis:
 ThesisUniversity of Florida.
 Bibliography:
 Bibliography: leaves 212217.
 Additional Physical Form:
 Also available on World Wide Web
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by John Curtis Bellum.
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 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.
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03173496 ( OCLC ) AAU6759 ( NOTIS )

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Full Text 
ErLECTRONIC AND DYNAMICAL, ASPECTS
Of DIATOMIC SYSTEMS
By
JOHN CURTIS BELLUM
A D:;SESTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UITIVERSITY OF FLORIDA
1976
Dedicated to
My Parents,
who, though familiar with little of what is reported here,
nonetheless know what I have been doing
and have supported me in it.
ACKN OWLEDGE MENTS
I want to express my deep appreciation to my advisor,
Professor David A. Micha, for the direction, support and
financial help he has provided me during this doctoral re
search. I am most grateful for his high caliber of scien
tific excellence and integrity, and for the great patience
he has shown in guiding me in research.
I owe a debt of gratitude to the numerous people, asso
ciated at one time or another with the Quantum Theory Pro
ject, who have been of special assistance and encouragement,
and whose influence has come to bear either directly or in
directly on this dissertation. Among these are several I
want to mention in particular. Professor N. Yncve Ohmn pro
vided me financial assistance during my first months at the
Quantum Theory Project, and has since maintained interest
in my work and progress. Professor Erkki J. Brandas and
Dr. Rodney J. Bartlett initially introduced me to research
in quantum chemistry, and generously made available to me
their expertise and enthusiasm. In connection with multi
plescattering and local exchange related matters, Profes
sor John W. D. Connolly. through lectures an? ready atten
tion to my inquiries, was of great help. Close associations
with Drs. Suheil F. Abdulnur, Poul JOrgensen and JianMin
Yuan, and Professor Manoel L. de Siqueira, have been per
sonally beneficial as well as significant to my overall
perspective in science. The able leadership of Professor
erCOlov Ldwdin as director of the Quantum Theory Project,
along with his nuTmerous lectures and also his interest in
philosopbtical considerations in science, have played an
important role in my graduate education.
As is the case with any undertaking, a program of
graduate studies provides a context and situation in which
one is able to grow personally in all respects. In this
regard I want to express my gratefulness to many friends,
both scientist and nonscientist alike, as well as to God,
my creator, by and in whom I exist.
PREFACE
I find it appropriate to make some remarks concern
jng the perspective and context in which the work reported
in this dissertation has been carried out.
Since its inception in the 1920's, Quantum Mechanics
has become well established as the suitable framework
within which to describe phenomena of a physical and chem
ical nature. Building upon only a few axioms, the formal
Quantum Theory manifests itself in the form of mathemati
cal equations, the solutions to which determine expressions
for calculating physically observable quantities. Confi
dence in Quantum Mechanics derives from the impressive
successes it has had in providing results in agreement with
experiments. However, in applying Quantum Mechanics to
describe actual, known, physical and chemical phenomena,
one quickly becomes aware of the fact that there are only
a few cases where an exact treatment has been possible.
In nearly all cases of interest, the mathematical equa
tions of the theory, though succinct in what they say, are
unmanageable to solve. Computational considerations,
therefore, have strongly influenced theoretical investiga
tions in physics and chemistry.
Basically two approaches have evolved. The socalled
ab initio calculations provide approximate solutions to
the "exact" quantum mechanical equations within a frame
work which, in principle, allows for the solutions to be
progressively improved upon to approach the "exact" solu
tions. One is presumably limited here only by the size
of the electronic computer available. On the other hand,
one may focus on the main features of some particular
physical phenomena of interest, and use the "exact" quan
tum mechanical equations only as a guide in order to ar
rive at approximate equations which mimic the important
aspects of the process being studied. These approximate,
or model, equations are many times only approximately solved!
Such apparently crude approaches require of the researcher
all of the physical intuition which can be mustered, in
order to properly assess the important features of the
physical situation and to approximate them reasonably well.
Nevertheless, much physical insight and many useful quan
titative results can be extracted from this point of view.
Indeed, the task of science is essentially to formulate
descriptive statements, both qualitative and quantitative,
which conform as nearly as possible to the laws and phenom
ona of nature as we observe them.
Finally, it should be mentioned that there are formal
results coining out of quantum theoxeticel investigations
which determine many characteristics of the "exact" and
approximate solutions, even before they have been calcu
laced. Both types of investigations described above rely
upon these formal results as well as upon each other. In
the dissertation which follows, research of the second
type mentioned above will be reported in the form of a
quantum mechanical investigation of electronic and dynam
ical aspects of diatomic systems.
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS . . . . . . . .. iii
PREFACE . . . . . . . . . . .
ABSTRACT. . . . . . . . . . . . x
CHAPTER
1. INTRODUCTION . . . . . . . ... .
1. A Formal Statement of the Problem. . . 2
2. Remarks Regarding Reference Frames . 6
3. Remarks Regarding the Wave Function. .. 17
4. The Coupled Equations and Coulomb and
BornOppenheimer Couplings .. . .. 19
5. Discussion . . . . . . ... 24
II. A STUDY OF SINGLEELECTRON AND TOTAL ENERGIES
FOR SOME PAIRS OF NOBLE GAS ATOMS. . . .. .28
1. Theoretical and Computational
Considerations . . . . . ... 30
2. Results. . . . . . .. . . . 39
3. Discussion .. . . . . . . 55
IIT. DIATOMIC MOLECULAR ORBITAL CORRELATION
DIAGRAMS FOR PENNING AND ASSOCIATIVE IONIZA
TION . . . . . . .. . . . 59
1. MO Calculations for He*+Ar and He+Ar . 61
2. Analysis of PI and AI Processes Based on
MO Correlation Diagrams. . . . . 74
3. Estimating MO Correlation Diagrams for
Diatomics. . . . . . . . ... 87
4. Discussion . . . . . . . . 110
viii
TABLE OF CONTENTS
(Continued)
Page
IV. A COUPLED CHANNELS APPROACH TO PENNING
IONIZATION OF Ar BY He*(ls2s,3S) . . . .
T. The Scattering Problem in Terms of
Discrete and Continuum Electronic States .
2. Discretization of the Continuum and
the Modified Coupled Equations . . .
3. Solution of the Modified Coupled
Equations. . . . . . . . .
4. Characteristics of PI and AI Processes
5. An Application of Discretizution to
PI and AI . . . . . . . .
6. Interaction Potentials for He' (ls2s,3S)
Ar and He + Ar+(3p5,2p). .. . .....
7. Parameterization of the Couplins. . .
8. Results from Coupled Channels Calcula
tions of He*(ls2s,3S) + Ar PI Collisions
9. Discussion . . . . . . . .
REFERENCES. . . . . . . . . . . .
SBGGPAPHICAL SKETCH . . . . . . . . .
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
ELECTRONIC AND DYNAMICAL ASPECTS
OF DIATOMIC SYSTEMS
By
John Curtis Bellum
August, 3976
Chairman: David A. Micha
Major Department: Physics
Diatomic systems are considered from the points of
view of their electronic structure and the dynamics of
motion of the heavy particles (nuclei) upon collision. In
Chapter I the electronic and nuclear motions are treated
formally by expressing the Schr6dinger equation for the
nuclei and electrons in independent variables in the body
fixed (BF), center of mass of the nuclei, frame, and then
introducing an expansion in terms of a complete set of
electronic states at each internuclear separation, R. Born
Oppenheimer and Coulomb couplings between the electronic and
nuclear motions are pointed out and discussed.
In Chapter II, atomatom interaction potentials and
electronic structure are investigated using the Multiple
Scattering (MS) calculational scheme with the socalled Xa
approximation to electronic exchange. The atom pairs,
aeHe, HeAr and ArAr are studied, and results are presented
for ground and excited state configurations. The computed
interaction potentials exhibit BoEnMayer repulsion, and the
calculations demonstrate how Xa orbital energies can be
used to predict crossings between interaction potentials.
An analysis is presented of the Xa theory and its usefulness
for the undsrstandinq of collision phenomena.
In Chapter III, Penning ionization (PI) and associative
ionization (AI) processes are considered in terms of molecu
lar orbital (MO) correlation diagrams. MO correlation
diagrams are calculated for He*(ls2s) + Ar(3p6) and He(ls2)
+ Ar (3p5) within the MSXe scheme for nonspinpolarized
and spinpolarized orbital. The ionization process is
discussed in terms of an Auger type mechanism involving
MO's which can be inspected in the unitedatoms (UA) limit
in a way which permits an analysis of the angular momentum
contributions of the emitted electron in the BF frame. MO
correlation diagrams are constructed based on atomic orbital
energies at the separated and unitedatoms limits, which
are determined from data available in the literature on
ground state atomic orbital energies. Estimated MO correla
tion diagrams are presented for He*(ls2s) + Ar(3p6),
+ Xr(4p6 ), + Hg(6s2), and Ne*(2p53s) + Ar(3p ),and in each
case an analysis is made of the angular momentum components
of the emitted electron. The results confirm that relatively
few such components are important for electrons emitted in
PI and AI. The UA analysis shows the importance of spin
polarized MO's, and also BornOppenheimer rotational
couplings, particularly between MO's which converge to the
same UA limit.
In Chapter IV consideration is given to the dynamics
involved in collisions Lonization processes. The formal
development of Chapter 1 is extended to include both discrete
and continuum internal electronic states. The resulting
continuously infinite set of coupled equations is then dis
cretized, leading to modified coupled equations for the
heavy particle motion. Discretization provides a suitable
framework in which to introduce physically reasonable ap
proximations which lead to a treatment of PI and AI in terms
of several (=20) twostate coupled equations. Application
is made to PI of Ar by He*(ls2s,"S), and the results show
that the approach includes the important dynamical features.
Partial ionization crosssections per unit energy, E, of
the emitted electron are calculated as a function of E, and
they show an s dependence in good qualitative and quanti
tative agreement with experimentally measured energy dis
tributions of emitted electrons. Partial crosssection
contributions for the heavy particles in specific angular
momentum states are also singled out." Their behavior as a
function of e, or of the angular momentum partial wave
number, shows structure which reflects regions of high
density of states in the continuum of final relative motion
of the heavy particles. The HeAr netastable and molecular
ion potentials are represented by a convenient functional
form describing atomatom int.raction potentials over the
entire range of R. In addition, the connection between the
decay width T and the coupling matrix elements between
discrete and continuum electronic states is used to make
reasonable estimates of the latter from semiempirical results
for P.
xiii
CHAPTER I
INTRODUCTION
The overall subject of this dissertation is the study
of electronic and dynamical aspects of diatonic systems.
In such a study it is the behavior of the electrons and
nuclei, during collision processes of the two atoms com
prising the diatomic, which is of interest. In this intro
ductory chapter the collision processes will be discussed
formally in terms of the Schrodinger equation which is sat
isfied by the wave function for the system of nuclei and
electrons. The Schr6dinger equation will be treated in the
bodyfixed, center of mass of the nuclei, frame by properly
transforming the Hamiltonian operator and wave function to
be expressed in this reference frame. The coupled equa
tions will be derived with attention focused on the various
sources of coupling between the electronic and nuclear mo
tions.
In the following chapter, matters specifically concern
ing the electronic structure of some diatoms comprised of
rare gas atoms will be considered, and possible applications
to collision processes discussed. In Chapter III special
attention will be given to chemiionization processes in
volving collisions in which one of the atoms is initially
in an excited state. Features of the electronic structure
of such collisions will be discussed, on the basis of which
an analysis of the angular momentum contributions to elec
trons emitted in such processes will be carried cut. Fi
nally, in Chapter IV, th: dynamics of a specific chsmi
ionization process, that of Penning ionization of Argon by
metastable Helium,will be treated by means of numerically
solving the coupled equations within a twostate approxima
tion. Total and partial crosssections obtained from the
calculations will be reported.
I. A Formal Statement of the Problem
In the quantum mechanical treatment of the two nuclei
and N electrons which constitute a diatomic system, the
Ilamiltonian operator expresses all of the energy contri
butions associated with the nuclei and electrons. The
actual form of the Hamiltonian depends upon the coordinate
frame with respect to which the positions of the nuclei
and electrons are located. The coordinate frame of most
practical use in terms of measuring the results of colli
sion events is one fixed in the laboratory, referred to as
the laboratoryfixed (LF) frame. The description of the
collision events, however, is most conveniently carried
out in a coordinate frame whose origin is fixed to the cen
ter of mass of the nuclei (CMN) of the two atoms which are
,olliding. The reader is referred elsewhere Csee, for
example, Pa6S and references therein, and Ju75) for a de
tailed discussion of various coordinate frames commonly
used and the form the Hamiltonian assumes under transfor
mation from one to the other. The formal development in
this chapter begins with the Hamiltonian expressed in the
socalled laboratoryfixed, center of mass of the nuclei
(LFCMN), frame, whose origin is fixed to the CMN and whose
axes remain parallel to those of the LF frame. Furthermore,
relativistic and masspolarization contributions to this
Hamiltonian are neglected (Pa68), and the coupling between
electronic spin and orbital angular momentum is assumed to
be small. In this LFCMN frame, R is the relative position
vector of the two nuclei labeled a and b, having masses m
a
and mb, respectively. The N electrons are located by the
set of spacespin coordinates {x.,i=.,N} = X, where
x. = (ri,s.), ri locating the spatial position and s. the
spin coordinate (a or 0) of the ith electron.
In terms of these variables, the LFCMN Hamiltonian
is expressed as
2
H(R,X) = (1/2m)VR + H(RX)
where m = m mb/(ma+mb) is the reduced mass of the nuclei,
and
N N
H e(R,X) = 1/2 (Z /r + /r )
e,(,2 r. _, a ia b io
i=l ri i=1 /
+ I/rij + Z Zb/R (2)
i
is the electronic Hamiltonian. In the usual sense, Za,
r. and r.. refer to the charge on nucleus a, the distance
:iaF 13
between the ith electron and nucleus a, and the distance
between the ith and jth electrons, respectively. The ex
pressions are in atomic units, where the unit of energy is
the Hartree, the unit of distance the Bohr radius (ao)
and the unit of mass that of the electron.
On the right side of Equation (1) the first term rep
resents the relative kinetic energy of the two nuclei,
which are the heavy particles taking part in an atomatom
collision. The terms on the right of Equation (2) repre
sent, in order, the kinetic energy of the electrons, the
Coulomb attraction energy of the electrons with the nuclei
a and b, the electronelectron Coulomb repulsion energy
and the Coulomb repulsion energy between nuclei a and b.
The description of a diatomic system can formally be
made in terms of the wave function, YCR,X), for the total
system of nuclei and electrons, which satisfies the time
independent Schradinger equation,
H(R,X)Y(R,X) = ET(R,X) ,
where E is the total energy of the system. In solving
Equation (3) it is important to pay attention to the angu
lar momentum' of the electrons and nuclei. The total or
bital angular momentum. K, is the sum of the nuclear and
( ) (e)
electronic orbital angular moment, and L where
Z(n) 
= x iVR (4)
and
N
+(e) N
L"'  r. x iV (5)
i=l r
2 +
The LFCMN components, K K and K as well as K = KK,
y z
all obey the usual commutation relations for angular momen
ta, and commute with the LFCMN Hamiltonian of Equation (1),
owing to its rotational invariance. Accordingly, the solu
tion, T(R,X), of Equation (3) is simultaneously an eigen
2
function of K and K with eigenvalues K(K+l) and M,
respectively, and the total orbital angular momentum and
its z component are constants of the motion. Because of
the as umption of negligible spinorbit coupling for the
electrons, the spin angular momentum has been left out of
this discussion for convenience, but could easily be in
cluded. So, with no loss of generality, the solution to
Equation (3) is classified according to the constants of
the motion, K and K and is written
Y(ax) = KM(R,X)
KM
SRemarks Regarding Reference Frames
At this point attention will be turned toward specif
cally pedagogical considerations, providing a reminder of
some basic concepts of a mathematical and physical nature
which are helpful in understanding the approach which will
be taken in solving Equation (3). The remarks which fol
low will serve to reiterate some key ideas which have long
been established (see, for example, Kr30). The ideas are
not easy to grasp, and are often passed over either in too
sophisticated or too cursory a way in the literature. No
table exceptions, however, can be found (Kr30, V151, Ho62,
Th6i, Th65, Pa68, Sm69).
As it stands, Equation (3) involves the Hamiltonian
and wave function, Yi', expressed as functions of R and
{ri} referred to the LFCMN frame. The axes of this frame
are labeled by x, y and z. The ith electron is thus lo
cated by ri having coordinates (xi,yi,z.). R is most con
veniently represented by its spherical polar coordinates,
R, 0 and ,. Therefore, in the LFCMN frame, (x.,yi.,z,
R.0,,) constitute an independent set of coordinates in
which to solve Equation (3).
In terms of these coordinates, the components in the
j(n) *(e)
LFCMN frame of t) and Le) (Equations (4) and (5)) ap
pear as
L(n) = i(sini/ae6 + cotocosfs9/9')
x
L(n) = i(cos +/9 + cotBsin0/ah)
Y
L(n) (7)
L = i /a (7)
z
and
N
(e) = i= (v/3z z.i/y.i)
L(e) = i (zi.3/xi xia/zi)
L(e) = i I (xiN/3y yi/3xi) 8)
2 Ci=l
The wave function of Equation (6) can also be written ex
plicitly in terms of these variables to read
SKMRW2) = YKM(R,O,I,xYi ,z ) (9)
Also, the first term on the right of Equation (1), express
ing the relative kinetic energy of the nuclei, appears as
follows (see, for example, Co62):
(1/2m)V2 = (1/2mR2 { 2/R(R23/DR)
+ cot63/oe + a2/Do2
+ sin2 e2/~ 2}
(10)
One can check that the terms within the brackets of Equa
tion C10), involving the angles, can be replaced by
I(n' (n)
Ln' according to Equation (7) (see, for example,
Ed60). It is important to keep in mind that partial de
rivatives, such as occur in Equations (7), (8) or (10),
depend upon which variables are actually independent of
one another during differentiation. Because the LFCM
frame constitutes an inertial reference frame, the set of
variables, (x ,yi,z ,R,e,4), of the electrons and nuclei
is indeed an independent set of variables. Therefore, for
example, 8/36 in Equation (10) means to differentiate with
respect to 8 while holding all other variables, (xi,Yi,zi,
R,4), fixed. Similar considerations hold in turn for each
variable in this set.
Now one would like to proceed to solve Equation (3)
by expanding the wave function of Equation (9) in a com
plete set of electronic wave functions at each R. It is
here that the need arises to express Equation (3) in terms
of variables referred to a coordinate frame in which the
internuclear vector R is fixed. This is due to the fact
that normally electronic wave functions are determined in
such coordinate frames. Such a reference frame, fixed to
the nuclei with origin at the CMN, will be called a body
fixed, CMN (BFCMN) frame. The key concept regarding elec
tronic wave functions is that they are usually calculated
under conditions where the set of electronic coordinates
in the BFCMN frame are treated as independent variables.
The BFCIN frame considered here shall have axes x',
y', z", where the 2" axis is along the internuclear vector
R. Consequently, the angles ( and 6 are the first two of
the Euler angles Csee, for example, Ju75) which rotate the
SFCMN frame into the BFCMN frame. The third Euler angle,
y, can be freely chosen since it only serves to define the
x' and y' axes, which can be arbitrarily set for a. diatomic
molecule (Ju75). Therefore, y is an auxiliary variable,
and as such will play only an indirect role in what follows.
Figure 1 shows the Euler angles (9y) by which the SFCMN
frame is rotated into the BFCMN frame.
The coordinates of the internuclear vector in the BF
CMN frame are simply (R,0,0). The ith electron, with co
ordinates (xi,yi,'i) in the LFCMN frame, has coordinates
(x,yl,z') in the BFCMN frame, specified for a given ori
entation (06,) of the internuclear vector by the unitary
transformation (see, for example, Ti64 and Ju75)
x. = Ex(COy) = (cosycosecos sinysinc)x.
+ (cosycosfsin + sinycost)yi cosysin9zi
y = y' (0Oy) = (sinvcosfcos{ + cosysinf)xi
wI 1
+ (sinycosesin + coscosi)yi + sinysinOz.
l
10
 y
The Euler angles (0y) reiating the SFCMN
frame (x,y,z axes) to the BFCfN frame
(x',y',z axes). y specifies the x' and v'
axes, which can be arbitrarily chosen and are
not shown. R labels the internuclear vector.
Figure 1.
11
z. = z'(00) = sinOcosix. + sin9siny.i + coszi (11)
Here attention has been drawn to the dependence of x ,y'
and zI on the angles 6 and y. The inverse transformation,
1
th
qjiving the coordinates (x.,yi,zi) of the i electron in
the SFCMN frame in terms of its coordinates (xi,yC,z)
in the BFCMN frame, is,
x. = (cosycosOcos sinysinp)xC
(sinycos6cosj + cosysinp)y + sin6cos6z'
y. = (cosycosOsin4 + sinycost)x1
+ (sinycosOsint + cosycos)y. + sinOsin2z
z. = cosysin9xC + sinysinly$ + cosOz (12)
With these expressions, the Schr6dinger equation of
Equation (3) in the LFCMN frame, where (x.,Yi,z.i R,,i)
are independent variables, can be rewritten allowing the
BFCMN electronic variables (x,,y',zi) to be treated as
independent variables (as they are in molecular electronic
structure calculations). The expression, "treated as in
dependent variables", in the preceding sentence speaks
to an important concept. To an observer in the BFCMN
12
frame, the electronic and nuclear coordinates are simply
x' ,y',z',R,0,0). But such an observer must keep in mind
that the BFCMN frame is not an inertial frame, and, by
Equation (11), (x,y',zf) have explicit (86y) dependence
for a given set of electronic coordinates in the LFCIN
frame.
In order for an observer in the BFCMN frame to use
the electronic variables Cx ,y',z ) as independent variables
in treating Equation (3), two new variables, 6' and 4',
may be introduced with two restrictions: (3.) the set
(xryt,z3,R,6',') must be an independent set of variables
for the observer in the BFCMN frame, and (2) 6' and #^
must be given by,
= ,
(13)
According to this equation, ("O') may seem to be redundant
variables, but this is not at all the case. The variables
(Cx,y,zt) are independent of (6'o') as far as the observer
in the BFCMN frame is concerned, and Equation (13) simply
specifies their values in terms of variables which are de
termined by an observer in the LFCMN (inertial) frame.
Equation (13) gives the relationships for 8' and "' just
as Equation (11) does for x',yv and z'.
Equation (3) will now be transformed so that an ob
Equation (3) will now be transformed so that an oh
33
server in the BFCMN frame could attempt to solve it in
terms of the independent set (x 'y!,z,R,6',I'). Sach a
transformation involves both the Hamiltonian operator (see
Equations (1), (2) and (10)) as well as the wave function
(see Equation (9)). First the Haiiltonian will be consid
ered.
For clarity, when performing partial differentiation
involving the independent variables Cx ,y,z i,R,9,4) the
symbol 3, which has already been used, will now specific
ally indicate that,when a variable of differentiation from
this set has been singled out, all the others are held
fixed. For instance,
a/as = a9/l{(xy ,z.),R,} (14)
where the variables held fixed are explicitly indicated.
Similarly, 3' will be used to indicate partial differen
tiation involving the independent variables (x',y',zi,
R,m e,''). By analogy with Equation (14) for instance,
= (15)
Then, if w is one of the variables Cxi,yi,zi,R,e6,),
14
3/3w t (+e'/w)V'/2e' + (U4t/2w)V'/24^
N
+ ((x /3w) 3'/@x + (3yC/Dw) 3'/y' *+ ( 3z/3w) '/9z) (16)
A similar relation holds for expressing 3'/Dw" as a linear
combination of partial derivatives involving the variables
(xi,yi,zi,R,6,4) where w' is cne oE the variables (x.,yi,
zt,R,9',O'). These are useful] expressions to keep in mind.
especially as to their meaning emphasized by using the 3
and 3' notation. This conceptual and notational viewpoint
follows the work of Kronig (Kr30).
Referring now to Equation (2), the kinetic energy of
th
the i. electron involves the operator,
i2 2 x 1 a2 2 2 2
V2 = 2/x2 + 2/Y2 + 2 /2z2 (17)
r ii i
On the basis of Equations (11) and (16), one can write
3/;xi (cosycosecos4 sinysin)}a/3x'
(sinycosOcosp + cosysin)3 '/2y' + sin8cos'/Dz: (18)
1 1
with similar expressions for 2/9yi and 3/2zi. Using these
expressions directly, one finds that
2 2 2/ y 2 +2 2/
3 /3x.+D /9y. + 2 /2z.
,2/ax 2 + 2a2/y2 + ,2 /az 2 = V2 (19)
i
15
The potential energy terms in Equation (2) involve only
the distances between electrons and nuclei, and therefore
are unchanged in going from the variables (xi,yi,zi,R,0, )
to the variables (X ,y ,z,R,6 ,'). Consequently, the
electronic Hamiltonian of Equation (2) takes The same form
for an observer in the BFCMN frame as for an observer in
the LFCMN frame.
Such is not the case for the first term of Equation
(1), which is expressed in the variables (R,6,<) in Equa
tion (10). By a straightforward, but tedious, application
of Equation (16), using Equations (11) and (12), one finds
that
(e) (e)
3/30 '/38' icosyL isinyLxe (20)
and
e) (e) (e)
3/3(  a'/8' icosO'L Ce + isin0'(cosyL sinyL )
2 x y
(21)
(e) (e) (e)
Here, L L and LI are the components of the elec
x y z"
tronic orbital angular momentum in the BFCMN frame. They
are expressed according to Equation (8) by replacing all
LFCMN quantities by their appropriate BFCMIN (primed)
counterparts. Equations (20) and (21), again with some
tedious algebra, lead to the following result for the an
gular terms in Equation (10):
C6
cot3/89 4 2/302 + sin 2 02/32
Scot8."3'/3ae + ^ /9ae0 + sin 6 '3 /WS
.e i:! e) + + iyL(e_ D /a0 kL'(e) 2
47a
2i(cot0/sinO)3'/'L ) cot2 .L ) 2 + (Le)
z 2 z
 (2/sin9) (cosyL ) sinyL ) (ie'/4g cosSL e) (22)
x Y z
(e (e) C (e)
Here, in the usual way, Le)" = LX + iL ). The R depen
dent terms in Equation (10) are unaffected in the transfor
mation from the LFCMN frame to the BFCMN frame. Conse
quently, replacing the angular terms of Equation (10) by
the right side of Equation (22) leads to the appropriate
Hamiltonian operator (see Equation (1)) which can be used
by an observer in the BFCMN frame in order to formally
treat the behavior of the electrons and nuclei.
The first three terms on the right side of Equation
(22) are similar to those found in the LFCMN Hamiltonian.
The remaining terms are those compensating for Coriolis ef
fects due to the fact that the BFCMN frame is not an in
ertial frame. In a sense, the inclusion of these Corolis
terms is the price paid by the BFCMN observer in order to
reckon (x ,y',z',R,6',0') as independent variables.
17
3. Remarks Regarding the Wave Function
The wave function of the total system of electrons and
nuclei (see Equation C91) mush also be properly transformed
and expressed in terms of appropriate functions in the BF
CMN frame. Here the approach of Davydov (Da65) is adopted.
Attention is drawn again to the fact that the wave
function of Equation (9) is an eigenfunction of the square
of the total orbital angular monentmn, K and its component
along the LFCMN Z axis, K with eigenvalues K(K+1) and M,
respectively. Following Davydov (Da65), if a coordinate
frame undergoes a transformation by rotation through Euler
angles (atS) to another coordinate frame, then an eigen
function in the first frame of K2 and K with eigenvalues,
respectively, K(K+1) and M, can be written as a linear com
bination of the (2K+1) such eigenfunctions in the rotated
frame, all of which are eigenfunctions of K2 with eigen
value K(K+I), and each of which is an eigenfunction of K z
with eigenvalue, A, among the possible values K,K1,....,K.
For the case considered here, of a transformation from
the LFCYN frame to the BFCMN frame through Euler angles
(Gy), the wave function of Equation (9) can be written
K
K (R,,,xiyiz D ((R,,,x y z)
A=K
(23)
THere the expansion coefficients, D (Oy), are the socalled
11,A
generalized spherical functions, or Dfunctions, and are
eigenfunctions of the symmetric top. A good discussion of
their properties is given by Edmonds (Ed60). In Equation
(23), .M A(R,0,0,x ',z') is an eigenfunction of KZ the
KA (R, ,xii,
component of K along the internuclear axis, with eigenvalue
t. The orbital angular momentum of the nuclei has no com
ponent along the internuclear axis, as can easily be veri
(e)
fied from Equation (4). As a consequence, Kz = LZ
It is at this point that it is convenient to intro
duce a complete set of electronic wave functions at each
internuclear separation, R. The electronic wave functions
employed can be any of the ones commonly calculated, where
the electronic variables, (x',yi,z), in the BFCMN frame
are taken as an independent set of variables. Such elec
tronic wave functions, whether of the singleconfiguration
or more elaborate configurationinteraction type, are clas
sified according to their component of electronic orbital
angular momentum along the internuclear axis. That is,
(e)
they are constructed as eigenfunctions of L e having ei
genvalues denoted by A. For each A, the complete set of
electronic wave functions, { nA(R,x ,y ,z')}, will be in
troduced at each R, and will be taken to be orthonormal.
M
Then each yK(R,0,0,x',y',z') of Equation (23) may be ex
panded in the set of electronic wave functions, {,nA}:
19
Y (R0,0,xC,y',z) = "'(R)R)/)0Rn (R,xC,yC,z ) (24)
1A i nh' nA
n
Substitution of Equation (24) into Equation (23)
gives a useful expansion for the total wave function of
nuclei and electrons in terms of functions of the elec
tronic variables C(x,yi,z) in the BFCMN frame:
YKM(R,8, ,x.,yiz ) = (1/R) (R)DK (ey)n (Rx',yzi).
KM n An
(25)
It should be emphasized that the equal sign in Equations
(23) and (25) means equality only in the sense that in
each case the function on the left of the expression,
where the variables are the coordinates of the electrons
and nuclei in the LFCMN frame, can be replaced by the
linear combination of functions on the right of the expres
sion, where the variables are the coordinates of the elec
trons and nuclei in the BFCMN frame. The presence of the
variables 0 and 9 in the Dfunctions on the right of Equa
tions (23) and (25) will be discussed presently.
4. The Coupled Equations and Coulomb and
BornOppenheimer Couplings
Recalling the previous discussion regarding the Haril
tonian operator, Equations (19) and (22) provide an observ
er in the BFCMN frame the appropriate Hamiltonian for
20
the nuclei and electrons under conditions where (x ,y ,z'
R,0',$') are independent variables. Using this Hamiltonian,
an observer in the BFCMN frame may now replace ( and e on
the right hand side of Equation (25) by <' and 8' according
to Equation (13). Then the right side of Equation (25) be
comes an appropriate expansion for the "transformed" wave
function, KM(R,O' ,,x',y',iz'l), which satisfies the "trans
formed" Schr6dinger equation. Thus, an observer in the
BFCMN frame may proceed to solve Equation (3) by relying
on Equations (19) and (22) and solving for the wave func
tion
KIM (K
Y M(Re ,',x',yZl)=(1/R) Tn (R)D ($ e'Y)nA (Rxy z.
i I An n M,n JA 1IJ(
An
(26)
In dealing with the operators in Equation (22), it
will be useful to rely on some of the formal properties of
the Dfunctions (Ed60). The Dfunctions, D (9y) are
formally associated with rotations through the Euler angles
(Jey), and these rotations are generated by the angular
4
momentum operator, N, whose components are
(n)
N (04y) = L i(cosj/sin6)/3y ,
x x
(n)
N (46y) = L i(sin4/sin)6/y ,
y y
N (46y) = L(n)
z z
21
where L(n), L(n) and L(n) are giver in Equation (7). The
x y z
2 2 2
Dtunctions are eigenfunctions of N N +N +N
x y z
N2(6Y)D (6) = K(K+41)DK (y) (28)
By expanding out N 2(Oy) using Equation (27), and recalling
thac the ( and y dependence of D CA(By) is in the factors
e and e Equation C28) can be rearranged as follows
(Ed60):
2 2 K
(O2/e2 + cotW/39 + sin2S /42)DK A((9y)
= {sin 2(A 2cosOMA) K(K+1)}D ,A (y) (29)
Other useful relationships involve the operators
N (tey) = N iN In particular, based on the properties
x y
of the Dfunctions (Ed60), it can be shown that
N. ( K (Y)q(0y) = At+l(O) (30)
where [1 = {K(K+1) A(CAl)}1. Furthermore, 3/38 can be
expressed as
3/ae = '(e N el% N) (31)
Using Equations (19), (22) and (31), the BFCMN Hamil
tonian can be expressed as follows:
H(R, 0',',x ^,y,z 2= (1/2nR2)D/9R(R2 9/R)
(1/2mR2) )cot9'a'/a6 + 2/82 + sin2 e'2 /at2
22
(e i 2 (e) () 2 (Ce) 2
ji(e 'L+ ) + e L_ ) (e N N (4 0"y)  ei N ( 0^e ) )
(L(e) 2 2i(cot6/Asin) cotO (L ) + (L
z z z
iY (e) e(e)
/sin ')(e L +e l'e) )(i'/ cosOl'L )}
+ He(R,xCyCz ) (32)
Now, in the usual way, the right side of Equation (26) may
be substituted into the "transformed" Schr6dinger equation,
(H E)NKM = 0 (33)
When this is done, many of the terms from the operators of
K
Equation (32) acting on functions D 5 n of Equation (26)
will combine and lead to simplifications. One can, for
example, compare the contributions from some of the opera
(e)
tors involving L e in Equation (32) with the terms in
volving A on the right side of Equation (29).
Upon multiplication of Equation (33) on the left by
D ,A' I'Y) *, (R,x ,yz) followed by integration over
M,A nA I i
the coordinates C'8'0yr xtytz2, the following set of
KwM
nA
{(/ d/dR2 (/2mR2) [2A2 K(K+1)] E n(R)
= ({< nH eli A> (I/2l~R22
+ (1/nm)< d/dRnA> + (/2m)6A,
+ (I/m)<~nhld/dlR/ Pn.A. + (i/2m)< ~nAd2/dR21 bn.A.>} AA
 23
2 (e)I KM+
(1/2mR 2)< ,i' I,AA +
_1I/2mR,2) KM (KM R) 34)
/ nA n'A AA A,A'1 n ( (34)
Here, the brackets indicate integration over electronic
KM+
variables. The coefficients g A are given by
KM+ 2 K K *K
gA = (2K+1) (8T2)1fd'sin6d6'dyD ^ 6Y)G M (6Y)
(35)
where
K +iy K
A6. y) = C (Msin 0 + Acote')DMA(POY)
+ (i/2)eZiy{A+e'DA+i ('6y) Ae'DK,Al(eY)} .
(36)
In Equation (34), all of the various couplings be
tween the nuclear and electronic motions are included ex
plicitly. The radial BornOppenheimer couplings appear
in the matrix elements between the electronic expansion
states over the d/dK' and d2 /dR2 operators, and reflect the
effect of the radial motion of the nuclei on the electronic
motion. As can be seen, radial BornOppenheimer couplings
exist only between electronic states having the same value
of A. The rotational BornOppenheimer couplings appear in
the matrix elements over the L(e) and (L(e)2 operators,
and reflect the effect of the rotational motion of the nu
clei on the electronic motion. A convenience of treating
the problem in the BFCMN frame is that the influence of
nuclear rotational motion on the behavior of the electrons
becomes expressed in terms of matrix elements between elec
24
tronic states over electronic angular momentum operators,
rather than nuclear angular momentum operators. As can be
(e)
seen, rotational coupling due to L e exists only between
electronic states having A values differing from one an
(e) )
other by il, whereas coupling due to (L( ') exists be
tween electronic states having the same A value. The so
called Coulomb coupling appears in the matrix element over
the electronic Hamiltonian in Equation (34), and exists
only between electronic states having the same value of A.
5. Discussion
Some brief remarks are in order regarding the consid
erations of this chapter. No qualifications have been
placed on the basis set of electronic wave functions,
{ nA}, used in the expansion of Equation (26), other than
that it be complete and orthonormal. Traditionally, ap
proaches to molecular electronic structure have tended to
focus on electronic states which are eigenfunctions of the
electronic Hamiltonian, Hel, and which therefore leave the
matrix of Hel diagonal. These are the socalled adiabatic
states which provide an adiabatic representation. In such
a representation all of the coupling between the electronic
states, associated with inelastic collisional processes,
rests in the BornOppenheimer terms of Equation (34). The
eigenenergies of Hel associated with these adiabatic states
25
obey the noncrossing rule, as is well known.
However, in treating atomic and molecular collision
processes, it is not at all clear that the set of adiabatic
states is always the most appropriate representation to use.
Stemming from the point of view emphasized by Lichten (Li63)
much consideration and discussion have resulted regarding
the importance of the socalled diabetic representations.
These representations are comprised of electronic states
which are not eigenfunctions of Hel. Consequently, the
diagonal matrix elements of these states with Hel need not
obey the noncrossing rule, and the offdiagonal matrix ele
ments may be appreciable. Compared to the BornOppenheimer
couplings, the Coulomb couplings in a diabetic representa
tion can often actually be the dominant source of coupling
associated with inelastic processes influencing the heavy
particle motion described by the coupled equations of Equa
tion (34).
Two particularly good discussions of these matters
have been made by Smith (Sn69) and Sidis (Si76). Here, it
is simply pointed out that the appropriateness of the adia
biatic or a particular diabetic representation depends upon
how successfully the dominant coupling terms can be identi
fied, as well as calculated or estimated. Radial Born
Oppenheimer couplings are difficult to calculate and nor
mally must be estimated. In addition, they are character
sized by singularities in regions of R near avoided crossings
of the associated adiabatic eigenenergies. The coupling
through Hel of diabetic states can be estimated, if not
often calculated. However, because of frequent lack of
information abcut the BornOppenheimer couplings, one can
not always be sure when the Coulomb couplings constitute
the dominant contribution in describing inelastic processes.
As can be seen from Equation (34), the rotational Born
2
Oppenheimer couplings have a R2 dependence. Therefore,
their contribution will be of increasing importance as dis
tances of closest approach of the nuclei become smaller.
The research related to atomic collision processes re
ported in the remainder of this dissertation has been car
ried out within the framework of diabatic representations.
In this connection, electronic states constructed as deter
minants of oneelectron molecular orbitals are particularly
useful. Furthermore, the behavior of the oneelectron mo
lecular orbitals and associated orbital energies can itself
provide information of use regarding inelastic processes in
atomatom collisions. Considerations along these lines are
pursued in Chapter II, illustrated by molecular orbital cal
culations on some rare gas diatomic molecules, and in Chap
ter III, where Penning and associative ionization processes
in thermal energy collisions of excited (metastable) rare
gas atoms with ground state atoms are discussed in terms of
molecular orbital correlation diagrams. An assessment of
the angular momentum contributions in the EFCMN frame of
the emitted electron in these processes is made based on
the correlation diagrams analyzed in their united atoms
limit. In this united atoms analysis the significance of
rotational BornOppenheimer couplings will need to be con
sidered, as has just been mentioned.
The topic of study in the final chapter is the dynam
ics involved in Penning ionization of Argon by metastable
Helium in thermal energy collisions. The approach will be
to solve numerically in a two state approximation the cou
pled equations of Equation (34). An interesting feature
of chemiionization is that the electronic state prior to
ionization is embedded in the continuum of electronic states
associated with the ionized electron. Thus, in the expan
sion of Equation (26), the sum over discrete electronic
states must be augmented by an integral over the continuum
electronic states. This feature will be dealt with in
Chapter IV. The solution of the coupled equations with
the appropriate boundary conditions for scattering will
lead to results for total and partial cross sections for
Penning ionization.
CHAPTER II
A STUDY OF SINGLEELECTRON AND TOTAL ENERGIES
FOR SOME PAIRS OF NOBLE GAS ATOMS
In this chapter* consideration will be given to
atomatom interaction potentials and electronic structure
pertinent to the description of collision events. Sev
eral features should characterize the method employed in
this type of treatment of the electronic structure of
atom pairs:
(a) The method should provide a description of
the interaction potentials for the ground
as well as excited states which govern the
motions of the heavy particles (nuclei)
during collisions.
(b) Details of the electronic structure, at least
of the type found within the selfconsistent
molecular orbital (MO) framework, should be
available so that processes related to elec
tronic excitations may be studied, especially
in the case of energetic atomatom collisions.
*This chapter is an essentially unaltered version of a
contribution (Be74a)to the Proceedings of the Interna
tional Symposium on Atomic, Molecular and Solid State
Theory and Quantum Statistics held at Sanibel Island,
Florida, 2026 January 1974, where a preliminary report
of the results was made.
28
29
(c) The method should be applicable to a variety
of pairs of atcmic neutral and ionic species
while at the same time it should involve only
a moderate degree of calculational effort and
complexity.
Statistical approaches, such as the ThomasFermi
Dirac method, have provided useful results by applying
freeelectron gas energy expressions in conjunction with
a molecular charge density taken as a superposition of
atomic charge densities. However, these approaches are
deficient in that they only describe the ground state in
teraction. They afford no information of the type men
tioned in point (b), and furthermore they are not appli
cable to situations where appreciable charge rearrange
ment occurs in the diatom, since there is usually no pro
vision for selfconsistency in the calculations.
These considerations have led to the use of the MSXa
method (Jo66, Jo73, Sl71a, Sl71b, S172) in this work. It
parallels the HartreeFock approach in that it provides
a oneelectron description with corresponding oneelectron
eigenvalue equations and eigenstates which are solved
selfconsistently. The method makes use of a convenient
local approximation to the exchange potential. Although
approximate, the treatment of exchange in the Xa approach
should be quite adequate in handling the short range part
of atomatom interactions, where Coulomb and exchange
forces between the electrons are the important ones.
The calculational procedure is furthermore based on
the "muffintin" approximation to the oneelectron poten
tial, which entails no additional computational complica
tions as the number of electrons being treated increases.
Thus the scheme has wide applicability.
The first application of the MSXa method to inter
action potentials of rare gas pairs was a calculation on
the NeNe system (Ko72, see also Tr73). In this chapter,
calculations performed on the atom pairs, HeHe, HeAr
and ArAr are presented, and their relevance to collision
phenomena is pointed out. At the outset, aspects of the
Xa theory and computational approach which directly re
late to the present work are described, and limitations
of the theory are considered. Results of the calcula
tions are then presented, and finally, a discussion of
the results and their significance is given.
1. Theoretical and Computational Considerations
Within the Xa formalism (S171a, S172), the total en
ergy EyX, of a system of N electrons is specified by a
set of spinorbitals, {ni}, according to the expression
EXC = n ndr u (r) V + X(2Za/rla )}u (rI)
1 a
+ /drl/dr2{2p(rl)p(r2)} /rl2
Slrd {pr)U O ) +P (rl)U I
S+l {P 1Xc 1 1 Xu 1
+ 2Z Zb/Rab (37)
a
Equation (37) is expressed in Rydberg units, and in the
usual sense, Za, Rab rla, and r12 refer to the charge
on nucleus a, the distance between nuclei a and b, the
distance between rl and the position of nucleus a and
 +
the distance between r, and r2, respectively. The elec
tronic charge density is
p(r,) = p r) + p (r)
1+ I
nu(rl)u(l) + nu (rlu (l (38)
comprised of the charge density of electrons with spin
up (denoted by +) and that of electrons with spindown
(denoted by 4), and
UO (r ) = 9a((3/4T)p ( ))1 (39)
is the Xa local exchange energy density, a being a multi
plicative factor. It should be emphasized that E is an
energy functional depending on the spinorbitals, ui, and
the occupation numbers, ni, and does not necessarily rep
32
3
resent an average value of a manyelectron wave function
over a Hamiltonian operator, as is the case in Hartree
Fock theory.
For a given assignment of the ni's, the u.'s are de
termined by making EXa stationary with respect to their
variation. This leads to a set of eigenvalue equations
for the spinorbitals. For the u 's of spinup, one has
(fl(rl) + fdr2{2p(r2)/r,2} + V x(r)ui(r = (r) ,
(40)
where
+ 2
flrl) 7 + (2Za/rla) (41)
a
and
Va ( ) = 2/3 Ue(l) (42)
A similar set of equations is obtained for the u.'s of
spindown. As is well known (S171a, S172), the interpre
tation of the Xa orbital eigenenergies, e., differs from
that of HartreeFock orbital eigenenergies, and is based
on the relation,
Ei = E Xa/ ni (43)
This condition between EX, the E 's and the n 's insures
that Fermi statistics holds within the framework of the
Xa description of a system of electrons; namely, the low
est value of EX for a system of electrons is achieved
33
when the spinorbitals of lowest eigonenergies are occu
pied. Concentrating on diatonics at fixed internuclear
distance, R, a state of the electronic system may be
identified by means of an assignment of the n 's. A con
venient way, then, of determining over which region of R
a particular Nelectron state is the one of lowest energy
is by looking at the behavior of the .i's for the occupied
and unoccupied orbitals of that state, and observing over
which region of R the .i's of the occupied orbitals are
the lowest ones. In the results which follow, this fea
ture will be demonstrated. Equation (43) is also the
basis for the familiar transition state approach (Sl7lb),
from which good approximations may be found to ionization
energies as well as excitation energies between electronic
states.
In practice Equation (40) is solved by the Multiple
Scattering (MS) method (Jo66, Jo73), with the potential
operator
V(r!) = fdr2 2p(r2)/r} + V r) = V(rl + VX(rl) (44)
approximated by a "muffintin" form, whereby it is av
eraged over angles within nonoverlapping spherical re
gions centered on the various nuclear sites and also oth
er sites in the molecule (outer sphere, empty spheres,
etc.), and volumeaveraged elsewhere. The u.'s are de
1
34
termined selfconsistently in terms of partial wave ex
pansions within the spherical regions, and expansions in
"multiplyscattered" waves elsewhere. Furthermore, at
each iteration of the selfconsistent procedure, the "muf
fintin" form of p (r), as found from the u.'s according
to Equation (38), is used to evaluate the "muffintin" po
tential as well as the "muffintin" approximation to EXa
of Equation (37). The reader is referred elsewhere (Co72)
for the details about these approximations as well as ways
of correcting for them (Da73, Da74a, Da74b). It suffices
here to say only that the effect of the "muffintin" ap
proximation in the MSXa evaluation of EXa is appreciable.
However, orbital energies e. for diatoms appear to be more
reliable than EXa within the MSXa approach (We73).
It has been shown (S174) that in obtaining the ex
change potential of Equation (42), one need not assume
that the electrons of an atom or molecule behave locally
like those of a freeelectron gas of ths same density.
Rather, one may assume a spherically symmetric "Fermi hole"
and apply dimensional arguments. Nevertheless, consider
able discussion has been devoted to shortcomings of ex
change potentials of the type in Equation (42) associated
with the finite numbers and inhomogeneous spatial distri
bution of electrons in atomic and molecular systems (Li70,
Li71, Li72, Li74, Ra73, Ra75). By means of a more careful
35
look at the exchange energy of a freeelectron gas of a
finite number of electrons in a finite volume, corrections
to the exchange potential have been derived by separating
out the contribution from the interaction of each electron
with itself (Li70, Ra73, Ra75); for a small number (<200)
of electrons the contribution trom this "selfinteraction"
becomes quite sizable. Estimates of the "selfinteraction",
based on the abovementioned corrections, show that in
atoms and molecules only about 85% of its contribution to
the exchange potential is included in V (Li72), while
Xc
the Coulomb potential, VC, in Equation (44) includes all
the "selfinteraction". This imbalance may be partially
remedied by an adequate choice of the a factor.
Now, a variety of ways have been suggested for spec
ifying the a factor of Equation (39) for a system of elec
trons (Sc72 and references therein). The values of a for
atoms, resulting from the various schemes, almost all dis
play the same trend; namely, for atoms of larger numbers
of electrons the values of a tend toward 2/3, the factor
appearing for a freeelectron gas, whereas for atoms with
few electrons, larger values of a result. Since the "self
interaction" contribution is large for few electrons, this
trend in a values has often been interpreted as reflecting
the required greater compensation for the deficit in "self
interaction" in the case of few electrons, the compensation
36
becoming less as more elections are involved (Li72). Thus,
despite the importance of the corrections which have just
been reviewed, the Xa exchange potential represents quite
well the exchange interaction in a system of electrons.
As long as the "muffintin" approximations are being made,
a treatment in terms of the Xa exchange potential alone,
with a commonly used value of a, is expected to be ade
quate for describing the shortrange interaction between
atoms, as mentioned at the beginning of this chapter.
The calculations performed here employed a double
precision (14 hexadecimal or roughly 16 decimal digits
available per number on an IBM370/165 computer) version
of the MSXa program, MUSCATEL. This precision was re
quired since the interaction energy, AE, is computed as
the difference between the total energy of the diatom
(in our case, EXa in the "muffintin" approximation) and
that of the two isolated atoms. For instance, the case
of ArAr at moderately large internuclear separations, R,
involves interaction energies seven orders of magnitude
smaller than the total energies used to determine the in
teraction, hence requiring at least eight significant
figures in the total energies. The total energies of the
isolated atoms were calculated using the HartreeFock
Slater (HFS) atomic program (He63, Za66).
For the homonuclear cases, HeHe and ArAr, the so
called "virial theorem" values of a for the atoms, as re
ported by Schwarz (Sc72), were used in all regions of the
molecules. For HeAr the respective atomic "virial the
orea" values were used in the spherical regions about the
atoms, and a weighted mean (weighted according to the num
ber of electrons of each atom) of the two values was used
elsewhere. The heteronuclear case of HeAr required a
choice of the radii to be used for the spherical regions
about the atomic sites. Contiguous spherical regions were
chosen in all cases, and the ratio between the He and Ar
sphere radii, used at all internuclear separations calcu
lated, were determined in the following way. Average ra
dii obtained from numerical atomic HartreeFock calcu
lations (Ma67a, Ma68) were used to estimate the size of
each of the isolated closed shell atoms, He and Ar, based
on the Is orbital of He, and the 3p orbital of Ar. The
values are, respectively, 0.92727 a and 1.66296 a The
O O
He and Ar sphere radii at different R were then chosen in
the ratio of these two characteristic charge extents.
From the same atomic calculations, the maximum values of
the He Is and Ar 3p radial probabilities occur, respec
tively, at about 0.55 a and 1.30 a so for internuclear
separations larger than about 1.85 ao, the above scheme
for selecting sphere sizes should serve well. Other con
siderations must be made for cases which do not involve
two closed shell atoms and where significant charge rear
38
rangement occurs in the diatom. Such a case is treated
in the following chapter where calculations are reported
on the excited HeAr diatom which separates at large R
1,3 6 1
to He* Cls2s, 'S) plus ArC3p ,S).
The selection of partial waves to be included in the
expansion of an orbital in the various regions depends
upon over which regions of the molecule the orbital tends
to be concentrated. For a very deep lying core orbital,
which is essentially of atomic character, only the partial
wave corresponding to that of the associated atomic orbi
tal was used in each appropriate atomic region, since par
tial waves of other R values give a negligible contribu
tion. For higher lying orbitals, appreciable contribution
from a number of partial waves, s, p, d, etc. may occur.
Such partial waves were included as long as their ampli
tudes were at least onehundredth the amplitude of the
dominant partial wave.
All of the calculations required no more than medium
size core on an IBM370/165 computer and the times per
iteration of the SCF procedure were about two seconds for
HeHe, five seconds for HeAr and between five and ten
seconds for ArAr, depending on the "goodness" of the
starting point for a calculation. Denoting the absolute
difference, occurring between the values of the "muffin
tin" potential of Equation (44) at one iteration and those
39
of the previous iteration, by AV, and the maximum value
4
of AV/V by s, the degree of selfconsistency c<104 was
achieved typically in 1520 iterations.
It should be pointed out that the relative error of
the MO wave functions is of the same order of iaagnitude
as that of the potential. Since the total energy is vari
ationally determined, and therefore accurate to second
order in the wave functions, the degree of selfconsis
tency we have used is sufficient to insure the accuracy
required in the total energies at large R.
2. Results
In Figure 2 are displayed the interaction energies,
AE, for the three diatoms, HeHe, HeAr and ArAr, as a
function of R, the internuclear separation. The interac
tion energies are shown for the states with the lowest
energy at large R (i.e., asymptotic ground states). The
interactions on this semilogarithmic plot are seen to be
quite linear, indicating the repulsion they show over the
investigated ranges of R is characteristically of the
BornMayer type, namely A exp(bR). This behavior, of
course, would break down at very small R, where the Cou
lomb repulsion between the nuclei becomes strongly domi
nant. The beginning stages of this other behavior is seen
in the HeHe case at R 0.7 a On the other hand, as R
0
Figure 2. Interaction energies, AE, for the pairs HeHe,
HeAr and ArAr in their separated atom ground
states. Calculated points are encircled. a.u.
of distance refers to the Bohr radius, ao
O
E (c.u)
1.0 2.0 3.0 40 5.0 6.0 70
R(o.u)
42
increases for a given interaction and approaches the van
der Waals radius, the actual interaction energy would pass
through zero, and its logarithm would asymptotically ap
proach () as R nears the point of zero interaction.
This pronounced deflection of the logarithm of AE away
from the BornMayer straightline behavior is not exhibited
by the calculated interaction energies in Figure 2, indi
cating that these "muffintin" interaction energies show
no tendency to describe the van der Waals attractive well.
It should be pointed out, though, that efforts made to
correct for the "muffintin" effects in the case of NeNe
have shown a well defined attractive region (Da73, Da74a,
Da74b). The procedure for calculating these corrections
is, however, nontrivial, and would be impractical for
the present purposes.
Therefore, BornMayer A and b parameters for the in
teractions of Figure 2 were determined by means of a
leastsquares fit to the calculated points over the re
gions of straightline behavior. The parameter b measures
the slope of an interaction as shown in Figure 2, and re
flects the "hardness" of the repulsion described. The
parameter, A, serves as a measure of the overall strength
of repulsion, being the value of In(AE) at R=0, although
it is somewhat unrealistic, since R=0 is an unphysical
separation at which to compare the "strength" of the repul
sion of a BornMayer type.
43
So, the A parameters of the interactions of Figure
2 are simply listed in Table I, and it is pointed out
that over the region of BornMayer behavior, they lead to
interaction energies for the three diatoms which overes
timate the repulsion, but obey the combination rules for
such interactions in that the HeAr BornMayer line falls
between those of HeHe and ArAr. Listed in Table I by
way of comparison are the b parameters of the interactions
of Figure 2 along with those determined from other theo
retical calculations and experiment, as indicated. The
ranges over which the listed parameters apply are shown
in parentheses. In general, the b parameters of the pres
ent work reflect "softer" repulsions than do the other
tabulated b parameters. However, the ranges of R values
of the present work extend to larger R, and it is possi
ble that "muffintin" effects lead to a proportionally
larger overestimation of the repalsion at large R, i.e.,
to smaller b parameters.
The case of HeHe is now considered in more detail.
In particular, the behavior of the MSXa eigenenergies is
looked at, providing a good example for diatomic interac
tions. As is well known, at large internuclear separation,
the lowest state of the HeHe diatom is the IZ+ state spec
g
ified by doubly occupied I and la molecular orbitals.
These two MO's are, at large R, essentially the gerade
44
I
04
0
0
H)
40
C)
a)
Ii
r
H
0
04
'4
rC
a)
EH
t1
a
H
'4
0 r
U
u
U 
Her
E
I 
0 CC
O
H
U
E
E
H
0
PC
 0
o
I 0
4a
0
Ho
I 0
C)
H
rl
I 0
rfi
rc co
'2*
oN to
LD
O)
I
CO
,r
0
Cr)
,0
o
0
ii
to
03
Ln
Ln
O
0
rC)
Si
O
I
i [
co
n I
o co
L00
o
N *
N
In
'lp
0 I
H 0
o
O0
0
N
o in
.0 I
c3
o
N
tO
ra
en
03
\D
'03
ca
to
Ln
.1 *
0
u
to
03 I
CO
0o
H *
r[
0
rC
'D Ln
N
LO
us
o
rHl*
en
0
03 N
en
I
Lo
to
H *
cN
03
4.r
0o
,l
IN
11
'4
PC
45
and ungerade combinations of the He Is atomic orbitals.
In Figure 3 is shown the plot of the eigenvalues of these
two orbitals as a function of R over the range 0.5 ao to
5.0 a In keeping with the idea of a correlation diagram,
0
arrows indicate the HFS atomic orbital eigenenergies; the
is orbital of He in its ground Is2 configuration, in the
separated atom CSA) limit, and the is, 2s and 2p orbitals
2 2
of Be in its ground Is 2s configuration in the united
atom [(A) limit. It is seen that cl, and E are nearly
g u
degenerate with aes of He at large R, as expected, and
separate as R decreases. Now, in the hA limit, the lo'
orbital correlates with the Be 2p atomic orbital. Thus
1' 2 2 2 2
the l' (l a 1 ) state approaches the excited Be is 2p
g g u o
atomic state in the UA limit. On the other hand the 2o
g
orbital correlates with the UA Be ..3 atomic orbital. It
is therefore the Z+ (la2ao ) state which in the UA limit
g g g
2 2
correlates with the is 2s Be ground state. It is of in
terest to determine at what internuclear separation the
1 + 2 2
E state specified by la 2a becomes lower in energy than
g g g
2 2
that specified by la la As discussed earlier in this
g u
chapter, one may proceed in two ways: (1) direct observa
tion of the interaction energies, AE, of the two states
as functions of R to see where they cross; or (2) observa
2 2
tion of the eigenenergies of, for instance, the Cl 22
g g
state to see where cla and 20 become the ones of lowest
g g
0c C
D 4 C 0
(N tn0
NX C '
Su 4oa
d o a)
1 CU
0 0H 0
rd C C U
C
SI H 3, C
WV 0) U
,fl U
roI 0 m
So 0 )J
U.C
D 'OO4 m0
U I (C 0
4J C) CU
00 CUC
t 0 04) M
,. .
C CU
(U (U 0
0'
r4
47
Si
a 
NN c o
o
o o
/.
? I 4 
3 
N 
. . I , I i P I i I , ,
i  oo .
value. Proceeding from 1.2 a to smaller R, the eigenen
2 2
erqies for the la 2a state have been plotted along with
g g
22 2
those of the 1a lIc state. For the l"2ag state, ,la
lies lower, to begin with, than ,2 and Fermi statistics
2 2
indicates that l1 21 is not the state of lowest energy.
Sgg
However, Cle is rising sharply as R decreases, and is
seen to cross above e2g between 0.5 ao and 0.6 a In
ward from this crossing the la and 2a orbitals have the
g g
1 2
lowest eigenenergies and hence the lc' 2o state has the
g g
lowest energy. For comparison, in the insert of Figure 3,
a plot with linear scales is shown of AE versus R for
these two states, which corroborates this behavior.
It should be noted that the interaction energies cross
in a very gradual manner whereas the eigenenergies cross
more sharply. Thus the eigenenergy behavior indicates
more clearly the position of the crossing, which is found
to be 0.53 a These results compare well with SCF results
reported on HeHe (Ma67c), where it was found that the en
2 2 2 2
ergies of the la la and la 22 configurations cross near
g u g g
0.6 a Also, in the accompanying 50 configuration natural
orbital iteration calculations, it was reported that the
2 2
l glou configuration was dominant beyond 0.7 ao. Though
the calculations reported here go inwards only to 0.5 ao,
2 2
the eigenenergies of the la 2a state are seen to be ap
g g g g
preaching the appropriate eigenenergies of ground state Be.
48
An investigation (Ya74) of the HeHe diatom, subse
quent to the one reported here CBe74a), hut closely paral
leling it, has been carried out in the HartreeFock ap
proximation. It is interesting to compare the MSXa and
HartreeFock results. In both approaches, the behavior
of the total energies of the. lc21a and 1 22o configura
g u g g
tions as R decreases from 0.6 a to 0.5 a shows that
o o
2 2
they cross very gradually, the energy of the lo 222 state
g g
becoming lower than that of the loa2la state at 0.56 a
g u o
in the HartreeFock case, and as has been seen here, at
0.53 a in the MSXa case. These values are in good agree
ment. However, in the HartreeFock approach, the deter
mination is based solely on the total energy curves of
the two states as they cross with nearly the same slope.
This is because in the HartreeFock approach there is no
immediate connection between the state of lowest energy
and the eigenenergies of the MO's associated with that
state. However, as has been discussed and demonstrated
here, such a connection can be made in the Xa approach on
the basis of Equation (43), which permits the detection
of crossings either by observing MO eigenenergies or
total energy curves.
For ArAr, at large R, the ground state is also
+ and is specified by the first five a and a MO's,
g g u
each being doubly occupied, and the first two 7 and T
u g
50
MO's, each having occupation number 4. For large R, or
bitals 10 to 5gu are formed from the appropriate com
g,u g,u
binations of Ar Is, 2s, 2Po, 3s and 3po atomic orbitals,
and the first and second gu orbitals from the appropri
ate combinations of Ar 2p and 3p atomic orbitals, re
spectively. In keeping with the Fermi statistics, the
eigenenergies of these occupied orbitals are the lowest
ones at large R, where we also find lying above them the
eigenenergies of the unoccupied 60 and 16 orbitals. A
number of states can be specified at smaller R by the vari
ous assignments of occupation numbers to the 5U 60 and
16 orbitals.
Calculations have been made on some of these states
and are displayed in Figure 4 on a semilogarithmic plot
of AE versus R (in Figure 4, maonp refers to 51, 609
and 16P). In the region of R shown, numerous crossings
can be seen, and they are all of a very gradual type.
11 2
The Uo and 6 curves are from nonspinpolarized MSXa
u g g
calculations, the corresponding spinpolarized calculations
reflecting only a small splitting scarcely noticeable on
such a graph as Figure 4. We can see that the SA ground
state, denoted by 2a in Figure 4, no longer is of lowest
energy for R less than about 3 a This can be confirmed
again, by looking at the eigenenergies for this state, and
in Figure 5 we display the highest of them versus R on a
Figure 4. Interaction energies, AE, for ArAr in states
where the highest orbitals have occupation
numbers as specified. Calculated points are
encircled, a.u. of distance refers to the
Bohr radius, a .
o
Ar Ar
Sm+ nr p n =2
m+n+p=2
2.0 2.5
R (au.)
Figure 5. Orbital eigenenergies for ArAr in its
separated atom ground state. Calculated
points are encircled. a.u. of distance
refers to the Bohr radius, a .
o
54
R (a u.)
20 40 60 8.0
C (a~u.)
3p Ar
3s Ar
55
loglog plot. The appropriate SA 3s and 3p eigenenergies
are also shown. The eigenenergy of the unoccupied 6g MO
(which, in the SA limit correlates with the unoccupied 4s
atomic orbital of ground state Ar, whose orbital energy
is 0.0023 a.u. and lies off the scale in the figure) is
seen to be descending rapidly as R decreases, crossing
the occupied So orbital eigenenergy sharply at 3 ao,
indicating that for R less than 3 ao, this state is in
deed no longer of lowest energy.
3. Discussion
The results which have been presented illustrate the
possibilities of the MSXa method in the study of interac
tions that play a role in collision events. The interac
tion potentials themselves render information on the Born
Mayer type repulsion and, in conjunction with results on
van der Waals attractions, enter into the calculation of
relative motion of colliding atoms. Implicit in the re
marks on crossings between interaction terms for various
states is that the states calculated in the Xc method
are diabetic in nature. Indeed this is the case since
each state is independently calculated after being speci
fied by an assignment of occupation numbers to the orbi
tals. So, while describing the dynamics of collision
events, we can expect the largest coupling between molecu
56
lar configurations to come from the electronelectron in
teraction. In principle, these interaction matrix ele
ments between determinantal wave functions comprised of
MSXo orbitals can be calculated, but in practice the
problem at present seems quite formidable.
The usefulness of the Xa orbital energies, i., in
showing where interaction potentials cross, has been shown.
Hence, critical distances of approach for the occurrence
of various electronic excitation phenomena can be deter
mined. Also of interest are crossings between eigenener
gies such as occurs between c5. and E2x in Figure 5, if
g u
one or the other of the involved orbitals were partially
occupied. Such crossings are of importance in electron
promotion mechanisms involved in energetic atomatom and
ionatom collisions.
Although the pairs studied here consist of closed
she' atoms, it is expected that the MSXa method, because
of its selfconsistent treatment, can handle as well the
repulsion in cases where sizable charge rearrangement
takes place. Of particular interest would be the mecha
nisms involved in Penning and associative ionization phe
nomena, where atom + excited atom and ion + atom interactions
are of importance. Here again, though, reliance upon ad
ditional results for describing the van der Waals region
would be needed. In Chapters III and IV the considerations
57
prompted by the research of this chapter will be applied
to an investigation of the electronic structure and col
lision dynamics involved in Penning ionization of Ar(3 )
by Ke*CIs2s,1,3S).
CHAPTER III
DIATOMIC MOLECULAR ORBITAL CORRELATION DIAGRAMS
FOR PENNING AND ASSOCIATIVE IONIZATION
In the previous chapter it has been pointed out that
an adequate understanding of atomic and molecular collision
phenomena requires information on electronic structure as
a function of the changing internuclear separations. Such
information is needed in order to describe inelastic pro
cesses involving electronic excitation and charge transfer
as well as ionization. In this regard, the usefulness of
oneelectron molecular orbital (MO) approaches in treating
electronic structure was emphasized. Such approaches afford
a selfconsistent calculational framework of minimal complex
ity which can treat ground as well as excited states. Chemi
ionization is a prime example of processes where electron
ically excited states play a crucial role. In this chapter*
the electronic structure involved in a collisional process
of this type will be considered.
Well known among chemiionization processes are Pen
ning and associative ionization (PI and AI) of the type
*A preliminary report of the results presented in this chap
ter was made at the International Symposium on Atomic,
Molecular and Solid State Theory, Collision Phenomena and
Computational Methods held at Sanibel Island, Florida,
1824 January 1976.
58
59
A* + B A + B + e (PI) and A* + B + AB + e (AI),
where A* is usually an atom in some metastable state and B
is an atom or molecule (Mu66, Mu68, Mu73, Ni73, Be70a,
Be70b, Ru72, Ma76). Experimental information for such col
lisions includes total ionization crosssections as a func
tion of collision energy (Ta72, Ch74, Pe75, 1175), angular
distributions of heavy particles (Ha73), and energy distri
bution (Ho70, Ho75, Ce71) and angular distribution (Ho71,
Eb74) of emitted electrons. Most theoretical effort has
been directed at determining the energy dependence of total
ionization crosssections and the angular distribution of
heavy particles (Na69, Mi70b, Mi71, 0172a). The angular
distribution of emitted electrons has been recently stud
ied within a semiempirical model based on MO correlation
diagrams (Mi75). One of the present concerns is to re
emphasize the usefulness of MO correlation diagrams as
they apply to angular distribution of ejected electrons
in PI and Al involving atomic collision partners.
In what follows, calculated as well as estimated MO
correlation diagrams are presented. The calculated results
are given first, where the electronic structure of
1,3 6 1
He*(ls2s, S) + Ar(3p S) has been considered within the
MSXa framework as described in the previous chapter. A
study has been made, in the nonspinpolarized approxima
tion, of the K ArLAr (lse) (o3sAr) (o3pAr) (w3pAr) (a2sHe)
tinofth K2 21Arr~~osr~
60
configuration of the excited HeAr diatom whose MO eigen
energies approach the atomic orbital eigenenergies of
He*ils2s) and Ar(3p ) at large internuclear separation,
and also the ground 2E state of the (HeAr) molecular ion,
whose MO eigenenergies approach the atomic levels of
HeCls2) and Ar +3p5) at large internuclear separation.
Some spinpolarized calculations are presented in the
neighborhood of a crossing exhibited by the nonspin
polarized calculations, along with some comments on the
conditions under which the noncrossing rule applies for
MO eigenenergies.
Keeping in mind the calculated results, consideration
is given to PI and AI processes in terms of MO correlation
diagrams. Analyzing the ionization on the basis of an
Auger type process, MO's can be identified, which, togeth
er with the continuum state of the emitted electron, are
involved in the process. Inspecting the unitedatoms lim
it in the centerofmass, bodyfixed frame then permits a
determination of the angular moment which contribute to
the continuum state of the emitted electron.
Next a procedure for estimating MO correlation dia
grams is described, which makes use of available data on
atomic orbital energies, and of two basic guidelines. Ap
plication is made to the collision pairs He*Cls2s) +
Ar(3p6), + Kr(4p6), + Hg(6s2), and Ne*(2p53s) + Ar(3p6).
The resulting estimated MO correlations for these systems
are then analyzed to determine the minimal set of angular
momentum values which are needed in each case to describe
emitted electrons in the bodyfixed frame. Finally, a dis
cassion is given of the results of this work and their
significance.
1. MO Calculations for He*+Ar: ar.d He+Ar+
The representative case of PI and AI in
He*(ls2s,' S) + Ar(3p S) collisions will be considered.
Here, calculations are needed for the excited states of the
HeAr diatom which separate appropriately to
Ie*(ls2s,1,3S) and Ar(3p6,1S), and for the states of
+ 2 1 + 5 2
(HeAr) which separate to HeCls,1 S) and Ar (3p5, P).
Following the work reported in the previous chapter, the
calculations are performed within the MSXa oneelectron
MO framework.
For convenience in this chapter, the set of oneelec
tron equations satisfied by the spin orbitals, ui, and ex
pressed in Equation C40), are written
h (ffl)Ui( = Eiui(') (45)
eff 1 i 1 ii 1'
+ 
Here, heff(r ) is the effective oneelectron Hamiltonian
for electrons of spinup, and according to Equation (40),
61
62
hf (r) = f] l + dr2{2p(r2)/rl2} + VtX (46)
where Vx (rl) is given by Equations C42) and (39), and
p(r), given by Equation (38), is the charge density com
prised of the contributions from electrons of spinup and
spindown. In the nonspinpolarized (NSP) approximation,
p () = (r) p ()/2 (47)
which means that the orbitals of spinup and spindown be
come identical, and each orbital can be considered as ac
commodating as many electrons of spinup as of spindown.
These calculations were carried out first with a NSP
treatment. As discussed in the previous chapter, each
selfconsistent calculation begins with a potential which
is the "muffintin" form of a superposition of atomic po
tentials centered at each atomic site of the molecule.
Therefore, for the HeAr excited state Ar(3p ) and NSP
He*(ls2s) HartreeFockSlater (HFS) CHe63) potentials were
used, and for the HeAr ionic state He(1s2) and NSP Ar(3p5)
HFS potentials were used. For both the excited molecule
and the molecular ion, the electronic states were specified
by occupying the MO's so that their eigenenergies were cor
rectly separating at large internuclear separations to the
corresponding atomic orbital eigenenergies of the above
mentioned HFS atomic calculations. In other words, the
63
boundary conditions of the PI and AI processes at large R
determined the appropriate excited and ionic states of the
HeAr molecule. Heteronuclear molecules require a choice
for the radii of the spherical regions centered at each
atomic site as mentioned in the previous chapter. There,
for the ground state HeAr molecule, the ratio of the He
to the Ar sphere radii was taken to be He/Ar
0.92727/1.66296 = 0.5576, where, e.g. e denotes the
average value of r for the Is orbital of He(s 2). The sit
uation for (HeAr) is roughly the same as for HeAr, since
a calculation of for Ar (3p5) (NSP) shows it to be
1.549 a So, for the molecular ion the ratio of 0.5576
was used at all R. On the other hand He*(ls2s) + Ar(3p )
is quite another case, since the 2s orbital of He*(ls2s)
is very diffuse. NSP calculations show of He*(ls2s)
to be about 4.546 a Because of this diffuseness, it was
decided to choose the ratio of the He to the Ar sphere ra
dii by finding which of its values minimized the total en
ergy of the specified excited state of HeAr at a fairly
large R, namely R = 9.0 a In this way, a ratio of 1.4
was found, which was then used at all R for the excited
state of HeAr. The values of the factor a in the various
"muffintin" regions of the molecule, for both the excited
as well as ionic states, were those used previously in the
ground state HeAr calculations, and the specific computa
64
tional details also remain as reported in the previous
chapter.
The results of the NSP MSXa calculations are displayed
in Figures 6 and 7, respectively, for the excited state of
the HeAr diatom and the ground state of the HeAr molecu
lar ion which are appropriate to PI and AI. Shown on log
log plots are the MO eigenenergies versus R for 2.3 a
o o
energies are labeled according to their symmetry and occu
pation number. In each case, the NSP HFS atomic orbital
eigenenergies for the separated atoms are shown at the
right of the plot, and the dashed lines indicate how each
MO eigenenergy is approaching properly its respective sep
aratedatoms (SA) limit. In the unitedatoms (UA) limit
the excited HeAr molecule is expected to approach
Ca*(3p54s23d), and the ground state molecular ion to ap
proach Ca C3p64s). The atomic orbital eigenenergies from
NSP HFS calculations on these united atoms are appropri
ately shown at the left of each plot, and, while the MO
eigenenergies may exhibit much structure between the re
gion of 2 a and the UA limit, the dashed lines at the
left of each plot show that these UA limits are not unrea
sonable. In both plots, the label for the a MO arising at
large.R from 3pAr appears above the MO label. In fact,
Figure 6. Molecular orbital correlation diagram from
nonspinpolarized MSXa calculations of the
HeAr diatom in the excited Z configuration
which separates at large R to He*(is2s) +Ar(3p6).
Calculated points are encircled.
0.07
0.08
0.09
0.15
0.20 
0.25
0.30
 'Her TrV A
_ He'Ar^ ,
0.401
i5 0.50
d
.c 0.60
0.70
1.50
3sCa'(3?4s3d)
0 2
3 4 5 6 7 8 910
2sHe(I s2s)
J3pAr(3p )
 sAr(3p)
 sHe(is2s)
00
R in ao
Molecular orbital correlation diagram from
nonspinpolarized MSXa calculations of the
(HeAr)+ molecular ion in the ground Z state
which separates at large R to He(ls2) +Ar+(3p5)
Calculated points are encircled.
Figure 7.
68
//I I I I i I II I //
0.07 
o0.0He
0.09 
0.10
0.1 5 
0.20 
0.25
3dCoa(3pF4s),
0.30
4sCc Sp(34s)
0.40
0.50
0.60
0 TO
0.80
0.90
1.00
3 pCa (3 ?4s)
1.50
3sCa (3g4s).
0 2
\

i f
  ^ s o
^ 
 7c
:^ ^
cr0
crz.4ASHe(Is2)
7TT
3pAr"(3 p5)
3sAr'(3 p5)
S. I I I Ii ,, 1 111,111
3 4 5 6 7 8910 (0
R in ao
~o~O~QO(M~b~~O'2
69
this ordering for those levels is valid only for R < 5 ao.
For R > 6 a the T level lies above the o level, but they
o
are too close together at large R to be distinguishable on
these plots.
In Figure 6 a crossing is shown near 3.5 ao occurring
between the doubly and singly occupied NSP a orbitals aris
ing respectively at large R from 3sAr(3p6) and NSP IsHe*(ls2s).
This crossing would appear to violate the noncrossing rule
for the MO eigenenergies, and warrants a detailed analysis.
The orbitals of a given symmetry are ordered according to
their eigenenergies, obtained selfconsistently from Equa
tion (45). This equation is an eigenvalue equation involv
ing an effective Hamiltonian determined at each R according
to Equation (46). To establish the noncrossing rule for
the eigenenergies of Equation (45), one expresses the effec
tive Hamiltonian at a supposed crossing, Rc, in terms of its
expansion about R = R +6R located a small distance, 6R,
from R :
c
hff(R) = heff(R) dhff/dR) 6R (48)
c
The noncrossing rule follows by noticing that
+
(dheff/dR)R 6R is a perturbation which lifts any degeneracy
c +
in the eigenvalue spectrum at R However, if hf (R) is
c eff
made to be discontinuous by choice, the conditions of the
noncrossing rule no longer apply.
Referring to Equation (46), the R dependence of the
effective Hamiltonian appears explicit in fl(l) (see
Equation (41)) and implicity in the charge density, p.
70
Showing the full R dependence, the effective Hamiltonian
may be written as heff{r,;R,p(r1,R)}. and
dhff/dR = heff/ Rl + (6hAff/6p)(dp/dR) (49)
The term, dp/dR, in Equation (49) can be seen, from Equa
tion (38), to involve derivatives of the orbitals and oc
cupation numbers with respect to R. As long as the n.
and u. are continuous in their R dependence, dhtff/dR will
be well behaved, and the noncrossing rule will hold. How
ever, if the occupation numbers are changed discontinously
in some region of R, then the noncrossing rule will no
longer be valid in that region. These considerations apply
as well to the NSP effective Hamilton and in fact, in the
case at hand, as shown in Figure 6, a discontinuous change
in occupation numbers does occur. To the right of the
crossing, the a orbitals, in order of increasing eigenen
ergy, have occupation numbers 1, 2, 2 and 1. To the left
of the crossing, they are 2, 1, 2 and 1. A further in
vestigation of the region of the crossing was made by do
ing spinpolarized calculations at R = 3.0 a 3.5 a
and 4.0 a in the case of the 3Z excited state which sepa
61 3
rates at large R to Ar(3p S) and He*(ls2s, S). In Fig
2
ure 8 attention is restricted to the levels of the NSP o
and a MO's which cross in Figure 6, and they are con
trasted with their spinsplit counterparts calculated in
I E
0) 3 O ) 01,) r
0. 4 ., I H
404 0 T 0 rd
CtiCJCyit OH
J) a4 a) Eo a 0
aU < rl u aQ
C)W4H3 U C)Id lA
0 0 0 0 0 P u
O ,t O H,Q
C")u C o )
 4I! 0 tnC 0 *H
,I r to m a) a)
0 0 0 H l *, iH V)
4 r M4 (d 4
SH4 X 4J O 0
: H 4J 0 r r
 I 4 U4 ) ,I Q)
0 U ( 01 4C U
CI0 1 00
CC HC)00C)H0
 (0d N OH M N ri
0 rl 4 a 0. C C 0
aH 40 iN H C u
r a) r i o *
1 C3 O00
*H C0 + C 0 C
r tX I (U 0) 1 C
OO 0 0 AO
iC 4 H H
0 o mn 41 
'IM 1 0 t4
0 0 3 *1 c 
t3CO W C0 O 1 
*H roC 0 04
p'a *H O 0 C
frd 0 ,C (
rH H L0D C 4 4) A
o a o a H 
1 04' 0 0rl
H H4 73 0 H 4
4 (r C H '4 0 a m
Ut) U M '014 C) 04
CO
0
H
11
72
CU)
ro
C,)
C'J
U,
to
a
a
U,
10
O Cj
C,)
Ui,
C,I
C,)
to
0
~ n
*O
CI3
L.
<
rn
Q.
ro
0L
0
C
IJ
n'o UI 9
 0
ro rO
0 a
to
73
the region 3.0 a
O O
limits.
Important to notice is that for R>3.5 ao the NSP a
orbital of He* is split considerably into its occupied com
ponent of spinup a and its unoccupied component of spin
down a, on out to the SA limit, where the splitting is be
3
tween the unoccupied is and the occupied la of He*(ls2s, S).
The o2 orbital from Ar is split only slightly into each of
its occupied spin components. Crossings between two orbi
tal eigenenergies of different spin components are permit
ted since each involves a different effective Hamiltonian
(see Equation (46)). Between 3.5 a and 4.0 a we find
such a crossing for the two spin components that split from
the NSP a2 orbital of Ar, and in fact, as R decreases, the
1 6 1
1 level from 3sAr(3p ,S) is decreasing in energy to pair
1 3
up with the occupied a level from Is He*(ls2s, S), where
61
as its spinup partner from 3s Ar(3p S) is rising to
pair up with the (empty) oa level from ls He*(ls2s, S).
In the UA limit, this empty oa level will correlate with
the partially occupied 3p atomic orbital of
Ca*(3p 4s23d,3L), where L here denotes one of the possible
orbital angular moment of the excited Ca atom.
Such a spinpolarized analysis as this confirmed the
choice of occupation numbers of the NSP calculations for
which the vacancy associated with the NSP o1 orbital is
74
correlating in the UA limit with the partially occupied
3p atomic orbital of Ca* (3p54s23d. This feature will
prove important for the considerations later on regarding
the angular momentum contributions to the continuum state
of the electron emitted during PI and AI. The crossing
shown in Figure 7, between the NSP singly and doubly oc
cupied a orbitals of the molecular ion near 5.5 ao, is
another example of a discontinuous change in occupation
numbers, which in this case is required to reach the appro
priate SA limit.
2. Analysis of PI and AI Processes Based on
MO Correlation Diagrams
Consideration is given now to how an MO correlation
diagram study, such as has just been described, can help
in understanding what takes place in PI and AI processes.
Collisional ionization occurs for R greater than the dis
tance of closest approach, which in the case of He* + Ar
is around 7 ao at thermal collision energies. Referring
to the calculated MO diagram in Figure 6, one can see that
in this range of R there is a vacancy associated with the
a MO arising from lsHe* at large R, above which are some
fully occupied MO's as well as the singly occupied 1 MO
arising from 2sHe* at large R. This situation suggests
that the ionization may proceed by an Auger type process.
Namely, two electrons in higher lying orbitals participate
75
in ionization as one of them is promoted to a continuum
state while the other drops to fill a vacancy, in this
case associated with the ao MO arising from SA lsHe*.
The process may be characterized as one in which initially
the two electrons are in MO's ul and u2, while finally
(after ionization) they are in MO's ul and u2, where u
designates the continuum state of the ionized electron,
having momentum k and angular momentum components tl.
If the wave function a(N) is associated with the
system of N electrons before ionization, and ,b(k,Nl) is
associated with the system of CN1) electrons plus the
emitted electron in its continuum state after ionization,
then, in the bodyfixed (BF) frame, the transition proba
bility for ionization involving these two wave functions
is expressed in terms of the interaction matrix element,
Va (R,k) B where
ba BF
Vba(Rk)F = BF (50)
Here H is the electronic Hamiltonian for the N electrons,
E is the total energy, and the brackets indicate integra
tion over electronic variables. Within a single determi
nant description,
electron orbitals ul and u2 of $a are replaced by uo and
u2 to obtain b". As a consequence Equation (50) reduces
to a sum of direct and exchange contributions which can
be written respectively, as
76
VD(R,k) = BF
VE(R,k) = BP (51)
where v = 1/rl2, the electronelectron Coulomb interaction
in atomic units.
In Figure 9a is shown a schematic MO correlation dia
gram for the He*(Cs2s) + Ar(3p6) case. The construction
of such estimated MO correlation diagrams will be discussed
in the next section. The MO energies at large R represent
levels at the time of ionization. For instance, referring
to Figure 9a, two electrons, one initially in o2s from He*
and the other in r3p from Ar could participate in an Auger
process whereby cne is promoted to a continuum state while
the other fills the Is vacancy from He*. That is,
ul = 22s, U2 = r3p, the continuum state ul = (k, , ml)
and u' = als.
MO correlation diagrams, such as shown in Figure 9a,
allow one to predict the minimal number of angular momen
tum contributions, ', which is necessary to describe the
continuum state of the emitted electron in the BF frame.
One proceeds by following the MO's involved in ionization
towards their UA limit. With the exception of the contin
uum state, u', the MO's correlate in the UA limit to atomic
orbitals of well defined angular momentum. That is, one
can write
54
0
4C4
F
In
4H
ON
(L)
a,
r
a!
cdwZ
rO.
Or
U
dip
rIFC
ca
'~
42C
78
LOD O ( n
C n
(u CL L V)
uN n rn 
\A
ir
C. Q,. C,
%T cl_
3 ai a a in
<  (I) Qo n
PA *
in
CL
o in a (n
901 3 m
/ / ^ / rs
a
H
4
co
C
C)
m
4
.L
0
u
'I
cz~
a,
4J
4
.0
$4
04
If)
0 04
Un
43
.14
a,
U,
C)
$41
80
CN C
+ +
U), 0 tr,
Lf
LO C C LOL
t tID tD
44
I I
C))
2:
cfl ct. l
I ~i CL
E)U)
LB
m
(I) +
cn ar(0
d (3)901
81
UA
u2 X2( 2,m2)
1 A x{(k,, km{) (continuum),
u A (Zm (52)
u2 X2(2'm2) (52)
where the X's refer to the UA (atomic) orbitals. Looking
also at the direct and exchange matrix elements in the UA
limit, Equation (51) becomes
VD(R,k) qA VUA(k) = BF
VE(R,k) UA VUA(k) = BF (53)
Next, using the notation of previous work by Micha
(Mi70a), the electronelectron Coulomb interaction is ex
pressed as an expansion in terms of its multipole compo
nents,
1/r2 = (1/r>) (r /r>)LPL(cosel2) (54)
L=0
and Equation (53) is written as follows:
VUA(k) = 6(sl,s) 6(s2,s) FL(1'2'12)AL(1'2'12)
L O0
VU (k) = 6(s;,s2) 6(ssl) L FL(1221)AL(1'2'21) (55)
L=0
The factors FL and AL are proportional to products of 3j
coefficients:
/(29+1) (22+1) (2 +1)(2 2+1) i L 9l
FL(12 12) = 2L+ I 0 x
0 0 0
82
1 .L 21
AL(1 212) = (l)m +2M (2L+l) m  m m m (56)
The RL1 '2'12) are Coulomb intearals involving the radial
parts of the orbitals in the UA limit, and depend on their
principal as well as angular momentum quantum numbers.
The presence of the 3j coefficients in the FL and AL fac
tors of Equation (56) reflects the coupling of the angular
moment of the electrons due to the 2L multiple component
of the electronelectron Coulomb interaction and allows
one to specify the ranges of values of k and ml for which
contributions will appear in the direct and exchange ma
trix elements of Equation (55). This is accomplished by
employing the selection rules for the 3j coefficients
(see, for example, Me66). In the UA limit, (1,ml),
(2,m2) and (2,m2) are known. Therefore, referring to
Equation (56), one of the 3j factors in FL specifies the
allowed range of L, and similarly, one of the 3j factors
in AL further specifies the range of M. Once the ranges
of L and M have been determined, the remaining 3j coeffi
cient factors, one in FL and one in AL, specify the ranges
of kl and ml for the continuum state of the emitted elec
tron.
Thus, for the direct matrix element of Equation (55),
L and M are restricted as follows:
S' < L < k+
2 2 2 2
(I+ +L) ever
m+M = m2 (57)
For each L and M possible from Equation (57), the remaining
two factors in FL and AL restrict ; and mj in similar fa
shion:
S1L _< < 1+L
(I+L+) even
ml+M = m (58)
Interchanging the indices 1 and 2 among the primed symbols
in Equations (57) and 158) provides corresponding expressions
for the exchange matrix element of Equation (55).
Thus far the discussion has only been in terms of the
interaction matrix element of Equation (50) between two de
terminantal wave functions, Da and ob, distinguished from
one another, respectively, by the MO's u1 and u2 before
ionization and uf and u2 after ionization.
Consideration is now given to the manifold of determi
nantal states a which is needed to represent the electron
ic state D. before ionization. Each of those states (a
has an angular momentum component along the molecular axis,
a, equal to the absolute value of the sum of axial angular
momentum components of the MO's from which the determi
nantal wave function is constructed. To the extent that
one may neglect rotational BornOppenheimer couplings,
only determinants 0a having A = A. are needed in repre
senting the state of the electrons prior to ionization,
where A. denotes the axial component of electronic angular
1
momentum in the SA limit of the incident channel. How
ever, BornOppenheimer couplings cannot be neglected in
2
the UA limit because of their R dependence (Sm69, Si76).
Hence, in order to properly carry out the UA analysis just
described one must include contributions from states a
for which a Ai = 0,1.
For example, according to the MO correlation diagram
of Figure 9a, one constructs the leading determinantal
wave function prior to ionization from the MO's shown at
the right which correlate to the occupied SA atomic orbi
tals, and observes that Ai = 0. As was mentioned earlier,
there is the possible case of an Auger type process in
which the participating orbitals ul and u2 are identified
with o2sHe* and ir3pAr, respectively. Not shown in Figure
9a are levels of MO's associated with unoccupied SA atomic
orbitals. Among these MO's there may be one whose axial
component of angular momentum differs by 1 from that of
an occupied MO with which it shares the same UA atomic
orbital limit. An example of such a case is the 72p MO
85
arising from the unoccupied SA 2p orbital of He*(ls2s),
together with the a2s MO of Figure 9a, both of which cor
relate in the UA limit to the singly occupied 3d atomic
orbital of Ca*(3p54s23d). Replacing the a2s MO of the
previously described Z determinant by this I2p MO would
result in a H determinant which is significant for the
present UA analysis due to BornOppenheimer couplings.
In the Z case, ul of the Auger type process would be iden
tified with a2s, in the I case with 72p. According to
Equation (52), the UA limit results in = 2,ml = 0 for
the E case and i = 2,ml = 1 for the H case. This means
that in applying Equations (57) and (58) for these two
cases, the restrictions on the values are the same for
both, while the mvalue restrictions involve m, = 0 for
Z and mi = 1 for H determinants.
In general, then, the UA analysis of the angular mo
mentum contributions to the emitted electron requires that
the initial electronic state prior to ionization be written
as a linear combination of such determinants;
S= a C (59)
a
Similar considerations hold after ionization, where a mani
fold of determinants Qb results, each differing from the
other by the particular continuum state ul associated with
it. The final electronic state then is written,
0f = bCb (60)
b
and the total transition probability for ionization is ex
pressed in terms of Vfi(R,k)BF, which is a linear combina
tion of interaction matrix elements of Equation (50):
V (R,k)BF= ICCaVba(Rk)BF (61)
a,b
According to this general description, Equations (57) and
C58) of the UA analysis may be applied using the. value
restrictions directly with values found from an MO corre
lation diagram such as in Figure 9a, but remembering that
the mvalue restrictions are weakened due to BornOppen
heimer couplings.
Of course, ionization occurs far from the UA limit,
and the values for ' which are obtained here are certain
ly not all which should be included, but they do constitute
the minimal set required for making a reasonable physical
description of the emitted electron. That such a minimal
set can be specified is important for the parameterization
of expressions at various levels of approximation by which
calculations of angular distributions of the emitted elec
trons can be made (Eb74, Mi75).
87
3. Estimating MO Correlation Diagrams for Diatomics
An analysis such as has just been outlined requires
only schematic correlation diagrams, which should, how
ever, be reliable in relating SA and UA limits of the
higher lying MO's. In this section, a procedure will be
described for estimating MO correlation diagrams, and ap
plied to the collision pairs He*(ls2s) + Ar(3p6), +
Kr(4p6), + Hg(6s2), and Ne*(2p53s) + Ar(3p6). Based on
these estimated MO correlation diagrams and the analysis
of the previous section, the minimal set of angular momen
tum contributions required to describe the emitted electrons
in PI and AI will then be determined.
In order to begin constructing estimated MO correla
tion diagrams, one must have the appropriate SA and UA
atomic orbital energy levels of the collision partners
both before and after ionization. The following scheme
has been found to be sufficiently reliable and simple to
apply. For the ground state levels of neutral atoms one
can use any of the results of the HartreeFock or Hartree
FockSlater calculations, which are available in tabulated
form in the literature CFi73, De73, Ma67a, C174). Further
more, one can rely on these calculated atomic orbital en
ergy levels for ground state neutral atoms in order to ob
tain the levels of the ground and excited state atomic ions
and of excited state neutral atoms. Clementi and Roetti

Full Text 
PAGE 1
ELECTSOHIC AND iiV/L^MICAI. AS^.'EC or DIATOMIC SYSTSMS By JOHH CURTIS BELLUM i D ; 3SERTATI0N PRESENTED TO THE GRADUATE COUNCIL 01 THE UNIVERSITY OF FLORIDA IK PARTIAlu FULFILLMETfT OF THE REQUIPEiMENTS FOR THI DEGRcJE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 197b
PAGE 2
Dedicated to My Parents, who, though familiar with little of v/hat is reported here, nonetheless know what J. have been doing and have supported ir;e in it.
PAGE 3
AC KN 0V7LEDGEMENTS I want to express my deep appreciation to my advisor. Professor David A. Kicha, for the direction, support and financial help he has provided me during this doctoral research. :C arA most grateful for his high caliber of scientific excellence and integrity, and for the great patience he has shown in guiding ine in research. I owe a debt of gratitude tc the numerous people, associated at one time or ancther with the Quantum Theory Project, v.'ho have been of specia] assistance and encouragement, and vrhose influence has come to bear either directly or indirectly on this dissertation. Among these are several I vrant to mention, in particular. Professor N. Yjigve Ohm provided me financial assistance during my first months at the Quantum Theory Project, and has since maintained interest in m.v work and progress. Professor Erkki J. Erandas and Dr. Rodney J. Bartlett initially introduced me to research in euantura chemistry, and generously made available to mie their expertise and enthusiasm. In connection with multiplescattering and local exchange related matters. Professor John VI. D. Connolly, through lectures and ready attention to my inquiries, was of great help. Close associations with Drs. Suheil F. Abdulnur, Poul Jiz5rger.sen and JianMin Yuan, and Professor Manoel L. de Siqueira, have been personally beneficial as well as significant to my overall
PAGE 4
oerapective in ijcience, The able leadership of Professor PexGlov Lovi'din as director of the Quantum Th30;ry Project, along vn'.th his nameroua lectures and also his interest in philosophical considerations in science, have played an iiaportant role in my graduate edvication. As is the case with any undertaking, a program of graduate studies provides a context and situation in which one is able to grow personally in all respects. In this regard I want to express ray gratefulness to many friends, both scientist and nonscientist alike, as well as to God, my creator, by and in whom I exist.
PAGE 5
PREFACE I find it appropriate to make some remarks concernjnq the perspective and context in which the work reported in thif.; dissertation has been carried out. Since its inception in the 1920 's. Quantum Mechanics has become well established as the suitable framework within which to describe phenomena of a physical and chemical nature. Building upon only a few axioms, the formal Quantum Tlieory manifests itself in the form of mather;.ati~ Cell equations, the solutions to which determine expressions for calculating physically observable quantities. Confidence in Quantum Mechanics derives from the impressive successes it has had in providing results in agreement with experiments. Hov^ever, in applying Quantum Mechanics to describe actual, known, physical and chemical phenomena, one quickly becomes aware of the fact that there are only a few cases v/here an exact treatment has been possible. In nearly all cases of interest, the mathematical equatioHc; of the theory ^ though succinct in what they say, are unmanageable to solve. Computational considerations, therefore, have strorgly influenced theoretical investigations in physics and chemistry. Basically two approaches have evolved. The socalled ab initio calculations provide approximate solutions to
PAGE 6
tke "exact" quantum mechanical equatioxis within a fraraework wiiich, in principle, allovs for the solutions to be progressively improved upon r.o approach the ''exact'' solutions. One is presumably limited here only by the size of the electronic computer available. On the other hand, one may focus on the main features of some particular piiysical phenomena of interest, and use the "exact" quantum mechanical equations only as a guide in order to arrive at approxim.ate equations V7hich m.iip.ic the important aspects of the process being studied. These approximate, or model, equations are many times only approximately solved! Such apparently crude approaches require of the researcher all of the physical intuition which can be mustered, in order to properly assess the .important features of the physical situation and to approximate them reasonably v/ell. Nevertheless, much physical insight and many useful quantitative results can be extracted fro.oi this point of view. Indeed, the task of science is essentially to formulate descriptive statements, both qualitativ'^a and quantitative, v^hicJi conform as nearly as possible to the laws and phenomena of nature as we observe them. Finally, it should be mentioned that the::e are formal results coming out of quantum theoretical investi.gations which determine many characteristics of the "exact" and approximate solutions, even before they have been calcuVI
PAGE 7
lated. EotK types of investigations described above rely upon these forinal reKultS as V7ell as upon each, other. In the dissertation which folicv/s, research of the second type, mentioned abova will be reported in the form of a quantuia laechanivjal investigation of electronic and dynamical aspects of diatomic systems.
PAGE 8
TABI,E OF C0?;!TENT3 PagG ACKtoOWLEDGEiyiEMTS . . iii PREFACE ,,...... . V ABSTl^ACT. X CHAPTER I. INTRODUCTION . 1 1. A Formal Statement of the Problem ^ 2. Remarks Regarding Reference Frames .... 6 3. Remarks Regarding the Wave Function. ... 17 4. The Coupled Equations and Coulomb and BornOppenheimer Coup] xngs ... 13 5. Discussion ...., 24 II. A STUDY OF SINGLEELECTRON AND TOTAL ENERGIES FOR SOME PAIRS OF NOBLE GAS ATOMS 2 8 1. Theoretical and Compucational Considerations 30 2. Results 39 3. Discussion 55 III. DIATOMIC MOLECULAR ORBITAL CORRELZ\TION DIAGRAMS FOR PENNING AND ASSOCIATIVE IONIZATION 53 1. MO Calculations for He*4Ar and He+Ar ... 61 2. Analysis of PI and AI Processes Based on KO Correlation Diagrams 74 3. Estimating MO Correlation Diagrams for Diatomics ^7 4. Discussion llO
PAGE 9
TABLE OF CONTENTS (Continued) P_aqe IV. A COUPLED CHANNELS APPROACH TO PENNING IONIZATION OF Ar BY He*(ls2s,2s) 114 1. The Scattering Problem in Terras ol E'iscrete and Continuum ElectronJc States . lib 2. Discretization of the Continuum and the Modified Coupled Equations 12S' 3. Solution of the Modified Coupled Equations 13 3 4. Characteristics of PI and AI Processes . .138 5. An Application of Discretination to PI and AI 147 6. interaction Potentials for Hf; Â• (ls2s , ^S) +Ar and He ~ Ar+ ( 3p'^ , 2p} . . 153 7. Parameterization of Lae Couplings. .... 166 8. Results from Coupled Channels Calculations of He*{ls2s,3s) + Ar :I Col] isions , 171 9. Discussion 205 REFERENCES , 212 BICGR.APHICAL SKETCH 218
PAGE 10
Abstract of Dissertation Presented bo the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Pli.ilosophy ELECTRONIC AND DYMAI4ICAL ASPECTS OF DIATOMIC SYSTEMS By John Curtis Bellum August, 1976 Chairman: David A. Micha Major Department: Physics Diatom.ic systems are cor.sic'erf:!d frovr. the points of vxew of their electronic structure and the dynamics of motion of the heavy particj.es (nuclei) upon collision. In Chapter I the electronic and nuclear motions are treated formally by expressing the Schrodinger equation for the nuclei and electrons in independent variah^les in the bodyfixed (BF) , center of mass of the nuclei, frame, and then introducing an expansion in terms of a com.plete set of electronic states at each internuclear separation./ R. BornOppenheimer and Coulomb couplings between the electronic and nuclear motions are pointed out and discussed. In Chapter II, atomatom interaction potentials and electronic structure are investigated using the Multiple Scattering (MS) calculational scheme with the socalled Xa approxim.ation to electronic exchange. The atom pairs,
PAGE 11
HeHe, no?.r and ArAr are studied, and results are presented for gxQund and excited state configurations. The computed interaction potentials exhibit Eo:ri'4ayer repulsion, and the calculations dcraonstrate hovj Xa orbital energies can be used to predict crossings between interaction potentials. An analysis is presented of the Xo. theory and its usefulness for the understanding of collxsioji phenoraenci. In Chapter HI, Penning ionization (PIi and associative ioni?ation (Al) processes art considerec 3.n terms of molecular orbital (MO) correlation diagrai.s. MO correlation diagrams are calculated for He* (ls2s) + Ar(3p^) and He (Is ) + Ar (3p ) v;ithin the MSXu scheuie for nonsuir.polarized and spinpolarized orbitalb. The ionizarion process is discussed in terms of an Auger type mechanism involving WO's which can be inspected in the unitedatoms (UA) limit i.n a way which permits an analysis of the angular momentum contributions of the emitted electron in the BF fram.e. MO correlation diagrams are constructed based on atomic orbital energies at the separatedand unitedatoms limits, which are determined from data available in the literature on ground state atomic orbital energies. Estimated MO correlation diagram.s are presented for He*(ls2s) + Ar ( 3p ), 6 2 5 6 + Kr(4p ), + Hg(6s ), and Ne*(2p 3s) + Ar ( 3p ) , and in each case an analysis is miade of the angular momentum components of the emitted electron. The results confirm that relatively few such components are important for electrons emitted in XI
PAGE 12
PI and AI. The UA analysis shows the importance of spinpolarizod MO ' s , and also BornOppenheimer rotational couplings, particularly between HO ' s which converge to the same UA limit. In Chapter IV consideration is given to the dynamici? inv'olved in collisionsl Lonisetion processes. The form^il development of Chapter I is extended to include both discrete and continuum internal el^^ctronic states. The resulting continuously infinite set of coupled equations is then discretized, leading to modified coupled equations for the heavy particle motion. Discretization pirovides a suitable framev/ork in which to introduce physically reasonable approximations which lead to a treatment of PI and AI in term.s of several (20) twostate cocpleo equations. Application is made to PI of Ar by He*(ls2s,"S), and the results show that the approach includes the important dynarriical features Â» Partial ionization crosssections per unit energy, Â£ , of the em.itted electron are calculated as a function of e, and they show an e dependence in good qualitative and quantitative agreement with experimentally measured energy distributions of em.itted electrons. Partial crosssection contributions for the heavy particles in specific angular momentum states are also singled outTheir behavior as a function of e, or of the angular momentum partial wave numJoer, shov.'s structure which reflects regions of high density of states in the continuum of final relative m.ot:ion
PAGE 13
of the hea\ry particles. The KeAr netastable and molecular ion potentials are represented by a corivenient functional forrn describing atomatom .int.racMon potentials over the entire range of R. In addition, the connection between the decay v/idth V and the coupling matrix elements between discrete and continuum electronic states is used to iriake reasonable estimates of the latter from semienipirical results for r.
PAGE 14
CPJAPTFR T. It^TROr.UCTION The overall subject of this dissertat5.on is the study of electronic and dynatnlcal aspects of diator.ic systems. l\i such a study it is the behavior of tie electroDs and nuclei, during collision processes of the tv/c atoms comprising the diatomic, which io of inberest. In this introductory chapter the collision processes v/ill be discussed formally in terms of the Schrodinger equation which is satisfied by the wave function for the systeia of nuclei and electrons. The Sciirodinger equation will be treated in the bodyfixed, center of mass of the nuclei, frame b^ properly transforming the Hamiltonian operator and v/ave. function to be expressed in this reference frame. The coux:)led equations will be derived with attention focused on the various sources of coupling between the electronic and nuclear notions . In the following chapter, matters specifically concerning the electronic structure of some diatoms comprised of rare gas atoms will be considered, and possible applications to collision processes discus:ied. In Chapter III special attention will be given to chemiionization processes involving collisions in which one of the atoms is initially 1
PAGE 15
2in an excited state. Features or the electronic structure of such coJ.lisicns will be clisjussed, on the basis of which an ciuaiysis of the angular norientuir. contributions to electrons einiCTzed in such processes will be carried out. Finally, in Chapter IV, tho dynamics oi a specific ch.uiiionization process, that of Fenniiig ionization of Argon by n',etastable Keliur'i,wili be treated by :neans of numerically solving the coupled equations within a twostate approxiraation. Total and partial crosssections obtained from the ciiculations will be reported. 1. a For mal Statement of the Problem In the quantum mechanical treatment of the two nuclei crnd N electrons v/hich constitute a diatomic system, the Ilamiltonian operator expresses all of the energy contributions associated with the nuclei and electrons. The actual form, of the Eamiltonian depends upon the coordinate frame with respect to v;hich the positions of the nuclei and electrons are located. The coordinate frame of m.ost practical use in terms of measuring the results of collision events is one fixed in the laboratory, referred to as the laboratoryfixed (LF; frame. The description of the collxsion events, however, is most conveniently carried out in a coordinate frame whose origin is fixed to the center of m.ass of the nuclei (CMN) of the two atoms which are
PAGE 16
3collidinq. The reader is referrec elsev/here is<^&, for example, Pa6B and references therein,, and Jul 5) fcr a detailed discussion of various coordinate frames coiPanorily used and the forra the Hamiltonian assumes under transformation from one to the other. The formal development in thin chapter begins with the Hamiltonian expressed in the socalled laboratoryfixed, center of mass of the nuclei (LFCMN) , frame, whose origin is fixed to the CMN and whose axes remain parallel to those of the IS' frame. Furtherinore, relativistic and masspolarization contributions to this Hamiltonian are neglected {Pa.68) , and the coupling between eJ.ectronic spin and orbital angular momentum is assumed to be small. lit this LFCMN frame, R is the relative position vector of trie two nuclei labeled a and b, having masses in and m, , respectively. The N electrons are located by the set of spacespin coordinates {x.,i=l,N} = X, where J. X. (r,,s.), r. locating the spatial position and s. the 1111 1 th spin coordinate (a or 3) of the i electron. In terms of these variables, the LFCMN Hamiltonian is expressed as H(R,X) = (l/2m)V^ + H^^(R,X) , (1) where m = m m, /(m +m, ) is the reduced mass of the nuclei, a b a b '^
PAGE 17
4H ,(R,X) = 1/2 y V"^ I (2 /r. + Z, /r.,) 1/2 1 r^ 1 i=i ""i ii + IWr .^ ZJ,^/R (2) is the electronic Kajniltonian, In the usual sense, Z , d r. and r. . refer to the charge on nucleus a, the distance ia ij between the i' electron and nucleus a, and the distance between the i^^ and j^" electrons, respectively. The expressions are in atoraic units, where the unit of energy is the Hartree,the unit of distance the Bohr radius (a^) and the unit of mass that of the electron. On the right side of Equation (1) the first term represents the relative kinetic energy of the two nuclei, which are the heavy particles taking part in an atomatom collision. The terms on the right of Equation (2) represent, in order, the kinetic energy of the electrons, the Coulorab attraction energy of the electrons with the nuclei a and b, the electronelectron Coulomb repulsion energy and the Coulomb repulsion energy betv/een nuclei a and b. The description of a diatomic system can formally be raade in terms of the wave function, ^ iR,X) , for the total system of nuclei and electrons, which satisfies the time independent Schrodinger equation, H(R,X)^(R,X) = E'i'(R,X) , (3)
PAGE 18
where E is the total energy of the system. In solving Equation (.3) it is importanL to pay attention to the angular moiTiea'CLiiT! of the electrons and nuclei. The total orbital angular moraentum. K, is the sum of the nuclear and electronic orbital angular momenta, L and L , v/here L^"^ =~R X iVj^ (4) and ^ =^ I r^ X iV^ . (5) i=l i The LFCMN components, K , K and K^ , as well as K = KK, all obey the usual commutation relations for angular momenta, and commute v;ith the LF~CMN Hamiltonian of Equation (1) , owing to its rotational invariance. Accordingly, the solution, '4'(P,X), of Equation C3) is simultaneously an eigenf unction of K' and K , v\'ith eigenvalues K(.I\+1) and M, respectively, and the total orbital angular momentum and its z component are constants of the motion. Because of the as 'amption of negligible spinorbit coupling for the ele;ctron3, the spin angular momentum has been left out of this discussion for convenience, but could easily be included. So, vv'ith no loss of generality, the solution to Equation (3) is classified according to the constants of 2 the motion, K and K , and is written z
PAGE 19
Remarks Regarding Reference Frames At this point attention v/ill be turned toward specifCrtlly pedagogical considerations, providing a reminder of some i>asic concepts of a mathematical and physical nature whi.ch are helpful in understanding the approach which will be taken in solving Equation (3) . The remarks which follow will serve to reiterate some key ideas which have long heen escablished (see, for example, Kr30) , The ideas are not easy to grasp, and are often passed over either in toe sophisticated or too cursory a v;ay in the literature. Notable e:cceptions, however, can ba found (Kr30, V151, Ho62, Th6i, Th65, Pa68, Sm69). As it stands, Equation C3) involves the Hamiltonian and v/ave function, H' expressed as functions of R and {r.} referred to the LFCMN frame. The axes of this frame are labeled by x, y and z. The i electron is thus located by r^ having coordinates (x.,y.,z.). R is most conveniently represented by its spherical polar coordinates, R, 9 and ({) . Therefore, in the LFCMN frame, (x^,y^,z^, R,9,'i) constitute an independent set of coordinates in which to solve Equation (3) . In terms of these coordinates, the components in the LFC?W frame of t^^^ and L^^^ (Equations (4) and (5)) appear as
PAGE 20
and Â•7L^^*^ :^ i(sin(l)8/3e + cot0co3()3/9(i)) X L^'^^^ = i(cos<)8/J9 + cotesln(i)3/3(;)) L^'"^ = i 5" (y.9/3z. z 9/3y ) X . , 1 1 1 1 1=1 L^^^ i.I (Z.3/3X. X..3/3Z.) L^^^ = i.I (x.3/3y. y.3/8x.) . (8) N L i1 The V7ave function of Equation (6) can also be written explicitly in terms of these variables to read " " ^ (9) ^KM^^'^^ = ^KM^^'^'^'^i'^i'^i^ KM AlsOj. the first term on the right of Equation (1) , expressing the x^elative kinetic energy of the nuclei, appears as follows (see, for example, Co62) : (l/2m)vj = ~(l/2mR^){3/3R(R^3/3R) R + cot63/39 + 3^/36^ + sin'^eaV^t^} . (10)
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8One can check theit the cerms within the brackets of Equation CIO) , involving the angie3, can be replaced by L^"'L^^'', according to Equation C7 ) (see, for example, Ed60) . It is important to keep in mind that partial derivatives, such as occur in Equations (7) , (8) or (10) , depend upon wliich variables are actually independent of one another during differentiation. Because the LFCriN frame constitutes an inertial reference frame, the set of variables, (x . ,y . , z . ,R, G ,0) , of the electrons and nuclei is indeed an independent set of variables. Therefore, for example, 3/3 6 in Equation (10) means to differentiate with respect to 8 while holding all other variables, (x.,y^,z., ]R,(f)) , fixed. Similar considerations hold in turn for each variable in this set. Now one would like to proceed to solve Equation (3) by expanding the wave function of Equation (9) in a complete set of electronic wave functions at each R, It is here that the need arises to express Equation (3) in terms of variables referred to a coordinate frame in which the internuclear vector R is fixed. This is due to the fact that normally electronic wi'.ve functions are d3termined in such coordinate frames. Such a reference frame, fixed to the nuclei, with origin at the CMN, will be called a bodyfixed, CMN (BFCMN) frame. The key concept regarding electronic v/ave functions is that they are usually calculated
PAGE 22
9uncler conditions v/here the set of electronic coordinates in the 3FCMN frame are treated as indepandent variables. The 3FCMN frame considered here shall have axes x', y', z', where the 2. '* axis is along the internuclear vector R. Consequently^ the angles cp and G are the first t'vvo of thi.e Euler angles Csee, for example, Ju7.5) which rotate the SFCMN franie into the BFCMN frame. The third Euler angle, Y, can be freely chosen since it only serves to define the x^ and y ' axes, which can be arbitrarily set for a diatomic molecule (Ju75) . Therefore, y is an auxiliary variable, and as such v/ill play only an indirect role in what follows, Figure 1 shows the Euler angles ((J)9y) by which the SFCMN frame is rotated into the BFCIW frame. The coordinates of the internuclear vector in the BF~ CMK frame are simply (RfO,0). The i electron, with coordinates (x.,v.,z.) in the LFCMN frame, has coordinates {kI ,y%z'') in the BFCMN frame, specified for a aiven orientation (9,4)) of the internuclear vector by the unitary trans forrnat ion (see, fojr example, Ti64 and Ju75) xT = y/. {^^y) = (cosycosecoscf; sinYsin4i)x. + (cosycosflsini) + sinycoscl.) y . cosysinOz^ V^ y'((t'OY) = (sinycosBcosc^ + cosysinc})) x . ' i ^ 1 ^ ' X + (sinycosOsincji + cosycoscj)) y . + sinysinOz.
PAGE 23
10y
PAGE 24
11%'. = z'((j)G) ^ sin9cosc)x. + sinGsincJiyj^ + cosG/.^ , (11) Here attention has been drawn to the dependence of x^,y.^ and s" on the angles cj) , 6 and yThe inverse transformation, giving the coordinates {x.,y.,z.) ot the i electron in the SFCMN frame in terms of its coordinates 'xr,yr,zp in the BFCMN frame, is, X. = (cosYoos9coscJ) sinYsin4))x7 X 1 (sinycosecos'^ + cosysinci)) y^ + sinBcosiizr y. = (cosycos9sin(i) + sinycostfi) x^ + (sinYCOsOsinft) + cosycoscj') y ' + sinesinct)2T z. cosysinexT + sinysinGyT + cos9z7 . (12) X i X 1 With these expressxons, the Schrodinger equation of Equation (3) in the LFCMN frame, wh^ra (x.,y.,z. R,e,(j)) are independent variables, can be rewritten a].lowing the BFCMN electronic variables Cx^',Y.',zO to be treated as j ^ X X independent variables (as they are in molecular electronic structure calculations) . The expression, "treated as independent variables", in the preceding sentence speaks to an important concept. To an observer in the BFCMN
PAGE 25
12frarae, the electronic and nuclear coordinates are simply CxT rY 'T , z r , R, 0, 0) . Emt ouch an observer must keep in mind that the BFQCN frame is not an inertial frame, and, by Equation {11}, (xr,y',2r) have explicit ((JjOy) dependence for a given set of electronic coordinates in the LFCMN frame. In order for an observer in the BFCMN frame to use the. electronic variables Cxr,yr,z.O as independent variable: in treating Equation C3) , two new variables, 6" and (j)', may be int:coduced with two restrictions: (1) the set (xT ryf, zr,R, ^,(j}') must be an independent set of variables for the observer in the EiPCMIT frame, cind (2) G ' and 4)' must be given by. (13) According to this equation, C^'il)^) may seem to be redundant variables, but this is not at all the case. The variables Cx^.yr,zr) are independent of C6 '(})') as far as the observer in the BFCMN frame is concerned, and Equation (13) simply specifies their values in terms of variables which are determined by an observer in the LFCMN (inertial) frame. Equation C13) gives the relationships for 9' and (j) ' just as Equation (11) does for xr,vr and zT. Equation (3) will now be transformed so that an ob
PAGE 26
13aerver in the BFCMN frame could att.eiiipt to solve ?..t m teinns of the independent set Cx'"y T jZ.^ ,R. 6 ^,(J> ') . Sach a cransformation involves both the Ilamiltonian operabor (see Equations (1), (2) and (10)) as v/ell as the wave function (iree Equation C'^)). First the Haii^iltonian will be considered . For clarity, when performing partial differentia cion involving the independent variables iy...y.,z.,R,B,(^), the symbol 9, which has already been used, will now specifically indicate that, when a variable of differentiation from thi't! set has been singled out, all the ethers are held fixed. For instance, 8/39 E 8/561^ ^^^^^_^^ , (14) v/here the variables held fixed are explicitly indicated. Similarly, 3'' will be used to indicate partial differentiation involving the independent variables (x',y^,zr, R, 9 ^ ,([)') . By analogy with Equation (14) for iristance, 3V39=. ^/^^'\u.l,yl,zp.R,,n *"' Then, if w is one of the variables (x . ,y . , z . ,R, 8 ,(i)) ,
PAGE 27
14N Â•f I { Ox;/9w) 3 V3x' + (3yr/8w) a V3y^ iO^; /3w) 9 "/9j^ ' i . (16) Jv, similar relation holds for expressing 3''/Dw'* as a linear coinbination of partial cleri.vatives involving the variables (x . ,y . ,z . ,R, 9, d>) where w' is cne ot: the variables (xr,yr, z;',R, 9 ',<})') . These are useful expressions to keep in niind.. especially as to their meaning emphasised by lising the 3 and 3^ notation. This conceptual and notational viewpoint follows the work of Kronig CKr30) Referring now to Equation (2) , the kinetic energy of . th the X electron involves the opsrator, 2 2 2 2 2 2 2 V = 3 /3xt + 3 /3y + 3 /SzT . (17) On the basis of Equations CH) and C16) , one can write 9/ox^ > (cosYcos9cos(! sinysincf)) 3 '/SxT (sinycosecoscj) + cosysincf)) 3 '/3y ' + sinecos()3 "/3;c^ , (18) with similar expressions for 3/3y, and 3/32^ . Using these expressions directly, one finds that 3^/3x^ + 3^/3y': + 3^^/32^ 1 ^ 1 1 3'^/3xr" + 3'^/3yr^ + 3'^/3zr^ = V^. . (19) "i
PAGE 28
Â•15Ths potential enargy terras in Equation (2) involve only the distances between electrons and nuclei, and therefore are unchanged in going from the variables (x . ,y . ,z . , R,G , cp) to the variables (x'T ,yr , z.'T ,R, 9 '^ ,(j) ') . Consequently, the electronic Hamiltoniar of Equation (2) talces the same form for an obsverver in the BFCMN frame as for an observer in the LFCMN frame. Such is not the case for the first tenr. of Equation (1), which is expressed in the /ariatles (R,6,i4)) in Equation (10). By a straightforward, but tedious, application of Equation C16) , using Equations CH) and (12), one finds that a/39 > d'/Se' icosyL isinyL .' (20) y X and 9/84 O V^tt"" icosG'L^?^ + isin6 " (cosyL^^ sinyL ? ) . z X y (21) Ce) (e) (e) Here, L^ ^ , L , and L\.' are the components of the elecX y z ^ tronic orbital angular momentum in the BFCMN frame. They are expressed according to Equation (8) by replacing all LFCMN quantities by their appropriate BFCMI'J (primed) counterparts. Equations (20) and (21), again with some tedious algebra, lead to the following result for the angular terms in Equation (10) :
PAGE 29
iGcot83/8e + 9^/36^ + sin ^eS^/a^^^ cot9"3''/9e'' Is'"^/ae"' + sin ~e''d' /d^'" 2i{cot9/sinO)d'/9(()'L^!^ cot^ClL^!^^ + (L^?^^ (2/sin9) (cosyL^^sinyL^^^^ ) (13 V9 ' cos8L^^^) , (22) Here, m the usual xvay, L^"^ = L^^^ Â± iL^^^. The R dependent terms in Equation (10) are unaffected in the transformation from the LFCMN frame to the BFCI4N frame. Consequently, replacing the angular terms of Equation (10) by the right side of Equation (22) leads to the appropriate Kamiltonian operator Csee Equation (1) ) which can be used by an observer in the BFCMW frame in order to formally treat the behavior of the electrons and nuclei. The first three terms on hhe right side of Equation (22) are sim.ilar to those found in the LFCMN Hamiltonian. The remaining terms are those compensating for Coriolis effects due to the fact that the BFC"Â«IN frame is not an iner tial frame. In a sense, the inclusion of these Cordis terms is the price paid by the BFCMN observer in order to reckon CxT ,y_^ , z T ,R, 9 ' , ') as independent variables.
PAGE 30
173 . Remarks Regard i ng the Wave P' unction Tlie wave function of the total systera of electrons and nuclei isaa Equation i3'i) mus'r. also be properly transfonned and expressed in temis of appropriate functions in the BFCMN fraT.e. Here the approach of Davydov (Da6 5) is adopted. Attention in drawn agarn to the fact thac the v^ave function of Equation (S) is an eigenf unction of the square of the total orbital angular moment uin, K , and its component along the LFCMN Z axis, K^ , with eigenvalues K(K+1) and M, respectively. Following Davydov CDa55) , if a coordinate freime undergoes a transformation by rotation through Euler angles (aS'i') to another coordinate frame, then an eigen1 function in the first frame of t"^ and K , with ed.genvalues , respectively, KCK+1) and M, can be written as a linear combination of the (2K41) such eigenf unctions in the rotated 2 frame, all of which are eigenf unctions of K v/ith eigenvalue KCK+1),and each of which is an eigenf unction of K ^ z with eigenvalue, A, among the possible values K,K1, . . . . ,K. For the case considered here, of a transformation from the LFCI4N frame to the BFCMN frame through Euler angles (iG'y) , the wave function of Equation (9) can be written \^^(R,G,*,x.,y,,z.) . I
PAGE 31
18K Here the expansion coef riciencs, ^li^yv ^'^'^'^^ ' ^^^ ^^'^ socalled generalized spherical function^!, oc Df unction.s , and are elgenfanctions of the symmetric top.. A gcod discussion of their properties is given by Edmonds (EdoO) . In Equation (23), ^!:\CR,0,0,x.r,i^;,zr) is an eigenfunction of K^.r the SS.!\. jL JL Jcomponent of K along the internuctear axis, with eigenvalue A. The orbital angular raomentAiin of the nuclei has no component along the internuclear axis, as can easily be veri(e) fied from Equation C4) . As a consequence, K^. ^ L^ ^ . It is at this point that it is convenient to introduce a complete set of electronic wave functions at each internuclear separation, R. The electronic wave functions employed can be any of the ones commonly calculated, where the electronic variables, (xr,yr,z:), in the BF~CMN frame are taken as an independent set of variables. Such electronic wave functions, whether of the singleconfiguration or more elaborate configurationinteraction type, are classified according to their component of electronic orbital angular mom.entum along the internuclear axis. That is, (e) they are constructed as eigenf unctions of !.Â„ ^ having eigenvalues denoted by A. For each A, the complete set of electronic wave functions, {"li^y^ (R,x^ ,yr , zp } , will be introduced at each R, and will be taken to be orthonorraal. Then each 'l*!?, (R, 0, ,xr ,yr ,z T) of Equation (23) may be exK.A 1X1 panded in the set of electronic wave functions, {^_j^^ =
PAGE 32
19'i'^,_(R,0,0,x^,y^,zp = I'^s5/J'f^)/^''nA^^'''i'^i'^P ' ^^'^^ Substitution of Equation (24) into Equation (23) gives a useful expansion fo:. the total wave function of nuclei and electrons in terms of functions of the electronic variables ixl ,y1 ,z'^) in the BFCMN fraine: V^^'^'^^'^i'^i^i^ = ^^/'''\K^^^^%.^^^''^^'n^^''Â•'i'yi'^V ^^" (25) It should be emphasized that the equal sign in Equations (23) and (25) means equality only in the sense that in each case the function on the left of the expression, where the variables are the coordinates of the electrons and nuclei in the LFCMN frame, can be replaced by the linear combination of functions on the right of the expression, where the variables are the coordinates of the electrons and nuclei in the BFCM^J frame. The presence of the variables (p and 6 in the Dfunctions on the right of Equations (23) and (25) will be discussed presently. 4^^ The Coupled Equat ions and Coulomb and BornOppenheimer Couplings Recalling the previous discussion regarding the Hamiltonian operator. Equations C19) and (22) provide an observer in the BFCMN frame the appropriate Hamiltonian for
PAGE 33
20the nuclei ar.d electrons under conditions where '^^ 'Y^''^^^' P., 9 ',({)''} are independent variables. Using thi.s Hamiltonian, an observer in the BFCMN frame laay nov/ replace (>> and G on the rig?it hand side of Equation (25) by 9" and G " according to Equation (13) . Then the right side of Equation (2::)) becomes an appropriate expansion for the "transformed" wave function, 7 (R, ' ,(j)',xr , yT, z :') , which satisfies the "transformed" Schrodinger equation. Thus, an observer in the BFCMN frame nay proceed to solve Equation (3) by relying on Equations (19) and (22) and solving for the wave function VM^^.^^^^^y^I)(l/R)I):<5J(R)Df (re'Y)<.^^(R,xr,yr,zj /\n (26) In dealing with the operators in Equation (22) , it will be useful to rely on some of the formaJ properties of the Dfunctlons (Ed60) . The Dfunctions, D' . (*9y) are formally associated with rotations through the Euler angles C
PAGE 34
Â•21whFre jJ^'^ . L^^^ and L^^^ aire aiver in Equatj.on (7). The XV z 2 2 "? '> Dt"unctions are eigenf unctions of N N +N +N : ^ X y z N'''(cJ)ey)DJ^,^^^((teY) = K(K41)dJj ^((t.6y) . (28) 2 By expanding out N (.(J)9y) using Equation (27) , and recalling K thcc the cp and y dependence of D . C4'9y) is in the factors '] P'f ^'^ '' A "v e " " "" and e"^ , Equation C28) can be rearranged as follows (Ed60) : (d^'/oQ^ + cot03/Be + sin~^e2^/a<^^)Dj^^ ^(t^ey) = {sin~'^0(A^ 2coseMA) K(K+1)}D^^ ^^(s'9y) (29) Other useful relationships involve the operators N^((fi0y) = N Â± iN . In particular, based on the properties ^ y of the Df unctions (Ed60) , it can be shown that N^(c))eY)4j^^(4)ey) = ^_,I^4;vÂ±i(4)6y) (30) v/here \^ = {.KCK+1) ACAÂ±1)}''. Furthermore, 3/89 can be expressed as 9/ae = ~(e"^'^N^ e^'^N_) . (31) Using Equations (19) , (22) and (31) , the BFCr4N Harailtonian can be expressed as follov/s: H(R, 0',(J)',xr,yr,zr) (l/2niR^) ?/9R(R^V9R) (l/2mR^) {cot6'3 V36' + a'^'/oB'^ + sin~^e ' 9 '^/3(j) "^
PAGE 35
22(L^'^^ )^ 2i(coh6Vsin9^)9V9Â«'L^!^ cot^e'(L^^^^+ (L,^?^^ "*" "el^'''i'^i'"? Â• (32) Now, in the usual way, the right side of Equation (26) may be substituted into the "transformed" Schrodinger equation, (H E)H^j.^ = . (33) When this is done, m.any of the terms from the operators of Equation (32) acting on functions D,, ,Â« , of Equation (26) M , A nA V7ill combine and lead to simplifications. One can, for example, compare the contributions from some of the opera(e) tors involving L^.' in Equation (3 2) with the terms involving A on the right side of Equation (29) . Upon multiplication of Equation (33) on the left by K* '^M,A'^*'^''''^S'A'^^'^i'^i'^P ' follo^'sd by integration over the coordinates ((j) ', 9 ', Y,xr ,y^, z^ , the following set of coupled equations for the radial functions i['Â™(R) results: {(l/2m)d^/dR^ (l/2mR^) [2A^ K(KH)] E}/^^(R) nA ^^J^^^nA'^elf^^A^^ " (l/2mR^^ <.^,  (L ^^^ ^ ^^ I .^.^ .> "* (l/n)<4.^.j'd/dRi0^.^.> .^ (V2m)<.Y,dVdR2l^^.^.>}6^^^.
PAGE 36
239 ('=]'" ^ KM+ Mere, the brackets indicate integration over electronic KM: AA KMÂ± variables. The coefficients g.'. are given by gf^:^= (2K41) (8f^) ^7d Â• (36) In Equation C34) , all of the various couplings between the nuclear and electronic motions are included explicitly . The radial BornOppenheiraer couplings appear in tha matrix e].ements betv/een the electronic expansion 2 2 states over the d/dl'. and d /dR operators, and reflect the effect of the radial motion of the nuclei on the electronic motion. As can be seen, radial BornOppenheimer couplings ex.1 3 1 only between electronic states having the same value of A. The rotational BornOppenheimer couplings appear in (e) ^ ig) ^ 2 the matrix elements over the L^ and CL ) operators, and reflect the effect of the rotational motion of the nuclei on the electronic motion. A convenience of treating the problem in the BFCMN frame is that the influence of nuclear rotational motion on the behavior of the electrons becomes expressed in terms of matrix elements between elec
PAGE 37
24" irronic states over electronic angular moiTientum operators, rcTther than nuclear angular rcoriientum operators. As can be (e) ^ seen, rotational coupling due to L^ exists only between electronic states having A values differing from one an(q) > other by Â±1, v/hereas coupling due to (L ^ ' ^yexists between electronic states having the same A value. The socalled Coulomb coupling appears in the raatrix element over the electronic Hamiltonian in Equation C34) , and exists only between electronic states having the same value of A. 5. Discussion Some brief remarks are in order regarding the considerations of this chapter. No qualifications have been placed on the basis set of electronic v/ave functions, {5'^,^}/ used in the expansion of Equation (26), other than that it be complete and orthonorraal. Traditionally, approaches to molecular electronic structure have tended to focus on electronic scates which are eigenfunctions of the electronic Hamiltonian, H , , and which therefore leave the matrix of H^^ diagonal. These are the socalled adiabatic states which provide an adiabatic representat_on. In such a representation all of the coupling between t.he electronic states, associated with inelastic collisional processes, rests in the BornOppenheimer terms of Equation (34) . The eigenenergies of H^^ associated with these adiabatic states
PAGE 38
25obey the noncrossing rule, as is v/ell kno\>m. Hovvsvar, in treating atomic and molecular collision processes, it is not at all clear that the seh of adiabatic states is always the most appropriate representation to use. Stemming from the poj.nt of view emphasized by Lichten (Li63) , much consideration and discussion have resulted regarding the importance of the socalled diabatic representations. These representations are comprised of electronic states v/hich are not eigenf unctions of H ^ Â„ Consequently, the diagonal matrix elements of thece states with H , need not obey the noncrossing rule, and the off diagonal matrix elements may be appreciable. Compared to the BornOppenheimec couplings, the Coulom.b couplings in a diabatic representation can often actually be the dominant source of coupling associated with inelastic processes inflviencing the heavy particle motion described by the coupled equations of Equation (34) . Two particularly good discussions of these matters have been made by Smith (Sn69) and Sidis (Si76) . Here, it is simply pointed out that the appropriateness of the adiabiatic or a particular diabatic representation depends upon how successfully the dominant coupling term.s can be identified, as well as calculated or estiiaated. Radial BornOppenheimer couplings are difficult to calculate and normally must be estimated. In addition, they are character
PAGE 39
26ized by singularities in regions cf R near avoided crossings of the associated adiabatic eigenenergies . The coupling through K ., of diabatic states can be estimated, if not ei ofc'^.n calculated. However, because of frequent lack of iaformation abcut the BornOppenheirner couplings, one cannot always be sure when the Coulomb couplings constitute the dominant contribution in describing inelastic processes. As carii be seen from Equation (34), the rotational Born9 Oppenhaimer couplings have a R '^ dependence. Therefore, their contribution v;ill be of increasing importance as distances of closest approach of the nuclei become smaller. The research related to atomic collision processes reported in the remainder of this dissertation has been carried cut within the framework of diabatic representations. In this connection, electronic states constructed as determinants of oneelectton molecular orbitals are particularly useful. Furthermore, the behavior of the one electron molecular orbitals and associated orbital energies can itself provide information of use regarding inelastic processes in atomatom collisions. Considerations along these lines are pursued in Chapter II, illustrated by molecular orbital calculations on some rare gas diatomic molecules, and in Chapter III, where Penning and associative ionization processes in thermal energy collisions of excited (metastable) rare gas atoms with ground state atoms are discussed in terms of
PAGE 40
27niolecular orbital correlation diagrams. An assessment of the anqular moinantuin contributions in the EFCMN frame of the emitted electron in these processes is made based on the correlation diagrams analyzed in their united atoms limit. In this united atoms analysis the significance of rotational BornOppenheimer couplings will need to be considered, as has just been mentioned. The topic of study in the final chapter is the dynamics involved in Penning ionization of Argon by metastable Helium, in thermal energy collisions. The approach will be to solve nuoierically in a two state approximation the coupled equations of Equation (34) . An interesting feature of chemiionization is that the electroni.c state prior to ionization is embedded in the continuiim of electronic states associated with the ioni^ied electron. Thus, in the expansion of Equation (26) , the sum over discrete electronic states must be augmented by an integral over the continuum electronic states. This feature will be dealt v/ith in Chapter IV. The solution of the coupled equations v;ith the appropriate boundary conditions for scattering will lead to results for total and partial cross sections for Penning ionization.
PAGE 41
CHAPTER II A STUDY OF SINGLEELECTRON AND TOTAL ENERGIES FOR SOME PAIRS OF NOBLE GAS ATOMS In this chapter* consideration vill be given to atoraatoir. interaction potentials and electronic structure pertinent to the description of collision events. Several features should characterize the raethod employed in this type of treatment of the electronic structure of atom pairs : (a) The method should provide a description of the interaction potentials for the ground as well as excited states which govern the motions of the heavy particles (nuclei) during collisions. (b) Details of the electronic structure, at least of the type found within the selfconsistent molecular orbital (MO) framework, should be available so that processes related to electronic excitations may be studied, especially in the case of energetic atomatom collisions. *This chapter is an essentially unaltered version of a contribution (Be74a) to the Proceedings of the International Symposium on Atomic, Molecular and Solid State Theory and Quantum Statistics held at Sanibel Island, Florida, 2026 January 1974, where a preliminary report of the results was made. 28
PAGE 42
29(c) The method should be applicable to a variety of pairs of a comic neutral and ionic species while at the same time it should involve only a moderate degree of calculational effort and complexity. Statistical approaches, such as the ThomasFermiBirac method, have provided useful results by applying freeelectron gas energy expressions in conjunction with a molecular charge deiisity taken as a superposition of atomic charge densities. However, these approaches are deficient in that they only describe the ground state interaction. They afford no information of tne type mentioned in point (b) , and furthermore they are not applicable to situations where appreciable charge rearrangement occurs in the diatom, since there is usually no provision for self consistency in the calculations. These considerations i.ave led to the use of the MSXa method CJ066, Jo7.3, S171a, S171b, S172) in this work. It parallels the HartreeFock approach in that it provides a oneelectron description with corresponding oneelectron eigenvalue equations and eigenstates which are solved self consistently . The method makes use of a convenient local approximation to the exchange potential. Although approximate, the treatment of exchange in the Xa approach should be quite adequate in handling the short range part
PAGE 43
30of atomatom .interactions, wtiere Coulonib and exchange forces between the electrons are the iraportant ones. The calculational procedure is furthermore based on the "wuf fintin" approximation to the oneelectron potential, v/hich entails no additional computational complications as the nvuiiber of electrons being treated increases. Thus the scheme has v/ide applicability. The first application of the FJSXa meth.od to interaction potentials of rare gas pairs was a calculation on the NeNe system (Ko72, see alr.o Tr73) . In this chapter, calculations performed on the atom pairs, HeHe, HeAr and ArAr are presented, and their relevance to collision phenomena is pointed out. At the outset, aspects of the Xa theory and computational approach which directly relate to the present work are described, and limitations of the theory are considered. Results of the calculations are then presented, and finally, a discussion of the results and their significance is given. 1 . Trieoretical and Computational Consideration s Within the Xa formalism (S171a, S172) , the total energy Ey.^ , of a system of N electrons is specified by a set of spinorbitals, {n.}, according to the expression _ T 'Xa In./dr^u.(r^)(V^ + I ("^ V^la^ ^^ ^^l^
PAGE 44
31+ ^ldr.^jdr^{2p (r ) p (r^) i/22 + y 2Z Z. /R , . (37) ^, a b ab a distance between r, and the position of nucleus a and > > the distance between r, and rÂ„, respectively. The electronic charge density is p(rj = p (r^) + p (x^) = I n.u*(?_^)u. (?^) 4I n.u*{?L)u^(?^) , (38) if 14comprised of the charge density of electrons with spinup (denobed by i) and that of electrons with spindown (denoted by 4) , and "xi^^'^l^ = 9a((3/4Tr)p^''^(?^))^/^ (39) is the Xa local exchange energy density, a being a multiplicative factor. It should be emphasized that E,, is an Xa energy functional depending on the spinorbitals , u., and the occupation numbers, n., and does not necessarily rep
PAGE 45
32resent an average value of a many electron wave function over a Harailtonian operator, as is th^ case in HartreeFock theory. For a qiven assiqnrpent of the n. 's, the u. 's are deterrained by making E stationary with respec't to their variation. This leads to a set of eigenvalue equations for the spinorbit. als . For the u ' s of spinup, one has [f^(?^) + JdT^[2p(r^)/r^^^} + V^^(?3))u.(r^) = ^i}\'^r^) . (40) where a and <.<^i' = ^/^ "L'^i' <Â«' A siiuilar set of equations is obtained for the u.'s of 1 spindown. As is well known (S171a, S172) , the interpretation of the Xa orbital eigenenergies , e., differs from that of HartreeFock orbital eigenenergies, and is based on the relation, e. = 9E^, /3n. . (4 3) 1 Xa 1 This condition between E^ , the e.'s and the n.'s insures Xa 1 X that Fermi statistics holds within the framework of the Xa description of a system of electrons; namely, the lov/est value of E^^^ for a system of electrons is achieved
PAGE 46
33when the spinorbitc.ls cf lowast. eigcrienergies are occupiedo Concentrating en diatomics at fixed internuclear distance, R, a state of the electroalc .s^steiP. may be identified by means cf an assignment of the n. 's. A convenient vray, then, of deteriiLining over which region of R a particular Nelectror. state is the one of lowest energy is by looking at the behavior of the Â£. 's for the occupied and unoccupied orbitals of that state, and observing over whxch region of R the e: , 's of t:he occupied orbitals are the lov/est ones. In the results which follow, this feature will be demonstrated. Equation (43) is also the basis for the familiar transition state approach (3171b) , from which good approximations may be found to ionization energies as well as excitation energies betv;een electronic states. In practice Equation (4 0) is solved by the Multiple Scattering (MS) method (Jo66, Jo73) , with the. potential operator V(J^) = /d?2{2p(r2)/r^2> ^ ^a ^^1^ = ^C^^l^ ^ ^a^^l^ ^''' approximated by a "muffintin" form, whereby it is averaged over angles within nonoverlapping spherical regions centered on the various nuclear sites and also other sites in the molecule (outer sphere, empty spheres, etc.), and volumeaveraged elsewhere. The u.'s are de
PAGE 47
34termined self consistently in terms of partial v/ave expansions within the spherical regions, and expansions in "multiplyscattered" v/aves elsewhere. Furthermore, at each iteration of the selfconsistent procedure, the "muffintin" form of p (r) , a^ found from the u.'s according to Equation (38) , is used to evaluate the "muffintin" potential as well as the "muffintin" approximation to E of Equation (37). The reader is referred elsev/here (Co72) for the details about these approximations as well as ways of correcting for them (Da73, Da74a, Da74b) . It suffices hex'e to say only that the effect of the "muffintin" approximation in the MSXa evaluation of EÂ„ is appreciable. Xa ^'^ However, orbital energies e. for diatoms appear to be more reliable than E within the MSXa approach (We73) . It hars been shown (S174) that in obtaining the exchange potential of Equation (42), one need not assume that the electrons of an atom or molecule behave locally like those of a freeelectron gas of th:same density. Rather, one may assume a spherically symrrLetric "Fermi hole" and apply dimensional arguments. Nevertheless, considerable discussion has been devoted to shortcomings of exchange potentials of the type in Equation (42) associated with the finite n;imbers and inhomogeneous spatial distribution of electrons in atomic and molecular systems (Li70, Li71, Li72, Li74, Ra73, Ra75) . By means of a more careful
PAGE 48
35look at the exch^.nge energy of a f reeelectror) gas of a finite numbGi of electrons in a finite volume., corrections to the exchange potential have been derj.ved by separating out the contribution from the interaction of each electron with itself CLi70, Ra73, Ra'>5) ; for a small number C<200) of electrons the contribution from this "self interaction" becoxnes quite sizable. Estimates of the "self interaction" , based on the abovementioned corrections, show that in atoms and molecules only about 85% of its contribution to the exchange potential is included in V (Li 72) , v^hile Xa the Coulomb potential, V^ , in Equation (44) includes all the "selfinteraction". This imbalance may be partially remedied by an adequate choice of the a factor. Now, a variety of ways have been suggested for specifying the a factor of Equation (39) for a system of electrons (Sc72 and references therein) . The values of a for atom,s, resulting from the various schemes, almost all display the same trend; namely, for atoms of larger numbers of electrons the values of a tend toward 2/3, the factor appearing for a freeelectron gas, whereas for atoms with few electrons, larger values of a result. Since the "selfinteraction" contribution is large for few electrons, this trend in a values has often been interpreted as reflecting the required greater com.pensation for the deficit in "selfinteraction" in the case of few electrons, the compensation
PAGE 49
36becoming less as more elections are involved (Li72) . Thus, despite the importance of the corrections which have just been reviewed, the Xa exchange potential represents quite wei'l. the exchange interaction in a system of electrons. As lonq as the "muffintin" approxim^ations are being made, a treatment in terms of the Xa exchange potential alone, with a commonly used value of a, is expected to be adequate for describing the shortrange interaction between atoms, as mentioned at the beginning of this chapter. The calculations perfoiraed here employed a double precision (14 hexadecimal or roughly 16 decim^al digi.ts available per number on an IBM370/165 computer) version cf the MSXa program, MUSCATEL. This precision was required since the interaction energy, AE, is computed as the difference between the total energy of the diatom (in our case, E in the "muffintin" approximation) and that of the two isolated atoms. For instance, the case of ArAr at moderately large internuclear separations, R, involves interaction energies seven orders of magnitude smaller than the total energies used to detenrdne the interaction, hence requiring at least eight significant figures m the total energies. The total energies of the isolated atoms were calculated using the HartreeFockSlater (HFS) atomic program (He63, Za66) . For the homonuclear cases, HeHe and ArAr, the so
PAGE 50
37called "virial theorem" values of a for the atoms, as reported by Schv/ar?; {3c72) , v^are used in all regions of the TTiolecuies. For HeAr the respective atomic "virial theorem" val^ies were used in the spherical regions about the atoms, and a v/eighted in.ean (weighted according to the number of electrons of each atom) of the tv/o values was used elsevhere. The heteronuclear case of HeAr required a choice of the radii to be used for the spherical regions about the atomic sites. Contiguous spherical regions v;eri chosen in all cases, and the ratio between the He and Ar sphere radii, used at all internuclear separations calculated, were determined in the following way. Average radii obtained from numerical atomic HartreeFock calculations (Ma67a, Ha6 8) were used to estimate the size of each of the isolated closed shell atoms. He and Ar, based on the Is orbital of He, and the 3p orbital of Ar. The values are, resoectively , 0,92727 a and 1.66296 a . The ' ^ o o He and Ar sphere radii at different R were then chosen in the ratio of these two chai'acteristic charge extents. From the same atomic calculations, the maximum values of the He Is and Ar 3p radial probabilities occur, respectively, at about 0.55 a and 1.30 a , so for internuclear GO separations larger than about 1.85 a , the above scheme o for selecting sphere sizes should serve well. Other considerations must be made for cases which do not involve two closed shell atoms and where significant charge rear
PAGE 51
38rangemsnt occurs in the diatom. Such a casa is treated in the following chapter v^here calculations are reported on the excited MeAr diatom which separates at large R to He* as2s,"' '"^S) plus Ar t3p^ , "S) . The selection of partial waves to be included in the expansion of an orbital in the various regions depends upon ov^er which regions of the molecule the orbital tends to be concentrated. For a very deep lying core orbital, v/hich is essentially of atomic character, only the partial wave corresponding to that of the associated atomic orbital was used in each appropriate atomic region, since partial waves of other Â£ values give a negligible contribution. For higher lying orbitals, appreciable contribution from a number of partial waves, s, p, d, etc. may occur. Such partial waves were included as long as their amplitudes were at least onehundredth the amplitude of the dominant partial wave. All of the calculations required no more than medium size core on an IBM370/165 computer and the times per iteration of the SCF procedure were about two seconds for HeHe, five seconds for HeAr and between five and ten seconds for ArAr, depending en the ''goodness" of the starting point for a calculation. Denoting the absolute difference, occurring betv/een the values of the "muffintin" potential of Equation (44) at one iteration and those
PAGE 52
39of the previous iteration, by AV. and the maxirraam value A of AV/V by c, the degree of self consistency Â£.<10 " was achieved typically in 1520 J.terations. It should be pointed out that the relative error of the MO wave functions is of the same order of magnitude as that of the potential. Since the total energy is variationally determined, and therefore accurate to second order in the v/ave functions, the degree of selfconsistency we have used is sufficient to insure the accuracy requj.red in the total energies at large R. 2. Result s In Figure 2 are displayed the interaction energies, AF, , for the three diatoms, HeHe, He~Ar and ArAr, as a function of R, the internuclear separation. The interaction energies are shown for the states with the lowest energy at large R (i.e., asymptotic ground states). The interactions on this semilogarithmic plot are seen to be quite linear, indicating the repulsion they show over the investigated ranges of R is characteristically of the BornMayer type, namely A exp(bR). This behavior, of course, would break down at very small R, where the Coulomb repulsion between the nuclei becomes strongly dominant. The beginning stages of this other behavior is seen in the HeHe case at R~ 0.7 a . On the other hand, as R o
PAGE 53
igure 2. Interaction energies, AE, for the pairs HeHe, HeAr and ArAr in their separated atom ground states. Calculated points are encircled. a.u. of distance refers to the Bohr radius, a .
PAGE 54
41 1.0 20 3.0 40 5.0 6.0 R(a.u)
PAGE 55
42increases for a given interaction and approaches the van der Waals radius, the actual interaction energy would pass through zero, and its logarithirt would asymptotically approach (~^) as R nears the point of zero interaction. This pronounced deflection of the logarithiTi of AE away from the BornMayer straightline behavior is not exhibited by the calculated interaction energies in Figure 2 , indicating that these "muffintin" interaction energies show no tendency to describe the van der Waals attractive well. It should be pointed out, though, that efforts made to correct for the "muffintin" effects in the case of NeNe have shown a well defined attractive region (Da73, i:'a74a, Da74b) . The procedure for calculating these corrections is, however, nontrivial, and would be impractical for the present purposes. Therefore, BornMayer A and b parameters for the interactions of Figure 2 were determined by means of a leastsquares fit to the calculalied points over the regions of straightline behavior. The parameter b measures the slope of an interaction as shown in Figure 2 , and reflects the "hardness" of the repulsion described. The parameter,. A, serves as a measure of the overall strength of repulsion, being the value of In(AE) at R=^0, although it is som.ev/hat unrealistic, since R=0 is an unphysical separation at which to compare the "strength" of the repulsion of a BornMayer type.
PAGE 56
Â•43So, the A parameters of the interactions of Figure 2 are simply listed in Table I, and it is pointed out that over the region of BornMayer behavior, they lead to interaction energies for the three diatoms which overestimate the repulsion, but obey the combination rules for such interactions in that the He~Ar BornMayer line falls between those of HeKe and ArAr. Listed in Table I by v/ay of comparison are the b parameters of the interactions of Figure 2 along V7ith those determined from other theoretical calculations and experiment, as indicated. The ranges over v.'hich the listed parameters apply are shown in parentheses. In general, the b parameters of the present v/ork reflect "softer" repulsions than do the other tabulated b parameters. However, the ranges of R values of the present work extend to larger R, and it is possible that "muffintin" effects lead to a proportionally larger overestimation of the repulsion at large R, i.e., to smaller b parameters. The case of HeHe is now considered in more detail. In particular, the behavior of the MSXa eigenenergies is looked at, providing a good example for diatomic interactions. As is well known, at large internuclear separation, the lov/est state of the HeHe diatom is the I state specy ified by doubly occupied la and la molecular orbitals. These two MO ' s are, at large R, essentially the gerade
PAGE 57
44w
PAGE 58
45ar.d ungerada combinations of the He Is atomic orbitals. In Figure 3 is shown the plot of the eigenvalues of these two orbitals as a function of R over the range 0.5 a^ to 5.0 a . In keeping with the idea of a correlation diagram,arrows indicate the HFS atomic orbital eigenenergies ; the 2 , . , Is orbital of He in its ground Is configuration, m tne separated atom (SA) limit, and the Is, 2s and 2p orbitals 2 2 of Be in its ground Is 2s configuration in the united atom CHA) limit. It is seen that c,^ and e^ are nearly ^g u degenerate with e of He at large R, as expected, and separate as R decreases. Now, in the UA limit, the la orbital correlates v^;ith the Be 2p atomic orbital. Thus ^o ] 49 2 2 2 the 'T(la la ) state aoproaches the excited Be Is 2p ggu ^ ^ ^ o atomic state in the UA limit. On the other hand the 2o g orbital correlates with the UA Be .. ^ atom.ic orbital. It is therefore the E (la~2a ) state which in the UA limit g g g 2 2 correlates with the Is 2s Be ground state. It is of interest to determine at what internuclear separation the 1 4 . 2 2 Z state specified by la 2a becomes lower m energy than 2 2 that specified by la 'la . As discussed earlier m this ^ ^ g u chapter, one may proceed in two ways: (1) direct observation of the interaction energies, AE, of the two states as functions of R to see v/here they cross; or (2) observa2 2 tion of the eigenenergies of, for instance, the la 2a g g state to see where e, and Â£. become the ones of lowest la 2a g g
PAGE 59
in u
PAGE 60
47o _ Â—5 , . . ^^r
PAGE 61
48vaiue. Proceeding from. 1.2 a to smaller R, the eigenen2 2 erqies for the la 2a ' state have been plotted along with those of the la la state. For the la ''2a'' state, e. g u g g la^ lies lower, to begin with, than e , and Fermi statistics 2 2 ^^ indj.cates that la 2a is not the state of lowest energy. g g However, e, is rising sharply as R decreases, and is seen to cross above eÂ„ betv.'een 0.5 a and O.G a . In2a o o g ward from this crossing the la and 2a orbitals have the ^ g ct 2 ? lowest eigenenergies and hence the Ic 2a ' state has the ^ g g lowest energy. For comparison, in the insert of Figure 3, a plot with linear scales is shov.n of AE versus R for these two states, which corroborates this behavior. It should be noted that the interaction energies cross in a very gradual manner whereas the eigenenergies cross more sharply. Thus the eigenenergy behavior indicates more clearly the posi.tion of the crossing, v;hich is found to be 0.53 a . These results compare well with SCF results reported on HeHe (Ma6 7c) , where it was found that the en2 2 2 2 ergies of the la la and la 2a configurations cross near g u g g 0.6 a . Also, in the accompanying 5 configuration naturalorbital iteration calculations, it was reported that the 2 2 la la , configuration was dominant beyond 0.7 a . Though the calculations reported here go inv/ards only to 0.5 a , 2 2 the eigenenergies of the la 2o state are seen to be apy y preaching the appropriate eigenenergies of ground state Be.
PAGE 62
Â•49An inve.stigation CYa.74) of Vae HeHe diatom, subsequent to the one reported Ix.ire CBe74a),but closely paralleling it, has been carried out in the HartreeFock approxiination. It is interesting to compare the MSXa and HartreeFock results. In both approaches, the behavior ''2 2 2 of the total energies of the lc"la and lo 2o conrigura^ g u g g tions as R decreases from G.G a to 0.5 a shows that o o 2 2 they cross very gradually, the energy of the Icr 2o state becoming lov/er than that of the ia'^la state at 0.55 a ^ g u o in the KartreeFock case, and as has been seen here, at 0.53 a in the MSXa case. These values are in good agreement. However, in the HartreeFock approach, the determination is based solely on the total energy curves of the two states as they cross with nearly the sam.e slope. This is because in the HartreeFock approach there is no immediate connection between the state of lowest energy and the eigenenexgies of the MO's associated with that state. However, as has been discussed and demonstrated here, such a connection can be made in the Xa approach on the basis of Equation (43) , which permits the detection of crossings either by observing MO eigenenergies or total energy curves. For ArAr, at large R, the ground state is also i; and is specified by the first five a and a MO's, g g u each being doubly occupied, and the first two tt and ir u g
PAGE 63
50HO's, each havincf occiipation nurrJoer 4. For large R, orbi^als lo to 5a are formed from the appropriate combinations of Ar Is, 2s, 2p , 3s and 3p^ atomic orbitals, and the first and second u orbitals from the approprig , u ate comhdnations of Ar 2p ana 3p c tomic orbitals, respectively. In keeping with the Fermi statistics, the eigenenergies of these occupied orbitals are the lowest ones at large R, where we also find lying above them the eigenenergies of the unoccupied 6a and 16 orbitals. A number of states can be specified at smaller R by the various assignments of occupation numbers to the 5a , 6a and 16 orbitals. g Calculations have been made on some of these states and are displayed in Figure 4 on a semilogarithmic plot of AE versus R (m Figure 4, o o 6 refers to 5a , 6a u g g u' g and 16^) . In the region of R shown, numerous crossings y can be seen, and they are all of a very gradual type. 11 2 The a a and 6 curves are from non spinpolarized MSXa u g g calculations, the corresponding spinpolarized calculations reflecting only a small splitting scarcely noticeable on such a graph as Figure 4. We can see that the SA ground 2 state, denoted by a in Figure 4, no longer is of lowest energy for R less than about 3 a . This can be confirmed o again, by looking at the eigenenergies for this state, and in Figure 5 we display the highest of them versus R on a
PAGE 64
Figure 4. Interaction energies, AE , for ArAr in states V7here the highest orbitals have occupation numbers as specified. Calculated points are encircled. a.u. of distance refers to the Bohr radius, a .
PAGE 65
52" !0 r Ar Ar 4 AE(a.a) 2.0 2.5 R (au.) 3.0 3.5
PAGE 66
Figure 5. Orbital eigenenergies for ArAr in its separated atom ground state. Calculated points are encircled. a.u. of distance refers to the Bohr radius, a .
PAGE 67
!0" R(au.) 2.0 40 60 8.0 1 Â— ri~7~r Â•io' (z (a.u.) iO^ 3p Ar 3s Ar 4crg ^ I I i Â«
PAGE 68
Â•55loglog plot. The appropri.ate SA 3s and 3p eigenenargies are also shown. The ei.genenergy of the unoccupied 6a MO y (which, in the SA limit correlates with the unoccupied 4s atomic orbital of ground state Ar, whose orbital energy is "0.0023 a.u. and lies off the scale in the figure) is seen to be descending rapidly as R decreases, crossing the occupied 5o orbital eigenenergy sharply at 3 a , indicating that for R less than 3 a , this state is indeed no longer of lowest energy. 3. Discussion The results which have been presented illustrate the possibilities of the MSXa method in the study of interactions that play a role in collision events. The interaction potentials themselves render information on the BornMayer type repulsion and, in conjunction with results on van der Waals attractions , enter into the calculation of relative motion of colliding atoms. Implicit in the remarks on crossings between interaction terms for various states is that the states calculated in the Xa method are diabatic in nature. Indeed this is the case since each state is independently calculated after being specified by an assignment of occupation numbers to the orbitals. So, while describing the dynamics of collision events, v/e can expect the largest coupling batv/een molecu
PAGE 69
Â•56i<5r configurations to coine from the electronelectron interaction. In principle, these interaction matrix elements between determinantal viave functions comprised of MSXa orbitals can be calculated, but in practice the problem at present seems quite formidable. The usefulness of the Xa orbital energies, Â£., in showing where interaction potentials cross, has been shown. Kence, critical distances of approach for the occurrence of various electronic excitation phenomena can be determined. Also of interest are crossings between eigenenergies such as occurs between e^ and Â£Â„ in Fiaure 5, if g u one or the other of the involved orbitals were partially occupied. Such crossings are of importance in electron promotion mechanisms involved in energetic atoraatom and ionatom collisions. Although the pairs studied here consist of closed sheM atoms, it is expected that the MSXa method, because of its self consistent treatment, can handle as v/ell the repulsion in cases where sizable charge rearrangement takes place. Cf particular interest would be the mechanisms involved in Penning and associative ionization phenomena, where atom + excited atom and ion + atom interactions are of importance. Here again, though, reliance upon additional results for describing the van der Waals region would be needed. In Chapters III and IV the considerations
PAGE 70
Â•57proiapted by the research of this chapter will be applied to an investigation of the electronic structure and collision dynaraics involved in Penning ionization of Ar(3 ) by Re*as2s/'^S) .
PAGE 71
CHAPTER III DIATOMIC MOLECULAR ORBITAL CORREK^.TION DTAGRAI4S FOR PENNING AND ASSOCIATIVE IONIZATION In the previous chapter it has been pointed out that an adequate understanding of atomic and molecular collision phenomena requires information on electronic structure as a function of the changing internuclear separations. Such information is needed in order to describe inelastic processes involving electronic excitation and charge transfer as well as ionization. In this regard, the usefulness of oneelectron molecular orbital (MO) approaches in treating electronic structure was emphasized. Such approaches afford a selfconsistent ca].cul;itional framework of minimal complexity which can treat ground as well as excited states. Chemiionization is a prime example of processes where electronically excited states play a crucial role. In this chapter* the electronic structure involved in e collisional process of this type will be considered. Well known among chemiionization processes are Penning and associative ionization CPI and AI) of the type ^'A preliminary report of the results presented in this chapter was laade at the International Symposium on Atomic, Molecular and Solid State Theory, Collision Phenomena and Computational Methods held at Sanibel Island, Florida, 1824 January 1976. 58
PAGE 72
59A* ^ B > A + b"^ + e~ (PI) and A* + B ^ AB"^ + e~ (AI) , where A^ is wsually an citom in some metastable state and B is an atom or molecule (Mu66, Mu68, Mu73, Ni73, BeVOa, Be'/Ob, Ru72, Ma76) . Experimental information for such collisions includes total ionization crosssections as a function of collision energy (Ta72, Ch74, Pe75, 1175), angular distributions of heavy particles (Ha73) , and energy distribution (Ho70, Ho75, Ce71) and angular distribution (no71, Eb74) of emitted electrons. Most theoretical effort has been directed at determining the energy dependence of total ionization crosssections and the angular distribution of heavy particles (Na69, Mi70b, Mi71, 0172a). The angular distribution of emitted electrons has been recently studied v/ithin a semiempirical model based on MO correlation diagram.s (Mi75) . One of the present concerns is to reemphasize the usefulness of MO correlation diagrams as they apply to angular distribution of ejected electrons in PI and Al involving atomic collision partners In what follows, calculated as well as estimated MO correlation diagrams are presented. The calculated results are given first, where the electronic structure of He*(ls2s,'"' S) + Ar ( 3p , S) has been considered within the KSXa framework as described in the previous chapter. A study has been made, in the nonspinpolarized approximation, of the K L (alsHe) ""^ (a3sAr) ^(a3pAr) (7r3pAr) ^ (a2sHe)
PAGE 73
60configuration of the excited HeAr diatom v/hose MO eigenenergies approach the atomic orbital eigenenergies of He* i_is2s) and Ar(3p ) at large internuclear separation, and also the ground ^1 state of the (HeAr) molecular ion, v;hose MO eigenenergies approach the atomic levels of 2+5 He Cls ) and Ar C3p ) at large internuclear separation Some spinpolarized calculations are presented in the neighborhood of a crossing exhibited by the nonspinpolarized calculations, along v;ith some comments on the conditions under which the noncrossing rule applies for MO eigenenergies. Keeping in mind the calculated results, consideration is given to PI and AI processes in terms of MO correlation diagrams. Analyzing the ionization on the basis of an Auger type process, MO ' s can be identified, v/hich, together with the continuium state of the emitted electron, are involved in the process. Inspecting the unitedatoms limit in the centerofm.ass, bodyfixed frame then permits a determination of the angular momenta which contribute to the continu'Jin state of the emitted electron. Next a procedure for estimating MO correlation diagrams is described, which makes use of available data on atomic orbital energies, and of two basic guidelines. Application is made to the collision pairs He* Cls2s) + ArOp*"), + Kr(4p^), + Hg{6s^), and Ne*C2p^3s) + Ar(3p^).
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~&; The resulting estimated MO correlations tor these systems are then analyzed to determine the minimal set of angular momantum I values whicli are needed in each case to describe emitted electrons in the bodyfixed frame Â„ Finally, a discussion is given of the results of this work and their significance. 1. MO Calculat i.ons for He*+ Ar a nd He+Ar The representative case of PI and AI in Fie*Cls23, S) + Ar{3p , S) collisions will be considered. Here, calculations are needed for the e:ccited states of the HeAr diatom which separate appropriately to He*(ls2s, ' S) and Ar(3p , S), and for the states of CHeAr) whJ.ch separate to HeCls , S) and Ar (3p , P) . Following the work reported in the previous chapter, the calculations are performed v/ithin the MSXa oneelectron MO framework. For convenience in this chapter, the set of oneelectron equations satisfied by the spin orbitals, u., and expressed in Equation C40) , are written ^eff^^l^i^'^l) = ^i^i^^^ Â• ^''^ Here, h'^^Cr, ) is the effective oneelectron Hamiltonian ef r 1 for electrons of spinup, and according to Equation (40) ,
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62v/here V (r, ) is given by Equations (42) and (39) , and Xo: 1 p Cr) , given by Equation (33), is the charge density comprised of the contributions from electrons of spinup and spindown. In the nonspinpolarized (NSP) approximation, p'^ih pN?) p(?)/2 , (47) which means that the orbitals of spinup and spindown become identical, and each orbital can be considered as acconimodating as many electrons of spinup as of spindown. These calculations v/ere carried out first with a NSP treatment. As discussed in the previous chapter, each selfconsistent calculation begins with a potential v/hich is the "m.uf fintin" form of a superposition of atomic potentials centered at each atomic site of the molecule. Therefore, for the HeAr excited state Ar(3p ) and NSP He*(ls2s) HartreeFockSlater (HFS) CHe63) potentials were 2 S used, and for the HeAr ionic state He Cls ) and NSP Ar(3p ) HFS potentials were used. For both the excited molecule and the molecular ion, the electronic states were specified by occupying the MO ' s so that their eigenenergies were correctly separating at large internuclear separations to the corresponding atomic orbital eigenenergies of the above mentioned HFS atomic calculations. In other v;ords , the
PAGE 76
63boundary conditions of the PI and AI processes at large R determined the appropriate excited and ionic states of the HeAr xiiolecule. Heteronuclear molecules require a choice for the radii of the spherical regions centered at each atomic site as mentioned in the previous chapter. There, for the ground state KeAr molecule, the ratio of the He to the 7ir sphere radii was taken to be ^^ig^^c./'^^ap^Ar" 0.92727/1.66296 0.5576, where, e.g. <^]_s>Ke denotes the 2 average valiie of r for the Is orbital of He (Is ). The sxtuation for (HeAr) is roughly the same as for IleAr, since + 5 a calculation of for Ar (3p ) (NSP) shows it to be 1.549 a . So, for the molecular ion the ratio of 0.5576 o r: was used at all R. On the other hand Ke*(ls2s) + Ar (3p ) is quite another case, since the 2s orbital of He* (ls2s) is very diffuse. NSP calculations show of He* (ls2s) to be about 4.546 a . Because of this diffuseness, it was o decided to choose the ratio of the He to the Ar sphere radii by finding which of its values minimized the total energy of the specified excited state of HeAr at a fairly large R, namely R = 9.0 a . In this way, a ratio of 1.4 was found, which was then used at all R for the excited state of HeAr. The values of the factor ct in the various "muffintin" regions of the molecule, for both the excited as well as ionic states, were those used previously in the ground state HeAr calculations, and the specific coraputa
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64tional details also remain as reported in the previous chapter. The results of the NSP MSXn calculations are displayed in r'igures 6 and 7, respectively, for the excited state of the HeAr diatom and the ground state of the HeAr molecular ion wliich are appropriate to PI and AI . Shown on loglog plots are the MO eigenenergies versus R for 2.3 a
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Figure 6. Molecular orbital correlation diagram from nonspinpolarized MSXa calculations of the HeAr diatom in the excited S configuration which separates at large R to He*(ls2s) +Ar(3p^) Calculated points are encircled.
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6 60.07 0.08 0.09 0.10 3dCa(3p'4l3d>Â— 43Ca"(3p^.l3d)Â— 0. !5 3 d 3pCo'(3p*4s^3dW 1.50 3sCtf(3p'4l3d)rVA 1 Â— III' nrrn PTT V HeAr Ly/J LJ_ 2sHe(is2s) I I I I I 3 4 5 6789 10 R in aÂ„ 00
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Figure 7. Molecular orbital correlation diagram from nonspinpolarized MSXa calculations of the (HeAr)"^ molecular ion in the ground E state which separates at large R to He(ls2) +Ar'^(3p^). Calculated points are encircled.
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Â•68r/A Â•0.07 0.08 0.09 0.10 0.15 0.20 0.2 5 3dCd*'{3ff^4sK 0.30' 4sCd^(3p'4sH ^ 0.40 Z3 d c 3pCa*{3f?4s)^' 1.50 3sCa*(3p*4s)HeAr^' 2 a' Â— '45Ar^(33) HsHeds^) .+ /Tr5l 3pAr"'(3p') _l \ I I I I i M I 1 I I I I ^/ ^/ 3 4 56789 10 CO R in a.
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69this ordering for those levels is valid only for R Â£ 5 a . For R > 6 a , the tt level lies above the a level, but they o are too close together at large R to be distinguishable on these plots. In Figure 6 a crossing is shown near 3.5 a occurring betv.'een the doubly and singly occupied NSP a orbitals arising respectively at large R from 3sAr(3p ) and MSP IsHe* Cls2s) This crossing would appear to violate the noncrossing rule for the MO eigenenergies, and v/arrants a detailed analysis. The orbitals of a given symmetry are ordered according to their eigenenergies, obtained self consistently from Equation (4 5) . This equation is an eigenvalue equation involving an effective Hamiltonian determined at each R according to Equation (4 6) . To establish the noncrossing rule for the eigenenergies of Equation (45) , one expresses the effective Hamiltonian at a supposed crossing, R , in terms of its expansion about R = R +6R located a small distance, 6R, c ' ' frora R : c ^eff ^\.^ = ^eff (^^ (^^eff/d^^R ^^ ' ^'^^ c The noncrossing rule follows by noticing that f (dh ^^/dR) 5R is a perturbation which lifts any degeneracy m the eigenvalue spectrum at R . However, if h ^^ (R) is ^ c ef f made to be discontinuous by choice, the conditions of the noncrossing rule no longer apply. Referring to Equation (46), the R dependence of the effective Hamiltonian appears explicity in f , (r, ) (see Equation (41)) and implicity in the charge density, p.
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Â•70Shovring the full R dependence, the effective Kamiltonian may be written as h _{r^ ;R, p (x* ,R) } .. and dh^^^/dR 3h^ff/aR!p + (6h^^^/6p) (dp/dR) . (49) The term, dp/dR, in Equation (4 9) can be seen, from Equation C3 8) , to involve derivatives of the orbitals and occupation numbers with respect to R. As long as the n. and u. are continuous in their R dependence, dh ^^/dR will 1 "Â• eff be well behaved, and the noncrossing rule v/ill hold. However, if the occupation niimbers are changed discontinously in some region of R, then the noncrossing rule will no longer be valid in that region. These considerations apply as well to the NSP effective Hamilton and in fact, in the case at hand, as shown in Figure 6, a discontinuous change in occupation numbers does occur. To the right of the crossing, the a orbitals, in order of increasing eigenenergy, have occupation numbers 1, 2, 2 and 1. To the left of the crossing, they are 2, 1, 2 and 1. A further investigation of the region of the crossing was made by doing spinpolarized calculations at R= 3.0 a , 3.5 a o o 3 and 4.0 a^ m the case of the Z excited state which sepa6 1 ? rates at large R to Ar C3p , S) and He*(ls2s, S) . In Fig2 ure 8 attention is restricted to the levels of the NSP a and a MO's which cross in Figure 6, and they are contrasted with their spinsplit counterparts calculated in
PAGE 85
"72O o QC ro ro Â•no Ul 5
PAGE 86
73tne recjion 3.0 a 3.5 a the NSP a orbital of He* is split considerably into its occupied component of spinup o and its unoccupied component of spindov;n Oo, on out to the SA limit, where the splitting is be3 tween the unoccupied lsÂ„ and the occupied Is of He*(ls2s; S) p ct The o^ orbital from Ar is split only slightly into each of its occupied spin components. Crossings between two orbital eigenenergies of different spin components are permitted since each involves a different effective Harailtonian (see Equation (46)). Between 3.5 a and 4.0 a^ we find such a crossing for the tv;o spin components that split from 2 the NSP a orbital of Ar, and in fact, as R decreases, the 1 fi 1 aÂ„ level from 3sAr(3p , S) is decreasing in energy to pair p 1 3 up with the occupied a level from Is Ke*(ls2s, S) , whereas its spinup partner from 3s Ar(3p , S) is rising to pair up with the (empty) ag level from lSoHe*(ls2s, S) . In the UA limit, this empty a^ level v/ill correlate v/ith p the partially occupied 3po atomic orbital of 5 2 3 Cd* (3p 4s 3d, L) , where L here denotes one of the possible orbital angular mom.enta of the excited Ca atom. Such a spinpolarized ana].ysis as this confirmed the choice of occupation numbers of the NSP calculations for which the vacancy associated with the NSP a orbital is
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74correlating in the UA. limit v.'ith the partially occupied 5 2 3p atomic orbital of Ca*(3p 43 3d). This feature will prove iiTiportanc for the considerations later on regarding the angular momentum contributions to the continuum state of the electron emitted during PI and AI . The crossing shown in Figure 7 , between the NSP singly and doubly occupied a orbitals of the m.olecular ion near 5.5 a , is ^ o another excimple of a discontinuous change in occupation numbers, which in this case is required to reach the appropriate SA limit. 2. Analysis of PI and AI Process e s Based on MO Correlation Diagrams Consideration is given now to how an MO correlation diagram study, such as has just been described, can help in understanding v;hat takes place in PI and AI processes. Collisional ionization occurs for R greater than the distance of closest approach, which in the case of He* + Ar is around 7 a at thermal collision energies. Referring to the calculated MO diagram in Figure 6, one can see that in this range of R there is a vacancy associated with the a MO arising from IsHe* at large R, above which are some fully occupied MO's as well as the singly occupied o MO arising from 2sHe* at large R. This situation suggests that the ionization may proceed by an Auger type process. Namely, two electrons in higher lying orbitals participate
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Â•7 5in ionization as one of them is promoted to a continuum state whj.le the other drops to fill a vacancy, in this case associated with the a MO arising from SA IsHe* . 7:he process may be characterized as one in v/hich initially the two electrons are in MO's u and u^ / v/hile finally (after ionization) they are in MO's uÂ£ and u^ , where uÂ£ designates the continuum state of the ionized electron, having momentum k and angular momentum components Â£Â£. If the wave function 6 (N) is associated with the 3. system, of N electrons before ionization, and 5, (k,Nl) is associated with the system of CN1) electrons plus the emitted electron in its continuum state after ionization, then, in the bodyfixed (BF) frame, the transition probability for ionization involving these two wave functions is expressed in terms of the interaction matrix element. ^ba^^'^^BF = "^^b^^'^^^^l^el^l^^'^'^^BF (50) Here H , is the electronic Hamiltonian for the N electrons, ex E is the total energy , and the brackets indicate integration over electronic variables. Within a single determinant description, d., will differ from $ in that the one '^ b a electron orbitals u, and u_ of $ are replaced by u' and 1 2 a ^ 1 11' to obtain $, . As a consequence Equation (50) reduces to a sum of direct and exchange contributions which can be written respectively, as
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76V^{R,k) g^ Vj^(Rjc) = g^, , (51) where v = l/r,Â„, the electronelectron Coulomb interaction in atoraic units. In Figure 9a is shown a schematic MO correlation diagram for the Ha* (ls2s) + Ar(3p ) case. The construction of such estimated MO correlation diagrams will be discussed in the next section. The MO energies at large R represent levels at the time of ionization. For instance, referring to Figure 9a, two electrons, one initially in a2s from He* and the other in irBp from Ar could participate in an Auger process whereby one is promoted to a continuum state while the other fills the Is vacancy from He*. That is, u.. = c2s , uÂ„ = Tr3p, the continuum state u' = (k, Â£w m:r) and u^ als. MO correlation diagraias, such as shown in Figure 9a, allow one to predict the minimal namÂ±>er of angular momentum contributions, ?. ', which is necessary to describe the continuum state of the emitted electron in the BF frame. One proceeds by following the MO ' s involved in ionization towards their UA limit. With the exception of the continuum state, n^ , the MO's correlate in the UA limit to atomic orbitals of well defined angular momentum. That is, one can write
PAGE 90
o Ul g rt Â• u Â•H o H P n Q) 5^ V4 O o to Â•H XI u o u Â— 3 m c u o H + 4'CN d to p* 0)
PAGE 91
Â•78< 'Si X VO
PAGE 92
u o e u en fd H 'O c o Â•H +> H O n ^ rHlT) CJ n QJ ^ O M e < Xi + p ^ rOCM e en Â•H rH CD 0) Â•H
PAGE 93
80+ < I en
PAGE 94
UA '^1 " >^1^S./Â™1^' UA "2 "^ ^2^^'2'^:>^ ' . UA X^" (k, Ji wHi,") (continuum). . UA . , , . . . U2 X2(^2'"^2^ (52) where the x'^ refer to the UA (atomic) orbitals. Looking also at the direct and exchange matrix elements in the UA limit. Equation (51) becomes V^(R,k) ^^ V^^(k) Bp Vj,(R,k) S^ V^^(k) = BF (53) Next, using the notation of previous work by Micha (MiVOa) , the electronelectron Coulomb interaction is expressed as an expansion in terms of its multipole components. and Equation C53) is written as follows: V^^(k) = S(s^,s^)&{s^,s^) I F^(1^2a2)A^(l'2a2) L=0 ,UA Vh:''(k) = S{s^,s^)&{s'^,s^) I F^(1^2'21)Aj^(l'2'21) L=0 (54) (55) The factors F and A are proportional to products of 3j 1j Li coefficients F^ (1'2^12) Ji v/(2!l^ + l) (2Â£2 + l) (25,^+1) (2Â£2+l) r^.2 L Â£2^ 2L+1 R_ (1'2'12)
PAGE 95
o Z ~ ^1 L ?, M m L Â£, m^ M m^ (56) The R, Cl'2^12) are Coulomb inteorals in^'olving the radial parts of the orbitals in the UA limit, and depend on their principal as well as angular nioTientum quantum numbers. The presence of the 3j coefficients in the F and A factors of Equation ib6) reflects the coupling of the angular momenta of the electrons due to the 2 multipcle com.ponent of the electronelectron Coulomb interaction and allows one to specify the ranges of values of Â£:r and m:j" for which contributions will appear in the direct and exchange matrix elements of Equation C55) . This is accom.plished by employing the selection rules for the 3j coefficients (see, for example, Me66) . In the UA limit, (Â£,,m,), il^,Ta ) and (Â£XfinO are known. Therefore, referring to Equation (56) , one of the 3j factors in F_ specifies the Li allowed range of L, and similarly, one of the 3j factors in A^ further specifies the range of M. Once the ranges of L and M have been determined, the remaining 3j coefficient factors, one in F and one in A , specify the ranges 1j Li of jiÂ£ and m" for the continuum state of the emitted electron.
PAGE 96
3 3ThuSf for tlie direct matrix element of Equation (55), L and M are rastri.cted as follov/s: (il^+Â£^+L'i ever* m'+M = m^ . (57) For eacrL L and M possible from Equation (57) , the remaining two factors in F_ and A, restrict 5,' and m:^ in similar fashion; I Ji,Li_< Ji < 5, +L (Â£ +LFÂ£'') even m'+M = m . (58) Interchanging the indices 1 and 2 among the primed s^mbols in Equations C57) and C58) provides corresponding expressions for the exchange matrix element of Equation (55) , Thus far the discussion has only been in terms of the interaction matrix element of Equation (50) between two determinantal wave functions, $ and ^,, distinguished from a D one another, respectively, by the MO's u, and u before ionization and u:^ and u.^ after ionization. Consideration is now given to the manifold of determinantal states $ which is needed to represent the electronic state &. before ionization. Each of those states $ X a has an angular m.omentum component along the molecular axis, [\ , equal to the absolute value of the sum of axial angular a
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84aomentura components of tlxe MO's from which, the determinantal wave function is constructed. To the extent that one may neglect rotational V3orn0ppenheimer couplings, only determinants ^ having A A. are needed in reprea a 1 senting the state of the electrons prior to ionization, v/nere A, denotes the axial component of electronic angular momentum in the SA limit of the incident channel. However, 3orn0ppenheimar couplings cannot be neglected in the UA limit because of their R dependence (Sm69, Si76) . Hence, in order to properly carry out the UA analysis just described one must include contributions from states $ a for which A A= 0,Â±1. a 1 For example, according to the MO correlation diagram of Figure 9a, one constructs the leading determinantal wave function prior to ionization from the MO's shown at the right which correlate to the occupied SA atomic orbitals, and observes that A= 0. As was mentioned earlier, there is the possible case of an Auger type process in which the participating orbitals u, and u are identified with a2sHe* and TT3p7ir, respectively. Not shown in Figure 9a are levels of MO's associated with unoccupied SA atomic orbitals. Among these MO's there may be one whose axial component of angular m.omentum differs by Â±1 from that of an occupied MO with which it shares the same UA atomic orbital limit. An example of such a case is the iT2p MO
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B5arising from th.e unoccupied SA 2p orbital of He* Cls2s) , together with the a2s MO of Figure 9a, both of v/hich correlate in the UA limit to the singly occupied 3d atomic 5 ? orbital of Ca* C3p 4s '3d). Replacing the a2s MO of the previously described Z deteriainant by this rT2p MO would result in a n detenninant which is significant for the present UA analysis due to BornOppenheimer ciioup lings . In the E case, u, of the Auger type process would be identified with a2s, in the n case with tt2p. According to Equation C52) , the UA limit results in Â£, = 2,m, = for the Z case and 5,, = 2,m, = Â±1 for the IT case. This means that in applying Equations (57) and (58) for these two cases, the restrictions on the Â£values are the same for both, while the mvalue restrictions involve m, = for y. and m, Â±1 for E determinants. In general, then, the UA analysis of the angular momentum contributions to the emitted electron requires that the initial electronic state prior to ionization be written as a linear combination of such determinants; Â• $. = Y'l' C . (59) 1 ^ a a a Similar considerations hold after ionization, where a manifold of determinants $ results, each differing from the other by the particular continuum state u^ associated v/ith it. The final electronic state then is written.
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860. = l^^C. , (60) f /; b b b and the total transition probability for ionization is expressed in terms of V^. iR^'k)^^, which is a linear combina13. Br tion of interaction matrix elements of Equation (50) : ^fi^'^^)BF= ^.^b^a^ba^^'^^BF ' ^^^^ a,b According to this general description. Equations (57) and (58) of the UA analysis may be applied using the 51 value restrictions directly with values found from an MO correlation diagram such as in Figure 9a, but remembering that the mvalue restrictions are weakened due to BornOppenheimer couplings. Of course, ionization occurs far from the UA limit, and the values for I' which are obtained here are certainly not all which should be included, but they do constitute the minimal set required for making a reasonable physical description of the emitted electron. That such a minimal set can be specified is important for the parameterization of expressions at various levels of approximation by which calculations of angular distributions of the emitted electrons can be made CEb74, Mi75) .
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873. Estimati ng MO Correlation Diacyrams for Diatoinics An analysis such as has just been outlined requires only schematic correlation diagrams, v/hich should, however, be reliable in relating SA and UA limits of the higher lying MO's. In this section, a procedure will be described for estimating MO correlation diagrams, and applied to the collision pairs He*Cls2s) + Ar(3p ), + KrC4p^), + HgC6s^), and Ne* (2p^3s) + Ar(3p^). Based on these estimated MO correlation diagrams and the analysis of the previous section, the minimal set of angular momentum contributions required to describe the emitted electrons in PI and AI will then be determined. In order to begin constructing estimated MO correlation diagrams, one must have the appropriate SA and UA atomic orbital energy levels of the collision partners both before and after ionization. The following scheme has been found to be sufficiently reliable and simple to apply. For the ground state levels of neutral atoms one can use any of the results of the HartreeFock or HartreeFockSlater calculations, which are available in tabulated form in the literature CFi73, Da73, Ma67a, C174) . Furthermore, one can rely on these calculated atomic orbital energy levels for ground state neutral atoms in order to obtain the levels of the ground and excited state atomic ions and of excited state neutral atoms. Clementi and Roetti
PAGE 101
have published results of HartreeFock calculations on ground states of atomic ions as well as neutral atoms for Z<54 CC174) . Comparison of these results shows that the spacing between the levels of any tvi/o of the higher occupied orbitals of the neutral atoms is very nearly preserved for the corresponding two orbitals of the atomic ions. This observed property of the HartreeFock orbitals lends itself well to a quite accurate determination of the levels of ground state ions for all of the atoms. One simply locates the level of the highest occupied orbital at the ionization potential of the atomic ion in question, below v/hich the next few levels are positioned according to their relative spacing in the corresponding neutral atom. Ionization potentials can be readily found, for example, in a CRC Handbook of Chemistry and Physics. The levels of excited states of neutral atoms and atomic ions involve somewhat more uncertainty in their determination. The essential requirement these levels should satisfy is that of being properly ordered with respect to energy. Within an atomic orbital framework, basically two types of excited states may occur. First are the excited states formed from the ground state by promotion of an electron Cs) from an inner shell to a higher level previously completely unoccupied in the ground state Csuch as CaC3p 4s ) > Ca* C3p 4s 3d)). Second are the excited states
PAGE 102
89formed from the ground state by promotion of an electron (s) from an inner shell to a higher level previously partially 10 2 2 occupied in the ground state (such as Pb(5d 6s 6p ) > 9 2 3 Pb* C5d 6s 6p ) ) . In this second case, the excited state only involves altering the occupation numbers of orbitals already partially occupied in the ground state, and, though levels will be shifting due to this, it is expected that the levels of the excited state will be well described by those of the parent ground state. However, in the first case mentioned above, the promoted electron (s) can be treated as moving in the field of the remaining electrons in the ionic state characterized by the appropriate vacancy (s). Therefore, the ionic core levels are determined as previously described, and then a version of Slater's rules (S160) , based on estimated screening factors, is applied to describe the level (s) of the promoted electron (s) in the presence of the ionic core. The construction of the schematic MO correlation diagrams will now be discussed. In order to proceed, the following fundamental guidelines are followed: (.a) Only those MO levels are considered which conform at the SA limit to given atomic orbital levels of either the reactant or product collision partners. (b) The MO levels, for both the excited molecule and the molecular ion, correlate from SA to UA limits in accordance with the noncrossing rule for spinpolarized orbitals.
PAGE 103
90The specific example of He* (ls2s) + Ar(3p ) will show how these guidelines are applied. Referring to Figure 9a, on the right are displayed the occupied levels of He*(ls2s) and Ar(3p ), determined as described earlier in this section. It is pointed out that the procedure for estimating levels of excited atoms leaves some uncertainty as to effects of spin splittings. normally, except for the high lying level of the promoted electron in an excited atom, splitting due to spinpolarization is expected to be sm.all compared to the relative spacings of the levels. However, Helium in its ls2s excited states is a rather special case since there are only two electrons. The estimates m.ade here of the occupied Is and 2s levels of excited Helium (based on estimated screening factors) are certainly reasonable, especially in relation to the 3s and 3p levels of Ar(3p ). But each of these occupied levels has associated with it the level of its unoccupied partner of opposite spin. The unoccupied Is level of He* may lie below or above the 3s level of Ar ( 3p ) , depending on whether there is a weak or a strong splitting of the two Is spin components (one occupied and one unoccupied) . Applying the guidelines (a) and (b) to the case of weak splitting, one vrould find the SA levels correlating to UA levels of 6 2 an excited state of Calcium denoted by Ca*(3s3p 4s 3d). That is, the unoccupied Is level, lying below the 3s level
PAGE 104
91of Ar C3p ) and obeying the noncrossing rule, would go to the partially occupied 3s level of Ca* C3s3p 4s 3d). On the other hand, in tlie case of strong splitting, the SA 2 5 2 levels correlate to UA levels of Ca* (3s 3p 4s 3d). That is, the unoccupied Is level, lying above the 3s level of Ar C3p ) , would go to the partially occupied 3p level of o c 2 Ca* C3s 3p 4s 3d) . The calculations reported earlier xn this chapter confirm that, of the two estimated possibilities, this latter situation is the case. In Figure 9a is displayed the MO correlation diagram of the occupied MO's for this latter case, where the two spin components of the a3s orbital branch, one approaching the UA 3s level and the other the UA 3p level, in the way required by the noncrossing rule and shown in Figure 8 for the calculated spinpolarized results. As Figure 9a shows, the MO's are designated at the right and left according to their large R and small R behavior, respectively, and are labeled by their appropriate SA and UA atomic orbital limits. The convention used for these schematic MO correlation diagrams is to designate by a single line the levels of both spin components (a and g) of an orbital, except for situations where one spin component is occupied and the other unoccupied Csuch as als{a)), or where the noncrossing rule requires that the two occupied spin components of a given orbital correlate each to
PAGE 105
92a different UA or SA atomic orbital (such as occurs for the a3s(a,3) levels). It is emphasized that tke ls2s state of excited Helium really provides an extreme example of the effects of spinsplitting due to the fact that only two electrons are involved. The discussion of He* Cls2s) + Ar(3p ) indicates that one could reasonably limit the possibilities to only tv;o. With the additional effort of performing spinpolarized calculations of the ls2s excited states of Helium, one then could resolve which of the two cases was applicable. Certainly calculations on SA and UA excited atoms lend themselves to a more definite estimation of MO correlation diagrams, but require a higher level of effort and would be helpful only for certain borderline cases. Continuing now with the example of He* (ls2s) + Ar(3p ), a determination is made of the various possible Auger processes which result in MO ' s for the molecular ion that correlate according to guidelines (a) and Cb) to the correct SA atomic orbital levels of the product channel. In the notation of the previous section, u will designate the a2s orbital Cor 7T2p orbital which shares the sam.e UA 3d orbital limit) , u^ the continuum state of the emitted electron, u can then possibly be a3p or Tr3p, which correlate, respectively, to UA 4s and 3p atomic orbitals as shown in Figure 9a. Therefore, if an electron in a3p participates in the Auger process, the resulting UA atomic ion from
PAGE 106
93which the MO ' s separate in the product channel would be + c + Ca (3p 4s) . A correlation diagram for He + Ar , v/here Ca"*'(3p^4s) is the UA ion, is not shown, but one may consuit instead the correlation diagram for He + Kr m Figure 10b, which will be discussed shortly, and which shows for the He* + Kr case the exactly analogous situation, where, in place of Ca"^ C3p 4s), Sr'^(4p 5s) is the UA ion. Referring now to the right of Figure 9b, the occupied SA levels of ground state Ar (3p ) all lie belov; the doubly 2 occupied Is level of He (Is ). This has been conrirmed by + 5 2 spinpolarized calculations on Ar (3p , P) . Therefore, + 6 the level of the empty 4s spin component of UA Ca (3p 4s) , which is split above its occupied counterpart, could not correlate in the SA limit to the partially occupied 3p lev+ 5 el of Ar (3p ) without violating the noncrossing rule with a spin component of the a orbital which separates to the Is level of He(ls ). Therefore, guidelines (a) and (b) do not favor an Auger process where uÂ„ is the a3p orbital. On the other hand, if uÂ„ is the iT3p orbital, then upon ionization + S 2 the appropriate UA ion will be [Ca ]*(3p*4s ), which is shown in Figure 9b, and from which the MO ' s do correlate 2 +5 to the atomic levels of He (Is ) and Ar (3p ) without violating guidelines (a) and (b) . The analysis of the angular momentum contributions to the emitted electron is, therefore, performed on the basis of both Auger type processes, with the awareness, however, that guidelines (a) and (b)
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94favor the one involving the 7r3p orbital. c This detailed discussion of the Ke* (ls2s) + Ar(3p ) case demonstrates just how one goes about analyzing PI and AI processes in terms of estimated MO correlation diagrams. In Figures 10 to 12 the results of similar analyses, per6 2 formed, respectively, on He*Cls2s) + Kr (4p ), + Hg C6s ), and Ne*(2p 3s) + Ar C3p ), are shown. Attention is drawn to the similarity between the results for excited Helium + Krypton of Figure 10a and those for excited Helium + Argon of Figure 9a. The MO ' s from the 4s and 4p levels of Kr C4p ) have near exact counterparts in those from the 3s and 3p levels of Ar(3p ) . So, the unoccupied Is level of He*Cls2s) is expected to lie above the 4s level of Kr. A comparison of Figures 10a and 9a suggests that He*Cls2s) in collision with Kr(4p ) goes through the same Auger type processes as it does in collision with Ar(3p ) However, in Figure 10b one sees that the SA 4p level of + 5 2 Kr C4p ) lies very close to the Is level of He (Is ), enough so that the splitting of the spin components of the 4pKr level could likely result in the level of the partially occupied component lying above the IsHe level. If it lies above, then the Auger type process favored for Kr would be different from that for Ar, and would permit the MO correlation diagram for He + Kr shown in Figure 10b; if it lies below, then the Auger type process favored for Kr would cor
PAGE 108
u o m e (0 u tr (d Â•H tJ C O H 4J (d rH (U M M O O H n3 +J H XI Si O U ^ U '' 0) M rH Â« O E + 'd 
PAGE 109
Â•96
PAGE 110
u
PAGE 111
98.Q 1. I X L'",
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99respond to the one we found favored for Ar, and would perrait an MO correlation diagram for He + Kr analogous to the one shovm for He + Ar in Figure 9b. Considering its borderline nature, the analysis of the angular momentum contributions to the emitted electron shall be made in both these cases for He*Cls2s) + Kr C4p ). 2 The MO correlation diagram for He* Cls2s) + Hg(6s ) shown in Figure 11a is the one which applies if the level of the unoccupied Is spin component of He* (ls2s) lies below the 5d level of ground state Hg. Auger processes involving 65d, Tr5d, or a5d MO ' s can be ruled out, and Figure lib shows the MO correlation diagram for the molecular ion for an Auger process involving the a6sHg MO, in terms of v/hich the analysis of the angular momentum components of the emitted electron is carried out. Only a and tt orbitals are occupied in the SA limit of Ne* + Ar, as seen in Figure 12a. This means no more than six electrons can correlate to the 3d orbital of the UA excited state of Nl. . As a result, a very highly excited state of Ni results at the UA limit. Shown in Figure 12a is the case of the iT2pNe* MO filled and the a2pNe* MO par1 3 tially filled, corresponding to a ' E molecular state. Also to be considered is the case of the 7T2p MO partially 1 3 occupied and the a2p MO filled, corresponding to a ' TI molecular state and resulting in an excited state of UA
PAGE 113
u o E as U tn nJ Â•H 73 C O Â•H +)
PAGE 114
101
PAGE 116
103cn IS) o o (/I o in X I 0) X o^ (3)901 CL VO CN O) VD O in J2) Q_
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u
PAGE 118
105en
PAGE 119
u o e u Â•H G O H +J RJ rH 0) o O H (Â« P Â•H XI U O M Â• rHlD O ro Q) Â— ' .H + O U e <: a + 0) +j . e a Â•H CM p Â— XI ri 0) fa
PAGE 120
107< I Z '. m in '^ VD Q. CL VD Cl ^"^ K^ Q(NJ ^^ CN Â— + + (D il_ Q) 2: < < :z: Cl Cl LO to CN ^n m Csi CS^ CN CL U vO o)9on
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108r c o O Mi denoted by Ni* (3p 3d 4s 4p 4d) . Auger processes involving aSp and TrSp MO ' s can be ruled out, whereas those involving the a3s MO are allowed, resulting in the MO correlation diagram for the molecular ion shown in Figure 12b. This analysis assumes that the levels of both spin compo5 nents of the 2p orbital of Ne* (2p 3s) lie below the 3s level of ground state Ar, which., according to Figure 12a, is not unreasonable. On the basis of the MO correlation diagrams presented in Figures 9 to 12, together with the analysis of the previous section, an evaluation of the minimal set of angular momentum contributions needed in describing the emitted electrons is made. These results for the various PI and AI processes are shown in Table II. The I values for He* + Ar and He* + Kr are tabulated according to which of the two previously discussed cases they belong. The favored case for He* + Ar , which is supported by the spinpolarized atomic calculations, is indicated. Because of the borderline nature of the He* + Kr estimates, both cases should be equally considered within this analysis. A feature common to all of the collision pairs studied is that relatively few angular momentum components are predicted as being necessary to reliably represent the angular behavior of the emitted electrons in the BF frame. This result lends additional support to existing evidence that few partial v/aves need be kept v/hen calculating electron
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109Table II. Minimal set of angular momentum components in the bodyfixed frame predicted for electrons emitted in PI and AI collisions Collision Predicted Pairs ^ Values He* + Ar 0, 2, 4 Cfavored) 1, 3 He* + Kr 0, 2, 4 1, 3 He* + Hg
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110angular distributions, which confirms experimental measurements CEb74, Mi75) . As can be seen, He* + Hg involves only even values of Z, indicating an angular distribution of emitted electrons symmetric about 9 0Â° in the BF frame. It should be pointed out that in all cases the Â£ value analyses in terms of the direct and exchange matrix elements of Equation C55) give identical results, so that both couplings must be equally considered in MO descriptions. 4. Discussion For the most part, the importance of MO correlation diagrams in understanding atomic collision processes has centered around the occurrence of crossings between MO eigenenergies which are associated with electron promotion mechanisms for electronic excitation or charge transfer (see, for instance, Ba72, Li67 and Ke74). In the present chapter, quite another application of MO correlation diagrams has been made. The model and calculated results correspond to a mechanism of ionization in PI and AI collisions associated with an Auger type process, rather than one associated with electron promotion. In the analysis of this chapter, the MO ' s obey the noncrossing rule for spinpolarized orbitals. However, the reliance upon adiabatic spinpolarized MO ' s is not to
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Illbe confused with the fact that the total system of electrons is represented by diabatic states constructed as determinants of the MO's, either before or after ionization. Therefore, coupling between these states derives from Coulomb as well as BornOppenheimer terms, as discussed in Chapter I . The work reported in this chapter has been prompted by that of Micha and Nakaraura O^lilS) . The two approaches lead to results which confirm one another, while at the same time they each emphasize a different point of view. Micha and Nakamura also consider adiabatic MO's, but inherent in their approach is to assume that the MO from which the emitted electron escapes has associated with it a width (or decay rate) for leakage into the continuum. There, the UA analysis of the angular momentum contributions to the emitted electron in the BF frame follows according to rules for autoionization. In the approach of the present chapter, em.phasis is placed on the Auger type process, and the UA analysis of the angular momentum contributions to the emitted electrons focuses on the direct and exchange matrix elements of the Coulomb coupling, with 2 equal importance placed on both. Also, owing to its R dependence, rotational BornOppenheimer coupling is found to be significant for properly assessing all of the angular momentum contributions. Such rotational BornOppenheimer
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112coupling is of particular importance when the two MO ' s , by v;hich tv/o initial or final determinantal wave functions differ, share the sarrie UA atomic orbital limit. In the present analysis, useful application has been made of estimated MO correlation diagrams. The estimated correlation diagrams themselves rely upon estimated atomic orbital energy levels for atomic ions and excited atoms. The procedure proposed for estimating levels of occupied atomic orbitals in atomic ions and excited atoms appears to be quite reliable, and requires only tabulated results for ground state atoms together with an application of Slater's rules based on estimated screening factors. However, uncertainty remains in the atomic level estimates in connection with splitting due to spinpolarization. With the additional effort involved in carrying out spinpolarized calculations on atomic ions and excited atoms, much of this uncertainty could be avoided, which would lead to even more reliable MO correlation diagrams. Nevertheless, the convenience of estimated MO correlation diagrams based on estimated atomic levels, and requiring no large scale calculations, makes possible ready application to a variety of collision partners. On the basis of the estimated MO correlation diagrams, together with the two mentioned f undair:ental guidelines, just a few of the several possible Auger processes may be singled out as
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113being iinporhant. In this connection, the diagrams for the molecular ion of the product channel are seen to play a necessary role.
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CHAPTER IV A COUPLED CHANNELS APPROACH TO PENNING IONIZATION OF Ar BY He*(ls2s,3s) Having dealt in the previous two chapters with matters specifically related to electronic structure during the course of a collision between two atoms, the focus of attention is now turned to the collision dynamics of the heavy particles (nuclei) . The particular case of PI of 3 Ar by He*(ls2s, S) will be considered in detail, and calculations of total and partial ionization crosssections will be reported. In order to properly describe the heavy particle motion during a process such as PI, one must keep in mind that initially, before ionization, the internal electronic state of the diatom is a discrete (metastable) state which is embedded in a continuum of internal electronic states associated with the free emitted electron together with the remaining bound N1 electrons after ionization. These features have been mentioned previously in this dissertation. So the first concern of this chapter will be to augment the formal development of Chapter I to include continuum as well as discrete electronic states in the expansion of Equation (26) . 114
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115Next, following an approach applied by Wolken (Wo75) to collisional dissociation of diatomic molecules by atoms, the continuum coupled equations will be discretized, resulting in a set of modified coupled equations analogous to Equation (34). Similar discretization techniques have been employed in connection with nuclear reactions, as described in the work of Bloch (B166) . The procedure for solving the modified coupled equations with scattering boundary conditions will be presented, pointing out the pertinent quantities for calculating partial and total crosssect ions . Next a qualitative discussion of the important features of PI and AI will be presented, follov/ed by an application of a discretization procedure to these processes, which entails an approximation of the set of coupled equations in terms of several twochannel coupled equations, The specifics of the actual calculations of PI of Ar by 3 He*(ls2s, S) will then be presented. Convenient functional forms are used to describe the interaction potentials associated, respectively, with the excited (metastable) state of the HeAr molecule in the initial channel, and the ground state of the (HeAr) molecular ion in the product channel. The parameters used in these potential functions are discussed in terms of the behaviors of the potentials in the crucial regions of short range repulsion, long range attraction, and near the potential minimum.
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116Based upon functional forms for the "width" , or decay rate into the continuum, which have been suggested in semiempirical approaches to PI and AI , it is possible to extract equally useful expressions for the coupling between the incident and product channels. Finally the results of the coupled channels calculations for PI in He*(ls2s, S) + Ar ( 3p ) collisions are presented and discussed. 1. The Scattering Problem in Terms of Discrete and Continuum Electronic States In the formal development of Chapter I, coupled equations, determining the heavy particle motion during a collision between tv/o atoms, were derived by appropriately expressing the Hamiltonian and wave function for the nuclei and electrons in independent variables in the BFCMN frame, and then introducing an expansion in terms of a complete set of electronic wave functions at each internuclear separation, R, according to Equation (26) . Although not explicitly stated in Chapter I, the treatment in terms of Equation (26) is carried out under conditions of a discrete basis set of electronic wave functions. Of course, a complete description of any system of electrons would include both the discrete as well as continuum parts of the spectrum. However, many inelastic collisional processes involve primarily transitions between bound electronic states (such as excitation or charge transfer) or transfer
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Â•117of energy from bound electronic states to motion of the heavy particles (such as electronic to translational or vibrational energy transfer) . So in practice, for these processes only a discrete set of electronic states need be considered. On the other hand, collisional ionization processes require of necessity that both discrete as well as continuum electronic states be included in representing the socalled internal states of the system during collision. To set the stage for the specific treatment in this chapter of the dynamics of the heavy particle motion in PI of Argon by metastable Helium, the formal development of Chapter I v/ill be extended to include both discrete and continuum electronic states. Keeping in mind PI processes, the continuum electronic states which will be considered are those associated with an electron escaping the region of the molecule along a direction specified by the unit vector e (in the BF frame) , and having kinetic energy e which is constrained by conservation of total energy during the collision. Physically, once ionization has occurred, the system of N electrons consists of the emitted electron (of energy e and direction e) in the presence of the remaining N1 electrons in some state of the molecular ion. The continuum state of the emitted electron can therefore be well represented by an appropriate linear combination of Coulomb v;ave functions.
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Â•118Designating the continuum state of the emitted electron C "^ by u ^i^) r one can write (see, for instance, Ku75) E , e u^ .(r) = I y* (e)u^. (?) , (62) e , e ^ ^ im elm C ^ wh^ .'e u n (r) is a Coulomb wave function of energy e with angular momentum partial wave values % and m in the BF frame. That is (see, for instance, Be57) , u^Â„ (?) = R Ax)Y^{i) . (63) e iLm r cl Â£m The work of the previous chapter centered around an analysis, in terms of molecular orbital correlation diagrams, of the minimum number of such partial wave contributions required to describe the emitted electron in the BF frame. The asymptotic behavior of the radial function, R . (r) , in Equation (63) is r^ sin(kr+6 +a ^ + kn(2kr)) RÂ„(r) /% "^ ^ ^ ^ , (64) eSi r^^ / TTk r where k = /2e is the wave number in atomic units, aÂ„ is the Coulomb phase shift of the ^ Coulomb partial wave, and 6 is the phase shift arising from the actual physical situation. Reference will be made later on in this chapter to Equation (64) . The continuum orbitals of Equation (62) are normalized to unity on the energy scale. By this is meant the following:
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Â•119:u^ Ju^^ ..> = fd? u^ .(r)u^. ..(r) = 6 (ee ') 6 (ee') . (65) Â£,e'Â£,e Â•' e,e e,e A good discussion of the normalization of continuum wave functions has been given by Bethe and Salpeter (Be57) . Confusion can easily arise regarding the normalization of continuum functions. In this work, the expression, ''normalized to unity on the energy scale", will be used, and it will refer to normalization according to Equation (65) . It is worth noting that continuum functions have units which differ from their bound function counterparts. That is, if a set of bound (normalized) functions, {u.}, are orthonormal , = (S . . is unitless, whereas, for ID ID continuum functions (normalized to unity on the energy scale), = 6(eÂ£'') has units of (energy) (as is readily seen since /dÂ£
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120electronic states introduced are those associated with a particular partial wave component of the emitted electron in the BF frame. Therefore, the A label of the continuum electronic states designates the component of angular momentum of the electrons along the internuclear axis associated with the (im) partial wave of the emitted electron and the remaining N1 bound electrons in a state with axial component of angular momentum, A. That is, A = m+A . + + The normalization conditions are taken as follows: nA n A nn AA nA ' nA nA ' nA <^ A^ 'A '^ = '5 (nn') 6, , . . (66) nA ' n A A A Here, Sdin') = 6(Ge')5 .8 .. X. , X, m,m Again, as in Chapter I, the discrete and continuum states introduced here have been left quite general, aside from the normalization conditions of Equation (66). In practice, however, N electron states constructed as determinants of oneelectron molecular orbitals constitute convenient and useful basis wave functions for both discrete and continuum states, as has been emphasized in Chapters II and III. Accordingly, the discrete electronic states can be described in the limited configuration space spanned by determinantal wave functions comprised of bound (normali7.ed) oneelectron molecular orbitals, and the
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121continuum electronic states can be described in the limited configuration space spanned by determinantal wave functions formed from N1 bound (normalized) oneelectron molecular crbitals together with a continuum orbital such as in Equation (63) , normalized to unity on the energy scale and orthogonal to the bound molecular orbitals (see, for instance, B166) . This chapter continues in the context of such diabatic representations, as discussed in the previous chapters, and applied to both the discrete and continuum electronic states. Before deriving a continuously infinite set of coupled equations for scattering involving discrete and continuum electronic states, some remarks, pertinent to the conditions under which collisional processes take place, should be made, especially in order to maintain continuity v/ith the formal development of Chapter I. In Chapter I, attention was focused on a total wave function for the system of electrons and nuclei prepared in a specific eigenstate )Â• ^(n) "*"(e) of the square of the total angular momentum, K = L + L , and its z (SF frame) component (see Equations (6) and (23)). In other words, the wave function of Equations (6) or (23) describes the electrons and nuclei where the electronic and nuclear angular momenta are coupled in such a way that 2 their state is an eigenf unction of K and K with eigenvalues K(K+1) and M, respectively. However, the conditions of an actual collision process are specified by the
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122asymp totic boundary conditions of the incident and final channels. The colliding atoms are characterized ideally by their relative translational mOiTientum and the internal electronic states in which they have been prepared. Before collision, the electronic states of the two atoms are essentially unperturbed atomic states, and the electrons of each atom together have some total angular momentum and component along the (asyir.ptotic) internuclear axis (which may or may not correspond to the SFCMN z axis) . A description of the incident relative translational momentum prepared for the tv70 atoms involves a linear combination over all angular momentum partial waves (eigenvalues of (L ) and L ) of the nuclei. Before collision, the electronic angular momentum of each atom together with the nuclear angular momentum are all uncoupled, and for each partial wave of the nuclear motion, the three combined result in a total angular momentum and z component contribution prior to collision. All partial waves of the nuclear motion are required to describe the relative motion of the two atoms before collision. Consequently, all possible values of the total (nuclear and electronic) orbital angular momenta and z components must be properly included in establishing the coupled equations for the heavy particle motion. That is, as the tv/o atoms collide, the electronic structure of the resulting diatom is determined by molecular states appropriately built from the two
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123atomic states, and characterized by an electronic angular momentum with some component, A, along the internuclear axis. This molecular electronic angular momentum couples with the angular momentum partial waves of the nuclear motion resulting in all possible values of the total angular momentum and its z component, (KM) . To each of these partial waves is associated a contribution in accordance with the appropriate angular momentum vector coupling coefficients. So, strictly speaking, the development of Chapter I constitutes only a starting point for a more complete treatment of the coupled equations, which would include all of the required coupling of electronic and nuclear angular momenta. However, by considering a particular partial wave component of the total orbital angular momentum during collision, the development of Chapter I establishes how the Schrodinger equation can be properly expressed in the BFCMN frame, involving electronic wave functions and coupling matrix elements calculated in the BFCMN frame. In the actual application to PI of Ar(3p , S) by He*(ls2s, S), which v:ill be eventually presented in this 3 chapter, the colliding atoms give rise to only a Z molecular term. Therefore, the electronic angular momentum in the incident channel is very small compared to the angular momentum of the nuclei, which typically has values
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124around 15 or 20. So only the nuclear angular momentum has to be considered in the incident channel. Upon ionization, the emitted electron escapes with BF angular momentum components (Â£m) , leaving the molecular ion in a Z or n state. The product Argon ion is in its P ground state. In the previous chapter, the analysis of the BF angular momentum components of the emitted electron indicated that only few partial waves contribute. This confirms other such analyses (Eb74, Mi75) , and according to the calculations of Micha and Nakamura (Mi75) , the emitted electrons are predominantly in Â£ = states, with proportionately smaller contributions from the next few partial waves accounting for the anisotropy in their angular dxstributions (Ho71, Eb74 and Ni73) . So, not only in the incident channel, but also in the product channel, the electronic angular ;.jmentum may be neglected in PI of Argon by metastable Helium. Therefore, the details of electronic and nuclear angular momentum coupling in deriving the coupled equations will not be dealt with here. However, as a reminder of the fact that angular momentum coupling is important for a general treatment of the coupled equations, appropriate notation will be suggested, even though a complete analysis has not been carried out. It will be recognized, for instance, that the wave function of Equation (26) corresponds actually to a situation where the two atoms
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125are initially prepared in atomic states which result in initial molecular states labeled (n.A.). Furthermore, for instance, the (KM) on the wave function of Equation (26) labels it as one of the contributing partial waves resulting from coupling of electronic and nuclear angular momentum partial waves in the SF frame, (L M ,L M ) , n n e e respectively. To the (KM) should be added the labels of the nuclear and electronic angular momenta which have coupled to constitute the (KM) partial wave. Accordingly, the index X = (L ,L ) will be introduced. In the case n e of a channel associated with a continuum electronic state, >>( + ) >It IS unaerstood that L = L +1, that is, the sum of e e the electronic angular momentum of the remaining ion and that of the emitted electron. Based on the preceding discussion, L << L and Â£ << L for the actual applicae n n ^'^ tion which will be carried out in this chapter. The foregoing remarks have been made only to highlight some additional considerations regarding atomatom collisions which are needed to place the formal development of Chapter I n perspective. Such details will not be needed in the application to PI of Argon by metastable Helium in this chapter. Serving as a guideline, however, the notation introduced here points to what may be expected from a more detailed analysis of electronic and nuclear angular momentum coupling. Returning now to the primary consideration of this
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126chapter, in place of Equation (26) the partial contribution to the total wave function of nuclei and electrons is expanded in terms of discrete and continuum electronic states as follows: 11 A A KM ,.. , r. . ,. ^,KM ^(^)j ^ nA 1 ^nAA ,n.A.A j. r\h Â± eAA ,n.A./ n 1 1 u L J(67) The variables are understood to be the independent set in the BFCMN frame discussed in Chapter I, and the tilde and primes designating, respectively, BFCMN functions and variables in Chapter I, are omitted in this chapter. On the left of Equation (67) , the partial contribution to the total wave function is labeled according to the molecular electronic state of the incident channel, n.A., and corresponds to one of the possible couplings of electronic and nuclear angular momentum, designated by KM KMX . The c:::pansion coefficients, i]^ . , ^ . , and '^ ^nAA ,n^A^X KM ip .,^ A ^/ specify the behavior of the heavy particles e ' ' i i at each R associated with transition during collision from the (KMX) component of the initial channel, n.A., to the (KMX') component of some other accessible channel, nA or nA, respectively. It should be noted that the (Â£m) indices of n = ielva) are absorbed in the (KMX) indices of KM U; . , ^ . .. As noted earlier m this chapter, the units ^eAX ,nj_Aj_X ^
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127of the continuuin states differ from those of the discrete 1/2 states by (energy) . Referring to the expansion of Equation (67) , one can see that a similar difference in Â• ^ ,_ ^ ^^ ,KM , ., ,KM units exists between the 4^ . . . . . and the u^,,. Â„ . . Â• eAa .n.A.A nAA ,n.A.A 11 11 The coupled equations satisfied by the heavy particle wave functions, ^^^.^^.y^.;, and '^^^^ . ^^ . y^ . A ' Â°' Equation (67) , are obtained essentially in the same way Equation (34) of Chapter I was obtained. By left multiplication of the Schrodinger equation (BF Harailtonian of Equation (32) and wave function of Equation (67)) by D ..{(t)Qy)<^ .^(R,r.) and integration over (4),e,Yfr^), one finds o o KM {(l/2m)d'^/dR K(K+l)/2mR + ^'^'I'^^'x' \n . A . X^^^ AA n 11 ^/^=fCA,Â„A<''> ^^Â°A,nAÂ™'*eAX,n,A,XÂ™) Â• <Â«Â«' Here, for convenience, the Coulomb and 3orn0ppenheimer couplings discussed in Chapter I have been introduced as follows : V^'^'fi ,(R) = 5,. ,<$ .,.h J$ > , (69) n A ,nA A ,A n A ' el' nA and
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128V^9,. ,(R) = V""^?. ,(R) + V''?^. (R) , (70) n A ,nA n A ,nA n A ,nA where ^"A',nA(^)=^A%A<^'A''(^/^)^/^^(^/2m)d2/dR20^^> ,^^_^^ and (referring to Equations (35) and (36) of Chapter I) V^?^. ,(R) = 6,. ,<<:> .,. (L^^^^/2mR^$ Â„> n A ,nA A , A n A ' ' nA 2 2 6.^.6. A /mR A , A n ,n Â«A'.A.l5A'I^*nAl''/2"'''l*nA> + <5,. , ,g^^,<$ .. JlJ^VzitiR^U .> (72) A ,A1^A A n A ' + ' ' nA are the radial and rotational BornOppenheimer coupling contributions. Expressions similar to Equations (69) , (70), (71) and (72) hold when the discrete labels, n' or n, or both n^ and n, are replaced by continuum variable counterparts, n and n'In Equations (69), (71) and (72), attention is drawn to the 6,^ , and 6.^ ,,, factors, A ,A A ,AÂ±1 which, though not seen explicitly in Equation (68) , are important for assessing which electronic states enter the coupled equations due to Coulomb and BornOppenheimer couplings. In similar fashion, multiplication of the Schrodinger K* * ,. equation on the left by D .^{(^Qy)^ ..^(R,r.) followed by
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129integration over {(p,Q,y,r.) leads to the resulting coupled equations : {(l/2m)dVdR^ K(K+l)/2mR^ + E}/^^.^.. ^ ^ ^ (R) ' i i = I (Hv^^yi ,(R) + v"^,. AR)}i>^:\. , ,(R) ^V'^*^' n A ,nA n A ,nA ^nAX ,n.A.A AA n 11 Wdc(V^?:ji,,,(R) +V^?,,_^,(R))*Â™, _Â„.,., (R)] . (73, Equations (68) and (73) formally constitute the coupled equations which determine the heavy particle motion associated with both discrete and continuum internal electronic states in atomatom collisions. 2. Discretization of the Continuum and the Modified Coupled Equations In order for the coupled equations of Equations (68) and (73) to be of any practical use, not only must the discrete set of electronic states (which in principle is infinite) be described in terms of relatively few, important, electronic states, but also the continuum electronic states must be treated in some manageable fashion. The terms involving integration over the energy, e, of the emitted electron, which correspond to internal state couplings of the boundcontinuum and continuumcontinuum type, mean that Equations (68) and (73) constitute
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130a continuously infinite set of coupled equations which must be suitably reduced to some tractable form (see, for example, B166, Wo74 and V7o75) . This may be accomplished by means of a discretization of the continuum with respect to the coupled equations of Equations (58) and (73) . The approach to discretization used here follows the work of Wolken in connection with gassolid energy transfer (Wo74) and collision induced dissociation in atomdiatomic molecule scattering (Wo75) , as well as that of Bloch (B166) in connection with nuclear reactions. The heavy particle wave functions associated with transitions into the continuum are formally expanded in a complete set of discrete functions of the continuum variable, e: oo ' 1 1 1=1 ' 1 1 The set of functions, {a }, is characterized by I a (e) a (e') =6 (ee") 1=1 /dea*(e)a^(e) = 6^ ^ . (75) One can see from Equation (75) that the functions, a , 1/2 have dimensions of (energy) ^ . Therefore, the effective KM heavy particle wave functions, \b^.,. , ,, introduced as lAX ,n . A . X 1 1
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131tl\e expansion coefficients in Equation (74), have dimensions compatible with heavy particle wave functions associated with bound statebound state transitions, KM iL) , , ^ fl 1 Â• The advantage to the expansion of Equation nAX ,n . A . A ^ ^ ^ 1 1 (74) is that integrals over the continuum variable, e, can be replaced by summations over the discrete set of functions of e. Formally, these are infinite summations. However, in the same way a small number of discrete electronic states are singled out as the important ones for some collision process. Equation (74) allows more easily the possibility of selecting a limited number of terms in the expansion which may be the most significant for the dynamics of a collision process. With the use of Equation (74) in Equations (68) and (73), and multiplying Equation (73) by a*^{e^) and integrating over Â£ ^ , the following set of coupled equations results : o o y KM {(l/2m)d /dR K(K+l)/2mR + E}ij> . ^ (R) = I{V^?'^1(R) t V^9 (R)}<^^^^ (R) ^ q Â»q q .q q/q^ .I(V,H, .VB?_^(K,)*Â™^,K, , (76, and
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1329 9 y KM { (l/2m)d /dR^ K(K+l)/2mR^ + E]i>^.^^{R} HÂ°:> Cq<'"''C,''^' j'^qÂ°:qÂ™^q',q"^'>c,Â™ '"' In these coupled equations abbreviated notation has been introduced; q = (nAX) labels a discrete electronic state and Q = (lAA) labels a "discrete", effective state associated with some region of electronic configuration space in the continuum. The q'q coupling terms in Equation (76) are simply those of Equations (69)(72). The other coupling terms are given as follov/s: ^Coul,BO,j^, = /de/aca*,(e)V^?);=:;^Â°(R)aj(.) , (78) where the coupling terms inside the integrals are determined according to Equations (69) (72), Equations (76) and (77) constitute a modified set of coupled equations resulting from the discretization procedure of Equation (74).
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Â•1333 . Solution of the Modified Coupled Equations As was pointed out in the previous section, the effective heavy particle wave functions associated with transition from a discrete electronic state to an effective continuum electronic state have dimensions compatible with those of the bound statebound state heavy particle wave functions. Therefore, the coupled equations of Equations (76) and (77) may be combined into a single set of coupled equacions corresponding to scattering in a "discrete" set of effective electronic states which span both discrete and continuum parts of electronic configuration space. One may introduce the label p which ranges over the previous lables q ,q2,..., Q, rQ2' Â• Then Equations (76) and (77) may be written as a single matrix equation: (l{ (l/2m)d^/dR^ K(K+l)/2mR^ + e} _ V^oul(R) v^O(R))/"(R) =0 . (79) Here, I is the unit matrix and ^Coul,BO^^^ = [V^Â°^1'^Â°(R)) , (80) Â— P, P where the coupling matrix elements are those of Equation (78) , and
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13^ In Equation (81), the index p' ranges only over the labels q ,q . . . of the discrete electronic states, whereas p ranges over all the labels (q.,Q)This is because in ionization processes the initial channel considered is associated v/ith a discrete electronic state from which transition to the continuum (or also to another discrete electronic state if ionization does not occur) takes place during collision. Therefore, only wave functions for heavy particle motion associated with boundtobound or boundtocontinuura transitions are considered. Equation (79) is solved by integrating the coupled secondorder differential equations for the solutions which fulfill scattering boundary conditions. In what follows, only socalled open channels of the heavy particle motion are considered. That is, only those discrete and effective continuum electronic states are considered for v/hich the relative kinetic energy of the heavy particles is asymptotically greater than zero. For channel p this asymptotic relative kinetic energy is expressed in terms of K , where P (l/2m)K^ = lim{E v'^Â°^^(R)} . (82) P Ryoo PP To treat the open channel asymptotic boundary conditions, the Ricatti Bessel functions, p^(x) = Ax/ 2 J 2^+1/2^^^ x^Â°o sin(x + Â£tt/2)
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135and Neuinann functions. v^(x) = AfV2 J_^_3^/2^''^ x^~ cos(x Â£Tr/2) , (83) S are introduced, along with the diagonal matrices M (R) Li C ^ and M^ (R) having diagonal elements y (k R) and n n ^ v,(k R) , respectively. Li P n ^ Then, upon integrating the coupled equations of Equation (79) into the asymptotic region (R >> 1) , the matrices, X,,., and Y^,. ,, are determined from the scattering Â—KM Â—KM ^ boundary conditions, n n for the solution, along with the derivative of the solution asymptotically. That is. n n n n .C V ^ ,KM Â„C , ,KM, J i Ml " n n X {(m" )'i"^'^ m!: {^t'')^ , (85) and n n n n r ,, S , . ,KM , S , KM, ., tag.. X { (M^ ) ^ n (^ ) i , (86) n n where here the prime denotes differentiation with respect
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136to R, as From XÂ„Â„ and Y^,,, the socalled R~matrix is calculated Â—KM Â—KM ^KM ^KMKM^ ' ^^^^ where <_ is the diagonal matrix having diagonal elements, K . The Smatrix can be obtained from the Rmatrix as P and the Tmatrix is related to the Smatrix: ^KM=  are KM the p *p' transition probability amplitudes as a result of collisional processes treated in terms of the "channels" of the effective coupled equations of Equation (79) . For electronic boundtobound state transitions (i.e., when both p and p' are among the electronic bound state labels, qir<3o/')f the Smatrix elements of Equation (88) maybe used directly for calculating partial, total, and differential crosssections. However, in the case of electronic boundtocontinuum state transitions, the physical process corresponds to transition from one of the discrete electronic states, labeled by p^ (=q'= (n'A'A ') ,
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Â•137to an actual continuurn state (corresponding to an electron emitted with energy e) , labeled by (rAA) . The Smatrix element (transition probability amplitude) corresponding to the physical process is therefore determined as the appropriate linear combination, based on Equation (74) , of Smatrix elements for transition from a bound (q = nAX) to a discretized continuum (Q = lAA) state. Using the full labeling notation, one therefore writes nAA,n^A^A' ^ ^ IAA,n^A'X' ^ KM T Similarly, nAA,n'Aa' = I a, (e) tJI^^ '^'^ '^ ^ . (91) KM ^ I KM This completes the formal development regarding collisions involving both discrete and continuum internal electronic states. For collisional transition between discrete internal electronic states, the Sand Tmatrix elements of Equations (88) and (89) are the necessary quantities for calculating partial, total and differential crosssections. In the case of collisional transition from a discrete internal electronic state into the continuum, however, the Sand Tmatrix elements of Equations (90) and (91) must be used. Furthermore, although the details are not included here, the earlier remarks regarding
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IBScoupling of electronic and nuclear angular momenta indicate that crosssections from initial states of the colliding atoms to final states of the atoms would be expressed in terms of appropriate combinations of the S^,., and S,,,. matrix elements. These comKM KM binations should be determined in the process of reconstructing assymptotic atomic states from appropriate linear combinations of molecular (KMAnA or KMAnA) states. At this point attention will be directed to the case of PI of Argon by metastable Helium, and the formalism presented thus far will serve as a convenient starting point for specific application to a collisional ionization process . 4. Characteristics of PI and AI Processes There are a number of important features which distinguish PI and AI , and are useful to keep in mind when considering the dynamics involved in these collisional ionization processes. Several of the features will now be discussed. Reference will be made to He* + Ar collisions to illustrate these characteristic features. In PI and AI , the fact that the electronically excited metastable has more energy of excitation than is required to ionize its collision partner may lead to quite dramatic effects since this excess energy of electronic excitation can be as much as an order of magnitude larger than the
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139low (thermal) energy of relative motion of the colliding J C atoms. For instance, in the case of He*(ls2s, S) + Ar(3p ), the excitation energy of He* is 19.81 eV (see, for example, Sc73) while the ionization potential of Ar is 15.755 eV (see, for example, Mo52) . Thus, in the absence of collision, the reactants (He*+Ar) have 4.055 eV (0.149 a.u.) more energy than the products (He+Ar ) . Typical collision energies in PI and AI of Ar by He* range, on the other hand, from 1 to 10 x 10 a.u., that is, around 2 orders of magnitude smaller than the assymptotic difference in channel energies. So, in PI and AI , the heavy particle motion primarily serves to bring the atoms close enough to permit the excitation energy of the excited metastable to be expended in ionizing its collision partner. In this connection, the ionization process was discussed in the previous chapter in terms of an Auger type mechanism within a molecular orbital framework. Because the relative energy of heavy particle motion is so small compared to the excess of electronic energy available for ionization, the BornOppenheimer coupling terms can be expected to be of negligible influence compared to the Coulomb coupling terms. In the previous chapter, it was only in the context of a united atoms analysis of molecular orbital correlation diagrams that the influence of rotational BornOppenheimer 2 couplings v;as taken into account, because of their R dependence.
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140A somewhat simplified, though very adequate, description of the dynamics of PI and AI collisions is one in which the heavy particle motion of the reactants is governed by an interaction potential of a discrete metastable molecular electronic state associated with the colliding metastable atom and its partner. In the product channel it is the interaction potential of a state of the molecular ion which governs the heavy particle motion. Most approaches to PI and AI processes have been in terms of semiempirical models relying on the width of metastable molecular states, r(R), for leakage into the continuum of electronic states in which they are embedded (Na69, Mi70b, Mi71, 0172a). In these semiempirical models, the transition into the continuum is assumed to be a FranckCondon type process, in which the relative motion of the heavy particles is unaffected by ionization. Much useful physical insight into PI and AI processes can be gained on the basis of these semiempirical approaches. They confirm that the emitted electrons in PI and AI are very energetic, as is known experimentally (see, for example, Ho70, Ho75, and Ce71) , and as FranckCondon type transitions would indicate. Figure 13 illustrates what basically occurs in PI and AI collisions. This figure is not just a sketch, but rather displays in quantitative detail the interaction potentials for PI and AI of Ar by He*(ls2s, S) which will be discussed and used
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Figure 13. Interaction potentials and coupling for PI and AI of Ar by the He*(ls2s, S). V^ and V2 label, respectively, the potentials for He*(ls2s,3s) + Ar and He + Ar"*" ( 3p5 , 2p) , as described later on in the chapter, and represented by Equation (107) with the parameters of Table III. V^2 labels a typical coupling matrix element (referred to the left scale shifted by 10"^ a.u.) between the discrete (raetastable) state associated with V^ and a continuum state associated with V2 and an emitted electron of some energy e, as described later on in the chapter and parameterized according to Equation (118) . The energy scale on the left i.ndicates the energy difference Eq between the separated limits of V^ and V2 E^ labels a typical thermal collision energy, and the arrows with e labels are discussed in the text and illustrate a few representative energies removed by the emitted electrons, leaving the nuclei with final relative energies Ef indicated on the right and measured along a scale similar to that on the left, but referenced to the separated limit of V2 Â• The lower dotted curve shows Ef (R) for FranckCondon transitions, as discussed in the text.
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142^ i I ' I I ' I ' I ' I
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143later on in treating the dynamics of the PI process. V^ labels the interaction potential of the metastable molecular state, and VÂ„ labels the interaction potential of the molecular ion. The energy scale on the left of the figure is in accordance with the difference EÂ„ in energy 3 of 0.149 a.u. between the separated reactants (He*(ls2s, S) c 4^9 + Ar(3p )) and separated products (He (Is ) + Ar (3p , P) ) . The zero of energy is arbitrary, and here assigned to the separated reactants. E. designates a typical thermal collision energy. V, Â„ labels a curve (measured against _3 the energy scale on the left shifted by 1 x 10 a.u.) which reflects the expected behavior as a function of R of the Coulomb coupling between the discrete metastable state and a continuum state consisting of the state of the molecular ion plus an emitted electron which has escaped with kinetic energy, e. The rate of decay into the continuum, used in semierapirical models, is proportional to v, Â„ . These details will be taken up later in this chapter. Here, it is simply pointed out that the behavior of V, Â„ in Figure 13 is in keeping with an increasing probability for transition into the continuum as the heavy particles approach their classical turning point of motion in the incident channel. A few representative values of the amount of energy, e, typically carried away the the emitted electron are indicated in Figure 13 by vertical arrows having lengths
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144equal to the e value labeling theia. The sample of z values shown is merely illustrative, and there is no connection between where an arrow appears along the R axis and the probability of an electron being emitted at that R having the energy v/hich labels the arrow. The main feature emphasized by the arrows is the range of final (asymptotic) energies of relative motion typically available to the heavy particles in the product channel. This range of energies can be measured along the energy scale at the lower right of the figure, which has the same units as the left hand scale, but references the zero of energy to the separated products, He+Ar . In the case z = E, the emitted electron happens to escape with exactly the energy the separated reactants have in excess of the separated products. Therefore, the final collision energy, E^ , in the product channel would be just the collision energy of the incident channel. That is, E^ = E . . The e, arrow f 1 b simply represents another typical value of e , in a case where the final collision energy, E^, is less than E.. As will be pointed out shortly, treating the transitions as FranckCondon in nature indicates that energies of the emitted electrons most probably have values in the neighborhood of z , leaving the products with less energy of relative motion asymptotically than the reactants had in the incident channel. The Penning limit z indicates the largest amount of energy an electron can remove for which
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145the heavy particles in the final channel can still (in principle) separate. The final channel is a socalled closed channel whenever an electron escapes with more energy than the Penning limit e. That is, they are left with less total energy than the separated products possess. The product channel is then characterized by the heavy particles being trapped in the potential well of the electronic state of the molecular ion. The arrow labeled e indicates such a case. It actually corresponds to the maximum amount of energy (classically) which an emitted electron could remove, and E_ is the final energy of relative motion of the heavy particles, which are left in some vibrational state of the product molecular ion. Indicated at the lower right of Figure 13 are the regions of PI and AI . PI, of course, occurs when emitted electrons remove energy less than the Penning limit e, leaving the heavy particles with translational energy assymptotically . AI occurs when e is greater than the Penning limit c, leaving the heavy particles in some bound vibrational state of the molecular ion. If the transitions were FranckCondon in nature at each R, then the amount of final energy, E^, left for the heavy particles upon transition occurring at each R would be as shown by the dotted curve in the bottom section of Figure 13. One can quickly arrive at an intuitive picture of the dynamics of PI and AI processes by inspecting this
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146"FranckCondon" curve. It can be seen that the range of E^ is basically from Eto E_^ = E.. The behavior of V, ~ f ^ Â£ f 1 12 indicates that the probability for transition increasingly favors transition at separations approaching the classical turning point of V, , i.e., R ranging from around 9a to 7a . Thus one can expect the most likely final collision energies of the products to be in the neighborhood of E^, as mentioned earlier. These remarks are, of course, qualitative, but do explain trends and are confirmed by experimental measurements. They are important as guidelines for applying the formalism presented earlier in this chapter to an actual PI or AI process. One can immediately recognize that the heavy particle motion in the product channel will be associated primarily with continuum internal states confined to a narrow region of the continuum variable, e. 3 6 In the case of He*(ls2s, S) + Ar ( 3p ), as has been seen, the range of Â£ values is of the same order as the amount of thermal collision energy of the reactants. This feature makes PI and AI well suited for discretization procedures discussed formally at the beginning of this chapter. It means that expansions, such as Equation (74), can be reliably truncated, focusing on important regions of e. One other aspect of PI and AI seen in this qualitative discussion is that the emitted electrons are very energetic. The conditions of the collisional process, in
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147fact, means they are more likely to escape with energy larger than the amount of excess energy available in the incident channel. As a consequence, the emitted electrons quickly escape the region of the molecular ion during collision, which means that the continuum states of the emitted electrons can well be described in terms of Coulomb waves (see Equations (62) and (6 3) ) . 5. An Application of Discretization to PI and AI With the general background of the preceding qualitative discussion of PI and AI processes in mind, coupled equations for these processes will be derived based on the discretization formalism of the first part of this chapter, The approach here follows the basic description of PI and AI just presented, where the incident channel is represented by a single discrete (raetastable) state embedded in the continuum consisting of a single final bound state of the molecular ion plus the emitted electron of energy e, Recalling the earlier remarks on nuclear and electronic angular momentum (L << L and Â£ << L ) , the e n n electronic angular momentum is neglected as far as the coupled equations for the heavy particles are concerned. Consequently the notation of the formal development at the beginning of this chapter reduces considerably. The discretized coupled equations of Equations (76) and (77) become simply
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148{ (l/2m)d^/dR^ L(L+l)/2mR^) + E V^ d^^^^'^'d d^^^ = I ^d,i""*i,d<'<' (Â«' and { (l/2m)d^/dR^ L(L+l)/(2mR^) + E V^ ^ (R) }4)^ ^ (R) V^ ^(R)^^ ^(R) + I V^ r(R)'l'T H^R) Â• (93) I,d ''d,d j^^ I, J M,d Here, d labels the discrete (metastable) state of the incident channel, and I labels discretized continuum states as before. The angular momentum of the nuclei is now denoted by L. BornOppenheimer couplings are neglected as previously discussed. So the V , , V and V coupling i,Cl C1,X L/U terms are the Coulomb couplings given according to Equation (78) , and V, , is the Coulomb coupling given according to Equation (69). In the treatment presented here, only continuum states associated with the Â£ = partial wave of the emitted electron are considered. At this point, a simple, but physically reasonable, set of discrete functions of the continuum variable, e, is introduced. Namely, the set of functions, (a } , used in the expansion of Equation (74) is taken to be a set of step functions in e as follows:
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149a^(c) = (Ae) 1/2 for Ey Â£ e ;^ ^T + 1 (94) otherwise Here the e scale has been divided into equal increments of magnitude Ae, and c = lAe locates the start of the I such increment. The basis set of step functions only partially satisfies the completeness relations of Equation (75) . That is. /dea (e) a (e) I, J (95) but I a (e) a (e') = < 1/Ae for Â£ and e ' both in some increment, J 1=0 (96) otherwise In terms of this stepfunction basis set, and referring to Equation (78) , the coupling terms V , , V , and V Ci,L J_,cl L,J of the coupled equations of Equations (92) and (93) are expressed as follows: 1/9 '^ + 1 V. j(R) = (Ae) "/^ / de d' el ' e 1/2 I + l V^^^(R) = (Ae)^/^ / de<$jH^J^^> 1 ^I+l ^J+1 V^ ^(R) (Ae) " / de' / de<^ .h $ > (97)
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150These coupling terms may be treated based on the following approximations. If the increment, Ae, is made small enough (Ae << 1) , then, ^I+l (Ae)"""^"^ / dE<0 Jh J$ > ' d' el ' Â£ ^I+l = (AG)"^/^Â«I>^H^J^ > / de = (AÂ£)^/2<0^H^J6^ > .(98) I Â£_ I As has been mentioned, the continuum wave functions here correspond to an emitted electron of energy e in the presence of a single bound state of the molecular ion. If r locates the continuum electron, then, for r > 'o (i.e., c ' ' c the continuum electron far removed from the molecular ion) , one may make the following approximation: H^^(R,?^)$^(R,?^_^,r^^co) .. {V"^(R) + e}^^(R) , (99) or <$ Jh 1$ > =^ {v'*'(R) + e'}6(e'E) . (100) e ' ei ' Â£ Here, V (R) is the interaction potential of the molecular ion. This approximation amounts to neglecting coupling between continuum electronic states. In view of the previously m.entioned fact that the continuum electronic states in PI and AI are those for v;hich the emitted electron is very energetic, and escapes the region of the molecular
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151ion quickly, the approximation of Equation (10 0) is a reasonable one. On the basis of Equation (100) , and for Ae small, the matrix elements, V^ ^, in Equation (93) can 1 , J be expressed as follows _^I + 1 ^J+1 ^ V^ ^(R) ^ (Ae) / de' / de{V (R) + e'}6(Â£"Â£) ^I J (Ae) / de'{V (R) + e'}6^ ^ 1 + '^^1 (Ae) {V (R) + Ej} / dÂ£^6j j = ^^ ^^^ + ^I^*^I,J (101) With these approximations, the coupled equations of Equations (92) and (93) reduce to {(l/2m)d^/dR^ L(L+l)/(2mR^) + E V^^^ (R) }if^^ ^^ (R) = I(AÂ£)^'^^<*dHei!\ ^'^id^^^ ' ^^Â°^^ and {(l/2m)d^/dR^ L (L+1) / (2mR^) + E V^(R) Sjl'^j ^ (R) = (AÂ£)^/^<(& h Jc[)^>i^^ .(R) . (103) e ' el' d d,d Recalling the previous qualitative discussion of PI and AI in terms of FranckCondon transitions, only a narrow region
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152of the e scale will be important for the collision dynamics. Therefore, depending on the size of Ac, Equations (102) and (103) will comprise a set of N coupled equations where N is on the order of 20. Furthermore, to a good approximation, this set of N coupled equations may be replaced by N sets of twochannel coupled equations essentially by assum.ing that each I^d transition occurs independently of the others. Thus, by a series of reasonable approximations in the context of a discretized set of coupled equations, it is possible to describe the collision dynamics of PI and AI in terms of several sets of tv/ochannel coupled equations: {(l/2m)d^/dR^ L(L+l)/2mR^ + E V^ d^^^^'^d d^^^ = (A^)'^'<^dl\ll^/'^I,d(^) ' { (l/2m)d^/dR^ L(L+l)/2mR^ + E v'^(R) ^jH\ ^^(R) = (Ae)^/2^$^ I"el'V'^d,d(^^ Â• ^^Â°^^ Though approximate, this approach goes beyond semiempirical model approaches, which treat the heavy particle motion only in the presence of the interaction potential of the incident channel metastable state. Here, for a given transition into the continuum, the heavy particle motion is considered in both the incident and final channels, and
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153these are appropriately coupled by the matrix element of the electronic Hamiltonian between the discrete and discretized continuum internal states. On the basis of the coupled equations of Equation (104) , not only can total ionization crosssections be calculated, but in the process, partial crosssections as a function of the energy of the emitted electron are obtained. Thus one can study quantitatively the details of the energy distribution of ionized electrons in PI and AI processes. In the remainder of this chapter, an application of the coupled equations of Equation (104) to PI of 3 Ar by He*(ls2s, S) will be presented. 3 6. Interaction Potentials for He*(ls2s, S) + Ar and He + Ar+(3p^,^Py Suitable interaction potentials for the reactants and products are of course required before one can proceed with solving the coupled equations of Equation (104) . In this work, convenient parameterized functional forms for these potentials are introduced which describe their behavior over the whole range of internuclear separations, R. An interaction potential is characterized by its behavior in three important regions of R; namely, the region of short range repulsion, the region of long range van der Waals attraction, and the region of the attractive potential well. These three behaviors should be reporduced as accurately as possible in representing an interaction
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154potential by a parameterized functional form. The short range repulsion typically has BornMayer Â— R /t) {Ae ' ) behavior. This is confirmed by information on the repulsion extracted from scattering experiments as well as by electronic structure calculations, as discussed in Chapter II (see, for instance. Table I). Note that in Â— bR Chapter II the form of the BornMayer potential, Ae , v;as used. So in this chapter, b refers to the inverse of the b of Chapter II. In Chapter II it was pointed out that the MSXot total energy for rare gas diatoms generally leads to an overestimation of the repulsion. However, the b parameters were of reasonable magnitude. Therefore, in this work, BornMayer b parameters for He*+Ar and He+Ar are determined, in the way described for th results reported in Table I of Chapter II, from the nonspinpolarized MSXa calculations on these systems reported in the previous chapter. The BornMayer A parameters, which are severely overestimated by the MSXa calculations on He*+Ar and He+Ar , are determined from other considerations, which will be discussed shortly. The appropriate long range van der Vaals behavior for He*+Ar is R , while for He+Ar , 6 4 both R and R behavior must be included, owing to the chargeinduced dipole interaction between Ar and He. Values for the C, and C. coefficients reported in the literature will be used here. A primary consideration in representing an interaction
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155potential is that of joining the short range repulsion to the long range van der Waals attraction in a way which adequately describes the region of the potential minimum. The important quantities with respect to the potential m.inimum are its depth, e , and its position, R . Often '^ m m the approach taken is to splice together, from one region to the next, potential functions appropriate for each region (see, for instance, Ch74 or Na75) . However, a single potential function valid over the entire range of R is convenient and desirable. In this connection, LennardJones (126) potentials have often been used for the interaction potential of the metastable state (Ro65) . However, from studies of collisional ionization of Ar by Ne*, it appears that the LennardJones (126) potential is too repulsive, which would be of steadily worsening consequences as the collision energy increases (0172b) . There are differing points of view, however, on this matter (Ne75) . On the other hand, Morse potentials which yield the proper potential well depth and position tend to produce too soft a repulsion (0172a) . Toennies (To73) found that reasonable potential well depths and positions could be obtained using a single potential function over the entire range of R taken simply as the sum of a BornMayer repulsion and the van der Waals attraction potentials. In the present work, a potential function similar to that of Toennies, yet m.ore flexible.
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156is introduced. The idea behind the potential function introduced here is to modulate the long range van der Waals 4 6 attractive potential (which has, for instance, R , R , _ Q R , etc. behavior) by a function which preserves the van der Waals behavior at large R, while allowing in the region of the potential well an inverse R behavior stronger than that provided by the van der Waals terms. The physical grounds for this feature is to permit a more appropriate representation of the attractive forces (which typically deviate, as R decreases, from the inverse R van der Waals behavior) , thereby describing the balance between attractive and repulsive forces which are responsible for the occurrence of potential energy minima. A function well siaited for such a modulation is one studied by Eckart (Ec30) to simulate a potential energy barrier: (RR )/B _ f^(R;RQ,8,E) = (1 + e " ) " (RR )/B X (1 + 3(1 + e "^ ) ") . (105) As can be seen, for R >> RÂ„ , f_ >1, for R << R, fÂ„ ^ 0, U Â£1 U E exponentially, and for 6 > , f ^ exhibits a local maximum E at R given by max ^ ^ (R^^^ R)/B = ln[(3+l)/(Bl)) . (106) 2 The value of f at R is (8+1) /4S. Denoting by Â£j in 3.x V, (R) the long range van der Waals attractive potential ,
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157the following potential function is introduced; V(R) = Ae ^/^ + V^^(R)fg(R;RQ,6,B) . (107) This potential function will be applied to both the 3 +52 . . He*(ls2s, S) f Ar and He + Ar ( 3p , P) potentials. Since the effect of the local maximum in f is to increase the inverse R dependence of V, (R) , especially in the region of the minimum, R , of V(R) , the position of the local maximum of fÂ„ will be taken to coincide with R . Therefore, E m according to Equation (106) , RÂ„ is determined by (R^Rq)/B = ln((6 + l)/(6l)) . (108) The b parameters in the BornMayer repulsion will be taken from the MSXa calculations reported in the previous chapter, as has just been mentioned. The exponential factor, B, in the Eckart potential will be taken equal to the BornMayer b parameter in each case. This is reasonable, since the BornMayer b parameter reflects the "softness" or "hardness" of the repulsion as described in Chapter II. The exponential behavior of the modulating Eckart function is therefore taken to be the same as that of the BornMayer repulsion. Finally, the BornMayer A parameter and the Eckart B parameter are determined by the potential well conditions, V(R ) = e ra m dV{R)/dRl = , (109: K. m
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158where, for both He*+Ar and He+Ar , e and R are taken mm from the literature. The conditions of Equation (109) specify 3 and A as follows: 3 = (2e Vd) 1 + /(e /d){(Â£ /d)l} m mm R /b A = be "^ ^ir^^m^^m/^ ' ^^^Â°^ wh ere d {V, (R ) + bV' (R )} and the prime denotes difIr m Ir m ^ ferentiation with respect to R. 3 For He*(ls2s, S) + Ar , the C, coefficient used here is D the one reported by Bell, Dalgarno and Kingston (Be58) . The values for R and g used are those found by Chen, mm ^ Haberland and Lee (Ch74) in an analysis of the angular distributions which they measured for He*+Ar collisions. They were able to fit parameters to a MorseSplinevan der Waals potential. The He*+Ar potential of Chen, Haberland and Lee is the one most reliably determined, although recent work by Nakamura (Na76) indicates that the potential suggested by Olson (0172a) may be more appropriate for He* (ls2s,"'"S) + Ar. Potentials for He+Ar have been determined from data on the ratio of AI to PI crosssections, together with considerations of energy balance based on FranckCondon transitions (Ch74, Pe75) . Recently, Nakamura (Na75) has reanalized these results within a semiempirical framework, and proposed a new He+Ar potential. Here the R and e ^ mm
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159values for Nakainura ' s He+Ar potential have been used, as v/ell as his value for the C, coefficient based on the SlaterKirkwood formula in terms of the polarizabilities of He and Ar . The He+Ar C. coefficient used is the accurate (experimental) value quoted by Dalgarno (Da62) . HartreeFock calculations have been made on He+Ar (Si71; see also Br75) , but, as for the MSXa calculations of Chapter III, these would lend information only on the repulsive behavior of the interaction potential. Table III lists the parameters used in the interaction 3 +52 potentials for He*(ls2s, S) + Ar and He + Ar (3p , P) . The corresponding interaction potentials are the ones shown in Figure 13, and earlier referred to in the qualitative discussion of PI and AI. Olson (0172a) was guided in describing the repulsive interaction of He*+Ar by comparing it to the behavior of the repulsion determined for Li+Ar using appropriate sum rules and the results for the repulsive interaction in homonuclear diatoms based on ThomasFermiDirac statistical treatment of the electronic energy (Ab69) . The MSXa results for He*+Ar are quite different, giving a value of b nearly twice that of the Li+Ar value (Ab69) . The MSXa value of b for He+Ar (0.532) is quite close to that calculated for He+Ar (0.5435) in Chapter II, supporting the similarity between the He+Ar and He+Ar repulsive interactions . Figure 14 displays the interaction potential for
PAGE 173
Â•160Table III. Parameters for the interaction potentials of He*(ls2s,^S) + Ar and He + Ar"*" ( 3p5 , ^P) as given by Equation (107) , where B = b and Rq is given by Equation (108) . 3 and A are determined by Equation (110) . The b parameters are from the MSXa calculations of the previous chapter. He* + Ar He + Ar"*" 4.39678 44.6527 0.9675 0.532 0.1583 X lO""^ 0.945 x lO"^"^ 9.4488^ 5.936^ 226.0^ 7.94^ A(a.u.)
PAGE 174
Figure 14. V(R) of Equation (107) for Ke*(ls2s, S) + Ar with the parameters of Table III. The various contributions to V(R) as discussed in the text are indicated. ^CHL ^^^ labels the corresponding potential proposed by Chen, Haberland and Lee (Ch74) , shown for comparison. The Eckart potential, fÂ£(R), is plotted against an arbitrary scale for clarity.
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1520.3 0.2 ro b X > 2 13 14 15 16
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1633 He* (ls23, S) + Ar represented by Equation (107) with the parameters of Table III. This is the potential labeled V. in Figure 13. In Figure 14, the various contributing terms of the potential of Equation (107) are shown explicitly. The deviation fi'om R behavior of the long range portion of the potential modulated by the Eckart potential, as well as the Eckart potential itself, are to be noted in particular. For clarity, the Eckart potential has been plotted in expanded fashion. The potential of Equation (107) shown in Figure 14 is based on well depth and position reported by Chen, Haberland and Lee (Ch74) . The comparison with their MorseSplinevan der Waals potential is quite good. The Eckart form matches their well depth conditions with a slightly stronger repulsion than provided by the Morse potential, and a slightly less attraction in the region just beyond the potential minimum. Figure 15 is a similar plot, showing the contributions to the potential of Equation (107) for He+Ar , and contrasting that potential with the one proposed by Nakamura (Na75) , which joins two functional forms describing different regions of R. Again, the comparison is good, the Eckart form showing slightly weaker repulsion for the well depth and position, and first less and then more attraction for R beyond the potential minimum. So the Eckart form of Equation (107) constitutes a very useful way to represent interaction potentials, giving
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165CJ O CO d (X) ^ OJ O CVJ O O O O
PAGE 179
166a single function which can be used over the whole range of R. Of course, the Eckart form, as well as other potentials describing repulsion by a BornMayer function, break down as R becomes very small. This has been mentioned already in Chapter II. In addition, it will be noted that the Eckart potential tends to suppress the long range inverse R terms as R ^ 0. However, the Eckart potential, though exponentially decreasing as R << RÂ„ , cannot quench the inverse R dependence of the long range terms. The concave curve at the lower right of Figure 15 is part of the Eckartmodulated long range term. This behavior is not at all crucial as far as describing the interaction potential is concerned, since the exponential repulsion strongly dominates the interaction potential for R sufficiently inside of R to accomodate the usual range of collision m ^ energies. The important modulation effect of the Eckart potential rather is in the region around the potential minimum. 7. Parameterization of the Couplings The interaction potentials proposed in the previous section for He*(ls2s, S) + Ar and He + Ar ( 3p , P) provide suitably parameterized expressions for V, j(I^) ^nd V (R) , respectively, appearing in Equation (104) . The remaining expressions needed to solve these coupled equations are 1 /2 the Coulomb coupling matrix elements, (Ae) '^'^'d^^el (5 >.
PAGE 180
167The approach here will be to rely on functional forms v/hich have been used in seraiempirical models to express the width T (R) associated with the rate of decay of the discrete (metastable) state into the continuum. The width F (R) for decay of a discrete state *^ into a continuum state degenerate with it, can be expressed as (see, for instance, Na69 or Mi70b) r(R) = 2tt<$^h^^^ , (111) and has units of energy in view of the fact that ^^ is a continuum state normalized to unity on the energy scale, as described earlier. If the transition has been of the FranckCondon type at separation R, then in Equation (111) , one must take e = V , , (R) V (R) . In seraiempirical d, d approaches, parameterized forms for r(R) have been proposed (Ho70, 0172a, Mi71, 1175, Na75) . The functional form adopted most often has been the simple exponential, namely, R/aÂ„ r(R) = A^e ^ . (112) This functional form appears to represent well the behavior of r(R) as determined from both experimental and theoretical analysis (see, for instance, Ni73 or Ho70, and references therein) . In view of Equations (111) and (112) , the coupling matrix element itself can be expressed in a simple exponential form, vv'hich should be a valid representation.
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168considering the extent to which Equation (112) has been successful in semiempirical approaches. In addition, the range of parameters, A^ and aÂ„ , suggested in various semiempirical analyses will then give an idea of the parameters suitable for an exponential representation of the coupling matrix element itself. When treating the matrix element, <$,h ^ I $ >, of ^ d' el ' e Equation (111), one must remember that it has an e dependence as well as an R dependence. However, in semiempirical approaches, F (R) is parameterized according to Equation (112) in order to account for transition at each R into continuum states of all possible e. Thus, when A and a_ are adjusted to give semiempirical results in agreement with experiment, they reflect primarily the behavior of F (R) of Equation (111) at an energy, e , near the most probable energy of the emitted electron (i.e., the peak of the electron energy distribution) . Clearly, in the coupled equations of Equation (104) , coupling matrix elements are required over the whole range of Â£ values, Â£ , which are important for PI and AI . So strictly speaking, one should first find the e dependence of the discrete to continuum coupling matrix elements, and then use Equations (112) and (111) at Â£ to determine the parameters of their R dependence. An assessment of this Â£ dependence may be made in terms of the Â£ dependence of the continuum orbital describing the emitted electron. Namely, referring to Equation
PAGE 182
169(64), it is seen that the continuum orbital of energy e, expressed according to Equation (62) , asymptotically has its dominant e dependence in the factor /2/iTk /T/^/ /2z. Its other e dependence appears in the argument of the sine function, which is not such a strong dependence since the sine function remains bound by Â±1 regardless of its argument. So, although the matrix element, ^"^^ I ^^i I "^^^ has its largest contribution from the region of configuration space near the molecule (where the emitted electron sees more than simply a Coulomb field) , the e dependence just mentioned should be appropriate. Basically, the assessment being made here is that, whereas the spatial behavior of the wave functions describing the emitted electrons near the molecule is not that of a Coulomb wave function, the emitted electrons are energetic and escape the region of the "molecule rapidly, so that their e dependence may be well represented on the basis of Equation (64) . Therefore, for each continuum state $ , a state $ is introduced according to the relation. $ = (/2A/^/27)i . (113) e e Based on Equation (64) , and recalling that Q has units 1/2 of (energy) compared to bound v;ave functions, one sees that the factor in front of ^j in Equation (113) has the 1/2 units (energy) . One may further wrxte.
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Â•170^> = (/27^/^/T?)<$^h^^5^> , (114) where the matrix element on the right side of the equation is expected to have only weak e dependence. It is this matrix element of weak e dependence whose R dependence will be here expressed in exponential form: R/a <$^h , U > = a e ^ . (115) d' el' e m Thus, using equations (111) , (112) , (114) and (115) based on a value, e , near the peak of the emitted electron energy distribution, one finds the appropriate expression by which A and a can be related to AÂ„ and aÂ„: 'mm r 1 R/a ^ 2R/a Aj,e ^ = (4//2i^)A^d ^ . (116) On the basis of Equation (116) , one sees that ^r^ = 2aÂ„ , m 1 A = ('^/2^/2)/Ar . (117) m o i R/a Recalling that A e expresses the weak e dependent part m of a coupling matrix element, the coupling matric elements of the coupled equations of Equation (104) are, therefore, written, (Ae)l/2^$^H^J0^_^> ^ (Ae)'^^(/2A/'^/TF^)A^e ^ . (118)
PAGE 184
171Values for A and a based on A^ and aÂ„ parameters from mm r r '^ semiempirical analyses can be found according to Equation (117). For He*(ls2s, S) + Ar PI and AI , typical values of the emitted electron energy are around 0.1505 a.u. (Ho69, Ho74, Ch74) . This is the value used in Equation (117) to compute values for A and a based on A^ and a^ as deterra m r r mined in several semiempirical analyses. The results are shown in Table IV. As can be seen, there is a wide range of A_ and aÂ„ values indicated by semiempirical calculations, mm I sr The results based on Olson's v;ork (0172a) deviate markedly from the others. However, Olson chose to vary only the exponential factor, aÂ„, in his work, leaving A = 1.0 a.u. Basically the trends set by semiempirical v/ork indicate an a value of about 0.7 a and A values in the range 20100 m o m ^ 8. Results from Coupled Channels Calculations of He* (ls2s, ^S) + Ar PI Collisions The heavy particle dynamics involved in PI of Ar by 3 He*(ls2s, S) can now be treated in terms of the coupled equations of Equation (104) by implementing the interaction potentials and couplings just presented. These interaction potentials and a typical coupling term are shown in Figure 3 13, and from now on the He*(ls2s, S) + Ar interaction potential will be referred to as V, , or the upper potential, 2 + 5 2 the He + Ar ( 3p , P) interaction potential as VÂ„, or the
PAGE 185
172Table IV. Ap and ap values of Equation (112) reported in several semiempirical analyses as indicated, and the Aj^ and a^ values of Equation (115) calculated by Equation (117) with e = 0.1505 a.u. Semiempirical Analysis
PAGE 186
173lower potential, and the coupling term as V, ^. The DeVogelaere numerical integration scheme, adapted from a version used previously in connection with other work (Mc71, Mc72) , was employed to integrate two linearly independent solutions from deep within the nonclassical regions inside the repulsive walls of both potentials out to the asymptotic region. There the boundary conditions for scattering are imposed and the Rm.atrix is calculated according to Equations (84) to (87) . One important characteristic feature of PI and AI is that the upper and lower potentials have repulsive walls which are quite far apart. As can be seen from Figure 13, the repulsive wall of V, falls in the range 6.5a to 8 a , ^ 1 ^ o o whereas that of V^ falls in the range 4.5 a to 5 . 5 a , for typical incident and final collision energies, respectively. This can lead to difficulties of a numerical nature which arise in the following way. The two linearly independent solutions which are integrated are both regular at the origin. That is, in the classically forbidden region, as R decreases sufficiently from the classical turning point of a given potential, the solution appropriate to that potential will decrease exponentially. As is normally the case, where the repulsive walls of the potentials for the various channels are in the same small range of R, all the linearly independent solutions, each appropriate for a given channel, can be specified
PAGE 187
Â•174at a single point inside the repulsive barriers by giving them a small value (for numerical purposes, zero) and differing values for their first derivatives. As the solutions are integrated from this starting point to larger R, they increase roughly exponentially (as they should) until the classical turning point is passed (which occurs nearly simultaneously for all solutions) , whereupon they exhibit typical oscillatory behavior for the classical region. For numerical reasons, even in such normal cases, the starting point cannot be chosen too deep into the nonclassical region, since, even if small derivative values are assigned, the exponential increase of the solutions sustained over too many integration steps numerically becomes unstable. Neither can the integration starting point be chosen too close to the classical turning point, for then a solution may be generated which is not regular in behavior at the origin. Numerical instability is nearly unavoidable, then, for cases such as shown in Figure 13, if a single starting point is chosen inside VÂ„ . The linearly independent solution appropriate to VÂ„ would behave properly at first, but the one appropriate to V, would be increasing over an unnecessarily large part of its nonclassical region, becoming numerically unstable. In addition, since the two solutions are coupled, the numerical instability of one would affect the other. To avoid such numerical problems, a procedure
PAGE 188
175was introduced for suppressing the solution appropriate to the potential having the outermost repulsive wall (V, in Figure 13) deep inside its nonclassical region where the integration for the solution appropriate to the potential having the innermost potential wall (V^ in Figure 13) begins. As the outward integration proceeds, suppression of the one solution continues until the point inside its potential wall is reached where it satisfies the same type of starting conditions as did the solution appropriate to the inner potential v/all. At that point, the suppressed solution becomes included in the numerical integration of the coupled equations with a suitable (nonclassical region) starting value and derivative. Such a procedure conforms to the condition that, for R deep inside the classical turning point of a given potential, the wave function for the heavy particle motion in that potential can physically be neglected, and has led to numerically stable integration of the coupled equations. A convenient way for simultaneously determining both suitable integration starting points and initial derivatives for a solution appropriate to a given potential was also incorporated into the calculation. The approach relies on making an estimate of the exponentially decreasing behavior of the solution, as R decreases beyond the classical turning point, in terms of a JWKB wave function in the nonclassical region of the potential barrier as approximated by a linear
PAGE 189
176functional form. Such JWKB wave functions can be expressed in terms of Airy functions in the limit of large argument in the classically forbidden region. An integration step size of 1/24 the smaller of the two channel wave lengths v;as found to give reliable numerical accuracy, yielding Rmatrix elements differing between integration steps by only a few parts in 10 once the integration had proceeded into the assymptotic region to R values between two and three times the potential well position of V . Each set of coupled equations of the type in Equation (104) corresponds to transition into the I discretized continuum state. That is, referring to Equation (94) , each set of coupled equations is characterized by electrons emitted with c in the I interval of the partitioned energy scale. According to the earlier qualitative discussion of PI and AI, one knows which range of the e scale should be important to the extent that FranckCondon transitions occur. However, this information is not "built" into the present coupled channels approach. Rather, for given interaction potentials and coupling, the dynamics as.ciated with electrons emitted with energy in the various ranges of e can be investigated, to see what range of e is indeed significant. For the twostate problems at hand, and referring to Equation (94) , the Sand Tmatrix elements of Equations (90) and (91) can be simply written
PAGE 190
177S e,d i:I I,d J,d (119) where S^ i and T^ i are the Sand Tmatrix elements of I,d I,d Equations (88) and (89) , which are found by satisfying scattering boundary conditions for the discretized set of (two state) coupled equations. In Equation (119) , it is assumed that e happens to be in the J interval of the e scale. The partial ionization crosssection per unit energy of the emitted electron for e 
PAGE 191
178In the present approach one may treat the integration in Equation (122) approximately as a sum, and write I, a^^(E) ^ I Acido^ ^^{E,e^)/de} I 1 {An/A I 1{2L+1) T^^^2 , (123) where I ^ refers to the last interval of the e scale within the Penning limit. The total PI crosssection of Equations (122) and (123) does not represent the entire crosssection contribution. Rather, it is the contribution from only one of the possible molecular states of the incident channel, arising from 3 He*(ls2s, S) + Ar, undergoing transition into one of the possible final continuum states, where the emitted electron has energy e and the state of the (HeAr) molecular ion + 5 2 separates to He + Ar ( 3p , P) . So, taking into account electronic spin and orbital angular momentum of the reactant and product atoms, the process involves 3 possible states initially and 6 finally. The total crosssection is obtained by averaging over initial states and summing over final states (see, for example, Mi75) . Inherent in the present approach is the assumption that V, and V of Figure 13 suitably represent any of the possible molecular states of the reactants or products which are compatible with their separated limits. This is a good assumption as far
PAGE 192
179as spin multiplet differences are concerned, but does not 2 properly account for the presence of the II molecular ion state, which was found to play a role in terms of the Auger type analysis of the previous chapter. More rigorously, 7 2 both the E and II states of the molecular ion should be included, which would involve at least sets of threestate coupled equations. However, in the present approach, the heavy particle dynamics are treated only in the presence of E states in the initial and final channels. Averaging over initial states and summing over final states introduces a factor of 6 which should multiply o^^ of Equations (122) and (123) in order to correctly assess the total PI crosssection in the framework of the present calculations. Of interest in the study of the dynamdcs of the heavy particle motion are the L partial crosssections per unit energy of the emitted electron for z^d. transition. These can be extracted from Equations (120) and (121) , and are written da^^^(E,e)/de (Ae) "^ (4u/k^) (2L+1)  T^^^I ^ , (124) where again e is assumed to be in the J interval. The L partial crosssection represents the contribution to the partial crosssection of Equation (121) due to L angular momentum partial wave of the heavy particles. That is, it is associated with the heavy particle motion in the effective potentials comprised of the bare interaction
PAGE 193
1802 potentials of Figure 13 plus the L(L+l)/2mR centripetal potential . In the calculations which will be reported here, Ae was chosen to be 2 x 10 a.u., since it provides a reasonably fine partition of the e scale. Such a Ac divides the FranckCondon range of electron energies (see Figure 13) into about 15 increments, which should be sufficient for probing the details of this region, which is anticipated to be of most importance. The partial crosssection per unit energy of the emitted electron (Equations (120) and (121)) is an important quantity to monitor. Its behavior as a function of e can be com.pared v/ith experimental measurements of the energy distribution of emitted electrons (Ho69, Ni73, Ch74, Ho74) , and therefore provides a good v/ay of demonstrating and checking the more detailed treatment of PI which is possible in the present approach. Figures 16 and 17 display the results of calculations of da _^, (E,Â£)/dÂ£ as a function of 4 e for collision energies E of 23.89 x 10 a.u. (65 meV) 4 and 15.0 x 10 a.u. (=41 meV) , respectively. These calculations v;ere performed with coupling parameters A and a in Equation (112) of 100.0 a.u. and 0.9 a , respectively. As seen in Table IV, this A value is the one extracted from m Nakamura ' s work, and the a value lies between the first m three and the one derived from Olson's work. In Figures 16 and 17, the smooth curves have been drawn through the
PAGE 194
Figure 16. Partial ionization crosssection per unit energy of the emitted electron (Equation . (121)) calculated for PI of Ar by He*(ls2s, S) at an incident collision energy E = 23.89 x 10"^ a.u. (=^65 meV) , and plotted against the emitted electron energy ((e E_) , top scale), or, equivalently , the relative collision energy in the final channel (E^, bottom scale) . The parameters of the coupling (Equation (118)) are Aj^ = 100 a.u. and a^^ = 0.9 ao . The dots are positioned at the midpoints of the increments Ae ( = 2 x lO"'^ a.u.) along the Â£ scale, as described in the text.
PAGE 195
18216 (Â£E^) (a.u. X10^^) 8 Â° 8 15 T Â— 1 Â— r in 5 O X fO 4 C\l O 03 3 Â— UJ D 2 Qc Â— A t^ 40 O
PAGE 196
Figure 17. Partial ionization crosssection per unit energy of the emitted electron (Equation (121)) calculated for PI of Ar by He* (Is2s,^S) at an incident collision energy E = 15.0 X 104 a.u. (==41 meV) . The details for this plot are otherwise the same as for Figure 16.
PAGE 197
184(cE Xa.u. X10 '^) 16 Â° 8 E^(a.u. XIO"")
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185do , (E,Â£)/de values of Equation (121) at the midpoint of e^d each increment of the e scale. The Â£ dependence shown in the figures does depend on the increment size Ae , but the value 2 x 10~ a.u. used here represents a fine enough partitioning of the e scale that the e dependence shown sould be expected to change only slightly upon refinement of Ae . In Figures 16 and 17, the bottom scales show the collision energy of the final heavy particle motion corresponding to each e of the emitted electrons, which is read along the top scales referenced to E , the difference in energy between reactants and products (recall that EÂ„ = 0.149 a.u.). It is seen that the Penning limit lies at the right edge of each plot, and each curve stops just short of the Penning limit, corresponding to the last increment before entering the AI region. In order to probe the AI region, the computer program used in these calculations would have to be modified to treat both open and closed channels. The behavior of the dependence shown in Figures 16 and 17 compares fairly well qualitatively with experimental measurements of electron energy distributions in PI and AI 3 of Ar by He*(ls2s, S) (see, in particular, Ho69 and Ho74) . The distributions in the figures peak for e larger than E , which is what would be expected (the reader is referred to the qualitative discussion earlier in this chapter) .
PAGE 199
Â•186However, the experimental distributions peak at values of 4 e larger still (i.e., (e Eq) 16 x 10 a.u.), favoring transitions of a FranckCondon type much more than these calculated results. Also, not surprisingly, the experimental distributions enter the AI region having proportionately much larger contribution than those of Figures 16 and 17. Nonetheless, these calculated distributions do diminish abruptly for Â£ < E., and the distribution at the lower collision energy shifts with respect to E in the right direction (albeit very slightly) , and becomes narrower, as shown by experimental results (Ho74) . The total PI crosssections, determined according to Equation (123) , for the calculations of Figures 16 and 17 2 are grossly overestimated, being about 4922 a at the 2 higher collision energy and 3809 a at the lov/er collision energy. This certainly indicates the possibility of an inappropriate choice of the coupling parameters A and a . Figure 18 shows results for da ^^(E,c)/dÂ£ calculated at 4 E = 23.89 X 10 a.u. (=65 meV) with coupling parameters A = 50.0 a.u. and a = 0.7 a , which, referring to Table m mo'' IV, are more in the range of the majority of the values extracted from semiempirical work. Figure 18 is plotted exactly as Figures 16 and 17. The e dependence shown in Figure 18 constitutes a definite improvement over that of Figures 16 and 17. The peak lies nearer the experimental peaks, corresponding to a predominance of FranckCondon type
PAGE 200
Figure 18. Partial ionization crosssection per unit energy of the emitted electron (Equation (121)) calculated for PI of Ar by He*(ls2s,3s) at an incident collision energy E = 23.89 x lO'^ a.u. (==65 meV) . The details for this plot are the same as for Figure 16, except here the calculations were performed with coupling parameters A = 50 a.u. and a^^ = 0.7 a^. The left and right arrows at the top edge of the graph locate peaks of distributions determined experimentally at E = 36 meV and E = 95 meV, respectively, and the center arrow marks the point midway between the other two, as described in the text.
PAGE 201
(eE^)(a.u. XIO'^)^ 8 8 16 Ej(a.u. X10 )
PAGE 202
189transitions. Also, the distribution shows proportionately large contribution as e enters the AI region. An interesting comparison can be made with experimental PI electron energy distributions reported by Hotop (Ho74) . He shov7s distributions measured at collision energies of 36 meV, 95 meV and 125 meV. The peaks of these three distributions, in order of increasing collision energy, shift in roughly linear fashion to higher relative e, being located at (e E^) values of about 35 meV, 55 meV and 70 meV, respectively. The calculated distribution of Figure 18 is 4 for a collision energy of 65 meV (23.89 x 10 a.u.), which falls nearly midway between 36 meV and 9 5 meV. So based on the trends shown by the experimental results of Hotop, the peak of the distribution at 65 meV should fall approximately between the peaks at 36 meV and 9 5 meV. Along the top right edge of Figure 18 are arrows, the left one locating the peak of the experimental distribution at 36 meV collision energy, the right one the peak of the experimental distribution at 95 meV collision energy. The calculated distribution does indeed peak nearly between the left and right arrows, indicating a good agreement with the experimental results of Hotop. The total PI crosssection for the calculations of Figure 18 is also quite reasonable, being 32.5 a (9.1 A ). This value compares well with reported experimental results (see, for instance, Mu68, Ru72 or Ni73) . However, little is to be gained by adjusting the
PAGE 203
190parameters A and a to reproduce certain total PI crosssection results. Rather, the calculated results of Figure 18 show that, with reasonable coupling parameters, A^ and a the present coupled channels approach provides a rein liable description of the PI process, reflecting well the behavior of the energy distribution of the emitted electrons, and resulting in a reasonable total PI crosssection. The A and a parameters of Figures 16 and 17 and those of m m '^ Figure 18 are quite different, and yet the results of both calculations are qualitatively good, with those of Figure 18 comparing well quantitatively with experiment. As has been mentioned, the L partial crosssection per unit energy of the emitted electron (see Equation (124) ) is an interesting quantity coming out of the calculations which can give insight into some of the details of the PI process. Figure 19 shows a plot, similar in all respects to Figures 16 to 18, of the L partial crosssection per unit energy of the emitted electron for L = 20, where the collision energy and coupling parameters are the same as for Figure 18. The structure exhibited by da ^^ (E,Â£)/de as e decreases from the Penning limit (i.e., as E^ increases from zero) can be understood by referring to Figure 20. In Figure 20 V^ of Figure 13 (He+Ar interaction potential) is plotted along with the effective potentials of the heavy particles in various angular momentum states characterized by the partial wave number, L, as indicated. Such effective
PAGE 204
Figure 19, L partial ionization crosssection per unit energy of the emitted electron (Equation (124) ) for the L = 20 angular momentum partial wave of the heavy particles, calculated for PI of Ar by He*(ls2s,3s) at an incident collision energy E = 23.89 x 10 '* a.u, (65 meV) . The details for the plot are otherwise the same as for Figure 18.
PAGE 205
192(Â€E^)(a.u. XIO"^) 16 8 T Â— I Â— r 1 Â— i Â— r Â— r 1 Â— r 8 1 Â— \ Â— r CO O 13 d CVJ o o o LÂ—J L__l. 40 i ^r^r^^ 32 E^(a.u. XIO"'')
PAGE 206
n3 0) in Oi P C Pi O J C m OJ g g +J ro P \ >l Ph O iH ^ fC no p., 4) + C M O + U t^ tr> w hi P H H rH rH nq o + s GOO. (ti 0) 5^ +J u c O H m ^^ CO W (D fO iTj (0 C Â•H X! IS 4J Â— C O rt3 H M Â•H O H IS fO P > > Q) CU 0) M OiPm > 41 nH ;d >, Sl O O O P C 3 1^ O H U rH U rt Id o en H H :3 aH U P M ^ M rO fd > O (D 41 &. Â•H fO en tn nj H (lJ(UcnPDj>iOQ)(ti 4t^ 0H>Ml>^ O G O Â•H 41 O (T3 U Q) 4J Â•H 0) (U ^ > (d Â•H .H P G O 13 41 13 >4 fd Sl C Q) 4J (U ^ ;3 0) H B (Â« 13 U Q) n3 CO 0)0) ^ Ji (U > ^ Â• 0) 44 QJ 4J 0) 54 H IT) x; ^ 44 4:^ :3MrH4)Og04J w (u w c d T5 CD (4 3 tn 0) ^ (UriiarHCcu^o rH tP OiH > Eh X^ >iH TJ (d W r4 Cm H C & tn rd O Â• * 3 a H OiiH xi 4) o H 44 w td (u a Â•H C (U H 44 (D > ^ oj n 4J td e (1) P: 4J M M U (U M ^ O O td H rH a> a< o Q^xi o) 0) V4 ;3 tn Â•H J4
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194(Â£OIXnD)U!A
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195potentials consist of the bare potential, VÂ„ , plus the rip (L) 2 centripetal potential, L (L+1) /2mR ' , and will be referred to as V2 , The plot in Figure 19 shows, as a function of e, the partial crosssection contribution for cfd transition when the heavy particles are moving in the L = 20 effective potentials initially and finally. Referring to Figure 20, when E^ is near zero (e near the Penning limit) , the heavy particles moving in VÂ„ see the sizable rotation barrier for L = 20, permiting only negligible crosssection as seen in Figure 19. But, as E_ increases (e decreasing), the partial crosssection contribution exhibits a typical pattern, characterized by pronounced peaks of contribution, the first one being quite large, and each successive one increasingly smaller, until finally negligible contribution 4 is found for E_ larger than 30 x 10 a.u. Looking at a particular partial crosssection as a function of E^ (i.e., of e) amounts to probing the final channel continuum for a given collision energy, and detecting regions of high density of states for a given angular momentum state of the heavy particles, reflected by the resonance peaks of cross4 section contribution. The large peak between 4 x 10 a.u. 4 and 6 X 10 a.u. in Figure 19, for instance, may well be due to orbiting resonances associated with the rotational barrier of V^ , which can be seen by referring to Figure 20. Actually, in studying the detailed structure of a
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195part?cal crosssection, such as for L = 20 in Figure 19, it may be advisable to refine the increment size, Ae, for a more detailed look at the structure. But for studies of the overall crosssection per unit energy, such as in 4 Figures 16 to 18, Ae = 2 x 10 a.u. is sufficient for assessing the e dependence. Not only may da ,(E,e)/de be looked at as a function of e for a given E and L, but it can be studied as a function of L for a given E and e. Such results are plotted in Figures 21 to 24. All of the results in these figures 4 are for a collision energy of 23.89 x 10 a.u. with A = 50 a.u. and a = 0.7 a . Each plot is labeled by the m m o '^ ^ final collision energy E,, prescribed by the fixed e value (i.e., E_ = E + E e and E = 0.149). Pronounced resonance peaks characterize the partial crosssections, this time as a function of L. The structure can again be understood by referring to Figure 20. The final collision energies corresponding to each plot are indicated in Figure 20 by the appropriate horizontal lines. Focusing attention on a single final collision energy, one can realize, that as L increases from 0, the specified collision energy will fall Â• near resonance peaks, of the sort seen in Figure 19 for L = 20, for some of the effective potentials, VÂ„ , and fall between such resonance peaks for other effective potentials. The largest L value for appreciable partial crosssection contribution for a given final collision energy can
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Figure 21. L partial ionization crosssections per unit energy of the emitted electron (Equation (124)) calculated for PI of Ar by He*(ls2s,^S) at an incident collision energy E = 23.89 x 10"^ a.u. (65 meV) , and plotted against the partial v/ave number L. The results shown are for a single energy of the emitted electron e, with the corresponding final collision energy (given by Ef = E + Eq e) as indicated in the figure. Eg = 0.149 a.u. and the results shown are from calculations with the coupling parameters A^ = 50 a.u. and am = 0.7 aÂ„.
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Â•198
PAGE 212
Figure 22. L partial ionization crosssections per unit energy of the emitted electron calculated for PI of Ar by He* (ls2s , ^S) , and plotted against L. The explanation regarding the indicated value of E^ as well as all other details for this plot is exactly as in the figure caption to Figure 21.
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Â•2005
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Figure 23. L partial ionization crosssections per unit energy of the emitted electron calculated for PI of Ar by He* (ls2s , ^s) , and plotted against L. The explanation regarding the indicated value of E^ as well as all other details for this plot is exactly as in the figure caption to Figure 21.
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202
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Figure 24. L partial ionization crosssections per unit energy of the emitted electron calculated for PI of Ar by He* (ls2s, ^s) , and plotted against L. The explanation regarding the indicated value of Ef as well as all other details for this plot is exactly as in the figure caption to Figure 21 .
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204CJ O X 4 b h \ CO c> O 4. E, = ZZXlO'^a.u Vi/ vy LU l..Â«s=: 10
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Â•205be readily estimated from Figure 20 by inspecting the classical turning points of the VÂ„^ for that collision energy, and seeing for which L they begin to fall beyond the range of appreciable coupling (a typical coupling is shown by the dotted curve in Figure 20) . The results shov/n in Figures 21 to 24 confirm such estimates, as can easily be checked. The trend as E^ increases is for the primary peaks of the structure to shift to higher L values, and in addition, for the number of peaks to increase. In Figure 21 there are two main peaks v/hereas in Figure 2 4 there are four. The results in each of the figures are characterized by a proportionately smaller and broader peak at low L values. In Figure 24, E^ is nearly as large as the initial collision energy for these calculations, and the partial crosssection contributions from all of the peaks have diminished appreciably. This, of course, would be expected from the E^ dependence of Figure 18. 9. Discussion Formal discretization procedures, similar to those introduced in connection with continua encountered in atom + diatom collisional dissociation (Wo75) as well as in nuclear reactions (B166) , have in this chapter been applied to the internal continuum electronic states associated with collisional ionization processes. Discretization permits one to focus more readily on the important regions
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20Gof the continuum as specified by the energy e of the emitted electron, and provides a suitable framework within which to introduce and test approximate treatments of the related collision dynamics. By implementing the basis set of step functions of e (Equation (94)), together with several physically reasonable approximations, it was suggested that the dynamics involved in PI and AI processes could be treated in terms of relatively few (=^20) twostate coupled equations, each associated with a particular small range of e defined by one of the step functions. The results of the application to PI of Ar by He*(ls2s,^S) have amply demonstrated that, though approximate, this approach includes the important dynamical features of collisional ionization processes, and permits a quite adequate treatment. It goes beyond usual semiempirical models in that the heavy particle motion is governed by the interaction potential of the final state of the molecular ion as well as that of the excited (metastable) molecular state. The process of discretization allov/s the heavy particle motion to be calculated and distinguished according to the amount of energy z removed by the emitted electron. Consequently, the partial ionization crosssection per unit energy of the emitted electron can be computed as a function of e, and can be compared with experimental measurements of the energy distribution of emitted electrons (Ho69, Ni73, Ho74) .
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207This represents an important additional level of detail in the description of collisional ionization which at the same tirae; provides a more stringent test of the theoretical approach than is afforded by calculations of total ionization crosssections only (see, for instance, Be74b) . The results for the e dependence of these distributions shown in Figures 16 to 18 show good qualitative agreement with experiment, and in the case of Figure 18, quantitative agreement is also good. The additional feature of being able to study the L partial ionization crosssections per unit energy of the emitted electron means that even more detail of collisional ionization processes can be monitored by looking at the crosssection contributions associated with the heavy particles moving in the various effective potentials associated with their relative rotational motion. Whether studied as a function of e (Figure 19) , or as a function of partial wave number L (Figures 21 to 24) , their behavior reflects resonances associated with regions of high density of continuum states for the various effective potentials. In the course of carrying out the present calculations 3 of PI of Ar by He*(ls2s, S) , a convenient functional form for atomatom interaction potentials has been introduced (see Equation (107) ) . It has the nice features of representing a potential by a single function over the entire range of internuclear separations, and of having the additional flexibility provided by modulating the long range
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208van der Waals contribution by the Eckart function. This form for atomatom interaction potentials is expected to be of quite general applicability. 3 The results reported here for PI of Ar by He*(ls2s, S) represent only a sample of what can be extracted from the calculations and analyzed as done for the results reported in P^igures 19 and 21 to 24. In addition, further studies of the dependence of the results on the parameters of the interaction potentials and couplings could be made. In the present approach, reliance was placed upon the expression for the decay width F (R) in terms of the coupling matrix element between the discrete and continuum electronic states (Equation (111)) in order to draw a convenient relation connecting the parameters of T (R) from semiempirical work with the parameters, A and a , of the coupling matrix '^ mm IT ^ elements (Equation (117)). In this v;ay, AÂ„ and aÂ„ values from semiempirical analyses served as useful guidelines for choosing A and a values. However, because the coupled ^ m m '^ channels calculations permit more detailed comparisons with experimLental results, they would perhaps be better able to specify suitable coupling parameters, A and a , for a given set of interaction potentials, thereby providing good estimates of AÂ„ and a^ according to Equation (118) . Strictly speaking, the coupling matrix elements used for the present calculations are between the discrete (metastable) state and continuum states associated v;ith the emitted electron in its
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209J!. = angular momentum state. Couplings to components of higher I should be included, but the results indicate this is not of serious consequence as far as the heavy particle dynam.ics are concerned. The most interesting extension of this work would be the modification of the existing program to include the boundary conditions for closed channels. Then the coupled channels calculations for AI processes could be carried out. This vi70uld make possible a comparison v/ith additional experimental results. For instance, the calculated distribution of partial ionization crosssection per unit energy of the emitted electron, such as shown in Figure 18, could be extended into the AI region, thereby allowing complete comparison with measured energy distributions of emitted electrons. It is anticipated that such calculated distributions in the AI region may reflect some structure associated with the vibrational states of the various effective potentials which are important. At any rate, analyses in the AI range of e could be made, similar to the one shown in Figure 19 for e in the PI range. That is, by inspecting Figure 20, one could quickly determine which of the angular momentum partial v/aves of the heavy particles in the final channel would result in effective potentials supporting vibrational states for e in the AI region. By singling out one such L partial ionization crosssection, and looking at its e dependence in
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210the AI range, one v/ould expect to see a structure characterized by peaks in the crosssection contribution, and reflecting the spectrum of vibrational states of the effective potential. Combining such a study in the AI range with one, such as shown in Figure 19, in the PI range, would allow one to observe, as e decreased from the PI range to the AI range, first the structure associated with regions of high density of states in the continuum of the heavy particle motion, including states trapped in the rotational barrier, follov/ed by the structure associated v;ith the bound vibrational states of the heavy particles. Furthermore, the relative amounts of crosssection contribution among the various vibrational states in the AI region would lead to information about the distribution among final vibrational states of the (HeAr) products in AI . Thir: would be of interest in assessing possible mechanisms of vibrational state population inversion which may result in AI of Argon by metastable Kelium. Calculations in the AI range would also permit the comparison between the total PI crosssection a and the total AI crosssection o . Experimentally, the total ionization crosssection, o_^ = a^^ + a,^, and the ratio, TI PI AI a /o , have been studied as a function of the incident collision energy, E (Ch74, Ho74, Pe75, 1175). Semiempirical analyses of these quantities have also been made (Mi71, 0172a, 0172b, Na75, Na76) . Such studies would be interesting
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211to carry out with the present approach, if it were extended to include ionization in the AI range.
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217(S171a) J.C. Slater and J.H. Wood, Int. J. Quantum Chem. 4, 3 (1971). (S171b) J.C. Slater, in Computational Methods in Band Theory , eds . P.M. Marcus, J.F. Janak and A.R. Williams (Plenum. Press, New York, 1971), p. 447. (S172) J.C. Slater, Adv. Quantum Chem. 6, 1 (1972). (S174) J.C. Slater, Quantum Theory of Molecules and Solids (McGrawHill Book Co., Inc., New York, 1974) , Vol. 4, Chap. 1. (Sm59) F.T. Smith, Phys . Rev. 179, 111 (1959). (Ta72) S.Y. Tang, A.B. Marcus and E.E. Muschlitz, Jr., J. Chem. Phys. 56, 1347 (1972). (Th61) W.R. Thorson, J. Chem. Phys. 34_, 1744 (1961) . (Th65) W.R. Thorson, J. Chem. Phys. 42, 3878 (1965). (Ti64) M. Tinkham, Group Theory and Quantum Mechanics (McGrawHill Book Co., New York, 1964), Chap. 5, p. 105. (To73) J. P. Toennies, Chem. Phys. Lett. 20, 238 (1973). (Tr73) S.B. Trickey, F.R. Green and F.W. Averill, Phys. Rev. B8, 4822 (1973) . (V151) J.H. Van Vleck, Rev. Mod. Phys. 23, 213 (1951). (We73) P. Weinberger and D.D. Konowalow, Int. J. Quantum Chem. S7, 353 (1973) . (Wo74) G. Wolken, J. Chem. Phys. 6^, 2210 (1974). (Wo75) G. Wolken, Jr., J. Chem. Phys. 63, 528 (1975). (Ya74) D.R. Yarkony and H.F. Schaefer, III, J. Chem. Phys. 6J,, 4921 (1974) . (Za66) R.N. Zare, Joint Institute of Laboratory Astrophysics Report No. 80 (1966, unpublished) .
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BIOGRAPHICAL SKETCH John Curtis Bellum war, born August 2 , 1945 in Lakeland, Florida. His parents are Clifford A. and Mabel I. Bellum. He grew up in Sarasota, Florida, attending the local schools there, and graduated from Riverview High School in 1953. In the fall of the same year he entered Georgia Institute of Technology, where he received the Bachelor of Science Degree in Physics, Coop Plan, in March of 1963. He then proceeded directly into graduate school in the Department of Physics of the University of Florida, employed as a graduate teaching assistant except during the summers of 1968 and 1970, when he was engaged in training as an ROTC cadet. He attended the 1970 International Sumir.er Institute in Quantum Chemistry, Solid State Physics and Quantum Biology, held in Uppsala, Sweden and Beitostjz^len, Norway, and returned to join the Quantum Theory Project of the University of Florida as a graduate research assistant in November of 1970. He then pursued studies as a full time student until the summer of 1974, which he spent on active duty in the United States Army. Since the fall of 19 74 he has continued studies as a part time student. John Curtis Bellum is a member of the Americal Physical Society and the American Scientific Affiliation. 218
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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. David A. Micha, Chairman .> 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. v^i^j^Jvu^ N. Yngvp Ohrn 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. 'Vv^^^ Joh/i W. D. Connolly Associate Professor of Physics
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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. ^/:V7 Thomas L. Bailey, III Professor of Physics ayid Electrical Engineering 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. .^. JA . ^^ AwvCvk ^ / PerOlov Lowdin Graduate Research 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. Charles P. Luehr Assistant Professor of Mathematics
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This dissertation was siibmitted to the Graduate Faculty of the Department of Physics in the College of Arts and Sciences and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August, 1976 Dean, Graduate School

