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MOLECULAR EXCITATION GROUP I

I I

CONDON LOCI OF DIATOMIC

MOLECULAR SPECTRA

bY

MARY FRANCES MURTY

https://ntrs.nasa.gov/search.jsp?R=19650016445 2020-06-07T00:40:41+00:00Z

CONDON LOCI OF DIATOMIC MOLECULAR SPECTRA

by

Mary Frances Murty

Submitted i n partial f u l f i l l m e n t

of t h e requirements f o r t he degree o f

Master of Science

Faculty of Graduate S tudies

The University of Western Ontario

London, Canada

1964

i

Approve'd for the

Department of Physics

by the Advisory Committee

Approved for the

Department of Physics

ii

ABSTRACT

The geometry i n t h e vtv" plane of Condon l o c i of diatomic

molecular band systems i s studied. Using both simple harmonic

and Morse p o t e n t i a l models i t i s shown t h a t f o r most band systems

t h e l a r g e s t con t r ibu t ion t o the ove r l ap i n t e g r a l of t h e v i b r a t i o n a l

wave func t ions of a t r a n s i t i o n i s from t h e coincidence of a s i n g l e

p a i r of ant inodes of t he wave funct ions. The coincidence of a

p a i r of terminal ant inodes desc r ibes t h e primary Condon locus ,

while t h e coincidence of a terminal and an i n t e r n a l ant inode

desc r ibes a subs id i a ry Condon locus i n cases where t h e l o c i are

nested.

A s i n g l e p a i r of coinciding ant inodes does n o t c o n t r o l t h e

magnitude of t h e overlap i n t e g r a l i n band systems where A r e i s

e i t h e r very small ( < 0.01 A ) or very l a r g e ( > 0.3 2). band systems are examined with

0 Thir teen

0 A re ranging from zero t o 0.4 A .

It i s shown t h a t t h e l o c a t i o n of s t rong bands i n t h e V*V'* p lane

can be p red ic t ed with reasonable accuracy without t h e computation

of Franck-Condon f a c t o r s .

iii

ACKNOWLED G W T S

I I I I I I I I I I I I I I I

The au tho r wishes t o convey h e r s i n c e r e thanks t o D r . R. W.

Nichol ls who suggested t h i s problem, f o r h i s supervis ion and

encouragement, and f o r making a v a i l a b l e h i s computations of Franck-

Condon f a c t o r s .

Grateful thanks a r e extended t o D r . P. A . F r a s e r and D r . H . I. S .

Ferguson f o r h e l p f u l discussions.

The au tho r a l s o wishes t o thank Dr. P. A . Forsyth, Head of t h e

Department of Physics, f o r making a v a i l a b l e t h e f a c i l i t i e s of t h e

department.

The a s s i s t a n c e of M r . J. Radema and M r . T . Davidson i n t h e

p repa ra t ion of diagrams, and of Miss K . T r i t j a k i n computational

work i s g r a t e f u l l y acknowledged.

This work has been supported by a r e sea rch c o n t r a c t from t h e

Department of Defence Production, and by research g r a n t s from t h e

Nat ional Research Council of Canada, t h e Nat ional Aeronautics and

Space Administration -&$-&349), t h e United S t a t e s A i r Force Office

of S c i e n t i f i c Research (AF-AFOSR62-236) and t h e United S t a t e s A i r

Force Cambridge Research Center (AF 1 9 (628)-2820). This support

i s g r a t e f u l l y acknowledged.

i v

TABLE OF CONTENTS

ABSTRACT ....................................................... iii

ACKNOWLEDGEMENTS ............................................... i v

. . . LIST OF TABLES ................................................. v ~ i i

LIST OF FIGURES ................................................ x

Chapter 1. The Franck-Condon P r i n c i p l e i n Diatomic Molecular Spectroscopy ....................................... 1

1.1 Introduct ion .................................. 1

1 . 2 The I n t e n s i t y D i s t r i b u t i o n i n Band Systems of Diatomic Molecules - Condon Loci .............. 4

1 . 3 The Franck-Condon P r i n c i p l e and Franck-Condon Fac to r s ....................................... 4

Chapter 2 . Concepts ........................................... 9

2 .1 Band I n t e n s i t i e s and Franck-Condon Fac to r s .... 9

2 . 2 Suggested I d e n t i t y of Condon Loci ............. 1 5

2 . 3 Geometry o f the Primary Condon Locus .......... 1 7

Chapter 3 . Condon Loci and the Simple Harmonic Molecular P o t e n t i a l Model .................................... 18

3 . 1 The Simple Harmonic O s c i l l a t o r Model of a Diatomic Mclecule ............................. 18

3 . 2 Reduction of Antinode P o s i t i o n t o Algebraic Form .......................................... 21

3 . 3 Coincidence of Antinode P a i r s ................. 23

3 . 4 Form of t h e Loci of Antinode Coincidence ...... 27

V

v i

3.5 Deta i led Examination of Representat ive Types o f Trans i t ion ........................... 31

2 3.5.1 MgH C217 + X2Z and A2F -+X L. ......... 31

3.5.la Comparison o f Overlap I n t e g r a l with Antinode Coincidence ...... 40

3.5.2 CO A TI 4 X Fourth P o s i t i v e System .. 42

3.5.3 C2 B% j X3;7 Fox-Herzberg System .... 46

1 1

3.6 Addit ional Remarks on t h e Geometry of t h e Primary Condon Locus .......................... 46

3.7 A Simple Method of Finding t h e Subs id ia ry Condon Loci ................................... 51

Chapter 4 . Condon Loci and t h e Morse P o t e n t i a l Model .......... 54

4 . 1 The Morse Model of a Diatonuc Molecule ........ 54

4.2 Wave Functions. Overlap I n t e g r a l s and Franck- Condon Fac tors ................................ 56

4.3 Reduction of Antinode Pos i t i on t o Algebraic Form .......................................... 56

4.4 Graphical Method f o r Locating t h e Coincidence o f Antinode P a i r s ............................. 57

4.5 Loci of Antinode Coincidence .................. 57

4.6 Deta i led Examination of Representat ive Types o f Trans i t ion ................................. 59

4.6.1 MgO B1t + X1 L Green System .......... 59

4.6.2 64

4.6.3 C 2 B k + X 3 E Fox-Herzberg System ..... 64

Chapter 5 . Conclusions ........................................ 70

CO A h 3 X I L Fourth P o s i t i v e System . .

Appendix1 . Fur ther Resul t s of t h e Examination of Condon Loci Using t h e Simple Harmonic O s c i l l a t o r Model ......... 72

A l . 1 Cl@ A1n 3 X'h ............................... 73

vii

A1.2 CO+ A2TT+X2L Comet Tail System ............ 76 A1.3 O2 B3z, 3 X 3 7 - Schumann-Runge System .... 82

A1.4 GaI A3n +XIL System A .................. 82 L g

Appendix 2 . Further Results of the Examination of Condon Loci Using the Morse Oscillator Model ................... 90 W2.1 NZ B2L+,t N,$L+ Rydberg System .......... 90

g

A2.2 MgO B1x+ Aln Red System ................... 94

A2.4 N i ‘C2z-t+_ N2X1 ‘7+ ........................ 100

A 2 . 5 $ D 2 T \ g t - N2X1‘+

A2.3 CO+ A 2 n -+X2 Comet Tail System ............ 94 & - g

- g ........................ 100

A2.6 Schumann-Runge System ...... 105 A2.7 GaI A% -P XIL- System A .................... 105

02 B3 L’ 3 X3 E- U 6

REFERENCES ..................................................... 112

VITA ........................................................... 114

LIST OF TABLES

Table 1.1 Band I n t e n s i t i e s of MgO B -+A ...................... 5

3 . 1 Empirical Constants Describing Antinode Loci ........ 22

3 .2 Data f o r T rans i t i ons Examined with SHO Model ........ 29

3.3 Observed Bands o f M@ C + X ........................ 32

3 .4 Observed Bands o f MgH A - + X ........................ 32

3 .5 Overlap I n t e g r a l s of MgH C 4 X ..................... 34

3 .6 Overlap I n t e g r a l s of MgH A -+X ..................... 34

3 .7 Exanunation of Contr ibut ions t o Overlap I n t e g r a l of MgH c “ X .......................................... 38

3 .8 Overlap I n t e g r a l s of CO A 4 X ...................... 43

3 .9 Observed Bands o f CO A - + X ......................... 44

3.10 Observed Bands of C2 B - + X ......................... 47

3.11 Overlap I n t e g r a l s of C2 B . - ? X ...................... 47

4 . 1 Data f o r Trans i t i ons Examined with Morse Model ...... GO

4 . 2 Rand I n t e n s i t i e s o f MgO B . . i X ...................... 61

4 .3 Franck-Condon Fac to r s f o r MgO B . - + X ................ 62

4 . 4 Franck-Condon Fac to r s for CO A P X ................. 65

4 . 5 Franck-Condon Fac to r s for C2 B - > X ................. 67

A 1 . 1 Observed Hands of CH+ A > X ......................... 74

A1.2 Overlap I n t e g r a l s of CH-‘ A : -X ..................... 74

A1.3 Examination of Contr ibut ions t o Overlap I n t e g r a l o f CH‘ A-+X ....................................... 77

v i i i

A1.4

A1.5

A1.6

A1.7

A 1 . 8

A2 . 1

A2.2

A2.3

A2.4

A2.5

A2.6

A2.7

A2.8

Band I n t e n s i t i e s of CO' A - t X ...................... 78

Overlap I n t e g r a l s of CO+ A F X ..................... 80

Band I n t e n s i t i e s of 02 B +X ....................... 83

Overlap I n t e g r a l s of GaI A Y X ...................... 86

Band I n t e n s i t i e s of G a I A . X ...................... 87

Franck-Condon Fac tors for N a B - . - N z X ................. 92

Franck-Condon Fac tors for MgO B A ................ 95

Band I n t e n s i t i e s of MgO B , - .A ...................... 96

+

Franck-Condon Fac tors f o r CO+ A - X ................ 98

Franck-Condon Fac tors f o r N$C .. -N2X ................. 101

Franck-Condon Fac tors for %D . N2X ................. 103

Franck-Condon Fac tors f o r 02 B .. X .................. 106

Franck-Condon Fac tors f o r G a I A-:X ................ 109

LIST OF FIGURES

Figure l . l a Franck-Condon Fac tor q v t v ~ ~ Surface f o r M g O (B1Z-.tA1Tl ) Band System ......................... 3

Condon Loci f o r MgO (B'L-tAlTT ) Band System ..... 3 l . l b

2 . 1 Vibra t iona l Wave Function, v -= 5 .................. 1 3

2.2 Overlapping of p a i r s o f ant inodes ................. 1 4

2.3 Reduced Vibra t iona l wave func t ion ................. 1 6

2.3a even v i b r a t i o n a l quantum number

2.3b odd v i b r a t i o n a l quantum number

3.1 Antinode l o c i of simple harmonic o s c i l l a t o r . . . . . . . 20

3.2 Nomenclature of ant inode coincidence on p o t e n t i a l ener,gy diagram .................................... 26

3.3 Pos i t ion ing of ant inode coincidence on v h l l plane f o r r < re ...................................... 26

3.4 Trans i t ion i l l u s t r a t j n g primary l o c u s ............. 28

' l l I e

3.5 Segments of primary locus ......................... 28

3.6 Loci of coincidence of ant inode p a i r s , MgH C -+X .. 35

3 . 7 Loci of coincidence of ant inode p a i r s , MgH A - t X . 36

3.8 Progression p l o t s of over lap i n t e g r a l of MgH C - b X . 37

3.9 45 Loci of coincidence of an t inode p a i r s , CO A -+X .. . 3.10 48

3.11 Primary Condon locus .............................. 50

Loci of coincidence of an t inode a p i r s , C2 R - + X ...

3.12 Approximate l o c i of coincidence of an t inode p a i r s , c0 A +x ......................................... 53

x

I I I I I I I I I I I I I I I I I I I

4 . 1

4 . 2

4 . 3

4 . 4

4 . 5

A l . 1

A1.2

A l . 3

A1.4

A2.1

A2.2

A2.3

A2.4

A2.5

A2.6

A2.7

xi

P o t e n t i a l energy o f a Morse diatomic molecule ...... 55

Antinode pos i t i ons of CO+ A and X states ........... 58

Loci of coincidence of antinode p a i r s , MgO B - X ... 63

Loci of coincidence of antinode p a i r s , CO A + X ... 66

Loci o f coincidence of antinode p a i r s , C2B-X ...... 68

Loci of coincidence of antinode p a i r s , C@ A . 4 X . . . 75

Loci of coincidence of antinode p a i r s , COf A 3 X . . . 81

Loci of coincidence of ant inode p a i r s , O2 B * X .... 84

Loci of coincidence of antinode p a i r s , GaI A + X ... 88

Loci of coincidence of antinode p a i r s , N$ BcN2X ... 93

Loci of coincidence of antinode p a i r s , M g O B + A ... 97

Loci o f coincidence of antinode p a i r s , CO' A + X ... 99

Loci of coincidence of antinode p a i r s , NfzC tN2X . . . . 102

Loci of coincidence of antinode p a i r s , NZDcN2X . . . . 104

Loci of coincidence of antinode p a i r s , O2 B - * X .... 107

Loci of coincidence of ant inode p a i r s , G a I A-+X ... 110

CHAPTER I

THE FRANCK-CONDON PRINCIPLE

I N DIATOMIC MOLECULAR SPECTROSCOPY

1.1 - In t roduc t ion

The i n t e r p r e t a t i o n of emission and absorpt ion i n t e n s i t y

measurements of atomic and molecular spec t r a i s an important

p a r t of contemporary astrophysics, space physics and l a b o r a t o r y

atomic and molecular physics.

S p e c t r a l i n t e n s i t i e s a r e con t ro l l ed , a p a r t from populat ion

and energy quantum f a c t o r s , by t h e t r a n s i t i o n s t r e n g t h S. In

t h e general case t h i s i s t h e square of t h e quantum mechanical

ma t r ix element of t h e d i p o l e moment,

In t h e case o f one r o t a t i o n a l l i n e of a diatomic molecular spectrum

i t i s

1

2

where S A ' J 1 i s t h e H"o-London f a c t o r o r l i n e s t r eng th , R e ( r )

i s t h e e l ec t ron ic t r a n s i t i o n moment which can be r e l a t e d t o t h e All J1.I

"band-system strength", and qv lv2, i s t h e Franck-Condon f a c t o r

o r band s t r eng th . The H'dnl-London f a c t o r s are we l l known simple

polynomials of J and A . The e l e c t r o n i c t r a n s i t i o n moment i s

t h e matrix element of t h e e l e c t r i c d i p o l e moment with r e s p e c t t o

t h e two e l e c t r o n i c wave func t ions

The Franck-Condon f a c t o r i s t h e square of t h e overlap i n t e g r a l of

t h e v i b r a t i o n a l wave func t ions involved:

This f a c t o r e x e r t s t h e dominating c o n t r o l on t h e d i s t r i b u t i o n of

i n t eg ra t ed band i n t e n s i t i e s over a band system.

f a c t o r array may be represented as a three-dimensional su r f ace (see

f i g . 1 , l a ) which i s s t rong ly undulatory with a s e r i e s of r i d g e s

o f t e n roughly symmetrical about t h e l i n e v1 = VI!.

r ep re sen t t h e p o t e n t i a l l y s t rong bands.

r i d g e s on t h e V ~ I I p lane i s a s e r i e s of nested open curves ( see

f i g . l . l b ) c a l l e d Condon l o c i .

The Franck-Condon

The r i d g e s

The p r o j e c t i o n o f t h e

It i s t h e purpose of t h i s t h e s i s t o d i s c u s s t h e circumstances

which control t h e geometrical p r o p e r t i e s of t h e l o c i . Such a study

has obvious a p p l i c a t i o n s t o t h e i d e n t i f i c a t i o n of molecular spec t r a

and t o the p r e d i c t i o n of t h e p o s i t i o n s of o t h e r s t r o n g bands of

a system which i s only known f r agmen ta r i ly .

3

I I I I I I I I I I I I I I I I

x L 0 .- E

t 5 m ii

0

E CI)

L Y- 0

0 U

4

1 . 2 The I n t e n s i t y D i s t r i b u t i o n i n Band System? of Diatomic Molecules - Condon Loci

If t h e observed bands of a system are placed i n a two-dimensional

array of v i b r a t i o n a l quantum numbers vt and V I ' (Deslandres Table),

i t i s noticed t h a t f o r n e a r l y a l l t r a n s i t i o n s t h e most i n t e n s e f a l l

on d i s t i n c t quasi-parabolic l o c i .

i n s i d e the o t h e r and are sa id t o be nested. The outermost l o c u s

r e l a t e s t o t h e s t ronges t bands of t h e system and i s known as t h e

primary Condon locus .

l o c i .

a r r a y of t h e MgO BIZ - AIlT system.

degenerate i n t o a s i n g l e diagonal l i n e .

t r a n s i t i o n s t h e l o c i appear as p a r a l l e l diagonal l i n e s , t h e main

diagonal (vf = VI! ) r e l a t i n g t o t h e most i n t e n s e bands.

The successive l o c i l i e one

The i n n e r l o c i are t h e subs id i a ry Condon

The l o c i a r e c l e a r l y i l l u s t r a t e d i n t a b l e 1.1 by t h e i n t e n s i t y

For some t r a n s i t i o n s t h e l o c i

Fu r the r , f o r a f e w o t h e r

The l o c i of i n t e n s e bands are c l o s e t o t h e l o c i of high Franck-

Condon f a c t o r s , t h e proximity being governed by t h e accuracy of t h e

v i b r a t i o n a l model used i n t h e i r c a l c u l a t i o n .

1 . 3 The Franck-Condon P r i n c i p l e and Frsnck-Condon Fac to r s

The mechanism of photodissociat ion of a diatomic molecule was

discussed q u a l i t a t i v e l y by Franck (1926) i n a s tudy of t h e s t r o n g e s t

bands of a system. He explained t h a t a spontaneous e l e c t r o n i c

t r a n s i t i o n a f f e c t s n e i t h e r t he p o s i t i o n n o r t h e momentum of t h e

n u c l e i d i r e c t l y , f o r t h e e l e c t r o n i c rearrangements occur much more

r a p i d l y than nuc lea r rearrangements.

nuc lea r separat ion and nuc lea r momentum remain unchanged i n a

Thus t h e in s t an taneous i n t e r -

5

Table 1.1

Band I n t e n s i t i e s of MgO B + A (Mahanti 1932)

V " V' 0 1 2 3 4 5 6 7 8 9 10 11

0 6 - 4 - - 3

I 5 2 3 I I

2 4 3 1 . 2 A.,l '1 I'\. 0

I / '\

3 3 3 2 i ' .\ 5.

4 2 ; 2 2 1 0 x . 0 ~'\ \

5 1 1 '1

0 \ '\

6 1\ 1 \

.'\ 7 1

8

9

10

11

\ 0 '\

1

0

\ 1, 1 0

\

1 1

1 2 1

13 ' 1

1 4

1 5

'J \

1

6

spontaneous t r a n s i t i o n . Such a t r a n s i t i o n i s represented ( f i g . 1.2)

as a v e r t i c a l l i n e from a po in t on t h e lower curve t o a p o i n t on t h e

upper curve on a p o t e n t i a l energy diagram.

on t h e p o t e n t i a l s are t h e c l a s s i c a l t u rn ing p o i n t s of t h e v i b r a t i o n a l

motion.

The r e spec t ive p o i n t s

Condon (1926) q u a l i t a t i v e l y extended t h e theory t o band systems

in emission showing t h a t t h e p re fe r r ed bands l i e on a quasi-parabolic

l ocus on t h e v lv t l plane ( t h e Condon Parabola). Jenkins e t a l . (1927)

found t h a t m o s t of t h e observed p predicted Condon Parabola.

i n s i d e t h i s locus, and no explanat ion was advanced a t t h e time.

Condon (1947) r e c a l l s t h a t following t h e i n t r o d u c t i o n of wave mechanical

i d e a s by Schrzdinger he was a b l e (Condon 1928) t o a s s o c i a t e t h e

i n t e n s i t y o f a band with t h e square of t h e overlap i n t e g r a l of t h e

v i b r a t i o n a l wave func t ions of t h e i n i t i a l and f i n a l s t a t e s , i . e .

I OLIJ w v 1 wvI1 d r 1 . Late r Bates (1952) suggested t h a t i t be

c a l l e d the Franck-Condon f a c t o r . It i s o f t e n denoted symbolically

by qvtV,, as i n eq. 1 .4 .

bands of NO l i e c l o s e t o t h e

A f e w bands however were found t o l i e

Franck-Condon f a c t o r arrays have been c a l c u l a t e d f o r a l a r g e

number of t r a n s i t i o n s f o r a v a r i e t y of molecular p o t e n t i a l f u n c t i o n s

using many d i f f e r e n t methods, e.g. by Hutchisson (1930) , WeWi (1934),

Gaydon and Pearse (1939), Pi l low (1949), Aiken (1951), Jarmain e t a1

(1955), Nicholls (1962a), Jarmain (1963) and F l i n n e t a1 (1964) t o

name o n l y a few. In a l l cases, except t hose s tud ied by Wehrli,

l a r g e values o f t h e f a c t o r were found t o l i e on nested l o c i and t o

7

L

c 0 .-

8

be associated with s t rong bands, t h e o u t e r one being c a l l e d t h e

primary Condon locus and t h e i n n e r ones subs id i a ry Condon l o c i ,

though i n sone cases t h e l o c i were confined t o t h e V I = VI! diagonal.

The study by Wehrli was of t h e simple harmonic o s c i l l a t o r model,

t h e i n i t i a l and f i n a l s t a t e s having t h e same equi l ibr ium i n t e r n u c l e a r

separat ion.

overlap i n t e g r a l of p a i r s of v i b r a t i o n a l wave func t ions of odd

d i f f e rence i n v i b r a t i o n a l quantum number.

t h i s s i t u a t i o n i s no t i n accordance with t h e c l a s s i c a l Franck-Condon

P r inc ip l e .

semi-quantum form of t h e p r i n c i p l e discussed i n t h i s t h e s i s .

I n t h i s ca se t h e r e i s complete c a n c e l l a t i o n i n t h e

Wehrli concluded t h a t

It fol lows t h a t t h i s case i s no t i n accordance with t h e

The following chap te r s of t h i s t h e s i s show t h e relevance of

t h e Franck-Condon f a c t o r t o t h e i n t e n s i t y of a s p e c t r a l band and

the d i s t r i b u t i o n of i n t e n s i t y i n a band system. The semi-quantum

form of the Franck-Condon P r i n c i p l e advanced t o account f o r t h e

geometrical a s p e c t s of t h e d i s t r i b u t i o n i n t h e vlvll plane i s d i s -

cussed, and examined f o r a number of band systems us ing simple

harmonic and Morse molecular models.

CHAPTER 2

CONCEPTS

2.1 Band I n t e n s i t i e s and Franck-Condon Fac to r s

The concept of t r a n s i t i o n p r o b a b i l i t y which c o n t r o l s s p e c t r a l

i n t e n s i t i e s i s reviewed i n t h i s chapter .

i n t e n s i t y i n a band system of a diatomic molecule i s c o n t r o l l e d

i n a l a r g e p a r t by t h e degree o f overlap between t h e v i b r a t i o n a l

wave func t ions of t h e upper and lower l e v e l s o f t h e t r a n s i t i o n as

asserted by t h e Franck-Condon P r i n c i p l e and explained i n t h e

in t roduc t ion .

The d i s t r i b u t i o n of

The i n t e n s i t y LL of r a d i a t i o n of t h e energy quantum Ed

emitted from a n o p t i c a l l y t h i n gas i s p ropor t iona l t o t h e p r o b a b i l i t y

of t h i s t r a n s i t i o n , t h e population of t h e upper state Nu , and t h e

energy quantum. Thus

IUL = A u L Nu E,L ’ ( 2 -1)

where AuL i s t h e E i n s t e i n A c o e f f i c i e n t , t h e spontaneous t r a n s i t i o n

p r o b a b i l i t y .

The E i n s t e i n A c o e f f i c i e n t i s def ined (see Al l en 1962) i n terms

of t h e l i n e s t r eng th SuL by

9

10

33 SUL - 6 4 T

3hc LlL gu *UL - -

where gu i s t h e s t a t i s t i ca l weight of t h e upper s t a t e , u, SuL i s

t h e frequency of t h e quantum of energy Ea, and SuL i s t h e t r a n s i t i o n

s t r eng th which may be separated i n t o p a r t s dependent on t h e e l e c t r o n i c

and v i b r a t i o n a l motion and t h e r o t a t i o n a l motion v i z .

'sv ?VI1 i s known as t h e band s t r e n g t h and i s given as

(2.5)

S J f h 1

a l l important t r a n s i t i o n s as simple func t ions of J and A (see

i s known as t h e Hkl-London f a c t o r and has been t abu la t ed f o r J" A,'

Herzberg 1950). However i n t h e d i scuss ion of i n t e g r a t e d band in t en -

SJ'*' and S J r A t a r e required. They are t h e JJJ J I I Aqt J ? J O I ~ l l

s i t i e s 2 s t a t i s t i c a l weights of t h e J1 and JlI levels r e s p e c t i v e l y and equal

t o 25, + 1 and 2J"J + 1, The s ta t i s t ica l weight of t h e upper s ta te ,

g,, may be expressed i n terns of t h e e l e c t r o n i c s ta t i s t ica l weight, d,:

g, = d, (251 + 1). (2.6)

Hence f o r one r o t a t i o n a l l i n e of a spectrum eq. 2.2 may be w r i t t e n

and an A c o e f f i c i e n t f o r a band de f ined as

11

(2.10)

where 3 v ? v l l i s an average frequency f o r t h e v ? v f r band. F rase r

(1954) showed t h a t although the e l e c t r o n i c t r a n s i t i o n moment v a r i e s

markedly with r, an r-centroid, FvtvlI may be def ined so t h a t

= Re2 (Fvtv1i) q v t v t i

where qvtvii i s t h e Franck-Condon f a c t o r .

The i n t e g r a t e d i n t e n s i t y of a band i s thus

(2.12)

(2.13)

where N v r i s t h e population o f t h e upper l e v e l V I . The Franck-

Condon f a c t o r v a r i e s from band t o band by many orders o f magnitude,

dominating the e f f e c t on the i n t e n s i t y of t h e d i s t r i b u t i o n of popu-

l a t i o n among t h e l e v e l s v ? and t h e v a r i a t i o n of t h e wavelength over

a band system.

The va lues of v ? and vrl cf s t rong bands de f ine t h e Ccndcn l o c i

on t h e vfv1I plane ( t a b l e 1.1 and f i g . 1.1).

a r e nested t h e o u t e r (primary Condon locus) i s c l o s e l y assoc ia ted

with t h e equivalence of turning p o i n t s of t h e c l a s s i c a l v i b r a t i o n a l

lk cases where t h e l o c i

motion because of t he l a r g e con t r ibu t ion t o t h e overlap i n t e g r a l

t h e al igned terminal ant inodes of t h e v i b r a t i o n a l wave func t ions

1 2

of

W V .

These funct ions, W

The terminal loops are considerably l a r g e r than t h e i n t e r n a l loops,

because W has g r e a t e r "wavelength" i n t h e t e rmina l region than

elsewhere.

than t h a t under an i n t e r n a l loop.

, are o s c i l l a t o r y with v+l ant inodes ( f i g , 2.1) .,

This means t h a t t h e area under a t e rmina l l oop i s g r e a t e r

The nomenclature used t o spec i fy an ant inode i s shown i n f i g . 2.1.

The antinode p o s i t i o n i s denoted by r with two subsc r ip t s .

i s e i t h e r 1 o r 2 depending on whether t he ant inode i s c l o s e s t t o t h e

l e f t o r right-hand tu rn ing p o i n t of t h e v i b r a t i o n a l motion. The

second subscr ipt i s t h e number, p, of t h e ant inode counting inwards

from t h e terminal ant inode. The terminal , o r primary, ant inodes

a r e thus rlY1 and r2,1; t h e secondary rlYa and r2,2; t h e p th

and r

The f i r s t

rlYp

2,P' An antinode of a v i b r a t i o n a l wave f u n c t i o n . o f one e l e c t r o n i c

s t a t e l y ing a t t h e same i n t e r n u c l e a r d i s t a n c e as one of ano the r state

w i l l make a l a r g e con t r ibu t ion t o t h e overlap i n t e g r a l ( f i g . 2.2a)

p a r t i c u l a r l y i f t h e loops a r e l a r g e and of comparable wavelength

( f i g . 2.2b).

a d d i t i o n a l l y augmented i f o t h e r ant inodes of t h e two wave func t ions

e x i s t i n t h e same region of i n t e r n u c l e a r s e p a r a t i o n ( f i g . 2 . 2 ~ ) .

The overlap i n t e g r a l of t h e two wave func t ions w i l l be

1 3

Fig.2.1 Vibrational wave function, v=5

14

Fig.2.2 Overlapping of pairs of antinodes (see text)

1 5

2 .2 Suggested I d e n t i t y of Condon Loci

The subsidiary Condon l o c i have a very similar geometrical appear-

ance t o t h e primary Condon locus. A s previously mentioned t h e primary

1.ocus i s t h e r e s u l t of t h e l a r g e c o n t r i b u t i o n t o the overlap i n t e g r a l

by t h e coincidence of terminal ant inodes. It i s suggested (Nicholls

1963a) t h a t t h e r e i s a similar a s s o c i a t i o n of ant inode p a i r s g iv ing

t h e systematic behaviour of the Franck-Condon f a c t o r s and t h e r e g u l a r

geometrical appearance of t h e subsidiary Condon l o c i . Each subs id i a ry

l o c u s i s thought t o be associated with t h e coincidence of a terminal

ant inode of one v i b r a t i o n a l wave func t ion with an i n t e r n a l ant inode

of a v i b r a t i o n a l wave funct ion of t h e o t h e r s t a t e of t h e t r a n s i t i o n ;

t h e p th locus being a s soc ia t ed with a p th antinode.

The geometrical study of Condon l o c i i s thus based on t h e cal-

c u l a t i o n of t h e l o c i in the vtvff plane of t h e coincidence o f ant inode

p a i r s given by t h e cond i t ion r f = rl' o r

r - r where p ~ 1, 2, 3 ... and denotes t h e

1 o r 2, p 1 o r 2, 1 I !I

1 o r 2, 1 1 o r 2, p

o rde r of t h e locus . The coincidence of p a i r s o f i n t e r n a l ant inodes

\I1 , p' = 2, 3 ... , p" = 2, 3 ... ) wilL n o t (r ' = r 1 o r 2 , p ' 1 o r 2, PI '

d e sc r ibe new l o c i on t h e v f v t l plane, but w i l l l i e c l o s e t o t h e l o c i

descr ibed above and give add i t iona l con t r ibu t ions t o t h e overlap

i n t e g r a l .

The above t reatment i s equivalent t o t r e a t i n g each v i b r a t i o n a l

wave func t ion as a s e r i e s of d e l t a func t ions o f u n i t value a t t h e

ant inodes, and zero elsewhere. This i s i l l u s t r a t e d by f i g . 2.3

16

V= 6

(a)even vibrational quantum number

v=5

(b) odd vibrational quantum number

Fig.2.3 Reduced vibrational wave functions

1 7

which shows both even and odd v i b r a t i o n a l wave func t ions thus reduced.

The wave func t ion i s now given by

W v --= 0 when r # r1 or 2J

v even 1 o r 2, p =- (-1) p1 when r = r

(2.14) v odd 1’ = (-1)’ when r = r

J P when r = r2 Ptl = (-1) v odd

Calculat ion of t h e coincidence of ant inodes w i l l i n general

y i e l d non-integral values o f v i b r a t i o n a l quantum numbers.

u n r e a l i s t i c bu t a high value of overlap i n t e g r a l can be expected i n

t h e surrounding region on t h e v f v t t plane.

These are

2.3 Geometry of t h e Primary Condon Locus

The geometry of primary Condon l o c i u s ing a simple harmonic

p o t e n t i a l has been discussed by Manneback (1951) and Nicho l l s (1962b),

They have shown t h a t t h e width of t he skewed d i sp laced parabola

r e s u l t i n g from t h e coincidence of t u rn ing p o i n t s i s a func t ion of t h e

equi l ibr ium i n t e r n u c l e a r separat ion, and t h a t t h e i n c l i n a t i o n of t h e

locus t o t h e main diagonal i s a func t ion of t h e harmonic o s c i l l a t o r

f requencies . No similar study has been made of subs id i a ry l o c i , o r

of t h e l o c i r e s u l t i n g from anharmonic p o t e n t i a l s due t o t h e a l g e b r a i c

d i f f i c u l t i e s .

ca l cu la t ed according t o Nicho l l s f suggestion (1963a) f o r a wide range

of A re and GA/aE values f o r both simple harmonic and Morse

p o t e n t i a l s .

We have however s tudied t h e gene ra l form of t h e l o c i

CHAPTER 3

CONDON LOCI AND THE SIMPLE HARMONIC MOLECK4R POTENTIAL MODEL

3.1 The Simple H a m n c O s c i l l a t o r Model of a Diatomic Molecule

The s implest poss ib l e assumption about t h e form of t h e v i b r a t i o n s

i n a diatomic molecule i s t h a t each atom moves towards o r awiy from

t h e other i n simple harmonic motion. Such a motion o f t h e two atoms

i s equivalent t o the harmonic v i b r a t i o n of a s i n g l e mass po in t

having the reduced mass ( E " ) of t h e molecule about an equi l ibr ium

pos i t i on ( r e ) .

(Pauling and Wilson 1935) and have been t abu la t ed by Gaydon and

Pearse (1939) and Pi l low (1949).

overlap i n t e g r a l based on SHO wave func t ions have been ca l cu la t ed

by Aiken (1951) and discussed by Manneback (1951).

harmonic o s c i l l a t o r model i s thus very s u i t a b l e f o r ou r study o f

t h e systematics o f ant inode coincidence and Condon l o c i .

The wave func t ions of t h i s v i b r a t i o n a r e we l l known

Sixty-four arrays of v i b r a t i o n a l

The simple

The Schr'ddinger wave equation of a simple harmonic system i s :

(3.1)

where x ~~ r - re , t h e displacement of t h e mass p o i n t from t h e

18

1 9

equi l ibr ium p o s i t i o n , and I) i s t h e v i b r a t i o n a l frequency of t h e motion.

The eigenvalues of .3 .1 are

E(v) = h 3 (V + 4) (3.2)

where v i s t h e v i b r a t i o n a l quantum number t ak ing i n t e g r a l va lues

(0, 1, 2 ...) only. The eigenfunctions of 3 . 1 a r e known t o be t h e

Hermite orthogonal func t ions (see Pauling and Wilson 1935)

(3 .3 ) 2

where '4 = & x

and H, = (-1) v e 5 2 d" e- 82 df2

n o m a l i s i n g cons t an t (3 .6 )

t h e Hermite poly- (3.7)

nomial of o r d e r v

The energy leve l diagram showing t h e p o t e n t i a l energy, t h e

v i b r a t i o n a l l e v e l s and t h e wave func t ions f o r a few l e v e l s o f a simple

harmonic o s c i l l a t o r are shown i n f i g . 3.1. Antinode p o s i t i o n s of

t h e wave func t ions are joined and t h e quasi-parabolic curves l a b e l l e d

wi th t h e appropr i a t e value of p (see a l s o f i g . 2.1).

It must be emphasized t h a t t h e simple harmonic o s c i l l a t o r model

i s of l i m i t e d app l i ca t ion .

motion of a diatomic molecule only c l o s e t o t h e equi l ibr ium p o s i t i o n

and a t low va lues of v i b r a t i o n a l quantum number.

It i s a r ea l i s t i c r e p r e s e n t a t i o n of t h e

20

1 1 1 1 1 , 1 1 1 1 , 1 1 , - 7 - 0 - 5 - 4 - a - 2 - 1 o I 2 3 4 s e 7

v- bp= a,lk I "P Fig 3.1 Antinode loci of slmple 'larrnoqqc oscillator

21

3 . 2 Reduction of Antinode Pos i t ion t o Algebraic Form

In o u r study it was necessary t o spec i fy a l g e b r a i c a l l y t h e p o s i t i o n

of an ant inode of a v i b r a t i o n a l wave func t ion i n order t h a t t h e coin-

cidence of p a i r s of ant inodes could be ca l cu la t ed . Attempts t o f i nd

a n a l y t i c a l r e l a t i o n s from the wave func t ions having proved f r u i t l e s s ,

empir ica l equat ions of t h e antinode l o c i shown on f i g . 3 . 1 have been

developed by Nichol ls (1963b). For va lues of p = 1, 2, 3 . O O 8 an

expression has been found o f the form

v - b P = a P Itl“P ’

o r

(3 .8 )

( 3 . 9 )

where ap, b der ived by f i t t i n g log-log p lo t s t o eq. 3 . 8 . From eqs. 3 .4 , 3 .5 and

and np a r e the empir ica l cons t an t s given i n t a b l e 3 . 1

3 .8 n 2 n -b, = (k We) (r - re) ,

“P 4 n 2 c r - where k =

h

( 3 .lo)

(3 .11)

The p o s i t i o n r o f t he pth antinode i s t h e r e f o r e

The former case of eq. 3 .12 (rl,p) a p p l i e s when r < re ( i . e . ant inodes

t o t h e l e f t ) and t h e l a t t e r (r

t h e r i g h t ) .

) when r > re ( i .e . an t inodes t o 2,P

22

Table 3.1

Empirical Constants Describing Antinode Loci

P bP

0

2

4

6

8

10

1 2

1 4

“P

0.809

1.80

2.29

2.79

3.03

3.42

3.51

3.57

nP

1.79

1.38

1 , 2 8

I .17

1.14

1.1

1 .o

1.0

23

3.3 Coincidence of Antinode P a i r s

It i s our a i m t o assess t h e v a l i d i t y of t h e approximate form of

t h e Franck-Condon P r i n c i p l e which considers t h e v i b r a t i o n a l wave

func t ion as a s e r i e s of d e l t a func t ions a t t h e ant inodes and p r e d i c t s

l a r g e values of t h e overlap i n t e g r a l when t h e r e i s alignment ( i .e .

equality of r) between one d e l t a func t ion of each 1eve.L of t h e t r a n s i -

t i o n .

plane i s known t o be associated with t h e alignment of primary antinodes.

This suggests t h a t t h e secondary l o c u s may h e a s soc ia t ed with t h e

alignment o f a primary and a secondary ant inode; t h e pth l o c u s with

t h e alignment of a primary and p th ant inode.

c a l c u l a t e t h e l o c i on t h e v f v r l plane of t hese cases of coincidence

and compare them with t h e Condon l o c i exhibi ted i n t a b l e s of overlap

The primary Locus of high Franck-Condon f a c t o r s on t h e v1v1I

We s h a l l t h e r e f o r e

i n t e g r a l s .

The p o s i t i o n of t h e p th ant inode of t h e upper s ta te i s r ' ( l e f t ) l , P 1

o r r I ( r i g h t ) which i s given by equat ion 3.12 2,P'

S i m i l a r l y t h e p th ant inode of t h e lower s t a t e i s a t

(3 .13)

(3.14)

The approximate form of t h e Franck-Condon P r i n c i p l e under considerat ion

r e q u i r e s t h e c a l c u l a t i o n of v t and va lues when t h e condi t ion 11

r (3 .15 ) I r 1 o r 2,pI 1 or 2,p"

24

i s applied, i . e .

The p th locus descr ibed above i s def ined by eq. 3.17 where p ' - 1,

pll =- p, and p' =- p, pt l = 1, p = 1, 2, 3 .... (This i s alignment of

a primary and a pth ant inode) . Eq. 3.17 then becomes

P , l o(. + -i O C L P + - K " V p , l JP o r -Are -K ' V i

1 where K -~

(k L+)i

V - b v r - a

(3.19)

(3.20)

(3.21)

and o c = - 1 . (3.22)

For p #l, eq. 3.19 con ta ins t h e e i g h t i d e n t i t i e s o f eq. 3.15.

n

Only

I t 1 six of these a r e phys i ca l ly achievable. When re < re t h e six cases a r e : I1

25 11 1

rl,l = r2,p C O'1 !JI "Lp ,Are I= + K I V: + K f l Vp

(3.23)

The untenable cases a r e

I 11 1 11

and '2,p = '1,J -

2 , l - r

I1 1 s ince we have s t i p u l a t e d r < re . The poss ib l e coincidence cases ,

a - f of eq. 3.23, a r e i l l u s t r a t e d i n the t r a n s i t i o n of f i g . 3.2 and

i n t h e Deslandres diagram o f f i g . 3.3 which shows the segmented l o c i .

e

The d i s t i n c t n e s s o f t h e segments of t h e l o c i has been exaggerated.

There i s however no d i scon t inu i ty of s lope a t t h e junc t ion of segments,

t h e s lope becoming r ap id ly zero o r i n f i n i t y very c lose t o t h e junc t ion .

n For an e l e c t r o n i c t r a n s i t i o n i n which re 7 rl f i g . 3.3 would be transposed as f o l l o w s :

t h e l e t t e r s on

a - e and b - f .

UPPER STATE\

26

temi na I anti nodes

of pm antinodes

I I C (

I" I e T

us of pth antinodes

Fig.3.2 Nnnenclature Of antinode coincidence on potential energy diagram for t& 6

Fig.3.3 Positioning of antinode coincidence on v ' v" plane for rg < r i I

27

For p = 1 eq. 319 conta ins t h e fou r i d e n t i t i e s of eq. 3.15.

I Three o f these a r e phys i ca l ly achievable.

cases a r e :

When r: < re t h e th ree

a and b) rl = rtl 1,l 1,l

c and d ) rl = rtl 1,l 2 , l (3.24)

I I e and f ) r1 = r 2 , l 2 , 1

These cases a r e i l l u s t r a t e d by t h e t r a n s i t i o n i n f i g . 3.4 and i n the

Deslandres diagram of f i g . 3.5.

3.4 The Form o f t h e L o c i o f Antinode Coincidence

The l o c i a r i s i n g from t h e coincidence of p a i r s f ant inodes has

been ca l cu la t ed f o r t he t r a n s i t i o n s l i s t e d i n t a b l e 3.2 by solving

eq. 3.19 numerically for p a i r s o f vIvtl values .

p a i r s of ant inodes alone con t ro l s t h e magnitude of t h e overlap i n t e g r a l ,

then t h e l a r g e s t values w i l l occur when te rmina l ant inodes co inc ide

(p = 1, i . e . eq. 3.24). There will be l e s s e r maximum values when a

te rmina l ant inode coincides with a subs id ia ry (p # 1 i n eq. 3.23).

Nichol ls (1963a) suggests t h a t t h i s descr ibes t h e p th Condon locus .

If coincidence of

The g raph ica l appearance of the so lu t ion t o eq. 3.19 i s l a r g e l y

dependent on t h e s i z e of b re . with A r e ranging from zero t o 0.397 A . Three cases r ep resen ta t ive

Of Smal l (o*049 A and 0.051 62>, medium (0.1069 i) and l a r g e (0.2233 i)

Eight t r a n s i t i o n s have been s tudied 0

0

28

\

d b car

L Fig.3.4 Transition i I lustmting primary locus

V" V '

a and b \ Fig.3.5 Segments of primary locus

29

4 Q, d d u)

0

Ob

9 9 a, a o 0

N a-l W r) W co M b 0 N N m

M

0 0 0 0 I I @?

0

r) a, 0 r)

I

a, m W r- Q, rl 0 r) 03 v) hl d

rl rl d rl

$ 0 I I I ?

O b = a , L

Lo LD 0 0 r) l-9 b b

rl

h

0 v) Q, rl

9 w a-l r) d r) CY

a, rl LD 0 u) r) u) d r) d r) 0 hl N v) W

u) hl ol OD r- LD W

k a) -a, P L N

4 ri rl ri d d d L a, X

E 0

pr) k r,

A 0

W rl

d v)

m r) r- c.1

I

V E b.

W d r4

r- u) m d rl

= w 3 3 x

5 ..+ 3

d I

5 hl

r) m d

hl

0 v) co rl

9 4 W W W W 0 u) pr) a,

r i e cd h

-

v) hl W 0 rl W 0 0 u) 10 d r- 4 rl rt

0 d b -a,

3

+

bl N

X

T

bl ri X

w ..+ ffl

I= N d

F ri b

d N r) r) 4 4 m m

r,

cd w cd Q

rl hl 0 0 r)

0 a, r? d

N a-l co rl u) 0 co W W co

9

rl d io v) a,

W

N co LD ?

0 d 0 0 <s d

tc u +

0 0 N N u V u 0

30

va lues o f Are

cases a re discussed i n Appendix 1.

are discussed i n d e t a i l i n s e c t i o n 3.5, t h e o t h e r

The r e s u l t s may be summarised as fol lows:

A t zero a re (eg. G a I Appendix 1 .4) t h e r e a r e no c and d segments,

but a and e, and b and f which a r e separated f o r a l l but t h e coincidence

of terminal ant inodes. The sec t ions a and e are co inc iden t due t o t h e

symmetry o f t h e ant inode l o c i , as a r e b and f . The primary locus

(p = 1) i s a s i n g l e l i n e on t h e vfv" plane c l o s e t o v f = v". The

secondary l o c u s (p = 2 ) c o n s i s t s of a l i n e on each s i d e of t h e primary

locus. The subsequent subs id i a ry l o c i (p = 3 , 4 ...) are p a i r s of

l i n e s external t o t h e lower o r d e r l o c i . There i s no n e s t i n g of l o c i .

A t s l i g h t l y la rger ' A re (eg. MgH sec t ion 3.5.1) segments c and d

a r e very sho r t , and sec t ions a and e s l i g h t l y separated, as are b and f .

The primary locus c o n s i s t s o f two l i n e s which are very c l o s e nea r

v f = VI! = 0 and open out a t l a r g e r v f and VI! . Each subs id i a ry locus

c o n s i s t s of two s l i g h t l y open three-segment l o c i , one each s i d e of

t h e primary locus.

A t l a r g e r Are (eg. CH+ Appendix 1.1 and CO s e c t i o n 3.5.2) t h e

l o c i become broader and c r o s s each o t h e r . The i n t e r n a l l imbs o f

t h e secondary l o c u s , f o r example, l i e f o r t h e most p a r t i n s i d e t h e

primary locus.

This t r end i s continued with i n c r e a s i n g A re (eg. CO' Appendix

1 . 2 and C2 s ec t ion 3 . 5 . 3 ) , segments c and d i n c r e a s i n g i n l eng th

and t h e i n t e r n a l l i m b s of each subs id i a ry locus c r o s s i n g i n s i d e t h e

region described by the primary locus .

31

A t very l a r g e A r e (eg. 02 Appendix 1.3) segments c and d occupy

m o s t of t h e v f v " a r r a y and each segment i s crossed many t imes.

I n t h e remaining p a r t of t h i s chapter t h e t h r e e r ep resen ta t ive

t r a n s i t i o n s a r e discussed i n d e t a i l . The suggest ions put forward

i n t h e i r d i scuss ion a r e a l s o based on the t r a n s i t i o n s discussed in

Appendix 1.

cidence a r e compared with t h e a r r ays of over lap i n t e g r a l and measured

band i n t e n s i t i e s where ava i lab le .

those ca l cu la t ed by Aiken (1951).

modulus compared with t h e i r neighbours a r e underlined and connected

by curves which a r e t h e Condon l o c i . The Condon l o c i a r e a l s o

ind ica t ed on the i n t e n s i t y a r rays .

The a r r a y s of ca lcu la ted l o c i o f ant inode p a i r coin-

The overlap i n t e g r a l a r r a y s a r e

The e n t r i e s which a r e of l a r g e

3.5 Detai led Examination of Representative Types of - Trans i t ion

3.5.1 MgH C2n+X2z -- and A2n + X 2 L

These two band systems a r e very s i m i l a r and will be examined 0 0

The A re values a re very small (0.049 A and 0.051 A toge ther .

r e spec t ive ly ) and have been chosen t o i l l u s t r a t e t h e v a l i d i t y of our

approximate form of t he Franck-Condon P r i n c i p l e i n t h i s region.

The systems occur i n the flame of burning magnesium i n a i r ,

and i n t h e absorp t ion spec t ra o f sunspots. No i n t e n s i t y measurements

a r e a v a i l a b l e but f i v e bands o f t he C + X system, and e i g h t bands of

t h e A - + X system a r e known ( t ab le s 3.3 and 3.4). The former l i e

i n t h e u l t r a - v i o l e t region of t h e spectrum ( A -2500 1) and t h e

l a t t e r i n t h e red ( A -6000 A ) . 0

32

Table 3 . 3

Observed Bands o f MgH C + X (Pearsc and Gaydon -1950)

o x x l X X X

2

Table 3 . 4

Observed Bands of MgH A+X (Pearse and Gaydon 1950)

VI'

v t 0 1 2 3

o x x x l X X X

2 x x 3

33

On t h e a r r a y s o f overlap i n t e g r a l ( t a b l e s 3 . 5 and 3 . 6 ) both

systems show a primary fklosedll Condon locus , t y p i c a l o f smallA re.

This locus l i e s a long the main diagonal f o r t h e A + X system, and

s l i g h t l y inc l ined t o higher v" f o r C 3 X .

between the two a r r a y s i s t h e occurrence o f a subs id ia ry locus of

high overlap i n t e g r a l e n t r i e s i n t h e C -+ X t r a n s i t i o n a r r ay .

l i e s d o n g the A v = v f - v" = -3 sequence.

Figs . 3 . 6 and 3 . 7 show the l o c i of ant inode coincidence ca lcu la ted

An i n t e r e s t i n g d i f f e r e n c e

This

by t h e method out l ined i n sec t ion 3 . 3 of t h i s t h e s i s .

e x h i b i t s l i g h t l y open l o c i of t he coincidence o f p a i r s of primary

ant inodes s t r add l ing the main diagonal . That of A -+X i s almost

symmetrical about t h e main diagonal, but t h a t of C + X i s s l i g h t l y

inc l ined t o lower VI!, i . e . i n t h e oppos i te d i r e c t i o n t o t h e overlap

locus . The l o c i of coincidence of te rmina l ant inodes with i n t e r n a l

an t inodes l i e ou t s ide t h e primary locus due t o t h e shor tness o f t h e

curved sec t ions c and d .

Both systems

The r e l a t i o n s h i p between t h e ca l cu la t ed l o c i and t h e Condon l o c i

i s no t immediately apparent and it i s necessary t o examine each con-

t r i b u t i o n t o t h e overlap i n t e g r a l .

i n t e g r a l f o r t h e C 3 X a r r ay a r e given i n f i g . 3 . 8 . The systematic

progression of t h e shape o f the p l o t i s due t o t h e c o n s i s t e n t na tu re

of t h e e n t r i e s on each diagonal. Table 3 . 7 i s an examination of each

e n t r y of t h e overlap i n t e g r a l a r r a y . The s igns of t h e wave func t ions

a t t h e r e l evan t ant inodes a r e given, t h e s ign of t h e product of t h e

an t inodes i n coincidence, and t h e s ign of t h e overlap i n t e g r a l .

v" progression p l o t s of overlap

34 Table 3.5

Overlap I n t e g r a l s of MgH C 3 X

,.I1

V I v . - 0 1 2 3 4 5 6 7

0 0.9717 -0.2353 -0.0116 0.0161 -0.0012 -0.0010 0.0002 0.0001

\ \ 1 0.2181 0.9161 -0.3344 -0.0164 0.0319 -0.0029 -0.0023 0.0005 -

\ \ 2 O.OE65 0.2867 0.8598 -0.4101 -0.0181 0.0498 -0.0053 -0.0042

- \ \ 3 0.0247 0.1434 0.3242 0.8032 -0.4726 -0.0171 0.0694 -0.0086

\ -0.5255 -0.0134 0.0902

\ 0*7467\ \

4 0.0084 0.0471 0.1937 0.3433

5 0.0025 0.0182 0.0711 0.2383 0.3492 0.6907 -0.5707 -0.0074 \

6 0.0008 0.0059 0.0303 0.0956 0.2775 0.3450 0.6355 ‘-0.6901 \

7 0.0002 0.0021 0.0107 0.0446 0.1200 0.3114 0.3325 0.5815

Table 3.6

Overlap I n t e g r a l s of MgH A-+ X

V 1’

V ’ 0 1 2 3 4 5 6 7

0 0.9710 -0.2384 0.0158 0.0050 -0.0011 -0.0000 0.0000 -0.0000

\ 1 0.2297 0.9140 -0.3332 0.0286 0.0097 -0.0025 -0.0001 0.0001

\ 2 0.0640 0.3089 0.8580 -0.4028 0.0420 0.0149 -0.0044 -0.0002

\ 3 0.0157 0.1069 0.3591 0.8033 -0.4588 0.0562 0.0205 -0.0067

\ 4 0.0039 0.0304 0.1456 0.3928 0.7498 -0.5055 0.0710 0.0263

\ 5 0.0009 0.0085 0.0466 0.1810 0.4151 0.6977 -0.5451 0.0865

6 0.0002 0.0022 0.0144 0.0637 0.2132 0.4289 0.6470 -0.5792

7 0.0001 0.0006 0.0041 0.0213 0.0815 0.2425 0.4359 0.5977

\

\

35

Fig.3.6 Loci of coincidence of antinode pairs,

MgH C-X

v'

I -

2-

3.

4.

5

6

7


MgH A-X

36

I

8

6

4

2

v' ( I 2 3 4 5 6 7

- -2

- -4

- I 4 - I

37

,8-

.6-

-4-

2-

V'

- .6

- .8

- I

v' (

- .2

- .4

- .6

- .8

-I

F~.3.8 Progression plots of overlap integral of MgH C-X

38

0 1 2 2 3 3 4 5

0 1 2 3 3 4 5 6 6 7 7

0 1 2 3 4 5 6 6 7 7 7

0 1 1 2 3 4

0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1

2 2 2 2 2 2 2 2 2 2 2

3 3 3 3 3 3

Table 3 . 7

Examination of Contr ibut ions t o t h e Overlap I n t e g r a l

of MgH C -P X

Sign of w v 1 yvll Close t o : Sect ion a t ant inode

Primary a , b & e,f + Primary e , f Secondary b Primary e,f Secondary f Primary e , f Tertiary b Te r t i a ry f

Primary a ,b Primary a , b Primary e,f Primary e,f Secondary b Secondary b Secondary f Secondary f T e r t i a r y b T e r t i a r y b T e r t i a r y f

Secondary e Primary a ,b Primary a,b Primary e,f Primary e,f Secondary b Secondary f Primary e , f Secondary f Te r t i a ry b Primary e,f

Secondary a Secondary e Primary a , b Primary a , b Primary a , b Primary e,f

+ + + + + + + + - - + + - - + + - - + - + + + + + + + + + +

+ - - - - +

system

Sign of

a t coincidence of ant inodes

y V I XWVll

3 f W V

- + + + + + + + + + +

- - + + + + + + + + +

39

5 3 6 3 6 3 7 3 7 3

Primary Secondary Primary Secondary Primary

+ + - - + + + - + +

0 4 1 4 1 4 2 4 3 4 4 4 5 4 6 4 7 4 7 4

T e r t i a r y Secondary Secondary Secondary Primary Primary Primary Primary Secondary Primary

+ + + - - -

+ - + - - + + + + +

+ + +

+ - - + + + + + + + + + + +

0 5 1 5 1 5 2 5 2 5 3 5 4 5 5 5 6 5 7 5

T e r t i a r y T e r t i a r y Secondary Secondary Secondary Secondary Primary Primary Primary Primary

- + + + - + + + + + +

- - -

- - + + + +

0 6 1 6 1 6 2 6 3 6 3 6 4 6 5 6 6 6 7 6 7 6

Tertiary T e r t i a r y T e r t i a r y Secondary Secondary Secondary Secondary Primary Primary Pr imary Primary

- + + +

t

+ + -

1 7 2 7 2 7 3 7 4 7 4 7 5 7 5 7 6 7 7 7

T e r t i a r y T e r t i a r y Secondary Secondary Secondary Secondary S e c onda ry Primary Primary Primary

+

+

40

3.5.la The Comparison of Ovv lap I n t e g r a l with Antinode Coincidence

The bands on t h e main diagonal a l l have lbrgc p c s i t i w cver lap

i n t e g r a l s ( t ab le 3.5). They a l l c lo se ly r ep resen t coincidence of p a i r s

of l e f t hand te rmina l ant inodes ( t a b l e 3.7), which g ive a p o s i t i v e

cont r ibu t ion t c t h e overlap i n t e g r a l ( e g . band 3, 3 ) .

of r i g h t hand p a i r s of te rmina l ant inodes a l s o g ives a p o s i t i v e con-

t r i b u t i o n (eg. band 4, 3) , but in this case A v # 0 so t h e contr ibu-

t i o n i s l e s s and no second arm of t h e primary Condon locus appears .

The coincidence

Below the main diagonal ( v f )VI*) a l l t h e overlap i n t e g r a l s

a r e of pos i t i ve s ign, while above ( v f 4 VI!) t h e r e a r e both p o s i t i v e

and negat ive values .

t h e overlap i n t e g r a l due t o p a r t i a l overlap between te rmina l ant inodes

i s g r e a t e r than t h a t due t o t h e overlap of a subs id i a ry ant inode with

a terminal antinode (eg. t h e value for 7 , 3 i s +ve, while -ve i s

expected i f the overlap i n t e g r a l i s dominated by t h e over lap of a

te rmina l and secondary an t inode) .

It i s suggested he re t h a t t h e con t r ibu t ion t o

The sequence A v = v f - VI! = -1, a l s o r ep resen t s c l o s e coin-

cidence of p a i r s of l e f t hand terminal ant inodes. The con t r ibu t ion

t o t h e overlap i n t e g r a l i s l a r g e and negat ive , t h e over lap i n t e g r a l

a r r a y showing t h e same f e a t u r e (eg. 2, 3) .

The coincidence of t h e r i g h t hand te rmina l ant inode of t h e upper

s t a t e with the r i g h t hand secondary an t inode of t h e lower s t a t e

adequately represents t h e form of t h e over lap i n t e g r a l s of t h e

A v = v f - vll = -2 sequence. This con t r ibu t ion which i s negat ive

appears t o be much l a r g e r than t h e p o s i t i v e con t r ibu t ion due t o t h e

p a r t i a l overlap of p a i r s of l e f t hand te rmina l an t inodes (eg. 1, 3 ) .

41

The ad jacent sequence A v = -3 i s notab le due t o t h e over lap

i n t e g r a l s of i t s members being l a r g e r than those o f t h e i r neighbours.

This i s the genera l c r i t e r i o n f o r t h e s p e c i f i c a t i o n of a Condon locus .

This diagonal i s assoc ia ted w i t h t h e p a r t i a l overlap of two p a i r s

of an t inodes ; t h e terminal l e f t ant inode of t h e upper s t a t e wi th the

secondary l e f t ant inode of t h e lower s ta te , giving a p o s i t i v e con-

t r i b u t i o n t o overlap i n t e g r a l ; and t h e te rmina l r i g h t ant inode of

t h e upper state with t h e secondary r i g h t ant inode of t h e lower state,

g iv ing a negat ive con t r ibu t ion t o t h e overlap i n t e g r a l (eg. 2, 5 ) .

The former case of coincidence appears t o g ive t h e l a r g e r con t r ibu t ion

as t h e overlap i n t e g r a l s a r e a l l p o s i t i v e . This i s probably due t o

t h e smaller value o f v1 involved and hence g r e a t e r s i z e of t h e loop

of t h e wave func t ion .

The sequence a v = V I - = -4 i s adequately represented by

t h e coincidence of l e f t hand primary and secondary ant inodes mentioned

i n t h e previous paragraph (eg, 1, 5 ) .

The apparent Condon locus a t A v = -3 must be concluded t o be

due t o t h e abnormally small overlap i n t e g r a l s a long t h e A v = -2

diagonal caused by the opposing con t r ibu t ions of t h e p a r t i a l overlap

of two p a i r s of ant inodes.

A similar study has been made of t h e A -P X t r a n s i t i o n , but from

it nothing can be added t o the comments above.

We a r e a b l e t o conclude t h a t t h e primary Condon locus i n t h e

case of small change i n in t e rnuc lea r separa t ion i s due t o t h e coin-

cidence of p a i r s o f terminal ant inodes. The s i z e o f o t h e r overlap

42

i n t e g r a l s on t h e array can be i n t e r p r e t e d as t h e overlapping of

o t h e r p a i r s of ant inodes.

t h e smallness of neighbouring overlap i n t e g r a l s , r a t h e r than t h e

magnitude of t h e i r own e n t r i e s .

Apparent e x t e r n a l Condon loci are dne ?x

3.5.2 CO AIU +.X1x Fourth P o s i t i v e System

This well-known system has been chosen t o i l l u s t r a t e t h e

v a l i d i t y of t h e approximation a t in t e rmed ia t e values of Are

(Are = 0.1069 8) . Over one hundred bands of t h i s system are found i n emission.

They l i e i n the u l t r a - v i o l e t region o f t h e spectrum (A-2500 1). Some of t h e sho r t wave l eng th bands a r e found i n absorpt ion.

The overlap i n t e g r a l a r r ay , t a b l e 3.8, of t h i s t r a n s i t i o n shows

t h e familiar nested Condon l o c i , t h e primary locus comprising t h e

l a r g e s t e n t r i e s . Primary, secondary and tertiary Condon l o c i are

a l s o c l e a r l y def ined on t h e array of observed bands of t a b l e 3.9.

The coincidence of ant inode p a i r s has been c a l c u l a t e d t o high

quantum numbers (v ' = 20, VI! = 30) and high ant inode numbers (p = 8) .

The calculated l o c i are given i n f i g . 3.9. The primary Condon l o c u s

i s c l o s e l y represented by t h e coincidence of t e rmina l ant inodes; t h e

secondary by t h e i n t e r n a l segments (a and f ) of t h e coincidence of

a terminal antinode (p = 1) of t h e v i b r a t i o n a l wave func t ion of one

s t a t e with a secondary ant inode (p = 2 ) of t h e o t h e r s ta te as suggested

by Nichol ls (1963a); t h e tert iary by a similar coincidence of primary

and t e r t i a r y antinodes.

43

Table 3.8

Overlap h t e g r a l s of CO A 4 X

V I t

V’ 0 1 2 3 4 5 6 7

0 0.3516 0.4593- 0.4683-0.4198 - 0.3462 <1995 0.2689 0.1427

1 -0.5496 -0.3718 -0.0926 0.1422 0.2851 0.3415 0.3363 0.2964 F y

2 0.5632 -0.0292 -0.3221-0.3217 -O..L607 0.0325 0.1853 0.2737

\ 1,

3 -0.4285 0.4003 0.2807 -0.0473 -0.2619 -0.2830 -0.1708 -0.0115

\ \ -- -

1

\ \

4 0.2483 -0.5192- 0.3256 0.1828 -0.0720 -0.2361‘ -0.2555

5 -0.1055 0.4086 ‘‘70,4437 0.2004 0.2752 0.1146 -0.0946 \> ‘. \ \ -\-------l

\ ~\ \

6 0.0270 -0.2189 0.4976 -0.2740 -0.2678 0.0161 0.2360 0.2236 \ \ \ \

7 0.0014 0.0722 -0.3362 0.5050 -0.0786 -0.2775 -0.1436 0.1158

44

v, hl

n

al rl

z

cr a 5 5 .rl k a rd W X a s 0 r) m rl

x b) c, VI w

W

a td m

.. ?

9 o?

r)

W rl

c

hl c.l

rl hl

0 hl

al rl

hl rl

rl rl

0 rl

X x x / ’

X x x x * / / x

X * x p s /’ x

x x

X x x y x

/ X , K x x

/ * x x /

X ’ X x x al x , k x x co x / x x x

45

X

t a 0 0 a

.- a a

L .- 0 0

Y- O

46

3.5.3 C2 B3n 3 X3n Fox-Herzberg System 0

This u l t r a - v i o l e t system of C2 ( A- 3000 A) has been chosen

t o i l l u s t r a t e t h e v a l i d i t y of t h e approximation a t l a r g e va lues of

re, ( Are = 0.2233 i). No i n t e n s i t y measurements are a v a i l a b l e hu t t h e f c u r t e e n obser-"red

bands ( t a b l e 3.10) f a l l on a broad curve a t low V I . The overlap

i n t e g r a l a r r a y ( t a b l e 3.11) e x h i b i t s broad nested Condon l o c i . It

appears t h a t t h e primary Condon l o c u s s u f f e r s a change i n d i r e c t i o n

a t v t = 6, vll = 0; t h e secondary a t V I = 6, vll = 2.

The primary Condon locus i s success fu l ly represented by t h e

ca l cu la t ed locus of t h e coincidence of t e rmina l ant inodes ( f i g . UO).

The sharp turn a t 6,O i s t h e junc t ion p o i n t of t h e curved s e c t i o n s

c ,d and a,b.

cu l a t ed sec t ions c,d and a of t h e coincidence of a terminal and

secondary antinode.

a t a junct ion po in t .

represented by t h e ca l cu la t ed locus invo lv ing a te r t ia ry ant inode.

The secondary Condon l o c u s i s represented by t h e ca l -

Again t h e sharp t u r n i n t h e Condon l o c u s comes

The te r t ia ry Condon l o c u s i s s i m i l a r l y w e l l

3.6 Additional Remarks on t h e Geometry of t h e Primary Condon Locus

The primary Condon locus proposed by Nicho l l s (1962b) a rose from

a simple considerat ion of t h e coincidence of c lass ical t u r n i n g p o i n t s

of a simple harmonic o s c i l l a t o r . The l o c u s i s descr ibed i n terms of

a s i n g l e skewed displaced parabola. On t h e o t h e r hand t h e l o c u s

descr ibed by eq. 3.24 i s more r e a l i s t i c as i t r e p r e s e n t s t h e coin-

cidence of terminal ant inodes of t h e v i b r a t i o n a l wave func t ions .

47

Table 3.10

V VI

0

1

2

3

7

Observed Bands of C2 B + X ( P h i l l i p s 1949)

V "

VI 0 1 2 3 4 5 6

0 x x x x x 1 x x x 2 x x x 3 x x 4 X

Table 3.11

Overlap In t eg ra l s of C2 B + X

0 1 2 3 4 5 6 7

0.0527 0.1146 0.1834 0.2484 0.3008 0.3356- 0.3513-0,3491 /

-0.1396 -0.2517 -0.3265 -0.3459 -0.3091 -0.2287 -0.1234 -0.0124

0.2540 0.3585 0.3391 0.0574 -0.0975 -0.2058--0.2514 /- /-

-0.3660 -0.3641 -0.1792 0.0498 0.2580 0.1989 0.0778

-0.0930 0.0813 0.1976 /-

0 . 4 4 1 7 / E -0.0724 -0.2554 /-

-0.4592 -0.0293 0.2670 0.2387 0.0243 -0.1712 -0.2308 -0.1530

I I /=- 0.4177 -0.1952 -0.2923 -0.0274 0.2077 0.2233 0.0671 -0.1143 1 \

-0.3352 0.3555 0.1420 -0.2033 -0.2325 -0.0203 0.1772 0.2077

43

I 2 3 4 5 6 7 v " \d ' I I I I 1 1 A

I -


Ce B-X

49

This l ocus c o n s i s t s of three d i s t i n c t s e c t i o n s which j o i n

smoothly. Their s lopes change r a p i d l y n e a r t h e junc t ion po in t s

giving an appearance of d i s c o n t i n u i t y of s lope. This p o i n t i s of

no p r a c t i c a l value as p o i n t s on t h e Condon l o c i a r e def ined only

a t i n t e g r a l values of V I and VI!. However, i t i s c l e a r l y n o t

f e a s i b l e t o desc r ibe t h e Condon l o c u s def ined by t h e t h r e e p a r t s

of eq. 3.24 by a s i n g l e equation.

The form of t h e locus has been shown i n f i g . 3.5 which i s

repeated he re ( f i g . 3.11) f o r convenience. We have seen t h a t t h i s

corresponds c l o s e l y t o t h e primary Condon l o c u s except when A r e

i s very s m a l l and when v i s l a r g e assuming t h a t a simple harmonic

p o t e n t i a l i s v a l i d .

concerning t h e two s t a t e s of t he t r a n s i t i o n may be found from t h e

t r i a f lg l e AOB ( f i g . 3.11).

It i s possible t h a t important information

The po in t A i s given by

I A r e l = ~1 v i a1 , (3.25)

The s i d e s of t h e t r i a n g l e a r e

oa -~ v1 Ll\r n l kn1/2 ad n1/2 e “1’

(3.26)

(3.27)

(3.28)

(3.29)

Equations 3.27 and 3.28 give a method of f i n d i n g A r e , a parameter

which i s n o t known f o r a l l t r a n s i t i o n s . The equi l ibr ium i n t e r n u c l e a r

50

0 - a

51

separa t ion i s usua l ly known f o r t h e ground s t a t e of a molecule from

t h e i n f r a r e d spectrum, hence the value of

may be found.

value a s t h e po in t s A and B a r e never c l e a r l y def ined on an i n t e n s i t y

re f o r t h e upper state

Such ca l cu la t ions have been found t o be of l i t t l e

a r r a y .

The r a t i o of t he s ides OA and OB of t h e t r i a n g l e g ives t h e

tangent of t h e angle AB0 o r the i n c l i n a t i o n of t h e locus i n t h e

v lvtt plane ;

t a n AB0 = (- La: ) n u 2 (3 .30) "E n l i s given a s 1 . 7 9 ( t a b l e 3 . 1 ) giv ing an i n c l i n a t i o n of (- 0;) 895

1 *e i n s t e a d of (%)0*5 a s found by Manneback (1951).

d e The width of t h e locus can be measured i n terms of t h e l eng th

of t h e s i d e AB of t h e t r i a n g l e (eq. 3.29) . This i s a func t ion o f A r e ,

/k, de1 andWell , not re alone a s i s genera l ly understood. We

propose t h i s l eng th as a r e a l i s t i c t r a n s i t i o n parameter which desc r ibes

t h e shape of t h e primary Condon locus .

3 . 7 A Simple Method of Finding t h e Subs id ia ry Condon Loci

In t h e i d e n t i f i c a t i o n of bands of a s p e c t r a l system t h e p o s i t i o n s

of t h e Condon l o c i a r e of g r e a t e r p r a c t i c a l i n t e r e s t t o many u s e r s

than t h e Franck-Condon f a c t o r s . We the re fo re suggest a simple method

of l o c a t i n g the Condon l o c i without t h e lengthy and c o s t l y computation

of wave func t ions and overlap i n t e g r a l s .

a parabol ic p o t e n t i a l .

For s i m p l i c i t y we chose

52

F i r s t l y t h e p o t e n t i a l curves of t h e upper and lower states must

be p l o t t e d on a s c a l e o f v ve r sus r. Using t h e equat ions

u = 2 IT' p c 2 2 we (r - re)' , (3 .31)

and E 7- h c we (V + 3) (3 .32)

f o r one s t a t e a p o i n t on t h e parabola i s found a t a convenient value

of v.

t h e p o t e n t i a l parabola i s e a s i l y constructed g raph ica l ly .

curve of the o t h e r s ta te i s similarly constructed.

1 Another po in t i s made a t v = --2, r := re. From t h e s e two p o i n t s

The p o t e n t i a l

The pos i t i ons of t h e ant inodes a r e now found f o r each value of v

by d iv id ing t h e width of t h e parabola by v and marking o f f t h e d i s t a n c e

between the p o i n t s on t h e parabola.

approximately r ep resen t t h e p o s i t i o n s of t h e terminal ant inodes,

The p o i n t s on t h e parabola

and t h e po in t s made between them rep resen t t h e i n t e r n a l ant inodes.

In t h i s way a l l t h e ant inodes of both states may be found.

A procedure described i n d e t a i l i n Chapter 4 s e c t i o n 4 .4 i s now

followed t o l o c a t e t h e values of v t and VI! a t t h e coincidence of

r e l evan t p a i r s of antinodes.

r ep resen t the Condon l o c i ca l cu la t ed by more s o p h i s t i c a t e d and c o s t l y

methods.

The l o c i drawn on t h e vW' plane c l o s e l y

Our method h a s been t r i e d f o r t h e GO Fourth P o s i t i v e System.

The r e s u l t s a re shown i n f i g . 3.12 which i s i n very good agreement

with f i g . 3 . 9 .

53

2 3 4 5 6 7 0 9 IO

Fig.3.12 Approximate loci of coincidence of antinode pairs,

CO A-X

CHAPTER 4

CONDON LOCI AM) THE MORSE POTENTIAL MODEL

4.1 The Morse Model of a Diatomic Molecule

The motion of a diatomic molecule can be safely represented

by a harmonic oscillator only close to the equilibrium position

and for low quantum numbers. The expression

- P ( r - r ) 2 U (r - re) = D (1 - e e ) (4.1)

proposed by Morse (1929) is a closer representation of the potential

at other values of r.

energy referred to the minimum (fig. 4.1) and

parameter given by

In this expression D is the dissociation

P is an anharmonicity

1 p = (2n2 c p/Dh)2 we . (4.2)

A solution of the one-dimensional Schrodinger equation when U ( r ) is

of the form of eq. (4.1) is the set of vibrational wave functions

(Morse 1929),

k-2V-1 -z/2 (k-2v-1)/2 qJ = Nv e Z

54

(4.3)

55

U

C r re

Fig.4.1 Potential energy of a Morse diatomic molecule

56

(4.7)

(4.8)

4.2 Wave Functions, Overlap I n t e g r a l s and Franck-Condon Fac tors

The Morse wave func t ions of many molecular s ta tes (Nichol ls 1962a)

up t o t h e highest v i b r a t i o n a l l e v e l s known have been ca l cu la t ed a t

0.01 A i n t e r v a l s a t t h e Computing Center of t h e Nat ional Bureau of

Standards, Washington.

pairs of wave func t ions of a t r a n s i t i o n were evaluated and squared.

These d a t a were k ind ly made a v a i l a b l e f o r t h e present work.

0

Overlap i n t e g r a l s between a l l poss ib l e

4.3 Reduct ionof Antinode Pos i t i on t o Algebraic Form

In an at tempt t o fol low t h e same procedure as s e c t i o n 3.2 t h e

wave funct ions w f o r t h e X 3 t

aga ins t r t o f i n d t h e p o s i t i o n s of t h e an t inodes , rp . were made t o f i n d a simple empir ica l a l g e b r a i c r e l a t i o n s h i p between

r and v i n analogy with t h e expression v - b = a

(eq. 3.8) found f o r t h e an t inodes of t h e harmonic wave func t ions .

No s h p l e expression was found which would apply t o a l l va lues

of a l l t h e bound v i b r a t i o n a l l e v e l s computed

and B3z s t a t e s of thc 02 molecule were p l o t t e d

Attempt s

P l q n p P

57

of v and p f o r a l l molecular s t a t e s .

t o r e s o r t t o a g raph ica l method f o r l o c a t i n g t h e coincidence of

ant inode p a i r s .

It was thus found necessary

4 . 4 GraLhical Method f o r Locating t_he Coincidence of Antinode P a i r s

A graph ica l method f o r l o c a t i n g t h e coincidence of t h e turning

po in t s of t h e classical motion of an anharmonic o s c i l l a t o r was used

by Condon (1926).

- ----

The method has been extended he re f o r use with

any d e s i r e d p a i r of antinodes.

were found from t h e ca l cu la t ed wave func t ions e i t h e r by p l o t t i n g

them o r by in spec t ion , and p lo t t ed on a graph o f v a g a i n s t r.

p o i n t s of each ant inode number p were joined by s t r a i g h t l i n e s

( f ig . 4 . 2 ) .

The p o s i t i o n s of t h e ant inodes

The

For each t r a n s i t i o n studied t h e ant inode p o s i t i o n s of t h e

upper s ta te were p l o t t e d on t r anspa ren t paper, of t h e lower s ta te

on opaque paper. The V I and v" va lues (not n e c e s s a r i l y i n t e g r a l )

corresponding t o t h e alignment of ant inode p a i r s could them be

e a s i l y found.

f o r t h e terminal ant inode of one s ta te with t e rmina l and subs id i a ry

ant inodes of t h e o t h e r s t a t e .

This alignment was found, as i n t h e harmonic case,

4.5 Loci of Antinode Coincidence

The l o c i of ant inode coincidence are again sis-segment curves

on t h e VIV" plane as shown i n f i g . 3 . 3 f o r p = 2, 3 ... and three-

segment curves f o r p = 1 (f ig . 3 . 5 ) . The terminology used i s t h a t

59

shown i n f i g s . 3 . 2 and 3 .3 . In t h e fol lowing sec t ion of t h i s chapter

t h e l o c i a r e examined f o r th ree t r a n s i t i o n s r ep resen ta t ive of small

(0.012 A ) , medium (0.1069 A) and l a r g e (0 .2233 8) A re. Our con-

c lus ions a r e based on these t r a n s i t i o n s and seven o the r s discussed

i n Appendix 2 .

0 0

The da ta f o r t h e t r a n s i t i o n s s tud ied a r e given i n t a b l e 4.1.

The l o c i of ant inode coincidence are compared with t h e Condon l o c i

found from t h e Franck-Condon f a c t o r arrays and t h e band i n t e n s i t y

a r r ays .

a r r a y s i s t h e power of t e n by which t h e e n t r y i s mul t ip l i ed .

Arrays ai€ measured emission band i n t e n s i t y are given where a v a i l a b l e

un le s s a l ready given i n Chapter 3 o r Appendix 1.

The negat ive number following each e n t r y i n t h e f a c t o r

4 . 6 Deta i led @amination o f Representat ive Types of T rans i t i on

4 . 6 . 1 MgO B1 + X1x Green System

Twenty-three bands of this system l y i n g i n t h e v i s i b l e

-5000 A ) a r e known i n emission. 0

region (

of a small A r e ( A re = 0.012 8 ) having a marked 0,O sequence.

The observed bands ( t a b l e 4.2) a l l belong t o t h e 0,O and 0 , l sequences.

The Condon locus of t h e Franck-Condon f a c t o r a r r a y ( t a b l e 4 . 3 ) i s

confined t o t h e main diagonal.

The system i s t y p i c a l

The ca l cu la t ed l o c i of ant inode coincidence a r e shown on f i g . 4 . 3 .

The subs id ia ry l o c i a r e i n p a i r s e x t e r n a l t o t h e primary locus which

l i e s about t h e main diagonal. The con t r ibu t ion t o t h e overlap

i n t e g r a l due t o t h e coincidence of te rmina l p a i r s of an t inodes

along t h e ca l cu la t ed primary locus add t o give high values a long

60

N N rr) 0 rl N

04

ha 9 9 9 Q 0 0 0

N W

P- a, N 3

I I I r!

m W 0 N

r) rr) r) N 00

rr)

0 0 cv.

P- m rr)

0 0 0 0 0

a, m rr) P- O N

rl

W a m rl a, d N W rl m 5 : P - I m. I 9

P- W P-

rl rr)

rl rl 9

rl rl rl rl

a, W

P- rr) N rl v)

W N

rr) r? d N P- N

P- P- v) ou - a J P- 0 P-

v) P- 0

0 u) rl rr) a, 10 d

rl rl rl rl rl rl rl N r l rl * Ip

OD

10 I

m

d rl

I M k W P

h W Lc E 0

P- m

d rl rl rl

I '9 u)

0 rr) rr) v) 9 rl rl d

v) 0 N v) W rr) N P- v)

r- d rr) rl rl rl 4

9 W

rr) m r) rl

? cv. a,

8 m 2 2 rr)

rl W

0 a, v) rl

? d d P- N d N 0

v) P-

OD rl u)

rl N

9 ? W * r r )

rl d

rr) d m m N rl - a J rl a, d

N

"! I ? E 0

3

W z rl 0 W u)

W m 0 rl rl m

cv. W rr)

v) d N rl N W W a, v) 0

rl N rl z 8

P- d

I M

rr) b 4

X

+ M

r) rl x x

b D M I = e m n

r? N

I S

rr) w m

s = h N u r l d

4 m U

e r r ) P - W

Z s " 9 . P - P -

z z .. .. N +N

rl d u) u)

W

a,

d m

ai W d

d m W a,

61

Table 4.2

Band I n t e n s i t i e s o f MgO B 3 X (Mahanti 1932)

V V I 0 1 2 3 4 5 6 7 8 9 10 11 1 2 13

0 10 4

1 1 9 4

3 \ 7 3

2 \8 3

6 3 \

4

5 5 2 \

9

10

11

1 2

13

4 1 \

3 1 \

2 0 '\

\ 2

2 \

\

\O \ '\O

62

n

cn d

I4 W t .,

x rrl 0 t- cu . t- I

\o

A 8

W

\D

A m

I 7-

.

F

I

v I 2 3 4 5 6 7 v'


MgO B-X

64

t h e main diagonal. Thus t h e primary Condon l o c u s i s we l l explained

as due t o t he l a r g e con t r ibu t ion of overlapping terminal loops of

t h e v i b r a t i o n a l wave func t ions .

4 . 6 . 2 CO A1n 9 -- X 1 z Fourth P o s i t i v e System

The simple harmonic o s c i l l a t o r model of t h i s system was

discussed i n Chapter 3 s e c t i o n 3 . 5 . 2 and t h e semi-quantum form of

t h e Franck-Condon P r i n c i p l e found t o apply w e l l wi th a medium value

of A re ( A r e =y 0.1069 A ) .

found i n t a b l e 3 . 9 . Franck-Condon f a c t o r s of t h e Morse model and

0 The p o s i t i o n s of observed bands w i l l be

t h e ca l cu la t ed l o c i of ant inode coincidence a r e given i n t a b l e 4 . 4

and f i g . 4 . 4 r e spec t ive ly .

The primary Condon locus l i e s very c l o s e t o t h e l o c u s of t h e

coincidence o f terminal ant inode p a i r s . The secondary Condon l o c u s

l i e s along the i n t e r n a l branches of t h e c a l c u l a t e d secondary locus .

The ter t iary and quaternary Condon l o c i a r e equa l ly w e l l given by

branches of t h e l o c i of coincidence o f t e rmina l and r e l evan t sub-

s i d i a r y antinode p a i r s .

4 . 6 . 3 C, B 3 n 3 X3n Fox-Herzberg Svstem

T h i s system was discussed i n Chapter 3 ( s e c t i o n 3 . 5 . 3 ) and 0

i s r ep resen ta t ive of l a r g e va lues of Are ( A r e := 0.2233 A ) . The

l o c a t i o n of observed bands i s t o be found i n t a b l e 3 . 1 0 . Franck-

Condon f a c t o r s f o r t h e Morse o s c i l l a t o r model a re given i n t a b l e 4 . 5 .

The calculated l o c i of ant inode coincidence are shown i n f i g . 4 . 5 .

The primary Condon locus i s we l l represented by t h e s e c t i o n s c , d and

e , f only of t h e locus of coincidence of t e rmina l ant inode p a i r s

ln W

n

a Q,

a

u I

CY

0 k E

I I

I

I

;

I I I I I I I I


GO A-X

O

7

'9

P I t t

3

67

68

m

cc 0

69

g iv ing t h e asymmetrical nature of t h e Condon locus.

Condon l o c i are equal ly w e l l represented by t h e d and f s e c t i o n s

of t h e ca l cu la t ed subsidiary l o c i .

The subs id i a ry

CHAPTER 5

CONCLUSIONS

The study of Condon l o c i of diatomic molecular s p e c t r a descr ibed

he re has shown t h a t t h e l o c a t i o n of s t rong bands of a system i s

l a r g e l y determined by t h e overlapping of p a i r s of ant inodes of t h e

v i b r a t i o n a l wave func t ions . A s s t a t e d by Condon t h e s t r o n g e s t bands

occur a t t h e condi t ion of m a x i m u m overlap of terminal ant inodes.

These bands d e f i n e t h e primary Condon locus . We have found t h a t

t h e subsidiary Condon l o c i a r e def ined by t h e maximum overlap of

a terminal antinode of one v i b r a t i o n a l s ta te and an i n n e r ant inode

of t h e other s ta te .

i n t e g r a l i s from a s i n g l e p a i r of overlapping loops of t h e v i b r a t i o n a l

wave funct ions.

Thus t h e l a r g e s t c o n t r i b u t i o n t o t h e overlap

In t h e extreme and unusual ca ses of very s m a l l and very l a r g e

equilibrium i n t e r n u c l e a r s epa ra t ion d i f f e r e n c e t h e s e gene ra l con-

c lus ions a r e no t app l i cab le .

i n t h e overlap i n t e g r a l between wave f u n c t i o n s with an odd d i f f e r e n c e

i n v i b r a t i o n a l quantum number. No nes t ed Condon l o c i can be def ined.

In t h e l a t t e r case t h e important v i b r a t i o n a l quantum numbers a r e so

high t h a t t h e i n d i v i d u a l loops of t h e wave func t ions are of minor

s ign i f i cance . Here nested l o c i a r e n o t c l e a r l y def ined. In both

I n t h e former case t h e r e i s c a n c e l l a t i o n

70

7 1

t h e s e cases t h e complete overlap i n t e g r a l must be considered i n

l o c a t i n g t h e s t rong bands of a system.

A t r a n s i t i o n parameter depending on t h e equi l ibr ium i n t e r n u c l e a r

separa t ion of t h e two e l ec t ron ic s t a t e s , t h e i r c l a s s i c a l v i b r a t i o n a l

f requencies , and t h e reduced mass o f t h e molecule has been def ined .

This parameter desc r ibes t h e shape o f t h e primary Condon locus .

A method has been described f o r f i n d i n g t h e approximate p o s i t i o n s

of t he Condon l o c i f o r t h e wide range of A r e i n which t h e maximum

con t r ibu t ion t o t he Franck-Condon f a c t o r i s from t h e overlapping

o f a p a i r of ant inodes of the v i b r a t i o n a l wave func t ions .

approach can be of use in the i d e n t i f i c a t i o n of observed bands of

a system, and i n t h e observation of unrecorded bands.

This

APPENDIX 1

Al Further Resu l t s o f t h e Examination of Condon Loci Using t h e

Simple Harmonic O s c i l l a t o r Model

The t r a n s i t i o n s discussed i n Chapter 3 ( s e c t i o n 3 . 5 ) a r e

r ep resen ta t ive of small, medium and l a r g e va lues of A re.

conclusions concerning them a r e based on a study of four more

The

t r a n s i t i o n s which are included i n t h i s Appendix. They are:

GaI A 3 n ; + XIL+ , CH+ AITT -? XlL+ ,

and

t h e molecular d a t a f o r which are given i n t a b l e 3 . 2 .

a r e t r e a t e d i n o rde r of i nc reas ing A r e bu t t h e appa ren t ly anomalous

case of zero A r e i s t r e a t e d l a s t .

The t r a n s i t i o n s

From t h i s s e c t i o n of t h e study w e base our conclusions t h a t t h e

approximation we have made t o the Franck-Condon p r i n c i p l e breaks

down a t very small and very l a r g e va lues of A re f o r t h e reasons

out l ined i n t h e r e l e v a n t s ec t ions .

I

72

73

1 + 0

Four bands l y i n g i n the v i o l e t region of t h e spectrum a r e

Al.l @ A1n -? X ( A r e 0.10356 A )

observed i n emission.

i n t a b l e Al.1, no i n t e n s i t i e s being a v a i l a b l e .

a s t rophys ica l i n t e r e s t because c e r t a i n l i n e s a re observed i n t h e

absorpt ion spec t r a of some s tars and i n t e r s t e l l a r space.

Their l o c a t i o n on t h e vtv" plane i s given

This system i s of

The overlap i n t e g r a l a r r a y ( t a b l e A1.2) shows a s l i g h t l y open

primary Condon locus. One branch l i e s along t h e A v = V I - vll = 1

diagonal and has negative overlap i n t e g r a l s .

t h e primary Condon locus i s inc l ined t o t h e main diagonal with

v" > V I , and has p o s i t i v e overlap i n t e g r a l s .

e x t e r n a l l o c i a long t h e hv = V I - vll - 3 and 6 diagonals .

A v - 4 diagonal has negative e n t r i e s , t h e b v - 6 diagonal p o s i t i v e .

The o t h e r branch of

There are two apparent

The

The Condon l o c i given i n t a b l e R1.2 correspond c l o s e l y t o t h e

l o c i of ant inode coincidence shown i n f i g . A l . l as fol lows:

(a) t h e A v ~ 1 branch o f the primary with coincidence between p a i r s

of l e f t hand terminal antinodes (segment a , b ) and coincidence

between r i g h t hand p a i r s of ant inodes (segment f ) t h e secondary

ant inode of t h e upper s t a t e , and t h e primary ant inode of t h e

lower s t a t e .

(b) t h e ) V I branch of t h e primary with coincidence between l e f t

hand p a i r s of ant inodes (segment a ) t h e primary ant inode of t h e

upper s ta te with t h e secondary ant inode of t h e lower s t a t e .

V V I

0

1

2

3

4

5

6

7

Table Al.1

0 X X

1 X

2 X

T a b l c A1.2

Overlap lntcj i ra ls of CH-' A -c X

74

0 1 2 % 4 5 6 7

0.13412 0.4317 0.2719 0.14130 0.0838 0.0450 0.0243 0.0128

0.1167 -0.66139 0.3185 0.3322-0.3947 0.2969 0.2203 0.1448

0.0411 0.2193 -0.71135 0. I342 0.1961 0.36132. 0.31713 0.2692 1

--- \

\ 1 -0.0324 0.0040 0.5230 -0.7062 0.0042 0.0503 0.3062 0.3041

\ 0.0019 -0.0701 0.0740 0,4200 -0.6565 -0.0737 -0.01306 0.2239

\ \ \ \ \

0.0052 0.0073 -0.1157 0.0696 0.5030 -0.5781 -0.1044 -0.2026

-0.0010 0.0128 0.0179 -0.1661 0.0507 0.5667 -0.41349 -0.0948

75

VI

i

z

3

4

5

6

7

1

9

H

Fig. A I . I Loci OF coincidence OF antinode pairs,

CH+ A + X

76

( c ) t h e A v = 4 diagonal with coincidence between l e f t hand p a i r s

of antinodes (segment b) t h e secondary ant inode of t h e upper

s t a t e w i t h the primary antinode of t h e lower s t a t e , and with

coincidence between r i g h t hand p a i r s of ant inodes (segment f )

t h e t e r t i a r y ant inode of t h e upper s ta te and t h e primary ant inode

of the lower s t a t e .

t h e A v = 6 diagonal i s a l s o a s soc ia t ed with t h e coincidence

between the l e f t hand p a i r s of ant inodes (segment b) mentioned

above and with t h e coincidence of l e f t hand p a i r s of ant inodes

t h e upper ant inode being t h e t e r t i a r y , t h e lower ant inode t h e

primary (segment b) .

(d)

Table Al.3 i s an examination of t h e s i g n of t h e c o n t r i b u t i o n

t o t h e overlap i n t e g r a l due t o t h e coincidence o f p a i r s of ant inodes

f o r v f = 0 - 7, vll = 0 - 2 . It i s clear t h a t t h e above comments

concerning t h e primary Condon locus a r e s u b s t a n t i a t e d , while t h e

apparent s u b s i d i a r i e s are due t o t h e c a n c e l l a t i o n ofyopposing con-

t r i b u t i o n s i n neighbouring e n t r i e s .

t o those reached i n t h e study of t h e MgH t r a n s i t i o n s discussed i n

sec t ion 3.5.1.

These conclusions a r e i d e n t i c a l

0 A1.2 g+

T h i s system i s of a s t r o p h y s i c a l importance as i t i s emitted by

A2n 3 X2L Comet Ta i l System (Are = 0.12862 A )

t h e t a i l s o f comets.

and i n t h e nea r u l t r a v i o l e t region of t h e spectrum ( A- 6000 - 3000 A )

a r e known i n emission.

Twenty-nine bands l y i n g throughout t h e v i s i b l e 0

Their i n t e n s i t i e s are given i n t a b l e A1.4.

77 Table Al.3

Examination of Contributions t o Overlap I n t e g r a l of C& A + X

0 0 1 0 2 0 3 0 4 0 5 0 5 0 6 0 7 0

0 1 1 1 1 1 2 1 2 1 3 1 3 1 4 1 4 1 5 1 5 1 6 1 6 1 6 1 7 1 7 1 7 1

0 2 1 2 1 2 2 2 2 2 2 2 3 2 3 2 4 2 4 2 5 2 5 2 5 2 5 2 6 2 6 2 7 2

Primary Primary Primary Secondary Secondary Tertiary Secondary Tertiary Tertiary

Primary e,f Primary e,f Primary a ,b Primary a ,b

Primary a ,b

T e r t i a r y f Primary a ,b

Tertiary f

Tertiary f

Secondary f

Secondary f

Secondary b

Secondary b

T e r t i a r y b Secondary b T e r t i a r y b Quarternary f

Primary Primary Secondary Primary Primary Secondary Primary Secondary

Secondary Tertiary Primary Secondary Secondary T e r t i a r y Secondary Secondary

Pl5ItElry

+ +

- +

+ + - + + + + + + + + + + + + + +

+ + + -

+ - +

+ + + - - + + + + - - + + + + + +

+ + + + + + - - + + + + + -+ - - +

78

Table U.4

Band I n t e n s i t i e s of CO+ A -+ X (Pearse and Gaydon 1950)

V 11

V’

0

1

2

3

4

5

6

7

8

9

10

11

0 1 2 3 4

3

8 /I-; 5

I 10

I I 8 2 3

I 2 1 6

I

\ 1 2 \ 1 \ 4 \ 4

1

2 \

\ 1 ,.,,

1

79

The observed bands dre confined t o the l e f t hnnd por t ion of t h e

Deslandrcs t a b l e and i t i s possibl c t o d i s t i n r u i s h t h e paths of

primary and secondary Condon l o c i .

The overlap i n t e g r a l a r r a y (ts b l e A l . 5) shows t h r e e Condon

l o c i nested i n t h e conventional manner t y p i c a l o f t h e medium

vri Lue. The l o c i a r e , however, no t symmetrical about t h e main diagonal ,

o r even roughly so , as much o f the l i t e r a t u r e i n f e r s (Herzbersr 1950

p. 198) .

cresyion from v" 2 - 5. From V I 2, v" 0 t h e locus drops

s t e e p l y t o v1 - 7 , v" - 2 . Thus t h e r e i s a d i s t i n c t "coriiertl a t

v1 ~ 2 . The subs id i a ry l o c i a r e of sjmilar shape.

A r e

The primary Condon l o c u s i s confined t o t h e v1 0 pro-

Fig. A 1 . 2 shows t h e loci of ant inode coincidence f r o m which t h e

Condon l o c i can be i n t e r p r e t e d .

t e rmina l ant inode p a i r s c lose ly f o l l o w s t h e primary Condon locus.

The tlcornerlf a t V I = 2, v" = 0 a l r e a d y mentioned i s c l o s e t o t h e

junc t ion of t h e branches a, b and c , d of t he c a l c u l a t e d locus.

The l o c u s of t h e coincidence of

The secondary Condon locus fol lows c l o s e l y t h e s e c t i o n s a , c

and d , and f of t h e l o c u s of coincidence between a secondary and a

terminal ant inode. The remaining coincidences of t h i s l ocus a r e

understood t o con t r ibu te l i t t l e t o t h e overlap i n t e g r a l due t o t h e

sho r tnes s of t h e overlapping regions of t h e ~ b r a t i o n a l wave func t ions

and t o t h e l a r g e d i f f e r e n c e i n loop s i z e s of t h e wave f u n c t i o n s

r e s u l t i n g from l a r g e d i f f e r e n c e i n v i b r a t i o n a l quantum number.

The te r t ia ry Condon l o c u s can be s i m i l a r l y i n t e r p r e t e d i n terms

of t h e l a r g e con t r ibu t ion t o the ove r l ap i n t e g r a l a t t h e coincidence

of a t e r t i a q ant inode and a terminal ant inode.

80

Table A1.5

V" V'

0

1

2

3

4

5

6

7

Overlap I n t e g r a l of CO+ A 3 x

0 1 2 3 4 5 6 7

0.2126 0.3394 0.4092-0.4251-0.4006- 0.3517 0.2924 0.2327 /--

/ -0.4041 -0.435$ -0.3051 -0.1100 0.0759 0.2137 0.2927. 0.3196

/ '\

/' 0.5173 0.2630 -0.0692 -0.2721--0,3042 -0.2119 -0.0646 0.0812

/' -0.5108 0.0670 0.3305 0.2459 0.0137 -0.1843 -0.2656 -0.2314

1.

\ \

\ -0.0604 0.2780 -0.4624 0.1390 0.2884 0.0290 -0.2146 -0.2125

E l

II

V

V'

I

z

3

4

S

6

7

a

9

Fig. A1.2 Lou OF coincidence OF antinode pairs.

CO+ A+ X

82

A1.3 O2 B 3 Z - -t X 3 r - Schumann-Runge System ( A r e = 0.397 1) 11 U

This extensive system of O2 exists throughout t h e u l t r a - v i o l e t

region of t h e spectrum i n both emission and absorpt ion.

emission bands up t o A 4372 a r e w e l l known. Continuum appears

below A 1759 A s e t t i n g t h e u l t r a - v i o l e t l i m i t t o spectroscopy i n

a i r . Most of

t h e observed bands f a l l on a primary Condon locus , with two bands

(2,15 and 2,16) poss ib ly i n d i c a t i n g a secondary Condon locus .

Twenty-nine

0

Measured band i n t e n s i t i e s a r e given i n t a b l e A1.6.

The overlap i n t e g r a l array f o r t h i s t r a n s i t i o n us ing t h e harmonic

o s c i l l a t o r model i s n o t a v a i l a b l e .

A re giving t h e s t ronges t bands a t high VI' v a lues f o r which t h e

harmonic o s c i l l a t o r model i s a very poor approximation.

as a matter o f i n t e r e s t , t h i s t r a n s i t i o n has been s tudied with both

The t r a n s i t i o n i s of very l a r g e

However,

a harmonic and an anharmonic p o t e n t i a l .

The c a l c u l a t i o n s of ant inode coincidence were c a r r i e d ou t t o

high quantum numbers ( V I = 20, VI! = 30) and high ant inode number

(p = 8) . The r e s u l t s a r e shown in f i g . A1.3.

Because of t h e above mentioned f e a t u r e s of t h i s t r a n s i t i o n it

i s n o t poss ib l e t o re la te t h e l o c i of ant inode p a i r coincidence with

t h e Condon l o c i .

t h e Franck-Condon P r i n c i p l e i s inadequate a t very l a r g e b re values .

It must be concluded t h a t t h e approximate form of

A1.4 GaI A 3 n -+ X 1 t System A ( A r e = 0)

0 Sixty bands i n t h e blue region ( -4000 A ) of t h e spectrum are

known t o belong t o t h i s system, which i s however overlapped by t h e

B +X system of t h e same molecule.

83

n

Q, rl m

z!

e X

CD rl .3

x 4 a,

N

0 r)

I a, cv)

I

b CD CD

I

co u)

0 rl N

85

This system was studied i n d e t a i l by Wehrli (1934) i n r e l a t i o n

t o t h e Franck-Condon Pr inc ip le .

i n order t o a s s e s s h i s conclusion t h a t only t h e complete quantum

mechanical a n a l y s i s of v ib ra t iona l overlap i s v a l i d when A re = 0,

t h i s system being one of the f e w with zero o r very near zero d i f f e r e n c e

i n equi l ibr ium i n t e r n u c l e a r separa t ion .

It has been included i n our study

The overlap i n t e g r a l a r r ay f o r zero A r e i s not a v a i l a b l e but

i s ind ica t ed q u a l i t a t i v e l y i n t a b l e U . 7 .

i s due t o t h e symmetry of the v i b r a t i o n a l wave func t ions .

e n t r i e s have zero overlap i n t e g r a l ( cance l l a t ion of odd and even

func t ions ) .

decreasing from t h e main diagonal.

The unusual appearance

Odd A v

Even A v e n t r i e s have f i n i t e over lap i n t e g r a l t h e value

Thus odd A v t r a n s i t i o n s a r e

forbidden; even A v t r a n s i t i o n s allowed.

The a r r a y of measured band i n t e n s i t y ( t a b l e A 1 . 8 ) shows that

t h e G a I t r a n s i t i o n A 3 X i s c lose ly one of zero change i n equi l ibr ium

i n t e r n u c l e a r separa t ion .

diagonals and no nested Condon l o c i can be drawn.

a s s o c i a t e t h e main diagonal with t h e primary Condon locus , and p a i r s

of ex te rna l d iagonals with the subs id ia ry l o c i .

The s t rong bands appear on t h e even b v

However we s h a l l

The coincidence of antinode p a i r s ca l cu la t ed by our a l g e b r a i c

method with A r e = 0 i s shown i n f i g . A1.4. There i s coincidence

of primary an t inodes along t h e l i n e c l o s e t o t h e main diagonal of

t h e a r r ay . It i s important t o note t h a t t h i s l i n e co inc ides with

t h e main diagonal only i f de1 = Wz. The i n c l i n a t i o n of t h e l i n e

t o t h e V I a x i s i s tan-1(We''/ae')nl/2. The coincidence of c e n t r a l

modes o r an t inodes , r a t h e r than of te rmina l an t inodes , desc r ibe t h e

d iagonals .

86

Table A 1 . 7

Overlap I n t e g r a l s of GaI A 4 X (Wehrli 1934)

v ” v f 0 1 2 3 4 5 6 7 8 9 10 11 1 2 13 1 4 1 5

0 x - x 1 - x - x 2 x - x - x 3 x - x - x - x 4 x - x - x - x 5 x - x - x - x 6 x - x - x - x 7 x - x - x - x - x 8 x - x - x - x - x 9 x - x - x - x - x

10

11

x - x - x - x - x x - x - x - x - x

legend X non-zero ove r l ap i n t e g r a l

- zero ove r l ap i n t e g r a l

Table 81.8

Band In tens i t ies of GaI A 3 X (Miescher and Wehrli 1934)

V " V I 0 1 2 3 4 5 6 7 8 9 10 11 1 2 1 3 1 4 1 5 1 6

0 1 0 5 2

i 8 9 5 2 0

2 4 9 8 5 2 1 1

3 1 0 7 4 3 0 1

4 7 7 2 4 0 1

5 3 6 6 3 0 1

6 4 6 1 3 0

7 6 1 4 0 1

8 5 1 3 1 1

9 3 2 2 0 1

10 1 2 0 1

11 1 2 0 1

1 2 1 1

1

87

\I'

1

2

3

4

5

6

7

8

9

I(

Fig. AI.4 Loci of coincidence of antinode poifs,

GaI A + X

\

'\

89

The p a i r of l i n e s i n f i g . A 1 . 4 one on each s ide of t h e main

diagonal , descr ibe t h e coincidence of a te rmina l ant inode of one

s t a t e with a secondary antinode of the o ther .

from the A v = 2 l i n e s of the overlap i n t e g r a l a r r ay , and cannot be

assoc ia ted with t h e secondary Condon locus .

s i m i l a r i t y between t h e l i n e s o f primary and t e r t i a r y ant inode coin-

cidence, and t h e h v = 4 diagonals .

These l i n e s a r e f a r

There i s a l s o l ack of

The e f f e c t of a very small Are on t h e l o c i of ant inode coin-

cidence has been inves t iga t ed .

A r e = 0.002 A y i e l d six-segmented l o c i with t h e c and d sec t ions

very s h o r t ( - 0.1 quantum number) and t h e a and e, b and f sec t ions

very close.

Calcu la t ions of the l o c i with 0

The na tu re o f the l o c i i s unchanged.

This system i s discussed f u r t h e r i n Appendix 2.7 where t h e

Morse model i s s tudied .

APPENDIX 2

A2 Further Results of t h e Examination of Condon Loci - Using t h e

Morse O s c i l l a t o r Model

This appendix r e l a t e s t o t h e results of Chapter 4. I t s arrange-

ment and purpose i s similar t o t h a t o f Appendix 1 and t h e conclusions

concerning t h e v a l i d i t y of our approximation t o t h e Franck-Condon

P r i n c i p l e a r e t h e same. Seven t r a n s i t i o n s a r e examined i n d e t a i l ;

t hese a r e :

1 i Ga I A3n' * X 2 , N 2 i o n i s a t i o n N2B L u t N2 X L Mgo BIL 3 A1n

+ 2 + 1 + g y

c o+ A2Tl+X2X

N2 i o n i s a t i o n N2C + 2 + f- N$'Z>

N2 i o n i s a t i o n N+D2 TI t X1 zi 2

B3L- -+ X 3 r - U g and 02

t h e molecular d a t a f o r which a r e given i n t a b l e 4.1.

0 A 2 . 1 Ng B2 rz +- N 2 X1zL Rydberg System (Are = 0.023 A )

The i o n i z a t i o n t r a n s i t i o n s of n i t r o g e n have been included i n our

study due t o i n t e r e s t i n g f e a t u r e s i n t h e i r Franck-Condon f a c t o r arrays.

90

91

Some of t h e t r a n s i t i o n s have been observed i n emission by Takamine

e t al. (1938).

observed i n t h e vacuum u l t r a - v i o l e t ( A - 700 - 800 2). been no v i b r a t i o n a l a n a l y s i s .

Five emission and t e n abso rp t ion bands have been

There has

The Franck-Condon f a c t o r array ( t a b l e A 2 . 1 ) shows a narrow

primary Condon locus l y i n g e n t i r e l y a long t h e main diagonal up t o

V I = VI! = 10.

a n i n n e r secondary locus runs from V I = vff = 1 2 .

l o c i are many e n t r i e s which a r e l a r g e i n comparison with t h e i r

neighbours.

t a b l e d e f i n i n g a d d i t i o n a l subsidiary Condon l o c i .

f a c t o r s involved are, however, very much smaller than those of t h e

gene ra l ly recognized Condon l o c i .

The locus broadens a t h ighe r quantum numbers and

External t o t h e s e

O f t h e s e many l i e along diagonals o f t h e Deslandres

The Franck-Condon

The ca l cu la t ed l o c i of antinode coincidence are shown i n f i g . A 2 . 1 .

The primary l o c u s i s unusual i n t h a t i t becomes narrow a t high quantum

numbers.

Their branches i n t e r s e c t considerably.

The subs id i a ry l o c i are i n p a i r s each s i d e of t h e primary.

The e,f branch o f t h e ca l cu la t ed primary l o c u s l i e s c l o s e t o t h e

main diagonal and hence t o t h e primary Condon locus up t o i t s p o i n t

of broadening. From V I = vll = 12 t h e secondary Condon l o c u s l i e s

a l o n g t h e main diagonal and may be a s soc ia t ed with t h e c a l c u l a t e d

primary locus.

p o i n t of broadening.

The a , b branch l i e s along t h e primary l o c u s from i t s

The e x t e r n a l Condon l o c i l i e a long branches of t h e c a l c u l a t e d

subs id i a ry l o c i . The ex i s t ence of t h e s e minor Condon l o c i can be

1 d d N I l l

m r - v in -i N a" d c o d d r ( m . . . I

. . . . . . . G 4 f pi m' m N N co m m in

mcc r- r - s a m m in v m~ N 1 l 1 0 1 l l 1 l 1 1 l

. . . . . . . . . . . . . d N d m m m d N * d N m c o

. . . . . . . . . . . . . n A * 7 9 4 GIN .F m r- v r l \ D .-(

I I I In r- N

1 1 1 N d d I I I I

N A N I l l -i r- tn N m U' .-I .o m tn I- I.

- 4 N m . . .

0 6 0 A . r m P N O V W N N o l m

$412

/ i A 41; . . .

m N 4

/ i - 4 N . i 1 1 I I

- I N N I l l

I- t- .D 0' 0 .i o m 0 oat- . . . TV In N

I i l I

0 - . . . . . . . . . . . u d d 9 N r- dl -+ N u 9 9 NJ 4 "14 In'

I I

I i

I I . . . SAco'NrnN

I I / N N - i N

I 1 1 1 t- tn I - r- d I- m .D in LV m r- co/o -r od 2' t-' ,n'

I I I - F I N

h 6

011 (0 0, rl

P 6 * 0 1 1 C Z

93

V"

Fiy. A2.I Loci OF Coincidence OF antinode pairs,

94

assumed t o b e due t o t h e con t r ibu t ions t o t h e overlap i n t e g r a l of

overlapping p a i r s of loops of t h e v i b r a t i o n a l wave func t ions .

A2 2 %OL---.---.--. BIL j AIU Red System (h1.e 0.127 8) Forty-six bands l y i n g i n t h e red region of t h e spectrum (h-6000 8)

a r e known i n emission.

def ined nested Condon l o c i on t h e Franck-Condon f a c t o r a r r a y ( t a b l e A2.2) .

The t r a n s i t i o n i s of medium A r e giving w e l l -

The s t ronges t bands a r e a l s o seen t o lie on the conventional form of

Condon locus ( t a b l e A2.3) .

The c a l c u l a t e d l o c i of t h e coincidence of ant inode p a i r s a r e

shown on f i g . A2.2. The primary Condon locus i s well represented

by t h e locus of t h e coincidence of terminal ant inode p a i r s . The

secondary Condon locus i s represented by t h e i n t e r n a l branches of

t h e locus of coincidence of a terminal and a secondary ant inode.

This case i s very similar t o t h a t of t h e CO Fourth P o s i t i v e

System.

l a r g e r A r e value.

The l o c i of t h e MgO system a r e s l i g h t l y broader due t o a

A2.3 C0'- A2Tl +X2z Comet T a i l System (Are 0.12862 8) The harmonic o s c i l l a t o r model o f t h i s system was s tudied i n

s e c t i o n A1.2 of t h i s t h e s i s . The measured band i n t e n s i t i e s w i l l be

found i n t a b l e A 1 . 4 . The Franck-Condon f a c t o r s f o r t h e Morse

o s c i l l a t o r model a r e given i n t a b l e A2.4 where broad nested Condon

l o c i a r e c l e a r l y def ined.

The ca l cu la t ed l o c i of ant inode coincidence a r e given i n f i g . A2.3.

A s i n the previous cases discussed t h e Condon l o c i of t h i s system can

be w e l l represented by se l ec t ed branches of t h e c a l c u l a t e d l o c i .

95

I

f 'a " i 4 ,lJ

k 0

m Lc 0

q I 0 q \J IC

n

N I N n

? r

(u

W In rn \D

N

I

e

x !U cn W

A .

96

Table A2.3

V " V'

0

1

2

3

4

5

6

7

8

9

10

11

1 2

13

14

1 5

Band In tens i t ies of MgO B 3 A (Mahanti 1932)

0 1 2 3 4 5 6 7 8 9 10 11

6 -

5

4

3

I I I

2 2 1

1 1 1 0 \

1\1 0

1

0 \

1 \

1 1 0 \

1 \

1 \

97

v’

i

2

3

4

5

6

7

Fig. A2.2 Loci OF coincidence OC onlitlode pairs,

MqO B + A

3 v)

t- o\

N

n

41 A I

9 I I-

P 9 N

I

5 ni cn

I %ga m

m T - r I / I /

x W 0

\D c'i

8 A

I & 0 k

m k 0 3 0 cd CU

Y 1 I

x W 2

W o\

/ SJX L n In$

4 y 2

/

N I ' v- ni l-

N

t- A ln

I .c

4 0 . . c n t - l n

\ /

99

U

F.

9

x

< 0 0

P

r +

c

.- Q Q

d a 0 C

0 5 h 0

d 4 C 3 rn 3 c .-

3 U 0

'3 0 -I

rr)

c\i 6 ci. iz

100

0 A2.4 N2' C2Z+ t- N2 XIZ: (&e = 0.164 A )

_I_ - The Franck-Condon f a c t o r a r r a y ( t a b l e A2.5) shows broad nested

Condon l o c i which are asymmetrical about t h e main diagonal of t h e

Deslandres t a b l e . A t high vt t t h e r e are a d d i t i o n a l e x t e r n a l Condon

l o c i l y i n g roughly along the diagonals . The Franck-Condon f a c t o r s

involved are very small, being much smaller than those of t h e i n t e r n a l

subsidiary l o c i .

The ca l cu la t ed l o c i of antinode coincidence are shown i n f i g . A2.4.

The primary Condon locus l i e s c l o s e t o t h e c,d and e,f branches of

t h e locus of t h e coincidence of terminal ant inodes.

e x i s t s a t h ighe r V I va lues than those considered.

only two of t h e sec t ions of t h e primary l o c u s exp la ins t h e asymmetrical

na tu re of t h e Condon a r r a y . The i n t e r n a l subs id i a ry Condon l o c i a r e

w e l l represented by t h e i n t e r n a l s e c t i o n s (d and f ) of t h e c a l c u l a t e d

l o c i .

The a , b branch

The appearance of

The e x t e r n a l Condon l o c i are w e l l represented by t h e e x t e r n a l

branches ( e ) of t he ca l cu la t ed l o c i .

The Condon l o c i ( t a b l e A2.6) of t h i s system a r e very broad and

far separated from the main diagonal of t h e Deslandres t a b l e .

calcula.ted l o c i of ant inode coincidence are shown i n f i g . A2.5.

The

The primary Condon l o c u s i s w e l l represented by s e c t i o n s c,d

and e,f of t he locus of terminal ant inode coincidence.

Condon locus l i e s c l o s e t o sec t ions c and f of t h e c a l c u l a t e d secondary

locus. The t e r t i a r y l o c u s l i e s c l o s e t o s e c t i o n s d of t h e c a l c u l a t e d

The secondary

r- 6-4

I r- In

0

m

Q 4

I

rl 4 4

in . m

In

I 9 Q r- N

9

-4 m I 4 r- m m

m .

I dl d

6 4

I 6 4 N In

in 4

I rr m r- cr

L n 4

I a .-I In rT.

r- I

-+ 0 0 N

'V

R J N 6-4

rr N

N 4

In

r- I N Ln m Q

. . . n J 4 m " " . G

,'I d . . 4 0 3

I

. . r - r r ) I

Q V I I

o m 4 M

rl I rr 0 9 r( . ,-I

. . 9 0

N I

m 4 In N

m N

I m * m r- m

4 0 c o d

m m or- * .

N /a m cn

N I

in 4 4 N

4

N I I' N 0 0

-1

m m 0 * in . Q

g 2: b4 C'I

c

0

102

CL O

x

N I

d u l 4 9

9 r- . d

P- N

9 N

v1 N

2.

rr n

h: h

r n

E

0 r

a r

r

u

U

c

r

N N :: N I

m

Q

m

a .

$1,. m'

l i i I 1. d

N m V I

o w N Q 4 4 0 4

" 2 ' " 7 -40 * a m - 4 o r - 4 w N m

I 8 9 4 m 4 4 0 N O

. *

. . 4 4

f 2 nil;

4 0 1 9 0

4 m 7 "It?' d 1

N I

-4 0 -4 m 4

m 0 r- 9 -4

In

N 8 In m 0 4 . 4

m 9 4 m N

01

N 8

m r- 0 -4

0 -4

m I 01 0 -w 4

r.i

T r- * 4 9

9

m m 0 o\ 0

m

ll

N I 0. m -? N

N I 9 m 20 N

* I 'x,

0 4

m

m I u3 0,

0 4

m I

m N 01 In

Q l

Ln m 9

m I r- N

rx, 4

I

N I

m 4 00 9

-4 .

3 In rl

m m

tn t- In

N I

m In 01 4

r.i

N # I- O 0 N

m

3 m m

m 9

-f tn 4

\ \

\ \

I N N m m N N N 8 N

N I

r- m m m m

N I In m * m m i

N

9 m In m . -4

m I 0. m m m In

N I 0 4 0 w

.

mi

* I 9 r- 0 4

4

m I 9 \o 9 t-

N

5 m N m W

r-

m I

m \o r- m Q

N I 9 * N m 4

m In 0 9 m d

N I 0 + 0 4

N

N I *

N Q m N

N I m

Q 01 m 4

N I a, 03 4 4

m Jn I

r- m L n e ni

m I

r- 0 4 N

r.i

m I 0 r- 9 00

m

N I N 01 * 0 1 . 4

N I 0 a\ In 9 ni

N I 4 m * m ni

N I

m * -? 0

m

9 * TP CJ

m

r- 0 m In -4

9

W In rl 0.

In I o\ m 9 o\

3 0 N m 0

d 0. 9 0 W

3 N 9 In m

a t- 4 N

m I 9 0 00 4

m CO m 9 *

m 0 9 r- 0

m I r- m * 01

m I 0

m m 4

m h

+, V d E

8 a d u I a

E

m r- t- In N

r- I In rl P- Ln

9

m 03 D In

In I t- W 9 m

In I

In 9 0

+ 0 In 0 0

4

m N Ln -4

* 0 In 0 4

* N rl 4 4

m I 0 .D m -4

m 01 Q r- rn

m I N m 01

m 0 m N '0 d

4)

7 r- m 0

I 4

r- 4 r- m Ln

-4

r- I

r- 0 ul Ln

d

Q 1

m r- N 4

4

Q I 9 N 00 * n;

9

N m 0

m

tn In

I 0 0 In In

4

In I r- m m * N

104

U

- 7

.- 0 0

-J

105

tert iary locus and c of t h e quaternary, and c l o s e t o s e c t i o n f of

t h e t e r t i a r y . The quaternary Condon l o c u s l i e s c l o s e t o s e c t i o n s d

and f of t h e ca l cu la t ed quaternary.

In cases where A r e i s l a r g e t h e s e c t i o n s c and d are l a r g e and

It i s t h u s these i n t e r s e c t o t h e r calculated l o c i on t h e v f v " plane.

d i f f i c u l t t o a s s o c i a t e a subsidiary Condon locus with any s i n g l e

ca l cu la t ed locus.

A2.6 02 B3L- + X 3 t - Schwnann-Runge System ( A r e = 0.397 g) U -

This system shows a confusion similar t o t h e N 2 i o n i s a t i o n

t r a n s i t i o n discussed in sec t ion A2.5.

system ( t a b l e A2.7) are very broad and crowded a t Idgh VI! (-15).

The pos i t i on ing of t h e l o c i i s ambiguous, and t h e s e a r e n o t u sua l ly

drawn f o r t h i s system.

The Condon l o c i of t h e 02

The ca l cu la t ed l o c i of antinode coincidence ( f i g . A2.6) show

t h a t t h e f i r s t f e w l o c i are recognizable i n t h e usua l manner, but

f u r t h e r subs id i a ry l o c i (- 4 and up) are much confused by t h e c ros s ing

of ca l cu la t ed curves.

The pos i t i on ing of observed bands ( t a b l e A 1 . 6 ) a l r e a d y discussed

i s i n good agreement with t h e Condon l o c i shown i n t a b l e A2.7 con-

f i rming t h e anharmonic na tu re of t h e p o t e n t i a l .

A2.7 GaI A3n + XIL System A (Are = 0.002 8) The simple harmonic model o f t h i s system has been discussed i n

Appendix 1.

Franck-Condon P r i n c i p l e i s inadequate a t zero A re with a symmetrical

It w a s pointed out t h a t t h e approximate form of t h e

(0 0 d

t-

i Q,

Pi

Q t3 n 'J i .

107

V

V'

2

4

6

8

IO

I2

uc

w

I8

20

Fig. A2.6 Loci OF coincidence of antinode pairs,

OL B - 4

108

p o t e n t i a l funct ion.

small value of A r e w i l l be s tudied i n t h i s s ec t ion .

The e f f e c t of imposing an anharmonicity and

Table A2.8 i s t h e Franck-Condon factor a r r a y of t h e Morse model

using measured cons t an t s and a Are of 0.002 8. l ocus l i e s a long t h e main diagonal up t o v1 VI! - 4 . From t h i s

po in t i t s branches diverge, and a secondary Condon l o c u s appears

a t v1 = VI! = 9 . e n t r i e s of l a r g e r Franck-Condon f a c t o r s than t h e i r neighbours.

e n t r i e s a r e o f very small value compared with those d e f i n i n g t h e

conventional Condon l o c i .

The primary Condon

To t h e high v1 s i d e of t h e main diagonal a r e i s o l a t e d

These

The ca l cu la t ed l o c i of ant inode p a i r coincidence a r e shown i n

f i g . A2.7.

value of A r e used.

each other due t o t h e l a r g e a n h a m o n i c i t y (

t h e upper s ta te . The po r t ion of t h e primary Condon l o c u s l y i n g along

t h e main diagonal i s w e l l represented by t h e coincidence of l e f t hand

p a i r s of terminal ant inodes.

continues t o r ep resen t t h e primary Condon locus .

of t h e Condon locus l i e s c l o s e t o t h e l o c u s of t h e coincidence of

r i g h t hand terminal ant inode p a i r s .

The s e c t i o n s c and d are very s h o r t due t o t h e small

The s e c t i o n s a and e, b and f d ive rge from

xe = 2 . 4 cm- l ) of

This coincidence a t h ighe r v va lues

The high V I ! branch

The high VI! branch of t h e secondary Condon locus i s w e l l represented

by p a r t of t h e sec t ion f of t h e coincidence of a secondary and primary

ant inode.

Franck-Condon f a c t o r s , cannot be represented according t o o u r scheme.

The a d d i t i o n a l high Franck-Condon f a c t o r e n t r i e s are probably due t o

t h e overlapping of terminal ant inodes with s u b s i d i a r y ant inodes.

The low VI' branch which i s n o t c l e a r l y de f ined by high

I

~

m I 0 In v) m v)

N I v) N d

rl “1

N

m 5? 10

m

N I d d

W

in

01

~

r i I m

m rl

r) t

N I

W rl P-

W t

0 1 I

P- m rl

d

In rl I

t- d

9

? N

v) d I m 0 m I v)

d rl I

m d m

d

d rl

1 v) rl 0

d

d rl

rl P- 0

d

‘?

?

“i

=1

01

I W d m

N

m rl

I

0,

0,

0, I m d rl

d

m I m 0 N m OD

m 1

v)

m

W I

ID W 0

N

W N t W

?

?

ON 01

I

c: P m

v) 0 In r- rl

m m 10 W

rl

r) I m

W PI

v)

?

@?

r? d

I r- r3 0

m ‘9

A I

W m t- v)

rl

rl rl I

W W rl

In 01

0 rl

I d d 0

10 t

Q, I

W W W

ID 9

m 0 m 1 C1

t- I t- d t- m m

W I

01) L D m P-

rl

v) I

m m m N

rl

d I

N LD t-

d

d . I N d

I

2 d

m I v) m rl

d

N I d

?

5: I N

N

r- P- O ? P-

3 --f

0 rt W N

N

m I m

r- N

d I

m N LD t- r- m

OD I 0 r- N

W s

t- I

m m

m

m

1

W

m 0 0

N ?

Ln I

m N N

rl t

v) I m m N

a, “?

P t m 0 m

v)

m N m m

N

N I m m

N

d

N 1

P-

d

rl

N LD m m ?I

I

t

2 ?

-t -I

1- m ID

rl

in rl I 0 N rl

v)

N d I m

W N

d

N rl I t- N 10

N

rl rl

v) LD rl

d

0 4 v) v) d

d

0 rl

I 10

01

0:

9

9

?

?

d ? W

OD I m a3 m 9 m

I- I

N m m v)

d

W I a, ‘0 P-

d

v) I

m LD LD

rl

v) I

W

I

01

5: 3 W

m I d m 0

rl

W I 0 t- m

P-

N I m 0 W

N

p!

?

?

1- I m

r- m t-

m

W I

d 0 d

d t

v) I v) d m 9 N

d 1

d N N

rl 9

d m m 0

d ?

m I

ID m P- r- d

m I m N m W

W

N I m

d v)

9 r a

N I m 0, LD

ID

rl I m

-3 d N

rl

‘9

IO

m iD a) P-

N

d I

v)

d

$ I

d I 10 a, u)

d ‘9

m I 10 N r-

rl ?

m I

OD m N

v) ?

N I

P- W v) r- ?I

N

2 t m

d

d I m

E 9 d

f I v

r- 4 v)

m I

0

rl

3 00.

m I t- m d 10

10

N I

N a, v)

d

m

N I

N m LD

m ?

N I

N t- d

t- 01

rt

=1 U> P-

rl

N I

P- d m

m @?

N

0, 4 t-

m 0 d

I . I m

N m

ri Y

cO I

d

m

ON 01

t- I

rl m 0

10 4

W I

rl :: 9 m

v) I

LD m LD m rl

r) I 10 N v)

a, ‘9

P t e4 OD d

d

m LD 0 m

N

rr)

rl v) P-

P-

N

m LD 0

m

N I

m m

m

d I

m N d

9

?

t

m

-?

01

r d

PI

N 1

r- N v)

N t

* I m

W LD

v) ?

1

r: 9

m m d

I” N

I

0 0 d I ?fi d

I

m

N I m

N N

4 ?

m I

v)

m

74 t

rl I m

W

/2 ap a t - P - d

d W 101

I

-;I/ 10 m v)

d t

c?

9

P t

e N N

d

t- t- v)

rl

d d t- cO

d

* t-

m N

v) I m m m

LD

N

I

d

P- I

P- 0 m

W I

m I

W rl m

W

m I 0 0 N P-

m

d rl

rl

rl

U>

p!

m

t

N d

0 ? rl

m I

N

t-

W I

N 0 LD

v)

d 1 m m

d

N

10 I r) m t-

3 “i

I

9

? m

W

W v)

W

m

m

d d

10

0,

d N m

m

01

m

-?

01

0 rl

d

W

N

in

?

=1

I rl I

N a,

? 2 Q d e d c . .

N d

m 1

d N m r- N

d I

rl t- N

rl

m

v) I

d

rl

z t

ID I P-

OD rl

N ?

v) I

P- d W v)

N

W I 0 rl * 10

m

P- I

P- N W

m I

v) I m

W 0

t- 9

d

d W v)

W ?

0 ?I

m r- m

N t

0 rl

LD P- d @!

0 r 4

N 3 W m 6-

10 I

N t- d I

d I

rl d m

m 9

N W N rl m

110

V'

\I'

2

4

6

8

io

12

Fig. A2.7 Loci OF coincidence OF antinodc p i ~ * : D ,

Gal A 4 Y.

111

The Franck-Condon f a c t o r and i n t e n s i t y a r r a y s a r e s i m i l a r only

i n t h e predominance o f t h e main diagonal .

f o r t h e lack of s i m i l a r i t y between t h e o t h e r d iagonals :

(a)

There a r e seve ra l reasons

t h e i n t e n s i t y a r r a y may not be r e a l i s t i c .

band system a r e overlapped, and t h e B system (B3T1+X L') of GaI

e x i s t s i n t h e same s p e c t r a l region. The i n t e n s i t y measurements

a r e only eye es t imates of dens i ty and t h e r e has been no r ecen t

study.

The Franck-Condon f a c t o r a r ray may not be r e a l i s t i c .

small non-zero value o f A r e was chosen which may not have been

c o r r e c t .

causes much change i n t h e form o f t h e a r r a y .

model used i n t h e ca l cu la t ions has a high anharmonicity cons tan t

in f luenc ing t h e form o f the a r r a y considerably.

The sequences of t h e

1

(b) A n a r b i t r a r y

However, it i s doubtful t h a t a small change i n A r e

The Morse p o t e n t i a l

It i s concluded t h a t e i t h e r t h e i n t e n s i t y measurements a r e

i n e r r o r , o r t h a t t he e l ec t ron ic s t a t e s a r e we l l represented by

parabol ic p o t e n t i a l s with v i r t u a l l y i d e n t i c a l equi l ibr ium p o s i t i o n s

( A r e = 0) a s discussed i n Appendix 1.

REFERENCES

Aiken, H . H . 1951. Harvard Problem Report No. 27, ItComputation of t he b t e n s i t i e s of Vibrat ional Spec t ra of Elec t ron ic Bands i n Diatomic Molecules (AF Problem 56) I t .

Allen, C.W. 1962. IIAstrophysical Quan t i t i e s " , 2nd e d i t i o n . Athlone Press, Univers i ty of London.

Bates, D.R. 1952. Mon. Not. Roy. Astron, SOC. 112, 614.

Condon, E.U. 1926. Phys. Rev. 28, 1182.


Condon, E.U. 1947. Am. J. Phys. l5, 365.

Estey, R.S. 1930. Phys. Rev. 35, 309.

F l inn , D . J . , Sp indler , R . J . , F i f e r , S. and Kelly, M. 1964. J. Quant. Spect. Rad. Trans. 4, 271.

Franck, J. 1926. Trans. Faraday SOC. 2, 536.

F rase r , P.A. 1954. Can. J. Phys. 2, 515.

Gaydon, A.G. and Pearse, R.W.B. 1939. Proc. Roy. SOC. 173A, 37 .

Headrick, L.B. and Fox, G.W. 1930. Phys. Rev. 35, 1033.

Hdbert, G.R. and Nichol l s , R.W. 1961. Proc. Phys. SOC. 78, 1024.

Herzberg, G. 1950. IfMolecular Spec t ra and Molecular S t r u c t u r e . I Spectra of Diatomic Moleculest1. 2nd e d i t i o n . D . van Nostrand Company, Inc. Princetown, New Jersey.

Hutchisson, E. 1930. Phys. Rev. 36, 410.

Jarmain, W.R. 1963. Can. J. Phys. 9, 1926.

112

111

The Franck-Condon f a c t o r and i n t e n s i t y a r r a y s are similar only

i n t h e predominance of t h e main diagonal.

f o r t h e l a c k of similarity between t h e o t h e r diagonals :

(a)

There a r e s e v e r a l reasons

t h e i n t e n s i t y a r r a y may not be real is t ic . The sequences of t h e

band system a r e overlapped, and t h e B system (B3n+X 1 L+) o f G a I

e x i s t s i n t h e same s p e c t r a l region.

are only eye est imates of dens i ty and t h e r e has been no r e c e n t

study.

The Franck-Condon f a c t o r a r r ay may n o t be real is t ic .

small non-zero value of A r e was chosen which may no t have been

c o r r e c t .

causes much change i n t h e form o f t h e a r r a y .

model used i n t h e ca l cu la t ions has a high anharmonicity cons t an t

i n f luenc ing t h e form of the a r r a y considerably.

The i n t e n s i t y measurements

(b) An a r b i t r a r y

However, i t i s doubtful t h a t a small change i n A r e

The Morse p o t e n t i a l

It i s concluded t h a t e i t h e r t h e i n t e n s i t y measurements are

i n e r r o r , o r t h a t t h e e l ec t ron ic s t a t e s are w e l l represented by

pa rabo l i c p o t e n t i a l s with v i r t u a l l y i d e n t i c a l equi l ibr ium p o s i t i o n s

( A r e = 0) as discussed i n Appendix 1.

REFERENCES

Aiken, H . H . 1951. Harvard Problem Report No. 27, Womputation of the I n t e n s i t i e s of Vibrat ional Spectra of E lec t ron ic Bands i n Diatomic Molecules (AF Problem 56) I ! .

Allen, C .W. 1962. "Astrophysical Quan t i t i e s " , 2nd ed i t i on . Athlone Press, Universi ty o f London.

Bates, D.R. 1952. Mon. Not. Roy. Astron, SOC. 112, 614.


Condon, E.U. 1928. Phys. Rev. - 32, 858.

Condon, E.U. 1947. Am. J. Phys. IS, 365.

Estey, R.S. 1930. Phys. Rev. 35, 309.

F l inn , D . J . , Spindler , R . J . , F i f e r , S. and Kelly, M. 1964. J. Quant. Spect. Rad. Trans. 4, 271.

Franck, J. 1926. Trans. Faraday SOC. 2, 536.

Fraser , P .A. 1954. Can. J. Phys. 32, 515.

Gaydon, A.G. and Pearse, R.W.B. 1939. Proc. Roy. SOC. 173A, 37.

Headrick, L.B. and Fox, G.W. 1930. Phys. Rev. 35, 1033.

Hdbert, G.R. and Nichol ls , R.W. 1961. Proc. Phys. SOC. 78, 1024.

Herzberg, G. 1950. IIMolecular Spectra and Molecular S t ruc tu re . I Spectra of Diatomic Moleculest1. 2nd e d i t i o n . D . van Nostrand Company, Inc. Princetown, New Jersey.

Hutchisson, E. 1930. Phys. Rev. 36, 410.

Jarmain, W.R. 1963. Can. J. Phys. 3, 1926.

1 1 2

113

Jarmain, W.R., F ra se r , P.A. and Nichol l s , R.W. 1955. Astrophys. J. - 122, 55.

Jenkins, F.A., Barton, H . A . and Mulliken, R.S. 1927. Phys. Rev. 30, 175.

Mahanti, P.C. 1932. Phys. Rev. 42, 609.

Manneback, C . 1951. Physica l7, 1001.

Miescher, E. and Wehrli, M. von 1934. Helv. Phys. Acta 1, 331.

Morse, P.M. 1929. Phys. Rev. - 34, 5 7 .

Nichol l s , R.W. 1960. Can. J. Phys. 38, 1705.

Nichol l s , R.W. 1962a. J. Quant. Spect . Rad. Transf. 2, 433.

~ i c h o l l s , R.W. 196210. Nature 193, 966.

Nichol ls , R.W. 1962c. Journ. Res. Nat. Bur. Stand. m, 227.

Nichol l s , R.W. 1962d. Can. J. Phys. 40, 1772.

Nichol l s , R.W. 1963a. Nature 199, 794.

Nichol l s , R.W. 1963b. presented a t C.A.P. Congress 1963.

Nichol l s , R.W. 1964a. Pr iva te communication.

Nichol l s , R.W. 1964b. Pr iva te communication.

Pauling, L. and Wilson, E.B. 1935. !!Introduction t o Quantum Mechanics!'. McGraw-Hill Book Company Inc o , New York and London.

Pearse , R.W.B. and Gaydon, A.G. 1950. !!The I d e n t i f i c a t i o n of Molecular Spectra" . Chapman and Hall, Ltd. London.

P h i l l i p s , J . G . 1949. Astrophys. J. 110, 73.

Pil low, M.E. 1949. Proc. Phys. SOC. m, 237.

Takamine, T. , Suga, T. and Tanaka, Y . 1938. S c i . Papers fit. Phys. Chem. Research Tokyo 34, 854.

Wehrli, M. von 1934. Helv. Phys. Acta 1, 676.

VITA

NAiME :

BORN :

EDUCATION :

DEGREE :

APPOINTMENTS :

PAPER GIVEN:

Mary Frances Murty

Ru i s l ip , Middlesex, England. 1938.

S a t t e r t h w a i t e School, Lancashire Meadowside, Ru i s l ip , England Northwood College, Northwood, England Ba t t e r sea Polytechnic, London Imperial Colkge, London Universi ty of Western Ontario, Canada

B.Sc. London 1961

Summers 1956 and 1957. Laboratory A s s i s t a n t , The Metal Box Company.

Summer 1958. Student employee, Road Research Laboratory, Department of S c i e n t i f i c and I n d u s t r i a l Research.

Summer 1959. Student employee, H i lge r and Watts Limited.

Summer 1960. Student employee, Ontario Research Foundation, Toronto.

Summer 1961. Laboratory A s s i s t a n t , The Metal Box Company .

1961 - 1962. Research A s s i s t a n t , Molecular Exc i t a t ion Group, Universi ty of Western Ontario.

1961 - 1964. Part-t ime Demonstrator, Department of Physics, Universi ty of Western Ontario.

"Condon Loci of Diatomic Molecular Spectral1, Tetra- Universi ty Colloquium a t t h e State Universi ty o f New York a t Buffalo, October 1963.

' 114

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