• TABLE OF CONTENTS
HIDE
 Front Cover
 Dedication
 Acknowledgement
 Preface
 Table of Contents
 Abstract
 Introduction
 A study of single-electron and...
 Diatomic molecular orbital correlation...
 A coupled channels approach to...
 References
 Biographical sketch














Title: Electronic and dynamical aspects of diatomic systems /
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 Material Information
Title: Electronic and dynamical aspects of diatomic systems /
Alternate Title: Diatomic systems, Electronic and dynamical aspects of
Physical Description: xiii, 218 leaves : ill., diagrs., graphs ; 28 cm.
Language: English
Creator: Bellum, John Curtis, 1945-
Publication Date: 1976
Copyright Date: 1976
 Subjects
Subject: Collisions (Nuclear physics)   ( lcsh )
Physics thesis Ph. D
Dissertations, Academic -- Physics -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 212-217.
Additional Physical Form: Also available on World Wide Web
Statement of Responsibility: by John Curtis Bellum.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00097498
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000180229
oclc - 03173496
notis - AAU6759

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Table of Contents
    Front Cover
        Page i
    Dedication
        Page ii
    Acknowledgement
        Page iii
        Page iv
    Preface
        Page v
        Page vi
        Page vii
    Table of Contents
        Page viii
        Page ix
    Abstract
        Page x
        Page xi
        Page xii
        Page xiii
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
    A study of single-electron and total energies for some pairs of noble gas atoms
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
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        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
    Diatomic molecular orbital correlation diagrams for penning and associative ionization
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
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    A coupled channels approach to penning ionization of Ar by He
        Page 114
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    References
        Page 212
        Page 213
        Page 214
        Page 215
        Page 216
        Page 217
    Biographical sketch
        Page 218
        Page 219
        Page 220
        Page 221
Full Text











ErLECTRONIC AND DYNAMICAL, ASPECTS
Of DIATOMIC SYSTEMS










By

JOHN CURTIS BELLUM


A D:;SESTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY











UITIVERSITY OF FLORIDA


1976























Dedicated to



My Parents,



who, though familiar with little of what is reported here,

nonetheless know what I have been doing

and have supported me in it.











ACKN OWLEDGE MENTS

I want to express my deep appreciation to my advisor,

Professor David A. Micha, for the direction, support and

financial help he has provided me during this doctoral re-

search. I am most grateful for his high caliber of scien-

tific excellence and integrity, and for the great patience

he has shown in guiding me in research.

I owe a debt of gratitude to the numerous people, asso-

ciated at one time or another with the Quantum Theory Pro-

ject, who have been of special assistance and encouragement,

and whose influence has come to bear either directly or in-

directly on this dissertation. Among these are several I

want to mention in particular. Professor N. Yncve Ohmn pro-

vided me financial assistance during my first months at the

Quantum Theory Project, and has since maintained interest

in my work and progress. Professor Erkki J. Brandas and

Dr. Rodney J. Bartlett initially introduced me to research

in quantum chemistry, and generously made available to me

their expertise and enthusiasm. In connection with multi-

ple-scattering and local exchange related matters, Profes-

sor John W. D. Connolly. through lectures an? ready atten-

tion to my inquiries, was of great help. Close associations

with Drs. Suheil F. Abdulnur, Poul JOrgensen and Jian-Min

Yuan, and Professor Manoel L. de Siqueira, have been per-

sonally beneficial as well as significant to my overall









perspective in science. The able leadership of Professor

er--COlov Ldwdin as director of the Quantum Theory Project,

along with his nuTmerous lectures and also his interest in

philosopbtical considerations in science, have played an

important role in my graduate education.

As is the case with any undertaking, a program of

graduate studies provides a context and situation in which

one is able to grow personally in all respects. In this

regard I want to express my gratefulness to many friends,

both scientist and non-scientist alike, as well as to God,

my creator, by and in whom I exist.













PREFACE

I find it appropriate to make some remarks concern-

jng the perspective and context in which the work reported

in this dissertation has been carried out.

Since its inception in the 1920's, Quantum Mechanics

has become well established as the suitable framework

within which to describe phenomena of a physical and chem-

ical nature. Building upon only a few axioms, the formal

Quantum Theory manifests itself in the form of mathemati-

cal equations, the solutions to which determine expressions

for calculating physically observable quantities. Confi-

dence in Quantum Mechanics derives from the impressive

successes it has had in providing results in agreement with

experiments. However, in applying Quantum Mechanics to

describe actual, known, physical and chemical phenomena,

one quickly becomes aware of the fact that there are only

a few cases where an exact treatment has been possible.

In nearly all cases of interest, the mathematical equa-

tions of the theory, though succinct in what they say, are

unmanageable to solve. Computational considerations,

therefore, have strongly influenced theoretical investiga-

tions in physics and chemistry.

Basically two approaches have evolved. The so-called

ab initio calculations provide approximate solutions to










the "exact" quantum mechanical equations within a frame-

work which, in principle, allows for the solutions to be

progressively improved upon to approach the "exact" solu-

tions. One is presumably limited here only by the size

of the electronic computer available. On the other hand,

one may focus on the main features of some particular

physical phenomena of interest, and use the "exact" quan-

tum mechanical equations only as a guide in order to ar-

rive at approximate equations which mimic the important

aspects of the process being studied. These approximate,

or model, equations are many times only approximately solved!

Such apparently crude approaches require of the researcher

all of the physical intuition which can be mustered, in

order to properly assess the important features of the

physical situation and to approximate them reasonably well.

Nevertheless, much physical insight and many useful quan-

titative results can be extracted from this point of view.

Indeed, the task of science is essentially to formulate

descriptive statements, both qualitative and quantitative,

which conform as nearly as possible to the laws and phenom-

ona of nature as we observe them.

Finally, it should be mentioned that there are formal

results coining out of quantum theoxeticel investigations

which determine many characteristics of the "exact" and

approximate solutions, even before they have been calcu-










laced. Both types of investigations described above rely

upon these formal results as well as upon each other. In

the dissertation which follows, research of the second

type mentioned above will be reported in the form of a

quantum mechanical investigation of electronic and dynam-

ical aspects of diatomic systems.













TABLE OF CONTENTS


Page

ACKNOWLEDGEMENTS . . . . . . . .. iii

PREFACE . . . . . . . . . . .

ABSTRACT. . . . . . . . . . . . x

CHAPTER

1. INTRODUCTION . . . . . . . ... .

1. A Formal Statement of the Problem. . . 2

2. Remarks Regarding Reference Frames . 6

3. Remarks Regarding the Wave Function. .. 17

4. The Coupled Equations and Coulomb and
Born-Oppenheimer Couplings .. . .. 19

5. Discussion . . . . . . ... 24

II. A STUDY OF SINGLE-ELECTRON AND TOTAL ENERGIES
FOR SOME PAIRS OF NOBLE GAS ATOMS. . . .. .28

1. Theoretical and Computational
Considerations . . . . . ... 30

2. Results. . . . . . .. . . . 39

3. Discussion .. . . . . . . 55

IIT. DIATOMIC MOLECULAR ORBITAL CORRELATION
DIAGRAMS FOR PENNING AND ASSOCIATIVE IONIZA-
TION . . . . . . .. . . . 59

1. MO Calculations for He*+Ar and He+Ar . 61

2. Analysis of PI and AI Processes Based on
MO Correlation Diagrams. . . . . 74

3. Estimating MO Correlation Diagrams for
Diatomics. . . . . . . . ... 87

4. Discussion . . . . . . . . 110


viii









TABLE OF CONTENTS
(Continued)


Page


IV. A COUPLED CHANNELS APPROACH TO PENNING
IONIZATION OF Ar BY He*(ls2s,3S) . . . .

T. The Scattering Problem in Terms of
Discrete and Continuum Electronic States .

2. Discretization of the Continuum and
the Modified Coupled Equations . . .

3. Solution of the Modified Coupled
Equations. . . . . . . . .

4. Characteristics of PI and AI Processes

5. An Application of Discretizution to
PI and AI . . . . . . . .

6. Interaction Potentials for He' (ls2s,3S)
Ar and He + Ar+(3p5,2p). .. . .....

7. Parameterization of the Couplins. . .

8. Results from Coupled Channels Calcula-
tions of He*(ls2s,3S) + Ar PI Collisions

9. Discussion . . . . . . . .

REFERENCES. . . . . . . . . . . .

SBGGPAPHICAL SKETCH . . . . . . . . .













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



ELECTRONIC AND DYNAMICAL ASPECTS
OF DIATOMIC SYSTEMS

By

John Curtis Bellum

August, 3976

Chairman: David A. Micha
Major Department: Physics

Diatomic systems are considered from the points of

view of their electronic structure and the dynamics of

motion of the heavy particles (nuclei) upon collision. In

Chapter I the electronic and nuclear motions are treated

formally by expressing the Schr6dinger equation for the

nuclei and electrons in independent variables in the body-

fixed (BF), center of mass of the nuclei, frame, and then

introducing an expansion in terms of a complete set of

electronic states at each internuclear separation, R. Born-

Oppenheimer and Coulomb couplings between the electronic and

nuclear motions are pointed out and discussed.

In Chapter II, atom-atom interaction potentials and

electronic structure are investigated using the Multiple

Scattering (MS) calculational scheme with the so-called Xa

approximation to electronic exchange. The atom pairs,









ae--He, He-Ar and Ar-Ar are studied, and results are presented

for ground and excited state configurations. The computed

interaction potentials exhibit BoEn-Mayer repulsion, and the

calculations demonstrate how Xa orbital energies can be

used to predict crossings between interaction potentials.

An analysis is presented of the Xa theory and its usefulness

for the undsrstandinq of collision phenomena.

In Chapter III, Penning ionization (PI) and associative

ionization (AI) processes are considered in terms of molecu-

lar orbital (MO) correlation diagrams. MO correlation

diagrams are calculated for He*(ls2s) + Ar(3p6) and He(ls2)

+ Ar (3p5) within the MSXe scheme for non-spin-polarized

and spin-polarized orbital. The ionization process is

discussed in terms of an Auger type mechanism involving

MO's which can be inspected in the united-atoms (UA) limit

in a way which permits an analysis of the angular momentum

contributions of the emitted electron in the BF frame. MO

correlation diagrams are constructed based on atomic orbital

energies at the separated- and united-atoms limits, which

are determined from data available in the literature on

ground state atomic orbital energies. Estimated MO correla-

tion diagrams are presented for He*(ls2s) + Ar(3p6),

+ Xr(4p6 ), + Hg(6s2), and Ne*(2p53s) + Ar(3p ),and in each

case an analysis is made of the angular momentum components

of the emitted electron. The results confirm that relatively

few such components are important for electrons emitted in









PI and AI. The UA analysis shows the importance of spin-

polarized MO's, and also Born-Oppenheimer rotational

couplings, particularly between MO's which converge to the

same UA limit.

In Chapter IV consideration is given to the dynamics

involved in collisions Lonization processes. The formal

development of Chapter 1 is extended to include both discrete

and continuum internal electronic states. The resulting

continuously infinite set of coupled equations is then dis-

cretized, leading to modified coupled equations for the

heavy particle motion. Discretization provides a suitable

framework in which to introduce physically reasonable ap-

proximations which lead to a treatment of PI and AI in terms

of several (=20) two-state coupled equations. Application

is made to PI of Ar by He*(ls2s,"S), and the results show

that the approach includes the important dynamical features.

Partial ionization cross-sections per unit energy, E, of

the emitted electron are calculated as a function of E, and

they show an s dependence in good qualitative and quanti-

tative agreement with experimentally measured energy dis-

tributions of emitted electrons. Partial cross-section

contributions for the heavy particles in specific angular

momentum states are also singled out." Their behavior as a

function of e, or of the angular momentum partial wave

number, shows structure which reflects regions of high

density of states in the continuum of final relative motion









of the heavy particles. The He-Ar netastable and molecular

ion potentials are represented by a convenient functional

form describing atom-atom int.ract-ion potentials over the

entire range of R. In addition, the connection between the

decay width T and the coupling matrix elements between

discrete and continuum electronic states is used to make

reasonable estimates of the latter from semiempirical results

for P.


xiii












CHAPTER I

INTRODUCTION



The overall subject of this dissertation is the study

of electronic and dynamical aspects of diatonic systems.

In such a study it is the behavior of the electrons and

nuclei, during collision processes of the two atoms com-

prising the diatomic, which is of interest. In this intro-

ductory chapter the collision processes will be discussed

formally in terms of the Schrodinger equation which is sat-

isfied by the wave function for the system of nuclei and

electrons. The Schr6dinger equation will be treated in the

body-fixed, center of mass of the nuclei, frame by properly

transforming the Hamiltonian operator and wave function to

be expressed in this reference frame. The coupled equa-

tions will be derived with attention focused on the various

sources of coupling between the electronic and nuclear mo-

tions.

In the following chapter, matters specifically concern-

ing the electronic structure of some diatoms comprised of

rare gas atoms will be considered, and possible applications

to collision processes discussed. In Chapter III special

attention will be given to chemi-ionization processes in-

volving collisions in which one of the atoms is initially









in an excited state. Features of the electronic structure

of such collisions will be discussed, on the basis of which

an analysis of the angular momentum contributions to elec-

trons emitted in such- processes will be carried cut. Fi-

nally, in Chapter IV, th: dynamics of a specific chs-mi-

ionization process, that of Penning ionization of Argon by

metastable Helium,will be treated by means of numerically

solving the coupled equations within a two-state approxima-

tion. Total and partial cross-sections obtained from the

calculations will be reported.



I. A Formal Statement of the Problem

In the quantum mechanical treatment of the two nuclei

and N electrons which constitute a diatomic system, the

Ilamiltonian operator expresses all of the energy contri-

butions associated with the nuclei and electrons. The

actual form of the Hamiltonian depends upon the coordinate

frame with respect to which the positions of the nuclei

and electrons are located. The coordinate frame of most

practical use in terms of measuring the results of colli-

sion events is one fixed in the laboratory, referred to as

the laboratory-fixed (LF) frame. The description of the

collision events, however, is most conveniently carried

out in a coordinate frame whose origin is fixed to the cen-

ter of mass of the nuclei (CMN) of the two atoms which are










,olliding. The reader is referred elsewhere Csee, for

example, Pa6S and references therein, and Ju75) for a de-

tailed discussion of various coordinate frames commonly

used and the form the Hamiltonian assumes under transfor-

mation from one to the other. The formal development in

this chapter begins with the Hamiltonian expressed in the

so-called laboratory-fixed, center of mass of the nuclei

(LF-CMN), frame, whose origin is fixed to the CMN and whose

axes remain parallel to those of the LF frame. Furthermore,

relativistic and mass-polarization contributions to this

Hamiltonian are neglected (Pa68), and the coupling between

electronic spin and orbital angular momentum is assumed to

be small. In this LF-CMN frame, R is the relative position

vector of the two nuclei labeled a and b, having masses m
a
and mb, respectively. The N electrons are located by the

set of space-spin coordinates {x.,i=.,N} = X, where

x. = (ri,s.), ri locating the spatial position and s. the

spin coordinate (a or 0) of the ith electron.

In terms of these variables, the LF-CMN Hamiltonian

is expressed as

2
H(R,X) = -(1/2m)VR + H(RX)


where m = m mb/(ma+mb) is the reduced mass of the nuclei,

and









N N
H e(R,X) = -1/2 (Z /r + /r )
e,(,2 r. _, a ia b io
i=l ri i=1- /


+ I/rij + Z Zb/R (2)
i
is the electronic Hamiltonian. In the usual sense, Za,

r. and r.. refer to the charge on nucleus a, the distance
:iaF 13
between the ith electron and nucleus a, and the distance

between the ith and jth electrons, respectively. The ex-

pressions are in atomic units, where the unit of energy is

the Hartree, the unit of distance the Bohr radius (ao)

and the unit of mass that of the electron.

On the right side of Equation (1) the first term rep-

resents the relative kinetic energy of the two nuclei,

which are the heavy particles taking part in an atom-atom

collision. The terms on the right of Equation (2) repre-

sent, in order, the kinetic energy of the electrons, the

Coulomb attraction energy of the electrons with the nuclei

a and b, the electron-electron Coulomb repulsion energy

and the Coulomb repulsion energy between nuclei a and b.

The description of a diatomic system can formally be

made in terms of the wave function, YCR,X), for the total

system of nuclei and electrons, which satisfies the time

independent Schradinger equation,


H(R,X)Y(R,X) = ET(R,X) ,










where E is the total energy of the system. In solving

Equation (3) it is important to pay attention to the angu-

lar momentum' of the electrons and nuclei. The total or-

bital angular momentum. K, is the sum of the nuclear and
( ) (e)
electronic orbital angular moment, and L where

Z(n) -
= x iVR (4)


and

N
+(e) N
L"' - r. x iV (5)
i=l r
2 +
The LF-CMN components, K K and K as well as K = K-K,
y z
all obey the usual commutation relations for angular momen-

ta, and commute with the LF-CMN Hamiltonian of Equation (1),

owing to its rotational invariance. Accordingly, the solu-

tion, T(R,X), of Equation (3) is simultaneously an eigen-
2
function of K and K with eigenvalues K(K+l) and M,

respectively, and the total orbital angular momentum and

its z component are constants of the motion. Because of

the as umption of negligible spin-orbit coupling for the

electrons, the spin angular momentum has been left out of

this discussion for convenience, but could easily be in-

cluded. So, with no loss of generality, the solution to

Equation (3) is classified according to the constants of

the motion, K and K and is written


Y(ax) = KM(R,X)
KM










SRemarks Regarding Reference Frames

At this point attention will be turned toward specif-

cally pedagogical considerations, providing a reminder of

some basic concepts of a mathematical and physical nature

which are helpful in understanding the approach which will

be taken in solving Equation (3). The remarks which fol-

low will serve to reiterate some key ideas which have long

been established (see, for example, Kr30). The ideas are

not easy to grasp, and are often passed over either in too

sophisticated or too cursory a way in the literature. No-

table exceptions, however, can be found (Kr30, V151, Ho62,

Th6i, Th65, Pa68, Sm69).

As it stands, Equation (3) involves the Hamiltonian

and wave function, Yi', expressed as functions of R and

{ri} referred to the LF-CMN frame. The axes of this frame

are labeled by x, y and z. The ith electron is thus lo-

cated by ri having coordinates (xi,yi,z.). R is most con-

veniently represented by its spherical polar coordinates,

R, 0 and ,. Therefore, in the LF-CMN frame, (x.,yi.,z,

R.0,,) constitute an independent set of coordinates in

which to solve Equation (3).

In terms of these coordinates, the components in the
j(n) *(e)
LF-CMN frame of t) and Le) (Equations (4) and (5)) ap-

pear as










L(n) = i(sini/ae6 + cotocosfs9/9')
x


L(n) = i(-cos +/9 + cotBsin0/ah)
Y

L(n) (7)
L = -i /a (7)
z


and

N
(e) = -i= (v/3z z.i/y.i)


L(e) = -i (zi.3/xi xia/zi)


L(e) = -i I (xiN/3y yi/3xi) 8)
2 Ci=l

The wave function of Equation (6) can also be written ex-

plicitly in terms of these variables to read

SKMRW2) = YKM(R,O,I,xYi ,z ) (9)


Also, the first term on the right of Equation (1), express-

ing the relative kinetic energy of the nuclei, appears as

follows (see, for example, Co62):


-(1/2m)V2 = -(1/2mR2 { 2/R(R23/DR)


+ cot63/oe + a2/Do2


+ sin-2 e2/~ 2}


(10)










One can check that the terms within the brackets of Equa-

tion C10), involving the angles, can be replaced by
I(n' --(n)
-Ln' according to Equation (7) (see, for example,

Ed60). It is important to keep in mind that partial de-

rivatives, such as occur in Equations (7), (8) or (10),

depend upon which variables are actually independent of

one another during differentiation. Because the LF-CM-

frame constitutes an inertial reference frame, the set of

variables, (x ,yi,z ,R,e,4), of the electrons and nuclei

is indeed an independent set of variables. Therefore, for

example, 8/36 in Equation (10) means to differentiate with

respect to 8 while holding all other variables, (xi,Yi,zi,

R,4), fixed. Similar considerations hold in turn for each

variable in this set.

Now one would like to proceed to solve Equation (3)

by expanding the wave function of Equation (9) in a com-

plete set of electronic wave functions at each R. It is

here that the need arises to express Equation (3) in terms

of variables referred to a coordinate frame in which the

internuclear vector R is fixed. This is due to the fact

that normally electronic wave functions are determined in

such coordinate frames. Such a reference frame, fixed to

the nuclei with origin at the CMN, will be called a body-

fixed, CMN (BF-CMN) frame. The key concept regarding elec-

tronic wave functions is that they are usually calculated










under conditions where the set of electronic coordinates

in the BF-CMN frame are treated as independent variables.

The BF-CIN frame considered here shall have axes x',

y', z", where the 2" axis is along the internuclear vector

R. Consequently, the angles ( and 6 are the first two of

the Euler angles Csee, for example, Ju75) which rotate the

SF-CMN frame into the BF-CMN frame. The third Euler angle,

y, can be freely chosen since it only serves to define the

x' and y' axes, which can be arbitrarily set for a. diatomic

molecule (Ju75). Therefore, y is an auxiliary variable,

and as such will play only an indirect role in what follows.

Figure 1 shows the Euler angles (9y) by which the SF-CMN

frame is rotated into the BF-CMN frame.

The coordinates of the internuclear vector in the BF-

CMN frame are simply (R,0,0). The ith electron, with co-

ordinates (xi,yi,'i) in the LF-CMN frame, has coordinates

(x,yl,z') in the BF-CMN frame, specified for a given ori-

entation (06,) of the internuclear vector by the unitary

transformation (see, for example, Ti64 and Ju75)


x. = Ex(COy) = (cosycosecos sinysinc)x.


+ (cosycosfsin + sinycost)yi cosysin9zi


y = y' (0Oy) = -(sinvcosfcos{ + cosysinf)xi
wI 1


+ (-sinycosesin + cos-cosi)yi + sinysinOz.
l





-10-


- -y


The Euler angles (0y) reiating the SF-CMN
frame (x,y,z axes) to the BF-CfN frame
(x',y',z axes). y specifies the x' and v'
axes, which can be arbitrarily chosen and are
not shown. R labels the internuclear vector.


Figure 1.





-11-


z. = z'(00) = sinOcosix. + sin9siny.i + coszi (11)



Here attention has been drawn to the dependence of x ,y'

and zI on the angles 6 and y. The inverse transformation,
1
th
qjiving the coordinates (x.,yi,zi) of the i electron in

the SF-CMN frame in terms of its coordinates (xi,yC,z)

in the BF-CMN frame, is,


x. = (cosycosOcos sinysinp)xC


-(sinycos6cosj + cosysinp)y + sin6cos6z'


y. = (cosycosOsin4 + sinycost)x1


+ (-sinycosOsint + cosycos)y. + sinOsin2z


z. = -cosysin9xC + sinysinly$ + cosOz (12)


With these expressions, the Schr6dinger equation of

Equation (3) in the LF-CMN frame, where (x.,Yi,z.i R,,i)

are independent variables, can be rewritten allowing the

BF--CMN electronic variables (x,,y',zi) to be treated as

independent variables (as they are in molecular electronic

structure calculations). The expression, "treated as in-

dependent variables", in the preceding sentence speaks

to an important concept. To an observer in the BF-CMN





-12-


frame, the electronic and nuclear coordinates are simply

x' ,y',z',R,0,0). But such an observer must keep in mind

that the BF-CMN frame is not an inertial frame, and, by

Equation (11), (x,y',zf) have explicit (86y) dependence

for a given set of electronic coordinates in the LF-CIN

frame.

In order for an observer in the BF-CMN frame to use

the electronic variables Cx ,y',z ) as independent variables

in treating Equation (3), two new variables, 6' and 4',

may be introduced with two restrictions: (3.) the set

(xryt,z3,R,6',') must be an independent set of variables

for the observer in the BF-CMN frame, and (2) 6' and #^

must be given by,


= ,
(13)



According to this equation, ("O') may seem to be redundant

variables, but this is not at all the case. The variables

(Cx,y,zt) are independent of (6'o') as far as the observer

in the BF-CMN frame is concerned, and Equation (13) simply

specifies their values in terms of variables which are de-

termined by an observer in the LF-CMN (inertial) frame.

Equation (13) gives the relationships for 8' and "' just

as Equation (11) does for x',yv and z'.
Equation (3) will now be transformed so that an ob-
Equation (3) will now be transformed so that an oh-





-33-


server in the BF-CMN frame could attempt to solve it in

terms of the independent set (x 'y!,z,R,6',I'). Sach a

transformation involves both the Hamiltonian operator (see

Equations (1), (2) and (10)) as well as the wave function

(see Equation (9)). First the Haiiltonian will be consid-

ered.

For clarity, when performing partial differentiation

involving the independent variables Cx ,y,z i,R,9,4) the

symbol 3, which has already been used, will now specific-

ally indicate that,when a variable of differentiation from

this set has been singled out, all the others are held

fixed. For instance,

a/as =- a9/l{(xy ,z.),R,} (14)



where the variables held fixed are explicitly indicated.

Similarly, 3' will be used to indicate partial differen-

tiation involving the independent variables (x',y',zi,

R,m e,''). By analogy with Equation (14) for instance,


= (15)



Then, if w is one of the variables Cxi,yi,zi,R,e6,),





-14-


3/3w -t (+e'/w)V'/2e' + (U4t/2w)V'/24^


N
+ ((x /3w) 3'/@x + (3yC/Dw) 3'/y' *+ ( 3z/3w) -'/9z) (16)



A similar relation holds for expressing 3'/Dw" as a linear

combination of partial derivatives involving the variables

(xi,yi,zi,R,6,4) where w' is cne oE the variables (x.,yi,

zt,R,9',O'). These are useful] expressions to keep in mind.

especially as to their meaning emphasized by using the 3

and 3' notation. This conceptual and notational viewpoint

follows the work of Kronig (Kr30).

Referring now to Equation (2), the kinetic energy of
th
the i. electron involves the operator,

i2 2 x 1 a2 2 2 2
V2 = 2/x2 + 2/Y2 + 2 /2z2 (17)
r ii i


On the basis of Equations (11) and (16), one can write


3/;xi (cosycosecos4 sinysin)}a-/3x'


(sinycosOcosp + cosysin)3 '/2y' + sin8cos'/Dz: (18)
1 1

with similar expressions for 2/9yi and 3/2zi. Using these

expressions directly, one finds that


2 2 2/ y 2 +2 2/
3 /3x.+D /9y. + 2 /2z.



,2/ax 2 + 2a2/y2 + ,2 /az 2 = V2 (19)
i




-15--


The potential energy terms in Equation (2) involve only

the distances between electrons and nuclei, and therefore

are unchanged in going from the variables (xi,yi,zi,R,0, )

to the variables (X ,y ,z,R,6 ,'). Consequently, the

electronic Hamiltonian of Equation (2) takes The same form

for an observer in the BF--CMN frame as for an observer in

the LF-CMN frame.

Such is not the case for the first term of Equation

(1), which is expressed in the variables (R,6,<) in Equa-

tion (10). By a straightforward, but tedious, application

of Equation (16), using Equations (11) and (12), one finds

that

(e) (e)
3/30 '/38' icosyL isinyLxe (20)


and

e) (e) (e)
3/3( -- a'/8' icosO'L Ce + isin0'(cosyL sinyL )
2 x y
(21)
(e) (e) (e)
Here, L L and LI are the components of the elec-
x y z"
tronic orbital angular momentum in the BF-CMN frame. They

are expressed according to Equation (8) by replacing all

LF-CMN quantities by their appropriate BF-CMIN (primed)

counterparts. Equations (20) and (21), again with some

tedious algebra, lead to the following result for the an-

gular terms in Equation (10):





--C6-


cot3/89 4 2/302 + sin 2 02/32


Scot8."3'/3ae + ^ /9ae0 + sin 6 '3 /WS


.e- i:! e) + + iyL(e_ D /a0 kL'(e) 2
4-7a

-2i(cot0/sinO)3'/'L- ) cot2 .L ) 2 + (Le)
z 2 z


- (2/sin9) (cosyL ) sinyL ) (ie'/4g cosSL e) (22)
x Y z




(e (e) C (e)
Here, in the usual way, Le)" = LX + iL ). The R depen-

dent terms in Equation (10) are unaffected in the transfor-

mation from the LF-CMN frame to the BF-CMN frame. Conse-

quently, replacing the angular terms of Equation (10) by

the right side of Equation (22) leads to the appropriate

Hamiltonian operator (see Equation (1)) which can be used

by an observer in the BF-CMN frame in order to formally

treat the behavior of the electrons and nuclei.

The first three terms on the right side of Equation

(22) are similar to those found in the LF-CMN Hamiltonian.

The remaining terms are those compensating for Coriolis ef-

fects due to the fact that the BF-CMN frame is not an in-

ertial frame. In a sense, the inclusion of these Corolis

terms is the price paid by the BF-CMN observer in order to

reckon (x ,y',z',R,6',0') as independent variables.




-17-


3. Remarks Regarding the Wave Function

The wave function of the total system of electrons and

nuclei (see Equation C91) mush also be properly transformed

and expressed in terms of appropriate functions in the BF-

CMN frame. Here the approach of Davydov (Da65) is adopted.

Attention is drawn again to the fact that the wave

function of Equation (9) is an eigenfunction of the square

of the total orbital angular monentmn, K and its component

along the LF-CMN Z axis, K with eigenvalues K(K+1) and M,

respectively. Following Davydov (Da65), if a coordinate

frame undergoes a transformation by rotation through Euler

angles (atS) to another coordinate frame, then an eigen-

function in the first frame of K2 and K with eigenvalues,

respectively, K(K+1) and M, can be written as a linear com-

bination of the (2K+1) such eigenfunctions in the rotated

frame, all of which are eigenfunctions of K2 with eigen-

value K(K+I), and each of which is an eigenfunction of K z

with eigenvalue, A, among the possible values K,K-1,....,-K.

For the case considered here, of a transformation from

the LF-CYN frame to the BF-CMN frame through Euler angles

(Gy), the wave function of Equation (9) can be written


K
K (R,,,xiyiz D ((R,,,x y z)
A=-K
(23)










THere the expansion coefficients, D (Oy), are the so-called
11,A

generalized spherical functions, or D-functions, and are

eigenfunctions of the symmetric top. A good discussion of

their properties is given by Edmonds (Ed60). In Equation

(23), .M A(R,0,0,x ',z') is an eigenfunction of KZ the
KA (R, ,xii,
component of K along the internuclear axis, with eigenvalue

t. The orbital angular momentum of the nuclei has no com-

ponent along the internuclear axis, as can easily be veri-
(e)
fied from Equation (4). As a consequence, Kz = LZ

It is at this point that it is convenient to intro-

duce a complete set of electronic wave functions at each

internuclear separation, R. The electronic wave functions

employed can be any of the ones commonly calculated, where

the electronic variables, (x',yi,z), in the BF-CMN frame

are taken as an independent set of variables. Such elec-

tronic wave functions, whether of the single-configuration

or more elaborate configuration-interaction type, are clas-

sified according to their component of electronic orbital

angular momentum along the internuclear axis. That is,
(e)
they are constructed as eigenfunctions of L e having ei-

genvalues denoted by A. For each A, the complete set of

electronic wave functions, { nA(R,x ,y ,z')}, will be in-

troduced at each R, and will be taken to be orthonormal.
M
Then each yK(R,0,0,x',y',z') of Equation (23) may be ex-

panded in the set of electronic wave functions, {,nA}:




-19-


Y (R0,0,xC,y',z) = "'(R)R)/)0Rn (R,xC,yC,z ) (24)
1A i nh' nA
n




Substitution of Equation (24) into Equation (23)

gives a useful expansion for the total wave function of

nuclei and electrons in terms of functions of the elec-

tronic variables C(x,yi,z) in the BF-CMN frame:


YKM(R,8, ,x.,yiz ) = (1/R) (R)DK (ey)n (Rx',yzi).
KM n An
(25)
It should be emphasized that the equal sign in Equations

(23) and (25) means equality only in the sense that in

each case the function on the left of the expression,

where the variables are the coordinates of the electrons

and nuclei in the LF-CMN frame, can be replaced by the

linear combination of functions on the right of the expres-

sion, where the variables are the coordinates of the elec-

trons and nuclei in the BF-CMN frame. The presence of the

variables 0 and 9 in the D-functions on the right of Equa-

tions (23) and (25) will be discussed presently.



4. The Coupled Equations and Coulomb and
Born-Oppenheimer Couplings

Recalling the previous discussion regarding the Haril-

tonian operator, Equations (19) and (22) provide an observ-

er in the BF-CMN frame the appropriate Hamiltonian for





-20-


the nuclei and electrons under conditions where (x ,y ,z'

R,0',$') are independent variables. Using this Hamiltonian,

an observer in the BF-CMN frame may now replace ( and e on

the right hand side of Equation (25) by <' and 8' according

to Equation (13). Then the right side of Equation (25) be-

comes an appropriate expansion for the "transformed" wave

function, KM(R,O' ,,x',y',iz'l), which satisfies the "trans-

formed" Schr6dinger equation. Thus, an observer in the

BF-CMN frame may proceed to solve Equation (3) by relying

on Equations (19) and (22) and solving for the wave func-

tion


KIM (K
Y M(Re ,',x',yZl)=(1/R) Tn (R)D ($ e'Y)nA (Rxy z.
i I An n M,n JA 1IJ(
An
(26)

In dealing with the operators in Equation (22), it

will be useful to rely on some of the formal properties of

the D-functions (Ed60). The D-functions, D (9y) are

formally associated with rotations through the Euler angles

(Jey), and these rotations are generated by the angular
4-
momentum operator, N, whose components are


(n)
N (04y) = L i(cosj/sin6)/3y ,
x x

(n)
N (46y) = L i(sin4/sin)6/y ,
y y


N (46y) = L(n)
z z





--21-


where L(n), L(n) and L(n) are giver in Equation (7). The
x y z
2 2 2
D-tunctions are eigenfunctions of N N +N +N
x y z

N2(6Y)D (6) = K(K+41)DK (y) (28)


By expanding out N 2(Oy) using Equation (27), and recalling

tha-c the ( and y dependence of D CA(By) is in the factors

e and e Equation C28) can be rearranged as follows

(Ed60):




-2 2 K
(O2/e2 + cotW/39 + sin-2S /42)DK A((9y)


= {sin -2(A 2cosOMA) K(K+1)}D ,A (y) (29)



Other useful relationships involve the operators

N (tey) = N iN In particular, based on the properties
x y
of the D-functions (Ed60), it can be shown that

N. ( K (Y)q(0y) = At+l(O) (30)


where [1 = {K(K+1) A(CAl)}1. Furthermore, 3/38 can be

expressed as

3/ae = '(e N el% N) (31)


Using Equations (19), (22) and (31), the BF-CMN Hamil-

tonian can be expressed as follows:

H(R, 0',',x ^,y,z 2= -(1/2nR2)D/9R(R2 9/R)


(1/2mR2) )cot9'a'/a6 + 2/8-2 + sin-2 e'2 /at2




-22-


(e i 2 (e) () 2 (Ce) 2
-ji(e 'L+ ) + e- L_ ) (e N N (4 0"y) -- ei N ( 0^-e ) )

(L(e) 2 2i(cot6/Asin) -cotO (L ) + (L
z z z

iY (e) e(e)
/sin ')(e L +e -l'e) )(i'/ cosOl'L )}

+ He(R,xCyCz ) (32)


Now, in the usual way, the right side of Equation (26) may

be substituted into the "transformed" Schr6dinger equation,

(H E)NKM = 0 (33)

When this is done, many of the terms from the operators of
K
Equation (32) acting on functions D 5 n of Equation (26)

will combine and lead to simplifications. One can, for

example, compare the contributions from some of the opera-
(e)
tors involving L e in Equation (32) with the terms in-

volving A on the right side of Equation (29).

Upon multiplication of Equation (33) on the left by
D ,A' I'Y) *, (R,x ,yz) followed by integration over
M,A- nA I i
the coordinates C'8'0yr xtytz2, the following set of
KwM
nA

{-(/ d/dR2 -(/2mR2) [2A2 -K(K+1)] E n(R)


= ({-< nH eli A> (I/2l~R22


+ (1/n-m)< d/dRnA> + (/2m)6A,
+- (I/m)<~nhld/dlR/ Pn.A. + (i/2m)< ~nAd2/dR21 bn.A.>} AA




- 23-


2 (e)I KM+
-(1/2mR 2)< ,i' I,-AA +


_1I/2mR,2) KM- (KM R) 34)
/ nA n'A AA A,A'-1 n ( (34)


Here, the brackets indicate integration over electronic
KM+
variables. The coefficients g A are given by

KM+ 2 K K *K
gA = (2K+1) (8T2)-1fd'sin6-d6'dyD ^ 6Y)G M (6Y)
(35)
where
K +iy K
A6. y) = C (Msin 0 + Acote')DMA(POY)


+ (i/2)eZiy{A+e-'DA+i ('6y) Ae'DK,Al(eY)} .
(36)
In Equation (34), all of the various couplings be-

tween the nuclear and electronic motions are included ex-

plicitly. The radial Born-Oppenheimer couplings appear

in the matrix elements between the electronic expansion

states over the d/dK' and d2 /dR2 operators, and reflect the

effect of the radial motion of the nuclei on the electronic

motion. As can be seen, radial Born-Oppenheimer couplings

exist only between electronic states having the same value

of A. The rotational Born-Oppenheimer couplings appear in

the matrix elements over the L(e) and (L(e)2 operators,

and reflect the effect of the rotational motion of the nu-

clei on the electronic motion. A convenience of treating

the problem in the BF-CMN frame is that the influence of

nuclear rotational motion on the behavior of the electrons

becomes expressed in terms of matrix elements between elec-





-24--


tronic states over electronic angular momentum operators,

rather than nuclear angular momentum operators. As can be
(e)
seen, rotational coupling due to L e exists only between

electronic states having A values differing from one an-
(e) )
other by il, whereas coupling due to (L( ') exists be-

tween electronic states having the same A value. The so--

called Coulomb coupling appears in the matrix element over

the electronic Hamiltonian in Equation (34), and exists

only between electronic states having the same value of A.



5. Discussion

Some brief remarks are in order regarding the consid-

erations of this chapter. No qualifications have been

placed on the basis set of electronic wave functions,

{ nA}, used in the expansion of Equation (26), other than

that it be complete and orthonormal. Traditionally, ap-

proaches to molecular electronic structure have tended to

focus on electronic states which are eigenfunctions of the

electronic Hamiltonian, Hel, and which therefore leave the

matrix of Hel diagonal. These are the so-called adiabatic

states which provide an adiabatic representation. In such

a representation all of the coupling between the electronic

states, associated with inelastic collisional processes,

rests in the Born-Oppenheimer terms of Equation (34). The

eigenenergies of Hel associated with these adiabatic states





-25-


obey the non-crossing rule, as is well known.

However, in treating atomic and molecular collision

processes, it is not at all clear that the set of adiabatic

states is always the most appropriate representation to use.

Stemming from the point of view emphasized by Lichten (Li63)

much consideration and discussion have resulted regarding

the importance of the so-called diabetic representations.

These representations are comprised of electronic states

which are not eigenfunctions of Hel. Consequently, the

diagonal matrix elements of these states with Hel need not

obey the non-crossing rule, and the off-diagonal matrix ele-

ments may be appreciable. Compared to the Born-Oppenheimer

couplings, the Coulomb couplings in a diabetic representa-

tion can often actually be the dominant source of coupling

associated with inelastic processes influencing the heavy

particle motion described by the coupled equations of Equa-

tion (34).

Two particularly good discussions of these matters

have been made by Smith (Sn69) and Sidis (Si76). Here, it

is simply pointed out that the appropriateness of the adia-

biatic or a particular diabetic representation depends upon

how successfully the dominant coupling terms can be identi-

fied, as well as calculated or estimated. Radial Born-

Oppenheimer couplings are difficult to calculate and nor-

mally must be estimated. In addition, they are character-










sized by singularities in regions of R near avoided crossings

of the associated adiabatic eigenenergies. The coupling

through Hel of diabetic states can be estimated, if not

often calculated. However, because of frequent lack of

information abcut the Born-Oppenheimer couplings, one can-

not always be sure when the Coulomb couplings constitute

the dominant contribution in describing inelastic processes.

As can be seen from Equation (34), the rotational Born-
-2
Oppenheimer couplings have a R2 dependence. Therefore,

their contribution will be of increasing importance as dis-

tances of closest approach of the nuclei become smaller.

The research related to atomic collision processes re-

ported in the remainder of this dissertation has been car-

ried out within the framework of diabatic representations.

In this connection, electronic states constructed as deter-

minants of one-electron molecular orbitals are particularly

useful. Furthermore, the behavior of the one-electron mo-

lecular orbitals and associated orbital energies can itself

provide information of use regarding inelastic processes in

atom-atom collisions. Considerations along these lines are

pursued in Chapter II, illustrated by molecular orbital cal-

culations on some rare gas diatomic molecules, and in Chap-

ter III, where Penning and associative ionization processes

in thermal energy collisions of excited (metastable) rare

gas atoms with ground state atoms are discussed in terms of










molecular orbital correlation diagrams. An assessment of

the angular momentum contributions in the EF-CMN frame of

the emitted electron in these processes is made based on

the correlation diagrams analyzed in their united atoms

limit. In this united atoms analysis the significance of

rotational Born-Oppenheimer couplings will need to be con-

sidered, as has just been mentioned.

The topic of study in the final chapter is the dynam-

ics involved in Penning ionization of Argon by metastable

Helium in thermal energy collisions. The approach will be

to solve numerically in a two state approximation the cou-

pled equations of Equation (34). An interesting feature

of chemi-ionization is that the electronic state prior to

ionization is embedded in the continuum of electronic states

associated with the ionized electron. Thus, in the expan-

sion of Equation (26), the sum over discrete electronic

states must be augmented by an integral over the continuum

electronic states. This feature will be dealt with in

Chapter IV. The solution of the coupled equations with

the appropriate boundary conditions for scattering will

lead to results for total and partial cross sections for

Penning ionization.













CHAPTER II

A STUDY OF SINGLE-ELECTRON AND TOTAL ENERGIES
FOR SOME PAIRS OF NOBLE GAS ATOMS


In this chapter* consideration will be given to

atom-atom interaction potentials and electronic structure

pertinent to the description of collision events. Sev-

eral features should characterize the method employed in

this type of treatment of the electronic structure of

atom pairs:

(a) The method should provide a description of

the interaction potentials for the ground

as well as excited states which govern the

motions of the heavy particles (nuclei)

during collisions.

(b) Details of the electronic structure, at least

of the type found within the self-consistent

molecular orbital (MO) framework, should be

available so that processes related to elec-

tronic excitations may be studied, especially

in the case of energetic atom-atom collisions.


*This chapter is an essentially unaltered version of a
contribution (Be74a)to the Proceedings of the Interna-
tional Symposium on Atomic, Molecular and Solid State
Theory and Quantum Statistics held at Sanibel Island,
Florida, 20-26 January 1974, where a preliminary report
of the results was made.


-28-




-29-


(c) The method should be applicable to a variety

of pairs of atcmic neutral and ionic species

while at the same time it should involve only

a moderate degree of calculational effort and

complexity.

Statistical approaches, such as the Thomas-Fermi-

Dirac method, have provided useful results by applying

free-electron gas energy expressions in conjunction with

a molecular charge density taken as a superposition of

atomic charge densities. However, these approaches are

deficient in that they only describe the ground state in-

teraction. They afford no information of the type men-

tioned in point (b), and furthermore they are not appli-

cable to situations where appreciable charge rearrange-

ment occurs in the diatom, since there is usually no pro-

vision for self-consistency in the calculations.

These considerations have led to the use of the MSXa

method (Jo66, Jo73, Sl71a, Sl71b, S172) in this work. It

parallels the Hartree-Fock approach in that it provides

a one-electron description with corresponding one-electron

eigenvalue equations and eigenstates which are solved

self-consistently. The method makes use of a convenient

local approximation to the exchange potential. Although

approximate, the treatment of exchange in the Xa approach

should be quite adequate in handling the short range part










of atom-atom interactions, where Coulomb and exchange

forces between the electrons are the important ones.

The calculational procedure is furthermore based on

the "muffin-tin" approximation to the one-electron poten-

tial, which entails no additional computational complica-

tions as the number of electrons being treated increases.

Thus the scheme has wide applicability.

The first application of the MSXa method to inter-

action potentials of rare gas pairs was a calculation on

the Ne-Ne system (Ko72, see also Tr73). In this chapter,

calculations performed on the atom pairs, He-He, He-Ar

and Ar-Ar are presented, and their relevance to collision

phenomena is pointed out. At the outset, aspects of the

Xa theory and computational approach which directly re-

late to the present work are described, and limitations

of the theory are considered. Results of the calcula-

tions are then presented, and finally, a discussion of

the results and their significance is given.



1. Theoretical and Computational Considerations

Within the Xa formalism (S171a, S172), the total en-

ergy EyX, of a system of N electrons is specified by a

set of spin-orbitals, {ni}, according to the expression



EXC = n ndr u (r) -V + X(-2Za/rla )}u (rI)
1 a









+ /drl/dr2{2p(rl)p(r2)} /rl2


Slrd {pr)U O ) +P (rl)U I
S+l {P 1Xc 1 1 Xu 1


+ 2Z Zb/Rab (37)
a




Equation (37) is expressed in Rydberg units, and in the

usual sense, Za, Rab rla, and r12 refer to the charge

on nucleus a, the distance between nuclei a and b, the

distance between rl and the position of nucleus a and
- -+
the distance between r, and r2, respectively. The elec-

tronic charge density is

p(r,) = p r) + p (r)


1+ I
nu(rl)u(l) + nu (rlu (l (38)

comprised of the charge density of electrons with spin-

up (denoted by +) and that of electrons with spin-down

(denoted by 4-), and

UO (r ) = -9a((3/4T)p ( ))1 (39)


is the Xa local exchange energy density, a being a multi-

plicative factor. It should be emphasized that E is an

energy functional depending on the spin-orbitals, ui, and

the occupation numbers, ni, and does not necessarily rep-





-32-
-3


resent an average value of a many-electron wave function

over a Hamiltonian operator, as is the case in Hartree-

Fock theory.

For a given assignment of the ni's, the u.'s are de-

termined by making EXa stationary with respect to their

variation. This leads to a set of eigenvalue equations

for the spin-orbitals. For the u 's of spin-up, one has


(fl(rl) + fdr2{2p(r2)/r,2} + V x(r)ui(r = (r) ,
(40)
where

+ 2
flrl) -7 + (-2Za/rla) (41)
a

and
Va ( ) = 2/3 Ue(l) (42)

A similar set of equations is obtained for the u.'s of

spin-down. As is well known (S171a, S172), the interpre-

tation of the Xa orbital eigenenergies, e., differs from

that of Hartree-Fock orbital eigenenergies, and is based

on the relation,

Ei = E Xa/ ni (43)


This condition between EX, the E 's and the n 's insures

that Fermi statistics holds within the framework of the

Xa description of a system of electrons; namely, the low-

est value of EX for a system of electrons is achieved





-33-


when the spin-orbitals of lowest eigonenergies are occu-

pied. Concentrating on diatonics at fixed internuclear

distance, R, a state of the electronic system may be

identified by means of an assignment of the n 's. A con-

venient way, then, of determining over which region of R

a particular N-electron state is the one of lowest energy

is by looking at the behavior of the .i's for the occupied

and unoccupied orbitals of that state, and observing over

which region of R the .i's of the occupied orbitals are

the lowest ones. In the results which follow, this fea-

ture will be demonstrated. Equation (43) is also the

basis for the familiar transition state approach (Sl7lb),

from which good approximations may be found to ionization

energies as well as excitation energies between electronic

states.

In practice Equation (40) is solved by the Multiple

Scattering (MS) method (Jo66, Jo73), with the potential

operator

V(r!) = fdr2 2p(r2)/r} + V r) = V(rl + VX(rl) (44)


approximated by a "muffin-tin" form, whereby it is av-

eraged over angles within non-overlapping spherical re-

gions centered on the various nuclear sites and also oth-

er sites in the molecule (outer sphere, empty spheres,

etc.), and volume-averaged elsewhere. The u.'s are de-
1





-34-


termined self-consistently in terms of partial wave ex-

pansions within the spherical regions, and expansions in

"multiply-scattered" waves elsewhere. Furthermore, at

each iteration of the self-consistent procedure, the "muf-

fin-tin" form of p (r), as found from the u.'s according

to Equation (38), is used to evaluate the "muffin-tin" po-

tential as well as the "muffin-tin" approximation to EXa

of Equation (37). The reader is referred elsewhere (Co72)

for the details about these approximations as well as ways

of correcting for them (Da73, Da74a, Da74b). It suffices

here to say only that the effect of the "muffin-tin" ap-

proximation in the MSXa evaluation of EXa is appreciable.

However, orbital energies e. for diatoms appear to be more

reliable than EXa within the MSXa approach (We73).

It has been shown (S174) that in obtaining the ex-

change potential of Equation (42), one need not assume

that the electrons of an atom or molecule behave locally

like those of a free-electron gas of ths same density.

Rather, one may assume a spherically symmetric "Fermi hole"

and apply dimensional arguments. Nevertheless, consider-

able discussion has been devoted to shortcomings of ex-

change potentials of the type in Equation (42) associated

with the finite numbers and inhomogeneous spatial distri-

bution of electrons in atomic and molecular systems (Li70,

Li71, Li72, Li74, Ra73, Ra75). By means of a more careful





-35-


look at the exchange energy of a free-electron gas of a

finite number of electrons in a finite volume, corrections

to the exchange potential have been derived by separating

out the contribution from the interaction of each electron

with itself (Li70, Ra73, Ra75); for a small number (<200)

of electrons the contribution trom this "self-interaction"

becomes quite sizable. Estimates of the "self-interaction",

based on the above-mentioned corrections, show that in

atoms and molecules only about 85% of its contribution to

the exchange potential is included in V (Li72), while
Xc
the Coulomb potential, VC, in Equation (44) includes all

the "self-interaction". This imbalance may be partially

remedied by an adequate choice of the a factor.

Now, a variety of ways have been suggested for spec-

ifying the a factor of Equation (39) for a system of elec-

trons (Sc72 and references therein). The values of a for

atoms, resulting from the various schemes, almost all dis-

play the same trend; namely, for atoms of larger numbers

of electrons the values of a tend toward 2/3, the factor

appearing for a free-electron gas, whereas for atoms with

few electrons, larger values of a result. Since the "self-

interaction" contribution is large for few electrons, this

trend in a values has often been interpreted as reflecting

the required greater compensation for the deficit in "self-

interaction" in the case of few electrons, the compensation





-36-


becoming less as more elections are involved (Li72). Thus,

despite the importance of the corrections which have just

been reviewed, the Xa exchange potential represents quite

well the exchange interaction in a system of electrons.

As long as the "muffin-tin" approximations are being made,

a treatment in terms of the Xa exchange potential alone,

with a commonly used value of a, is expected to be ade-

quate for describing the short-range interaction between

atoms, as mentioned at the beginning of this chapter.

The calculations performed here employed a double

precision (14 hexadecimal or roughly 16 decimal digits

available per number on an IBM-370/165 computer) version

of the MSXa program, MUSCATEL. This precision was re-

quired since the interaction energy, AE, is computed as

the difference between the total energy of the diatom

(in our case, EXa in the "muffin-tin" approximation) and

that of the two isolated atoms. For instance, the case

of Ar-Ar at moderately large internuclear separations, R,

involves interaction energies seven orders of magnitude

smaller than the total energies used to determine the in-

teraction, hence requiring at least eight significant

figures in the total energies. The total energies of the

isolated atoms were calculated using the Hartree-Fock-

Slater (HFS) atomic program (He63, Za66).

For the homonuclear cases, He-He and Ar-Ar, the so-










called "virial theorem" values of a for the atoms, as re-

ported by Schwarz (Sc72), were used in all regions of the

molecules. For He-Ar the respective atomic "virial the-

orea" values were used in the spherical regions about the

atoms, and a weighted mean (weighted according to the num-

ber of electrons of each atom) of the two values was used

elsewhere. The heteronuclear case of He-Ar required a

choice of the radii to be used for the spherical regions

about the atomic sites. Contiguous spherical regions were

chosen in all cases, and the ratio between the He and Ar

sphere radii, used at all internuclear separations calcu-

lated, were determined in the following way. Average ra-

dii obtained from numerical atomic Hartree-Fock calcu-

lations (Ma67a, Ma68) were used to estimate the size of

each of the isolated closed shell atoms, He and Ar, based

on the Is orbital of He, and the 3p orbital of Ar. The

values are, respectively, 0.92727 a and 1.66296 a The
O O
He and Ar sphere radii at different R were then chosen in

the ratio of these two characteristic charge extents.

From the same atomic calculations, the maximum values of

the He Is and Ar 3p radial probabilities occur, respec-

tively, at about 0.55 a and 1.30 a so for internuclear

separations larger than about 1.85 ao, the above scheme

for selecting sphere sizes should serve well. Other con-

siderations must be made for cases which do not involve

two closed shell atoms and where significant charge rear-





-38-


rangement occurs in the diatom. Such a case is treated

in the following chapter where calculations are reported

on the excited He-Ar diatom which separates at large R
1,3 6 1
to He* Cls2s, 'S) plus ArC3p ,S).

The selection of partial waves to be included in the

expansion of an orbital in the various regions depends

upon over which regions of the molecule the orbital tends

to be concentrated. For a very deep lying core orbital,

which is essentially of atomic character, only the partial

wave corresponding to that of the associated atomic orbi-

tal was used in each appropriate atomic region, since par-

tial waves of other R values give a negligible contribu-

tion. For higher lying orbitals, appreciable contribution

from a number of partial waves, s, p, d, etc. may occur.

Such partial waves were included as long as their ampli-

tudes were at least one-hundredth the amplitude of the

dominant partial wave.

All of the calculations required no more than medium

size core on an IBM-370/165 computer and the times per

iteration of the SCF procedure were about two seconds for

He-He, five seconds for He-Ar and between five and ten

seconds for Ar-Ar, depending on the "goodness" of the

starting point for a calculation. Denoting the absolute

difference, occurring between the values of the "muffin-

tin" potential of Equation (44) at one iteration and those





-39-


of the previous iteration, by AV, and the maximum value
-4
of AV/V by s, the degree of self-consistency c<10-4 was

achieved typically in 15-20 iterations.

It should be pointed out that the relative error of

the MO wave functions is of the same order of iaagnitude

as that of the potential. Since the total energy is vari-

ationally determined, and therefore accurate to second

order in the wave functions, the degree of self-consis-

tency we have used is sufficient to insure the accuracy

required in the total energies at large R.



2. Results

In Figure 2 are displayed the interaction energies,

AE, for the three diatoms, He-He, He-Ar and Ar-Ar, as a

function of R, the internuclear separation. The interac-

tion energies are shown for the states with the lowest

energy at large R (i.e., asymptotic ground states). The

interactions on this semi-logarithmic plot are seen to be

quite linear, indicating the repulsion they show over the

investigated ranges of R is characteristically of the

Born-Mayer type, namely A exp(-bR). This behavior, of

course, would break down at very small R, where the Cou-

lomb repulsion between the nuclei becomes strongly domi-

nant. The beginning stages of this other behavior is seen

in the He-He case at R 0.7 a On the other hand, as R
0




























Figure 2. Interaction energies, AE, for the pairs He-He,
He-Ar and Ar-Ar in their separated atom ground
states. Calculated points are encircled. a.u.
of distance refers to the Bohr radius, ao
O

























E (c.u)


1.0 2.0 3.0 40 5.0 6.0 70
R(o.u)





-42-


increases for a given interaction and approaches the van

der Waals radius, the actual interaction energy would pass

through zero, and its logarithm would asymptotically ap-

proach (--) as R nears the point of zero interaction.

This pronounced deflection of the logarithm of AE away

from the Born-Mayer straight-line behavior is not exhibited

by the calculated interaction energies in Figure 2, indi-

cating that these "muffin-tin" interaction energies show

no tendency to describe the van der Waals attractive well.

It should be pointed out, though, that efforts made to

correct for the "muffin-tin" effects in the case of Ne-Ne

have shown a well defined attractive region (Da73, Da74a,

Da74b). The procedure for calculating these corrections

is, however, non-trivial, and would be impractical for

the present purposes.

Therefore, Born-Mayer A and b parameters for the in-

teractions of Figure 2 were determined by means of a

least-squares fit to the calculated points over the re-

gions of straight-line behavior. The parameter b measures

the slope of an interaction as shown in Figure 2, and re-

flects the "hardness" of the repulsion described. The

parameter, A, serves as a measure of the overall strength

of repulsion, being the value of In(AE) at R=0, although

it is somewhat unrealistic, since R=0 is an unphysical

separation at which to compare the "strength" of the repul-

sion of a Born-Mayer type.




-43-


So, the A parameters of the interactions of Figure

2 are simply listed in Table I, and it is pointed out

that over the region of Born-Mayer behavior, they lead to

interaction energies for the three diatoms which overes-

timate the repulsion, but obey the combination rules for

such interactions in that the He-Ar Born-Mayer line falls

between those of He-He and Ar-Ar. Listed in Table I by

way of comparison are the b parameters of the interactions

of Figure 2 along with those determined from other theo-

retical calculations and experiment, as indicated. The

ranges over which the listed parameters apply are shown

in parentheses. In general, the b parameters of the pres-

ent work reflect "softer" repulsions than do the other

tabulated b parameters. However, the ranges of R values

of the present work extend to larger R, and it is possi-

ble that "muffin-tin" effects lead to a proportionally

larger over-estimation of the repalsion at large R, i.e.,

to smaller b parameters.

The case of He-He is now considered in more detail.

In particular, the behavior of the MSXa eigenenergies is

looked at, providing a good example for diatomic interac-

tions. As is well known, at large internuclear separation,

the lowest state of the He-He diatom is the IZ+ state spec-
g
ified by doubly occupied I and la molecular orbitals.

These two MO's are, at large R, essentially the gerade






-44-


I


04






0
0
-H)


40
C)

a)




-Ii
r



H




0
04
'-4


rC





















a)










EH


t1


a
H
'4






0 r
U






u
U -
Her
E-


I --
0 CC










O
H


















U
E







E-


H
0


PC


- 0
o
I 0
4a

0











Ho








I 0






C)
H













rl
I 0
rfi


rc co
'2*




oN to
LD
O)


I-





CO










,---r
0











Cr)
,-0
o

0










ii
to







03
Ln

Ln
O











0

rC)



Si
O


I

i [


co
n I

o co









L00
o
N *
N






In

'-lp















0 I
H 0














o
O0
0






N







o in
.0 I














c3
o
N








tO




ra




en
03


\D



'03-
ca
to
Ln
.--1 *







0
u


to
03 I
CO
0o
H *
r-[







0
rC
'D Ln
N

LO



us
o
rHl*
en





0

03 N
en
I
Lo
to
H *
cN


03
4.r

0o
,--l
IN

1-1


'-4
PC





-45-


and ungerade combinations of the He Is atomic orbitals.

In Figure 3 is shown the plot of the eigenvalues of these

two orbitals as a function of R over the range 0.5 ao to

5.0 a In keeping with the idea of a correlation diagram,
0
arrows indicate the HFS atomic orbital eigenenergies; the

is orbital of He in its ground Is2 configuration, in the

separated atom CSA) limit, and the is, 2s and 2p orbitals
2 2
of Be in its ground Is 2s configuration in the united

atom [(A) limit. It is seen that cl, and E are nearly
g u
degenerate with aes of He at large R, as expected, and

separate as R decreases. Now, in the hA limit, the lo'

orbital correlates with the Be 2p atomic orbital. Thus
1-'- 2 2 2 2
the l' (l a 1 ) state approaches the excited Be is 2p
g g u o
atomic state in the UA limit. On the other hand the 2o
g
orbital correlates with the UA Be ..3 atomic orbital. It

is therefore the Z+ (la2ao ) state which in the UA limit
g g g
2 2
correlates with the is 2s Be ground state. It is of in-

terest to determine at what internuclear separation the
1 + 2 2
-E state specified by la 2a becomes lower in energy than
g g g
2 2
that specified by la la As discussed earlier in this
g u
chapter, one may proceed in two ways: (1) direct observa-

tion of the interaction energies, AE, of the two states

as functions of R to see where they cross; or (2) observa-
2 2
tion of the eigenenergies of, for instance, the Cl 22
g g
state to see where cla and 20 become the ones of lowest
g g






















0c C



D 4- C 0

(N tn0

NX C '


Su 4oa
d o a)

--1 CU
0 0H 0



rd C C U

C-

S-I -H 3, C
WV 0) U



,fl -U

ro-I 0 m



So 0 )J

U.C
D 'OO4 m0











U I (C 0

4J C) CU
00 CUC
t 0 04) M
-,. .
C CU



(U (U 0













0'
-r4




-47-






Si















a -

NN c o
o
o o
/.


?- I 4- -
3 -
N -






. . I , I i P I i I , ,
i- -- oo .











value. Proceeding from 1.2 a to smaller R, the eigenen-
2 2
erqies for the la 2a state have been plotted along with
g g
22 2
those of the 1a lIc state. For the l"2ag state, ,la

lies lower, to begin with, than ,2 and Fermi statistics
2 2
indicates that l1 21 is not the state of lowest energy.
Sgg
However, Cle is rising sharply as R decreases, and is

seen to cross above e2g between 0.5 ao and 0.6 a In-

ward from this crossing the la and 2a orbitals have the
g g
1 2
lowest eigenenergies and hence the lc' 2o state has the
g g
lowest energy. For comparison, in the insert of Figure 3,

a plot with linear scales is shown of AE versus R for

these two states, which corroborates this behavior.

It should be noted that the interaction energies cross

in a very gradual manner whereas the eigenenergies cross

more sharply. Thus the eigenenergy behavior indicates

more clearly the position of the crossing, which is found

to be 0.53 a These results compare well with SCF results

reported on He-He (Ma67c), where it was found that the en-
2 2 2 2
ergies of the la la and la 22 configurations cross near
g u g g
0.6 a Also, in the accompanying 50 configuration natural-

orbital iteration calculations, it was reported that the
2 2
l glou configuration was dominant beyond 0.7 ao. Though

the calculations reported here go inwards only to 0.5 ao,
2 2
the eigenenergies of the la 2a state are seen to be ap-
g g g g
preaching the appropriate eigenenergies of ground state Be.


-48-










An investigation (Ya74) of the He-He diatom, subse-

quent to the one reported here CBe74a), hut closely paral-

leling it, has been carried out in the Hartree-Fock ap-

proximation. It is interesting to compare the MSXa and

Hartree-Fock results. In both approaches, the behavior

of the total energies of the. lc21a and 1 22o configura-
g u g g
tions as R decreases from 0.6 a to 0.5 a shows that
o o
2 2
they cross very gradually, the energy of the lo 222 state
g g
becoming lower than that of the loa2la state at 0.56 a
g u o
in the Hartree-Fock case, and as has been seen here, at

0.53 a in the MSXa case. These values are in good agree-

ment. However, in the Hartree-Fock approach, the deter-

mination is based solely on the total energy curves of

the two states as they cross with nearly the same slope.

This is because in the Hartree-Fock approach there is no

immediate connection between the state of lowest energy

and the eigenenergies of the MO's associated with that

state. However, as has been discussed and demonstrated

here, such a connection can be made in the Xa approach on

the basis of Equation (43), which permits the detection

of crossings either by observing MO eigenenergies or

total energy curves.

For Ar-Ar, at large R, the ground state is also

+ and is specified by the first five a and a MO's,
g g u
each being doubly occupied, and the first two 7 and T
u g





-50-


MO's, each having occupation number 4. For large R, or-

bitals 10 to 5gu are formed from the appropriate com-
g,u g,u
binations of Ar Is, 2s, 2Po, 3s and 3po atomic orbitals,

and the first and second gu orbitals from the appropri-

ate combinations of Ar 2p and 3p atomic orbitals, re-

spectively. In keeping with the Fermi statistics, the

eigenenergies of these occupied orbitals are the lowest

ones at large R, where we also find lying above them the

eigenenergies of the unoccupied 60 and 16 orbitals. A

number of states can be specified at smaller R by the vari-

ous assignments of occupation numbers to the 5U 60 and

16 orbitals.

Calculations have been made on some of these states

and are displayed in Figure 4 on a semi-logarithmic plot

of AE versus R (in Figure 4, maonp refers to 51, 609

and 16P). In the region of R shown, numerous crossings

can be seen, and they are all of a very gradual type.
11 2
The Uo and 6 curves are from non-spin-polarized MSXa
u g g
calculations, the corresponding spin-polarized calculations

reflecting only a small splitting scarcely noticeable on

such a graph as Figure 4. We can see that the SA ground

state, denoted by 2a in Figure 4, no longer is of lowest

energy for R less than about 3 a This can be confirmed

again, by looking at the eigenenergies for this state, and

in Figure 5 we display the highest of them versus R on a

































Figure 4. Interaction energies, AE, for Ar-Ar in states
where the highest orbitals have occupation
numbers as specified. Calculated points are
encircled, a.u. of distance refers to the
Bohr radius, a .
o
















Ar Ar

Sm+ n-r p n =2
m+n+p=2


2.0 2.5


R (au.)



































Figure 5. Orbital eigenenergies for Ar-Ar in its
separated atom ground state. Calculated
points are encircled. a.u. of distance
refers to the Bohr radius, a .
o





-54-


R (a u.)
20 40 60 8.0


C (a~u.)


3p Ar


3s Ar




--55-


log-log plot. The appropriate SA 3s and 3p eigenenergies

are also shown. The eigenenergy of the unoccupied 6g MO

(which, in the SA limit correlates with the unoccupied 4s

atomic orbital of ground state Ar, whose orbital energy

is -0.0023 a.u. and lies off the scale in the figure) is

seen to be descending rapidly as R decreases, crossing

the occupied So orbital eigenenergy sharply at 3 ao,

indicating that for R less than 3 ao, this state is in-

deed no longer of lowest energy.



3. Discussion

The results which have been presented illustrate the

possibilities of the MSXa method in the study of interac-

tions that play a role in collision events. The interac-

tion potentials themselves render information on the Born-

Mayer type repulsion and, in conjunction with results on

van der Waals attractions, enter into the calculation of

relative motion of colliding atoms. Implicit in the re-

marks on crossings between interaction terms for various

states is that the states calculated in the Xc method

are diabetic in nature. Indeed this is the case since

each state is independently calculated after being speci-

fied by an assignment of occupation numbers to the orbi-

tals. So, while describing the dynamics of collision

events, we can expect the largest coupling between molecu-





-56-


lar configurations to come from the electron-electron in-

teraction. In principle, these interaction matrix ele-

ments between determinantal wave functions comprised of

MSXo orbitals can be calculated, but in practice the

problem at present seems quite formidable.

The usefulness of the Xa orbital energies, i., in

showing where interaction potentials cross, has been shown.

Hence, critical distances of approach for the occurrence

of various electronic excitation phenomena can be deter-

mined. Also of interest are crossings between eigenener-

gies such as occurs between c5. and E2x in Figure 5, if
g u
one or the other of the involved orbitals were partially

occupied. Such crossings are of importance in electron

promotion mechanisms involved in energetic atom-atom and

ion-atom collisions.

Although the pairs studied here consist of closed

she' atoms, it is expected that the MSXa method, because

of its self-consistent treatment, can handle as well the

repulsion in cases where sizable charge rearrangement

takes place. Of particular interest would be the mecha-

nisms involved in Penning and associative ionization phe-

nomena, where atom + excited atom and ion + atom interactions

are of importance. Here again, though, reliance upon ad-

ditional results for describing the van der Waals region

would be needed. In Chapters III and IV the considerations





-57-





prompted by the research of this chapter will be applied

to an investigation of the electronic structure and col-

lision dynamics involved in Penning ionization of Ar(3 )

by Ke*CIs2s,1,3S).













CHAPTER III

DIATOMIC MOLECULAR ORBITAL CORRELATION DIAGRAMS
FOR PENNING AND ASSOCIATIVE IONIZATION

In the previous chapter it has been pointed out that

an adequate understanding of atomic and molecular collision

phenomena requires information on electronic structure as

a function of the changing internuclear separations. Such

information is needed in order to describe inelastic pro-

cesses involving electronic excitation and charge transfer

as well as ionization. In this regard, the usefulness of

one-electron molecular orbital (MO) approaches in treating

electronic structure was emphasized. Such approaches afford

a self-consistent calculational framework of minimal complex-

ity which can treat ground as well as excited states. Chemi-

ionization is a prime example of processes where electron-

ically excited states play a crucial role. In this chapter*

the electronic structure involved in a collisional process

of this type will be considered.

Well known among chemi-ionization processes are Pen-

ning and associative ionization (PI and AI) of the type


*A preliminary report of the results presented in this chap-
ter was made at the International Symposium on Atomic,
Molecular and Solid State Theory, Collision Phenomena and
Computational Methods held at Sanibel Island, Florida,
18-24 January 1976.


-58-





-59-


A* + B A + B + e (PI) and A* + B + AB + e (AI),

where A* is usually an atom in some metastable state and B

is an atom or molecule (Mu66, Mu68, Mu73, Ni73, Be70a,

Be70b, Ru72, Ma76). Experimental information for such col-

lisions includes total ionization cross-sections as a func-

tion of collision energy (Ta72, Ch74, Pe75, 1175), angular

distributions of heavy particles (Ha73), and energy distri-

bution (Ho70, Ho75, Ce71) and angular distribution (Ho71,

Eb74) of emitted electrons. Most theoretical effort has

been directed at determining the energy dependence of total

ionization cross-sections and the angular distribution of

heavy particles (Na69, Mi70b, Mi71, 0172a). The angular

distribution of emitted electrons has been recently stud-

ied within a semiempirical model based on MO correlation

diagrams (Mi75). One of the present concerns is to re-

emphasize the usefulness of MO correlation diagrams as

they apply to angular distribution of ejected electrons

in PI and Al involving atomic collision partners.

In what follows, calculated as well as estimated MO

correlation diagrams are presented. The calculated results

are given first, where the electronic structure of
1,3 6 1
He*(ls2s, S) + Ar(3p S) has been considered within the

MSXa framework as described in the previous chapter. A

study has been made, in the non-spin-polarized approxima-

tion, of the K ArLAr (lse) (o3sAr) (o3pAr) (w3pAr) (a2sHe)
tinofth K2 21Arr~~osr~





-60-


configuration of the excited He-Ar diatom whose MO eigen-

energies approach the atomic orbital eigenenergies of

He*ils2s) and Ar(3p ) at large internuclear separation,

and also the ground 2E state of the (He-Ar) molecular ion,

whose MO eigenenergies approach the atomic levels of

HeCls2) and Ar +3p5) at large internuclear separation.

Some spin-polarized calculations are presented in the

neighborhood of a crossing exhibited by the non-spin-

polarized calculations, along with some comments on the

conditions under which the non-crossing rule applies for

MO eigenenergies.

Keeping in mind the calculated results, consideration

is given to PI and AI processes in terms of MO correlation

diagrams. Analyzing the ionization on the basis of an

Auger type process, MO's can be identified, which, togeth-

er with the continuum state of the emitted electron, are

involved in the process. Inspecting the united-atoms lim-

it in the center-of-mass, body-fixed frame then permits a

determination of the angular moment which contribute to

the continuum state of the emitted electron.

Next a procedure for estimating MO correlation dia-

grams is described, which makes use of available data on

atomic orbital energies, and of two basic guidelines. Ap-

plication is made to the collision pairs He*Cls2s) +

Ar(3p6), + Kr(4p6), + Hg(6s2), and Ne*(2p53s) + Ar(3p6).










The resulting estimated MO correlations for these systems

are then analyzed to determine the minimal set of angular

momentum values which are needed in each case to describe

emitted electrons in the body-fixed frame. Finally, a dis-

cassion is given of the results of this work and their

significance.



1. MO Calculations for He*+Ar: ar.d He+Ar+

The representative case of PI and AI in

He*(ls2s,' S) + Ar(3p S) collisions will be considered.

Here, calculations are needed for the excited states of the

He-Ar diatom which separate appropriately to

Ie*(ls2s,1,3S) and Ar(3p6,1S), and for the states of
+ 2 1 + 5 2
(He-Ar) which separate to HeCls,1 S) and Ar (3p5, P).

Following the work reported in the previous chapter, the

calculations are performed within the MSXa one-electron

MO framework.

For convenience in this chapter, the set of one-elec-

tron equations satisfied by the spin orbitals, ui, and ex-

pressed in Equation C40), are written

h (ffl)Ui( = Eiui(') (45)
eff 1 i 1 ii 1'

+ -
Here, heff(r ) is the effective one-electron Hamiltonian

for electrons of spin-up, and according to Equation (40),


-61-





-62-


hf (r) = f] l + dr2{2p(r2)/rl2} + VtX (46)



where Vx (rl) is given by Equations C42) and (39), and

p(r), given by Equation (38), is the charge density com-

prised of the contributions from electrons of spin-up and

spin-down. In the non-spin-polarized (NSP) approximation,


p () = (r) p ()/2 (47)


which means that the orbitals of spin-up and spin-down be-

come identical, and each orbital can be considered as ac-

commodating as many electrons of spin-up as of spin-down.

These calculations were carried out first with a NSP

treatment. As discussed in the previous chapter, each

self-consistent calculation begins with a potential which

is the "muffin-tin" form of a superposition of atomic po-

tentials centered at each atomic site of the molecule.

Therefore, for the He-Ar excited state Ar(3p ) and NSP

He*(ls2s) Hartree-Fock-Slater (HFS) CHe63) potentials were

used, and for the He-Ar ionic state He(1s2) and NSP Ar(3p5)

HFS potentials were used. For both the excited molecule

and the molecular ion, the electronic states were specified

by occupying the MO's so that their eigenenergies were cor-

rectly separating at large internuclear separations to the

corresponding atomic orbital eigenenergies of the above

mentioned HFS atomic calculations. In other words, the





-63-


boundary conditions of the PI and AI processes at large R

determined the appropriate excited and ionic states of the

He-Ar molecule. Heteronuclear molecules require a choice

for the radii of the spherical regions centered at each

atomic site as mentioned in the previous chapter. There,

for the ground state He-Ar molecule, the ratio of the He

to the Ar sphere radii was taken to be He/Ar

0.92727/1.66296 = 0.5576, where, e.g. e denotes the

average value of r for the Is orbital of He(s 2). The sit-

uation for (He-Ar) is roughly the same as for He-Ar, since

a calculation of for Ar (3p5) (NSP) shows it to be

1.549 a So, for the molecular ion the ratio of 0.5576

was used at all R. On the other hand He*(ls2s) + Ar(3p )

is quite another case, since the 2s orbital of He*(ls2s)

is very diffuse. NSP calculations show of He*(ls2s)

to be about 4.546 a Because of this diffuseness, it was

decided to choose the ratio of the He to the Ar sphere ra-

dii by finding which of its values minimized the total en-

ergy of the specified excited state of He-Ar at a fairly

large R, namely R = 9.0 a In this way, a ratio of 1.4

was found, which was then used at all R for the excited

state of He-Ar. The values of the factor a in the various

"muffin-tin" regions of the molecule, for both the excited

as well as ionic states, were those used previously in the

ground state He-Ar calculations, and the specific computa-





-64-


tional details also remain as reported in the previous

chapter.

The results of the NSP MSXa calculations are displayed

in Figures 6 and 7, respectively, for the excited state of

the He-Ar diatom and the ground state of the He-Ar molecu-

lar ion which are appropriate to PI and AI. Shown on log-

log plots are the MO eigenenergies versus R for 2.3 a

o o

energies are labeled according to their symmetry and occu-

pation number. In each case, the NSP HFS atomic orbital

eigenenergies for the separated atoms are shown at the

right of the plot, and the dashed lines indicate how each

MO eigenenergy is approaching properly its respective sep-

arated-atoms (SA) limit. In the united-atoms (UA) limit

the excited He-Ar molecule is expected to approach

Ca*(3p54s23d), and the ground state molecular ion to ap-

proach Ca C3p64s). The atomic orbital eigenenergies from

NSP HFS calculations on these united atoms are appropri-

ately shown at the left of each plot, and, while the MO

eigenenergies may exhibit much structure between the re-

gion of 2 a and the UA limit, the dashed lines at the

left of each plot show that these UA limits are not unrea-

sonable. In both plots, the label for the a MO arising at

large.R from 3pAr appears above the MO label. In fact,




























Figure 6. Molecular orbital correlation diagram from
non-spin-polarized MSXa calculations of the
He-Ar diatom in the excited Z configuration
which separates at large R to He*(is2s) +Ar(3p6).
Calculated points are encircled.
















-0.07
-0.08
-0.09


-0.15

-0.20 -


-0.25
-0.30


- '--He-r Tr-V -A
_ He'-Ar^ -,


-0.401--


i5 -0.50
d
.c -0.60
-0.70


-1.50
3sCa'(3?4s3d)-


0 2


3 4 5 6 7 8 910


-2sHe(I s2s)


-J3pAr(3p )


-- sAr(3p)

- -sHe(is2s)




00


R in ao
































Molecular orbital correlation diagram from
non-spin-polarized MSXa calculations of the
(He-Ar)+ molecular ion in the ground Z state
which separates at large R to He(ls2) +Ar+(3p5)
Calculated points are encircled.


Figure 7.





--68-


//-I- I I I i I II I -//--
-0.07- -
-o0.0-He-
-0.09 -
-0.10-



-0.1 5 -


-0.20 --


-0.25
3dCoa(3pF4s)-,
-0.30
4sCc Sp(34s)-
-0.40


-0.50
-0.60
-0 TO
-0.80
-0.90
-1.00


3 pCa (3 ?4s)-
-1.50

3sCa (3g4s)-.


0 2


-\



-
-i f
- - -^ s-- o-
^- -
- 7c
-:^ ^


cr0


crz.4ASHe(Is2)


7TT


-3pAr"(3 p5)




-3sAr'(3 p5)


S. I I I Ii ,, 1 111,111
3 4 5 6 7 8910 (0
R in ao


~o~O~-Q-O-(M~b-~~--O'2




-69-


this ordering for those levels is valid only for R < 5 ao.

For R > 6 a the T level lies above the o level, but they
o
are too close together at large R to be distinguishable on

these plots.

In Figure 6 a crossing is shown near 3.5 ao occurring

between the doubly and singly occupied NSP a orbitals aris-

ing respectively at large R from 3sAr(3p6) and NSP IsHe*(ls2s).

This crossing would appear to violate the non-crossing rule

for the MO eigenenergies, and warrants a detailed analysis.

The orbitals of a given symmetry are ordered according to

their eigenenergies, obtained self-consistently from Equa-

tion (45). This equation is an eigenvalue equation involv-

ing an effective Hamiltonian determined at each R according

to Equation (46). To establish the non-crossing rule for

the eigenenergies of Equation (45), one expresses the effec-

tive Hamiltonian at a supposed crossing, Rc, in terms of its

expansion about R = R +6R located a small distance, 6R,

from R :
c

hff(R) = heff(R) dhff/dR) 6R (48)
c

The non-crossing rule follows by noticing that
+
(dheff/dR)R 6R is a perturbation which lifts any degeneracy
c +
in the eigenvalue spectrum at R However, if hf (R) is
c eff
made to be discontinuous by choice, the conditions of the

non-crossing rule no longer apply.

Referring to Equation (46), the R dependence of the

effective Hamiltonian appears explicit in fl(l) (see

Equation (41)) and implicity in the charge density, p.





-70-


Showing the full R dependence, the effective Hamiltonian

may be written as heff{r,;R,p(r1,R)}. and


dhff/dR = heff/ Rl + (6hAff/6p)(dp/dR) (49)


The term, dp/dR, in Equation (49) can be seen, from Equa-

tion (38), to involve derivatives of the orbitals and oc-

cupation numbers with respect to R. As long as the n.

and u. are continuous in their R dependence, dhtff/dR will

be well behaved, and the non-crossing rule will hold. How-

ever, if the occupation numbers are changed discontinously

in some region of R, then the non-crossing rule will no

longer be valid in that region. These considerations apply

as well to the NSP effective Hamilton and in fact, in the

case at hand, as shown in Figure 6, a discontinuous change

in occupation numbers does occur. To the right of the

crossing, the a orbitals, in order of increasing eigenen-

ergy, have occupation numbers 1, 2, 2 and 1. To the left

of the crossing, they are 2, 1, 2 and 1. A further in-

vestigation of the region of the crossing was made by do-

ing spin-polarized calculations at R = 3.0 a 3.5 a

and 4.0 a in the case of the 3Z excited state which sepa-
61 3
rates at large R to Ar(3p S) and He*(ls2s, S). In Fig-
2
ure 8 attention is restricted to the levels of the NSP o

and a MO's which cross in Figure 6, and they are con-

trasted with their spin-split counterparts calculated in



















I- E
0) 3 O ) 01,) r
0. --4 ., I H

404 0 T 0 rd
Cti-CJCyi-t OH
J) a4 a) Eo a 0
aU < rl u aQ


C)W4-H3 U C)Id lA
0 0 0 0 0 -P u

O ,t-| O H,-Q
C")u C o )
- 4-I! 0 tnC 0 *H


-,I r to m a) a)
0 0 0 H -l *, iH V)
4- -r -M4 (d 4
S-H4 -X 4J O 0



: -H 4J 0 r -r
- -I 4- U4 ) ,I Q)

0 U ( 01 4-C U
C-I0- 1 0-0

CC HC)00C)-H0
- (0d N OH M-- N r-i
0 rl 4 a 0. C -C 0
aH 40 iN -H C u
r- a) r i o *
-1 C3 O00

-*H C0 + C 0 C
r- tX I (U 0) -1 C
OO 0 0- AO
iC 4 -H -H

0 o mn 4-1 -
'IM 1 0 t-4
-0 0 3 *1 c --
t-3CO W C0 O 1 -
*-H roC 0 04
p'a *H O 0 C
frd 0 ,C (
r-H -H L0D C 4 4-) -A
o a o a -H --

1 04-' 0 0r-l
-H H4 73 0 -H 4
4 (r C -H '4 0 a m
Ut) U M '014- C) 04


CO

0

-H
1-1





--72-


CU)
ro
C,)
C'J
U,
to

a
-a
U,


10
O Cj

C,)


Ui,




C,I

C,)


to





0
~- n
*O

CI3


L.
<


rn


Q.
ro
0L


0


C


IJ


n'o UI 9


- 0

ro rO


0 a

to




-73-


the region 3.0 a O-- O
limits.

Important to notice is that for R>3.5 ao the NSP a

orbital of He* is split considerably into its occupied com-

ponent of spin-up a and its unoccupied component of spin-

down a, on out to the SA limit, where the splitting is be-
3
tween the unoccupied is and the occupied la of He*(ls2s, S).

The o2 orbital from Ar is split only slightly into each of

its occupied spin components. Crossings between two orbi-

tal eigenenergies of different spin components are permit-

ted since each involves a different effective Hamiltonian

(see Equation (46)). Between 3.5 a and 4.0 a we find

such a crossing for the two spin components that split from

the NSP a2 orbital of Ar, and in fact, as R decreases, the
1 6 1
1 level from 3sAr(3p ,S) is decreasing in energy to pair
1 3
up with the occupied a level from Is He*(ls2s, S), where-
61
as its spin-up partner from 3s Ar(3p S) is rising to

pair up with the (empty) oa level from ls He*(ls2s, S).

In the UA limit, this empty oa level will correlate with

the partially occupied 3p atomic orbital of

Ca*(3p 4s23d,3L), where L here denotes one of the possible

orbital angular moment of the excited Ca atom.

Such a spin-polarized analysis as this confirmed the

choice of occupation numbers of the NSP calculations for

which the vacancy associated with the NSP o1 orbital is





-74-


correlating in the UA limit with the partially occupied

3p atomic orbital of Ca* (3p54s23d. This feature will

prove important for the considerations later on regarding

the angular momentum contributions to the continuum state

of the electron emitted during PI and AI. The crossing

shown in Figure 7, between the NSP singly and doubly oc-

cupied a orbitals of the molecular ion near 5.5 ao, is

another example of a discontinuous change in occupation

numbers, which in this case is required to reach the appro-

priate SA limit.



2. Analysis of PI and AI Processes Based on
MO Correlation Diagrams

Consideration is given now to how an MO correlation

diagram study, such as has just been described, can help

in understanding what takes place in PI and AI processes.

Collisional ionization occurs for R greater than the dis-

tance of closest approach, which in the case of He* + Ar

is around 7 ao at thermal collision energies. Referring

to the calculated MO diagram in Figure 6, one can see that

in this range of R there is a vacancy associated with the

a MO arising from lsHe* at large R, above which are some

fully occupied MO's as well as the singly occupied 1 MO

arising from 2sHe* at large R. This situation suggests

that the ionization may proceed by an Auger type process.

Namely, two electrons in higher lying orbitals participate




-75-


in ionization as one of them is promoted to a continuum

state while the other drops to fill a vacancy, in this

case associated with the ao MO arising from SA lsHe*.

The process may be characterized as one in which initially

the two electrons are in MO's ul and u2, while finally

(after ionization) they are in MO's ul and u2, where u

designates the continuum state of the ionized electron,

having momentum k and angular momentum components tl.

If the wave function a(N) is associated with the

system of N electrons before ionization, and ,b(k,N-l) is

associated with the system of CN-1) electrons plus the

emitted electron in its continuum state after ionization,

then, in the body-fixed (BF) frame, the transition proba-

bility for ionization involving these two wave functions

is expressed in terms of the interaction matrix element,

Va (R,k) B where
ba BF

Vba(Rk)F = BF (50)



Here H is the electronic Hamiltonian for the N electrons,

E is the total energy, and the brackets indicate integra-

tion over electronic variables. Within a single determi-

nant description,
electron orbitals ul and u2 of $a are replaced by uo and

u2 to obtain b". As a consequence Equation (50) reduces

to a sum of direct and exchange contributions which can

be written respectively, as




-76-


VD(R,k) = BF


VE(R,k) = BP (51)

where v = 1/rl2, the electron-electron Coulomb interaction

in atomic units.

In Figure 9a is shown a schematic MO correlation dia-

gram for the He*(Cs2s) + Ar(3p6) case. The construction

of such estimated MO correlation diagrams will be discussed

in the next section. The MO energies at large R represent

levels at the time of ionization. For instance, referring

to Figure 9a, two electrons, one initially in o2s from He*

and the other in r3p from Ar could participate in an Auger

process whereby cne is promoted to a continuum state while

the other fills the Is vacancy from He*. That is,

ul = 22s, U2 = r3p, the continuum state ul = (k, , ml)

and u' = als.

MO correlation diagrams, such as shown in Figure 9a,

allow one to predict the minimal number of angular momen-

tum contributions, ', which is necessary to describe the

continuum state of the emitted electron in the BF frame.

One proceeds by following the MO's involved in ionization

towards their UA limit. With the exception of the contin-

uum state, u', the MO's correlate in the UA limit to atomic

orbitals of well defined angular momentum. That is, one

can write



























5-4
0
4C4


F


















In





4-H



ON
(L)
a,
r-




a!







cdwZ

rO.

Or-
U-
dip
rIFC




ca




'-~-
42C





-78-










LOD O ( n

C n
(u CL L V)














uN n rn -



\A
ir
















C. Q,. C,









%T cl_
-3 ai a a in

















< -- (I) Qo- n







PA *
in
CL

o in a (n

901 3 m
/ / ^ / rs



























a

H




-4

co




C
C)



m
-4
.L




0
u

'-I








cz~
a,
4-J
4
.0
$4

04


If)
0 04







Un
4-3






.1-4






a,
U,

C-)
$41





-80-


CN C

-+ +

U), 0 tr,
Lf









LO C C- LOL
t t-ID tD
44









I- I
C))


2:-
cfl ct. l








I ~i CL







E)U)
LB

m
(I) +
cn ar(0
d --(3)901




-81-


UA

u2 X2( 2,m2)

1 A x{(k,, km{) (continuum),

u A (Zm (52)
u2 X2(2'm2) (52)

where the X's refer to the UA (atomic) orbitals. Looking

also at the direct and exchange matrix elements in the UA

limit, Equation (51) becomes

VD(R,k) qA VUA(k) = BF


VE(R,k) UA VUA(k) = BF (53)

Next, using the notation of previous work by Micha

(Mi70a), the electron-electron Coulomb interaction is ex-

pressed as an expansion in terms of its multi-pole compo-

nents,


1/r2 = (1/r>) (r /r>)LPL(cosel2) (54)
L=0

and Equation (53) is written as follows:

VUA(k) = 6(sl,s) 6(s2,s) FL(1'2'12)AL(1'2'12)
L O0

VU (k) = 6(s;,s2) 6(ssl) L FL(1-221)AL(1'2'21) (55)
L=0

The factors FL and AL are proportional to products of 3-j

coefficients:

/(29-+1) (22+1) (2 +1)(2 2+1) i L 9l
FL(12 12) = 2L+ I 0 x



0 0 0





-82-


1 .L 21
AL(1 2-12) = (-l)m +2M (2L+l) -m - m -m m (56)






The RL1 '2'12) are Coulomb intearals involving the radial

parts of the orbitals in the UA limit, and depend on their

principal as well as angular momentum quantum numbers.

The presence of the 3-j coefficients in the FL and AL fac-

tors of Equation (56) reflects the coupling of the angular

moment of the electrons due to the 2L multiple component

of the electron-electron Coulomb interaction and allows

one to specify the ranges of values of k and ml for which

contributions will appear in the direct and exchange ma-

trix elements of Equation (55). This is accomplished by

employing the selection rules for the 3-j coefficients

(see, for example, Me66). In the UA limit, (1,ml),

(2,m2) and (2,m2) are known. Therefore, referring to
Equation (56), one of the 3-j factors in FL specifies the

allowed range of L, and similarly, one of the 3-j factors

in AL further specifies the range of M. Once the ranges

of L and M have been determined, the remaining 3-j coeffi-

cient factors, one in FL and one in AL, specify the ranges

of kl and ml for the continuum state of the emitted elec-

tron.









Thus, for the direct matrix element of Equation (55),

L and M are restricted as follows:

S'- < L < k-+
2 2 2 2

(I-+ +L) ever


-m+M = -m2 (57)

For each L and M possible from Equation (57), the remaining

two factors in FL and AL restrict ; and mj in similar fa-

shion:

S1-L _< < 1+L

(I+L-+) even

ml+M = m (58)
Interchanging the indices 1 and 2 among the primed symbols

in Equations (57) and 158) provides corresponding expressions

for the exchange matrix element of Equation (55).

Thus far the discussion has only been in terms of the

interaction matrix element of Equation (50) between two de-

terminantal wave functions, Da and ob, distinguished from

one another, respectively, by the MO's u1 and u2 before

ionization and uf and u2 after ionization.

Consideration is now given to the manifold of determi-

nantal states a which is needed to represent the electron-

ic state D. before ionization. Each of those states (a

has an angular momentum component along the molecular axis,

a, equal to the absolute value of the sum of axial angular










momentum components of the MO's from which the determi-

nantal wave function is constructed. To the extent that

one may neglect rotational Born-Oppenheimer couplings,

only determinants 0a having A = A. are needed in repre-

senting the state of the electrons prior to ionization,

where A. denotes the axial component of electronic angular
1
momentum in the SA limit of the incident channel. How-

ever, Born-Oppenheimer couplings cannot be neglected in
-2
the UA limit because of their R dependence (Sm69, Si76).

Hence, in order to properly carry out the UA analysis just

described one must include contributions from states a

for which a Ai = 0,1.

For example, according to the MO correlation diagram

of Figure 9a, one constructs the leading determinantal

wave function prior to ionization from the MO's shown at

the right which correlate to the occupied SA atomic orbi-

tals, and observes that Ai = 0. As was mentioned earlier,

there is the possible case of an Auger type process in

which the participating orbitals ul and u2 are identified

with o2sHe* and ir3pAr, respectively. Not shown in Figure

9a are levels of MO's associated with unoccupied SA atomic

orbitals. Among these MO's there may be one whose axial

component of angular momentum differs by 1 from that of

an occupied MO with which it shares the same UA atomic

orbital limit. An example of such a case is the 72p MO





-85-


arising from the unoccupied SA 2p orbital of He*(ls2s),

together with the a2s MO of Figure 9a, both of which cor-

relate in the UA limit to the singly occupied 3d atomic

orbital of Ca*(3p54s23d). Replacing the a2s MO of the

previously described Z determinant by this I2p MO would

result in a H determinant which is significant for the

present UA analysis due to Born-Oppenheimer couplings.

In the Z case, ul of the Auger type process would be iden-

tified with a2s, in the I case with 72p. According to

Equation (52), the UA limit results in = 2,ml = 0 for

the E case and i = 2,ml = 1 for the H case. This means

that in applying Equations (57) and (58) for these two

cases, the restrictions on the -values are the same for

both, while the m-value restrictions involve m, = 0 for

Z and mi = 1 for H determinants.

In general, then, the UA analysis of the angular mo-

mentum contributions to the emitted electron requires that

the initial electronic state prior to ionization be written

as a linear combination of such determinants;

S= a C (59)
a


Similar considerations hold after ionization, where a mani-

fold of determinants Qb results, each differing from the

other by the particular continuum state ul associated with

it. The final electronic state then is written,











0f = bCb (60)
b

and the total transition probability for ionization is ex-

pressed in terms of Vfi(R,k)BF, which is a linear combina-

tion of interaction matrix elements of Equation (50):

V (R,k)BF= ICCaVba(Rk)BF (61)
a,b

According to this general description, Equations (57) and

C58) of the UA analysis may be applied using the. value

restrictions directly with values found from an MO corre-

lation diagram such as in Figure 9a, but remembering that

the m-value restrictions are weakened due to Born-Oppen-

heimer couplings.

Of course, ionization occurs far from the UA limit,

and the values for ' which are obtained here are certain-

ly not all which should be included, but they do constitute

the minimal set required for making a reasonable physical

description of the emitted electron. That such a minimal

set can be specified is important for the parameterization

of expressions at various levels of approximation by which

calculations of angular distributions of the emitted elec-

trons can be made (Eb74, Mi75).




-87-


3. Estimating MO Correlation Diagrams for Diatomics

An analysis such as has just been outlined requires

only schematic correlation diagrams, which should, how-

ever, be reliable in relating SA and UA limits of the

higher lying MO's. In this section, a procedure will be

described for estimating MO correlation diagrams, and ap-

plied to the collision pairs He*(ls2s) + Ar(3p6), +

Kr(4p6), + Hg(6s2), and Ne*(2p53s) + Ar(3p6). Based on

these estimated MO correlation diagrams and the analysis

of the previous section, the minimal set of angular momen-

tum contributions required to describe the emitted electrons

in PI and AI will then be determined.

In order to begin constructing estimated MO correla-

tion diagrams, one must have the appropriate SA and UA

atomic orbital energy levels of the collision partners

both before and after ionization. The following scheme

has been found to be sufficiently reliable and simple to

apply. For the ground state levels of neutral atoms one

can use any of the results of the Hartree-Fock or Hartree-

Fock-Slater calculations, which are available in tabulated

form in the literature CFi73, De73, Ma67a, C174). Further-

more, one can rely on these calculated atomic orbital en-

ergy levels for ground state neutral atoms in order to ob-

tain the levels of the ground and excited state atomic ions

and of excited state neutral atoms. Clementi and Roetti




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