Energy transfer in molecular collisions

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Energy transfer in molecular collisions
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viii, 152 leaves : ill. ; 28 cm.
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Vilallonga, Eduardo F ( Eduardo Fermin ), 1953-
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Collisions (Nuclear physics)   ( lcsh )
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Molecular dynamics   ( lcsh )
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Thesis:
Thesis (Ph. D.)--University of Florida, 1981.
Bibliography:
Includes bibliographical references (leaves 147-151).
Statement of Responsibility:
by Eduardo F. Vilallonga.
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Typescript.
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Vita.

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Full Text













ENERGY TRANSFER IN MOLECULAR COLLISIONS


BY

EDUARDO F. VILALLONGA





























A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1981

































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~To my parents.
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To my parents.
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ACKNOWLEDGEMENTS


I wish to express my deepest appreciation and gratitude to my

advisor, Professor David A. Micha, for his dedication, support and

encouragement. I thank him for suggesting the study of the problems

addressed in this dissertation, for many invaluable discussions, and

for providing me many opportunities to broaden my experience as a

scientist.

I am indebted to Professor Thomas L. Bailey for introducing me

to the fascinating subject of atomic and molecular collisions. I

wish to express to him my deepest appreciation and gratitude for his

support during my first two graduate years and for his constant

interest and encouragement.

I would like to thank all the faculty members of the Quantum

Theory project and of the Department of Physics, all of whom have con-

tributed to my development as a student and as a scientist. In

particular, I thank Professors Per-Olov L8wdin and Charles F. Hooper,

Jr., for providing me the opportunities to attend the International

Summer School in Uppsala, Sweden,and the NATO Advanced Study Institute

in Cortona, Italy, respectively.

I am deeply indebted to three very special persons, my parents

and Kristin V. Bjorn. Much of the work presented here might not have

been possible without their unwavering encouragement, patience and

understanding.


iii

















TABLE OF CONTENTS


Page

ACKNOWLEDGEMENTS. . . ... iii

ABSTRACT. . . .vi

INTRODUCTION. . . .. 1

CHAPTER

I THE CORRELATION FUNCTION APPROACH . 6
1. The System Hamiltonian. . 7
2. The Differential Cross Section. .... 8
3. Time-Evolution of the Correlation Function and
Moments of the Energy Transfer. 12
4. Cumulant and Moment Expansions. ... 15
5. Short-Time Expansions . .. 17

II THE MANY-BODY CORRELATION FUNCTION APPROACH 20
1. Outline of the Many-Body Approach 22
2. Vibrational and Rotational Correlations 26
3. Cumulant Expansion of the Vibrational Cor-
relation. . . ... 28
4. Evaluation of the Displacement-Displacement
Correlation Function. . ... 34
5. Effects of Molecular Anharmonicity in Hyper-
thermal Collisions. . .. 42
6. Short-Time Approximation to the Rotational
Correlation . . 43

III MULTICENTER POTENTIAL ENERGY SURFACES .. 48
1. Single-Center vs. Multicenter Expansion 48
2. Atom-Pair Potentials for Ion/Linear Molecule
Interactions . .52
3. Multicenter Potential for the System Li /CO 57
4. Multicenter Potential for the System Li /CO2. 73












TABLE OF CONTENTS (Continued)


Page

IV ROTATIONAL AND VIBRATIONAL ENERGY TRANSFER IN
HYPERTHERMAL COLLISIONS OF Li+ WITH N2, CO
AND CO 91
1. The Atom-Pair Correlation Function of a
Diatomic Molecule. . ... .91
2. Rotational and Vibrational Energy Transfer
in Hyperthermal Collisions of Li+ with N2
and CO . . 94
3. Evaluation of the Atom-Pair Correlation
Function of a Linear Triatomic Molecule. .. 102
4. Rotational and Vibrational Energy Transfer
in Hyperthermal Collisions of Li+ with CO 105
5. Effects of Molecular Anharmonicity in N2, CO
and CO2 . . 113

V A SEMICLASSICAL CORRELATION FUNCTION APPROACH TO
MOLECULAR COLLISIONS . ... .119
1. The Semiclassical Approximation. ... 121
2. The Differential Probability of Energy
Transfer . ... .125
3. Expansion of the Intermolecular Potential in
Vibrational Displacements. ... .130
4. The Evolution Operator in the Interaction
Representation . 131
5. Vibrational and Rotational Correlation 136
6. The Vibrational Correlation. ... 138
7. The Rotational Correlation ... .140
8. Discussion . ... 145

BIBLIOGRAPHY . . ... .. 147

BIOGRAPHICAL SKETCH. . . .. 152
















Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



ENERGY TRANSFER IN MOLECULAR COLLISIONS

By

Eduardo F. Vilallonga

June, 1981

Chairman: David A. Micha
Major Department: Physics

Vibrational and rotational energy transfer in hyperthermal molecu-

lar collisions are investigated by means of time-correlation function

methods. Within this approach, the collisional differential cross

section is expressed as the Fourier transform of the time-correlation

function of the transition operator, which evolves in time in accord-

ance with the internal motions of the noninteracting molecules. The

correlation function includes from the outset the experimental averages

over initial distributions, so that only quantities directly related

to experimental measurements need be calculated. The present approach

allows the systematic introduction of approximations to the transition

operator, which are based on models of the intermolecular forces and

of the collision dynamics; the resulting correlation functions are

readily evaluated for most polyatomics without need of internal-state

or partial-wave expansions.
















In particular, a many-body description of the dynamics, together

with a multicenter representation of intermolecular forces, shows that,

at hyperthermal energies, large-angle atom-polyatomic scattering is

related to the correlation of the positions of pairs of atoms that

constitute the target. The atom-pair correlation functions are

evaluated by means of cumulant expansions and of Green-function

techniques. The multicenter representation of the intermolecular

potential is investigated, and multicenter surfaces are developed

for the interaction of Li with CO and with CO2. Combining these

developments, we calculate differential cross sections for the

vibrational and rotational excitation of N2, CO and CO2 in collisions

with hyperthermal Li+ ions. Comparing these results with those

obtained in experiments, we gain understanding of the features of

the potential surface and of the dynamics that govern energy transfer

processes.

We also develop a new semiclassical description of molecule-

molecule collisions. Whenever the relative motion of the polyatomics

may be described classically, we find that the differential probability

of energy transfer is given by the Fourier transform of the time-

correlation function of quantal transition amplitudes; the correlation

function evolves in time in accordance with the Hamiltonians of the

noninteracting polyatomics and describes the response of the molecules

to the collision. The correlation function is evaluated for large
















polyatomics by means of a procedure which includes the intermolecular

potential to infinite order. We thus obtain simple analytical expres-

sions for the differential probability of energy transfer in terms

of action integrals; these integrals correspond to the actions that

the intermolecular forces exert on the atoms of the polyatomics.


viii



















INTRODUCTION


During the past few years, there has been an explosive develop-

ment of theoretical approaches to energy transfer processes in

molecular collisions, generally in the thermal collision energy regime.

These approaches range from fully quantal (Lester, 1976), in which the

wave-function of the system is expanded in a more or less complete set

of internal vibrationall and rotational) states of the colliding

molecules, to completely classical (Porter and Raff, 1976), in which

all the degrees of freedom of the system are described by Hamilton's

equations of motion. Halfway between these two extremes lie the various

semiclassical and time-dependent methods (McCann and Flannery, 1975,

Miller, 1975, Child, 1976, Billing, 1978) that treat the relative

motion of the collision partners classically while describing the in-

ternal motions by a wave-function expanded in internal states. Ex-

tensive reviews on these various approaches can be found in the book

edited by Miller, 1976, in the monograph by Gianturco, 1979, and in

the articles by Clark et al., 1977, by Dickinson, 1979 and by Micha,

1981a.

Concurrently with these theoretical developments, the progress of

crossed molecular beams techniques (Toennies, 1976, Faubel and Toennies,

1977) has made available a wealth of new experimental information in

atom-molecule collisions at hyperthermal energies (1 to 10 eV). The

1







2

systems which have been studied in detail include, among others,

Li ions in collision with N2 and CO (Bottner et al., 1976, Eastes

et al., 1979a), CO2 and N20 (Eastes et al., 1977), CH4 (Eastes et al.,

1979b) and SF6 (Ellenbroek et al., 1979). Techniques based on time-of-

flight or velocity analysis of the scattered ions have produced de-

tailed velocity distributions of the products as functions of col-

lision energy and scattering angle. These distributions show rich

and varied structure due to the collisional excitation of the vibra-

tional and rotational states of the target.

In spite of the recent theoretical advances outlined above, these

new measurements present a challenge to theoretical interpretation.

The fully classical approaches, which can give accurate values of

average energy transfers, cannot describe the quantum-mechanical na-

ture of the target, which is clearly shown by the experiments. Further-

more, for systems of more than about three or four atoms, several dyna-

mical approximations must be introduced in order to decrease the num-

ber of Hamilton's equations which must be integrated numerically-

(Schatz, 1980). The quantal and semiclassical methods can, in principle,

provide very detailed state-to-state transition probabilities. How-

ever, they become computationally impractical for energies above .1 eV

because the number of internal states that are energetically accessible

is unmanageably large. This necessitates the introduction of rather

drastic computational approximations whose dynamical consequences are

not yet fully understood. These approximations include the orbital-

sudden or coupled states (McGuire and Kouri, 1974, Pack, 1974), the

energy-sudden (Khare, 1977 and 1978), the infinite-order-sudden

(Secrest, 1975, Goldflam et al., 1977) and the effective potential methods










(Rabitz, 1976). Furthermore, the ultimate test of accuracy of a

theoretical model lies in a successful comparison with experimental

results, which usually correspond to averages over initial distributions

of internal states. The theoretician is forced to question the effi-

ciency of state-expansion approaches when he is faced with the prospect

of averaging over coarse experimental distributions the detailed re-

sults so laboriously obtained.

A complete description of molecular collision processes also re-

quires knowledge of the potential energy surface of the colliding part-

ners. Except for a limited number of small systems, accurate quantum-

chemical calculations of potential energy surfaces are beyond the reach

of present-day computational resources. This does not present a serious

difficulty for the description of thermal collisions, because they are

mediated mostly by the long and intermediate range forces. These

forces can be accurately determined by combining the results of per-

turbation expansions (Hirschfelder et al., 1967, Chap. 13) with ex-

perimental measurements of the molecular properties (electrostatic multi-

pole moments, polarizabilities, etc.) of the separate colliding partners.

However, as the collision energy increases above a few tenths of an

electron-volt, the projectiles probe the inner regions of the potential

surface. At short distances, the usual perturbation expansions diverge,

and accurate models for the forces have not yet been found.

We thus find that the description of hyperthermal collisions in-

volving polyatomics presents a double challenge to the theoretician,

due to the lack of knowledge of the intermolecular forces involved,

and due to theinapplicability of the existent dynamical models. The










purpose of this dissertation is to develop and apply new theoretical

approaches to the study of vibrational and rotational energy trans-

fer in molecular collisions. We will consider in particular the

range of collision energies above 1 eV but below the threshold for

electronic excitation or dissociation of the colliding partners.

In view of the previous discussion, these approaches must satisfy the

following requirements:

1) The internal motions of the colliding molecules must be

described quantum mechanically.

2) Expansions in internal states must be avoided; instead, we

will focus our attention on the many-body nature of the system.

3) The new models should be interpretable in terms of molecular

forces and dynamical processes so that we can gain understanding of the

physical mechanisms involved.

4) The models should allow us to include averages over initial

distributions of internal states without greatly increasing compu-

tational costs, in order to permit comparison with experimental measure-

ments.

During the course of our investigations, we have found that the

first two requirements lead us to express the collisional differential

cross section in terms of the time-correlation function of the transition

operator, as is shown in Chap. I; this function includes from the out-

set the averages over initial states, so that only quantities directly

related to experimental measurements need be calculated. The present

approach allows us to systematically introduce approximations to the

transition operator, which are based on different models of the inter-

molecular forces and of the collision dynamics. In particular, a many







5



body description of the dynamics, together with a multicenter

representation of the forces, shows that the large-angle scattering

of hyperthermal projectiles is related to the correlation of the

positions of pairs of atoms of the isolated target (Micha, 1979a). In

Chap. II, we study these atom-pair correlation functions and extend

them to include the effects of anharmonicity in the vibrational motion

of the polyatomic. In Chap. III, we investigate the multicenter re-

presentation of intermolecular forces and develop multicenter potential

surfaces for the systems Li /CO and Li /CO2. Next, combining the de-

velopments of the previous two chapters, we calculate differential cross

sections for the vibrational and rotational excitation of N2, CO and CO2

in collisions with hyperthermal Li ions. By comparing these results

with the experimental measurements (Bottner et al., 1976, Eastes et al.,

1977), we gain understanding of the relative importance of the features

of the potential surface and of the dynamics that govern energy transfer

processes. Finally, in Chap. V, we develop a new semiclassical de-

scription of molecular collisions. Whenever the relative motion of

the molecule may be described classically, we find that the differential

cross section is related to the time-correlation of quantal transition

amplitudes that evolve in accordance with the internal Hamiltonians of

the isolated molecules. This correlation function completely describes

the response of the molecules to the collision; furthermore, it is

readily evaluated for a wide range of polyatomics, without need of in-

ternal-state expansions.




















CHAPTER I
THE CORRELATION FUNCTION APPROACH


We consider collisions between two arbitrarily large molecules,

A and B, each having closed electronic shells. In order to single out

the mechanisms responsible for rotational and vibrational energy trans-

fer, we consider the range of collision energies below the thresholds

for electronic excitation and dissociation of the molecules. We

are thus interested in process of the type

A(a) + B(B) + A(a') + B(3').


Initially A and B are in the internal states (rotational and/or

vibrational) denoted by the sets of quantum numbers a and B, re-

spectively. During the collision an amount of energy is transferred

between them into internal motions, promoting A and B to their final

internal states, which we indicate by a' and B'.

For the system Hamiltonian specified in Sec. 1, doubly-differential

cross sections may be related to the time-correlation function of the

collisional transition operator, as is shown in Sec. 2. In the fol-

lowing section, we examine the evolution of the correlation function and

relate its time-derivatives at time equal zero to the moments of the

energy transfer. In Sec. 4, we introduce the method of cumulant ex-

pansions which provides a powerful tool for evaluating the correlation

functions. Finally, in Sec. 5, we show that internal motions that are


1











slower than relative ones lead to a Gaussian distribution of energy

transfer.

Although we focus our attention on molecular scattering, we should

emphasize that the analysis presented in this chapter applies, quite

generally, to collisions between any two many-body systems whose

interactions can be described by a single potential energy surface.

These include, for example, scattering of molecules from surfaces of

solids.


1. The System Hamiltonian

In the coordinate system fixed to the center of mass of the

pair (A,B), indicating operators with carats, the Hamiltonian of the

collision is

H = K + HM + V. (1.1.1)


Here, K is the operator for the kinetic energy of relative motion

given by
2 2
K 2M 2 (1.1.2)
2M -2
@R

where I is the position of the center of mass of the projectile A

with respect to that of target B, M = MAMB/ A+ MB) is the reduced

mass of the system with MA and M the masses of the respective molecules.

The Hamiltonian that describes the internal motions of the separated

molecules, HM, is the sum of the internal Hamiltonians of each iso-
A
lated molecule (HA and H B respectively). The operator V represents
+-M
the intermolecular potential which generally depends on R and on r =
-A -B +X
{r r 1, where r denotes the positions of the atoms of molecule X =

A or B with respect to the center of mass of X.










The motion of the noninteracting molecules is described by

(K + HM) kv >= [2k2/(2M) + E ] >,


MiM -3/2 ik-R --Mi



-*M -A -B
< r 1v> = ,


< r Iv>,


(1.1.3a)


(1.1.3b)


(1.1.3c)


in which ik is the relative momentum, v represents the collection of

vibrational and rotational quantum numbers of both molecules,

E = E + EB, E. and E are the energies of the internal states

Ia> and 1[> of A and B, respectively.


2. The Differential Cross Section

The scattering rate or transition probability per unit time be-

tween the free states defined in eqs. (1.1.3)is given by(Newton, 1966,

chap. 8)

R(k'v' + 1) = (2/) < '' T(+) (E) I >, 12. 1


where final and initial quantities are indicated by primed and unprImed

22
symbols, respectively, and E = 2 k /(2M) + E is the total energy of the

"(+
system. The transition operator for the collision, (+), satisfies the

Lippman-Schwinger equation for outgoing-wave boundary conditions


^(+) ^ ^(+) (+)
T (E) = V + VG (E)T(E),
o
^(+)
where G is the propagator for the free motion given by
o
a(+) -1
Go (E) = (E + in K HM) .


(1.2.2)


(1.2.3)


Experimental measurements usually correspond to thermal averages over
(X)
initial distributions of internal states with w(M (T ) with
ct X


\ ... .











temperature TX for each collision partner X = A or B. Furthermore,

experiments usually do not select the final internal states of the

molecules. In these cases, we must consider the total rate for scat-

tering between initial and final moment,


R ot( k) = (27r/%) w (T ,T) I<''~'I )(E)Ik>2 6(E E ) (1.2.4)
tot tV Ad B

where the delta function insures conservation of the total energy

of the system.

Typical rotational excitation energies of most molecules (except

those that contain several hydrogen atoms) are of the order of a few

thousands of an electron volt. At hyperthermal collision energies, the

sum over final states in eq. (1.2.4) contains a very large number of

terms, even for small values of energy transfers. This also occurs in

atom surface-scattering due to the low values of phonon excitation

energies. However, the sum over final states may be formally elimi-

nated by the following procedure (Micha, 1981a).

Introducing the integral representation of the delta function,
00dt
6(E E') = 2B exp[-i(E E')t/Y], (1.2.5)


into the matrix elements of the transition operator of eq. (1.2.4) and

using eq. (1.1.3a) yields


= fdtexp(-iEt/) -- 2 rrt
x exp(-ifHt/) M kv> 6
(1.2.6)









where the energy transfer is given by = 2 [k2 (k') 2]/(2M).

The matrix element of eq. (1.2.6) implies an integration over the
M
relative coordinate R and over the internal coordinates r Before

proceeding further it will prove convenient to explicitly factor

these two integration by defining the operator


T-* = , (1.2.7)
k'k

which acts only on the internal coordinates of the molecules. Sub-

stituting eq. (1.2.7) into eq. (1.2.6) and the latter into eq. (1.2.4),

the sum over final states may be eliminated by means of the com-

pleteness relation of the internal states Iv'>. One thus arrives

at the following expression for the total scattering rate,

+ 4 m dt ^ t ^dt
Rtot(k',k)=(27/%) 4 2- exp(-ict/M) <>,
-oo k'k k'k
(1.2.8)

where the double brackets indicate the quantum mechanical and

thermal averages over initial states,

<<* .>>= w(TA) wB) (TB). (1.2.9)
a,B
The time dependence of the transition operator is given by the

Hamiltonian of the noninteracting molecules (indicated explicitly by

the subscript M), as shown below


T (t)M= exp(iHMt/M) Tk, exp(-iMt/). (1.2.10)










The doubly differential (energy-angle) cross section for

scattering into a unit solid-angle Q accompanied by a transfer of

energy E may be calculated from the ratio of the transition rate to

the incident flux (Micha, 1979a), to give

do (22/ i)4 2 0 dt
dE-= (27T/4 M2(k'/k) f 2 exp(-iEt/)<>


(1.2.11)

where the final momentum is related to the energy transfer by
2 2 '
k'=(k2-2Me/2 )4. The cross section is thus expressed as the

Fourier transform of the time-correlation function of the transition

operator, with the time dependence determined by the internal motions

of the isolated molecules. Approximationsto the cross section in

terms of correlation functions have been previously used to describe

scattering of cold neutrons from solids (Marshall and Lovesey, 1971)

and from liquids (March and Tosi, 1976), and atoms from polyatomics

(Micha, 1979a and 1981a) and from solid surfaces (Micha, 1981b,c).

The present expression is more general than these models and is

therefore applicable to a much wider range of many-body systems.

The direct evaluation of the time-correlation function of eq.

(1.2.11) is as difficult as finding the solution to the many-body

problem which is implicitly contained in the transition operators.

At first sight, it seems that the sums over final states have been

removed at the expense of introducing an added time dependence (due

to the complicated internal molecular motions) into the already

intricate transition operators. However, the correlation function

approach allows the development of dynamical approximations to the










transition operator in a systematic way, as is shown in the following

chapters. These approximations, based on the relevant features of

the intermolecular potential surface and on the different types of

molecular motions, will allow us to understand the physical mechanisms

that govern energy transfer processes. The correlation function

approach has the additional advantage of describing the collision

in terms of time dependent Heisenberg operators which may be easily

interpreted in terms of the classical dynamical variables of the

colliding molecules.


3. Time-Evolution of the Correlation Function and
Moments of the Energy Transfer

Other quantities usually measured by experiments are the moments

of the energy transfer. These measurements are related to quantities

that can be calculated theoretically, in the following way. For

given values of the initial and final moment, the nth order moment-

average of the energy transfer with respect to the cross section is

defined by

=[f ded2/(dQdF)]-1 J de En d2a/(dde) (1.3.1)
-CO -CO

For the sake of brevity, we introduce the following notation for

the correlation function,

S(t) = <>, (1.3.2)
k'k k'k k'k M
and for its Fourier transform,
r^ 7dt
S ()= exp(it/$) C (t), (1.3.3a)
kCk -(t k'k

C. (t) = f de exp(ict/%) C, (E), (1.3.3b)
k'k










and we drop the subscripts k and k' while keeping in mind that the

following derivations correspond to fixed values of the initial and

final moment. Differentiating eq. (1.3.3b) with respect to time

shows that the moments of the energy transfer are related to the time

derivatives of the correlation function by


n (-i1)n 3n C(t) (1.3.4)
C(0) atn t=-0
and to the transition operator by

n t ^ n ^ ^t ^ -1
= <<(0)([H ,) T(0)J>><< (0)T(0)>> (1.3.5)



through differentiation of eq. (1.2.10). This last equation pro-

vides the bridge between calculated values and experimental measure-

ments.

However, the moments of the energy transfer, by themselves,

do not provide information about the cross section in the most

practical way. This can be seen by expanding the correlation

function in a Taylor series in time, about t=0, and equating the

time-derivatives to the moments of the energy transfer by eq. (1.3.4).

One thus obtains an expansion for the cross section in moments of the

energy transfer, but each term of this series diverges because it

contains the Fourier transform of tn. The origin of this divergence

is further clarified by expressing the time dependence of the cor-

relation function, as follows (1).

Differentiating eq. (1.2.10) with respect to time, yields

Heisenberg's equation of motion for the transition operator,

T (t)M i [H MT(t) M] (1.3.6)
M- M M
Dt i

1) The following derivation is based on class notes of Prof.
D. A. Micha.










We now define the Liouville superoperator for the internal motions,

L ,by its action on any arbitrary operator A which acts on the in-

ternal coordinates of the molecules; this is given by

LM = [H A], (1.3.7a)


and, more generally,

L n A= ([HM,)nA]. (1.3.7b)
M M

The time evolution of the transition operator may now be rewritten

as

DT (t)M i L T(t)M (1.3.8)
M= MM

which has the formal solution

T(t)M = exp(iLMt/A) T (0). (1.3.9)


Replacing eq. (1.3.9) into eq (1.3.2) yields the following ex-

pression for the correlation function,

C(t)= C(O)<><>. (1.3.10)
c~t) C(O~

The last two factors in the above equation are formally equivalent

to the average of the operator exp(iLMt/%) over the "scattering"

states defined by Iv>S=T(0) v>. Denoting this average by the sub-

script S, we write

C(t) = C(0)<>S. (1.3.11)


A comparison of eqs. (1.3.4) and (1.3.8) shows that


= <>S;


(1.3.12)










therefore, expanding the cross section in moments of the energy trans-

fer is equivalent to an expansion of the exponential

exp(iLt/i) in powers of the exponent. We are not surprised that

this expansion diverges when it is applied for all values of time.

In order to take full advantage of the theoretical and experi-

mental information provided by the moments of the energy transfer,

we must introduce the cumulant expansion methods which have been

widely used in statistical mechanics (Kubo, 1962, Munster, 1969).

These techniques will also provide us with powerful tools for

evaluating the correlation functions that we will encounter in the

following chapters.

4. Cumulant and Moment Expansions

For any operator A and complex number x, the average of

exp(xA) is defined in terms of the moment averages of A, <>, by


<>= x x<>/n!. (1.4.1)
n=0

Truncating the above series to a finite number of terms might give a

very poor (even divergent) approximation to <> for some

values of x. We look for an alternate expansion which will hope-

fully give a better approximation when truncated. In particular, we

write

<>= exp xn< n=l
where the <> are certain averages of the operator A called

cumulant averages (Kubo, 1962), which are implicitly defined in terms

of the moments by eqs. (1.4.1) and (1.4.2).







16

An explicit relation between a cumulant of a given order and the

moments and cumulants of lower orders may be obtained by the following

procedure (Munster, 1969). Differentiating eqs. (1.4.1) and (1.4.2)

once with respect to x and equating their right hand sides yields

00 00
Sxnl<>/(n-l)!= xn-l An->><
>/[(n-j)!(j-1)!],
n=l n,j=l (1.4.3)


where the equality follows by changing the dummy index of

summation n to n-n+ j. Comparing powers of x in the last equation

and solving for <>C gives the desired relation,
n-1
n ^n n-1 ^n-j ?j
<< >>= <
>- < >><>C. (1.4.4)
j=l

For example, the first four cumulants are

<
> = <>, (1.4.5a)

2 2 2
<
>-<> (1.4.5b)

3 ^3 ^ ^2 3
<
>-3<><>+ 2<> (1.4.5c)

^4 ^4 ^3 ^ 2 2 2 4
<
> = <>-4<><>+ 3<>(3<> -<>)- 6<>.

(1.4.5d)
In particular, letting x = it/j and A =L M, we can express

the correlation function of eq. (1.3.11) in terms of the cumulant

averages ofLM, as follows:

C(t) = C(0) exp (it/M)n< CS/n!. (1.4.6)
n=l
In analogy with eq. (1.3.12) we define the cumulant averages of the

energy transfer by CS which are related to the moment

averages as shown below


n-i
n < n> n n-> > (1.4.7)
< C =l -1
j=1










Finally, substituting eq. (1.4.6) into eq. (1.2.11), we obtain
2 m m
d = (2/) M2(k'/k)C(O) f t exp[-iet/(+ (i t/l n> /n!].
n=l
(1.4.8)
For given values of the initial and final moment, this last

equation provides a convenient parametrization of the cross section

in terms of the averages of the energy transfer. As the cumulant

averages correspond to differences between numbers of the same

order of magnitude, the cumulant series will converge very rapidly.

Furthermore, the above expression allows us to obtain well-behaved

approximations to the cross section by truncating the series at any

term of even order which is negative. The significance of these

approximations is considered in the following section.


5. Short-Time Expansions

We now apply the techniques introduced above to systems in which

the transfer of energy occurs in a time interval much smaller than

the typical periods of the internal molecular motions.

For example, even at collision energies below the threshold for

vibrational excitation, the rotational motion of most molecules is

much slower than the relative motion of the colliding partners (Micha,

1979b). This is also the case in atom-surface scattering at hyper-

thermal energies because the vibrational periods of phonons are much

longer than collision times (Micha, 1981c). In these cases, the cor-

relation between the two transition operators that determine the cross

section is significant only during a short period of time. We are










then justified in retaining only the lowest powers of t in the

cumulant expansion of the correlation function and writing


C(t) Z C(0) exp[it/1 t2/(2%~2)]. (1.5.1)


The applicability of the short-time approximation to a specific

system can be judged by calculating or measuring the cumulants which

have been omitted in the above equation. Alternatively, we can

systematically develop improved approximations by including the

higher order cumulants.

Substituting the correlation function of eq. (1.5.1) into

eq. (1.2.11) and evaluating the Fourier transform (Churchill et al.,

1974), we obtain the following expression for the cross section,
2
da 2 1- 2 2
d2d I(Q)(27C)-2 exp[-(e-<>) 2/(2< >C)], (1.5.2a)


I(0) = (2/M) 4 M2 (k'/k) C(O). (1.5.2b)


This result shows that collisions in which the transfer of energy

is much faster than the internal motions are characterized by a

normalized Gaussian distribution in the energy transfer. This holds

true for all scattering angles, regardless of the nature of the

collision partners. The distribution is centered about the average

energy transferred in the collision, , and broadened by an amount
2 2 2
C= (< >- <6> ) 2 The nature of the colliding species deter-

mines the values and the dependence on scattering angle of the in-

tensity I(Q) and of the location and width of the distribution.

These conclusions are supported by several experimental measurements

of atom-molecule (Bottner et al., 1976, Eastes et al., 1977 and

1979a,b) and atom-surface (Micha, 1981c) collisions.











We should emphasize the differences between the short-time and

the various sudden approaches which have been widely used to describe

molecular scattering. The latter are based on different approximations

to the dynamics by which the energy is transferred. In contrast,

throughout the present derivation, we have made no approximations

to the dynamics of the collision. The intensity, location and width

of the distribution of eqs. (1.5.2) depend on the exact transition

operator. In particular, the coordinate-sudden approach (also called

fixed-nuclei and adiabatic, see Chase, 1956) assumes that the in-

ternal degrees of freedom are frozen during the duration of the

collision. In the short-time approximations these degrees of free-

dom evolve in time, albeit slowly, as can be seen in eq. (1.5.1).

On the other hand, the energy-sudden (Khare, 1977 and 1978) and

infinite-order-sudden (Secrest, 1975, Goldflam et al., 1977) methods

fix the internal energy of the molecules throughout the collision;

in the short-time approximation, the internal energy can and does

change during the collision, as prescribed by the exact transition

operator.

















CHAPTER II
THE MANY-BODY CORRELATION FUNCTION APPROACH

In this chapter we develop and study the time-correlation

functions that are appropriate to describe hyperthermal collisions

in which a light projectile is deflected into large scattering

angles. In order to maintain the formalism at the clearest level

possible, we take the projectile A to be a structureless atom (neutral

or ionized), while the target B remains a general N-atom polyatomic.

Furthermore, we focus our attention on scattering processes in which

the amount of energy transferred is considerably smaller than the

relative collision energy.

The standard parameterizationof atom-molecule potential energy

surfaces is based on an expansion in a basis-set of functions of

angles centered at the center of mass of the target (see Staemler, 1975,

Parker et al., 1976 and Thomas et al., 1978 for examples). However,

when the collision energy is much larger than the typical well-depths

of the potential, large-angle scattering is due to the deflection of

the projectile from the steeply repulsive inner region of the surface

(McDowell and Coleman, 1970, Chap. 1), which is of a multicenter

nature. In these cases, the potential is better represented by ex-

panding about the atomic centers of the target, as follows (Micha,

1979a):

V(R,r ) =Iv (R,r ). (2.1)
a










Here, R is the position of A with respect to the center of mass

of the target and r ={ r a = 1 to N} is the collection of the
a,
coordinate vectors of the atoms that constitute the polyatomic,

referred to the center of mass of B. The two-body potential v
a
represents the interaction between the (A,a) atom pair and depends

on the electronic distribution of A and that of the valence state of

atom a in the molecule. In general, at short distances, the pair

potentials are steeply repulsive due to the overlapping of the

electronic clouds, while in the long-range region they fall off as

inverse powers of the relative distance between A and a. We post-

pone a detailed study of the pair potentials to a later chapter and

concentrate here on the collision problem.

The many-body representation of the potential shown above

suggests that atom-molecule scattering processes should be inter-

preted in terms of the dynamics of a many-body system; this approach

has resulted in the recent development (Micha, 1979a) of a quantal

many-body theory of atom-molecule collisions. Such a description,

briefly reviewed in Sec. 1, leads to a simple model for vibrational

and rotational energy transfer processes, in which the cross section

is expressed in terms of atom-pair correlation functions of the

isolated target molecule. These correlation functions can be readily

evaluated for polyatomics whose vibrational motions arise from har-

monic forces (Micha, 1979b). This model has been successful in de-

scribing the large-angle scattering of Li+ from N2 and CO (Micha

et al., 1979) and from CO2 (Vilallonga et al., 1979). However,

recent quantal calculations for the system He/CO2 indicate that










molecular anharmonicity may considerably influence vibrational energy

transfers at thermal collision energies (Clary, 1980). Therefore,

we extend the atom-pair correlation functions to encompass anhar-

monic force-fields, so that we can ascertain the effects of molecular

anharmonicity in hyperthermal collisions. In order to focus our

attention on the vibrational dynamics of the target, in Sec. 2 we

decouple the vibrational and the rotational correlations from each

other. By means of the cumulant expansion methods introduced in

Chap. I, we express the vibrational correlation in terms of the dis-

placement-displacement correlation functions, as shown in Sec. 3.

These simpler functions are readily obtained from the corresponding

double-time Green functions (Zubarev, 1960), which in turn may be

evaluated from their hierarchy of equations of motion. In Sec. 4,

this set of coupled differential equations, which is of infinite

size, is decoupled by a linear procedure that contains anharmonic

forces to infinite order. From this development, we conclude that

anharmonic intramolecular forces cause a shift in the energy transfer

spectrum and in the cross sections for vibrational excitation, as dis-

cussed in Sec. 5. Finally, in Sec. 6, we complete the evaluation

of the atom-pair correlation function, by means of a short-time

approximation to the rotational motions.


1. Outline of the Many-Body Approach

The many-body transition operator presented in eq. (1.2.2) can

be expressed in terms of the two-body potentials va of eq. (2.1), by

means of the formalism of multicenter scattering (Rodberg and Thaler,

1967, chap. 12). This leads to the final-channel decomposition










( (a)
T = ) (2.l.a)
a

^(a) A (b)
T(a) = + G T(b) (2.1.1b)
a a o
b#a
= v + v G T (2.1.1c)
a a a o a
A(a)
where (a) describes the process in which the last interaction is

mediated by va, while in T the only interaction is through v In
a a a
particular, iterating eq. (2.1.1) gives the multiple-collision ex-

A(a)
pension 2.1.2 that describes the operator T(a) in terms of successive

collisions between the projectile and each atom of the target.

Two assumptions (Micha, 1979a), that are appropriate to ex-

perimental conditions of high relative energies and large scattering

A(a)
angles, greatly simplify the evaluation of (a). When the wave-

length for relative motion is much shorter than the distance be-

tween scattering centers, one may assume that only single collisions

between the projectile and each atom of the target are significant, so

that T T In order to calculate T the N + 1 body problem must
a a
still be solved because the propagator G involves the motions of all

the atoms of the target. However, substantial simplifications occur

whenever the energy transfer is impulsive; this constitutes the

second assumption.

In an impulsive collision between the projectile A and target

atom a, a large force acts on a for a very short period of time. The

position of a does not change but its kinetic energy in the initial

state of B jumps to the new value that corresponds to the final

state the molecule. Energy and momentum are transferred to the target

through the interaction between the pair (A,a), while the remaining

N-l atoms only provide the restoring forces on a that determine its






24

momentum distribution within the polyatomic. When the collision

energy E is much larger than the energy transferred e, the part of

T that acts on the internal coordinates ofB is given by (Micha, 1979a)
a

= T (k',)exp(iK r ), (2.1.3a)
a a a

T ( ',k) = < |t (E ) > (2.1.3b)
a a a a a

Here, T is the transition amplitude for the collision of the atom
a
pair (A,a) evaluated at their relative kinetic energy E = m E/(M + m )
a a a
and relative momentum MK = ikm /(M + m ), and t is the two-body
a a a a
transition operator for the potential v ; whenever e< a a
independent of the internal state of the polyatomic. As a result of

the collision, atom a at position r absorbs an amount of momentum
a


Under these assumptions, the cross section of eq. (1.2.11)

factors into quantities that depend on two different types of motion,

as follows:
2
dS= (27/T) M2(k'/k) I Tb (t',k)T (k',)S(ba)(,E), (2.1.4a)
a,b

S (ba) ,) = exp(-ict/j)< r (2.1.4b)
The first factor, T (V',k), involves only the relative motion of the
a
(ba)
pair (A,a). The second factor, S (K,E), is the Fourier transform

of the atom-pair time-correlation function (Van Hove, 1954) and de-

pends only on the internal dynamics of the isolated target.

The correlation function for a pair of different atoms (a/b) con-

tains phases that depend on the initial values of the internal dynamical

variables of the target. Experimental measurements usually correspond










to initial averages over random phases so that the terms of eq. (2.1.4b)

that have afb average to zero. In these cases, the cross section is

given by
do 2 a (aa) +
d = a (k ,), (2.1.5a)
d~dE ak
a

S(,k) = (2/1)4 M (k'/k) IT (k',ik) 2 (2.1.5b)
a a

where a is an effective two-body cross section for the (A,a)-pair

collision. Hence, the contribution of atom a to the overall cross

section may be interpreted as the product of the probability that a

will deflect the projectile from k to k', given by a multiplied by

the probability that the molecule will absorb energy and momentum

through this atom, given by S(aa). The self-correlation function S(aa)

(K<,) completely describes the response of the target atom a to the

transfer of momentum %K and energy e in the impulsive collision,

regardless of the nature of the projectile atom.

The two-body cross-sections oa are, in principle, off-energy-

shell quantities that must be obtained from the T These can be
a
evaluated using any of several well-known techniques which need not

be repeated here (see, for example, Brumer and Shapiro, 1975, Beard,

1979 and Kuruoglu and Micha, 1980). However, for the experimental

conditions of quasielastic scattering,I t lt'| so the a can be

obtained from standard two-body elastic cross sections.

Due to the factorization of internal from relation motion just

described, from here on we need to consider only the internal dynamics

of the target molecule. In order to focus our attention on the

vibrational dynamics, it will prove convenient to decouple rotational

and vibrational motions, as follows.











2. Vibrational and Rotational Correlations

Working in the center of mass of the isolated polyatomic, for

each atom a of B we introduce its equilibrium position d and its
a
displacement from equilibrium u so that r = d + u The con-
a a a a
editions of zero total linear and angular moment lead to the con-

straints (Wilson et al., 1955, Chap. 11)


m = a (2.2.1a)
a


m u xd = (2.2.1b)
aa a
a

which introduces N tr=5 or 6 relationships among the vibrational dis-

placements for linear and nonlinear molecules, respectively. This

leaves NV = 3N-Ntr free vibrational coordinates in the body-fixed (BF)

reference frame. The atom-pair correlation function (APCF) can now

be written as


(aa) + + + +^
F (K,t) = <>VR
a a a R a M VR
(2.2.2)
For most polyatomics (except those that contain several hydrogen

atoms), rotational motions are much slower than vibrational ones, in-

dicating that the vibrational and the rotational correlations should

be studied separately from one another (Micha, 1979b). This can be

done by writing the internal Hamiltonian as H = H + HR and re-
M V R
quiring that the vibrational (H ) and rotational (H ) Hamiltonians
V R
satisfy [H ,H ] = 0. This means that [H ,u ] a 0 because the u
depend on the orientation of the molecule with respect to the
depend on the orientation of the molecule with respect to the










spaced-fixed (SF) coordinate system. These commutation relations

cause the APCF to factor as follows,


(aa) ^(aa) .
(K,t) = exp[-iT-d (0)] F (K,t)exp[i-1a (t)R]>> (2.2.3a)
a V a R R

(aa) +t + +t +i +
F (K,t) = <>V. (2.2.3b)
a a M V

^(aa)
The vibrational correlation of (aa) can now be calculated in the
V
BF frame, in which case it depends on the set of Euler angles

r ={a, B, y} that determine the orientation of this frame with re-

spect to the SF one. These angles vary in time in accordance with

the rotational Hamiltonian of the polyatomic. However, when this

motion is much slower than the vibrational one, we need only con-

sider the vibrational correlation for fixed orientation of the

molecule. Hence, within the vibrational APCF of eq. (2.2.3b) we may

approximate r(t) = F(0), so that u (t) u (t) Evaluating the
R a Mi a V
time derivatives of eq. (2.2.3b), one can easily show that this is

equivalent to neglecting the rotational energy transferred through

the torques that act on the vibrational displacements; this is

justified whenever u l< a a
citations well below the dissociation thresholds.

The effects of anharmonic intramolecular forces are now entirely

contained in the vibrational APCF of eq. (2.2.3b). For harmonic force-

fields, this function may be readily evaluated by obtaining the

u (t) from second quantization methods (Micha, 1979b). In contrast,
a V
anharmonic forces introduce two main difficulties into the evaluation

of the APCF. Firstly, the time-evolution of anharmonic motions











usually cannot be solved in a closed form. However, the evolution
4-
of the operators u is of central importance, because they determine.
a
how energy is absorbed by and distributed within the polyatomic.

Therefore, we must develop approximations to the vibrational dynamics

that contain the features of the evolution of the u which are most
a
important to the collisions of interest; this is the subject of

Sec. 4. The second difficulty resides in the calculation of quantal

and statistical averages of the complicated exponential operators;

however, these averages may be systematically approximated in the

following way.


3. Cumulant Expansion of the Vibrational Correlation

In order to abbreviate the notation, let xt be the component of
-t4-+
u (t) in the direction of the momentum transfer K, so that

K *u (t) =KX and omit the subscript V from the vibrational time
a t
dependence and from the average over initial states. The evaluation

of <> is complicated by the fact that, in general,



exp(-iKx ) exp(iKk t)exp[-iK(x0-xt)], (2.3.1)


because [ ,H ] # 0. We introduce the superoperator 0 which, when

acting on a product of powers of x0 and xt, orders all the powers of

x0 to the left of all the powers of xt, i. e. Oxtx0 = x0xt. The

APCF may now be written as


F (aa) t) = <>, (2.3.2)










where the average of the ordered exponential operator is defined in

terms of the moment-averages of the ordered exponent, as follows:


<> = (-iK)n<<(x )n>>/n!. (2.3.3)
n=0
From studies of the scattering of cold neutrons from crystals

(Marshall and Lovesey, 1971), it is well known that the terms of

eq. (2.3.3) which contain the product x0t, with A + m 1 = n,

describe the simultaneous n-tuple excitation of the vibrational

states of the target. In these cases, only single excitations are

usually observed so the APCF is approximated by the second moment.

However, in atom-polyatomic collisions, multiple excitations are

readily observed in the experiments mentioned in the Introduction.

Hence, it will prove more convenient to express the APCF in terms of

the cumulant averages of the exponent, << (x0-x )n>>, as follows

(aa) exn <<(^ ^ /nn>>c.
F( (K,t) = exp (-iK)n<(x0 ) > /n!. (2.3.4)
V n= t C
n=l

Comparing the coefficients of powers of (-iK) in eqs. (2.3.3,4),

as done in Chap. I, Sec. 4, shows that the cumulants are related to

the moment averages by
n-1

<(0-t )n>C = <> <<(x0-xt n- (x t)>> C
j=1
(2.3.5)
Due to the invariance of the trace of a product of operators with re-

spect to a cyclic permutation of their order, <>=<>, so the
O tf
first few cumulants are










<>C = 0, (2.3.6a)

2 r2
<<(x0-xt) C = 2(<< >>-<>) (2.3.6b)

3 n -2
C = xt>>-<)' (2.3.6c)

4 4 2 2
<<(x0 >>C = <<(xxt) >>-<<(x-t)>> (2.3.6d)

Equations (2.3.4,5) express the APCF in terms of the simpler dis-

placement-displacement correlation functions (DDCF) of the type
9Lm
<> with Z and m integers.

By means of a theorem due to Bloch (Messiah, 1961, Chap. 12),

one can readily show that, for harmonic motions, all the cumulants of

order higher than two are identically zero. Therefore, these cumu-

lants are explicitly proportional to the strength of the anharmonic

forces. However, in the usual experimental distributions of initial

states, only the lowest vibrational states are significantly popu-

lated. In these cases, the displacements are small and vibrational

motions are approximately harmonic, so that we may write

(aa) 2
F() (,t) Z exp{-<<[i.a (0)] >>+<<[Z-. (0)][ (t)]>>}. (2.3.7)
V a a a

If necessary for highly excited targets, this approximation may be

systematically improved by including the higher order cumulants in-

dicated in eq. (2.3.4). We note that the above equation still con-

tains anharmonic forces to infinite order, within the time dependence

of the u and within the average over initial states. Furthermore,
a
eq. (2.3.7) includes multiple excitations, as can be seen from the

power-series expansion of the exponential function.










We must now stress a very important point of the present de-

velopment. The cumulant series of the APCF corresponds to an ex-

pansion of the operator Oexp[-iK(x0-xt)] in terms of its exponent;

this is not equivalent to expanding the vibrational time-evolution

operator exp(iLVt/l) in powers of time. Therefore, eq. (2.3.7)

does not correspond to a short-time approximation to the vibrational

motions. In this case, a short-time approximation would not be valid

because vibrational periods are comparable to collision times; this

is further confirmed by the rich vibrational structure observed in

the experiments, which cannot- be described by the Gaussian distri-

butions characteristic of slow internal motions.

Before proceeding to the evaluation of the DDCF, the vibrational

Hamiltonian must be specified. The potential is usually given in

terms of the internal coordinates of the molecule s., i=l to N

(Wilson et al., 1955, Chap. 8), which are related to the cartesian

displacements (Ua) E=x,y,z,by the linear transformation


(u) = a~(r)s.. (2.3.8)


Here, the coefficients Ca i depend on the orientation of the BF frame

with respect to the SF one. Using eq. (2.3.8), the kinetic energy

of vibration can be expressed in terms of the internal coordinates, as

follows:


K 1 i Ds m. s. (2.3.9a)
V 2 -i j J
i,j

m = mac a.c a.. (2.3.9b)
ij a a a
a S










We take the vibrational potential V (s) to be a general function of

the s. which satisfies the conventions V (0) = 0 and (3V /s) =

Wilson et al., 1955, Chap. 2). In order to single out the effects

of anharmonicity, we separate VV into a harmonic part Vh and an an-

harmonic one Vanh, by writing


VV = Vh + Vanh, (2.3.10a)


V = s.k..s./2, (2.3.10b)
h J a3
i,j

k 2 V (2.3.10c)
ij 9s.Ds |s= .
1 J
Using eq. (2.3.8) we could express <<[-. U (0)][K U (t)]>>
a a
in terms of the correlation functions of the internal coordinates and

then proceed to evaluate the latter. However, in order to facilitate

a comparison of the results of this study with those of harmonic vi-

brations, it will prove convenient to work with the set of coordinates

that diagonalize the harmonic part of the Hamiltonian. These co-

ordinates, denoted by Qj, j=l to NV are defined by the orthogonal

transformation (Marion 1971, Chap. 13)


s = ajQ, (2.3.11)


where the coefficients a.. satisfy the set of linear equations
1j

S(ki- r )ai = 0, (2.3.12)
i ] j

with the normalization condition

ai j mi a = i, (2.3.13)
i,R










and with the frequencies w. given by the roots of the characteristic

equation


det{ki. w2 i }= 0. (2.3.14)
iit

Substituting eq. (2.3.11) into eqs. (2.3.9,10) yields for the

vibrational Hamiltonian


V= Hh + Vanh(Q), (2.3.15a)
o2 h 2a
S= (P + )/2, (2.3.15b)
i

where P. = -i3/QQ. is the canonical momentum conjugate to the co-
1 1
ordinate Q.. Combining the transformations defined in eqs. (2.3.8,11)

we write


S= .(r)Qi, (2.3.16a)
a ai i


( ai) = CaZ ()a i (2.3.16b)


hence, the APCF of eq. (2.3.7) becomes


F(aa)(,t) = exp{ (K- C .)(. C ai)[->+
V aj at j
i,j

+ <>]}. (2.3.17)


The evaluation of the displacement-displacement correlation

functions < is the subject of the next section. We

emphasize, however, that the methods presented below are applicable

to the evaluation of correlation functions of any vibrational displace-

ments that are related to the u by a linear transformation. The
a










present choice of the Q. is based only on the desire to compare the

results obtained from anharmonic force-fields with those of harmonic

ones. From a practical point of view, the coordinates that diagonalize

the harmonic part of the Hamiltonian are useful when anharmonic

couplings between Qi and Q. with i j j are small, and when breakup

channels need not be considered. If the intercoordinate couplings

are large, or if one is interested in dissociative processes, a de-

scription in terms of the correlation of the internal coordinates s.

is more appropriate. Furthermore, the following procedure may be

readily extended to the evaluation of correlation functions of the
^-^m
type <> with Z,m>l.


4. Evaluation of the Displacement-Displacement Correlation Function

Heisenberg's equation of motion for the displacement Q.,

AA
J- Qi(t) = [H V Qi(t)], (2.4.1)
Dt

leads to a second order inhomogeneous differential equation that

describes the time evolution of the displacement-displacement cor-

relation function (DDCF) <> (in this section we continue

to omit the subscript V from the vibrational time-dependence and from

the average over initial states). In order to solve this equation, one

would need to specify two boundary conditions (for example, the value

of the correlation function and of its time derivative at t=t'), which

require knowledge of averages of operators over the initial states.

For anharmonic Hamiltonians, the calculation of these averages re-

quires considerable computational effort; hence, we will obtain the

DDCF from the corresponding double-time Green functions, which al-

ready incorporate boundary conditions.









For any operators A and B, the double-time retarded (+) and

advanced (-) Green functions are defined by (Zubarev, 1960)


< ) = iO[+(t-t')]<<[A(t),B(t')]>>, (2.4.2)


where 0 is the Heavyside unit step-function. Differentiating

eq. (2.4.2) with respect to time using eq. (2.2.1),and evaluating

the necessary conmutators, yields the equation of motion for the

displacement-displacement Green function (DDGF). This is

2 2 2 W (W)
(2/9t2 + mi)<> <> =

= -4 6ij 6(t-t'), (2.4.3)



where f. =-V /anhQ. is the anharmonic force that acts on the co-

ordinate Qi. Following the same procedure, one can write an equation

of motion for the Green function <> which in turn

involves Green functions of higher derivatives of the anharmonic

potential. One thus develops an infinite hierarchy of coupled dif-

ferential equations which cannot be solved unless it is closed by a

decoupling approximation.

In studies of anharmonic vibrations of crystals, several de-

coupling schemes have been proposed (for examples, see Kascheev and

Krivoglaz, 1961, Thompson, 1963, Maradudin and Ambegaokar, 1964,

Pathak, 1965). These are based on polynomial-type anharmonicities,

so that the third term on the left of eq. (2.3.3) is a sum of higher

order DDGF of the type < then develops the hierarchy of equations of motion for these functions,
then develops the hierarchy of equations of motion for these functions,










which in turn involve DDGF of even higher order. At some level, the

hi-erarchy is decoupled by either neglecting the high-order DDGF, or

approximating them by products of time-independent averages multiplied

by DDGF of lower order. For molecular vibrations, these procedures

have the following disadvantages: 1) many anharmonic potentials of in-

terest (eg. Morse-type) cannot. be expressed as a finite-degree poly-

nomial in the displacements, 2) even for polynomial anharmonicities,

the factorization of the high-order Green functions is not uniquely

defined, so that different factorization prescriptions lead to dif-

ferent results, 3) on the other hand, the neglect of higher-order DDGF

introduces dynamical approximations that are not well understood.

However, we can develop an alternative decoupling procedure that does

not suffer from these drawbacks, as follows.

As discussed in the previous section, in the usual experimental

distributions, the vibrational displacements are small. In these

cases, we may approximate their dynamical correlation with an effective

harmonic one, by means of the linearization


<> (. b i> ( (2.4.4)

hence, all the dynamical effects of the anharmonic forces are contained

in the decoupling parameters b. Further on we will fix the bi9 to

reproduce the dynamical features of the exact DDCF during the brief

collision times characteristic of impulsive energy transfer. On the

other hand, when the population of excited initial states is large,

eq. (2.4.4) may be generalized to










<> (2) <>() +
1 J ii j A


+ (3) <>(t) +
tm im


+ b)<< (t) (t)Q (t);Q.(t')>>(+. '
i mn Z m n 3
mn
(2.4.5)

which corresponds to approximating the anharmonic correlation by means

of effective harmonic, cubic, quartic, and so on, correlations. The

(n)
decoupling parameters b) may then be fixed to reproduce the
im...
exact values of the dynamical properties of the DDCF that are most

relevant to the collisions of interest.

In order to maintain the formalism at the clearest level pos-

sible, from here on we consider the case of zero anharmonic couplings

between Qi and Q. for i#j, so that b..=5..b.. We discuss the effects
i J 1iJ ij3 1
of nonzero intercoordinate couplings, at the end of this section.

Introducing the Fourier transform of the DDGF, defined by
(t);Q d i(t-t')(_
<> = J dw e G (w), (2.4.6)
1 ij

iij
the linearized equation of motion is readily solved for G. (w),

giving

(+) 2 -2
G. () = S6../[2r (a2 w.)], (2.4.7)
S13 1

where i.=(w. b.) is the effective vibrational frequency of the
1 1 1
coordinate Q.. For a Boltzmann distribution of initial states, the
1Fourier transform Ji( of Fourier transform J..(G) of Q.(t')Q.(t), defined by
1J J 1









00
Qt) -im(t-t')
<> = de J..( ), (2.4.8)


(+)
may be obtained from G. (w) by means of the relation (Zubarev, 1960)
ij


J..(_) = i e lim [G (W+ i)-G (a -i )]. (2.4.9)
1J 2sinha(a) n-G ij



Here, a(w) = Wm/(2kB T), kB is Boltzmann's constant and TV the

temperature of the distribution. Substituting eq. (2.4.7) into

(2.4.9) and using the expression


lim (y in)-1 =-Py1 if 6 (y), (2.4.10)



where P denotes the principal value, we find that the spectral

function J.. (w) is given by

-U. a.
j .(w) = b6..[e 6(w-.) + e 1 6(+ti )]/(4i.sinha.), (2.4.11)
i3 1] 1 i 1 1

in which a. = M ./(2k T). Finally, evaluating the Fourier transform
i 1 BV
indicated in eq. (2.4.8), we arrive at the following expression for

the DDCF,


<>=iij [exp(-a.-iw.t)+exp(a.+iTi.t)]/(4i.sinha.). (2.4.12)
J 1 1J 1 1 1 1 1 1


The decoupling coefficients may now be fixed to reproduce the

dynamical properties of the exact DDCF that are most relevant to the

collisions of interest. In the present case, impulsive energy trans-

fer occurs during a brief period of time centered about t=0. Hence,

we choose the b., or equivalently the i., to yield the exact values












of the first few time-derivatives of the DDCF at the instant of

collision. Differentiating eq. (2.4.8) with respect to time. shows

that the b. thus chosen will conserve the first few frequency-moments

of the spectral function. From eq. (2.4.1) we see that the exact

derivatives are given by

an =Q( it>n ^ n^
"< =(i/>) <>. (2.4.13)
,tn t=0


Evaluation of the necessary commutators yields an expression for the

first derivative that, at t=0, is independent of the dynamics of the

anharmonic forces; this is


S<< Q(0)Qi(t)>> = i6 ../2, (2.4.14)
t=0 1


which coincides with the expression obtained by differentiating the

approximate DDCF of eq. (2.4.12). However, the second derivative is

exactly given by

2 2 A2
2 <> = 6 (<>.Q.Q>>, (2.4.15)
2 3 t=0 13 3 1 1 1
dt

which depends explicitly on the anharmonic forces. On the other hand,

a second differentiation of eq. (2.4.12) yields

2 ^ ^
2 < at2 3 t=0 1j 21 1
d t










Hence, equating the right hand sides of eqs. (2.4.15,16) defines

the effective frequencies w. by the relation
1

= ^ 2 ^ ^
Jw.cotha. 2( i<>- <>). (2.4.17)



Substituting into eq. (2.4.12) the w. thus obtained, we find that
1
the approximate DDCF contains the dynamical effects of the anharmonic

forces to infinite order. Furthermore, for harmonic motions,
^2
<>=(m./4)cotha. and f.=0,so the present development includes
1 1 1 1
harmonic vibrations as a specific case.

Replacing eq. (2.4.12) into eq. (2.3.13) and rearranging terms,

we obtain the following expression for the APCF.

(aa) 2 -OCi -il.t
FV (,t) =[exp{-6' ai)2 [cosh a. + (e +
i



a. + i1.t
+ e )/2]/(2 .sinha.)}. (2.4.18)
i 1

Using the generating function of the modified Bessel functions of

the first kind (I ) (Abramowitz and Stegun, 1972),
n



exp[x(y + l/y)/2] = y I (x), (2.4.19)
n=-on

we finally arrive at


(aa) (aa) + +
F (Kaa t) = p (K;F)exp (-in. Vt), (2.4.20a)
n n
n












(aa) + -
p (K;F) =n I (X .)exp(-n.a. X cosh a.), (2.4.20b)
n. aii i ai 1
n i



X ai = '[K .(r)] /(2w .sinha.), (2.4.20c)



in which we have introduced the abbreviations n={n.} and
1

V= (Wi.}. Comparing these last equations with the ones corresponding

to harmonic vibrations (eq. (4.16) in Micha, 1979b), we find that the

APCF for harmonic and anharmonic motions coincide in form; however,

for anharmonic molecules, the normal frequencies m. must be replaced
1
by the effective ones w. obtained from eq. (2.4.17).
1
Nonzero intercoordinate anharmonic couplings cause mixing of

the normal coordinates Qi'. which is reflected in the off diagonal

parameters biz of eq. (2.4.4). In this case, the linearized equations

of motion for the DDGF lead to the following matrix equation for

their Fourier transforms.



(w2 1 2 + b) G( (w) = 1 l/(27), (2.4.21)
rb 'V ^ 'I lb

(+) (+)
where G (w) = {G. (w)}, b={b..} and OV ={ 6 .ij.}. Solving for
% r\ ij ru ij i
G (w), shows that the effective frequencies are given by the roots of
2 2
the determinant of (w 1-0 +b). Hence, choosing the b.. to reproduce
'. "'V r" 1J
the exact second derivative of the DDCF at t=0 leads to a matrix

equation for b analogous to eq. (2.4.17). Based on the present de-

velopment, the effects of molecular anharmonicity in hyperthermal col-

lisions may be interpreted in the following way.












5. Effects of Molecular Anharmonicity in Hyperthermal Collisions

In analogy to eq. (2.1.4b), the Fourier transform of the

vibrational APCF,

(aa) (;) dt (aa)
Sr) =exp (-iet/() F-(K,t) = (2.5.la)
V 2 exr V
-00


(aa)
= p (K;r) 6(n. w + e), (2.5.1b)
Sn
n

corresponds to the probability that the molecule will absorb vi-

brational energy and momentum in a collision involving atom a, for

the fixed orientation of the principal axes of the polyatomic in-

dicated by r. This has the form of 6-function peaks, located at the

vibrational energy transfer e =-n V where n. is the change in

quantum number (initial minus final) of the ith effective normal mode;

+ (aa) -
the probability for the transition n is given by p (K;.r)- .In
n
practice, these peaks are broadened into Gaussian functions by the

rotational correlation, as is shown in the following section. In

light of this analysis, the effects of molecular anharmonicity in

hyperthermal atom-polyatomic collisions may be summarized as follows.

1) Anharmonic forces cause a shift in the spectrum of the vi-

brational energy transfer, from the set of harmonic frequencies m.
1
to the effective ones W..

2) Equation (2.4.17) shows that this shift is related to

the work done by the anharmonic forces on the vibrational displacements.

3) The change in the vibrational frequencies is accompanied by

a corresponding change in the probabilities for vibrational excitation,

as indicated in eqs. (2.4.20b,c).










4) In contrast to the harmonic case, the effective frequencies

depend on the temperature of the distribution of initial states.

This is due to the fact that the anharmonic forces affect all the

states that are present in the distribution.

The magnitudes of the effects depend on the specific nature and

strength of the intramolecular forces of the target and require the

evaluation of the averages indicated in eq. (2.4.17). These may be

calculated by standard numerical procedures, or by finite-temperature

perturbation theory, when the temperature of the initial distribution

is sufficiently low. The application of this development to specific

targets will be presented in Chap. IV. We now proceed to consider

the rotational correlation, in order to complete the evaluation of

the APCF.


6. Short-Time Approximation to the Rotational Correlation

The rotational correlation of heavy molecules has been previously

evaluated by means of a Taylor-series expansion of the logarithm of the

APCF (Micha, 1979b). In this section, we present an alternative ap-

proach to the evaluation of rotational correlation, which is based

on the cumulant expansion techniques developed in Chap. I. Although

the two approaches are completely equivalent, we feel that the

present one allows a more unified understanding of the role of

correlation functions in the description of many-body collisions.

Substition of eqs. (2.4.20) into eq. (2.2.3) gives the following

expression for the complete self-correlation function









(aa) + (
F(aa) (,t) = F (,t) exp(-in w t)V (2.6.1a)
+ n
n

(aa) (aa) ^ ex (t
F (K,t) = <

>
a K- aR R
n II

(2.6.1b)

where the time dependence of F(aa) is due only to the rotational
n
motion of the molecule. Following the procedure described in Chap. I,
A
Sec. 3, we find that the time evolution of exp(-iK*-d ) is given by
a

exp[i.da (t)R] = exp(itLR/7) exp[i 'a(0)]. (2.6.2)


Here, LR is the Liouville superoperator for rotational motions,

which is defined by its action on an arbitrary operator A, as follows


L R = ([R,)nA]. (2.6.3)


Equation (2.6.1b) can now be rewritten as


(aa) (t) (aa) (aa)) +^
F (t) = P a(K) {<

n n n


x exp[-i-- a(0)]>>R (paa) ; (0))>>R, (2.6.4a)
n



p (). <

>R. (2.6.4b)
n n


Whenever the rotational motion of the molecule is much slower than

its vibrational one, we perform a short-time approximation to the

rotational correlation of each vibrational transition. As discussed

in Chap. I, Secs.4,5, this corresponds to expanding the second fac-

tor of eq. (2.6.4) in terms of the cumulant rotational averages of LR

and retaining only the first two cumulants, so that










(aa) (aa) 2 2 2
Fa (K,t) P (K) exp [it<> /R t <>C/(2 )]. (2.6.5)
n n


The lowest-order cumulants are determined from eq. (1.3.5) to be


(aa) ^ +, 1 (aa)
<> = <

> /P (K),
R CR a R a R
n n

(2.6.6a)


2 (aa) ^ )[H R ((aa)
<> (K) -
R CR 'a RR R
n n


<>2R* (2.6.6b)



Replacing eq. (2.6.5) into eqs. (2.6.1) we obtain the following

expression for the APCF,


(aa) -(aa) 2 2/2
F (K,t) = > P (K)exp[i(<>C-n w V )t/-< + n
nI
(2.6.7)

Finally, evaluating its Fourier transform, as indicated in

eq. (2.1.4b), we arrive at



(aa) 1 (aa) >>
S (a ,) Y- P( (K) exp {-[(< 4v' n
n

E)/(2<> (2.6.8)
R CR R CR



Examining this last equation we find that, for a given value of the

momentum transfer K, each atom of the target contributes a group of











normalized Gaussian peaks to the differential cross section. Each

peak corresponds to the vibrational excitation indicated by the set

of changes in quantum numbers n, and is centered at the vibrational

energy transferred in.IV plus the average rotational energy trans-

ferred to this vibrational transition <> ; furthermore, each
R CR
2
peak is rotationally broadened by an amount 2<> Equations
R CR"
(2.6.6) show that the rotational energy transfer and the width con-

tributed by each atom are functions of the momentum transfer (hence

of collision energy and scattering angle) and depend on the vi-

brational transition indicated by n. The probability that a collision
4-
involving atom a will excite the vibrational transition n is given
(aa)
by P which is also a function of the momentum transfer.
n
However, if atom a is at the center of mass of the molecule,

so that d =0, then instead of eqs. (2.6.7) and (2.6.8), we have
a

F(aa) 4 (aa) 4
F (K,t) = P (K) exp(-in-wvt), (2.6.9a)
4 n
n
(aa) (aa) 4
S (aaK,) = Pa (K) 6(E+ tn v), (2.6.9b)
+ n
n
which shows that such an atom does not absor-brotational energy

but does contributeto the vibrational excitation.

As discussed in Sec. 1, the correlation functions given by eqs.

(2.6.8,10) together with eqs. (2.4.20b,c) and (2.6.6) completely de-

scribe the response of the target to an impulsive collision, regard-

less of the nature of the projectile. The nature of the projectile

atom determines the two-body cross sections of eq. (2.1.5), in

accordance with the atom-pair potentials. Therefore, before







47


proceeding to the applications of the many-body theory, we must con-

struct the multicenter representation of the intermolecular potential

for the collision systems of interest; this is the subject of the

following chapter.

















CHAPTER III
MULTICENTER POTENTIAL ENERGY SURFACES


In this chapter we investigate the multicenter representation

of the intermolecular potential energy surface (PES) for nonreactive

atom-molecule systems. Our aim is to develop a parametrization of the

PES which is appropriate for the study of hyperthermal molecular

collisions. In the first section, we weigh the relative advantages

of the single-center and multicenter representations. In the second

section, we develop a parametrization for ion/linear molecule sur-

faces in terms of atom-pair potentials that are based on the known

general properties of intermolecular forces. Next, we evaluate the

atom-pair potentials that reproduce the quantum chemical PES of the

system Li /CO. In the final section, we study the system Li /CO2,

for which no quantum chemical information is yet available. Using

a simple model of short-range forces plus experimental information

on the molecular properties of CO2, we construct a model PES for this

system. Both surfaces developed here will be used in subsequent

atom-molecule scattering calculations.


1. Single-Center vs. Multicenter Expansion

The standard procedure used to describe the interaction between

an atom A and a molecule B consists of assuming the Born-Oppenheimer

separation of nuclear and electronic motions (Tully, 1976) and then

calculating the lowest PES for a set of conformations, by means of

more or less accurate quantum-chemical methods; for examples see

48











Staemmler,1975, Parker et al., 1976 and Thomas et al., 1978.

Working in the body-fixed (BF) coordinate system determined by the

principal axes of B, one thus obtains a table of the intermolecular

potential V for several values of the relative position of A, denoted

by R, and of the positions of the constituent atoms of B, which we
+B +
indicate with r = {r ,aB}.. In order to facilitate scattering
a
calculations, one must usually construct a functional representation

for the surface. The most common of these is the single-center ex-

pansion, in which the dependence on the orientation of R is described

in terms of a set of basis functions of angles, centered at the center

of mass of the molecule, as follows:

.B -B
V(R,r ) = (R,r )Y(n,). (3.1.1)
n
4->
Here, n,E are the polar and azimuthal angles of R in the B1 system,

respectively, and the Y are the elements of the basis set. The radial
n
coefficients V are fitted to reproduce the values of the PES at the

tabulated configurations.

For atom-molecule scattering in the thermal collision energy

regime, such a single-center expansion has several advantages. Since

low energy scattering is mediated by long-range forces (McDowell and

Coleman, 1970) and these forces vary slowly with the orientation of
4->
R, usually only the first few terms of eq. (3.1.1) need be included

in scattering calculations. On the other hand, the PES can be re-

presented to any arbitrary degree of accuracy by increasing the size

of the angular basis set. Furthermore, if the radial dependence of

the coefficients V is judiciously chosen, these can be related to

molecular properties (electrostatic moments, polarizabilities, etc.)










of the isolated collision partners (Hirschfelder et al., 1967),

this can be used to check the accuracy of the fitting procedure,

or to supplement the quantum-chemical information when only a few

points of the surface are available.

However, as the collision energy increases above a few hundredths

of an electron volt, the single-center expansion, although accurate

in principle, presents rather serious disadvantages. At hyperthermal

energies and for small impact parameters, the projectile probes the

inner region of the PES which is of a multicenter nature, so the

single-center representation loses physical meaning. Furthermore,

as the short-range intermolecular forces depend strongly on the

orientation of R, a very large number of terms are necessary for

eq. (3.1.1) to converge. These drawbacks will be further clarified

by the example presented in the next section.

Alternatively, the multicenter nature of the intermolecular

forces can be better represented by expanding the potential about each

atom of the target, as follows:

S+B +* -
V(R,r ) = v (R,r ), (3.1.2)
a
where the sum runs over the N atoms the polyatomic. Here, v
a
corresponds to the interaction between the (A,a) atom-pair which de-

pends on the electronic distribution of A and on that of atom a in

the valence state of the target. Hence, v is a function of the
a
orientation of R with respect to r and of the magnitude of R-r .
a a
In general, for small A-a separation, the atom-pair potentials are

steeply repulsive due to the overlapping of the electronic clouds










of A and a; in the long-range region, the va fall off as inverse

powers of the A-a distance due to the interactions of the charge

distributions of A and a. For example, the PES corresponding to an

atomic ion with charge qA and a molecule could be modeled, most simply,

by the sum of the pair potentials


va(R,r ) = A exp(-B aR-lr) + Aq a (3.1.3)

in which the fractional atomic charges qa reproduce the experimental

values of the N-lowest electrostatic multiple moments of B. The co-

efficients A and B (both positive) can be obtained by fitting to a

previously calculated PES or from combination rules (Gaydaenko and

Nikulin, 1970).

More, generally, eq. (3.1.2) is the leading term obtained by

expanding the potential in contributions from two atom, three atom,

and so on, atom-clusters as indicated below (Micha, 1979a)

SRB ) Q(2) (,+~ 2 (3) -+ -
V ) = va (R,r) + v (Rr ar ) + (3.1.4)
a a Recent attempts have been made at understanding the properties of

large polyatomic systems in terms of contributions of the constituent

atomic centers (Bader, 1980). We hope that, in the future, such studies

would allow the determination of the atom-cluster potentials from

first principles. However, for our present purposes, we will con-

sider the atom-pair potentials to be functions of r and r with para-
a
meters that will be adjusted to reproduce a previously calculated PES

and/or the known molecular properties of the separated collision

partners. As we shall see in the following section, by judiciously

choosing the form of the v the contributions of the higher-order
cluster potentials can be made negligible small.
cluster potentials can be made negligibly small.











2. Atom-Pair Potentials for Ion/Linear Molecule Interactions

As with any other fitting procedure, the functional form of the

Va greatly influences the accuracy of the representation and the num-

ber of parameters required. In order to allow for a physically mean-

ingful interpretation of the pair potentials and to reduce the number

of parameters to a minimum, the v should be chosen in accordance with
a
the known general properties of intermolecular forces (Hirschfelder

et al., 1967, Margenau and Kestner, 1971). For the interaction be-

tween two closed-shell molecules, these properties may be summarized

as follows.

When the relative distance R is much greater than the lengths

over which the electronic clouds of each species are appreciable, the

intermolecular potential can be written as a sum of the following three

contributions:

1) the electrostatic interaction between the permanent multi-

pole moments of the charge distributions of each molecule,

2) the induction contribution, due to the interaction between

the permanent moments of one species and those induced in the other

one, and

3) the dispersion or Van-der-Waals forces, which may be in-

terpreted as arising from the interactions between the induced

moments of each species. However, when at least one of the molecules

has a net permanent charge, the magnitudesof the dispersion terms are

much smaller than the electrostatic and induction ones, so that

dispersion forces may be neglected.











In order to maintain the notation at the clearest level possible,

from here on we take B to be a linear polyatomic at its equilibrium

configuration (r = d ). Hence, the long-range potential for an atom-

ic ion A and B is given by the following single center expression,

SB qq q (cosB2)/R2 3
V(R,r ) qA /R + q (cosn)R + qAQBP2(cosn)/R

q9[aB/2 + ABP2(cosn)]/R4 + .. (3.2.1)

in which we have explicitly indicated the first three leading terms

of the electrostatic forces, the leading induction contribution (of
-4
order R ), and we have neglected the dispersion forces that are of

order R6 and smaller. Here, P (x) = (3x2 1)/2 is the second order

Legendre polynomial, qA is the charge of the ion, while the molecular

properties of B are the net charge qB, the electrostatic dipole

PB and quadrupole Q moments, and the isotropic a = (a + 2a )/3
SB B zz xx

and anisotropic A = (a a )/3 polarizabilities; a and a
B zz xx zz xx
are the components of the static dipole-polarizability tensor parallel

and perpendicular to the molecular axis, respectively. Equation (3.2.1)

follows from a perturbation-theoretical approach, in which the electro-

static interaction between the charge distributions of each species

is expressed in terms of a multiple expansion (Margenau and Kestner,

1971, Chap. 2). Such an expansion is valid only when the distance

between the centers of charge of each distribution is much larger

than the range of the individual densities (Jackson, 1975, Chap. 4).

Therefore, eq. (3.2.1) loses physical meaning whenever R r ; this
is reflected in the unphysical divergence for small R.
is reflected in the unphysical divergence for small R.











When the relative distance decreases, the overlap of the

electronic clouds of each species causes the intermolecular forces

to become steeply repulsive, Hence, the small R behavior of the

potential depends very strongly on the details of the electronic dis-

tributions. However, when the molecular orbitals of each species

are described in terms of Slater-type orbitals centered on each

nucleus, the overlap will be an approximately exponential function

of the A-a internuclear distance. Slater-type bases are most commonly

used in quantum-chemical calculations for atoms and for linear mole-

cules (Pople and Beveridge, 1970); in these cases, the short-range

behaviour of the potential is of the form

+ +B I
V(R,r ) A aexp(-B R-r ). (3.2.2)
R r
a

Here A and B are positive coefficients that are independent of the
a a

magnitude of R, but they may be functions of the orientation of R with
+
respect to r.
a

We thus find that the single-center representation of eq. (3.2.1)

must be smoothly transformed into a multicenter one, as the relative

distance decreases. The transformation must avoid the unphysical

divergences of the multiple expansion and must accurately reproduce

the intermediate region of the surface which includes the potential

wells. One might attempt to carry out this transformation by

multiplying the right-hand side of eq. (3.2.1) by a switching

function that tends to one for large R, and to zero whenever R r
a
Unfortunately, the second requirement would force the switching

function to have an extremely complicated dependence on the orientation










of R. However, this difficulty may be readily avoided by extending

the multicenter representation to large R, as follows.

Assuming that the molecular properties of B can be expressed in

terms of the individual contributions of each atom, we expand the

long-range region of the potential about the atomic centers and write

+ +B +
V(R,r ) = v (R,r ), (3.2.3a)
a a
a

+ (SR) + + f (LR) r (.
(R,r) = a (R ) + (R,r (R,r ), (3.2.3b)
aa a a a a

(LR) + 3
vaLR) (R,r ) = C 0a/R + C (cosn )/R + C P (cosn )/R +
a a 10a a 21a a(Ca) a 32aP2 a a


+ [C40a + C42aP2(cosna)]/Ra, (3.2.3c)

-4 -> ->
where R = R-r is the position of the ion with respect to target
a a
atom a and na= arcos[R *r /(R r )] is the angle between and r .
a a a aa a a
The terms of v LR) represent the interaction of the charge of A with
a
the first three electrostatic moments and polarization of the

electronic distribution about center a, respectively. In

(SR)
eq. (3.2.3b), v corresponds to the short-range repulsive forces
a
and fa is a switching function that eliminates the singularity of

v at R = r The detailed form of f is not crucial, pro-
a a a
vided it satisfies the following requirements: 1) f and its
a
gradient with respect to R must be continuous, 2) f a 0
a.- +
R-r
and 3) f 1 One such function is, for example, a
R-t

f (R,r ) = O(R c )exp[-Y (R c ) ], (3.2.4)
a a a a a a ar


where 6 is the Heavyside unit step-function. Here, c is a radius
a










about atom a within which the long-range pair potential is zero, while

Y determines how fast v LR) is turned off as the A-a distance de-
a a
creases. We may complete the representation by taking the repulsive

potentials to be

(SR) + +
v (R,r ) = A exp(-B R ), (3.2.5)
a a a aa

with A and B positive and independent of R. If necessary for a
a a
specific system, the v(SR) may be rendered more flexible by letting
a
(SR)
A and B depend on n or by expanding v is a basis-set of
a a a a
functions of the orientation of R .
a
Expressing IR-r in a Taylor series in r about R, we find

that eqs. (3.2.3) and (3.2.1) coincide as R + m, whenever the co-

efficients C satisfy
nZa

A B = C10a (3.2.6a)
a

qA;B = l (C10ara C 2laa/ra), (3.2.6b)


QB = (ClOarl + 2C21ara C2a) (3.2.6c)
qB = + /r (3.2.6d)
AB 10a + 21a ra a
a

qQ = (C r + 2C r + C ), (3.2.6c)

2


qa 2= C4a. (3.2.6e)
a

If the Cn a are fit to a previously calculated PES, these equations

provide us with an independent check of the fit, as the Cna should

yield the correct values of the molecular properties of B. On the

other hand, if a quantum-chemical PES is not available, eqs. (3.2.6)

together with experimental values of the molecular properties may be

used to partly determine the atom-pair potentials.











In the multicenter representation defined by eqs. (3.2.3-5),

the set of parameters that must be adjusted to reproduce the tabulated

PES is {p.} = {A ,B ,c ,C ,C ,C ,C C for a=l to N .
i a a a a 10a 21a 32a 40a' 42a
Although at first sight this set seems rather large, the above analysis

shows that this is the smallest possible set that is required by the

physical nature of ion/linear molecule forces. Furthermore, the pi

generally depend on the internal configuration of the polyatomic, and

hence on r. However, for a specific molecule, some of the long-range

parameters CnZa may be negligibly small. In particular, the electro-

static interaction between the charge of A and a permanent multiple

moment of B may be represented, at large R, as the sum of the forces

between the charge of A and the lower-order multipoles of the target

atoms, as shown in eqs. (3.2.6a-c); this usually involves cancellations

between the terms contributed by each center. As the A-a distance

decreases, the atomic contributions become large, due to the 1/R

dependence, and may not cancel to yield a correct description of the

intermediate region; this indicates that such atomic multipoles

should be made vanishingly small. We will encounter examples of this

phenomenon in the following sections.


3. Multicenter Potential for the System Li /CO

We now develop the multicenter representation of the intermolecular

potential for the Li /CO system. This is one of the few many-electron

systems for which a detailed PES has been calculated quantum chemi-

cally, at the configuration-interaction level of accuracy

(Thomas et al., 1978). This surface has been tabulated for a wide










range ofvalues of I, at the equilibrium configuration of CO

(Ira-rb1=1.224 X); this will allow us to judge the applicability

of the multicenter expansion for small and large R. Unfortunately,

only a few points have been calculated for nonequilibrium con-

figurations of the molecule; therefore, throughout this section,

we fix the target atoms at their equilibrium positionsand take the

-B
potential parameters to be independent of r

Due to the nonlinear dependence of the potential on several

of the parameters, the usual least-squares optimization (Doren and

McCracken, 1972, Chap. 7) would lead to a system of sixteen coupled

transcendental equations in the p. whose solution represents the best

values of the parameters. However, such a system can only be solved

by iterative methods (Ostrowski, 1966, Sec. 25) whose convergence

is impracticably slow due to the large dimensionality of the problem.

In order to avoid this inconvenience, we optimize the linear (C a)

and nonlinear (B ca, ya) parameters in two separate steps, as

follows.

In the first step, the Cna are obtained by a linear least-

squares fit to the long-range region of the PES. In order to test

the correctness of the functional representation chosen for the

v (LR), we do not a priori constrain the C to reproduce the known
a n~a
molecular properties of CO. Hence, we allow the Ca to vary freely

in order to minimize the root-mean-squared relative error of the

representation; for large R, this error is given by

-2 +- +B B -* +B 2
e = {[VT(Rr ) VRr )]/[N VT(,r )]} (3.3.1)
->F










Here, the sum runs over the N points of the surface for which
4-
R (LR)
R >3.7 X, V is the tabulated value and VF = vR). The
a
optimum values of the Cna are those that satisfy the system of

linear equations


2
3e / C = 0, (3.3.2)
na

2 2 2
together with the conditions @ e /DC >0.
nla
Solving eqs (3.3.1) for all the Cnla indicated in eq. (3.2.3c)

gave C0a, C21, and C32 of the order of 6 e2 n, n=0,1,2,

respectively, with opposite signs for each atom. These values, although

of unphysically large magnitudes, lead the v(LR) to cancel each other
a
in order to reproduce the tabluated PES. Examining the resulting fit,

we concluded that, for the Li /CO system, the long-range potential

could be reproduced with fewer parameters than those indicated in

eq. (3.2.3c). Repeating the fitting procedure while omitting each

time one pair of the C10a C21a, and C32a, we found that the C21a

were redundant. The optimum values of the parameters are shown in

table 3.1. These reproduce the long range region of the PES with an

r. m. s. error e = 0.85% over the 25 points available (four equally-

spaced values of R from 3.7 1 to 5.3 1 and one at R = 7.9 X, each

for the five orientations n= 0, 45, 90, 135 and 180 deg). Throughout

the remainder of the optimization procedure, the Cna were kept fixed.
nta
In the second step, using the full multicenter potential of

eqs. (3.2.3-5), for a given set of values of the nonlinear parameters

Ba, ca, and y a, we least-squares fitted the Aa to the short-range

region of the tabluated surface for which VT > 0. This fit was










repeated for several sets of B c and y each time obtaining

a new A Finally, we chose the B c y and the corresponding A
a a a a a
that yielded the smallest r. m. s. error in the short and intermediate

ranges; these values are also shown in the table below.


Table 3.1.


Ion/atom potential parameters for Li /CO


Li /C


d ()
a
A (eV)
a
B(A -1)

c (A)
a
ya -1)

C 0a(eVA)

C21a(eV 2)

C32a(eVA 3)

C 40(eVA 4)

C 42(eV 4)
42a


0.6447z

1086.33

4.25

1.38

1.38

-0.189892

0

-5.70326

-11.4071

-1.07765


-0.4835i

1191.92

4.63

1.48

2.27

0.193190

0

-0.951450

-2.54812

-4.90754


Substituting the CnZa into eqs. (3.2.6) yields the molecular

properties of CO that correspond to the present representation;

these are in good agreement with the experimental ones (Thomas et al.,

1978), as is seen in table 3.2. This comparison further confirms the

validity of the multicenter expansion for large R.


Li+ /










Table 3.2 Molecular properties of CO2






Present Fit Exact


qB(e) 2.3 x 104 0

B (eX) -0.0150 -0.0236

QB(eA2) -0.4646 -0.4666
o 3
aB (A) 1.939 1.947

AB( 3) 0.415 0.325




Figures 3.1 and 3.2 show the full multicenter potential (eqs.

(3.2.3-5) together with the parameters of table 3.1) and the quantum-

chemical values. We find that the multicenter representation re-

produces the tabulated values with an r. m. s. error e = 4.4% over

the 55 points available. The largest errors occur near the zeroes

of the potential where its absolute value is less than 0.03eV. We

also find a small discrepancy (15%) near the bottom of the potential

wells at the two orientations n=0 and 180 deg. This discrepancy is

most probably due to the simple angular dependence isotropicc) that

(SR)
we have chosen for the v( The errors could be greatly reduced
a
(SR)
by expanding v in a basis of functions of n However, we
a a
feel that the increase in computational effort resulting from a

larger set of parameters is not justified for the description of

hyperthermal collisions.


































Figure 3.1 Comparison of the multicenter representation (curves)
of the Li+/CO potential with the quantum chemical values
(points).










10-



8-



6-


W 4



2-



0-



-0.4-


I
I
I


1
I

I
I
I

SI

rI I

I


Li+/CO


-ml'--


-0

=45

=90


R(A)


N X




^


































Figure 3.2 Comparison of the multicenter representation (curves)
of the Li+/CO potential with the quantum chemical values
(points).









Li+/CO


10-
j-


8-


6-


=18 0


? = 1350







R (A)
4 5


3


I



1
I
I
I


ci,


4-


2-










Excepting the two regions just mentioned, the multicenter re-

presentation gives a very accurate description of the Li /CO sur-

face over the four-decade range of magnitudes of the potential, with

a minimum of adjustable parameters. The single-center expansion de-

veloped by Thomas et al., which is of comparable accuracy, required

fifty parameters, in contrast to the sixteen used in the present one.

Furthermore, the multicenter representation allows us to understand

the physical origin of the parameters, even in the short-range regions.

On the other hand, the ion/atom potentials just obtained may be

utilized to gain insight into the physical structure of the CO mole-

cule. Recognizing the dangers of extrapolating excessively from in-

direct results, we propose the following interpretation to the

differences between the Li /C and Li /O potentials.

A comparison of the short-range regions of the v shown in
a
fig. 3.3, suggests that the electronic density about carbon extends

farther from its nucleus than does that of oxygen. This observation

is consistent with the signs of the C10a which indicate a slight ex-

cess of electrons on the C-center and a corresponding deficit on the

O-one. This causes the Li /0 interaction to be repulsive at large R,
ag
while the Li /C potential is attractive for large distances, as is

seen if figs. 3.4 and 3.5, respectively. Comparing the magnitudes

of the C32a, we conclude that the charge distribution about C is more

anisotropic than the one about 0, and that both distributions are

prolate spheroidal. Finally, the C40 indicate that the electrons

around C are more polarizable than those about 0; however, the

anisotropy of the polarizability of 0 is quite large, so that, at
































Figure 3.3 Short-range region of the ion/atom potentials of Li /CO.
















--- Li /C


8-




6




4-




2-


' I I


1.00


1.25


Ra (A)


Li /O--


1.50
1.50






















C)


I 1 1 1

- 0 0d


(A)D A


0

II

P--













L0n -
0
Cr


0

II
F`


I I I
-- O -DA
o 5 o;

(A)D^


0


-J


0
0
0)
II
\


d


I
C\J
0


I i










an=90 deg. the Li /O potential is repulsive at all distances. We

expect that these conclusions would be confirmed by the configuration-

interaction charge density of the isolated CO molecule; unfortunately,

the wave-function of CO is not available to us at the present time.


4. Multicenter Potential for the System Li+/C02

Due to the large computational effort required by the number of

electrons in CO2, a quantum chemical PES for the Li+/C02 system has

not been calculated. Until such a surface becomes available, we

propose a multicenter potential which is based on the electron-gas

model of short-range forces (Gordon and Kim,1972) and on the long-

range interaction between the charge of Li and the known electro-

static multipoles and polarization of CO2.

Briefly, the uniform-electron-gas model of the interaction be-

tween two closed-shell atoms and/or molecules A and B is based on

the following assumptions. Firstly, the potential is written as the

sum of four terms that represent the kinetic energy of the electrons,

the electrostatic interactions between the charge distributions of A

and B, the electron exchange and correlation effects, respectively.

Each term is separately calculated as an integral over all space of

the corresponding energy density, which is a functional of the

electronic density of the system. The electrostatic energy-density

functional may be readily obtained by considering the Coulombic inter-

actions between the charge distribtuions. However, the kinetic,

exchange and correlation density functionals are taken to be those of a

uniform gas of electrons (Slater, 1968, Carr et al., 1961,










Carr and Maradudin, 1964). Furthermore, the electronic density of

the A-B system is approximated by the sum of the densities of the

isolated partners, regardless of their separation. Hence, if the

density of each species is known, the calculation of the PES is

reduced to the numerical evaluation of the integrals of the energy-

density functionals.

This model has been found to provide a reasonably accurate de-

scription of the short-range repulsive forces in atom-atom (Kim and

Gordon, 1974a, b) atom-molecule (Green et al., 1975, Parker et al.,

1976, Davies et al., 1979) and molecule-molecule (Parker et al., 1975)

systems, at a small fraction of the computational cost of even the

most approximate quantum-chemical methods. Unfortunately, the

model does not correctly describe the long-range forces, for the

following reasons. Due to the additive nature of the approximate

total density, the correlation effects depend on the overlap between

the individual densities. Hence, for large R, the correlation con-

tribution decreases exponentially and does not reproduce the
-2n
Van-der-Waals potential which is of order Rn, n 3. This de-

ficiency may be remedied by smoothly switching from the electron-

gas correlation potential to the dispersion one, as R increases

(Parker et al., 1976). In the Li /CO system, we expect that

the dispersion forces are much smaller than the induction ones, due

to the net charge of the ion. However, by assuming that the

electronic densities of each species remain unchanged for all dis-

tances, the model excludes the rearrangement of the charge distribution

of CO2 caused by the charge of Li and hence the induction forces.









Nevertheless, we may obtain a realistic model of the Li +/C02 PES

by combining the short-range results of the electron-gas method with

the long-range multicenter expansion studied in the previous section,

as shown below.

Electron-gas calculations were performed for the collinear

(n= 0 deg) and perpendicular (n = 90 deg) configurations of Li +/C02,

using the program developed by Green and Gordon, 1974. These calcu-

lations incorporated the correction due to Rae, 1973, that avoids

overcounting of the electrons in the density functionals. The

electronic density of CO2 was approximated by that of the self-

consistent-field (SCF) molecular wave-function calculated by McClean

and Yoshimine, 1967, p. 192. The equilibrium C-0 bond distance was

fixed at the value that corresponds to the SCF wave-function

(1.1348 X) instead of the experimental one (1.161 X, in Eastes et al.,

1977), because the experimental bond-length represents a slightly

stretched CO2 molecule in the SCF approximation. The Li ion was also

described by an SFC wave-function (Clementi, 1965, table 03-01).

The numerical integration of the energy density functionals was

carried out by means of three-dimensional Gaussian quadratures, the

number of quadrature points was gradually increased until the cal-

culated potential converged to 1%, at all the values of 1. Due to

the approximate nature of the model, we felt that the increase in

computational cost required for a greater accuracy was not justified.

Hence, the number of points ranged from (56, 80, 1), at n=0deg, to

(24, 40, 24),at n=90 deg, for the (X, y, @) spheroidal coordinates,

respectively.










Table 3.3 Electron-gas Li /C02 potential.


R V(R,0 deg) R V(R,90 deg)
(Bohr) (Hartree) (Bohr) (Hartree)


3.7444 0.81826 1.60 0.87678

3.8444 0.63961 1.80 0.62150

3.9444 0.52927 2.00 0.44061

4.0444 0.40132 2.20 0.31309

4.1444 0.32335 2.40 0.22474

4.2444 0.25550 2.60 0.16382

4.3444 0.20132 2.80 0.12175

4.4444 0.15878 3.00 0.09251

4.5444 0.12499 3.20 0.07347

4.6444 0.09931 3.40 0.05994

4.7444 0.07718 3.60 0.04937

4.8444 0.01634 3.80 0.04059

4.00 0.03335



The results of the above calculations are shown in table 3.3

(the odd-looking scale chosen for R at n= 0 is due to the face that

the coordinate most appropriate to short-range interactions is the

separation between the ion and the nearest atom of CO2). These re-

sults correspond to the so called Hartree-Fock model potential, which

is the sum of the kinetic, electrostatic and exchange contributions.










The electron-gas correlation is not presented here, due to the in-

accuracies of the model previously discussed. However, the correlation

effects were calculated to be of the order of 1% of the "Hartree-Fock"

potential, and hence comparable to the numerical errors of the quad-

ratures. In order to facilitate future scattering calculations, the

short-range potential was fitted to a multicenter expansion,as follows.

A plot of the logarithm of the potential versus the distance

between Li and the nearest molecular atom yielded an approximately

straight line, for the collinear configuration. This indicated that

the Li /O potentials, v1 and v3, could be represented by the single

exponential of eq. (3.2.5). However, an analogous plot for the per-

pendicular configuration showed a steep straight line for small R,

that smoothly changed to a shallower slope for larger distances. This

graph suggested that two exponential terms were required for the Li /C

potential. The parameters of the ion/atom potentials were adjusted

to reproduced the tabulated electron-gas surface, by a procedure

analogous to the second step described in Sec. 3.3. Briefly, for

fixed values of the exponential coefficients B1 = B3, B2 and B ,

the preexponential factors A =A3, A2 and A2, were simultaneously

least-squares fitted to all the points of the tabulated surface.

This fitting was repeated for several sets of values of

B1 = B3, B2 and B finally selecting the exponential coefficients

and corresponding preexponential ones that yielded the lowest r. m. s.

relative error. These are shown in table 3.4 (below), and they re-

produced the surface with an r. m. s. error e = 0.43% over the 25









tabulated points; the largest error (3.6%) occurred at

R = (4.8444 a ro= 0 deg). These short range ion/atom potentials may

now be combined with the known long-range forces of the Li+/CO2 system,

as follows.

Without further knowledge of the properties of the exact charge

distribution of CO2, there is some ambiguity in the determination of
(LR) (LR)
the va because only lim v may be determined from experimental
aR a
R- coa
measurements of CO2. In particular, the Li -charge/CO2-quadrupole

interaction, may be represented in several different ways that are in

accordance with eq. (3.2.6c). For example, Eastes et al., 1977,

assumed fractional charges qa located on the atomic centers of the

molecule. The values q = q3 = -0.331 e and q2 = +0.662 e reproduce

the experimental quadrupole moment QB = -0.90 eL 2, at large dis-

tances. However, for small R, these charges yield values for the

perpendicular Li /C02 interaction that are much larger than those

calculated above, because the Coulomb potentials q q /R do not

cancel in the appropriate way. Similarly, one may set the qa equal

to zero, and consider fractional atomic dipoles pa (note that =
a P2
due to the symmetry of oxygen); however, such a model is also in dis-

agreement with the above short-range results. Therefore, we presently

assume that q = 0 and P = 0, for all a, and take nonzero atomic
a a
quadrupoles Qa, such that Qa = QB We must emphasize that this choice
a
is not unique; a combination of small atomic charges, dipoles and

quadrupoles may not be excluded as a possibility. On the other hand,

the electronic clouds about each of the atomic centers are polarized

by the net charge of the ion; hence, we represent the isotropic and










and anisotropic polarizabilities of CO2 by the sum of the atomic

polarizabilities a and A respectively. Furthermore, since the num-
a a
ber of electrons in C and in 0 are comparable, we let Qa = Q/3,

a = a /3 and A = A /3, with aB = 2.602 3 and AB = 0.6931 A3

(Buckingham and Orr, 1967). Finally, the resulting v(LR) are switched
a
off at small A-a distances as in eqs. (3.2.3b,4) using the switching

parameters c and ya of C and of 0 determined in the previous section.
a a
The parameters for the long range ion/atom potentials of Li /CO2 are

also included in the table below.


Table 3.4 Ion/atom potential parameters for Li /CO2.





Li+/C Li+/O


A (eV) 650.697 1340.16
a
A2(eV) 17.5648
-1
B ( -) 4.346 4.856
a
B2( -1 1.455
2
c (A) 1.38 1.48
a
-1
c ( -1) 1.38 2.27
a
C0a(eVA) 0 0

C21a(eV2) 0 0

C32a(eVA3) -4.32 -4.32
o4
C 40a(eVA ) -6.246 -6.246

C 42a(eVA4) -3.327 -3.327
42a










The complete model potential (eqs. (3.2.3-5) together with

table 3.4) is presented in figs. 3.6,7. Examination of the short-

range region shows the large anisotropy of the repulsive forces,

which is due to the many-body nature of the target. In this region,

the accuracy of the model is determined only by that of the electron-

gas approximation because the v(LR) are switched off. By design,
a
the present model reproduces the exact long-range forces, as deter-

mined by the properties of CO2. A judgement of the validity of this

surface in the intermediate regions must be postponed until a quantum-

chemical PES becomes available. The latter region includes the

potential wells and is thus sensitive to the atomic quardupoles and

polarizabilities, which are not uniquely defined by QB, aB and AB.

However, we expect this model to provide at least a qualitative de-

scription of the eact PES for intermediate R; in particular, the dis-

appearance of the wellatg= 0 deg as n-*90 deg, seen in fig. 3.7,

clearly reflects the change in sign of the Li+ charge/CO2-polarization

interactions. Finally, in figs. 3.8-10 we present the ion/atom

potentials. Examination of their short-range behaviour (fig. 3.8)

indicates that the charge distribution about the C-center extends

farther away from its nucleus than do those of the 0-centers. Con-

sequently, the larger extent of the Li /C repulsion causes the wells

of v2 to be much shallower than thoseof v1 and v3, as is seen in

figs. 3.9,10.

Having determined multicenter potentials for Li /CO and Li+/CO2,

we may now proceed to study energy transferprocesses in these systems

by means of the many-body theory developed in Chap. II.

































Figure 3.6 Short-range region of the Li/C model potential.
Figure 3.6 Short-range region of the Li /CO model potential.
2








Li /C02


---o=00


R (A)


=45


I0-



8-



6-



4-



2-



0-


1.5


77=90.




































-l

*d
4-J


O
4-
0






0
1r








+
a











CI-

0












0
1-






4-)
ct
0


a)



b0





















C\j
0
C.)


0
CD
It1


0
o

II
F`


(A-)
(As) A


0

II
F'


































Figure 3.8


Short-range region of the ion/atom potentials of the
Li+/CO2 model surface.









10-
0 Li +/C02



8-


-Li /C
S6-



4-

Li /O 0

2-



0-A^-------- ^
1.00 1.25 1.50
Ra(A)












































































I-q U







o
0

Ca)












01
4-1 CM
CO 0



(U

4-JC







C)
m





88


LO











O O
0







00
*I-II1









0 Co
SII












0 0^

na~)00










































r-4





0
C1




0

+
*-






0
-41


e















10


I0
t0 ca
H *H




,-4 c0





I


cO +
Srj






r4
4-4
J 0
*r-1 *H

'0 l-













a)(
F









LO


o<
0
nr


C\J
0
C)


0

.-
.
_I


0
0

II


C\J


I I l
-0 7

(A9) DA


0

II
0
,-
)
















CHAPTER IV
ROTATIONAL AND VIBRATIONAL ENERGY TRANSFER IN HYPERTHERMAL
COLLISIONS OF Li+ WITH N2, CO AND CO2



The many-body theory of atom-molecule collisions that was presented

in Chapter II is applied here to the study of rotational and vibrational

energy transfer in the scattering of Li+ ions from N2, CO and CO2. In

Section 1 we evaluate the quantities that are required for the calcula-

tion of the atom-pair correlation function (APCF) of a diatomic target.

In Section 2 we present the theoretical results for the systems Li /N2

and Li /CO, and compare them with the experimental measurements of

Bottner et al., 1976. In the following section, we briefly outline the

steps of the evaluation of the APCF of a linear triatomic molecule.

Theoretical probabilities of vibrational excitation and average energy

transfers for the system Li /CO2 are presented in Section 4. Comparing

these results with those of experiments (Eastes et al., 1977) and analyzing

the theoretical model, we gain insight into the dynamical processes and

intermolecular forces that dominate atom-molecule scattering in the hyper-

thermal energy regime. Finally, in Section 5 we calculate the effects of

molecular anharmonicity on the vibrational energy transferredto the tar-

gets N2, CO and CO2.

1. The Atom-Pair Correlation Function of a Diatomic Molecule


The evaluation of the APCF of equation (2.6.8) requires knowledge
4->
of the equilibrium positions of the atoms (d ) and of the matrix elements

(C ai) of the transformation between Cartesian vibrational displacements
ai










(ua) and normal coordinates (Qi). For a diatomic molecule, equation

(2.2.1a) shows that there is only one vibrational degree of freedom, and

that d1 = -m2dl2d/mB and d2 = mldl2 d/mB; here m is the mass of the target

atom a, mB is the mass of the diatomic, dl2 is the equilibrium bond-length,

and d is a unit vector in the direction of the molecular axis, from atom 1

to atom 2. From the normal-mode analysis outlined in equations (2.3.8-15)
'-2+ 4 12
we obtain C11 = -[m2/(mlm) ] d and C21 [m/1(m2mB)] d, and for the har-

monic vibrational frequency, w1 = [kllmB/(m m2)] where kll is the quad-
11 12 11
ratic force-constant.

The rotational energy transfers and widths require the commutators

([i ,) exp (-iK *d )], for n = 1 and 2. These are readily evaluated for

a linear polyatomic in the coordinate representation, where the rotational

Hamiltonian is given by (Edmonds, 1974)

2 2
H 2[ ( -cos2 ) I (4.1.1)
R 21 [(cosB) ((cos)B) 2

4->
Here, a and B are the azimuthal and polar angles of d in the space fixed

(SF) reference frame, respectively, and I = m d2 is the moment of in-
B a a
a
ertia of the polyatomic. Taking the z-axis of the SF frame along the

direction of K, so that K-d = Kda cosB, and performing the necessary
a
differentiations, yields


[HR, exp(-iK d)] = (a /2IB) exp(-iKd cosB)x


x{K2d2 (1 cosi2)- 2iKd [cosB (1 cos2 B)3/9(cos 2). (4.1.2)
a a

Whenever Kda >> 1, only the highest power of Kda need be retained in the

above commutator; in these cases, one has that


([,)n exp (-i a