 Front Cover 
 Dedication 
 Acknowledgement 
 Preface 
 Table of Contents 
 Abstract 
 Introduction 
 A study of singleelectron and... 
 Diatomic molecular orbital correlation... 
 A coupled channels approach to... 
 References 
 Biographical sketch 

Full Citation 
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 ) nonfiction ( marcgt ) 
Notes 

Thesis: 
ThesisUniversity of Florida. 

Bibliography: 
Bibliography: leaves 212217. 

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 

Downloads 

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 singleelectron and total energies for some pairs of noble gas atoms
Page 28
Page 29
Page 30
Page 31
Page 32
Page 33
Page 34
Page 35
Page 36
Page 37
Page 38
Page 39
Page 40
Page 41
Page 42
Page 43
Page 44
Page 45
Page 46
Page 47
Page 48
Page 49
Page 50
Page 51
Page 52
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
Page 64
Page 65
Page 66
Page 67
Page 68
Page 69
Page 70
Page 71
Page 72
Page 73
Page 74
Page 75
Page 76
Page 77
Page 78
Page 79
Page 80
Page 81
Page 82
Page 83
Page 84
Page 85
Page 86
Page 87
Page 88
Page 89
Page 90
Page 91
Page 92
Page 93
Page 94
Page 95
Page 96
Page 97
Page 98
Page 99
Page 100
Page 101
Page 102
Page 103
Page 104
Page 105
Page 106
Page 107
Page 108
Page 109
Page 110
Page 111
Page 112
Page 113
A coupled channels approach to penning ionization of Ar by He
Page 114
Page 115
Page 116
Page 117
Page 118
Page 119
Page 120
Page 121
Page 122
Page 123
Page 124
Page 125
Page 126
Page 127
Page 128
Page 129
Page 130
Page 131
Page 132
Page 133
Page 134
Page 135
Page 136
Page 137
Page 138
Page 139
Page 140
Page 141
Page 142
Page 143
Page 144
Page 145
Page 146
Page 147
Page 148
Page 149
Page 150
Page 151
Page 152
Page 153
Page 154
Page 155
Page 156
Page 157
Page 158
Page 159
Page 160
Page 161
Page 162
Page 163
Page 164
Page 165
Page 166
Page 167
Page 168
Page 169
Page 170
Page 171
Page 172
Page 173
Page 174
Page 175
Page 176
Page 177
Page 178
Page 179
Page 180
Page 181
Page 182
Page 183
Page 184
Page 185
Page 186
Page 187
Page 188
Page 189
Page 190
Page 191
Page 192
Page 193
Page 194
Page 195
Page 196
Page 197
Page 198
Page 199
Page 200
Page 201
Page 202
Page 203
Page 204
Page 205
Page 206
Page 207
Page 208
Page 209
Page 210
Page 211
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
plescattering and local exchange related matters, Profes
sor John W. D. Connolly. through lectures an? ready atten
tion to my inquiries, was of great help. Close associations
with Drs. Suheil F. Abdulnur, Poul JOrgensen and JianMin
Yuan, and Professor Manoel L. de Siqueira, have been per
sonally beneficial as well as significant to my overall
perspective in science. The able leadership of Professor
erCOlov Ldwdin as director of the Quantum Theory Project,
along with his nuTmerous lectures and also his interest in
philosopbtical considerations in science, have played an
important role in my graduate education.
As is the case with any undertaking, a program of
graduate studies provides a context and situation in which
one is able to grow personally in all respects. In this
regard I want to express my gratefulness to many friends,
both scientist and nonscientist alike, as well as to God,
my creator, by and in whom I exist.
PREFACE
I find it appropriate to make some remarks concern
jng the perspective and context in which the work reported
in this dissertation has been carried out.
Since its inception in the 1920's, Quantum Mechanics
has become well established as the suitable framework
within which to describe phenomena of a physical and chem
ical nature. Building upon only a few axioms, the formal
Quantum Theory manifests itself in the form of mathemati
cal equations, the solutions to which determine expressions
for calculating physically observable quantities. Confi
dence in Quantum Mechanics derives from the impressive
successes it has had in providing results in agreement with
experiments. However, in applying Quantum Mechanics to
describe actual, known, physical and chemical phenomena,
one quickly becomes aware of the fact that there are only
a few cases where an exact treatment has been possible.
In nearly all cases of interest, the mathematical equa
tions of the theory, though succinct in what they say, are
unmanageable to solve. Computational considerations,
therefore, have strongly influenced theoretical investiga
tions in physics and chemistry.
Basically two approaches have evolved. The socalled
ab initio calculations provide approximate solutions to
the "exact" quantum mechanical equations within a frame
work which, in principle, allows for the solutions to be
progressively improved upon to approach the "exact" solu
tions. One is presumably limited here only by the size
of the electronic computer available. On the other hand,
one may focus on the main features of some particular
physical phenomena of interest, and use the "exact" quan
tum mechanical equations only as a guide in order to ar
rive at approximate equations which mimic the important
aspects of the process being studied. These approximate,
or model, equations are many times only approximately solved!
Such apparently crude approaches require of the researcher
all of the physical intuition which can be mustered, in
order to properly assess the important features of the
physical situation and to approximate them reasonably well.
Nevertheless, much physical insight and many useful quan
titative results can be extracted from this point of view.
Indeed, the task of science is essentially to formulate
descriptive statements, both qualitative and quantitative,
which conform as nearly as possible to the laws and phenom
ona of nature as we observe them.
Finally, it should be mentioned that there are formal
results coining out of quantum theoxeticel investigations
which determine many characteristics of the "exact" and
approximate solutions, even before they have been calcu
laced. Both types of investigations described above rely
upon these formal results as well as upon each other. In
the dissertation which follows, research of the second
type mentioned above will be reported in the form of a
quantum mechanical investigation of electronic and dynam
ical aspects of diatomic systems.
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS . . . . . . . .. iii
PREFACE . . . . . . . . . . .
ABSTRACT. . . . . . . . . . . . x
CHAPTER
1. INTRODUCTION . . . . . . . ... .
1. A Formal Statement of the Problem. . . 2
2. Remarks Regarding Reference Frames . 6
3. Remarks Regarding the Wave Function. .. 17
4. The Coupled Equations and Coulomb and
BornOppenheimer Couplings .. . .. 19
5. Discussion . . . . . . ... 24
II. A STUDY OF SINGLEELECTRON AND TOTAL ENERGIES
FOR SOME PAIRS OF NOBLE GAS ATOMS. . . .. .28
1. Theoretical and Computational
Considerations . . . . . ... 30
2. Results. . . . . . .. . . . 39
3. Discussion .. . . . . . . 55
IIT. DIATOMIC MOLECULAR ORBITAL CORRELATION
DIAGRAMS FOR PENNING AND ASSOCIATIVE IONIZA
TION . . . . . . .. . . . 59
1. MO Calculations for He*+Ar and He+Ar . 61
2. Analysis of PI and AI Processes Based on
MO Correlation Diagrams. . . . . 74
3. Estimating MO Correlation Diagrams for
Diatomics. . . . . . . . ... 87
4. Discussion . . . . . . . . 110
viii
TABLE OF CONTENTS
(Continued)
Page
IV. A COUPLED CHANNELS APPROACH TO PENNING
IONIZATION OF Ar BY He*(ls2s,3S) . . . .
T. The Scattering Problem in Terms of
Discrete and Continuum Electronic States .
2. Discretization of the Continuum and
the Modified Coupled Equations . . .
3. Solution of the Modified Coupled
Equations. . . . . . . . .
4. Characteristics of PI and AI Processes
5. An Application of Discretizution to
PI and AI . . . . . . . .
6. Interaction Potentials for He' (ls2s,3S)
Ar and He + Ar+(3p5,2p). .. . .....
7. Parameterization of the Couplins. . .
8. Results from Coupled Channels Calcula
tions of He*(ls2s,3S) + Ar PI Collisions
9. Discussion . . . . . . . .
REFERENCES. . . . . . . . . . . .
SBGGPAPHICAL SKETCH . . . . . . . . .
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
ELECTRONIC AND DYNAMICAL ASPECTS
OF DIATOMIC SYSTEMS
By
John Curtis Bellum
August, 3976
Chairman: David A. Micha
Major Department: Physics
Diatomic systems are considered from the points of
view of their electronic structure and the dynamics of
motion of the heavy particles (nuclei) upon collision. In
Chapter I the electronic and nuclear motions are treated
formally by expressing the Schr6dinger equation for the
nuclei and electrons in independent variables in the body
fixed (BF), center of mass of the nuclei, frame, and then
introducing an expansion in terms of a complete set of
electronic states at each internuclear separation, R. Born
Oppenheimer and Coulomb couplings between the electronic and
nuclear motions are pointed out and discussed.
In Chapter II, atomatom interaction potentials and
electronic structure are investigated using the Multiple
Scattering (MS) calculational scheme with the socalled Xa
approximation to electronic exchange. The atom pairs,
aeHe, HeAr and ArAr are studied, and results are presented
for ground and excited state configurations. The computed
interaction potentials exhibit BoEnMayer repulsion, and the
calculations demonstrate how Xa orbital energies can be
used to predict crossings between interaction potentials.
An analysis is presented of the Xa theory and its usefulness
for the undsrstandinq of collision phenomena.
In Chapter III, Penning ionization (PI) and associative
ionization (AI) processes are considered in terms of molecu
lar orbital (MO) correlation diagrams. MO correlation
diagrams are calculated for He*(ls2s) + Ar(3p6) and He(ls2)
+ Ar (3p5) within the MSXe scheme for nonspinpolarized
and spinpolarized orbital. The ionization process is
discussed in terms of an Auger type mechanism involving
MO's which can be inspected in the unitedatoms (UA) limit
in a way which permits an analysis of the angular momentum
contributions of the emitted electron in the BF frame. MO
correlation diagrams are constructed based on atomic orbital
energies at the separated and unitedatoms limits, which
are determined from data available in the literature on
ground state atomic orbital energies. Estimated MO correla
tion diagrams are presented for He*(ls2s) + Ar(3p6),
+ Xr(4p6 ), + Hg(6s2), and Ne*(2p53s) + Ar(3p ),and in each
case an analysis is made of the angular momentum components
of the emitted electron. The results confirm that relatively
few such components are important for electrons emitted in
PI and AI. The UA analysis shows the importance of spin
polarized MO's, and also BornOppenheimer rotational
couplings, particularly between MO's which converge to the
same UA limit.
In Chapter IV consideration is given to the dynamics
involved in collisions Lonization processes. The formal
development of Chapter 1 is extended to include both discrete
and continuum internal electronic states. The resulting
continuously infinite set of coupled equations is then dis
cretized, leading to modified coupled equations for the
heavy particle motion. Discretization provides a suitable
framework in which to introduce physically reasonable ap
proximations which lead to a treatment of PI and AI in terms
of several (=20) twostate coupled equations. Application
is made to PI of Ar by He*(ls2s,"S), and the results show
that the approach includes the important dynamical features.
Partial ionization crosssections per unit energy, E, of
the emitted electron are calculated as a function of E, and
they show an s dependence in good qualitative and quanti
tative agreement with experimentally measured energy dis
tributions of emitted electrons. Partial crosssection
contributions for the heavy particles in specific angular
momentum states are also singled out." Their behavior as a
function of e, or of the angular momentum partial wave
number, shows structure which reflects regions of high
density of states in the continuum of final relative motion
of the heavy particles. The HeAr netastable and molecular
ion potentials are represented by a convenient functional
form describing atomatom int.raction potentials over the
entire range of R. In addition, the connection between the
decay width T and the coupling matrix elements between
discrete and continuum electronic states is used to make
reasonable estimates of the latter from semiempirical results
for P.
xiii
CHAPTER I
INTRODUCTION
The overall subject of this dissertation is the study
of electronic and dynamical aspects of diatonic systems.
In such a study it is the behavior of the electrons and
nuclei, during collision processes of the two atoms com
prising the diatomic, which is of interest. In this intro
ductory chapter the collision processes will be discussed
formally in terms of the Schrodinger equation which is sat
isfied by the wave function for the system of nuclei and
electrons. The Schr6dinger equation will be treated in the
bodyfixed, center of mass of the nuclei, frame by properly
transforming the Hamiltonian operator and wave function to
be expressed in this reference frame. The coupled equa
tions will be derived with attention focused on the various
sources of coupling between the electronic and nuclear mo
tions.
In the following chapter, matters specifically concern
ing the electronic structure of some diatoms comprised of
rare gas atoms will be considered, and possible applications
to collision processes discussed. In Chapter III special
attention will be given to chemiionization processes in
volving collisions in which one of the atoms is initially
in an excited state. Features of the electronic structure
of such collisions will be discussed, on the basis of which
an analysis of the angular momentum contributions to elec
trons emitted in such processes will be carried cut. Fi
nally, in Chapter IV, th: dynamics of a specific chsmi
ionization process, that of Penning ionization of Argon by
metastable Helium,will be treated by means of numerically
solving the coupled equations within a twostate approxima
tion. Total and partial crosssections obtained from the
calculations will be reported.
I. A Formal Statement of the Problem
In the quantum mechanical treatment of the two nuclei
and N electrons which constitute a diatomic system, the
Ilamiltonian operator expresses all of the energy contri
butions associated with the nuclei and electrons. The
actual form of the Hamiltonian depends upon the coordinate
frame with respect to which the positions of the nuclei
and electrons are located. The coordinate frame of most
practical use in terms of measuring the results of colli
sion events is one fixed in the laboratory, referred to as
the laboratoryfixed (LF) frame. The description of the
collision events, however, is most conveniently carried
out in a coordinate frame whose origin is fixed to the cen
ter of mass of the nuclei (CMN) of the two atoms which are
,olliding. The reader is referred elsewhere Csee, for
example, Pa6S and references therein, and Ju75) for a de
tailed discussion of various coordinate frames commonly
used and the form the Hamiltonian assumes under transfor
mation from one to the other. The formal development in
this chapter begins with the Hamiltonian expressed in the
socalled laboratoryfixed, center of mass of the nuclei
(LFCMN), frame, whose origin is fixed to the CMN and whose
axes remain parallel to those of the LF frame. Furthermore,
relativistic and masspolarization contributions to this
Hamiltonian are neglected (Pa68), and the coupling between
electronic spin and orbital angular momentum is assumed to
be small. In this LFCMN frame, R is the relative position
vector of the two nuclei labeled a and b, having masses m
a
and mb, respectively. The N electrons are located by the
set of spacespin coordinates {x.,i=.,N} = X, where
x. = (ri,s.), ri locating the spatial position and s. the
spin coordinate (a or 0) of the ith electron.
In terms of these variables, the LFCMN Hamiltonian
is expressed as
2
H(R,X) = (1/2m)VR + H(RX)
where m = m mb/(ma+mb) is the reduced mass of the nuclei,
and
N N
H e(R,X) = 1/2 (Z /r + /r )
e,(,2 r. _, a ia b io
i=l ri i=1 /
+ I/rij + Z Zb/R (2)
i
is the electronic Hamiltonian. In the usual sense, Za,
r. and r.. refer to the charge on nucleus a, the distance
:iaF 13
between the ith electron and nucleus a, and the distance
between the ith and jth electrons, respectively. The ex
pressions are in atomic units, where the unit of energy is
the Hartree, the unit of distance the Bohr radius (ao)
and the unit of mass that of the electron.
On the right side of Equation (1) the first term rep
resents the relative kinetic energy of the two nuclei,
which are the heavy particles taking part in an atomatom
collision. The terms on the right of Equation (2) repre
sent, in order, the kinetic energy of the electrons, the
Coulomb attraction energy of the electrons with the nuclei
a and b, the electronelectron Coulomb repulsion energy
and the Coulomb repulsion energy between nuclei a and b.
The description of a diatomic system can formally be
made in terms of the wave function, YCR,X), for the total
system of nuclei and electrons, which satisfies the time
independent Schradinger equation,
H(R,X)Y(R,X) = ET(R,X) ,
where E is the total energy of the system. In solving
Equation (3) it is important to pay attention to the angu
lar momentum' of the electrons and nuclei. The total or
bital angular momentum. K, is the sum of the nuclear and
( ) (e)
electronic orbital angular moment, and L where
Z(n) 
= x iVR (4)
and
N
+(e) N
L"'  r. x iV (5)
i=l r
2 +
The LFCMN components, K K and K as well as K = KK,
y z
all obey the usual commutation relations for angular momen
ta, and commute with the LFCMN Hamiltonian of Equation (1),
owing to its rotational invariance. Accordingly, the solu
tion, T(R,X), of Equation (3) is simultaneously an eigen
2
function of K and K with eigenvalues K(K+l) and M,
respectively, and the total orbital angular momentum and
its z component are constants of the motion. Because of
the as umption of negligible spinorbit coupling for the
electrons, the spin angular momentum has been left out of
this discussion for convenience, but could easily be in
cluded. So, with no loss of generality, the solution to
Equation (3) is classified according to the constants of
the motion, K and K and is written
Y(ax) = KM(R,X)
KM
SRemarks Regarding Reference Frames
At this point attention will be turned toward specif
cally pedagogical considerations, providing a reminder of
some basic concepts of a mathematical and physical nature
which are helpful in understanding the approach which will
be taken in solving Equation (3). The remarks which fol
low will serve to reiterate some key ideas which have long
been established (see, for example, Kr30). The ideas are
not easy to grasp, and are often passed over either in too
sophisticated or too cursory a way in the literature. No
table exceptions, however, can be found (Kr30, V151, Ho62,
Th6i, Th65, Pa68, Sm69).
As it stands, Equation (3) involves the Hamiltonian
and wave function, Yi', expressed as functions of R and
{ri} referred to the LFCMN frame. The axes of this frame
are labeled by x, y and z. The ith electron is thus lo
cated by ri having coordinates (xi,yi,z.). R is most con
veniently represented by its spherical polar coordinates,
R, 0 and ,. Therefore, in the LFCMN frame, (x.,yi.,z,
R.0,,) constitute an independent set of coordinates in
which to solve Equation (3).
In terms of these coordinates, the components in the
j(n) *(e)
LFCMN frame of t) and Le) (Equations (4) and (5)) ap
pear as
L(n) = i(sini/ae6 + cotocosfs9/9')
x
L(n) = i(cos +/9 + cotBsin0/ah)
Y
L(n) (7)
L = i /a (7)
z
and
N
(e) = i= (v/3z z.i/y.i)
L(e) = i (zi.3/xi xia/zi)
L(e) = i I (xiN/3y yi/3xi) 8)
2 Ci=l
The wave function of Equation (6) can also be written ex
plicitly in terms of these variables to read
SKMRW2) = YKM(R,O,I,xYi ,z ) (9)
Also, the first term on the right of Equation (1), express
ing the relative kinetic energy of the nuclei, appears as
follows (see, for example, Co62):
(1/2m)V2 = (1/2mR2 { 2/R(R23/DR)
+ cot63/oe + a2/Do2
+ sin2 e2/~ 2}
(10)
One can check that the terms within the brackets of Equa
tion C10), involving the angles, can be replaced by
I(n' (n)
Ln' according to Equation (7) (see, for example,
Ed60). It is important to keep in mind that partial de
rivatives, such as occur in Equations (7), (8) or (10),
depend upon which variables are actually independent of
one another during differentiation. Because the LFCM
frame constitutes an inertial reference frame, the set of
variables, (x ,yi,z ,R,e,4), of the electrons and nuclei
is indeed an independent set of variables. Therefore, for
example, 8/36 in Equation (10) means to differentiate with
respect to 8 while holding all other variables, (xi,Yi,zi,
R,4), fixed. Similar considerations hold in turn for each
variable in this set.
Now one would like to proceed to solve Equation (3)
by expanding the wave function of Equation (9) in a com
plete set of electronic wave functions at each R. It is
here that the need arises to express Equation (3) in terms
of variables referred to a coordinate frame in which the
internuclear vector R is fixed. This is due to the fact
that normally electronic wave functions are determined in
such coordinate frames. Such a reference frame, fixed to
the nuclei with origin at the CMN, will be called a body
fixed, CMN (BFCMN) frame. The key concept regarding elec
tronic wave functions is that they are usually calculated
under conditions where the set of electronic coordinates
in the BFCMN frame are treated as independent variables.
The BFCIN frame considered here shall have axes x',
y', z", where the 2" axis is along the internuclear vector
R. Consequently, the angles ( and 6 are the first two of
the Euler angles Csee, for example, Ju75) which rotate the
SFCMN frame into the BFCMN frame. The third Euler angle,
y, can be freely chosen since it only serves to define the
x' and y' axes, which can be arbitrarily set for a. diatomic
molecule (Ju75). Therefore, y is an auxiliary variable,
and as such will play only an indirect role in what follows.
Figure 1 shows the Euler angles (9y) by which the SFCMN
frame is rotated into the BFCMN frame.
The coordinates of the internuclear vector in the BF
CMN frame are simply (R,0,0). The ith electron, with co
ordinates (xi,yi,'i) in the LFCMN frame, has coordinates
(x,yl,z') in the BFCMN frame, specified for a given ori
entation (06,) of the internuclear vector by the unitary
transformation (see, for example, Ti64 and Ju75)
x. = Ex(COy) = (cosycosecos sinysinc)x.
+ (cosycosfsin + sinycost)yi cosysin9zi
y = y' (0Oy) = (sinvcosfcos{ + cosysinf)xi
wI 1
+ (sinycosesin + coscosi)yi + sinysinOz.
l
10
 y
The Euler angles (0y) reiating the SFCMN
frame (x,y,z axes) to the BFCfN frame
(x',y',z axes). y specifies the x' and v'
axes, which can be arbitrarily chosen and are
not shown. R labels the internuclear vector.
Figure 1.
11
z. = z'(00) = sinOcosix. + sin9siny.i + coszi (11)
Here attention has been drawn to the dependence of x ,y'
and zI on the angles 6 and y. The inverse transformation,
1
th
qjiving the coordinates (x.,yi,zi) of the i electron in
the SFCMN frame in terms of its coordinates (xi,yC,z)
in the BFCMN frame, is,
x. = (cosycosOcos sinysinp)xC
(sinycos6cosj + cosysinp)y + sin6cos6z'
y. = (cosycosOsin4 + sinycost)x1
+ (sinycosOsint + cosycos)y. + sinOsin2z
z. = cosysin9xC + sinysinly$ + cosOz (12)
With these expressions, the Schr6dinger equation of
Equation (3) in the LFCMN frame, where (x.,Yi,z.i R,,i)
are independent variables, can be rewritten allowing the
BFCMN electronic variables (x,,y',zi) to be treated as
independent variables (as they are in molecular electronic
structure calculations). The expression, "treated as in
dependent variables", in the preceding sentence speaks
to an important concept. To an observer in the BFCMN
12
frame, the electronic and nuclear coordinates are simply
x' ,y',z',R,0,0). But such an observer must keep in mind
that the BFCMN frame is not an inertial frame, and, by
Equation (11), (x,y',zf) have explicit (86y) dependence
for a given set of electronic coordinates in the LFCIN
frame.
In order for an observer in the BFCMN frame to use
the electronic variables Cx ,y',z ) as independent variables
in treating Equation (3), two new variables, 6' and 4',
may be introduced with two restrictions: (3.) the set
(xryt,z3,R,6',') must be an independent set of variables
for the observer in the BFCMN frame, and (2) 6' and #^
must be given by,
= ,
(13)
According to this equation, ("O') may seem to be redundant
variables, but this is not at all the case. The variables
(Cx,y,zt) are independent of (6'o') as far as the observer
in the BFCMN frame is concerned, and Equation (13) simply
specifies their values in terms of variables which are de
termined by an observer in the LFCMN (inertial) frame.
Equation (13) gives the relationships for 8' and "' just
as Equation (11) does for x',yv and z'.
Equation (3) will now be transformed so that an ob
Equation (3) will now be transformed so that an oh
33
server in the BFCMN frame could attempt to solve it in
terms of the independent set (x 'y!,z,R,6',I'). Sach a
transformation involves both the Hamiltonian operator (see
Equations (1), (2) and (10)) as well as the wave function
(see Equation (9)). First the Haiiltonian will be consid
ered.
For clarity, when performing partial differentiation
involving the independent variables Cx ,y,z i,R,9,4) the
symbol 3, which has already been used, will now specific
ally indicate that,when a variable of differentiation from
this set has been singled out, all the others are held
fixed. For instance,
a/as = a9/l{(xy ,z.),R,} (14)
where the variables held fixed are explicitly indicated.
Similarly, 3' will be used to indicate partial differen
tiation involving the independent variables (x',y',zi,
R,m e,''). By analogy with Equation (14) for instance,
= (15)
Then, if w is one of the variables Cxi,yi,zi,R,e6,),
14
3/3w t (+e'/w)V'/2e' + (U4t/2w)V'/24^
N
+ ((x /3w) 3'/@x + (3yC/Dw) 3'/y' *+ ( 3z/3w) '/9z) (16)
A similar relation holds for expressing 3'/Dw" as a linear
combination of partial derivatives involving the variables
(xi,yi,zi,R,6,4) where w' is cne oE the variables (x.,yi,
zt,R,9',O'). These are useful] expressions to keep in mind.
especially as to their meaning emphasized by using the 3
and 3' notation. This conceptual and notational viewpoint
follows the work of Kronig (Kr30).
Referring now to Equation (2), the kinetic energy of
th
the i. electron involves the operator,
i2 2 x 1 a2 2 2 2
V2 = 2/x2 + 2/Y2 + 2 /2z2 (17)
r ii i
On the basis of Equations (11) and (16), one can write
3/;xi (cosycosecos4 sinysin)}a/3x'
(sinycosOcosp + cosysin)3 '/2y' + sin8cos'/Dz: (18)
1 1
with similar expressions for 2/9yi and 3/2zi. Using these
expressions directly, one finds that
2 2 2/ y 2 +2 2/
3 /3x.+D /9y. + 2 /2z.
,2/ax 2 + 2a2/y2 + ,2 /az 2 = V2 (19)
i
15
The potential energy terms in Equation (2) involve only
the distances between electrons and nuclei, and therefore
are unchanged in going from the variables (xi,yi,zi,R,0, )
to the variables (X ,y ,z,R,6 ,'). Consequently, the
electronic Hamiltonian of Equation (2) takes The same form
for an observer in the BFCMN frame as for an observer in
the LFCMN frame.
Such is not the case for the first term of Equation
(1), which is expressed in the variables (R,6,<) in Equa
tion (10). By a straightforward, but tedious, application
of Equation (16), using Equations (11) and (12), one finds
that
(e) (e)
3/30 '/38' icosyL isinyLxe (20)
and
e) (e) (e)
3/3(  a'/8' icosO'L Ce + isin0'(cosyL sinyL )
2 x y
(21)
(e) (e) (e)
Here, L L and LI are the components of the elec
x y z"
tronic orbital angular momentum in the BFCMN frame. They
are expressed according to Equation (8) by replacing all
LFCMN quantities by their appropriate BFCMIN (primed)
counterparts. Equations (20) and (21), again with some
tedious algebra, lead to the following result for the an
gular terms in Equation (10):
C6
cot3/89 4 2/302 + sin 2 02/32
Scot8."3'/3ae + ^ /9ae0 + sin 6 '3 /WS
.e i:! e) + + iyL(e_ D /a0 kL'(e) 2
47a
2i(cot0/sinO)3'/'L ) cot2 .L ) 2 + (Le)
z 2 z
 (2/sin9) (cosyL ) sinyL ) (ie'/4g cosSL e) (22)
x Y z
(e (e) C (e)
Here, in the usual way, Le)" = LX + iL ). The R depen
dent terms in Equation (10) are unaffected in the transfor
mation from the LFCMN frame to the BFCMN frame. Conse
quently, replacing the angular terms of Equation (10) by
the right side of Equation (22) leads to the appropriate
Hamiltonian operator (see Equation (1)) which can be used
by an observer in the BFCMN frame in order to formally
treat the behavior of the electrons and nuclei.
The first three terms on the right side of Equation
(22) are similar to those found in the LFCMN Hamiltonian.
The remaining terms are those compensating for Coriolis ef
fects due to the fact that the BFCMN frame is not an in
ertial frame. In a sense, the inclusion of these Corolis
terms is the price paid by the BFCMN observer in order to
reckon (x ,y',z',R,6',0') as independent variables.
17
3. Remarks Regarding the Wave Function
The wave function of the total system of electrons and
nuclei (see Equation C91) mush also be properly transformed
and expressed in terms of appropriate functions in the BF
CMN frame. Here the approach of Davydov (Da65) is adopted.
Attention is drawn again to the fact that the wave
function of Equation (9) is an eigenfunction of the square
of the total orbital angular monentmn, K and its component
along the LFCMN Z axis, K with eigenvalues K(K+1) and M,
respectively. Following Davydov (Da65), if a coordinate
frame undergoes a transformation by rotation through Euler
angles (atS) to another coordinate frame, then an eigen
function in the first frame of K2 and K with eigenvalues,
respectively, K(K+1) and M, can be written as a linear com
bination of the (2K+1) such eigenfunctions in the rotated
frame, all of which are eigenfunctions of K2 with eigen
value K(K+I), and each of which is an eigenfunction of K z
with eigenvalue, A, among the possible values K,K1,....,K.
For the case considered here, of a transformation from
the LFCYN frame to the BFCMN frame through Euler angles
(Gy), the wave function of Equation (9) can be written
K
K (R,,,xiyiz D ((R,,,x y z)
A=K
(23)
THere the expansion coefficients, D (Oy), are the socalled
11,A
generalized spherical functions, or Dfunctions, and are
eigenfunctions of the symmetric top. A good discussion of
their properties is given by Edmonds (Ed60). In Equation
(23), .M A(R,0,0,x ',z') is an eigenfunction of KZ the
KA (R, ,xii,
component of K along the internuclear axis, with eigenvalue
t. The orbital angular momentum of the nuclei has no com
ponent along the internuclear axis, as can easily be veri
(e)
fied from Equation (4). As a consequence, Kz = LZ
It is at this point that it is convenient to intro
duce a complete set of electronic wave functions at each
internuclear separation, R. The electronic wave functions
employed can be any of the ones commonly calculated, where
the electronic variables, (x',yi,z), in the BFCMN frame
are taken as an independent set of variables. Such elec
tronic wave functions, whether of the singleconfiguration
or more elaborate configurationinteraction type, are clas
sified according to their component of electronic orbital
angular momentum along the internuclear axis. That is,
(e)
they are constructed as eigenfunctions of L e having ei
genvalues denoted by A. For each A, the complete set of
electronic wave functions, { nA(R,x ,y ,z')}, will be in
troduced at each R, and will be taken to be orthonormal.
M
Then each yK(R,0,0,x',y',z') of Equation (23) may be ex
panded in the set of electronic wave functions, {,nA}:
19
Y (R0,0,xC,y',z) = "'(R)R)/)0Rn (R,xC,yC,z ) (24)
1A i nh' nA
n
Substitution of Equation (24) into Equation (23)
gives a useful expansion for the total wave function of
nuclei and electrons in terms of functions of the elec
tronic variables C(x,yi,z) in the BFCMN frame:
YKM(R,8, ,x.,yiz ) = (1/R) (R)DK (ey)n (Rx',yzi).
KM n An
(25)
It should be emphasized that the equal sign in Equations
(23) and (25) means equality only in the sense that in
each case the function on the left of the expression,
where the variables are the coordinates of the electrons
and nuclei in the LFCMN frame, can be replaced by the
linear combination of functions on the right of the expres
sion, where the variables are the coordinates of the elec
trons and nuclei in the BFCMN frame. The presence of the
variables 0 and 9 in the Dfunctions on the right of Equa
tions (23) and (25) will be discussed presently.
4. The Coupled Equations and Coulomb and
BornOppenheimer Couplings
Recalling the previous discussion regarding the Haril
tonian operator, Equations (19) and (22) provide an observ
er in the BFCMN frame the appropriate Hamiltonian for
20
the nuclei and electrons under conditions where (x ,y ,z'
R,0',$') are independent variables. Using this Hamiltonian,
an observer in the BFCMN frame may now replace ( and e on
the right hand side of Equation (25) by <' and 8' according
to Equation (13). Then the right side of Equation (25) be
comes an appropriate expansion for the "transformed" wave
function, KM(R,O' ,,x',y',iz'l), which satisfies the "trans
formed" Schr6dinger equation. Thus, an observer in the
BFCMN frame may proceed to solve Equation (3) by relying
on Equations (19) and (22) and solving for the wave func
tion
KIM (K
Y M(Re ,',x',yZl)=(1/R) Tn (R)D ($ e'Y)nA (Rxy z.
i I An n M,n JA 1IJ(
An
(26)
In dealing with the operators in Equation (22), it
will be useful to rely on some of the formal properties of
the Dfunctions (Ed60). The Dfunctions, D (9y) are
formally associated with rotations through the Euler angles
(Jey), and these rotations are generated by the angular
4
momentum operator, N, whose components are
(n)
N (04y) = L i(cosj/sin6)/3y ,
x x
(n)
N (46y) = L i(sin4/sin)6/y ,
y y
N (46y) = L(n)
z z
21
where L(n), L(n) and L(n) are giver in Equation (7). The
x y z
2 2 2
Dtunctions are eigenfunctions of N N +N +N
x y z
N2(6Y)D (6) = K(K+41)DK (y) (28)
By expanding out N 2(Oy) using Equation (27), and recalling
thac the ( and y dependence of D CA(By) is in the factors
e and e Equation C28) can be rearranged as follows
(Ed60):
2 2 K
(O2/e2 + cotW/39 + sin2S /42)DK A((9y)
= {sin 2(A 2cosOMA) K(K+1)}D ,A (y) (29)
Other useful relationships involve the operators
N (tey) = N iN In particular, based on the properties
x y
of the Dfunctions (Ed60), it can be shown that
N. ( K (Y)q(0y) = At+l(O) (30)
where [1 = {K(K+1) A(CAl)}1. Furthermore, 3/38 can be
expressed as
3/ae = '(e N el% N) (31)
Using Equations (19), (22) and (31), the BFCMN Hamil
tonian can be expressed as follows:
H(R, 0',',x ^,y,z 2= (1/2nR2)D/9R(R2 9/R)
(1/2mR2) )cot9'a'/a6 + 2/82 + sin2 e'2 /at2
22
(e i 2 (e) () 2 (Ce) 2
ji(e 'L+ ) + e L_ ) (e N N (4 0"y)  ei N ( 0^e ) )
(L(e) 2 2i(cot6/Asin) cotO (L ) + (L
z z z
iY (e) e(e)
/sin ')(e L +e l'e) )(i'/ cosOl'L )}
+ He(R,xCyCz ) (32)
Now, in the usual way, the right side of Equation (26) may
be substituted into the "transformed" Schr6dinger equation,
(H E)NKM = 0 (33)
When this is done, many of the terms from the operators of
K
Equation (32) acting on functions D 5 n of Equation (26)
will combine and lead to simplifications. One can, for
example, compare the contributions from some of the opera
(e)
tors involving L e in Equation (32) with the terms in
volving A on the right side of Equation (29).
Upon multiplication of Equation (33) on the left by
D ,A' I'Y) *, (R,x ,yz) followed by integration over
M,A nA I i
the coordinates C'8'0yr xtytz2, the following set of
KwM
nA
{(/ d/dR2 (/2mR2) [2A2 K(K+1)] E n(R)
= ({< nH eli A> (I/2l~R22
+ (1/nm)< d/dRnA> + (/2m)6A,
+ (I/m)<~nhld/dlR/ Pn.A. + (i/2m)< ~nAd2/dR21 bn.A.>} AA
 23
2 (e)I KM+
(1/2mR 2)< ,i' I,AA +
_1I/2mR,2) KM (KM R) 34)
/ nA n'A AA A,A'1 n ( (34)
Here, the brackets indicate integration over electronic
KM+
variables. The coefficients g A are given by
KM+ 2 K K *K
gA = (2K+1) (8T2)1fd'sin6d6'dyD ^ 6Y)G M (6Y)
(35)
where
K +iy K
A6. y) = C (Msin 0 + Acote')DMA(POY)
+ (i/2)eZiy{A+e'DA+i ('6y) Ae'DK,Al(eY)} .
(36)
In Equation (34), all of the various couplings be
tween the nuclear and electronic motions are included ex
plicitly. The radial BornOppenheimer couplings appear
in the matrix elements between the electronic expansion
states over the d/dK' and d2 /dR2 operators, and reflect the
effect of the radial motion of the nuclei on the electronic
motion. As can be seen, radial BornOppenheimer couplings
exist only between electronic states having the same value
of A. The rotational BornOppenheimer couplings appear in
the matrix elements over the L(e) and (L(e)2 operators,
and reflect the effect of the rotational motion of the nu
clei on the electronic motion. A convenience of treating
the problem in the BFCMN frame is that the influence of
nuclear rotational motion on the behavior of the electrons
becomes expressed in terms of matrix elements between elec
24
tronic states over electronic angular momentum operators,
rather than nuclear angular momentum operators. As can be
(e)
seen, rotational coupling due to L e exists only between
electronic states having A values differing from one an
(e) )
other by il, whereas coupling due to (L( ') exists be
tween electronic states having the same A value. The so
called Coulomb coupling appears in the matrix element over
the electronic Hamiltonian in Equation (34), and exists
only between electronic states having the same value of A.
5. Discussion
Some brief remarks are in order regarding the consid
erations of this chapter. No qualifications have been
placed on the basis set of electronic wave functions,
{ nA}, used in the expansion of Equation (26), other than
that it be complete and orthonormal. Traditionally, ap
proaches to molecular electronic structure have tended to
focus on electronic states which are eigenfunctions of the
electronic Hamiltonian, Hel, and which therefore leave the
matrix of Hel diagonal. These are the socalled adiabatic
states which provide an adiabatic representation. In such
a representation all of the coupling between the electronic
states, associated with inelastic collisional processes,
rests in the BornOppenheimer terms of Equation (34). The
eigenenergies of Hel associated with these adiabatic states
25
obey the noncrossing rule, as is well known.
However, in treating atomic and molecular collision
processes, it is not at all clear that the set of adiabatic
states is always the most appropriate representation to use.
Stemming from the point of view emphasized by Lichten (Li63)
much consideration and discussion have resulted regarding
the importance of the socalled diabetic representations.
These representations are comprised of electronic states
which are not eigenfunctions of Hel. Consequently, the
diagonal matrix elements of these states with Hel need not
obey the noncrossing rule, and the offdiagonal matrix ele
ments may be appreciable. Compared to the BornOppenheimer
couplings, the Coulomb couplings in a diabetic representa
tion can often actually be the dominant source of coupling
associated with inelastic processes influencing the heavy
particle motion described by the coupled equations of Equa
tion (34).
Two particularly good discussions of these matters
have been made by Smith (Sn69) and Sidis (Si76). Here, it
is simply pointed out that the appropriateness of the adia
biatic or a particular diabetic representation depends upon
how successfully the dominant coupling terms can be identi
fied, as well as calculated or estimated. Radial Born
Oppenheimer couplings are difficult to calculate and nor
mally must be estimated. In addition, they are character
sized by singularities in regions of R near avoided crossings
of the associated adiabatic eigenenergies. The coupling
through Hel of diabetic states can be estimated, if not
often calculated. However, because of frequent lack of
information abcut the BornOppenheimer couplings, one can
not always be sure when the Coulomb couplings constitute
the dominant contribution in describing inelastic processes.
As can be seen from Equation (34), the rotational Born
2
Oppenheimer couplings have a R2 dependence. Therefore,
their contribution will be of increasing importance as dis
tances of closest approach of the nuclei become smaller.
The research related to atomic collision processes re
ported in the remainder of this dissertation has been car
ried out within the framework of diabatic representations.
In this connection, electronic states constructed as deter
minants of oneelectron molecular orbitals are particularly
useful. Furthermore, the behavior of the oneelectron mo
lecular orbitals and associated orbital energies can itself
provide information of use regarding inelastic processes in
atomatom collisions. Considerations along these lines are
pursued in Chapter II, illustrated by molecular orbital cal
culations on some rare gas diatomic molecules, and in Chap
ter III, where Penning and associative ionization processes
in thermal energy collisions of excited (metastable) rare
gas atoms with ground state atoms are discussed in terms of
molecular orbital correlation diagrams. An assessment of
the angular momentum contributions in the EFCMN frame of
the emitted electron in these processes is made based on
the correlation diagrams analyzed in their united atoms
limit. In this united atoms analysis the significance of
rotational BornOppenheimer couplings will need to be con
sidered, as has just been mentioned.
The topic of study in the final chapter is the dynam
ics involved in Penning ionization of Argon by metastable
Helium in thermal energy collisions. The approach will be
to solve numerically in a two state approximation the cou
pled equations of Equation (34). An interesting feature
of chemiionization is that the electronic state prior to
ionization is embedded in the continuum of electronic states
associated with the ionized electron. Thus, in the expan
sion of Equation (26), the sum over discrete electronic
states must be augmented by an integral over the continuum
electronic states. This feature will be dealt with in
Chapter IV. The solution of the coupled equations with
the appropriate boundary conditions for scattering will
lead to results for total and partial cross sections for
Penning ionization.
CHAPTER II
A STUDY OF SINGLEELECTRON AND TOTAL ENERGIES
FOR SOME PAIRS OF NOBLE GAS ATOMS
In this chapter* consideration will be given to
atomatom interaction potentials and electronic structure
pertinent to the description of collision events. Sev
eral features should characterize the method employed in
this type of treatment of the electronic structure of
atom pairs:
(a) The method should provide a description of
the interaction potentials for the ground
as well as excited states which govern the
motions of the heavy particles (nuclei)
during collisions.
(b) Details of the electronic structure, at least
of the type found within the selfconsistent
molecular orbital (MO) framework, should be
available so that processes related to elec
tronic excitations may be studied, especially
in the case of energetic atomatom collisions.
*This chapter is an essentially unaltered version of a
contribution (Be74a)to the Proceedings of the Interna
tional Symposium on Atomic, Molecular and Solid State
Theory and Quantum Statistics held at Sanibel Island,
Florida, 2026 January 1974, where a preliminary report
of the results was made.
28
29
(c) The method should be applicable to a variety
of pairs of atcmic neutral and ionic species
while at the same time it should involve only
a moderate degree of calculational effort and
complexity.
Statistical approaches, such as the ThomasFermi
Dirac method, have provided useful results by applying
freeelectron gas energy expressions in conjunction with
a molecular charge density taken as a superposition of
atomic charge densities. However, these approaches are
deficient in that they only describe the ground state in
teraction. They afford no information of the type men
tioned in point (b), and furthermore they are not appli
cable to situations where appreciable charge rearrange
ment occurs in the diatom, since there is usually no pro
vision for selfconsistency in the calculations.
These considerations have led to the use of the MSXa
method (Jo66, Jo73, Sl71a, Sl71b, S172) in this work. It
parallels the HartreeFock approach in that it provides
a oneelectron description with corresponding oneelectron
eigenvalue equations and eigenstates which are solved
selfconsistently. The method makes use of a convenient
local approximation to the exchange potential. Although
approximate, the treatment of exchange in the Xa approach
should be quite adequate in handling the short range part
of atomatom interactions, where Coulomb and exchange
forces between the electrons are the important ones.
The calculational procedure is furthermore based on
the "muffintin" approximation to the oneelectron poten
tial, which entails no additional computational complica
tions as the number of electrons being treated increases.
Thus the scheme has wide applicability.
The first application of the MSXa method to inter
action potentials of rare gas pairs was a calculation on
the NeNe system (Ko72, see also Tr73). In this chapter,
calculations performed on the atom pairs, HeHe, HeAr
and ArAr are presented, and their relevance to collision
phenomena is pointed out. At the outset, aspects of the
Xa theory and computational approach which directly re
late to the present work are described, and limitations
of the theory are considered. Results of the calcula
tions are then presented, and finally, a discussion of
the results and their significance is given.
1. Theoretical and Computational Considerations
Within the Xa formalism (S171a, S172), the total en
ergy EyX, of a system of N electrons is specified by a
set of spinorbitals, {ni}, according to the expression
EXC = n ndr u (r) V + X(2Za/rla )}u (rI)
1 a
+ /drl/dr2{2p(rl)p(r2)} /rl2
Slrd {pr)U O ) +P (rl)U I
S+l {P 1Xc 1 1 Xu 1
+ 2Z Zb/Rab (37)
a
Equation (37) is expressed in Rydberg units, and in the
usual sense, Za, Rab rla, and r12 refer to the charge
on nucleus a, the distance between nuclei a and b, the
distance between rl and the position of nucleus a and
 +
the distance between r, and r2, respectively. The elec
tronic charge density is
p(r,) = p r) + p (r)
1+ I
nu(rl)u(l) + nu (rlu (l (38)
comprised of the charge density of electrons with spin
up (denoted by +) and that of electrons with spindown
(denoted by 4), and
UO (r ) = 9a((3/4T)p ( ))1 (39)
is the Xa local exchange energy density, a being a multi
plicative factor. It should be emphasized that E is an
energy functional depending on the spinorbitals, ui, and
the occupation numbers, ni, and does not necessarily rep
32
3
resent an average value of a manyelectron wave function
over a Hamiltonian operator, as is the case in Hartree
Fock theory.
For a given assignment of the ni's, the u.'s are de
termined by making EXa stationary with respect to their
variation. This leads to a set of eigenvalue equations
for the spinorbitals. For the u 's of spinup, one has
(fl(rl) + fdr2{2p(r2)/r,2} + V x(r)ui(r = (r) ,
(40)
where
+ 2
flrl) 7 + (2Za/rla) (41)
a
and
Va ( ) = 2/3 Ue(l) (42)
A similar set of equations is obtained for the u.'s of
spindown. As is well known (S171a, S172), the interpre
tation of the Xa orbital eigenenergies, e., differs from
that of HartreeFock orbital eigenenergies, and is based
on the relation,
Ei = E Xa/ ni (43)
This condition between EX, the E 's and the n 's insures
that Fermi statistics holds within the framework of the
Xa description of a system of electrons; namely, the low
est value of EX for a system of electrons is achieved
33
when the spinorbitals of lowest eigonenergies are occu
pied. Concentrating on diatonics at fixed internuclear
distance, R, a state of the electronic system may be
identified by means of an assignment of the n 's. A con
venient way, then, of determining over which region of R
a particular Nelectron state is the one of lowest energy
is by looking at the behavior of the .i's for the occupied
and unoccupied orbitals of that state, and observing over
which region of R the .i's of the occupied orbitals are
the lowest ones. In the results which follow, this fea
ture will be demonstrated. Equation (43) is also the
basis for the familiar transition state approach (Sl7lb),
from which good approximations may be found to ionization
energies as well as excitation energies between electronic
states.
In practice Equation (40) is solved by the Multiple
Scattering (MS) method (Jo66, Jo73), with the potential
operator
V(r!) = fdr2 2p(r2)/r} + V r) = V(rl + VX(rl) (44)
approximated by a "muffintin" form, whereby it is av
eraged over angles within nonoverlapping spherical re
gions centered on the various nuclear sites and also oth
er sites in the molecule (outer sphere, empty spheres,
etc.), and volumeaveraged elsewhere. The u.'s are de
1
34
termined selfconsistently in terms of partial wave ex
pansions within the spherical regions, and expansions in
"multiplyscattered" waves elsewhere. Furthermore, at
each iteration of the selfconsistent procedure, the "muf
fintin" form of p (r), as found from the u.'s according
to Equation (38), is used to evaluate the "muffintin" po
tential as well as the "muffintin" approximation to EXa
of Equation (37). The reader is referred elsewhere (Co72)
for the details about these approximations as well as ways
of correcting for them (Da73, Da74a, Da74b). It suffices
here to say only that the effect of the "muffintin" ap
proximation in the MSXa evaluation of EXa is appreciable.
However, orbital energies e. for diatoms appear to be more
reliable than EXa within the MSXa approach (We73).
It has been shown (S174) that in obtaining the ex
change potential of Equation (42), one need not assume
that the electrons of an atom or molecule behave locally
like those of a freeelectron gas of ths same density.
Rather, one may assume a spherically symmetric "Fermi hole"
and apply dimensional arguments. Nevertheless, consider
able discussion has been devoted to shortcomings of ex
change potentials of the type in Equation (42) associated
with the finite numbers and inhomogeneous spatial distri
bution of electrons in atomic and molecular systems (Li70,
Li71, Li72, Li74, Ra73, Ra75). By means of a more careful
35
look at the exchange energy of a freeelectron gas of a
finite number of electrons in a finite volume, corrections
to the exchange potential have been derived by separating
out the contribution from the interaction of each electron
with itself (Li70, Ra73, Ra75); for a small number (<200)
of electrons the contribution trom this "selfinteraction"
becomes quite sizable. Estimates of the "selfinteraction",
based on the abovementioned corrections, show that in
atoms and molecules only about 85% of its contribution to
the exchange potential is included in V (Li72), while
Xc
the Coulomb potential, VC, in Equation (44) includes all
the "selfinteraction". This imbalance may be partially
remedied by an adequate choice of the a factor.
Now, a variety of ways have been suggested for spec
ifying the a factor of Equation (39) for a system of elec
trons (Sc72 and references therein). The values of a for
atoms, resulting from the various schemes, almost all dis
play the same trend; namely, for atoms of larger numbers
of electrons the values of a tend toward 2/3, the factor
appearing for a freeelectron gas, whereas for atoms with
few electrons, larger values of a result. Since the "self
interaction" contribution is large for few electrons, this
trend in a values has often been interpreted as reflecting
the required greater compensation for the deficit in "self
interaction" in the case of few electrons, the compensation
36
becoming less as more elections are involved (Li72). Thus,
despite the importance of the corrections which have just
been reviewed, the Xa exchange potential represents quite
well the exchange interaction in a system of electrons.
As long as the "muffintin" approximations are being made,
a treatment in terms of the Xa exchange potential alone,
with a commonly used value of a, is expected to be ade
quate for describing the shortrange interaction between
atoms, as mentioned at the beginning of this chapter.
The calculations performed here employed a double
precision (14 hexadecimal or roughly 16 decimal digits
available per number on an IBM370/165 computer) version
of the MSXa program, MUSCATEL. This precision was re
quired since the interaction energy, AE, is computed as
the difference between the total energy of the diatom
(in our case, EXa in the "muffintin" approximation) and
that of the two isolated atoms. For instance, the case
of ArAr at moderately large internuclear separations, R,
involves interaction energies seven orders of magnitude
smaller than the total energies used to determine the in
teraction, hence requiring at least eight significant
figures in the total energies. The total energies of the
isolated atoms were calculated using the HartreeFock
Slater (HFS) atomic program (He63, Za66).
For the homonuclear cases, HeHe and ArAr, the so
called "virial theorem" values of a for the atoms, as re
ported by Schwarz (Sc72), were used in all regions of the
molecules. For HeAr the respective atomic "virial the
orea" values were used in the spherical regions about the
atoms, and a weighted mean (weighted according to the num
ber of electrons of each atom) of the two values was used
elsewhere. The heteronuclear case of HeAr required a
choice of the radii to be used for the spherical regions
about the atomic sites. Contiguous spherical regions were
chosen in all cases, and the ratio between the He and Ar
sphere radii, used at all internuclear separations calcu
lated, were determined in the following way. Average ra
dii obtained from numerical atomic HartreeFock calcu
lations (Ma67a, Ma68) were used to estimate the size of
each of the isolated closed shell atoms, He and Ar, based
on the Is orbital of He, and the 3p orbital of Ar. The
values are, respectively, 0.92727 a and 1.66296 a The
O O
He and Ar sphere radii at different R were then chosen in
the ratio of these two characteristic charge extents.
From the same atomic calculations, the maximum values of
the He Is and Ar 3p radial probabilities occur, respec
tively, at about 0.55 a and 1.30 a so for internuclear
separations larger than about 1.85 ao, the above scheme
for selecting sphere sizes should serve well. Other con
siderations must be made for cases which do not involve
two closed shell atoms and where significant charge rear
38
rangement occurs in the diatom. Such a case is treated
in the following chapter where calculations are reported
on the excited HeAr diatom which separates at large R
1,3 6 1
to He* Cls2s, 'S) plus ArC3p ,S).
The selection of partial waves to be included in the
expansion of an orbital in the various regions depends
upon over which regions of the molecule the orbital tends
to be concentrated. For a very deep lying core orbital,
which is essentially of atomic character, only the partial
wave corresponding to that of the associated atomic orbi
tal was used in each appropriate atomic region, since par
tial waves of other R values give a negligible contribu
tion. For higher lying orbitals, appreciable contribution
from a number of partial waves, s, p, d, etc. may occur.
Such partial waves were included as long as their ampli
tudes were at least onehundredth the amplitude of the
dominant partial wave.
All of the calculations required no more than medium
size core on an IBM370/165 computer and the times per
iteration of the SCF procedure were about two seconds for
HeHe, five seconds for HeAr and between five and ten
seconds for ArAr, depending on the "goodness" of the
starting point for a calculation. Denoting the absolute
difference, occurring between the values of the "muffin
tin" potential of Equation (44) at one iteration and those
39
of the previous iteration, by AV, and the maximum value
4
of AV/V by s, the degree of selfconsistency c<104 was
achieved typically in 1520 iterations.
It should be pointed out that the relative error of
the MO wave functions is of the same order of iaagnitude
as that of the potential. Since the total energy is vari
ationally determined, and therefore accurate to second
order in the wave functions, the degree of selfconsis
tency we have used is sufficient to insure the accuracy
required in the total energies at large R.
2. Results
In Figure 2 are displayed the interaction energies,
AE, for the three diatoms, HeHe, HeAr and ArAr, as a
function of R, the internuclear separation. The interac
tion energies are shown for the states with the lowest
energy at large R (i.e., asymptotic ground states). The
interactions on this semilogarithmic plot are seen to be
quite linear, indicating the repulsion they show over the
investigated ranges of R is characteristically of the
BornMayer type, namely A exp(bR). This behavior, of
course, would break down at very small R, where the Cou
lomb repulsion between the nuclei becomes strongly domi
nant. The beginning stages of this other behavior is seen
in the HeHe case at R 0.7 a On the other hand, as R
0
Figure 2. Interaction energies, AE, for the pairs HeHe,
HeAr and ArAr in their separated atom ground
states. Calculated points are encircled. a.u.
of distance refers to the Bohr radius, ao
O
E (c.u)
1.0 2.0 3.0 40 5.0 6.0 70
R(o.u)
42
increases for a given interaction and approaches the van
der Waals radius, the actual interaction energy would pass
through zero, and its logarithm would asymptotically ap
proach () as R nears the point of zero interaction.
This pronounced deflection of the logarithm of AE away
from the BornMayer straightline behavior is not exhibited
by the calculated interaction energies in Figure 2, indi
cating that these "muffintin" interaction energies show
no tendency to describe the van der Waals attractive well.
It should be pointed out, though, that efforts made to
correct for the "muffintin" effects in the case of NeNe
have shown a well defined attractive region (Da73, Da74a,
Da74b). The procedure for calculating these corrections
is, however, nontrivial, and would be impractical for
the present purposes.
Therefore, BornMayer A and b parameters for the in
teractions of Figure 2 were determined by means of a
leastsquares fit to the calculated points over the re
gions of straightline behavior. The parameter b measures
the slope of an interaction as shown in Figure 2, and re
flects the "hardness" of the repulsion described. The
parameter, A, serves as a measure of the overall strength
of repulsion, being the value of In(AE) at R=0, although
it is somewhat unrealistic, since R=0 is an unphysical
separation at which to compare the "strength" of the repul
sion of a BornMayer type.
43
So, the A parameters of the interactions of Figure
2 are simply listed in Table I, and it is pointed out
that over the region of BornMayer behavior, they lead to
interaction energies for the three diatoms which overes
timate the repulsion, but obey the combination rules for
such interactions in that the HeAr BornMayer line falls
between those of HeHe and ArAr. Listed in Table I by
way of comparison are the b parameters of the interactions
of Figure 2 along with those determined from other theo
retical calculations and experiment, as indicated. The
ranges over which the listed parameters apply are shown
in parentheses. In general, the b parameters of the pres
ent work reflect "softer" repulsions than do the other
tabulated b parameters. However, the ranges of R values
of the present work extend to larger R, and it is possi
ble that "muffintin" effects lead to a proportionally
larger overestimation of the repalsion at large R, i.e.,
to smaller b parameters.
The case of HeHe is now considered in more detail.
In particular, the behavior of the MSXa eigenenergies is
looked at, providing a good example for diatomic interac
tions. As is well known, at large internuclear separation,
the lowest state of the HeHe diatom is the IZ+ state spec
g
ified by doubly occupied I and la molecular orbitals.
These two MO's are, at large R, essentially the gerade
44
I
04
0
0
H)
40
C)
a)
Ii
r
H
0
04
'4
rC
a)
EH
t1
a
H
'4
0 r
U
u
U 
Her
E
I 
0 CC
O
H
U
E
E
H
0
PC
 0
o
I 0
4a
0
Ho
I 0
C)
H
rl
I 0
rfi
rc co
'2*
oN to
LD
O)
I
CO
,r
0
Cr)
,0
o
0
ii
to
03
Ln
Ln
O
0
rC)
Si
O
I
i [
co
n I
o co
L00
o
N *
N
In
'lp
0 I
H 0
o
O0
0
N
o in
.0 I
c3
o
N
tO
ra
en
03
\D
'03
ca
to
Ln
.1 *
0
u
to
03 I
CO
0o
H *
r[
0
rC
'D Ln
N
LO
us
o
rHl*
en
0
03 N
en
I
Lo
to
H *
cN
03
4.r
0o
,l
IN
11
'4
PC
45
and ungerade combinations of the He Is atomic orbitals.
In Figure 3 is shown the plot of the eigenvalues of these
two orbitals as a function of R over the range 0.5 ao to
5.0 a In keeping with the idea of a correlation diagram,
0
arrows indicate the HFS atomic orbital eigenenergies; the
is orbital of He in its ground Is2 configuration, in the
separated atom CSA) limit, and the is, 2s and 2p orbitals
2 2
of Be in its ground Is 2s configuration in the united
atom [(A) limit. It is seen that cl, and E are nearly
g u
degenerate with aes of He at large R, as expected, and
separate as R decreases. Now, in the hA limit, the lo'
orbital correlates with the Be 2p atomic orbital. Thus
1' 2 2 2 2
the l' (l a 1 ) state approaches the excited Be is 2p
g g u o
atomic state in the UA limit. On the other hand the 2o
g
orbital correlates with the UA Be ..3 atomic orbital. It
is therefore the Z+ (la2ao ) state which in the UA limit
g g g
2 2
correlates with the is 2s Be ground state. It is of in
terest to determine at what internuclear separation the
1 + 2 2
E state specified by la 2a becomes lower in energy than
g g g
2 2
that specified by la la As discussed earlier in this
g u
chapter, one may proceed in two ways: (1) direct observa
tion of the interaction energies, AE, of the two states
as functions of R to see where they cross; or (2) observa
2 2
tion of the eigenenergies of, for instance, the Cl 22
g g
state to see where cla and 20 become the ones of lowest
g g
0c C
D 4 C 0
(N tn0
NX C '
Su 4oa
d o a)
1 CU
0 0H 0
rd C C U
C
SI H 3, C
WV 0) U
,fl U
roI 0 m
So 0 )J
U.C
D 'OO4 m0
U I (C 0
4J C) CU
00 CUC
t 0 04) M
,. .
C CU
(U (U 0
0'
r4
47
Si
a 
NN c o
o
o o
/.
? I 4 
3 
N 
. . I , I i P I i I , ,
i  oo .
value. Proceeding from 1.2 a to smaller R, the eigenen
2 2
erqies for the la 2a state have been plotted along with
g g
22 2
those of the 1a lIc state. For the l"2ag state, ,la
lies lower, to begin with, than ,2 and Fermi statistics
2 2
indicates that l1 21 is not the state of lowest energy.
Sgg
However, Cle is rising sharply as R decreases, and is
seen to cross above e2g between 0.5 ao and 0.6 a In
ward from this crossing the la and 2a orbitals have the
g g
1 2
lowest eigenenergies and hence the lc' 2o state has the
g g
lowest energy. For comparison, in the insert of Figure 3,
a plot with linear scales is shown of AE versus R for
these two states, which corroborates this behavior.
It should be noted that the interaction energies cross
in a very gradual manner whereas the eigenenergies cross
more sharply. Thus the eigenenergy behavior indicates
more clearly the position of the crossing, which is found
to be 0.53 a These results compare well with SCF results
reported on HeHe (Ma67c), where it was found that the en
2 2 2 2
ergies of the la la and la 22 configurations cross near
g u g g
0.6 a Also, in the accompanying 50 configuration natural
orbital iteration calculations, it was reported that the
2 2
l glou configuration was dominant beyond 0.7 ao. Though
the calculations reported here go inwards only to 0.5 ao,
2 2
the eigenenergies of the la 2a state are seen to be ap
g g g g
preaching the appropriate eigenenergies of ground state Be.
48
An investigation (Ya74) of the HeHe diatom, subse
quent to the one reported here CBe74a), hut closely paral
leling it, has been carried out in the HartreeFock ap
proximation. It is interesting to compare the MSXa and
HartreeFock results. In both approaches, the behavior
of the total energies of the. lc21a and 1 22o configura
g u g g
tions as R decreases from 0.6 a to 0.5 a shows that
o o
2 2
they cross very gradually, the energy of the lo 222 state
g g
becoming lower than that of the loa2la state at 0.56 a
g u o
in the HartreeFock case, and as has been seen here, at
0.53 a in the MSXa case. These values are in good agree
ment. However, in the HartreeFock approach, the deter
mination is based solely on the total energy curves of
the two states as they cross with nearly the same slope.
This is because in the HartreeFock approach there is no
immediate connection between the state of lowest energy
and the eigenenergies of the MO's associated with that
state. However, as has been discussed and demonstrated
here, such a connection can be made in the Xa approach on
the basis of Equation (43), which permits the detection
of crossings either by observing MO eigenenergies or
total energy curves.
For ArAr, at large R, the ground state is also
+ and is specified by the first five a and a MO's,
g g u
each being doubly occupied, and the first two 7 and T
u g
50
MO's, each having occupation number 4. For large R, or
bitals 10 to 5gu are formed from the appropriate com
g,u g,u
binations of Ar Is, 2s, 2Po, 3s and 3po atomic orbitals,
and the first and second gu orbitals from the appropri
ate combinations of Ar 2p and 3p atomic orbitals, re
spectively. In keeping with the Fermi statistics, the
eigenenergies of these occupied orbitals are the lowest
ones at large R, where we also find lying above them the
eigenenergies of the unoccupied 60 and 16 orbitals. A
number of states can be specified at smaller R by the vari
ous assignments of occupation numbers to the 5U 60 and
16 orbitals.
Calculations have been made on some of these states
and are displayed in Figure 4 on a semilogarithmic plot
of AE versus R (in Figure 4, maonp refers to 51, 609
and 16P). In the region of R shown, numerous crossings
can be seen, and they are all of a very gradual type.
11 2
The Uo and 6 curves are from nonspinpolarized MSXa
u g g
calculations, the corresponding spinpolarized calculations
reflecting only a small splitting scarcely noticeable on
such a graph as Figure 4. We can see that the SA ground
state, denoted by 2a in Figure 4, no longer is of lowest
energy for R less than about 3 a This can be confirmed
again, by looking at the eigenenergies for this state, and
in Figure 5 we display the highest of them versus R on a
Figure 4. Interaction energies, AE, for ArAr in states
where the highest orbitals have occupation
numbers as specified. Calculated points are
encircled, a.u. of distance refers to the
Bohr radius, a .
o
Ar Ar
Sm+ nr p n =2
m+n+p=2
2.0 2.5
R (au.)
Figure 5. Orbital eigenenergies for ArAr in its
separated atom ground state. Calculated
points are encircled. a.u. of distance
refers to the Bohr radius, a .
o
54
R (a u.)
20 40 60 8.0
C (a~u.)
3p Ar
3s Ar
55
loglog plot. The appropriate SA 3s and 3p eigenenergies
are also shown. The eigenenergy of the unoccupied 6g MO
(which, in the SA limit correlates with the unoccupied 4s
atomic orbital of ground state Ar, whose orbital energy
is 0.0023 a.u. and lies off the scale in the figure) is
seen to be descending rapidly as R decreases, crossing
the occupied So orbital eigenenergy sharply at 3 ao,
indicating that for R less than 3 ao, this state is in
deed no longer of lowest energy.
3. Discussion
The results which have been presented illustrate the
possibilities of the MSXa method in the study of interac
tions that play a role in collision events. The interac
tion potentials themselves render information on the Born
Mayer type repulsion and, in conjunction with results on
van der Waals attractions, enter into the calculation of
relative motion of colliding atoms. Implicit in the re
marks on crossings between interaction terms for various
states is that the states calculated in the Xc method
are diabetic in nature. Indeed this is the case since
each state is independently calculated after being speci
fied by an assignment of occupation numbers to the orbi
tals. So, while describing the dynamics of collision
events, we can expect the largest coupling between molecu
56
lar configurations to come from the electronelectron in
teraction. In principle, these interaction matrix ele
ments between determinantal wave functions comprised of
MSXo orbitals can be calculated, but in practice the
problem at present seems quite formidable.
The usefulness of the Xa orbital energies, i., in
showing where interaction potentials cross, has been shown.
Hence, critical distances of approach for the occurrence
of various electronic excitation phenomena can be deter
mined. Also of interest are crossings between eigenener
gies such as occurs between c5. and E2x in Figure 5, if
g u
one or the other of the involved orbitals were partially
occupied. Such crossings are of importance in electron
promotion mechanisms involved in energetic atomatom and
ionatom collisions.
Although the pairs studied here consist of closed
she' atoms, it is expected that the MSXa method, because
of its selfconsistent treatment, can handle as well the
repulsion in cases where sizable charge rearrangement
takes place. Of particular interest would be the mecha
nisms involved in Penning and associative ionization phe
nomena, where atom + excited atom and ion + atom interactions
are of importance. Here again, though, reliance upon ad
ditional results for describing the van der Waals region
would be needed. In Chapters III and IV the considerations
57
prompted by the research of this chapter will be applied
to an investigation of the electronic structure and col
lision dynamics involved in Penning ionization of Ar(3 )
by Ke*CIs2s,1,3S).
CHAPTER III
DIATOMIC MOLECULAR ORBITAL CORRELATION DIAGRAMS
FOR PENNING AND ASSOCIATIVE IONIZATION
In the previous chapter it has been pointed out that
an adequate understanding of atomic and molecular collision
phenomena requires information on electronic structure as
a function of the changing internuclear separations. Such
information is needed in order to describe inelastic pro
cesses involving electronic excitation and charge transfer
as well as ionization. In this regard, the usefulness of
oneelectron molecular orbital (MO) approaches in treating
electronic structure was emphasized. Such approaches afford
a selfconsistent calculational framework of minimal complex
ity which can treat ground as well as excited states. Chemi
ionization is a prime example of processes where electron
ically excited states play a crucial role. In this chapter*
the electronic structure involved in a collisional process
of this type will be considered.
Well known among chemiionization processes are Pen
ning and associative ionization (PI and AI) of the type
*A preliminary report of the results presented in this chap
ter was made at the International Symposium on Atomic,
Molecular and Solid State Theory, Collision Phenomena and
Computational Methods held at Sanibel Island, Florida,
1824 January 1976.
58
59
A* + B A + B + e (PI) and A* + B + AB + e (AI),
where A* is usually an atom in some metastable state and B
is an atom or molecule (Mu66, Mu68, Mu73, Ni73, Be70a,
Be70b, Ru72, Ma76). Experimental information for such col
lisions includes total ionization crosssections as a func
tion of collision energy (Ta72, Ch74, Pe75, 1175), angular
distributions of heavy particles (Ha73), and energy distri
bution (Ho70, Ho75, Ce71) and angular distribution (Ho71,
Eb74) of emitted electrons. Most theoretical effort has
been directed at determining the energy dependence of total
ionization crosssections and the angular distribution of
heavy particles (Na69, Mi70b, Mi71, 0172a). The angular
distribution of emitted electrons has been recently stud
ied within a semiempirical model based on MO correlation
diagrams (Mi75). One of the present concerns is to re
emphasize the usefulness of MO correlation diagrams as
they apply to angular distribution of ejected electrons
in PI and Al involving atomic collision partners.
In what follows, calculated as well as estimated MO
correlation diagrams are presented. The calculated results
are given first, where the electronic structure of
1,3 6 1
He*(ls2s, S) + Ar(3p S) has been considered within the
MSXa framework as described in the previous chapter. A
study has been made, in the nonspinpolarized approxima
tion, of the K ArLAr (lse) (o3sAr) (o3pAr) (w3pAr) (a2sHe)
tinofth K2 21Arr~~osr~
60
configuration of the excited HeAr diatom whose MO eigen
energies approach the atomic orbital eigenenergies of
He*ils2s) and Ar(3p ) at large internuclear separation,
and also the ground 2E state of the (HeAr) molecular ion,
whose MO eigenenergies approach the atomic levels of
HeCls2) and Ar +3p5) at large internuclear separation.
Some spinpolarized calculations are presented in the
neighborhood of a crossing exhibited by the nonspin
polarized calculations, along with some comments on the
conditions under which the noncrossing rule applies for
MO eigenenergies.
Keeping in mind the calculated results, consideration
is given to PI and AI processes in terms of MO correlation
diagrams. Analyzing the ionization on the basis of an
Auger type process, MO's can be identified, which, togeth
er with the continuum state of the emitted electron, are
involved in the process. Inspecting the unitedatoms lim
it in the centerofmass, bodyfixed frame then permits a
determination of the angular moment which contribute to
the continuum state of the emitted electron.
Next a procedure for estimating MO correlation dia
grams is described, which makes use of available data on
atomic orbital energies, and of two basic guidelines. Ap
plication is made to the collision pairs He*Cls2s) +
Ar(3p6), + Kr(4p6), + Hg(6s2), and Ne*(2p53s) + Ar(3p6).
The resulting estimated MO correlations for these systems
are then analyzed to determine the minimal set of angular
momentum values which are needed in each case to describe
emitted electrons in the bodyfixed frame. Finally, a dis
cassion is given of the results of this work and their
significance.
1. MO Calculations for He*+Ar: ar.d He+Ar+
The representative case of PI and AI in
He*(ls2s,' S) + Ar(3p S) collisions will be considered.
Here, calculations are needed for the excited states of the
HeAr diatom which separate appropriately to
Ie*(ls2s,1,3S) and Ar(3p6,1S), and for the states of
+ 2 1 + 5 2
(HeAr) which separate to HeCls,1 S) and Ar (3p5, P).
Following the work reported in the previous chapter, the
calculations are performed within the MSXa oneelectron
MO framework.
For convenience in this chapter, the set of oneelec
tron equations satisfied by the spin orbitals, ui, and ex
pressed in Equation C40), are written
h (ffl)Ui( = Eiui(') (45)
eff 1 i 1 ii 1'
+ 
Here, heff(r ) is the effective oneelectron Hamiltonian
for electrons of spinup, and according to Equation (40),
61
62
hf (r) = f] l + dr2{2p(r2)/rl2} + VtX (46)
where Vx (rl) is given by Equations C42) and (39), and
p(r), given by Equation (38), is the charge density com
prised of the contributions from electrons of spinup and
spindown. In the nonspinpolarized (NSP) approximation,
p () = (r) p ()/2 (47)
which means that the orbitals of spinup and spindown be
come identical, and each orbital can be considered as ac
commodating as many electrons of spinup as of spindown.
These calculations were carried out first with a NSP
treatment. As discussed in the previous chapter, each
selfconsistent calculation begins with a potential which
is the "muffintin" form of a superposition of atomic po
tentials centered at each atomic site of the molecule.
Therefore, for the HeAr excited state Ar(3p ) and NSP
He*(ls2s) HartreeFockSlater (HFS) CHe63) potentials were
used, and for the HeAr ionic state He(1s2) and NSP Ar(3p5)
HFS potentials were used. For both the excited molecule
and the molecular ion, the electronic states were specified
by occupying the MO's so that their eigenenergies were cor
rectly separating at large internuclear separations to the
corresponding atomic orbital eigenenergies of the above
mentioned HFS atomic calculations. In other words, the
63
boundary conditions of the PI and AI processes at large R
determined the appropriate excited and ionic states of the
HeAr molecule. Heteronuclear molecules require a choice
for the radii of the spherical regions centered at each
atomic site as mentioned in the previous chapter. There,
for the ground state HeAr molecule, the ratio of the He
to the Ar sphere radii was taken to be He/Ar
0.92727/1.66296 = 0.5576, where, e.g. e denotes the
average value of r for the Is orbital of He(s 2). The sit
uation for (HeAr) is roughly the same as for HeAr, since
a calculation of for Ar (3p5) (NSP) shows it to be
1.549 a So, for the molecular ion the ratio of 0.5576
was used at all R. On the other hand He*(ls2s) + Ar(3p )
is quite another case, since the 2s orbital of He*(ls2s)
is very diffuse. NSP calculations show of He*(ls2s)
to be about 4.546 a Because of this diffuseness, it was
decided to choose the ratio of the He to the Ar sphere ra
dii by finding which of its values minimized the total en
ergy of the specified excited state of HeAr at a fairly
large R, namely R = 9.0 a In this way, a ratio of 1.4
was found, which was then used at all R for the excited
state of HeAr. The values of the factor a in the various
"muffintin" regions of the molecule, for both the excited
as well as ionic states, were those used previously in the
ground state HeAr calculations, and the specific computa
64
tional details also remain as reported in the previous
chapter.
The results of the NSP MSXa calculations are displayed
in Figures 6 and 7, respectively, for the excited state of
the HeAr diatom and the ground state of the HeAr molecu
lar ion which are appropriate to PI and AI. Shown on log
log plots are the MO eigenenergies versus R for 2.3 a
o o
energies are labeled according to their symmetry and occu
pation number. In each case, the NSP HFS atomic orbital
eigenenergies for the separated atoms are shown at the
right of the plot, and the dashed lines indicate how each
MO eigenenergy is approaching properly its respective sep
aratedatoms (SA) limit. In the unitedatoms (UA) limit
the excited HeAr molecule is expected to approach
Ca*(3p54s23d), and the ground state molecular ion to ap
proach Ca C3p64s). The atomic orbital eigenenergies from
NSP HFS calculations on these united atoms are appropri
ately shown at the left of each plot, and, while the MO
eigenenergies may exhibit much structure between the re
gion of 2 a and the UA limit, the dashed lines at the
left of each plot show that these UA limits are not unrea
sonable. In both plots, the label for the a MO arising at
large.R from 3pAr appears above the MO label. In fact,
Figure 6. Molecular orbital correlation diagram from
nonspinpolarized MSXa calculations of the
HeAr diatom in the excited Z configuration
which separates at large R to He*(is2s) +Ar(3p6).
Calculated points are encircled.
0.07
0.08
0.09
0.15
0.20 
0.25
0.30
 'Her TrV A
_ He'Ar^ ,
0.401
i5 0.50
d
.c 0.60
0.70
1.50
3sCa'(3?4s3d)
0 2
3 4 5 6 7 8 910
2sHe(I s2s)
J3pAr(3p )
 sAr(3p)
 sHe(is2s)
00
R in ao
Molecular orbital correlation diagram from
nonspinpolarized MSXa calculations of the
(HeAr)+ molecular ion in the ground Z state
which separates at large R to He(ls2) +Ar+(3p5)
Calculated points are encircled.
Figure 7.
68
//I I I I i I II I //
0.07 
o0.0He
0.09 
0.10
0.1 5 
0.20 
0.25
3dCoa(3pF4s),
0.30
4sCc Sp(34s)
0.40
0.50
0.60
0 TO
0.80
0.90
1.00
3 pCa (3 ?4s)
1.50
3sCa (3g4s).
0 2
\

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

