Low ENERGY RESONANT ELECTRON
surname FROM HOMONUCLEARf '
DIATOMIC MOLECULES wnn PARTIAL

BORN—OFFENHEIHER BREAKDOWN

Thesis for the Degree of Ph. D.
MICHIGAN‘STATE UNIVERSITY
KENNETH PENFIELDV WINTERS

1972

 

 
 
 
 

LIBRARY

Michigan State
Lhfivcnfiq;

 

. ' in ,

“fag“ ‘ it
' I. '. 7 . '
' ‘ thesis enutled

(
r

This is to certify that the

LoiiBnergy Resonant Electron Scattering from
" Homonuclear Diatomic Molecules with
Partial Born-Oppenheimer Breakdown
presented by

Kenneth Penfield Winters

has been accepted towards fulfillment
of the requirements for

Ph.D. Jegree in Jhxsinal Chemistry

 

 

 

 

 

 

  
     

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, LIBRARY smozns

 

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ABSTRACT
LOW ENERGY RESONANT ELECTRON SCATTERING FROM

HOMONUCLEAR DIATOMIC MOLECULES WITH
PARTIAL BORN-OPPENHEIMER BREAKDOWN

By

Kenneth Penfield Winters

In some homonuclear diatomic molecules, such as H;
and N,, there appears to be a single-particle resonance
in the low energy (1-10 ev) elastic and inelastic electron
scattering, although there is no stable negative ion
capable of being formed. I have developed a method of
treating these resonances with a one-determinant Hartree-
Fock wave function, with the molecular part frozen in
the undistorted ground state.

First, I use potential scaling to find a bound state
function for the negative ion. Then, considering the
combination of the bound orbital for the scattering
electron and a plane wave to be a zeroth-order wave-
function, I obtain the asymptotic form of the first-order
wave-function. This is a first-order estimate of the
asymptotic form of the scattering function, from which
I obtain estimates of the scattering amplitudes and

phase shifts for elastic scattering and for vibrational

excitation.

 

 

Verv s:
the result
shows a de
amplitudes

io not ref

 

Kenneth Penfield Winters

Very simple trial calculations were done for H2, but

the results were inconclusive. The elastic scattering

shows a definite resonance, but the calculated scattering

amplitudes,for both the elastic and inelastic scattering.

do not reflect eXperimental results.

LOW ENERGY RESONANT ELECTRON SCATTERING FROM
HOMONUCLEAR DIATOMIC MOLECULES WITH
PARTIAL BORNvOPPENHEIMER BREAKDOWN

BY

Kenneth Penfield Winters

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Chemistry

1972

To
(and in spite of)

Morgan de Mara ap Ries

He
hit

-.- war-nan“
g;
mama rat—riffwj

 

I uist.
Nazarcff,
assistanc
Censidere

I owe
have Hive
“i h'hich
Patient.

Vanv o
5‘11}? 3 f8“
Wider; Yr
“ith the
COEnUtEr

D

 

ACKNOWLEDGMENTS

I wish to acknowledge a special debt to George V.
Nazaroff, my major professor, who has provided great
assistance in many discussions on the various points
considered in this dissertation.

I owe a different kind of debt to my parents, who
have given various sorts of assistance, not the least
of which is the IBM Selectric which I am using at this
moment.

Many other people should be thanked, but I can name
only a few: Dr. Harrison, who consented to be second
reader; Yndvrei Hibaru and Caellyn y'Vearn, who helped
with the typing; the MSU Chemistry Department and
Computer Center; and all those who, for their own
reasons, preferred to use the CDC 6500 while I was

using the CDC 3600.

iii

h ‘. puns-In

E-‘L‘

 

 

LIST OF P

INIRODUC.

  

Chapter I
th

Chapter

Y
A

Chapter 1
Sta

Chapter I

Chapter "

 

Chapter \'
Phal

of 0v

8 TWO

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . v
LIST OF FIGURES . . . . . . vi
INTRODUCTION . . . . . . . . 1
Chapter I. A One Determinant Wave Function for

the System . . . . . . . 4
Chapter II. The Variational Integral, I? . 10

Chapter III. The Variational Equation for the
Scattering Function . . . . 46

Chapter IV. The form of the Scattering Function 52
Chapter V. The Bound State Wave-Function . 62

Chapter VI. The Scattering Amplitudes and
Phase Shifts . . . . . 82

BIBLIOGRAPHY . . . . . . . 110
APPENDIX A Determinant and Minors of the Matrix

of Overlap Integrals . . . . 112
APPENDIX B A One Determinant Approximation to

a Two Determinant Function . . . 125
APPENDIX c Evaluation of PJMji w . . . 129

APPENDIX D Listing of Computer Programs . 138

iv

TABLE 1:

TABLE 2:

 

TABLE 1:
TABLE 2:

LIST OF TABLES

FIRST ORDER MINORS . . .
SECOND ORDER MINORS . .

 

FIGURE 1
Ex-

FIEURE 2

 

LIST OF FIGURES

FIGURE 1: SCATTERING AMPLITUDE AS A FUNCTION OF
ENERGY . . . . . . . . . . 107

FIGURE 2: PHASE SHIFT AS A FUNCTION OF ENERGY 108

vi

..
. _ v2.29

Fur—.5

 

 

In th
from sma
involvin
be quite

Tezion.

 

there 15
Neither
State of

In th
and the
icn. F0
interaCt
be Stron

a b0Und

INTRODUCTION

In the scattering of low energy (l-lOev.) electrons
from small diatomic molecules, the inelastic scattering
involving vibrational excitation would be expected to
be quite small unless there is a resonance in this energy
region. In several such molecules, H, and N, for instance,
there is an apparent resonance structure at low energies.‘
Neither the ground state nor any reasonable excited
state of these molecules form a bound negative-ion state.

In the case of 02. the attraction between the molecule
and the electron is strong enough to form a stable 0;
ion. For the similar but less electronegative N2, the
interaction potential of the molecule and electron should
be strongly attractive, but not strong enough to form
a bound NZ ion. If this ion were fairly long-lived,
the inelastic scattering would be closely related to
the vibrational states of the ion. The process would
be divided into three parts: First, the electron is
captured. Then, the negative ion exists for a period
of time in a particular vibrational state. Finally,
the electron escapes and the molecule is left in some
vibrational state. For this description to be good,
the ion must exist long enough that the Born-Oppenheimer
coBrdinate separation for the nuclear and electronic

1

 

coErdina
vitratic

In .‘\3

 

 

long enc
exists f
sciecule
are not
Which thl
tion. T
Scatteri;
State,

ID dc]
other Si!
in which
Shay-3th

the CODY:

 

w‘lti‘lin I.
VaVe fun:
thOUQh dl
“Enron.
To deSCI‘i
I will Us
icn fu“ct
consideri
State COr

n1

To tes

2
coordinates can be applied, leading to well defined
vibrational states.

In N; the resonant state does not appear to exist
long enough for this to be the case. The negative ion
exists for about the period of one vibration of the
molecule.2 Thus, the capture and escape of the electron
are not separate events, and there is no period during
which the Born-Oppenheimer separation is a safe assump-
tion. The breadth of the resonant structure in H2
scattering suggests that this is also a very short-lived
state.

To describe the inelastic scattering from H2,N2, and
other similar molecules, I will deveIOpe a formalism
in which I do not make the Born-Oppenheimer coBrdinate
separation between the coBrdinates of the nuclei and
the cobrdinates of the scattered electron. I will work
within the structure of a one-determinant Hartree-Fock
wave function. This is possible because the molecule,
though distorted by the presence of the scattered
electron, remains electronically in the ground state.
To describe the resonant part of the scattering function,
I will use potential scaling to obtain a bound negative
ion function. I will then obtain the scattering by
considering it to be a perturbation of a zeroth-order
state consisting of the bound resonant function and a
plane wave.

To test the formalism, I have made highly simplified

calculat

at the e

‘ C.J.S

’ A.Her
4a(

 

3
calculations for H2. The results of these are presented

at the end of Chapter VI.

' G.J.Schulz,Physical Review 135,A988 (1964).

2 A.Herzenberg and F.Mandl,Proc.Roy.Soc.(London)A270,
48 (1962).

_s
l ,-'_

 

Chapter
I de

the tota
z-conpons
which ar:
angular r
scatterei
Y'I

Each
C05rdinat

SRatial C

 

reSPGCt t
SITuCted l
RDieCule
is a {UTIC

electron

Chapter I. A One Determinant Wave Function for the SyStem.
I define V§(1,2...N,N+1,R) as the wave function of

the total system with total angular momentum J and
z-component MJ=M. I? is a sum over j and B of functions
which are also eigenfunctions of molecular rotation
angular momentum,j, and orbital angular momentum of the
scattered e1ectron,£.

PM

J=z

j ng£(l,2...N+l,R) (1.1)

It

Each ngt is a function of the spatial and spin
coordinates of the N+l electrons and of the relative
Spatial coBrdinates of the nuclei, anti-symmetrized with
reSpect to the electronic coBrdinates. They are con-
structed by writing, first, a product of the target
molecule electronic functions and a spin-orbital which
is a function of the coBrdinates of the scattering

electron and of the nuclear coardinates:

Jth
Product

JMjn

w = xl(1)x2(2).-oxN(N)xN.1 (N+1.fi) (1.2)

I now operate on the product function with the anti-

symmetrizer,vQI, to give the desired function:

Jth

M
ij£(1,z...N+1,fi) = 'kaProduct

(1.3)

‘35-?“ '

 

Note

J,. 34.1

and obser

\I

R.

In sol

to be USe

notation:

\l

 

 

 

 

 

Jth
N+1

Jz, j”, and I” operators. Define also

Note that x (N+1,R) is an eigenfunction of the J2,

xfiTltN+1.§) = .22 xfiT{”(N+1.R) (1.4)
Jo

and observe that

L43

= I J4‘xl(1)-.-XN(N)X§T{£(N+1,R)

Jo;Q

= efilxlcl)...xN(~) ,Zg x§T{“(N+1.fi)
J.
- 54 xlcl)...xN(N)x§T1(N+1.i) . (1.5)

In some cases I will find a uniform notation for orbitals
to be useful, therefor I will define the following alternate

notation:
X‘iJM32(k.B)E ng(k.R)E xifk),i§N,all.IMLL£. (1.6)

I can now write:

N+l .
M , JMJB .
V ' .;4 ll - (1.R) (1.7)
J jge 1-1 x,

N+1
, $4 11} xg”(i,§) . (1.8)
1:

 

Writ:
I

that BOT?
for the

nuclei.

hm .
X311 (1"

fu.ction

C1ebsch-(

 

 

 

1 am

6
Writing the wave-function as I have, I am assuming
that Born-Oppenheimer separation is a good approximation
for the relative motions of the target electrons and the
nuclei. It is for this reason that I do not write
X§T{£(1,R) as the product of an electronic and a nuclear

function. I eXpand it as a sum of such products utilizing

Clebsch-Gordon coefficients:1

JM‘I . elec +
xN.i - v% m C(JlemjmnM) ¢k£m£(rN+l)a(N+l)
2 j
I‘IUC
evjmj (R) . (1.9)

where v is the vibrational quantum number.

I am using two systems of codrdinates in this problem.
One is the molecular coSrdinate system with the origin at
the center of mass of the molecule and the z-axis fixed to
the molecular axis. This is not an inertial frame of
reference, and a rigorous treatment of a problem in such
a cobrdinate system would require the inclusion of
centrifugal and coriolus "forces". The other coBrdinate
system is the external system with the same origin as the
molecular system, but with the z-axis fixed in some
arbitrary direction. The 32, 32, I“, 22, 32, and 32
Operators are defined in the external coBrdinate system.
The expansion in (1.9) requires that ¥N+1 and R be eXpressed

in the external coordinate system. Electronic angle

cofirdinates in this non-rotating frame of reference will

be marlea

Note
which his
C0152}.

The I.
demands c
eigenfunc
{(1%) an

eigenfun:

IIJ’I) ap

I

e
°k

and

there the
undistuTt.
that a CI}
for each ,

an nOt

initial 51

Can

 

be marked with a prime.
Note also that all Clebsch-Cordon coefficients for

which m£+mj#M vanish. In some cases I will write

C(jszj. mlmz) for C(jijzj. mxmz m1+m2).

elec

The k subscript on ¢ is an energy variable and

depends on v and j. The electronic function is an
eigenfunction of the 1’ and 22 operators with eigenvalues
2(2+1) and m, respectively. The nuclear function is an

eigenfunction of the j2 and jz operators with eigenvalues

j(j+1) and mj respectively. I define them by

JM.Vojom.

elec + J. + . - 8 +
¢ (r) ' f (r) Y (9 ) = f (r)Y (9')
ktm, vjmj,2m2 2m, 2m,
(1.10)
and ¢3§§ (I) - XM(R) ij. (QR) , (1.11)
J J

where the Xv(R) are the vibrational functions of the
undisturbed diatomic molecule and the f8 are unknown. Note
that a different set of solutions, fB(r), will be obtained
for each initial state, Specified by Vo.jo.mjo.J. and M.

I am not assuming a given initial to but will take the
initial state to be an incoming plane wave.

I can now write the total wave function as

 

I car.

Share If;

«i
t .

 

 

I? = Q4‘xl(l)...xN(N+l) Z C(jiJ,mjm2M)

JLlemj

f8(rN+1)Xv(R)“(N+1)Y2m£(“'N+1)ijj(9R) '
(1.12)

I can also write

13“- e‘lecl)...xN(N) I f8(rN.1)xv(R)a(N+1)

31v
1 (Q. Q ) (1 13)
JMje N+1' R ' '
q'
0 = . .
where fijJMj (Q N+1’QR) X C(JBJ,mjm£M)Y£m (Q N+1)
m m. 2
i J
I

A

This angle function is an eigenfunction of J2,Jz,j2,

and 22 with eigenvalues of J(J+1),M,j(j+1). and 1(1+1).

respectively, but is not an eigenfunction of jz or £1 .
M

J
terms of functions all of which are known except for the

Whether I use (1.12) or (1.13) I can express V in

f8 . To determine the f8 , I will derive an expression for

the integral

M M A M
IJ <VJ|}(-E|wJ> (1.15)

and then
arbitrar
original

states I

the re Ev
internuc
Electron
eNation
rOtation

1 M,E p.
Chap)

 

and then for the variation in I? corresponding to an
arbitrary variation in the fa. If the molecule is
originally in the v0 and jo vibrational and rotational

states then
2
E = E + E + E].o + 1/2ko , (1.16)

where EM01 is the electronic energy at equilibrium
internuclear distance, R0, as given by the molecular
electronic wave function actually used in the above
equations, and BW and Ejo are the vibrational and
rotational energies of the v0 and jo states, respec-

tively.

‘ M.E.Rose,E1ementary Theory of Angular Momentum,
ChapterIII,John WileyESons, Neinork, 1957}

 

Charter

 

 

as

 

Chapter II. The Variational Integral, 1g.

 

 

To find I? I must know/% . In this case it is
A N+1 N+l N+1
H: -1/2 2 V121 - [ rz + r2 ] + I: .1?!—
u=l u=l uo uB u>v uv
1 2
+V(R)- ‘Z'JfivR , (2.1)

where a and B designate the na>nuclei, “N is the
reduced mass of the nuclei, and Z is the nuclear charge.
The electronic function and the nuclear-electronic attrac-
tion (except for the scattered electron) are evaluated at
R=Ro. The actual variation in molecular electronic energy
with R is combined with coulombic repulsion in V(R).

The kinetic energy operator, -7%V2 , can be written

as

 

 

1 2 -1 i 2 d ] £2
- v = r + e (2.2)
Ti Zur2 g?» d? Zurz

Using atomic units, for an electron, u=1. This
formulation of V2 is not entirely correct for the
molecular electrons, since their angular coBrdinates

are not expressed in an inertial frame of reference.

The neglected coriolus and centrifugal forces, however,
are very small near the center of the coBrdinate system,
becoming significant only for the scattered electron
when it is far from the molecule (I have already

tacitly assumed this by writing the vibrational functions
as XV(R) rather than va(R)') ; and the angular

10

i- 1.".- Tim“

{1:} ~“.. ~

 

coordina
an inert

Defi

)

'43

we»

 

11

co3rdinates of the scattered electron are eXpressed in
an inertial frame of reference.

Define the operators

 

 

“ . B _ z _ Z _ Z
f(u,R) 1/2 vu 7.. .17... (2.3)
no uB
“ -1 d ad
NR) = [ R ] + V(R) (2.4)
2U R2 ER HR
N
I can now expresst -E in the form
A N+1 A N+1 1 A 1‘2
fl-E- Z f(u;R)+ 2-17—— +f(R)+ I -E .
u=l u>v uv ZuNR2
(2.5)

Note that the V2 parts of the ‘f(u;R) and the
1/ruv (unless u = N+1 ) depend only on electronic
codrdinates, and that f(R) and 32 depend only on nuclear
coardinates. In addition, when f(u;R) acts on a molec-
ular spin-orbital, it is taken to be f(u) = f(u;Ro),
which is independent of R. Thus, the Z/rN+1.a,
erN+1,B’ and l/rN+l,v terms are the only ones depend-

ing on both.

Using (2.5) I obtain the following eXpression for

J1

 

Any

Where 8

new inVe

inteill‘al

 

“111 Fro-

12

M N+l x N+l

I = M M

J Z 2 <V . |f(u°R)|V . > +
jj'tl' u=l J32 ’ JJ'R' ugv

1
WlelT VIij'£‘> + (WM j2|f(R)IYM j'l'>

?2
M
I

M .
R. IJj'2'> * ‘ijel

I <ij1'

~EI‘PM-..> .
ZUN Jj 2

(2.6)

Any of the above integrals may be represented by

the expression

M _
IJjj'oe'a = < jlloa (" V FIIij 1' electronic >§
coordinates
(2.7)

where Oa(u,v;R) is the appropriate Operator. I will
now investigate the properties of the electronic
integral above. Here the notation in (1.6) and (1.7)

will prove useful:

N+1 .
13'“ = 94 E1 xg’MJ’Lcifi) (2.8)

so that

M
1(R)Jjj'£t' ‘Vszlo'VJj'e"e1ec.

_- .1:

V- .- Wf‘a‘

 

who re [I

“.1pr

 

13
N+1 . A NT]. 1 9
a (.54 IT; ngJ£(i,R)I0I.7Q IT; ngj I (10§)>elec
I 1'
”+1 . A "*1 I O
. (.zITl X‘iJMMCipFIIOI #1. g 11.1 X?” l (i'ipelec

N+1 . A I ' "
" ‘IT1 ximomrlol u xJ’” " II (in...

(2.10)

where " XJMj'l'“ (R) is the determinant function:
i O O C 0 £0

x‘IMI'WzJ) eaves) . . . as: (2.70

Ilr””"'n(fi>ss , - ° 2

x‘IMj'utuoij) iniju'wufii) . . . xg’fi'VmAJ) I

 

 

(2.11)
I also define the electronic overlap integrals:
Jij'tz' = Jth JMj'z'
Du,v (i) - <xu (1.1m):v (1.13»elec
(2.12)

and the matrix of the overlap integrals

“355

Thi
Since 01
Orbita1
and g.

In;
If u is

 

 

14

I o 0 JM 0 1" JH 'lt' I
”ITjj 11 (g) p 51 f (I) . . . 01.11, (R)

DgMij'lt'(§) DJij'tt' (m . . . D2 Mfij'ii'd
DJij'll'(§) -

= L .
O

 

 

o 0 ' M 'll'
ngfijlzt (I) DJ"552“' (R) . . . n§,{iu,1 (R) J

(2.13)

This eXpression is not as complex as it appears
since only the overlap integrals involving the scattering
orbital actually depend on the various superscripts

and R. Thus,

Jij'zz' ,
nu.v (I) euv , u,v 5 N. (2.14)

. . Jij'nz' Jij'tt'
In addition, Du,N+l (R) and DN+l,u (R) are zero

if u is even due to the a—Spin of the scattering orbital.

Thus:

DJij'z.‘

I cai

terms 0 f

and

 

DJij'l£'(§) .

D

 

Jij'zz'

15

JM °'2z'
”1,§11 (fi)

DJij'zz'(§)

3,3‘1

 

DJij'tl'(§)6

N+1,N+1 ll'éjj' J

(2.15)

I can eXpress the first and second order minors in

terms of the unsigned minors:

IIDJW'“'(§)II“p a (-1)“*PIIDJ”55""(§)H fip

and

llDJij'2£'(§)" uv,pq E

(2.16)

.., ,
(-1)u*V+p*qP(u>v)P(p>q)I]DJMJJ 1‘ (§)" a§qu

(2.17)

i
x
N
'u ”IQ"! U 3‘2“

.‘-..n~a-;u~': .' -,

 

The
matrice
column
DJij':
equals
is fals
0nd ord
Minors
are def

Pet

when 0

Expand;

 

16

The unsigned minors are the determinants of the
matrices obtained by deleting the indicated row and
column (or rows and columns) from the overlap matrix
DJij'zl'(§). P(statement) is a parity Operator which
equals +1 if "statement" is true and -1 if "statement"
is false. Explicit eXpressions for the first and sec-
ond order minors will be developed in an: appendix.
Minors of the determinant wave function leJMj'z'“ (i)
are defined analogously to those of the overlap matrix.

Returning to eQuation (2.10), consider the case

A

when 0 = D.. Then

N+1

a Jth . JMj'l'
I(§)Jij'£2' < 51 Xi (10§)| H X H (i) >elec
(2.18)
Expanding IijMj'L'”(§) along row 1 gives
N+1 N+l
JM 2 JM l
1(R)Jij.,,. = < TT' x J (i. fi)lx J (1.fi)l p21

ngj'2'(1.§)lli“j'“'H 1P(fi) >

(2.19)

N+1N+ .., ,
- < ”IT2 xJMj2(i.§)l NZ DJMJJ 22 (ii)

17

n xJMj'£"II ”(m >

(2.20)

 

 

JM"'££' JM"'£2'
D1.iJ (fi) D1,§11 (fl)
xgw'z'czm mm.)
= < TT xiMJ£(§)l . .
i=2 . .
xiMj'£'(N+1,§) . . . xfifi'“'(~+1,fi)
(2.21)

Expanding this determinant successively along the

2ndthrough N+1St rows gives

M M
I(§)Jij'2£' (ijklej'£'>elec

= H DJij'““'(fi)l

 

. (2.22)

I will next consider the case where O =1/ruv. Then
N+1
Jsz . 1
1(fi) .. = (IT x. (1 (bl...—
JMJJ'££. 1‘1 1 . ruv
JMj'l'
llx Il(fi)>elec (2.23)
th

Expand the determinant wave-function along the u and

Vth (OHS

Where

SUbstit

 

18

Vth TOWS:

N+1 .
JM 2 . 1
1m)...“ TT x.3(1fi)—
JMJJ'zz i_, 1 . ‘ruv|
N+1 .' g '1'
Z ngJ 2 (u.fi)ijJ 2 (v.§)
p.q
‘0 1
leJM3 1 lluv'pq(fi)>elec (2.24)
N NTI v v
= < TT J”3‘(1 fi)l Z 33%)) 1“ (i)
i#u9v qu pq
JMj'i' uv pq
||x II ' (§))e1ec (2.25)

where

Jij'li' fi
V,pq ( )

<xflMj2(1.§)x3Mj£(2.§)

M°'9,' M°'2.'
I 1 IxJ J (1.fi)xj 3 (2.§)>

1‘
i2 D

(2.26)

 

elec

I can change the sum over p and q to a sum over p>q by

substituting

Jij'zz' i
uv,pq ( ) r12 2

 

g <xfl“52(1.fi)x3”j‘(2.fi)l 1 (l-P, )

IxJMj"'(i.fi)x§M5'£'(2.fi)> (2.27)

p elec

 

P

where .

EXpandi
eaCh rot

coérdin

So that.
I

 

19

where P exchanges the coordinates of electrons 1 and
12

2. Thus

N*1 -°c v
= < .IT' fo5‘(1.fi)I Z gJM’J 2‘ (i)
V

I i ..
( )JMJJ'iz' p>q uv,pq

JMj't' uv pq
llx n ' (10>elec (2.28)

Expanding the second order determinant successively along
each row and integrating over the appropriate electronic

coordinates I find that

N+1
< TT JMjl . JMj'i' uv pq a
ifu,v Xi (1.fi)| llx H ’ (§)>elec
."'9.9.'
H DJMJ’ (2)” "V'pq . . (2.29)

So that,

M
fll

I(§)Jij'££' ‘ ”th'% Jj £'>e1ec

a Z gJij'R2'(§)H DJij'££'(§)H uv,pq .
p>q uvaq

(2.30)

Next consider the integral for which 0—- f(u;§):

 

Expand ‘

 

JEfinin;

s. nOt

20

N+1 . x
= < 1T ng3£(i.§)lf(u;§)l

1(E) ..
JMJJ'zz' i=1

IIXJMj'£'||(fi)> (2.31)

elec

Expand the determinant function along the uth row:

N+1 .
. JM 2 .
I(§)Jij!2£' 3 < Eu Xi J (lbfi)

. A N*1 -v v
|<X3M31(u.fi)lf(u;fi)l 2 x3”’ ” (°»§)>u
p=1

JMj'z' up
'IX ll (§)>other electrons (2.32)

Defining

fJij'22.(§) E (X3Mj2(1.§)|E(1;§)IXgMj'£'(1'§)>elec

Up

(2.33)

and noting that

8
elec

N+1 . -c v
< I}; rim-”(1.2): n W” 1 ll "Mb
1 LI.

00' '
II'DJMJJ 2‘ mu“p .

 

 

I
I obtai

Next, q

 

ExFan

TOVS

in

£11
12’:

21

I obtain
' _ M a , M
1(R)J\’jj'9£' -’ (ij9‘|f(u,-‘§)IWJJ..2'>

N1 ~-' . "0 I
= E {3:33 22 (fi)“DJMJJ £2 (§)HUP . (2.34)
p=1 _

Next,'eva1uate the integral when O-l/R’:

1(fi)Jij'£l'

”*1 M‘ _ 1 JM"£'
< 1T x‘i’"”(1.fi)I-§—;I le ‘3 "(fibelec .

i=1

(2.35)

Expanding the determinant function successively along

rows 1 through N+1, I find that

In?)J

= 1 Jij'gg'
ij'u' 'R—z-UD (FUN . (2.36)

The remaining integral from equation (2.7) is

N+1 .
I R . . = J”) 0 - A Jh1' 0 £0
.( )JMJJ0£2v < Elxi (1,r{)|f(R)l ”X J |l(fi)>elec

(2.37)

By the procedure used in (2.10) I can show that

 

 

I sepa:

““9 Sec

 

22

M

M 5
- (ijilf(R)lej'g'>elec

1(fi)Jij'££'

Ntl -
l. A Jh! '1' .
= <nxJ‘J“u(§)If(R)IIT&xi J (1.§)>clcc

(2.38)

I separate (2.38) into two separate integrals

I(mJij'fil'

 

. N+l .
_ M '2' -
<HXJMJRIHEI 1 9_[R29—](TT XJ J (1’§)>e1ec

zuNR2 dR dR i=1 1

. N+1 .

M 2 ' J” '1' °

. (“XJ J u(§)!V(R)|TT1x, J (I-E)>elec
1:

(2.39)
The second integral can be evaluated immediately:
JMiz fi ”*1 JMi'l' . fl
<|x ' fl( )IVCR)|1:1xi - (1. )>elec
"' i
= V(R)HDJ“33 1“ (fi)n. (2-40)
To evaluate the first integral I note that
x+1 ,., , N . d JM.,£,
i—‘TT xi“) 1 (1.2) = TTixi(1) 3E xN.i (H+1.fi) .
1:

dR i=1

(2.41)

 

 

I def:

N in (

Thus.

 

23

This is a consequence of the fact that the molecular
electronic wave functions are derived with the assumption
of Born-Oppenheimer separation. Thus the first integral

(which I will call 11) can be written

N+1
JM 2 N+1 J
11 = < 2 ”X J n 'P(fi)x M11<~+1. a): 2
p: ZuNR

 

1212‘

ngj' £'(i.§) 9—[R2L XJMj‘£.(N+1,§)] >
1 dR dfi

(2.42)
I define

“32:1”? “'00

 

Xp zuNR2 dR an

JM" ' .
xN.{ z (1.15»elec (2.43)

Integrating over the cobrdinates of electrons 1 through

N in (2.42) and substituting from (2.43), I obtain

”*1 °" ' N M 22'
11 - z n DJMJJ “3 (3)" P' *ln‘f;1133'(n) .
n=1

(2.44)

Thus,

M * M
1(fi)Jij,,,, = <ij,|£(fi)|ij,,,>e1ec

24

... . N+1 ... . .
= wmllnrm'JJ ‘2 (§)H + 21 "DJ”J’ ‘1 (§)IP'N 1
p8

Défiiijj'““'cfi) . (2-45)

Substituting eXpressions from (2.34), (2.30), (2.45),

(2.36), and (2.22) for the integrals in equation (2.6),

I obtain an expression for lg:

N+1 N+1 .., , .., ,
I 1 21 Idfi fJMJJ ll (§)”DJMJJ £1 (§)Iup
u: p:

N*1 N+1 I

+ Z Z

u>v p>q

dfi gjfifg;““'(fi)HDJij'“"(fi)u"V'Pq

* 2 Jdfi ”gfifijj'll'(§)nDJij'“"(fi)Pp'“*1

+ [dfi V(R)ImJ”jj'“"(fi)H

+ Jdfi ['(j’12] _l ”DJij'22'(§)”

ZuN R2

.., ,
+ Jdfi (~E)IIDJ”JJ ‘1 (§)H . (2 .46)
I can combine the last three terms to give

N+1 . ..
M JM '22' “P JM '22'
I . dfi Jj i f 3’ (i)
J jg'zz' [ pin I Im ( )u up

N+l N+1
JM'°'££' b°"2£'
* 2 I Jdfi "D 3’ (fi)n“v'pq gflJ’gq (fl)
U>V p>q 0

(‘01..

 

 

and it
indivi
below,
SUpers
terms
GXpres

is R i

 

25

N"). ~-. 3 t 0
. 2 Jan IIDJMJJ ‘2 (ibllp'N’1 Dé2§f¥33 “2 (i)
p

+ JdfiuDJij'“£'(fi)u [V(R) + iLilll.-E]] .

ZuNR2

(2.47)

The determinant of the matrix of overlap integrals
and its minors can be expressed explicitly in terms of
individual overlap integrals as in tables (1) and (2)
below,which are derived in an appendix. I can delete
superscripts which apply only to the N+1St orbital from
terms not involving the N+1St orbital, with i where the
expression depends only on R, and R where the expression
is R independent. Equation (2.47) then becomes:

M
IJ

622'

N 0
JM 2
jj'zt' [ “£1 Jdfi {DN+i.N+1(§)6W

N
DJM 2 JM '2'
gun 3 (fi)0 3 (3)} fuu

i N+l ,i i ,N+l
N dfi DJsz (§)DJMj'£'(§)f
ugp DN+l,u p,N+l Up
_§ dfi DJsz (§)fJMj'£'(§)
u_1DN+l,u u,N+1
g dfi “JMj‘£'(§)fJMj£ (a)
n.1Du,N+1 N*l,u

 

 

 

 

a _D .,. 19‘
-

{1‘4

26

TABLH 1: FIRST ORDER MINORS

 

 

Jsz

<N+1 =u DN+1,N+1

(mesjjmm.

N .. ..
_ .5 DJMJJ'2£'(fi)DJMJJ'Rfl'(§)
1 u

N+1,i i,N+]
<N+1 #u,<N+l D;T{i;££'(§)DgT§2;RQ'(§)
<N+1 =N+1 -D§T%f;gl'(fi)
=N+1 <N+1 -D%T§ii££'(§)

=N+1 =N+1 1

 

 

 

 

27

 

 

 

 

 

 

 

TABLE 2: SECOND ORDER MINORS
J31 ' 22'
u F q "D jj (fiJHuV.PQ u>v,p>q
<N+l
. . Jfljn ‘
U V DN*I.N*1(R)Ogg'5jj'
N , bJHij'nz'(R)DJHjj'm'(R)
iEU.V Nfl'i i,3+1
. Jij'tl' * Jijvzzv g
u fv N*1'v (R)Dq,N+l (u)
a . 1M55'11' * Jujjvzzv »
v DN+],u (R)Dq.fi+1 (R)
. _ Jij'fifi' + Jij'in' »
<N+1 u DN+1,V (R)DP,N+1 (R)
v . Jij'ii' * Jijvzzv :
<x+l,fu v DN+l,u (runny+1 (L)
-N+1
. - Jij'zz'
V DN+1,u (p)
, - J“5j'£2'
u DN+l,v (R)
=N+l
= Jij'tz'
v Dq,N+1 (h)
. , Jij'nn'
<N+1 v Dp,N+1 (R)
=N+1 Iv 1

 

 

I

8’

28

1 Jujj'an'
+ Jdk rN+1';\'+] (is)

\’ O

‘ JMit
+ ; Jdfi {DN+1,N+1(K)6W6£1'
u>V

N
M11 2 1“ j ”9 1
' ';U V DN+i’ i(fi)ni, N+1 (L)) guv,uv

N .
.Jkij Q ilfij' 9-
+u§v q Jdfi DN+1.V(§)DQ. N*1 (F) guv uq
O
‘ g din J““ i DJ”5'£' K
u>v>q ‘ DN+1,u( ) q,N+1 ( ) guv,vq
_ g Jdfi DJMjP (§)DJMj'£'(fi)
p>u>v N+1,v p,N+1 guv,pu
+ g dfi DJsz K DJMj'n K
N+1,u( ) p,N+1 ( ) guv ,pv

N ., ,
_ Z Jdfi DJ.1j9 (fl) pJMJ R (a)

N+1,u ’uv,N+1 v

N ., ,
+ 2 jar nJMil (a) gJMJ z (i)

u>v N+1,v uv,N+l u
+ g dfi DJMj 2'(§) JMjn F
v>q ,N+1 QN+1 v,vq (‘3
_ § dfi n JMj (fi) JMja (fl
p>v p N+1 8N+1 v,pv )

N

J\1jj' 112'
+v§1 fdfi? gN+1 v ,N+1 v (fi)
(Z)JM j99

‘ Jdfi DN+1,N£1 (EJGfi 99'

In
an ast

summat

 

 

29

N .
' p21 Ia vilifpcméfada g cm

JM' JM 2
* Jdfi {DN+ifN+l(fi)6jj 522' Z DN+i,( i

 

JM°'£' +1
Di,&+1 (§)} [V(R) * is: R2) ' E] ] .
N

(2.48)

In the above equation the four terms labeled with
an asterisk can be combined after rearranging indices of

summation, as can those labeled with a dagger. Thus:

622'

N
M ' JMjn
IJ jg'li' [ ugl Jdfi {DN+1,N+1(§)5fi.

u DN+l,i ,N+l

N .
DJM 1 JM 'l' .
'1; 3 (fi30 3 (fi)} fuu

d3 DJsz (§)DJMj'£'(§) fup

u ép N+1,u p,N+1
N .
JM 2 JM '2'
- “£1 Idfi {DN+i.u(§)fu.&*1 (i)

O I J ’ ‘ O l.
Maximum; am + [cm £16.33an m

+ § Jdfi {DJM5 5‘ 1(§)5jj

u,v No1, N+ 522'

N
JM 1 JM '2'
' 1;” v DN+i ,i(§)Di §.1 (3)} guv,uv

30
N

.IHj 0, I'fij' S’.‘
+ Z Jdfi DN+1’V(I\)DQ’ N+1 (K) gUV,uq
UVQ
(311%)

N - c I
+ ; INN {n M"" (K) ’“5 9 (i) + D39§+§ (i)
U V .

IM+1 v Juv, N+l u

gm j 9. (T5) }

N+1 u,uv

9n+1 v,h+1 v

4.
“~42

Id dfi «JHjj'lflr' (ii)
1

V

4.

Id}? D(2)J51jj22('jJ]5)6

N+1,N+1612'

N a , .
- {1 dez’ D'ng’ p(R),,(2)J1j 1 (R)

 

_ N+l, ,N+1
p...
- N
J31” + -DJ.\1j2 mj
+ Idfi {DN*1.N+1(R)655'622' jgvnN+1,i(fi)Di,N+lv(fi)}
[van + j'(j'+12) _ 1:) ] .
ZuNR

(2.49)

I now expand the scattering spin-orbital so as to explic-

itly express its dependence on nuclear angles, where

. _ . JMjl .
convenient. The integral DN+1,N+1(§) can be written as

follows:

JMj L
DN+1 N+l(§) ' X. . ' C*(j£J,mjm£M)
vv mjmj mgmz

31
C(jEJ,mj'mz'M)<fB(r;)Y£m£(Ql')a(1)

a .
lfvvmjcm£o(rl)Y£m£0(nl )a(1)>lL;(R)Y3mj(nR)

KV'(R)ijj.(QR)

a z |C(j2J m.m M)|’6 <f8(r)l
vv'mjmj'mzmz' ’ J l mzmz'

r’lf5.mj.m2.(r)>r x3(R)KM.(R)Y3mj(QR)ijj(QR)

(2.50)

JM,voj¢m.

Jo
'm.'m '
J

where f5
2 v'jm.',£m
J

z. . Note that if mz-ml'
then mj-mj'. The subscripts 1 and r on the integrals
indicate integration over all coordinates of electron ]
and over only the radical coordinate of.electron 1, re-
spectively.

Integrating over 9 the nuclear angle coordinate:

RD

JMjl a . 2
Jana DN+1.N+1(§) vé'm [C(JlJ,m M-m M)|

<£8(r)lrzlffi.(r)> szgmuwm (2.51)

The characteristics of the Clebsch-Gordon coéfficients

allow me to simplify this equation, producing

32

JM 2 8 B
JdQR DN+i N+1(fi) ‘ ”é. <f (r)|r’|fv.(r)>

R2X;(R)lv.(R) . (2.52)

. JMjl JMj'
Now consider the product DN+1 i(§)Di N+l (fi) . It can

be expanded as

DJMjl fi JMj' 2' fl 8
DN+1 i( )Di N+1 ( ) vé'm.m£m.'m£' C* (le, ,mjm
J J

2”)

C(j'R'J,mj'mz'M)<fB(r)Y2m£(n')o(l)Ixi(l)>

yang,“ ijR)‘ x (1)If§.j..g m£.(r)Y£.m£.(n')a(1)>

21v. (R)Yj.mj.(nR) . (2.53)

But (9') must be eXpressed in terms of the molecule-
2

centered angle 9, and the molecular orientation QR . It

Yzm

then becomes

. 2.
ng2(nl ) a g Dmm2(-¢R'-eR'¢R)Ylm(Ql) (2054)

Thus the electronic integrals in (2.53) which are functions

of “R can be written in more convenient form as

(xi (1)|f5' ‘j '2' m2'(r)Y2 mvaQ')a(1)> =

"a
v nt‘flx
. v . - ll rr

 

Subsf

m05t

integ

33

c
5' Dfi'm£'(-¢R'-eR’¢R)<xi(1)IfS'j'l'mz'cr)

Yzomv(n)a(1)> (2055)
Substituting this expression into (2.53) gives

DJMj‘ (§)DJ"5'“'(§) - Z C‘(j2J, m.

”n+1 ,i i ,N+1 . . . .
VV mjmnmmj ml m

j “2”)

|
C(j'z'J,mj'm£'M) Dém;(-¢R'-eR'¢R) Dé'm£'(-¢R'-6R’¢R)

<f“ (mgm (n)a(1)lx (1)><x (1)If§. .. m£.(r)Yx. .m.(n)
“(13>Y3mj(fln)Yjvmjv<9R)x£(R)X»'(R) . (2.56)

If I now integrate both sides of this equation over 9R ,
most of the terms on the right can be written outside the

integral. I obtain

Jan DJMjl. (in)JMj z (fi) - I

R N+1,i 1 ,N+l . , . .
vv mjmzmmj m1 m

C‘(j£J,m mnM) C(j'2'J,mj'mL'M)<fB(r)th(9)a(l)|x1(l)>

5

<xi(1)I£§.j...ml.(r)v..m.(n)u(1)>xgcn)xv.(R)
2 v
Idaknmm"'¢n"°R-¢n) ”fixmL'(-¢R.-6R.¢R)

Y.
3m.

J(9R)Yj'mj0(nk) o

(2.57)

 

 

In orde
above e

can mak

34

In order to evaluate the nuclear angle integral in the
above eXpression, I must rearrange it considerably. I

can make use of the following relationships:’

1/2 .
Yimj 9R'¢R) ' [£}%l] ij0(¢R'eR"¢R) (2.58)

and

_. , . ‘.
Dg.m(aex) - e 1“ ° d;.m(e) e ‘m*

-. . O. '
- e 1my dinv('8) e 1m a

= Dim.(x-Ba) (2.59)

Now I can rewrite the spherical harmonics in (2.57) as

1/2 .
ngj(eRp¢R) ' [£%%l} ngjg('¢R,-6R,¢R) and
-. 1/2 .

The angle integral can now be written as

H
III

a 20
n [an Dam (-¢R.-6R.¢R)Dm.m£.(-¢R.-eR.¢R)ngjcnR)

Yj.mj.(nR)

Jan {iv—1- {are}

1/2

 

 

I can no

Di.

HIE

allOVing

 

 

1an

This an;

¢R and e

lamediat

35

L.

Dm'ml'('¢R"6R'¢R)ngj('¢R"9R'¢R)

z a j! a
Dmm (-¢R’-6R'¢R)D0mj'(-¢R'-6R’¢R) (2061)
I can now use the identity:”

J1 jz . . . ..
Dugm, DU2m2 g C(J13239Ul U2 UI+U2)

C(j.jzj;ml m2 mnmzmalwhmm2 , (2.62)

allowing me to write (2.61) as:

i 2
Jan Dm;;(‘¢Rs'eRo¢R) Dm'm£'(-¢R'-eR’¢R)

ngjcnR)Yj.mj.cnR)

l/Z
C(2'jj1,m'0m')

= (ii-:11” [#1

_2
ijz

C(z'jj1,m£'mjm£'+mj)C(£j'jz,mOm)
C(lj'jZszmj'mg+mj') Jan Di: m2! m£'+mj('¢R"eR’

¢R)Di2;;+mj'(-¢R'-eR'¢R) (2063)

This angle integral can be written as two integrals over
oR and 6R respectively. The first can be evaluated

immediately

 

 

To evalu

dj.
turn

I_

 

 

 

36
Zn w
R2 I doeim'¢ e'i(m£'+mj)¢ e'im¢ ei(m2+mj')¢ I sinede
0 0

j: . J: _
dm' m£'+mj( e)dm m£+m.'( 6)

- R26
m 2

sinede dj1 -6
',m+mj-mj'-m£+m£' m' m '+mj( )

jz -
dm m +m.'( 6) (2.64)
1 J
To evaluate the remaining integral, I expand the dim.(6).

dg:m -m '-m +m '
j j

- jz -
2 2 om£'+m.( 9) dm.m£*mj'( 6)

- {(j,+m+mj-mj'-m£+m£')! (jl-m-mj+mj'+m£-m2')!

(j,+m£'+mj)! (j‘-m£'-mj)! (j,+m)! (jz'M)!

(j2+m2+mj')l (jg-mi-mj')!}1/2

X (-l)k‘+k2/{(ja-m-mj+mj'+ml-m2'-kx)I

kxkz

(j.+m£v+mj-k.)z (k1+m-mj'-m2) k1! (jg-m-kz)!

(j2+m£+mj'-kz)l (k2+m-m£-mj')! kzl}

2(j,+j.-k.-k.-m+m.'+m )
[COS %] J 2'

2(k1+kz+m-mj'-m£)

[sin ;] (2.65)

37

This integral can be evaluated by making the substitution

a = 6/2 and integrating by parts.

i [ 9]2(j1+j2’k1‘kz-mfimj'+m2)
C05 7
0

[sin 2 sine d0

w/Z
= 4 I (cosa)2(j1‘32-k1-k2-m+mj.+m£)+l
0

6)2(k;+k2+m-mj'+m )

2(k1+k2+m-mj'-m2)+1

(sina) da

(J1*j2-k1'k2-m+mj'+m2)l (k1+k2+m-mj'-m l

2)
(JI‘J2‘1JJ

(2.66)
I now define
j 2 mj mg gm =
In “ 26m' m+m -m '-m +m '
j' 2' mj' m2. m! p j j 2 2
. l/Z

[2%%l] Z C(2'jj1,m'0) C(l'jj1,m2' m°)
jsz J

C(£5'52.m°) C(lj'jzm2 mj')
{(J1*m+mj'mj'-m£+m2')! (jx-m-mj+mj'+m£-m£')!

(jx+m£'+mj)l (jg-m2'-mj)! (j2+m)! (jg-m)!

38

(52+m2+mj')t (Jz-mzrijI 1’2

Z (-l)k”k2 / {(j;~m-mj+mj'+m£-mt'-k1)l

klkz

(jl+m£'+mj-k;)! (k1+m-mj'-m2)! k1! (jzrm-ka)l

(j2+m£+mj'-kz)! (k2+m-m‘-mj')l k2! }

(J1‘J2'k1'kzrm*mj'*mz)!
(jx*52*1)1

 

(k;*kz+m-mj'+ml)l

(2.67)

The angle integral in the right side of (2.57) can now be

written as
dn Dz '(-¢ -6 ¢ )D" (-¢ -6 ¢ )Y* (a )
R mm’v R' R‘ R m'mz' R' R’ R jmj R
j 1 m. m2 m
Yj,m ,(QR) - R2 1‘2 3 (2.68)
j j! 20 mj! m2" m!

I now define

2 m
13” [j mj 2 m l E C*(j2.J,mj mi M)
j! z! mj. mg! m!
j 2 m. mL m ‘
C(j'n'J.m3' “2' M) 1‘2 3 (2.69)
1' £0 mj' mg, m

so that I can now write equation (2.57) as

39

JM 2 JM°'£'
Jan DN+{’1(§)D1,§+1 (K) - vé'mjl mm vm 'm' <f8(r)Y£m(fl)
2 j 2

3(1) IXi(1)><Xi(1) IfS'j mimic (T)Y£cmc (9)0(1)>X$(R)

j 2 m. m m
xv.(R) R2 1%” .. ' 3' 2' . (2.70)
J 2 mj m, m
The nuclear angle dependence of the other terms in
equation (2.49) is similar to that of the second term

evaluated above, and can be integrated similarly. I

can therefore write equation (2.49) in the form:

I? - jzév'm m [IC(j2J,mjm,M)|2 JR’dR<fB(r)|r2|fs,(r)>
j 2
‘ N N ,1
2.(R)x..(n) [ E fun + ugv guv,uv +ch) + léé—g} - E]
N
1 B 2 8 d d
- ——— dR <£ r r If .(r)> * R ——R’——» .(R)
ZuN J ( )| v ‘Xv( )dR dRX” ]
IJM j 9. mj mg! m ]
n

 

+ E
' v t I t t I
j 22 vv mjmj mzml mm

3" 2' mj' mg! ml

N
B
[ - 1;“ Ida x3(R)xv.(R) <f (r)Y,m(n)a(1)Ixi(1)>
(xicx)125.,.,.m£.v,.m.(n)a(1)>fuu R’
N

B
4- ugp JdR 13(RJXVoCR)<f (T)Y2m(fl)a(1)lxu(1)>

. B
<xpcl)If..j...m2.Y..m.(a)a(1)>£up R“

40

N
- E I dRX:(R)xv,(R)[<fB(T)Y2m(Q)“(1)IXU(1)>(XU(1)I

u l

I?(1;R)If3.j.,.m£.v,.m.(a)a(1)> + <rscr)v,mcn)a(1)l

‘ 8
lf(1;R) IXU(1)><Xu(1) IIV'j'R'lng'Yfi'm' (9)0(1)>] Ra

Z

4.

r1a(R) rIB(R)

 

 

+ [ang3(p)gv.(R) <f“(r)v,m(9)u(1)l

IfStjvlvm£v(r)Y£vmt(Q)a(1)> R2

N N
B
- ”Ev I de3(R)xv.(R) igu V <r (r)Y,m(n)a(1)lxi(1)>

(xi(1)lfg'j'fl'ml'(r)Y£'m'(Q)a(1)> guv,uv R2
. N B
* X J dR13(R)xv.(R)<f (r)Y£m(Q)a(l)va(])><xq(])I

uvq
(all?)

8 32
lfvtjnfitm2|(r)yzumt(Q)a(1)> fluv’uq k

s
+ “3v Jan [<f (r)y,m(n)a(1)lxv(1)><xu(1)xv(1)I

1
FT?(1 P12)lf

S'j'l'mz'(rIJYk'm'(nl)a(1)xu(z)>

+ <f3(r.)x,.m.(n.)a(i)xu(2)I;%;(1-P..)I
Xu(1)xv(1)><xv(1)IfSIjtplmla(T)Y£Im;(n)a(1)>]

x;(R)xv.(R) R2

41

N
v21 JdR.x;(R)x,.(R) <facr.)x,m(n.ia(1)xv(z)

+

I?%;(1'P12)Ivaj122ml.(TI)Y£,m,(Q1)a(1)xv(1)>R2

N
l 8

<x,(1)If§.j.,.cr)Y,.m.cn)a(1)> [§RR*§Ex,.cn)]

N

121 IdR‘L:(R)x~v(R)-<f8(r)Y1m(Q)a(1)|xi(1)><xi(1)|

 

I£5.,...m2.v,.m.£fl)ac1)> [ vcn) + 1'15'*11-- E ]R* ]

ZuNRz
+ j; JdR <f8(1)| - %’§?r’§? +~£Léilljfg,(1)>
vv'
x3(R)xv.(R) R2 (2.71)

The last term of the above equation and the term containing
2 Z . . Jij'tt'
FIZTRT.+ FIETRT are a decomp051tion of the fN+l,N+l (fl)
term.
The vibrational functions are defined as an ortho-

normal set of solutions of the equation

[ ' 21%.} $15231! ” Vm " Ev ] 260*) - 0. (2.72)

I can use this identity to eliminate the derivatives
with respect to R from equation (2.71). I will assume

that the j(j+l) terms can be neglected, since the rotational

 

.._,..,....III.II..r .!
1 v

50

42

energy is very small compared to the vibrational energy.
I can now perform the integration over R in most of the
integrals in equation (2.71).

Also observe that

HMO]. 3 <94 X‘(1)X2(2)oooXN(N)| fiMOl I flXICl)

X2(2)o--XN(N)> ' uEl fuu + “Ev guv,uv (2.73)
so that
N N
Ev ' E + g fuu + “Ev guv,uv = Ev' E + EMol
. Ev- 5V0 - 1/2 kgo (2.74)

Equation (2.71) can now be rearranged an simplified to

the following:

. 1 d d
I = Z |C(32J m.m M)|2 <f8(r)| [ - a—rza—
lemjm2 ’ J l I. r r

. £L§zll . r2{nv-nv°- 1/2 x30} ] |f8(r)>

' 2 m. m m
+ Z IJFI [J J 2 ]

jj'vv'mjmj'm m 'mm' 9

j' 2' m.' m ' m'

2 2 j 2

[ é, JR.K;(R)xvv(R) [ <f8(r)ng(Q)°(1)l

43

Z Z 8
W 4' Wvaujcp‘!mol(r)ngml (9)0(1)>

N z 2
- I; <fB(rlY,m(9)a(1)lxu(1)><xu(1)I;IZTE7 +-;;Erfiyl

u=l
'fgvjuzcm£v(r)yltmv(9)0(1)> ] dR

+ $1 [“8” )Y (‘7 )a(1) (HI 1 (1 P (6
U=1 1 2m 1 xu $1; - 12)! julum2.(r1)

anmc(91)a(1)xu(2)>

- <f9(r)v,m(n)ac1)Ixu(1)><xu(1)|f§.,.m .(r)v,.m.cn)
l

oz(l)>(Ev-IE)

-<fB(r)Y,mca)a(1)I?(1)lxu(1)><xu(1)lr?.,.n .(r)
' 2

szmt(9)a(1)>

-<£B(r)Ym(Q)a(1)lxu(1)><xu(1)I - .3... 9.1.29...

2r2 dr dr

3::(9,'+1) B
+ 2r2 lfj'l'1n£'(r)ygvm.(Q)a(1)>

N B B
+ vgu [ <£ (r)Y£m(Q)a(1)lxu(1)><xv(1)lfj.£.mfiu(r)

chfi.(9)a(1)> fuv

44

-<£B(r)Y£m(Q)a(1)lxu(1)><xu(1)lf§.£.m£.(r)Y,.m.(fl)
a(1)> fvv
+<f8(r)Y,m(Q)a(1)lxv(1)><xu(1)xv(2)|;%;(1-P12)|
If§.,.m2.(rl)v,.m.(al)a(1)xu(2)>
+<fB(r1)Ylm(91)a(1)xu(2)l;%;(1-P12)|xu(1)xv(2)>

<xv(1)If§.,.m£.(r)v,.m.(n)a(1)>

N
* a
p;V,u <f5(r)Y2m(Q)a(1)IXV(1)><xp(1)lfj.£.m£.(r)

ngmv(Q)a(1)> guv,Up

N

- pin <f5(r)Y,m(n)a(1)va(1)><xv(1)Ir§.,.m£.(r)
#v
Y,.m.(n)a(1)> aup’up ] ] ] . (2.75)

Equation (2.75) will serve as an expression for the

. H A : M . . .
integral <W.,j9’l7(-I.WJJ. > . This W111 be used in chapter

'2'
(III) to develop a variational equation for the scattering
functions f8, which are assumed to be the only unknown

factors in the equation.

1 Rose.op.cit. p.60.

2 12059.0 .95. p.58.

45

Chapter III. The Variational Equation for the Scattering
Function.

I will now determine the equation which the scattering
functions f8 must satisfy in order that the variation
of the integral 1’} . <‘PI7?-E|‘P> , minus the surface term,
will be zero with respect to an arbitrary small variation
in the £8. The functions which satisfy this equation will
be taken to be the solutions which I am seeking. Ignoring
second order terms, the variation is given by the sum
of the variation from the left and the variation from

the right. The variation from the left is as follows:

I“ = X <6f8(rx)l
‘ ijj'i'v'mijmj'mg'

 

1 d 1 + «(2+1) + _ _ , ,.
[ - 7 U7TT1'&?T' “—3 [8v 5&0 1/~ RSOJTI}

6..,6

J) Ql'évv'émjmj'énzmg' free space part

+ interaction

2 Z ,2
jn x;(R)xv.(R){ <Y£H(Q‘)I?TFTWT + ?:?TVT

N
[\V'N'(Q’)>QI ' Z (qu(91)fl(1) vibrational
”=1 coupling
through
I > z 2 nuclear.
Xu (Xulriaiu) + rzrlhi attraction

46

47

IYP'ITI' (531)fl(])>] JR
+ 2 6vv' [ (Y2m(91)a(l)xu(2)I?%Z(I'P‘2)|ngmt(91)

coulomb and
a(1)xu(2)> exchange integral

- <Y,m(n.)a(1)lxu><xulv,.m.(9.)a(1)>(£;-E)

exchange

-<Y,m(n.)a(1)l§(1)Ixu><quY,.m.(a.)a(1)>

 

_ _ 1 d 2d £'(2'f1)
(Y2m(n‘)a(1)lxu><xul 2r; Hr;rl dr; + Zrl‘
|Y,,m,(Ql)a(1)> one-electron
Operator with
exchange

+ Vgu [ <Y,m(nx)a(1)lxu><vaY,.m.cnx)a(1)> ruv

-<Y,m(9.)a(1)lxu><xulY..m.(n.)n(1)> rvv

I
* <Y2m(91)a(1)lXV><Xu(1)XV(2)|;T;{1'P12)

coulomb and
|Y9.m.(91)a(1) X (2)> exchange integral
' ‘ u with second
exchange

48

+ <Y,mca.)a(1)xu(2)I;%;(1-P.z)Ixu(1)xv(2)>
(Xv|Y£omv(nl)a(1)>

N
3

N
- pin <Y,m(n.)a(1)va><xVIY,.m.(n.)a(1)> gup.up ]

{v

1 ] Ifg'j'l'm£'(rl)>rl (3.1)

I will make a convenient abbreviation by taking the
entire quantity within the outermost parentheses in
equation (3.1) and deleting it, writing only the paren-
theses enclosing the parameters on which it depends.

Equation (3.1) can then be written

M 8
I g i (éf (TIJI
left J . .

32VJ'2'v'mjmfimj'm2'

j 2 v mj m2

' t I I i
3' 2 v mj m,

6
B
'fv'j'2'm£'(r‘)>r,
(3.2)

The variation from the right will be

IN

Gright J a éleft

*

If8(r)>,] + [

|6f8(r)>r ]

I can now write the total variation as

'M M
61d Gleft IJ
5 IM
left J

B .
t i
la (r)>r

I do not want,
equal to zero, but

term. In place of

M

M
J a 6left

61 IJ

K
When I set 613

O

+

+

6

6

49

M * 1 d d
IJ [ jzv <6f8(r)| - 7 Errzd?
jév <f8(r)| - % §;r2§; I

right

left

I

M
IJ

M a
J

4.

j2v

(3.3)

[ «new! %§;r‘§,—.I

- <f8(r)I agape; |6f8(r)>r] (3.4)

however, to set the complete variation

rather the variation minus the surface

61

+

'M

J

6

I define:

left

I

M a
J

(3.5)

equal to zero, I obtain the equation

which the radial scattering functions f8(r) must satisfy.

61M = 0

50

j 2 v m. m

. z <6f5(r,)| [ . J 1
jj'22'vv'mjmj'm2m2' j' 2' v' mj'ml'
le'j'2'(rl)>rl + <6f8(rl)l

!
2m2

j 2 v m. m
[ ., 2, , J. 2"] |f\8)'j'2'(r!)>;l
J v m. m
j 2

' '22'vv'm.m.'m
JJ J J

(3.6)
But this equation is equivalent to
I? - 2 Re [ <6fB(r)|
O I t 0 O
jj 22 vv mjmj mam2
. 2 .
J v mJ ml lfBo'v .(l‘)>
j! 2! V'm'm' V32
. j 2
(3.7)

where Re(x+iy) = x.

Since the f8(r) are arbitrary complex functions, the

above equation implies that

j 2 v m m

3'2 -8 ,
J'2'v'mj'mg' 5' 2' av m.'m,' 'fv'j'n'(r)’ 0
J

(3.8)

for all v, j, 2, mj, and mi.

51

To use this equation, I need some explicit form for
the f8(r) containing variational parameters. These can
then be evaluated by requiring them to satisfy (3.8).
A very important factor which is pertinent to the form
of these scattering functions is that I am using a one
determinant wave function for a system in which resonant
scattering is dominant. Such systems must normally be
described by a two or more determinant system, but since
this is a single-particle resonance where the molecule
is not electronically excited, I can use a one-determinant
function. I must then compromise between the goals of
having the correct asymptotic form for the wave function
and having the polarization of the target well described.
This difficulty will be further discussed in Chapter IV

and Appendix B.

Chapter IV. The form of the Scattering Function.
In scattering of an electron by a molecule, the

complete electronic wave function can be written in the

form
M .
W(N+l) a Z 154 violecule(N)w§catter1ng(1) (4.1)
i=1
Molecule

where the W (N) are the wave-functions of the

1
various states of the molecule and 64 is the antic

symmetrizing operator. If correlation within the molecule

is neglected, the V¥OIBCUIC

(N) are the one determinant
Hartree-Fock wave functions. The wgcattering(1) are scat-
tering spin orbitals which I will assume to have a-spin.
This method is generally impractical because of the
need for a large number of molecular excited state functions
which are not in general known. If there are no molecular
excited states near or below the energy of the scattered

electron, a reasonable approximation of the above Function

can be made by
wow) = 24 211300410) (4.2)

where V?(N) is a distorted ground state function. To
obtain the correct assymptotic form I will split this

function into two parts

mm = a‘lcmewSm + cvpminm ) (4.3)
52

53

i . .
where 24(N) is the undistorted ground state function,
WP(N) is a distorted ground state function, 65(1) is a
scattering function, and wB(l) is a bound state function.

The asymptotic form is now correct:

v(N+1) +-.f§WM(N)wS(l) as rscat +~ . (4.4)

I choose the incoming electron beam to be travelling
along the z-axis of a non-rotating spherical coardinate
system centered at the center of mass of the molecule, so

that I can write 68 as
w = (e1kz + e1kr f(r.6.¢) ) = ( 220 izczn+1)j,(kr)
p,(cose) + eikr f(r,e,¢) ) (4.5)

where f(r,6,¢) + %f(8.¢) as r+~. I can write the Legendre

polynomials in terms of spherical harmonics.
l/2
(22+l)P£(cose) = (4a(22+l)) Y£0(6,¢) (4.6)

I can also expand f(r,6,¢) in terms of radial functions

and spherical harmonics as

00 +2
f( ,6,¢ 8 f Y 6, 4.7
r ) 220 m§_2 2m(r) ,m( 4) ( )

'where f£m(r) + %f2m as r+m. Equation (4.5) now can be

S4

expressed as

on

x5 - £20 (ii/2mm j,(kr)Y,’o(9.¢)
ikr *2
. e g 1 f,m(r)y,m(e.¢) (4-3)
m--

For a spherically symmetrical interaction f2m = O
for m f 0, but for scattering from molecules this is
not generally the case. The complete electronic wave

function is now of the form:
mu) - 64(2”(N)ws(1) + cipmwBun (4.9)

. .ein(N) {o (ii/TFTZITIT'j,(kr)Y£o(9.¢)
2'8

eikr +2
+ r mz-2 f,m(6,¢)) as r + m. (4.10)

 

I also wish to consider vibrational and rotational
excitation. If the diatomic molecule is initially in the
state specified by vo,jo,mjo, then the asymptotic form
of the function will be

mum) + 342M(N) E0 Mm 3-20. r)

r+ 2- v°j°

2,003.40),W (12in .m, . (6R'¢R)

55

f .
2m,vojomjo,vjmj

Ymcemmcmjmj(9R.¢R)) (4.11)

where xv(R) and ij.(eR’¢R) are the nuclear vibrational
and rotational functions, respectively. Let kg 5 kvojo°
So that the complete function is compatible with this

asymptotic form, I replace 68(1) with

I115(1,ii) - {0(12/1—17'2-171. + j£(kv°jor)Y£0(e,¢)

2:
on ,k 1' +2, +j
(my. (6 ¢ ) +2 2 e1 v) I
xvo Jomjo R. R v j-O mz-2 mj--j

fm2,vojomjovjmj(r)Y2m(e’¢)xM(R)ijj(6R’¢Rx’ (4.12)

I now have a scattering function consisting of a plane
wave with the molecule in the initial state and outgoing
waves with the molecule in all the accessible vibrational

and rotational states. This can be written in the form

S S
w (1.2) = vgmj xVOjomjo,,jmj(1)x,(R)ijj(0R.¢R)

(4.13)

where the electronic functions are

56

S .2 .
XV°5°m5,-ijj(1) £20 (1 “12(21+15 32(kVojoT)Y£0(9.¢)
. 2
k .r +
6 6.. 6 + e1 v; 2 f . . (r)
vvo Jjo mjmjo m--2 m2,vojomjo,Vij
Y£m(6,¢))a(l) . (4.14)

Since I expect Born-Oppenheimer breakdown to be important
in the resonant scattering process, I will also include
molecular vibrational and rotational functions in the

bound state orbital:

B B
Cv(1)= C. x- (1) (MY (6 4)
vjmj vjmj Vij 29v jmj R’ R ’
where
B
rxvjmj(1) + 0 as r + m . (4.15)
The Cvjm. are unknown constants giving the weighting

factors for the different molecular states. It would

be possible to obtain vibrational functions for the
bound state function YP(N)wB(l) and use them for the
.l~(R) in this expression. The regular molecular vibra-
tional functions are a complete set and may also be used.
'The vibrational functions and their coEfficients will

‘be discussec more fully in the following chapter.

The total wave function can now be written

57

M .
mini) -- z (m (foojom
Vij jo

vjm.

vjm.(1) I C
J J

24 ‘f’p(N)x5jmj(1))xv(R)ijj(0R.¢R)- (4.16)

I now wish to determine a reasonable one-determinant
approximation to the above two-determinant electronic
function which will be easier to work with. I have
shown in Appendix B that the best function which can
be obtained without doing most of the work required for
the two-determinant function is obtained by neglecting
the polarization and replacing WP(N) by WM(N).

The effects of the polarization are approximated by
potential scaling when the bound state Spin-orbital is

calculated in Chapter V. The wave function I am now

using is

2(~+1.2) -- 24va z (xiojom, m.m
)2ij Jo' J

B

+ Cvjmjxvjmj(1))gv(R)ijj(6R,¢R) (4.17)

By writing a set of different bound state functions

xgjm , I am assuming some breakdown of the Born-Oppen-
j
heimer separation between the nuclear motion and the

motion of the scattered electron. This is reasonable

for molecules such as N2 where the N; resonance is believed

58

to exist about the same length of time as the vibrational
period of the szolecule.l The rotational motion, however,
is much slower and I cannot justify Born-Oppenheimer

breakdown with respect to rotational motion.2 Equation

(4.17) now becomes

«6w am i
Vij J0 J

+c vxB , (1))xv(n)vjmj(ek.6R) (4.18)

Using equation (4.14) this is equivalent to

v(N+1,2) = 642M(N) I ( Z (in/TFTZITTT

vjmi 2‘0
32(k0 T)Y20(99¢)5vv06jj06mjmio
. +2
+ lkv r . . Y o a 1
J m§-£ m,’voJomjo.,ij(r) 2m( .6)) ( )
n
*C Vx v (1))gv(R)ijj(0R.¢R) (4.19)

This function still represents scattering of a plane wave.

To conveniently handle the rotational functions I need to
express V as a combination of states which are eigenfunctions
of total angular momentum, J, z-component, M, electronic

angular momentum, 2, and rotational angular momentum, j.

59

That is, electrons in a particular Spherical wave scattering
from a molecule in a particular rotational state, with

the system in a particular total angular momentum 5 ate.

These can be obtained as

vJMji JMj2

(N+1,i) . P IW(N+1,2)> (4.20)

JHj2

where P is a projection operator for the JHj2 state.

Then

wJMj2(N+1'E) = pJMj2‘gwa(N) Z (1)

(X3 ° m va
ijj 0J0 jo. Jj

+ c vxB v(l))gM(R)ngj(0R.¢R)

JMj2 M a r r S B
.. J4!) w (a) 2 (x . a (1) + C x (1)}
vth ‘Vojomj°,VJHB v V

X6(R)Yfmj(°R'¢R) (4.21)
The angle function for state 5Mj2 can be written

JH’2 . .
23 J (0.6.0R.¢R) = mg. chnJ.mij)Y,m(o.6)vjmi(0R.6R)
J .

(4.22)

JMj2

So P can be written

6O

pJMJ‘ = I IccjnJ.mjm”)" ‘Y2m(e'¢)Yij(eR.¢R)>

mm.
J

<Y,m(e.¢)vjmjceR.6R)I (4.23)

In Appendix C the resulting function is shown to be:

vJM5”(N+1,i) = v”(N) Z C(j2J,mjm£M)2

vmgmj

 

.2 .,
1 /4n(22+1) J£(k°r)6mjmj06mgoévvoéjj0

ik .r
+ e V] f2m£.\)ojomjo.\)jm.(r) + E 6i.(2‘20)/2

J

C (r) I(jmj2m2mo) Y 2(e'¢)Yij(eR'¢R)

vifui 2m

a(1)gv(R) . (4.24)

I(jm 2m£mo) is an angular integral which can be written
entirely in terms of the five given quantum numbers.
The expression is given in Appendix C. The numbers 20 and
mo are the angular quantum numbers for which Y£omo has
the same symmetry as the bound part of the scattering
function.

Before the scattering function can be evaluated, the

bound state part of the function must be known. This

will be evaluated in Chapter V.

61

‘ Herzenberg and Mandl, gp.cit.

2 A.S.Davydov,Quantum Mechanics,p.503,1968.

 

Chapter V. The Bound State Wave-Function.

The bound state function will describe the negative
ion which is formed temporarily. This is the N+1St

orbital of a one-determinant function:
B . B -
v (M1,?!) . Ax‘mXZm... xN(N)xN,1(N+i,2)
(5.1)

Orbitals 1 through N are the orbitals from the unperturbed
ground state Hartree-Fock function for the neutral
molecule. Using the standard Hartree-Fock approach we
would write the integral <WB|NJWB> - E and then vary the
function xN+1 to obtain the minimum energy, with the
constraint that XN+l remains normalized and orthogonal to

the other orbitals.

In many cases this is not a good method. First, I am
using an unpolarizable molecular function, frozen in its
unperturbed form. Thus, the N+lSt electron "sees" a
much weaker potential than the actual one. Second, the
actual potential is not strong enough to have a bound
state in many molecules. ”2 and N2, two of the molecules
I am most interested in, do not have bound negative ion
states. Thus, if Xg+1 is varied without restriction, it will

approach the wave function of a free electron at zero energy.

62

63

One solution to this problem is to restrict the variation.
The orbital is not allowed to approximate a free electron
wave-function, and a bound state function results. The
form of the function depends on the exact restriction

on the variation. A similar method, also commonly used,
is to vary the function only until the energy becomes
relatively stable with reSpect to further variation, not

to a rigorous minimum. This method is open to the same

criticisms as the first.

Another method is to alter the interaction between
the scattered electron and the molecule so that there
is an actual bound state solution. The function may
then be varied without restriction to get the function
for this bound state. Increasing the apparent nuclear
charge or decreasing the electronic repulsion will make
the potential of the molecule more attractive, but
either of these destroysthe electrical neutralitv of the
molecule. Thus any finite change in either will add a
l/r coulombic tail to the molecular potential and create
an infinite number of bound states. I can avoid this
bv increasing the nuclear and electronic parts of the
interaction bv the same amount. I have assumed that the
scattered electron "sees" an attractive potential of the
same order of magnitude as that necessary for the
existence of a bound state. If this assumption is

justified, multiplying the potential by a reasonably

64

sized number should produce an equation with a bound

state solution. This is the function which I will use

for x£+1 . Part of this potential scaling amounts to a
replacement for the polarization potential which I lost

by freezing the core. The rest converts the resonant state,
which has an energy above zero, to a bound state which

can be obtained more easily.

First, I need the original Hartree-Fock equation
for x§+1 , with the unscaled potential. I must

evaluate

a . <vB(N+1)tf(|vB(N+i)> (5.2)

wt:‘ :01‘ ~~o-o‘ : - ‘L- g» a -‘ ‘..- '- .r. 1 . -u.0-..- ..
aux.) aucvgaui a.) but; sum; as aJ . UA'ITUDJCL All utiuuuauu

(2.46), with two changes. First, this integral involves

;( instead of ILE so the last term in (2.46) is absent.

Second, I need not consider the various angular momentum

states designated by the J,M,j',j,2',and 2 superscripts.
Here the minors of the overlap matrix are much simpler

since the bound state function is normalized and orthogonal

to the other orbitals. The N+lSt orbital is

xfi.1(1;2) - é Cvxg.1.v(l)gv(R)ijj(flR) (5.3)

B
N+l,v

unknown electronic functions, and the Eu and ij are

. J
known nuclear Vibrational and rotational functions.

where the Cv are unknown constants, the x are

65

The normalization condition is

B B
*
Ev. Cucv' ‘ <xN+l,v(l)lxN+l,v'(l)>elec 16(R)

ijj(nR)Xv'(R)ijj(nR)>nuclear E 1 (5'4)

but the vibrational and rotational functions are ortho-

normal so this is equivalent to

2 B B 2
g lcvl (XN+l,v(1)'XN+l,v(1)>elee ‘ 1 (5.5)

I also require that

B B :
<xN+l,v(l)|xN+1,v(l)) - 1 , (5.6)
so that
zlcvl2 a 1 (5.7)
v

Define the pure electronic overlap integral, between

the various electronic functions:

vv' : B B
DN+1,N+1 ‘ (XN.1.v(1JIXN,1.v.(1)> ~ (5.8)

1TH: overlap integral for the resonance function with

itself is thus:

66

_ . vv'
DN+1 N+1(§) 2. cvcv'DN+l Nflxv(R)ij.(nR)
' vv ’ J

ch(R)ijj (QR) o (5.9)

The other overlap integrals are

Duv(§) 5 6UV’ unless u=v=N+l . (5,10)

The determinant of overlap integrals is
ln(i)n = DN+1,~+1(§) . (5.11)

Its minors are

l
1 qu+1

. -r UV:
ID(kNI éuvx

 

DN+1'N+1(2) u-N+1 (5.12)

k

and
f

1 qu+1 u>v,p>q

+ uv m ,.
HD(R)N ' éupévqx

u=N+l U>V.P>q

 

3
. DN+1,N+1(L)

(5.13)

I define the nuclear kinetic energy term,

67

 

(2) : vv'
DN+1,N+1(§) “ vé. CSCv'”N+1,N+1Kv(R)Y§mj(QR)
-1 d 2d
——R ——. (R)Y. (a ) (5.14)
ZuNR2 dR dR XV' ij R

the electronic kinetic energy and nuclear attraction

term,

fn+1,N+1(fi) 5 Ev, Cscv'fer,Ntl(R)lv(R)ij.(9R)
J
xv.(R)ijj(9R) (5.15)

and the coulombic and exchange term,

. . a E Y .*. . V”.
21“}: Ugm‘rl "(15) (H). rVrV' gl‘i’l U.N*l U
*
XV(R)ijj(nR)XV'(R)ijj(QR) (5016)

In these expressions, the one-electron integral is

vv' B ‘ + B
{N*1,N*1(R) <XN+1.V(1)|f(l'k)lxN+1,v'(l)>

(5.17)

- B - 2 a B
(XN*I.V(1)I l/ZV IXN*I.V'(1)> ' (XN*I,V(])I

68

Z 2 B
—— + --—-|x~.1 ,.(1)> (5.18)
rla(R) r18(R) .

. 1‘va - Vvv.(R) (5.19)
where Tvv' is the kinetic energy and Vvv'(R) is the

nuclear attraction as a function of internuclear distance;

and the coulombic and exchange integral is

vv' = B 1 -)
3N§1 U.N*1 U ' (XN+1.V(1)XU(2)I?T:(1 I12)|

XN*1,V'(1)XU(Z)> - (5.20)

The energy is now given by

N
E . I Jdk fun DN+1'N+1(R)

+
M

Id“ guv,uv DN+1,N+1(R)
u>v

62 0(31 N+1(ii)

ZpNR2

i
+ I62 [111111 + V(R)] DN+1’N+1(B)
Id“ fn+1, N+l(fi)

+
ha

u=l J62 gN,1 u,N+l u(ii) (5.21)

69

From the normalization condition, equation (5.5), and

the definition of DN+1,N+1(§)' equation (5.9):

J62 DN,1.N,1(2) - 1 (5.22)

I can combine the third :nulfourthintegrals as

 

I ' I . Cscvcnxri'N+1 Jdfi Xv(R)ij (QR)
vv J
1(i+1) . V(R) - 1 9.4229. ,(R)Y. (a )
[szRz ZuNR2 dR dR] Xv ij R

(5.23)

Deleting the rotational energy term, as in Chapter II,

c . vv'
I Ev. Cucv'DN+l,N*lévv'Ev
. 2 1'
I ICvl L, (5.24)

V

th

where Ev is the energy of the v vibrational state.

Recall that the xg+1 V(1) are normalized, although not
9

orthogonal: equation (5.6). The fifth term is

[an rN+1.N,l(fi) . 66' 636v. jai f§3l.N+1(R)

70

MR“)...

(9 ) IRJY- (9 l
J R xv jmj R

. z CGCv' IR’dRXv(R)(T

vu' vuv'(R)) xWU”

vv'-

' I lcvl2 Tvv - 66. Cscv' JRZdRXv(R)Vvv'(R)Xv'(R)
v

(5.25)

The last integral is

6 vv'
1 cvcv' gN+l u,N+l u

Jdfi gN+1 u,N+l u(§) ” vv.

Idfi XV(R)Y3mj(nR)XV'(R)ijj(nR)

VV

2
g 'Cv' “n+1 u,N+l u . (5.26)

Equation (5.21) now becomes

N N
r=zf+Ig +2ICI2E+ZICI2
“=1 UU U>V UV.UV V V V V V
TW - Ev. CCCv. Jdeva(R)Vvv,(R)xv,(R)
. j I |c |=gVV (5 27)
uul v V 'N+l u,N+1 u

I can now make an arbitrary variation of x8

N+l . The

resulting variation in energy will be

71

I 2 2
6E é Bvdlcvl + E rvv élcvi

1 B
* I [CV '2 6<XN*1W (1)1 'IVzIXN*1.v(l)>

- g“. [(6c;)cv. JR’dev(R)V.6e(R>x.v(R)

. c;(6cv.) IR’dRXv(R)VVV,(R)Xv,(R)

Z 2
B
v v v N+1,v “(R) r18(R)

lxg,1,v.(1)>xv.(R)]

. I l élc I’ ”V
“:1 v gN+1 u ,N+1 u
N . 2 B l _
+ i I Icvl 5<XN,1.V(l)xu(2)|;T;(1-P.2)
u-l v
I B (1) (2)> (5 28)
XN+l,v xu ‘°
This can be written as
GE = { (6C*)C (E + T + g VV )
v v v v VV Usl gN+l u,N+l u

B 1 B
+ g JCVIZ <6XN+1.v(l)1'2V2IXN+1.v(1)>

+ 2 (GXN+1'V(1)XU(2)l*l”(]“P12)'X\+]N (1)

72

xu(2)>]

- I [ (6C;)Cvc JRZdeV(R)vVV'(R)xV'(R)
vv'

2 2
B ____.
2
+ C3Cv' JR deV(R)<6XN+1.V(1)Ir1 (R) + r18(R)
a

 

Ix:,1.v(1)>xv.(R)}

+ Complex conjugate (5.29)

The variation must be constrained so that x£+l remains
normalized. This can be done by using Lagrange multipliers.

The variation of the normalization condition is

‘ B B
I [ (5C3)Cv<xN,1.v(l)lxN.1.v(1)>

2 B B
4' lcvl <6XN+1.V(1)|XN+1'V(1)>]
+ Complex conjugate = 0 (5~30)

There is an additional normalization restriction:

B B
<x~+1.v(1)'XN+l.v(l)> B 1 (5.6)

This gives the conditions

73

B .
<6X:+1,v(1)'xN+l,v(l)> + Complex conjugate . 0

for all u. (5.31)
Substituting (5.6) and (5.31) into (5.30), I find

* a.
g (6c3cv + Cv6Cv) o (5.32)

I can combine (5.29),(5.32), and (5.31) using
Lagrange multipliers:

N
VV
é 6C3Cv(Ev + Tvu + “£1 gN+1 u,N+1 u)

N

' B l B
+ g Icvlz [ <6xN.1.v(1)l-IV’IXN.1’v(1)> + “£1

<6x§,1.v(1)xu(2)l;%;Il-sz)Ix:,1.v(1)xu(2)> ]

Z 2

B
+ C‘C IRZdR (R)<6x (1)|~——___ + ._____
v v' 2.(-v N+l,v rla(R) r18(R)

|XN*1,V(1))x-V' (R) ]

t
+ Ac g GCVCV

74

B B
* 2 AV<GXN+1.V(1)IXN+1.\’(1))
v
0 Complex conjugate - 0 (5.33)

For my present purposes I will seek only a rough

approximation to the solution for the above equation.

I can expand x8 as

B . .E

xN.1.,(1) 0 f,i(r)v,o.2,.m°(n)a(1) (5.34)

12

An arbitrary function, f(r,9,¢), with the same angular
symmetry as the spherical harmonic Y£°m°(0,6), can be

expanded as

"MS

f(r,9,¢) = fi(r)Y (8.¢) . (5.35)

£o+21,mo

i 0

Since I am concerned with the effect of the bound state
function on low energy scattering, I will use only the

first two terms of this sequence:

x§.1.,(1) = (fv0(r)Y,°m°(9) + f.1(r)Yx,.z,m.(0))a(1)

(5.36)

I shall also assume that the correlation between

the electronic and nuclear motion is primarily angular

7S

and not radial. Qualitatively, the effect of the Y2.+2,mo
term is to shift the electron density toward the molecular
axis and away from the plane perpendicular to the axis,

or vice versa. The assumption I am making is that the
difference in the electronic functions for different
vibrational states can be fairly well approximated by

this shift along the axis. Thus, the radial functions

are the same for all v, only the relative importance of
the two terms changes. This has the consequence that

x§.1'v . (f “(r)c,o ,0 m. (n) + f1(r)c,1Y ,0 .2 m. (9))a(1)

(5.37)

where fo(r) and f1(r) are normalized functions.

Then,
XN+1(1; fl) . X cv(cvor 0(r)Y£0 mo(a) + Cv1f1(r)
V

Y,°.2.m,(n))a(1) 46(“3ijj(“al (5.38)
The overall trend for the Cu should be a decrease
as v increases. I will choose a set of Cu to approximate
this trend rather than attempt a direct solution of
equation (5.33). For a given diatomic molecule, with

a given observed resonant energy, there is a limit to

76

the vibrational states energetically accessible.

I will designate the first state not energetically

available by vm. I choose the Cv to be

Cv a C(v-vm) . (5.39)
Since I Cv2 a l , (5-7)
v

1'1/2 (5.40)

”m 2
C - I X (v-vm)
v-O
I have chosen the proportionality in (5.39) to be an
approximate fit to the trend observed by Schulz for
vibrational excitation of N2.1 The angular correlation
described by CvO and Cvl is prooably a small ettect

relative to the approximation made for the Cu‘ Therefore

I will let

C a C

v0 and Cu 8 CI for all v. (5.41)

0 1

B _
Let XN+1(1) ' (Cofo(’)Y2.mo(9) i C1f1(r)Y£o+2.mo(Q)) “(1)
(5.42)
The constants Co and C1 and the functions f0 and f1 will

be calculated at equilibrium internuclear distance. The

effect of Born-Oppenheimer breakdown is now included

77

entirely in the coefficients cv. The N+1St orbital is

“OW
x2.1(1;i) - (Cofo(r)Y..m.(9) . clflcr)v,.,2.m,(n))

«(1) g C(v-vm)xv(R)ijj(0R) (5.43)
Equation (5.33) can now be greatly simplified: The terms
involving variation of the Cu are zero since the Cu are
now fixed. Since the nuclear attraction is evaluated
at R=Ro it no longer couples vibrational states and
the CVCS, factor in this term is replaced by ICVIZ.
. B B
Since the YN+l,v are all replaced by xN+l , we can
factor out and delete the term 2 [Cv|2. Equation (5.33)
v

is now reduced to

Z 2

B l
<6x , (1)] — V2 + A + -—_—____ -._______
N 1 [ 7 [ r1a(R.) T18(Ro)

 

N
l B
+ Complex conjugate = 0 (5.44)
A z 2 § I 1 )
Let V(1) = - -——-- + <x (2) ?-(1'P12
r1a r18 u-l u 1"

lxu(2)>2 (5045)

78

Fortunately, V(l) is a Hermitian operator and equation

(5.44) can be replaced by
1 i . B (s 46)
[-7V. + A + V(1))xN,1(1) - o .

As I mentioned at the beginning of the chapter, in many

cases equation (5.46) does not have a bound state solution.

I will write instead

(-§vi + x + 79(1))x:,1(1) = o (5.47)

For some minimum value of 1, this equation will have

a bound state solution at zero energy.

I have calculated an approximate Hz function and an
approximate H; function using single center orbitals
with non-integral principle quantum number.2 The programs
used to optimize the functions will be described in

Appendix D.

The H; function which I used was
?(1,2) =‘J4¢1(1)a(1)¢,(2)8(2) (5.43)

Using one single-center orbital, I obtained the lowest

energy for

79

¢1(1) = 2.8684 r°23 e'l'22 r

Yoo(e,¢) (5.49)
at R=1.ZS (all quantities in atomic units.)

Using the HE function
W(1.2.3) = e1¢1(1)a(1)¢1(2)6(2)¢2(3)a(3) (5.50)

I obtained a bound state function

682 8-.607 r 1.852

¢2(1) = ( .3803 r' + 5.2957 r

e'“52 r ) Y10(9.¢) (5.51)

When 7 is set at 1.824, this function gives the lowest
energy, -.0000554 Hartree, or -.00151 av. The actual energy
is .289 Hartree, or 7.85 ev. I can interpret this relation-
ship as follows: The expectation value of the orbital energy
is

E - <¢2|T + V|¢2> a 7.85 e.v. (5.52)

When I increase 7, I increase the (negative) expectation
value of the potential energy. If the virial theorem
holds for relative changes in potential and kinetic
energy, then increasing 6 by the factor y will produce
zui adjustment in the orbital leading to an increase in
kinetic energy which will cancel half of the potential

energy change. That is the potential energy is

80
increased by
VY - (7-1) <¢zl§l¢z> = (Y'l) v (5.53)
and the kinetic energy increases by
T = -1V = 1’7 <¢ |G|¢ > (5.54)
Y IY T 2 2

so that the change in total energy is

TY + vY = I § -.§ - 1 + y J v = Igl.v (5.55)

but this change in energy is the energy of the original

resonance, since I am changing the energy to zero, and

I can now infer that the energy of the resonance is

approximately

E 3 1'* v (5 56)
_T_ ‘

I know that

<¢2|T + yv|¢2> = 0. (5.57)

Subtracting (5.52) from (5.57)

81
(y-1)v = <¢2|(y-1)§|¢2> = -7.85 e.v. (5.58)

and Er: 3.9 e.v. Within experimental error, this is a

perfectly reasonable energy.3

‘ Schulz,G.J.,Phys Rev,125,229,(1962).

2

Joy,H.W. and Parr,Robert G.,J Chem Phys,£§,448,(1958).

3 Schulz,G.J.,Phys Rev,135,A988,(1964).

Chapter VI. The Scattering Amplitudes and Phase Shifts

The radial scattering function corresponding to a
given partial wave, final state, and initial state has
been designated fB(r), where 8 refers to all the.appropriate
quantum numbers. Comparing equations (1.9), (1.10),
(1.11), and (5.42) with (4.52), I find that the form ?)
of this function is

JM,Vojom.

f8(r) - f J°(r) - C(ju,M0) i‘flm

 

vymj,£m£

‘r

j£(kor)6Mm. 6 6

Jo ”V0 110 + C(JlJ’mjmiM)

ik .r

e v3 (r) + C(le,misz)

£m2,vojomjo,vjmj

1

iEO 6i,(£-£o)/2 Cifi(r)C(v-vm)1(imjzm,m.)

(6.1)
Note that Ev-EVO- 1/2k3 - - l/Zk: (ignoring, as before,

the rotational energy term). From Chapter III, these

functions must satisfy

1 d 2d 2 2+1 1 2 z]
vjlmimg [ [ 7 3r) 3" 7 V ‘ 33 ”V
6 ,6 ,6 .

82

83

j l m. m m
* 2 13M[ J in mi ] [ IRZX;'(R)XV(R)

O ' O I 0
mm 3 1 mi m,

[ <Y,.m.(flx)| TIETRT * FIETRT 1Y;m(91)>

N
Z Z
- <
“21 yl'm'(n‘)°(1)lxu)<xul?I;TFT * FIETFT1

Y£m(91)0(1)> ] dR

N
+ 821 svv. [ <Y,.m.(n.)a(1)xu(2)I;%;(1—P:z)

IY,m(nn)a(1)xu(2)>
' (Ygtmu(91)0(1)lXu>‘Xu1Y£m(91)0(1)>(5v’5)

- <Y,.m.(flx)a(1)|E(l)lxu><xulY,m(nx)a(l)>

1 d d
- (Y2,m,(0.)a(l)lxu><xu|'2?f a-fi-riar,’

+ £§%;lllY2m(ai)a(l)>
l
N
+ vgu [ <Y£.m.(91)0(1)IXU><XV|Y1m(nl)°(1)) fuv

. (Yl,m,(a,)a(l)lxu><xu|Y,m(Ri)u(l)> fvv

+ <v,.m.(n.)a(1)va><xu(1)xv(2)I;%;(1-P.z)

84

IY,m(8.)a(1)xu(2)>
’ <Y£.m.(fli)utl)xu(2)|;%;(1-P12)lxu(1)xv(2)>
(XV|Y£m(Q')a(1)>
N
+ p;V.u <Y£,m,(fll)a(1)'Xv><Xp|YLm(fli)0(1)>8uv’up

N
- Z <Y£,m.(!2))a(l)lxv><xlegmml)°(l)> gup,up J

p<u
iv
fJM ,Vojom.
} jo (1‘)) s 0
fvjmj ,lm,
for all j',£',v',mj',m,' . (6.2)

I will assume that the resonant scattering ( 2-2.,
£o+2,g,+4,,,, m-mo ) is very large compared to the non-
resonant and ignore the non-resonant terms as a first
approximation. The resonant function is approximately
described in the vicinity of the molecule by the bound
state orbital, xB, which is orthogonal to the molecular
orbitals. Therefor I will make the further approximation
that the resonant scattering functions are orthogonal
to the molecular functions. (In the cases of H2 and N2
this is rigorously true due to symmetry.) Equation (6.2)

is now approximated bY

 

6..,6 ,6 ,6 ,6 ,
J] £2 vv mjmj m£m£

+ Z I

mm'

JM[-‘ 2 "‘j "‘2 ’“
o

R2 *,(R R
5' 2, “‘5' "‘2' W] [I x. )xv()

(Yg'm' (91)|""z—'—' + Z

r1a(R) —-—-—r18(R)IY£m(Qi)>QldR

N
1
* “El va' <Y£0m|(91)a(l)xu(2)'FT?(1‘P12)
JM,v.j.mj.
Iv,m(n.)a(1)xu(2)> If (r)> = o
vjm.,lm
3 l
for all j',v', and mj' and for 2'220,2o+2,...m,'-mo

(6.3)

Note that the mixing of vibrational states occurs entirely
in the nuclear attraction term.

I can use the usual expression for -l— in the last

T12
integral:
n
°° +n r
1 Z 2 41! < *
.._.... I z—I-———I- Y (91)Y (92)
r12 n-O mna-n n+ r>n+ nmn nmn

(6.4)

86

Similar derivations yield expressions for l/rla
and 1/r18 . These expressions are simpler because the

coBrdinate system is fixed to the molecular axis.

a 4" 1/2 R<“
W18 " {0 [m fair Ynom‘) (6'5)
n.
>
and
on 1/2 R“
>

where R< and R> are the lesser and greater of rland R/Z,

respectively. Multiplying by Z and adding (6.6) to (6.5);

 

 

n
r2 * r2 a 22 E [ZégT]1/2 -E:7T Y 0(9‘)
la 18 n-O R)n n
even

(6.7)

Substituting these expressions into equation (6.3),

I obtain

6 6 6

ij'avv' zz' mjmj' m m '

l l

* JM j 1 mj m, m J [ w
i In [ 5' 2' mj' m ' m' 22 Z

even

87

4 1/2 2 R<n
[HTI] JR K;.(R)XV(R)E—H7T<Y£,m,(n1)[yno(g,)
>

Y2m(n))>n dR

on +31

N
+ 2 8W, 2 z g_3<v,.m.m.)a(1)xu(2il

u-l n80 mn'-n

n
T<

—-n—+r Ynm (Qle‘m(02)(1-,P..)IY mmml)

] J” fJM ,vojom.
x (2)> 5° °(r)> - o
vjmj , 2mz

(6.8)

The angle integral in the nuclear attraction terms can

now be evaluated,

1/2
(anl Z£+1 J
n +

(ylvmu(nl)'Yno(QI)Y2m(Ql)> : [
C(nlz',0mm') C(nlz',000) (6.9)

Substituting this expression into equation (6.8), and
separating the integral into two parts for 2r<R and Zrkk

gives me the following expression

X [ ['12- a—ng— ‘0' +11 1‘1 - é-kérz]

'vtm.m,
J J

88

6.. 6 6 6 6
j)‘ vv' tl' mjmj' mzmx'

+ 2 IJM I j 2 mj m, m ] I 22 E 6

mm' 9 j' 1' m.' mz' m' n=O
3 even

mm'

r1 n+2
C(nli',0m)C(n££',00) [ I Efig—fi:r X3.(R)XV(R)dR
0 r,

Zn+ln
+ ]——_T‘—- x;.(R))(V(R)dR

ZrlR
N ” +n 4n

+ Z 6W. I Z rm <Y,.m.m.)a(1)xu(2)|

n-O m --n
n

_::T_1. Yn'm‘ (9.)Yn*n-(nz)(1 P.2)IY,m(m)a(1)

} }lf fJM ,Vojomj
xu (2)> °(r)> a 0
vj m. J, 8m2

(6.10)
Since I am interested in the vibrational excitation only
and not the rotational , I shall set f8(r)=0 for jfjo

and mjfmjo and replace kVj with kv‘

For convenience, I shall define the following operator

[ 1 d 2d + £(l+l)
__2___

Hvtm,,wnm,v(rx) 3 ' 2 air a?
- 1 k’r’ 6 6 6
2 v vv' 22' mzml'

89

J“ jo 1 mi m, m g
* 2 IQ [ '0 22 2 6mm'

mm' j' 1' m.' mg' m' n-O

J Zr even

‘ Rn+2
C(nlfi',0m)C(n£9.',00) I WT x3.(R)
0 2 T;
21141). n
l
lV(R)dR + I _EETT_—lx;'(R)lv(R)dR
2r:

N m +n 4n

+ z 6vv. z z 731T <Y,.m.(n.)a(1)xu(2)l

U'l n30 mun-“
r n
;-%:T Ynmn(fli)Y;mn(92)(l-Plz)|Y£m(Q,)a(1)
>
Xu”)> ] J (6.11)

I can now write equation (6.10) in the form

A . ik r
v§m£Hv£m£.vo£vmnn(r)' C(Jogj,mjosz) e V

B
lm90V0jom' 8Vj0m°(r) + C(Jon’MO1) 1

Jo Jo

JZHIZITTT J£(er)6Mmj06VVo + C(JolJ,mjom£M)

Cifi(r)C(v-vm)1(jomjozm mo)>

6i,(2-t.)/2 n

=0 (6.12)

To obtain the scattering amplitude and phase shift,

90

I will consider the scattering to be a perturbation.
First, I will construct a zero order wave function, 6°,
consisting of the incoming wave and the bound state
function, and a zeroth order Hamiltonian, (1°. of which
6° is an eigenfunction. Then, I will consider the difference
between2Q, defined in equation (6.11), and7(° to be a
perturbation, write the resulting equation for the first
order wave function, and salve it by means of a Green's
function. This proceedure will not, however, produce the
correct inelastic scattering since the Green's function
will not produce an outgoing wave at the correct energy.
To resolve this difficulty, I will include an arbitrarily
small, well behaved, term, eC(jo£J,mjomlM) eiva/(1+kvr),
c>0, representing an outgoing wave at the correct energy.
Since the bound state function falls off exponentiall)
at large r, the Green's function at infinity will have
the correct form in the limit as 6+0, and the first order
inelastic scattering will be correct.

The complete zeroth order wave function is

o . . .9. .
wvlm£(r) C(JoiJ,MOM) 1 l3n(ZI+I) 38(kvr)6Mm

Jo

VVo

1

10mm”) igo Gil(£-£0)/2

* C(jon,m Cifi(r)

C(v-vm)l(jomjonm mo)

9.

91
ik r

+ eC(j.1J,mjom,M) 13TH (6.13)

Equation (6.12) now becomes

A

. ik r
vém HV£m£.V'£'m£'(r) |C(JolJ.mj°m£M)e V
l

o c o
(f£m£.vojomjoevjomjo(r) Evr+1 ) + wv2m2(r)>

. 0 (6.14)

The zeroth-order Hamiltonian is

 

 

,. 11" '(r) 11° ' (r)
(“62%”) E - %%?rzg? * 5: $3231”) + 332:”)
(6.15)
so that
173,,m£(r)wg,m£(r) -= o . (6.16)
Let

7(v2m£,v'£'m£'(r) - 7(v8m2, v'l'ml ,(r) ;(v2,m2(

(6.17)

92

be assumed to be a perturbation. Then

vim (“MM“) * AHVLM£,V'2'm£'(r) )Whmlh)

+ 11;,m (r) + A w (r) + ... > - o

z vlm,

giving (6.18)
2 7a,”, (rwgmlm - 0 . (6.19)

Emf vim, (r)wv£m£ (r) ' ' v§m£7(vlm£,v'l'm£'(r)

v3£m£(r) , (6.20)

and so forth.
The outgoing wave in the scattering function can

be represented as

. ik r
C(joiJ,mj°m2M) e v f ,vjom. (r)

£m2,v.j.m. Jo

Jo

- vazm£(r) + lzwszm£(r) + ... (6.21)

eikvr

r+1
e to be arbYtrarily small, although not zero.

The term ck

 

is not included since I am considering

To obtain an approximate expression for the first-
order scattering, I will separate the set of coupled

equations represented by equation (6.20). I am assuming

93

vim £(T)Wv2m£m 291§2m2 v Hg m £'(T)Wv£m£(r)

for all vim£,v'i'm£' . (6.22)

Let the potential energy term in the zeroth order Hamil-

tonian be represented by

 

2 W0 (T) W0 (T)
v (r) - 1— ”% + r-:-’-£!‘+—-— (6.23)

“*”z 2 1° (r) (r)

vimz wvim,
then
. 1 d 2d

7(3i1n£( 3 ‘ z ‘8?" a? * V626,”) . ‘6'“)

The Green's function for if“ will satisfy

vim2

' [’ %§?r’g— vvim (’)]G(T|r' ) ' 6(r- r ) . (6.25)

The solution to equation (6.22) can be written in terms

of the Green's function:
“0
I ‘_ 0
13,, (r) (C(rlr')Hvim£,v'i'm£'(r')wvim£(r')dr'
0
(6.26)

I need an expression for G(r|r') so that I can perform

the necessary integration. From (6.25),

(6.27)

I can represent the Dirac delta function by the following

integral

 

8(?-¥') . 1 I eik°(r"') dk . (6.28)
(2")3

Integrating both sides of the equation over angles

. ++,
relative to r-r ,

 

 

 

2 ikIr-r'l

4n6(r-r') - r 13 d: (6.29)

4n2iIr-r'l k

Integrating over angles relative to k,

+00
2 o
6(r-r') = r I kelklr'r'| dk . (6.30)
8n2iIr—r'l

.00

I am interested only in the assymptotic behavior of the
scattering function. I will find directly the Green's
function in the limit as r+~ which will give the assymp-

totic form of the scattering function directly.

8°(rlr') - lim C(rlr') (6.31)

r-ree

95

Consider the limit of Vvim,(r) as r+~:
'krfl

* r2 . e1 v
Vvim,(r) [ T’[Ji(k°r)6vv, * E Evr ]

ik r '

* r[ji(k°r)6vvo * ESE‘¥' )“(jz(k°r)dvvo
V

9' -E——r— (6.32)

eivaJ'l
V

rk .
+ [ - 12"!!- went”.(1+1)u/2)8Wo - e7?- elkvr ]

1kvr -l

[E-:-?cos(kor-(i+l)n/2)6W0 + 62E:?—

 

 

 

(6.33)
-> - 1 r’k’ (6 34)
I v '
I also need the following expression
O»
l d 2d r2 J ikIr-r'l
- r ke dk
213? a? 8nziIr-r'l _m
+m
. rzk2 - Hrik r2 I kcikIr-r'ldk
2 8n2ilr-r'l -m
(6.35)

So that

96

 

 

 

 

 

 

4’” .
w | r2 k elkIT-r'I dk -
G (T T') ' 2 2
3"“ . (“Ti "3"?" 12L‘r'm
(6.36)
- 1 I“ k efl‘lr‘i'I dk
4n2i _m(k:-k2+4ik/r)|r-r'|
ik Iror'l ik r .
a _ e v + - e v e-ikvr' (6.37)
4nlr-r'l 4nr

Note that, while this becomes the usual Green's
function in the limit at large r, there is a difference
at small r which in fact automatically selects the outgoing
Green's function without requiring or allowing an arbitrary
choice from several possible integration contours, leading
to either incoming or outgoing waves.

The assymptotic form of the first-order scattering

O O O m
function can now be given in terms of G (rlr'):

la a - m I A. o
l”vimlh‘) (If (rlr )Hvimzfi'i'mg'(r.)wvimz(r')dr'

AI

ik r a .
e V -ik r'
= I e v 7’Ivim ,v U9 m .(r )wvlmz (r')dr'

 

 

4nr 0 i i
(6.38)
Writing out this integral explicitly:
ik r m .
- k r' i(i+1)
(r) a e V e 1 v I-————— -‘V (r')
wviml 4wr 2 vimg

97

 

1 2 2]
- —k r' 6 ,6 ,6 ,
2 v vv 2% mam,
jo l m. m m w
+ Z IJM 3° i 22 Z 6 ,
v Q 'I t I c c mm
mm 3 2 mj m, m n80
Zr' +2 even
0 v Rn a R
C(nU. ,Om)C(n29. .00) W lV'( ) fi
0 ii
2
2n+lr.n . §
gv(R)dR + W xv,(R)XV(R) dR , f
2r' ’
,
N on +n .;
a r
+ 2 6vv' Z W (YP'm'(Q')a(1) ,
usl n80 mn--n 2n+l ' L,
n
r

(92)(1‘P12)|Y£m(nl)

< a
r) n

a(1)xu(2)> I J [ C(j.2J,M0M)i‘/TFTIITT)

j£(kor')6Mm. GvVo + C(jonJ,mjom,M) E

J0

61.(1‘lo)/2 Cifi(r')c(v'vm)1(jomjozmgmo) dr'

(6.39)

In general, the scattering function would be evaluated
by a numerical integration of the right side of equation
(6.39). As a specific example, I will use the H; bound
state function obtained in Chapter V and find the scattering
functions for the p-wave scattering with vo-O and v80,l,2.

For H}, 20-1 and mo-O. I will choose the following more

98

or less arbitrary values for as yet undetermined quantum

numbers:
v' - v. - 0
j' ' i0 3 0
m.' - m. . 0

J J0
J I 2' I i I 1

and M - ml' - m2 - 0 . (6.40)
I find that
10 0 1 0 0 m
In - O , mfm' . (6.41)
0 l 0 0 m'

Equation (6.39) now becomes

ik r a . .
¢v10(r) = e v A e 1kvr I (l-Vv10(r') ~1/2 kSr")6vo

 

4nr

10 o 1 o o m
+ i In 2 X C(nll,0m)
m o 1 o o m n-0,2

Zr' n+2

C(nll,00) I I 3%:TETT X:(R)xv(R)dR

” Zn+1r .n
+ J '—_'T—-‘X *(R)KM(R)dR
2’2 RD

m +n 4n
<Y1m(9;)a(l)xu(2)
n-O mn--n 2n+1

+ 6v0

 

n

I‘
Ifi-q Ynmn(n.)rgmn(m)(1-Pn) lvlmmnul)
>

xu(2)> ] ] C(Oll,00) [i/T2?jl(kor')6v0

99
+ C(v-vm) Cofo(r') 1(00100) ] dr' (6.42)

Numerically evaluating the following eXpressionsqaccording

to equations (2.67) and (2.69( for 13” and (4.49) and

(4.50) for 1(00100):

10 0 l 0 0 m '

IQ - l/6w m-O,tl (6.43)
0 1 O 0 m

1(00100) a .1424 (6.44)

Evaluating the various Clebsch-Gordon coefficients,l

C(Oll,00) . 1
C(le,00) - - /77§

C(Oll,0m)C(Oll,00) . 1 m=0,11
C(le,0m)C(le,00) - -1/5 m-tl
C(le,0m)C(le,00) = 2/5 m-o (6.45)

I can evaluate the integrals over R by substituting
harmonic oscillator functions for va(R), producing
integrals expressible as incomplete gamma functions.
The only m dependance in the nuclear attaction terms
is in the Clebsch-Gordon coefficients. Summing them

over m gives

100

+1
X 1 C(011,0m)C(011,00) . 3
m.-

2 1 C(le,0m)C(le,0m) - o (6.46)
ml-

Thus only the n=0 term is non-zero and the eXpression

becomes
2!" m
lln [ET-I R21;(R)XV(R)dR + 2 IRX:(R)KV(R)dR ]
r 0 2r
5 Nuclear Attraction (6.47)

Making the substitution of harmonic oscillator functions:

 

1/2 1/2
szgmmwk) - [“57" [fr-37"

V .!

H,(MR-R«))H,.(¢H(R-Ro)) e'B‘R'MV #5.
(6.48)

R. is the equilibrium internuclear distance, 1.40 Bohr.
The constant B-liE/h where m is the reduced mass for
the nuclear motion and k is the force constant for the
vibrational motion. In atomic units, 8-18.38512. The

nuclear attraction term is now

101

 

1/2 “(2r"R°)
Nuclear Attraction - l ’ 5 1 [ l7! Hv(Y)
‘II 2 V! 1' 'fERo
.y2 m 1 -y
e dy + ZJE —————— H (y) e dy
v’ER * "
fE(Zr'-Ro) ° "
(6.49)

where y-/F(R-Ro).
The first integral can be written in terms of incomplete
gamma functions since

b
Jy“ e‘dey - 11, ( r(£‘.§l,a ) - r(fl§l,b )) (6.50)

a
In the second integral, I must first use the binomial

exnansion on l/lFRo-y :

__1___.__1_[1-_X_._Xi_- ...] (6.51)
/FR0*Y JERo /ER0 3R02

At Y='/ER6 this sum diverges, representing a singularity

in the original expression. This singularity, however,

is non-physical, deriving from the use of a symmetric
potential. If I keep the first eight terms in the expansion,
the error is negligible between R-Ro/Z and R-SRo/Z, which

is the region where the vibrational functions have
Significant amplitude, and the resulting expression

is well behaved at y--/FRo .

102

The resulting expressions for the nuclear attraction
term are different for 2f<Ro andibd>Ro since the incomplete

gamma function r(a,x) is only defined for x30.

To be able to write the scattering integrals, I
need the value of vm and the corresponding value of C.
I have chosen vm to be 3 since no appreciable excitation
of the third vibrational level is found at the resonance
peak for v-l excitation.’ I now refer to equation (5.40)

and find that Call/T4 .

The inelastic scattering integrals can now be Written

 

 

as
w ik)r m ..
W110(r) ' e I e 1k,r' (Nuclear Attraction)
4ur 0
2x,1424 ( .3808 r,.682 6-.607r'
(I?
. 5.2957 r'1°352 8-2.752r' ) dr' (6.52)
and
m ikzr m .- '
$210(r) a e e 1k2r (Nuclear Attraction)
4nr

.1424 .3808 r..682 e-.607r'

+ 5.2957 r.1.352 e’2‘752" ) dr‘ (6.53)

103

For the elastic scattering expression 1 must also

evaluate the coulombic and exchange integral.

2 an +n

I e g 6% 2 z 4"

u-l n-O mn--n 2n+l

 

<Y1mcn.)a(1)xu(2)l

n
1‘

;_%:I Ynmn(n,)y3mn(nz)(1-P.z)IY1m(n.)a(1)xu(2)>
>

(zilzij,(kr) + iialili Cof0(r))
(6.54)
where
x‘(2) - 41(r.)Y00(a.)a(2)
x2(2) - ¢l(r2)Y00(92)3(2) (6.55)

Separating the coulombic and exchange parts and summing

over molecular orbitals;

a: +11
2 4n
2

I :-

 

<Y (Q)|Y (Q)Y (9)>
m 66 n-O mn--n Zn+l 1m nmn 1m

11
r

<
(Y00(fl)Ynmn(Q)7Y00(Q)><¢)(r2) ;—fi:r
>

l¢l(rz)>¢°(rx)

104

 

1 co +11 4“
- X— Z 2 <Y (um um (01>
m '0 mn8-n 2n+1 1m mun 00

T
<Y00(n)rnmn(n)|y1m(n)><¢‘(r.)|;—%:T

>

lw°(rz)>9,(r.) (6.56)

Performing the summations and angle integrals indicated

above,

1 1
I - F<¢l(rz)I?:I¢!(T2)>W°(T1)

r<

 

' -l <¢1(r2)|

2
6n r>

|W°(T2)>¢!(r1) (6.57)

The integral for the elastic scattering can now be

 

written:
m ikor m -- . ,2
W010(T) - e I e lkor [ 1 ° (£—-¢°(T')"
4nr 0 2

+ rw°(r')')/w°(r') - l kzr'2 + Nuclear
2
Attraction + %‘¢1(r2)7?§|¢1(r2)>
r 9 (r')
- —l<¢ (rz)|-—i|w°(rz)>—l———— ) v°(r')
6w ‘ r)2 w°(r')

(6.58)

105

The coulombic integral and the part of the exchange
integral coming from the bound state function can be
evaluated using incomplete gamma functions, but the
part of the exchange integral involving the Bessel
function was integrated numerically.

The numerical integration of the scattering integrals
was carried out on the CDC 3600 computer at the Michigan
State University Computer Center. The programs and
original output are reproduced in Appendix D.

The value of the scattering integral is a complex
number which I will call A. The asymptotic form of

6‘ is therefor:
w” - Ak/4n eikr /kr (6.59)

The asymptotic form for the complete function is thus

e-i(kr+n/2) e+i(kr-n/2)

 

 

 

 

 

o 1 ,
Wr+m wr+m /; kr . kr 6v0
eikr
* Ak/4n (6.60)
kr
e-i(kr+n/2) e+i(kr-n/2+6)
I f1? 6V0 + 8&- (6.60)

kr kr

It follows that the scattering amplitude, a”, and

the phase shift, 6, are given by

106

Aki z

(6.61)
V0 46/?

 

 

 

and

 

6 - ~i ln{6 + Aki

v0 4 l?) + i 1n(a) . (6.62)
n

The amplitudes and phase shifts have been calculated
for fifteen energies from k-.20 to k-.90 (in atomic units),
that is,from .57 to 12.0 e.v. The results of these
calculations are presented in graph form in Figures
1 and 2, following.

The results show a definite but very broad resonance
in the elastic scattering. They also indicate a degree
of coupling of vibrational states, in spite of the
making of a series of rather drastic approximations.
I can state, based on my results, that there is vibrational
excitation of hydrogen at low energies due to the
breakdown of the Born-Oppenheimer separation between
the nuclear and electronic motions.

There is ample opportunity to make higher quality
calculations using the formalism which I have deve10ped
here. Whether such calculations will produce better
results is a question that cannot be answered until
such calculations are done. I believe that the method

is capable of producing good results.

107

 

.10

 

 

 

flu ‘.IO 1;

.n

1.. ';4.—‘"‘.1~o‘".u"‘..

" 5:" ' 7

 

 

 

 

 

 

 

emmm m

Ufll+n_L(..( "1.7.01.0.

 

k ('+l~;¢ ““1113

SCATTERING AMPLITUDE AS A FUNCTION OF ENERGY

FIGURE 1:

 

 

 

 

 

 

 

 

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0
00
0
0
Hh—t—o-o-—-b~~'-
0
00
: '1 , "
4 .g., ‘-
u—od-O—c—r-n 7
0
'0
If
‘r
0

 

 

 

 

 

i

0
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0

.

r.

0

l

0

0

0

I

0 . ‘ . ‘ .

-aasiuiflus_-s
...’;; .

0

I

i

0

0

I

 

 

 

 

 

 

 

108

 

 

 

 

 

 

 

 

 

 

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+0000 II 0.0 0 f 000 .0 000.... .0k'00 fi0 0.7 0 l 00 .- 0 II . . 04 0| 070 0 0 100 0 T0 0 . 00000.L
.
0 0000 Y 0 00. I 0 000000 . .0000 0 0 A 00 00 W I .0 0| 0. 0 .f 0 A . A.I g 0 0000. (0-00? .. 4.0000 0 ..0 0 It
. .
0000 #0 o 00 1 00L 00.06-.. .0100! 7. 0. 0. 000.0 0. 0 0| 0 .000 t 0 .0 - 0 .04 0.00 00.010000 01 0|I00J0-000000 I .- #00 000: I0 000001
I 0 0 O 0 0 0.00000 +0.00 1 0 l 0 00. 00 I l * 000 I0- 0 I; . . 00 -0 V 10.0 0.001. 0- .000 ..4 000000.01. 0 0.00 00+ 7100.. 0000 WI .0-00001L
. ,

 

 

 

al.0-0 0.0.-. .0. .0-J-tl-IPILI: 0 0000. . 06.0.00.0.0H.-.0000. .|.0|v0|00 I fi.0000|. +0700 02—.00 L00 .(Y00I0II..LYIu-.00lv0¢- 0... 0 .....01 0.00.0
0000* 00.00 I 00 A -

 

 

 

..‘IL‘

.700
uu|0s)

.5‘7'

((100 M‘oc

k

7“

PHASE SHIFT AS A FUNCTION OF ENERGY

FIGURE 2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

: I . I «00.00 I00 0000 000II.I. 0- .0000 000000 00.00 0000' ..0 00000101010000th . . 0.0.0 0 000.00 4* 000-0. 0-00I0.-.Il
0 0.01.40-.. . 0|0-lI-0l. LL10 - 0 0000A . 00004 .001 .. H I u -0 ~00 .00 .0 , .9070 0.? r b p > 0 w ..-. . .0 _ 10.0 -0 -0 000 00
040000. .6000 fili n _ .1 . . . o . c ... .. h v0.4 ,6. . , . .. L 01 VIL+| . . I

. e M l p . .

 

 

\H...€.m. vhUVnK

 

.7‘”

109
‘ Rbse,gn.cit.,p.39.

2 G.Herzberg,Molecular S ectra and Molecular Structure,
pp. 98 E 532, VanNostrand, New Ybrk,TT95[0.

’ Schulz,gE.cit.

BIBLIOGRAPHY

BIBLIOGRAPHY

A.S.Davydov,0uantum Mechanics,Addison-Wesley,Reading,
Massachuse s, 9

G.Herzberg, Molecular S ectra and Molecular Structure,
VanNostrand ,New‘YorEETTyfif).

 

A. Herzenberg and F. Mandl, Proc. Roy. Soc. (London)AZ70,
48 ,(1962).

H.W.Joy and R.G.Parr,J.Chem.Phys.£§,448,(1958).

 

M. E. Rose ,Elementary Theor of Angular Momentum, John
Wiley E’Sons, New YorE,El§37).

C.J.Schulz,Phys.Rev.125,229,(1962).
G.J.Schulz,Phys.Rev.135,A988,(1964).

General References

M. Abramowitz and I. A. Stegun, Handbook of Mathematical
Functions ,Dover,New York ,IIUESI.

 

A. M. Arthurs and A. Dalgarno, Proc. Roy. Soc. (London)AZS6,
S40 ,(1960).

M.J.W.Boness and J.B.Hasted,Phys.Letters £l,526,(1966).
T.R.Carson,Proc.Phys.Soc.(London)A67,909,(1954).
J.T.Dowell and T.E.Sharp,Phys.Rev.167,124,(1968).

I. Bliezer H. S .Taylor, and J. K. Williams J. Chem. Phys. 47,
2165 ,(1967).

H. Evring, J. Walter, and G.E. Kimball,Quantum Chemistr ,
John Wiley 6 Sons ,New York,(1944).

 

D.E.Golden,Phys.Rev.Letters l7,847,(1966).

H. G. M. Heideman, C. E. Kuyatt, and G. E. Chamberlain ,J. Chem.
Phys. 44,355 ,(1966).

110

111

M.Krauss and F.H.Mies,Phys.Rev.A l,1592,(1970).

L.D.Landau and E.M.Lifshitz,guantum Mechanics Non—
Relativistic Theor ,Addison- es ey,Réading,Hassa«
chusetts,(1965).

 

 

H.S.W.Massey and R.O.Rid1ey,Proc.Phys.Soc.(London)A69,
659,(1956).

A.Me$siah,anntum Mechanics,John Wiley 6 Sons,New York,
(1964).

 

S.Nagahara,J.Phys.Soc.(Japan)§,165,(1953).

S.Nagahara,J.Phys.Soc.(Japan)2,52,(1954).

Yu.D.Oksyuk,Soviet Physics JETP £1,873,(1966).

K.0midvar,"Theory of the 25 and 2p Excitation of the
Hydrogen Atom Induced by Electron Impact" (NASA TN
D-Zl4S),0ffice of Technical Services,Department of
Commerce,Washington,D.C.,(1964).

R.G.Parr and H.W.Joy,J.Chem.Phys.£§,424,(1957).

G.J.Schulz,Phys.Rev.116,1141,(1959).

H.S.Taylor,G.V.Nazaroff, and A.Golebiewski,J.Chem.Phys.i§,
2872,(1966).

H.S.Taylor and J.K.Wi11iams,J.Chem.Phys.i§,4063,(1965).

APPENDICES

APPENDIX A

Determinant and Minors of the Matrix of Overlap Integrals

I. Expansion of determinants.
Let u be an NxN matrix. The (Nth order) determinant

of this matrix is designated by "U“ . I can write "0|

lSt order determinants by expanding along

th

in terms of N-
a row or column of the matrix. Expanding along the i

row will give

N
IUII - X 0111‘” u jffinull . (M)
3'1
“U";j is a first order unsigned minor of "U", and is the
(N-lSt order) determinant of the matrix obtained by

th th column from U. The minor

deleting the i row and j
with the sign term included is the signed minor,flUfllJ.
Minors are henceforth assumed to be signed minors unless

stated otherwise.

IIUIIij - (-1L)1°”5IIUII,’},3i (M)
N .

IIUII = .21 ":5 nunILj . (M)
J‘

The first order minor can itself be expanded:
112

113

. + j . .
nunlj - on1 3 [ 2 (~1>““Ukrl lulh’lfi‘

2‘1
Nvl .
k+2 kt
. 225 (-1) Uk..+1I uUlfijflM

if k<i or,

. . j-l .
nun“j - (-1)1*5 [ 2§1 c-1)**‘uk.1,.l unlfillh‘

N-l

k+£ ' k2
+ 2;). c-n 0......1I mm. ]

if k>i. (A.4)

These eXpressions can be simplified by substituting the
second order unsigned minors for the first order minors

of the first order minors. The second order minors are
defined analogously to the first order minors but with

two rows and two columns removed. The distinction is

that k and L in equation (A.4), where used as superscripts
on the minors, refer to the row and column numbers in the
first order minors, not the original determinant.

‘ .. . J . .
Ilulll-1 = (~1)i*’ [ i (-1)k"uk.IUI‘k"”

l-l MM
N . .
k+£-1 1k )2
+ Z t-l) U IUH '

if i>k or

114

k-1+£ ik,j2

. . - . '-1
1 J a . 14‘] J -
HUM ' ( 1) £21 ( 1) ukznuuMM

k-1+£-1 ik,j£

+ ('1) URLHUHMM )

if i<k. (A.5)

In equation (A.S) all indices refer to the row and column
numbers in the original matrix. I can simplify this
equation further by using P( ) operators, where
P("statement") - +1 if "statement" is true and -1 if

"statement" is false:
1 j 1+5 N k+£ . . ik jl
nun - = (-l) 1;. (-1) P(1>k)P(J>2) URZHUHMM'
J
(A.6)

But the signed second order minor is

"Unik.jn . (,1)i+j+k+2 p(i>k)P(j>£) “U|;;,jt

50
i i N ik jt
"UH ’~ - 2;. Uklflun ' . (A.7)
J

and

115

N N ik 52
”U“ = ,2 1;, u..uk2uun ' (A.e)
J

J=1 13

II. The determinant of the matrix of overlap integrals.

The matrix of overlap integrals is

 

 

F‘ T
1 0 0 . . . n1,N+l
0 1 0 O O O O
D g 0 0 1 o e o DS’N+1
D 0 D . . .
L N+1,1 N+1,3 0N”.N+1 j
(A.9)

Expanding along row 1,

N+l

l l 1
”DH = "D" ' + 01.N+1HDH ' (A.10)

1,N+l

EXpanding "DH along column 1,

l N+l 1 N+1°N+l 1 1 N+l;l N+1
"on ' = nNe1.1unu ' ' ' = —DN+1.1uvu - .

116

but ”Dnl'N+1;1’N+1 = 1 so

1 1
IIDI = llnll ’ ° D I

N+1,1’1,N+1

Expanding IIDIII’1 along row 3

1 1 1 3;1 3 1 3;1 N+1
IIDII ' - IIDI ' ' + DLNHIIDII ' '
1 3;1 N+1 l 3 N+l-1 N+1 3
and ”D" n D . DN*1,3"D" e o o s O . -D
' 3
so unu -nnn1'3'1' - 0

Continuing this process, I eventually obtain

, _ N-1
”D” . "D|l1.3.oeN’1,l.3.oeN 1 z D
181
odd

where

leeeN°1.leeeN°1
IIDII ’

 

(A.11)

(A.12)

N+1,3

N+1,1“1,N+1 ' DN+1,SDS,N+1

(A.13)

N+131D1,N+1

(A.14)

 

,N+l

117

- DN+1’N+1 (A.15)

and

N

“D“ ' DN+1,N+1 ‘ E DN+1,i Di,N+1 (A'lb)

The terms in the above summation with i even are all
zero, so that the restriction can be dropped without

affecting the result.

III. The first order minors.

The first order minors can be divided into three
classes. In the first, both indices are less than N01;
in the second,one is N+1; and in the 1ast,both are N+l.

Class 1: unli'j i,j<N+l
If i-j then “D"i'j - flDHi’i. To evaluate "Dui’i
compare it to "Dfl and note that performing a series of
expansions analogous to those in section II produces

the same result except for the term 'DN+l,i Di,N+l

which is missing.

N

i i
“D" ' ' Dn+1,N+1 ' jgi DN+l,i Di,N+l

(A.17)

Note that if i is even the missing term is zero and

then ”uni-i - "on.

118

th

If ifj deleting the i row will leave D as
o N*1,i

th

the only non-zero element in the 1 column, and

expanding IIDIII’j gives

HDni'j - D HDu1.N+1:i.1 (A.18)

N+l,i

Deleting the jth column likewise leaves Dj N+l as the
D

h

only non-zero element in the jt row, so that

1 ° i n+1 '-' i n+1
"Ml ’3 ' DN+1.i-D5.N+1 '93 ' ’J’J' ' (A.19)
Now HDIi-”*1tj35ti»N+1 . -"D.i.j.N+1;j.i.N+1
I +aniDjDN+1;ipj.N*1 a 1 SO
i j _
"D" ' DN+l,i Dj,N+1 - (A.20)

Note that if either i or j is even llDIl’J - 0.

Class 2: IDlli'm1 and IDINH’i

th

Deleting the 1 row or column leaves D . or
N+1,1

Di N+1 , respectively, as the only non-zero element
C

th

in the i column or row, so that

i N+1 i Nsl°N+l i
IDII ' - am; an" - ' .

119

and HDIN+1,i _ Di.N*1"D”N+l,i;i,N+l . (A 21)
Now "D"N+l,i;i,N+l . "Dni,N+1;N+l,i . _"D"i,N+l;i,N+l
- —1 so that

HDHi’N+1 ' ’DN+1'1
and unuN*1-i - 'Di,N+l . (A.22)
Note that if i is even IIDIIi'INH1 - HDIINH’i - 0.

Class 3: ”DnN*1'N’1

By inspection

HDHN‘1'N*1 - 1 (A.23)

Section IV: The second-order minors.

The second-order minors can be grouped according to
the number of distinct numbers represented by the four
indices. It is meaningless to write a second-order
minor with both row indices the same or with both column
indices the same, since the same row or column cannot
be deleted twice. With this restriction, the four indices

may represent two, three, or four distinct numbers.

120

For the second-order minors, flD|"°'kz

, I will need
to consider only those minors for which n>v and k>£.
In the evaluation of these I will sometimes need to
use minors not obeying this convention, but will always
end up with expressions using only minors which do
obey it.
A. Suppose that the four indices represent three distinct
numbers.
Case I: all indices are less than N+l.

nun“k2 - "qu”z

unn"“'k" .

IDIIW'“2 . or

IIDIIW’kv (A.Z4)
To evaluate "DI”V'ul, I expand it along the 2th row.

h

The only non-zero element in the it row is D, N+1 so
. D

IIDIIW'” - D I“"‘“‘"‘”“1 (A.25)

2,N+1"D'

th

Next, eXpand the third order minor along the v column,

which has DN+l,v as its only non-zero element:

121

”Unuv2,nzN+1 _ D IDlnv£N+1,u£N+lv (A.26)

N+l,v

and "DI”va’l'"”N’1” - 1 (A.27)

so that

HDI"“'”‘ - n (A.28)

N+l,v D£,N+l

To evaluate "Dfluv'ku, expand successively along the

kth row and vth column giving:
k vk k N+l
"nu”“- ” - Dk.~+1flDl“ - "
uka+l kuN+1v
' Dk,N+1 DN+1,v"D" ' (A°29)
I - D (A030)

N+l,v Dk,N+l

SimilarlY. f0r "Dfluv’vzz

uv vi a uv2N+1 v2N+lp
non ' m.’“.l DN.1.uunu -
a - DN+1,u D£,N+l (A.3l)
And for "nfluv,kv;
nv kv _ uka+l ka+lu
" D" . Dk ’N+IDN+1 ' u" D" .

122

. DN+l,u Dk,N+l (A.32)
Case II: one index is N+l.
”DHuV'ki- “D"N+lv,kv .
"D"N+lv,v£ .
”Duuv,N+1V . or
IlDIIW’N+1u . (A.33)
To evaluate flDflN+1V'kV, expand along the kth row, which

has Dk,N+1 as its only non-zero element:

N+lv kv N+lvk ka+l
"DH ' - Dk,N+1"D| ' (A.34)

but ||D|N+1Vk'RVN+1 - - ||D||N+IVk'N+1vk - -l,since ka+l is

an odd permutation of N+lvk, and

N+lv kv
HDH . - -Dk,~+1 . (A.35)
The other three cases can be evaluated similarly:
N+lv v2 N+lv£ V£N+l
”D” . . D2,N+l"u| . . Dt,N+1 (A.36)

123

IlDHuv,N+lv - D lqu+1,N+1vu

. -D (A.37)

N+l,u“DI N+l,u

"DHHV,N+IU . D IDHUVN*I,N*IHV

I D (A.38)

N+l,vl N+l,v

Case III: two indices are N+l.

N+1v.N+12(V,,).

The N+l,N+l minor, however, is the determinant of the

The only example of case III is "D"
NXN unit matrix so

unuN*1”'N*1£ - 5 - o (A.39)
v,2

B. Suppose the four indices represent only two distinct
numbers.

Case I: qu+l.
IIDIIW'k2 = IIDII‘N'”v (A.40)

u u a _ -
Now "D“ a DN+1,N+1 1;” DN+l,i Di,N+l . Deleting

the vth row and column from this minor removes the
DN+l,v Dv,N+l term leaving
“V “V a -
”D' ' DN+l,N+l 1;” v DN+l,i Di,N+l (A'41)
D
Case 11: u-N+1.
Then
”Duuv,k£ g "D”N*1V,N+1V ‘ 6 ‘ 1 (A.42)

v,v

124

C. The remaining case which must be considered is that

uv,kt may all be different.

th th rows from the matrix leaves

th

Deleting the u and v

DN+1 u as the only non-zero element in the u column
0

(which is not deleted) and DN+1 v as the only non-zero
0

th column (which also is not deleted).

th

element in the v

EXpanding along the u column,

kl vN+l kl
HDH”“' ' ”n.1,aflDl” ' ” . (A.43)
The minor "DH”VN‘I’kL“ still contains the vth column,

but the N+lSt row has been deleted, including DN+1 v .
D

and the vth column now has no non-zero elements.
Thus
and Inuuvtk“ - o (u,v,k,£ all different) (A.44)

The results of the above derivations are compiled

in Table I and Table II on pages 23 and 24.

APPENDIX B

A One Determinant Approximation

to a Two Determinant Function

The two determinant function I wish to approximate
is

w(N+1.fi) = § ( V”(N)xS (1)
Vlllj

vnjomjgwjmj

+C

vjmj YP(N)xgjmj(l)) Xv(R)ijj(eR'¢R)’

(8.1)
The undistorted ground state function is written
v“(N) = 64xi(1)X2(2)... xN(N) (3.2)

where the Xi are the various normalized Hartree-Fock

orbitals. The distorted function can be written

WP(N) = e4x;(1)x;(2)... x((~) (8.3)

125

126

where the xi are the distorted orbitals. Let
6xi 5 xi - X, (3.4)
represent the distortion. Then
vpcu) - s4(x,(1)+ax,(1))(x2(2)+6x,(2))...
(xN(N)+6xN(N))

N
a SAT-1' (Xi(i)+5Xi(i)J (805)

1-1

I ¢I¥T ( ) at? 11'
g ‘ . + 6 e . o .
1.1 x1 1 =1 XJ(JJ i“(xlhl

3
J

*Gxi(i)) (3.6)

The first term is the same as VM(N), so that

ij VoJomj°,VJm.

W(N+1,§) - X [ $4W”(N)(xs (1)
j J

N N
+ c (1)) + cvjmj 54321 ijIJ) TI

vjm. xvjm. i j

J J

(xi(i)+6xi(i))x3jmj(1)] xv(n)vjmj(ek.¢n)

(B.7)

127

Let V‘(N) 2 $4M II (x (i)+eax (1)) (8.8)
i-l

where 05:51, and N: is the necessary renormalization

constant. Then v1(N) - YP(N) and YO(N) - v”(N).

If I write

oVij(l)

c e S
x (mm!) ‘5'") e4? (mewmj.

. Cvjmjx 35m (1))XV(R)ijj(eR’¢R) (3.9)
the error which I have introduced is the difference
between the original two-determinant function and

the new one-determinant function. This difference is

V(N+l,§) - v‘(N+1,§) - {[54 I (l-N )v” (N)
Vij

N
- eNc jgl 6x (j) TTj (xi(i)+€6xi(i)) ]

s
(XVojomjo.vjm.(1) + cvjm. xvjmj (1)) + c jm

J J J

N N
. . . B
.ai jg] 5xj(3) {$3 (xi(1)+5xi(1))x,,mj(1)

] ;,(R)Y m (6R.¢R) (8.10)
J

This can also be written:

128

‘1’(N+l,§) - wemuj) - Z I .94[ (l-NeHMm)

vjmj
N N . .
' EN ejzl 5Xj(j) {I1 (xi(1)+c6xi(1)) ]
S
ngjgmj.'vjmj(1) ’0 CVjfllj fl[ (1- N a)?” (N) +
N N . 2
(1.6515) jg]. 5Xj(J) gj Xi(1) ‘* (l-E NE)
1 N ()6 (k) 11" ( ]
6 . ° . + o e a
5-1 ij x, j xk ifj,k x, 1)
ngij) ] KV(R)ijJ-(6R'¢R) (Boll)

For an optimal one-determinant approximation epsilon
could be included as a variational parameter. This,
however, still requires obtaining the distortions éxi
and once these are obtained we might as well use the
two-determinant function, since finding the dxi is the
major part of that problem. Thus, I will set 5-0 and

obtain

“m.m - V°(N+1.§) - Z J24)?” (N)(xs

vaj

(1)

vojgmjo, ,vjmj

* Cvjmjx vjmj (1))XM(R)ijj (eRu¢RJ (8.12)

(1))

vjm. xvjm.

F4?"(N) g S . (1) + c
3' J'

v mj (XVUJOmjooVij

XV(R)ij.(eR'¢R) (8-13)
3

APPENDIX C

Evaluation of PJMJI V

I wish to obtain

WJMjl(N*l,N) - PJ"3” v"(N) (1)

S
v§vmj (XVOJOmj.pvJ.mj
+ cvx5(l))xv(k)yj'mj(9R'¢R)

_ Jth M s
p v (N) v§'m (Xv.j.mj..vj'm.(1)

j J

+ Cexg(1))X»(R)thnj(°a»¢nl (6.1)

where

C £J M 2 Y 6 Y 8 .¢ >
m.m, I (5 .mjm, )l l lm.‘ .t) ,mjc R R)

(Y2m£(e'¢)yjmj(eR'¢R)l o (C~2)

To do so I must evaluate the integral

F...

- M
I : <Y,m(e.¢)ijj(eR.¢R)lv (N) vj
J

129

130

(xgojomjo.v5.mj'(1) + cvx§(1))r,(n)rj.mj'(9R.¢R)

M co
a (Y (6, Y. (6 .¢ ) Y (N
2m, ¢) jmj R R I ) vjzmj. [ 12:0

(1"m5,.(k.r)v,.,(e.¢)amcj

'jo
ik r *g'
6 + " f . r)
mjimjo e v) "(NZ-.20 fi'm',vojomjo,\)j'm3(

Y,.m.(e.¢))a(1) + Cvx3(1) ]

XV(R)Yj'm (6R'¢R) . (C03)

j.

Most parts of this expression may be evaluated immediately,

giving

voajjoém.m.

I . vM(N) 2 [ ( ifxrrrzrrrr 51minsv J J
0

V

ik r
6 + e vj f . (r)]a(l)
mRO £m,,vojomj°,vjmj

B
+ .,Z (Ylm (e'¢)ij.(eR'¢R)'Cvxv(l)
J mj' l J

Yj'mj,(eR'¢R)’ JKV(R) (C04)

To evaluate the remaining integral, I expand xg(l)
in Spherical harmonics starting with Y where 2.

2(HM
and m, are determined by the symmetry of the orbital.

131
An arbitrary function, f(r,6,¢), with the same angular
symmetry as the spherical harmonic Y, m(6, ¢J can be

expanded as

f(r.e.¢) - {0 f (1’12. .2, m(e.¢) (c.5)
1.0

I arbitrarily take m.;0. Since I am summing over m,

states of the scattering electron this will not affect

the results.

Cvxg(1) . g cvifvi(r)r,°,21.m.(5,5)a(1) (C.6)

The tilda on 6 and e indicates angles relative to the
molecular axis. I now have to write these functions
in terms of functions of the angles of the external

coordinate system.

” ” 2 +2i ,
y,°,21.mo(e.¢) - 2' ”min. (-¢R.eR,eR)r,o,Zi.m.(e,¢)
(c.7)

The last term of equation (C.4) can now be rewritten

as

2 I Cvifv1(r)a(1)<y9m(e'¢)yjmj(6R'¢R)l

jOmj' im'

D1°;21( ¢R.6R.¢R)Y,, .21 m.(e.¢)vj. .m..(eR.eR)>
J

132

' 1 Ceif v1(r)a(1)a

ij'mj‘m' fl? 6‘ ‘°’Zi
(ijj(eR'¢R)inapo('ek’¢R’GR)Yj'mj'(6R'¢R)>
(C.8)
' mjij. E 6i.(z-z.)/2 Cvi 1vi(r)5(1) I £%%1 )1/2
[ Z%%:1 )1/2 [Djjo(¢n'° R’O)Damo(’¢R’GR'¢R)
Di;.;(¢R,6R,0)d¢RsineRd6R . (c.9)
Since Dg0(a,e,0) - D;o(o,8,y) (c.10)
and 93.;(a,e,y) a D;1m(-a,8,ey) , (c.11)

I can rewrite the right side of equation (C.9) as

C f ‘+ - 1/2
r a 2 l (2 '+1

m. 'm'
J

JD;:.O(-¢R1BR.¢R)DWFO ( ¢R'e R’¢R)DJ. o(‘ ¢Rse R’¢R)
J J

d¢Rsin6RdeR (C.12)

- 1/2
= a ((21+1)(ZJ'+1))
mjij' £6 1a (1 loJ/ZC vi fvi“) (1-J

133

Z C(j'lj1,m.'mzm.'+m)C(j'£j1,moOmo)
Jl J J

O
JDj;'+ .m.( ¢R’9R'¢R)ngo(’¢Ro9R.¢R)d¢RsineRdeR ,2

'1

(C.13)

This integral can be written,
3 2w

- J: J 'imo¢
JdGRSIDBR d o(6R) ldok e R
O

mj”'1»"‘o(eR)dmj

ei(mj'+m2¢R e-imjoR
" j
I 2fl6( Mg+qp(m +m0 ) ldBR Sinekd m h.*g'm (6R)

dgjo(°R) - (c.14)

Now, dj’,
mj +qcmo

(9R)d;.0(ok) ‘
J
{(51+mo)!(ji'mOJ!(j1*mj'+QJ!(jx'mj'QJ!(j*ij1

(5°mj)!j!j!11/2 z. (_1)k1+k2
k1k¢
{(j‘.m5..qfk’)!(j’*m°’li!(k1+mj'+quo)!ki!

(j-mj-kz)!(j-kz)!(kz+mj)gk2!}‘l [sin(eR/2)]Zkl*2k=

134

[cos(6R/2))Zj 1+mo-mj' -ni-2k1+2j ~mg+mj 0+m‘i2k1 3
(Cols)

So that

w

- j 5 .

laen SlneR am;.,q2m.(ek)dmjocea)
{(5!)2(jt*mo)l(jn’mo)!(51*mj*mo)l(jn'mj'mo)!

- 1/2 _ k +k
(j+mj)!(j mj)'} k: ( 1) ‘ 2

I 2

{(jl'mj'mo'k1)l(j1*mo'k1)l(k1*m;)!k1!

(5-mj-kz)!(5-kz)!(kz+mj)!kz!}’1

H
de sine '(cose ,2)2(Jl+J-kl'k2)(sine /2)2(kl*k2)
R R R R
O
(C.16)
Integrating by parts
2(k1*k2)

u
ldeRsineR(coseR/2)2(J‘+J'k"k’)(sinBR/Z)

.(j:+j-kn-k2)!(kx+kz+2)!
I Z (CO17)
(jn+j*2)!(kx+kz+1)

 

135

Equation (c.16) becomes
I
[den ’inekdg;'+qtm.(°n)d%jo(°n) ' {(5:*mo)!
(5"m°)!(51*mj*m0)!(jn‘mj°mo)!(5*mj)l(j-mj)!}1/2

j! Z (~1)k“k=2(j,+j-k.-k.)z(k,+k,+z)s
k}k1

{(jx'mj'mo'kx)’(jt‘mo'kl)!(kx*mj)!k11(j'mj'k2)!
(j-kz)!(kz+mj)!kzl(j;+j+2)!(k;+k;+1)}'1
(qus)

I can now replace (C.8) with

g 2 cv‘ifvitrwumn (e.¢)vjmj(ok.¢R)l

j mj' im' "&

Dlo*2i

m.m. ('¢R'6R'¢R)Yz.+21.m'(6’°)Yj'mj'(°R’¢R)>

' jimj. g 51.(2-2.)/2 Cvi‘v1(’)°(1)

. .1/2 1/2
2 1 2 ' 1 0 ° 0 O
I j‘ ) gj * ) 2‘ C(j LJ;,mj mzmj +qg

C(j.£jlgm00m0)6(m.c.m)(m.+m°) {(jx‘mo)!
J 2 J

136

(jg-mg)!(j;+mj*mo)!(jn-mj-mo)l(j+mj)!

(j'”5)'}1/2 51/(j‘+j+2)' RX 2(-1)"“k2

(5t’5'kn'kz)!(kn‘kz*2)! {(jl'mj‘mo'kx)!
(j;+m.+k1)!(k;¢mj)!k;!(j-mj-kz)!(j.k;)!

(k;+mj)!kz!(kx+kz+l)}’l (c.19)
E E Gi,(£-£o)/Z Cvifvi(r)a(1)l(jmj£m£mo) (C.20)

So that
<ng (o.¢)Y.m (9R.¢R)Iw”(N) 1 (x3 . m v..m (1)
2 35 mi W in” 5
.2.
+ Cvxgflnxvmwj.mj(eR,¢R)> - Wm) 2 [ 1
V1fllZI+Is j£(k°r)6mjmj06m£06vv06jjo

+ ik .r f . . . r + 6.
e VJ £m£.VOJOmJ-ogv3mj ( ) E 19(£'£O)/2

.(r)I(jmj2m£mo) ] a(1)xv(R) (C.21)

cvifvx

Thus the desired wave function is

137

WJMj£(N+1,§) - 2 C(sz,mjm£M)2Yzm£(9.¢)

mzmj

yjmjceR.¢R)w“(~) é [ i2/TfiTYTTTT j£(k.r)

“k .r
a a 6 5.. +e1v3 f . (r)
mjmj. mgo vvo JJo Em£,VoJoijijj

+ E 6i.(2-z.)/2 Cvifvi(’)l(jmj””t”°) J

(1(1)de

= v”(N) { C(sz,mjsz)2 [ izltiTZITTT'

vmzmj

j£(kor)6 5 + eikvj’

6 6..
mjmjo mzo vv. 3].

.f

f2m£,vojomj.,vjmj(r) + E 61,(2-£.)/2 CV1 vi(r)

1(jmjzm2m0) ] nglwmfljmj(6R.¢R)a(1)xv(R) -

(C.23)
‘ Rose,gg,cit..p.60.

2 ibid.,p.58.
3 ibid.,p.sz.

I.

10
20

40

APPENDIX D

Listing of Computer Programs

The program used to find the H1 wave function was:

FROGRAM ENHAPZ

EN! I 1.23 I Z! I [022

n-I.zs

DO IOOKI|02

ALPHA-IoBQOOoOOIDK

PRINT 3009

DO tOolIIOS

EN?I|.84+!I¢O|

PfltNT 40

DO lOoJcto?

22-J.I°l 0'0?

EIEOEL(FNIOEN?QZIOZPODQALPHA!

PRINT ?OoEoZ?oFN20ALDHA

FORMAT‘30XOF?OQHOGFIK03)

FORMAT‘CI'PI X'ENFQGY‘IJX'Z?‘13X9FN7’.I ‘x‘ALW‘.IOX.RI”BOJ’
FORMAT ( X)

END

FUNCTION EOEL(EN10FN2021OZZORQALPHA)

TYPE REAL INBETA

ENSIEN!+EN2 3 25-21422 I RHOIOZl'D s 9H02-2909 I Al-zoenl
A2-2GEN2 I 82=A2+l s C2-AZ-? s DZIA2+2 3 63-4263 sBl-AIOI
GEMS-GAMthENs-loOo)SGHPIGAMINCBPOOo)185"'N$‘lS'ENS'CEVS-3!.GENS
EONE I 2202?6(EN?+a)/cA?O(A?-l)3

ETUOA I GAMINCAZORH02i/GH2

ETwoa - (l-GAMthn?onH0?I/Gn2)/RH02

ETUOC - 9H02-DH07IGAMthC?09H02)/(pofiocn?)

ETUOO I od'(O?On?-GAMIN‘F309H02)/Gn?)/RHOP'.3

ETHREE I 2'2!‘lNfl€TA(Z?/ZSOD20AI)IFNI

EFOUR I 2.2?IINRETA(Zl/ZSOB!oA?)/EN?
EFIVE-¢?OZI)Iinli¢?!2?)OGH?*GENS'BSOINHFTACofioENS+zoENS-|)ltcnnth
caxoo.).on?n259o(20r~s+1a-1.s:

ETHO I 4‘22.(ETWOA+ETWOB+ETHOC+ETWOD) S CPEP . ETHPFEOEFOUQ

EDEL I FONS OALPHAGCEPEP-ETUO-EFIVE)
END

PE‘L FUNCTION INFETI (XOPOO’

BIO S NIO

PNIp+N I CNIN

lF‘O-NOFOQOo) GO TO ?

Bast-1)IoNIXG.DN/¢0N.GAM:N (O-NOOQ)IGAM1N (CN+1.0.))
8.6+8A s N-N+1 8 s-na':o.~.a/n s IF (Sisocfnlol 102
lNBETA-B'GAMIN ¢P+OoOo)/GAMIN (9.0.)

END

138

II.

IO

00

IS

20

29

139

The program used to find the R} wave

FDOGGAM FNFINO

TYDF DOUflLF 7

COMMON ll/C(?0I)0FN(?00I

COMMON I 2 I FDFLAOFOFLRFALOALPHAQQ
COMMON I J / FINTOFKINETICOM

COMMON I I I loJoMJ

COMMON I FIX I 2(?04)
ENNIloJoKILIIFN(IoJ)+FN(KoLL
IZ¢IOJOK0LIIZIIoJIOZCKoL)

“OIOZS I “-0

[3-0

M330

00 I00 II-IOO

DO IOoJJIIOZ
EN‘JJOIII-C‘JJOIIIOZIJJOIII.O

CONTINUF

ENIIOIIIIOZJ I ZIIOII‘IO?Z D C(IoIIIIo
EN(20IIIIOS7? I 2(POIIIOO“?7
ENC2021I20960 I 2(?02)I206‘5
EN‘ZOJIIIOOOIOQ ZI?OBIIBQ?68
ClzolI-Q7IQI7Q $ C(ZOPIIOAJQZ76 D C‘ZOBII'OOSBZOI
ALPHAIIOBIB

JIZ

JI3

II? IJI!

FFIFACTOPIII

CIIO])-FFIC(IOII I C(IOPIIFFCCII0?I
C‘IOJIIFF0CII031 I C(IOII-FF'CCIoIl
CALL ENERGYJ

D?IEDELA

M2.”

MSIMJ

JIJ-l

IFtJoEOoO) JCS

FP-FACTOP‘II

CIIOIIIFF.C(IOII 8 C¢I0?I-FFOC(I02I
C(IOSIIFFOC(IOJI I C(IoIIIFFICCIoII
CALL ENFQGYJ

OIEDELA

PDINT 500

function was:

ST?
57'

IO
I0

I' ‘ISoLTo?I pnINT IOOOFOFLQFALOFDFLAOALPHAOROI‘CIKOLIOLIIOOIOKIIO IOO

2?).(CENIKOL).L.1enjoK-IQPIQC(ZIK9L30L-1o‘IOK-loz)
ISII
IIIO
NED!

IF (4.50.1) ~E-2

"I."

Hanna

CALL ENERGYJ

IIIO

DIIEOELA

EN(I¢JI I ENC!0J)O.!OINE
ZIIOJIIZ¢I0J3+0100NE
CALL COPY

I1III+I

IF (EDELAoLToDII GO TO I!
CDELAIOI
E~IIOJIIENIIOJI‘OI..NE
Z‘IOJIIZCIOJI‘OI"NE

IF (IIOFOOII GO TO 20

GO TO 20

DIIEDELA
ENIIOJIIENIIOJI-019.NF
ZIIOJIIZIIOJI-OIC'NF
CALL COFT

IF (FOELAOLTQDII GO TO 20
EOFLAIOI

ENIIOJI I EN‘IOJIOOI..NE
zcloqgnz¢lo4)+.:oouc
I200

20
29

20

3O

50

60

I00

200

300
ADO
500

I0

140

Dl-EOELA
ENlloJ) I ENIIOJI¢.1IONE
ZIIOJIIZCIoJI-ozIINE

CALL COpT

I?II?¢I

IF IFDFLAOLToDII GO TO 30 3O
EOFLAIOI

ENIIOJIIFNIIOJI'oIIINE

ZIIOJIIZIIOJI+II"NF

u: ”2.50.” 60 TO so 5°

co TO 60 6°
DIITDCLA

r~(|.4)¢rn(loJ)-ol"NF

ZtIoJI-7(I0J)+ol¢INF

CALL (DDT

Ir (FDCLAOLToDII no To 50 9° .
FNIIOJI I rNIIoJIOII"NE

ZIIOJII/IIOJI-ol"'f

CALL COPT

mum wommmmrunola 30°
M'IM-MI

MQIMJ-MA

IFIIMQoGToIIIORoIIPOGTOIIIORQIIIoOToIII GO TO GO

NEINFII

IF INEOLTIO‘ GO TO GO ID
MpaM-M?

MfiIM3-MR

DOINT POOONOM90M30M5 QOO
INN? qoo ‘ goo
CALL PSIGQAPH

I3II3+|

PRINT IOOOIS IIIIIQOO
IFID?.GT.EOFLAI I3IO

IFIDZOGTQEDFLAI 02IEOFLA

PRINT IOOOIJ IIIIIAOO
IFIIJOLTOEI GO TO 9 5

PRINT QOOOIJ IIIIIQOO
FORMATI/O TRUF ENFOGY I O'IHIOOIOXOADJUSTED ENEPGY I QFlzoB OIOX
?IALPHA I OF7voIOXCP I .F601/l. C(IOJI I ‘HFISOOI. ENIIOJI
3- IRFISQOIO ZIIOJ’ I 'RFISOOIII

'OPMAT I. FNEDGY3 CALLED .Iq. TIMFS ITOTALICI. FNFQGYS CALLED CI!
2. TIMFS (THIS TIMF)./. COPT CALLFO .Iq' TIMES (TOTAL)./. COPT
3 CALLED .III TIMES (THIS TIMEIII

FORMAT (I MIOIRQ7XOM1IGIRO10X9NEIOIQOIOXGIBIOISQTXOIAICIEIOIOI

FORMAT IIISI IIIIIIII
FORMAT (01¢)

CONTINUE IIIIIIII
END

SUBROUTINE COD?

TYPE OOUBL! 2

COMMON II/C(204)0FN¢2043

COMMON I 2 I FDFLAOFDFLREALQALDHAQD

COMMON I I I IoJoM3

COMMON I Fix I 2(20A)

DIMENSION cpxcn)

MJIM3+I

FF-FAcTonct)

C(IOIIIFFICIIOII I C(IOZIIFFOCIIOZI

CIIOJIIFFICIIOJI I C(loA)IFFICIIoA)

CALL ENCRGYS

DIIEDELA -
IIIO

I2IO

NCIZ ‘ ' -
CQIIIIIIQ

CRI‘ZIICIIO?)/Ctlol)
CRICJIICIIOJIIC(IOII ' 1 "" r
CQIIOI'CIIo‘I/CIIOII

OIDI

CRIISI-CRIIJI

IF IIZOEOOOI CPICJ)ICPIIJ)+0100NC
IF (IZOEOOII CRICJIICPIIJI°II.'NC
CIIOIIICRIIII I C(lo?)-CRII?I I C(IOJIICRIIJI I CIIOAIICRIIAI '- "
'FIFACTORIII

C(IotIIFFOCIIotI I C(IOZIIPFOCIIoz)
C(loJIIFFOClIoai I CIIoIIIFFIC(IOII
CALL ENFRGYJ

DIIEDELA

141

IIIII+I "”‘”

IPIDIILTODI GO TO IO II
DIID
CRIIJIICRIISI ' ”
tail!
I'IIIOEOIII GO TO IO IO

C(IOIIICRIIII I C(IOZIICQII?I I C(IoJI-CR|(3I I CIIQQIICfijIAI
FFIFACTORIII

C(ICIIIFFOCIIOII I C(IO?)IFF'CIIO?I

CtloaburFICIIoEI I C(IoII-FFGC‘IOAI

CALL ENFRGYJ

NCINC+I

IIIO I l?-0

IFINCILToGI GO TO IO IO
POINT IOOOCDELREALQEDFLAOALPHAon.I(C(KoLIoLIloAIoKIIozIOCIENIKoLIo
?L'I"I‘KII0?IOI(ZIKOLIOLIIOIIOK=|Q?)

I00 FORMATI/G TQUC ENERGY I IFI5ODOIOX§ADJU$7ED ENERGY I G'IZI9 OIOX

QIALPHA - ~r7.4oloxon - «75.3/Io C(loJ) I '8Fl506l' EN(IOJI
3n IOFI506/I 2¢IOJI I ‘0F1506l/I
END

SURROUTINF PSIGDADH

TYDF oounLr 2

OIMFNSION OLOC73lonOLDI71)

COMMON II/CCPoanrNIPOAI

COMMON I fix I Z!P.4)

A(noC)-CII¢.I~n+.n)InAMIN(n+1.0o)oo.fi 51!
AIIA(?'rN(?0130907(POIII

azuntporntrorsorl7t?o?)!

A1.A(;otNtno3)o?.zt?oJ))

SUMIISUnyuhUMupnfluMauo.

SUMIR-SUM?3-uuua.a.

DO IOoII|o7\ IO

93-II-IIIOo?
DELTA-o?
IFIIoLTo??I n1III-II.OI
IPCIoLTo?2| DELTA-o!
lPIloGFoJR) "acct-23.0.4
lPIloGFoS?) OFLTAIoI
PSIIIAIOQROOtFN(?OII-IIOEXDt-n302€?oli)
PS!?-A?On300¢FN¢20?)-IIOEXPt-nanz¢7opgg
PSIJIASORIOOIFNttofli-|IIEXPI-RJOZI?03II
RPIIPSIIIQJ
n92-PSI?lR3
RPS-9513093
SUMIISUMIIDELTACDDI'DDI
SUMZISUMZIDFLTA‘DP?I992
SUM3ISUM3+O€LTACPP1¢PP1
SUMI?ISUMI?+OFLTAInPgCOP?
SUMISISUM13+OFLTAIIPI0993
SUM?3ISUM?3+DELTAI9920993
PSIAIC(?0IIIPSII¢C(?0?IIDSI2IC(ZofiI’DSIJ
RPo-DSIIIRJ
SUMO-SUMAODELTAGGPAIDDA
DIIV-PSII-OLOIII
fiOIFF-PPA-POLDIII
OLDCIIIPSIA
ROLotlI-ppo

IO POINT |OO¢93095I1099!OPS!#oDP?oPSljonfiaoPSlaoRflqooIF'QODIFF
ODINT ?DOOD1OSUM1oSUM?oSUM30$UM4
PRINT JOOoSUM12.SUM13oSUM?3

IOO FOQMAT (F7.3¢S(FI?¢I¢FIOQSII

ZOO FORMATI/F70304(I2XQF1OQIII

300 FORMAT I/I SUMIQI'FIOQSIO SUMIJI‘FIOoSII SUM23-.'I°OSI
END
SUBROUTINE ENEPGVI
TYPE DOUBLE 2
COMMON It/Ct?o¢)oFN(aoAI
COMMON I 2 I EDELAoEOELnEALoALPHAon
COMMON I 3 I EINToFKlNETICoM
COMMON I PIX / 2(z.4)

ZZCIoJOKoLIIZIIOJI+ZCK0LI 57'
RHOCIOJOKOLIIQI'DUZZIIOJOKOLI 57'
ENNIIOJQKOLIIFN(IoJI+FN¢KoLI “ ‘ 37'
AIBQCIICIIIosIH+05IlGAMINIB+I00-).905 ‘ 57'
MIM+I

EINTIO

DO IOOKIIOI

I00

I00
200
300

20

IO

30

142

IF I C(20KIOEOoOoI GO TO 10

"DO IOoLIIoA ’ ‘—' ' "' “" ‘“"—__"‘””

IF I C(20LI0EOoOoI GO TO IO
REPSUM IO

00 ZOoIIIoa I“ A».. -Wr__v

IF (CIIOIIoEOoOoI GO TO 20
DO 200JI104

IF IC‘IOJIQEOoOoI GO TO 20 - - --

REPSUN IREPSUM +C([OIIOCIIQJICAIENNIIOIotoIIoZZIIOIOIOIII

2'AIENNII0J010JI02211oJoIoJII*PEp(IoJoKoLI
CONTINUF ‘-

EINTIEINT+CI?0K)Ic(20L)IA(ENNIZoxofioKIOZZIZQKOZOKII'AIENNIZOLOZOLI
ZOZZIZOLOZOLIIGIPEPSUM-ATTQIKOLII

CONTINUE - ' ‘ ‘

EKINETICIO

DO JOQKIIIQI

IF ICI20KIIIEOOOOI GO TO 30 V _ T.
00 BOOLIIIIQ

IF ICI?OLIIOEOIOOI GO TO 30

PANT] I ZIPOLII‘ZIPOLII‘ENI?OKIICIENI?0KII'II‘2‘ZI20LII'ZI20KII J

zisnczoLx)lENcaoK:)+Z(poK1302(2oK1)¢EN(2oL1)9(EN¢poL|)-lI-?922(2.K1 PART|
JoZoLITIOz PART:
PART? - AIENN(2.K1o2oLx)ozz¢2.K1ozoLlI’loaozicunczoxa.2.L1)I¢ENNC2 PART:
zoKIoaoLns-n) PAnrz
Pants - CIZoKiI'CIZOLII'AIENN(?0K|¢20K|)oZZ(?oK1020K|))9A(ENN(20LI PART;
2020L1I022(?0L!o?oL1)) PART:

{KINETIC I EKINFTIC-PARTalpARTZIPARTI

CONTINUF

EOFLPEAL IFKINFTIC+EINT ' ’
EOELA I CKINETIC+ALPHAIEINT

END

FUNCTION QFPCIOJOKOLI

TYPE DOUHLF 2

COMMON II/CtzoIIoFN(?oII

COMMON / FIX I thoA)

TVPF DEAL IMHFTA

zzca.n.c.n)-/¢A.n)+7¢c.0T 57'
OHO‘AORQCODII.G'D'7ZIA000CODI STF
FNNIAOHQCODIIFN‘AONIIFNICIDI STF
FNCIIOJQVOLI I FNfi-VNHIIOJOVOL) ST?

an‘IIOJOKoLI I INNVTOI27IIIJOVILI/ZCQCNNIIoJoRoLI‘IorNCIIoIQKOLIIDTF
"FTSIIOJ~VIL)olrrrTAI?z¢toJoxoL1IznorNNtIoJ.r.L).flornc¢IoJor.L)-1)nrr

AcnoC)-(II(.q-n¢.q)InAM|M(uoxoo.loo.a 51;
[NS I CNIIOIIOCNIIOJIOFNIPGKIOFH(?QLI .

25 I Z!:.I)¢7¢1oJ)¢z¢nov’+th.L)

arm - poananopoL) Human-InoJT-uzznuuuno I"
aAlENNI?oKo?oLI027(?0K0,0LIII..?

REP: I zouETAttoIoch) IIACENNCPQKQZOLI-I022(29K920LII. 5

?A¢ENN(IoIoloJIoZZ(IolotoJIII003

9:93 - Ito/1IBET3190KOIOJI Itacruutuolo?oL)-7022(IotozoL)IO
,AIFNNI?OKOIOJI+IOZ7IPOK0IOJIII'O?

“FPO I Ila/1ODFT1IIOI0?OLI IIAIFNNI?QK9IoJI¢chZI2oKoIoJII.
?A¢FNNIIOI070LI+1022(10I0?0LIII.'2

DEPIREP|+RE92+9693¢QEPI

END

FUNCTION ATTPIKOLI

TYPE DOUBLF 2

COMMON l!/C(?odIoFN(?oII

COMMON I 2 I EDCLAOEOFLREALOALPHAoR

COMMON I FIX I 2(20‘)

22 I 2C?0KI+Z(20L)

RHO I 05.9.12

CNN I EN¢2¢KI+FN(POLI

N"N-‘Uhl

nfi’I
Hell
“6’!
REF?
HE’S
REFI
“6"
REHO

ATTR I ?I22"FNNOIGAMINIENNQQHOIOIIQHOG(CANIN(ENN+IoOoI‘GAMINIENN+ ATTfl I
21ODHO!)¢.IOQHOOOPIGAMINCENN-zoRHO)+oIIPHO..JI:GAMINtENN+JoOoI-GAMI ATTR z
3~¢z-+3.nn0333 AT?“ 3

END

FUNCTION FACTODIII ' " “
TYPE DOUBLE 2

COMMON IIIC¢?ooIoEN(2oII

COMMON / FIX I 2(2oai

ENN‘IOJOKOLIIFNIIOJIIFNIKOLI STF
ZZIIOJOKOLIIZIIOJIOZIKOLI ST?
AIBOCIIC.‘¢ISIFOOII/GAMINIF¢IoOoI.’05 STF
CCIJOKIOCIIOJI.CIIOKI STF

IFICCIIOZIOEOOOI GO TO 2 2

 

rll..I I (III-II. I]

143

FACT|ICCIQIIICIIoPIIA|PNNII0I9I0lIoZZIloIoloIIIOAIINNCIODQIOIIQZII
QIOEOIOPIIIIACINNUlu|oI0II022CIo|oloDIIIIP|

IPCCCIIODIoEOIOT an TO 3 3
FACTZICIIOII'ClIo‘)IA(fNN(IOIOIOIIOZZIIoiolo|$IIathNCI030I933922(
zloaoIOJII/IA(FNN(I01OIOJIOZZIIOIOIOJII..2I

|F(CC(104I.EOoO) GO TO A I
FACT3-c(to:)oc(10¢)ca¢FNN(toxotoggoZZtl.‘o|ogg)IA(ENN([o‘cloaonZI
ztooolo4)3I(A(FNN(xo1o1c4)022(loIOIOIIIOOZ)

IF!CC(203).EO¢O) GO TO 5 5
FACTa-CCIoZIOct103)§A(ENN([020102)oZZ‘IchIoszGACENNIIoaoIOJIOZZI
2I03‘I03II/‘AIFNNIIO?OI03IOZZII020I03II..2I

IFICC(2.4).EO¢OI GO TO 6 6
FACTSICIIO?)ICII0A39AIFNNIIO?QIO?IoZZIIo?oI02II’AIFNNIIOAOIOIIOZZI
ZIvoloA))/(A(ENN([020I04)022(loZoloAIIII?)

IFICCIBOQIIEOOOI GO TO 7 7
'ACTOICIIOJIICIIOAIIAIFNNIIOJOIOGIoZZIIofloIojIIOAIENNIIvoIoAIoZZI
2IvoloAIIIIAIFNNIIoaoloAIQZZIIODQIOAII..?I
FACTTICIIOIIIO?+C(I0?I.OP+C(I03I'O?‘CCloIIIoz
FACTQIzOIFACTI+FACTZ+FACTJOFACTIIFACTSOFACTOI
FACTORI(FACT74FACTOICCI-ISI

END

144

III.The program used to find the scattering amplitudes

and phase shifts was:

DROGQAM NUI

TYDE COMFLEX SUHOPARTZQINCQXQDSIIoDSIzoDSIJQVPvoEXCNoCOUL
RHIHIR‘R STF
HUI?

NUII

NUIO

PQIIOI

BIIfloSRfiI

BSOIA028778

DIOR-60002flo

BSOQ7I36003IH

HIOQJI21603I3

D?I09641806

P?2I03QR°4?3

BSOQIIHSOQ?O.?

DSOQHIBSOQP.B§093

HSOQOIBSOQSI'?

DSOQTIBSORJ'BSORA

DO acct-10:5

RIO.

UNIII5¢0059I

SUMIO.

CI. 0038080380

C?I 5020563707

CAI 0.076IIOOO

C7II0772‘539.I073?OWOB

IO CONTINUF

IFIROLTIIOI FPSIQOI

IF (RoLTooOOOII EPSIOOOI

RIR+EDS

RKIUNIR

SRISINIDKI

CRICOSIRKI

TIISR/PK-CR

JIITIIPK

JEISD-JI'?

J3I(?-°KO’?I'JI'?°J?

IF (NUoFOoO) GO TO 60

IFINUOEOQIII 60 TO A0

IF (NUoEOozoI GO TO 50

GO IPIOoGTo 07) GO TO 61

rgpn.aso.pao(g/(poQ).aztncn)+|/Pno(pocl13-6!(aoPI-tlasonicl‘209!¢l
2/85092O(?OG(3)-Glt3o93I-I/HSORJ'GI(I.R)+l/RSOPAO(?IG(5I-Gl(5099I-I
JIBSOQSIGII609I+IIBSOR6.(P'G(7I-GI(TORII-I’95097.GI‘O‘PIII
GO TO 80

61 IFIRIGTOIO‘I GO TO 62
TERMIBSOI920(1/(QO?I'(?'G(II-GIIIoRIIOI/QQ*IGIIIQRI-I/RSOQ'GI‘ZOQI
2+1/950Q2IGI(309)-1/85093IGI(don)¢|/RSOQQIGI(IoRI-I/OSORSGGI(609I+I
3/85096IGI(TORI-I/BSORTOGI(Hon)II

GO TO 80

62 CONTINUE
TERMIBSOIP2OGIII/Z
GO TO 80

I0 IFIRoGTo O7I GO TO A!

TERM-BSOIP22II-Gl(2.9)IP+9IRRI(GI(IoR)-1InSOR0(;¢c(3)-Gl(3.9)aol/n
zsonzicx(A.9)-1Inso93-(2Oatsy-ot(Ion,)+:/asonaost(gong-gI850P50(zOG
3(7)-GI¢7.R)Tot/nsonoocl(noPI-xIaSOn70(ZOG(9)-Gl(009!!!)

GO TO 70

A: [PIPIGToIoAI GO TO 42

TERM-950.9?PI(-GII?QDI/D+2/DQO(GII?oP)-I/RSOQ.GI(309)01/99097’5I(C
PoRI-I/nSOQSOGI(Son)+1/n509406l(609)-1/RSOWIIGI(709)41/RSOQ6'6I(aoR
SI-I/HSORTGGIlOoRIII

GO TO 70

on CONTINUE
TERM-O "
GO TO 70

9 s
a...

I. “

« rO-ulw.

‘3

 

SO

51

52

80

70

.0

.0

OOO

30

IO

20

145

IPIQoGTo 07) GO TO SI
TERM-.HODSOOP??CIIla-(2'6!(noRI-GIIIoRI3+?IROOI406IJI-?°CIIaoflIOOI
?(|oR)-?¢G¢II-I/nSOOOI7CGI(sons-6|(pow)3+I/R5092-(4OGISI-206I(Gonoo
36l¢3¢9)-?¢GI‘)I-I/RQON1GI?OGII6oQ)-Gl(Con))0l/HSODO’IO'GC7)-?'Gli7
aoPI+GII:oO)-?¢GIKII-I/nQODROI?OGIInoOI-GIIAoD))sa/nsonaocguccq)—7o
SGI(°QQI+GI(799)-?.G(7)I-I/nSOQTGI296!(:OoRI-GIIHQRIIII

GO To 70

IFIn.GToIaAI no To 5?
TF9M-.R‘HSO¢P?P'II/RII4‘G(1)-?.GII1oRI+GI¢1901-?OGIIII42/DROIZOGII
?aoRI-GIIIoPI-IinsonitrsnI(A.9)-GIIrow))oI/nsonyee7onltfion)-GI(3.n)
3)-I/nSOO1'(?OfiI(ficQI-GIIQOD)IeI/flSODaII7'filI7onj-GIIQQD))-]/RSOQSO
QIpnfilInonI—GII600))+I/n50fl6.‘?061(QQRI-Gl‘Tofl)I-l/050n7.‘?‘6l‘1009
5)-GIIAoDIIII

GO To 70

CONTINUF

TFQM-O

00 To 70

CONTINUF

XI‘OeOIQI’DK
PSII-IoocP.)5C7'JI#1066.(CI‘RO'oflnfiirxpt-.607|PI+C?'N"I.387’FXPI-
27.7%?09)I
PsI?-Io.or.IOCTOJ?¢1ece0ICI09'0ohflfi'FXpI-oAOT'QIOI.682-o6070DI*C?0
poo-I.nnpOFXFI-?oTQ?ODIOIIoFNP-?07%7ODII
PfiIa-IogoI.)or70J1o10reoIClino-can70rpr-onovingoI-.An7¢o1In/?-.oo
9?O.68?OR¢I.OOTOflIOO?I¢CPOCIIBS?OoAR?/'-?07%?OIolfla;§ot?0782'fli.070
3D '
VHeIPSISoflSIzI

EIC?I1OCOOI|/D .020ICI9IoP?T.|I-?o0I?IOIGAMINI?.n]?sOoi-GAHINI?.O|
??olofl?7OQ)I+C‘O1oflTfiiOI-Ooon?)0(GAMINIc.0a900oI-GAMINIO.OR?.3.07?O
‘n$"+9 C¢C|.Ioflfl700¢oOHflI'GANIN(.0030!ofl?7.n)‘C?O1007?OOI¢I0082,96
OANINIIOOR?01007?OQIII

AQCXCHIDQUN’
COUL-I/DO(7.40OOI-I.06)O(GAMINIIoOfiOOQI-GAMINI1.460?.QOOR)I¢?.OOOO
pI-o06396ANINI.4607oeaOnIIODSII-A/a-tXC?/6
INc-ICOUL+II/InoIaIQOP6R1RGI.77?4R15)0Trnu+1.-.qenx.O2)OPs|1-vngoc
2FXPI-XIOEDS

Go To an

CONTINUF

XI‘OOOO-'C’.'N.Q
Ffl-CIOPOOoenr0FXOI-o607.DI¢C?0R0¢Ioafl79£xp(-7.7q?09:

INC - Coornocrxpcx:ap2p¢759uza.Iagqq;65.¢rps

'F‘NUOEOOI’ INC'491NC

CONTINUE

SUM-SUMolNC

IFIOoLToSoOII GO TO IO

CONTINUE

DARTZISUMOUNOIoooIo)/IA030I4I%926HQOI07724538)

Ir INUoEOoo.I paaTa-pAnTaog

AMP-cABSIDAnTa)

PAnfz-‘OooIoIOCLOGIAMfi/PAOT2I/JoI4I592654

AMP-AMPOOz

POINT aOOoNUoUNoAMPoPAOTz

FORMAT (OH NUIOl2.IOXOJHUNIoFSQQOIOXsOHAMPuoEtSOTOIOXOCPHASE SH]
ZFTIDoCIFIOoSoFIOoSIoIOH TIMES PI/II

CONTINUE

END

FUNCTION GIN)

GIGAMINIoSUNoOo)

END

FUNCTION GIINoxI

Gl-GAMINIoS'NcleoSGSI "IoO-ZOXIOOPI

END

COMOLEX FUNCTION EXCH (QPQUN)

TYPE REAL LARGE

R-eOOOI

EXCH-IOooOo) _ _
CONTINUE

QKODOHN

59-SINIQKIIRK-cOSIBKI

CIQCIoTTZOGJGOIOTSZOSOO

Ex-ExOI-I02209I

IF InoLEoflfli SMALL: 900.23’SROC/99902

IF (9.67099! SMALL. Root-ZoTTIosn.COQP

LARGE-EXOSMALLGoOOI/HN

IF (PoGTooII LARGEIIOOLARGE

IF ‘RoLTooOSILADGEIoI.LAPGE

EXCHIEXCH+LAQGFGIOoOIoI

IF (RQLT0003)GO TO ?0

IF ‘POGTOOOQOO°) PI90001

IF (RoLTeoI) DIQ+OOOI

IF InoLTo9I Go TO IO

CONTINUE

If CQ.LT0.0J’PIR+QOOOI

IF IRoLTeeOSOOI) GO 10 IO

END

IV. GAMIN(ALPHA,X):

of the form GAMIN(ALPHA,X) are in fact calls to a library

146

In the above programs variables

subroutine which calculates and returns the value of

r(a,x), the incomplete gamma function.

The same subroutine

can be, and is, used with x-O to calculate the gamma

function wherever it is needed.

V. Data:
amplitudes
‘1. OH.
‘J' hfi'
‘0' ok-
UU' In.
\UI bu-
HUI wu-
nu- a“.
‘9. .1
10' U”.
‘U' UNI
‘9' bk.
VUI_ Lu.
‘0' un-
_ HUI. the
‘0' an.

The original output, listing the scattering

and phase shifts, is here reproduced:

0-30
c,2!
0.20
6.3,
00‘.
00‘,

c.90

:.*o
.06,

0.70

0",
0.90
3.05

Al'-

‘n"

An!-

1".

1.004;:o:-..;
0.099300‘...;
9.‘3649’4°°°3
‘o‘737’oo-oui
Sol!!2051°n31
3-:oz°no?'nos
2.3“:352-ooi
?.‘393052-001
2.7.3377‘000‘
3.6490923'001

’.167Oo¢J-co:

7.4003882-001
I.o400051oooo
1.421115:oooo

10"9’..7....

Paola IIIIII
rugs: .ulf'.
runsi salty.
'"*si 3‘1'1-
'“I‘g SHIIT'
Pupsi 8H]!!-
vӎsi s""1-
ruas: ‘ul'T.
Pw0§§ IwIIYO
PNQSk Sulfi-

r"figi gfllrve

ruas§_1ulrt.
puOsi sultr-
rHIIE untri-

rwal BuIrII

'|og7860
-|,22102
O..27,!O
.0.5J099
of.cov;7
-l.¢97;3
...sgaos
...?zilt
.‘.I)O.g
'00’6’0:

..93072

-Jflfl”

0,7973.

e.v4277

:
Cubic!)

tints PI
t'u‘: r'
vans: r:
t'u's p'
runs: at
tint: on
I'n‘t P'
tunes PI

TIN!‘ PI

TIMES Pl

I'N‘I "

VIHCI 'l

TInEI rI
IIHII pl

VIII. Pl

\J'

VU'

Vu-

VUI

NUI

\Ul

\J'

VUI

VU‘

‘0'

\UI

\UI

VIII

VUI

NUl

NUI

VUI

Nd-

NUI

NU.

VUI

VUI

VUI

\UI

\Ul

‘0'

\UI

UNI

Hu-

UN!

UNI

UNI

UNI

UNI

UNI

UNI

INI

UNI

HNI

UNI

SNI

UNI

UNI

UNI

UNI

UNI

In.

UNI

UNI

In.

UVI

uNl

UII

0.23

0.?5

0.3.

0.35

0.‘0

0.‘5

0.50

0.55

0.60

0.70

0.75

0.05

0.90

0.25

0.30

0.45

0.00

0.55

0.60

0.69

0.70

0.75

0.00

0.05

°.°0

IMP.

AHPI

InPn

AnPI

AIPI

\HPI

.nIo

AnPI

An!-

AnPI

AnPe

AnPo

IHPI

tnPn

ArPI

AHPI

Aura

AnPe

AnPe

In!-

An!-

AHPI

AnPe

AMP.

AnPI

IHPI

AnPI

An!-

AIPI

AnPI

147

Io’I!!I!!°!!!
7.12‘7944-00’
1.0251240000.
1.3989499-00I
3.0l86231-000
9.2907630-000
?.e339384-ouo
8.4236690'000
6.0674250-00I
4.7040309-000

2.914050I-000

6.3100870'000
7.17044930000
(.0747209-00!
9.0200I730000
4.1600703001!
6.023052l'018
9.7490i01'013
1.3010289001!
1.02070200012
9.30773100012
3.0‘05792'012
3.0232271'012
‘o72‘7315-012
Z.7673024¢012

(.900Q706'0l2

0033050090012
9.9009050-012

Io1I72904-01I

1.8072650'031'

Degas
Pufilt
pagan
PIASR
PIIS:
PEAS:
PIGS:
PHASE
ruAsE
PHASE

PHASE

PI590
PHQII
'NISI
PIAQI
'"900
m' as
'"PSE
PIISE
PNlSE
PNOSE
PNISE
PIISE
PIAIE
PIAIE

wuasc

PHASE
Pwaee
Pwawi

PHONE

IIIP'I
yuIFTu
SNIIVI
SHIP!-
SHIP!-
salt:-
SHIV!-
SHIP:-
SNI'II

INIITe

SHIP!-

SII'VI
IHIPYI
l
SNITV.
,NI'VI
IIIPVI
’NI'YI
IIIFT.
III'VI
SHIV!-
!H|'Ve
IHI'VI
)NI'VI

IHI'VI

unlit.

SHIFII:

,NI'VI

fiNlPVI

bulfVI

INI'VI

'UIIIIII
oo.sssil
-I.94213
-I.50vis
00.55017
-o,soaio
-l.srozi
-d.57724
-I.setzs
-l.5912|

-o.svo:1

..0I0,33
....‘23‘

....t".

00.02041

0.53!!!

0.93076

0.54003

0.50953

0.55400

0.55571

0.56264

0.90014

0.9091?

0.57I7I

0.9749.

0.!7’6!

I.!7eei

0.9777:

9.000...

VIII!

VIII!

VIIE!

VIIII

VIII!

IInEI

TIME!

IInIl

IIuEI

VIII!

VIII!

VIII!

IInII

VIII!

VIII!

VII!!

VII!!

VIII!

VII!!

VIII!

VINE!

VIII!

VIIE!

VIII!

IIuEs

VIII!

VIII!

VIII!

VIII!

VIII!

PI

PI

PI

PI

PI

PI

PI

PI

n

PI

PI