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Calculation of matrix elements for diatomic molecules Buckmaster, Harvey Allen 1952

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(1S"L CALCULATION OF MATRIX ELEMENTS FOR DIATOMIC MOLECULES BY . H a r v e y A l l e n B u c k m a s t e r A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n t h e Dep a r t m e n t o f MATHEMATICS We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e s t a n d a r d r e q u i r e d f r o m c a n d i d a t e s f o r t h e d e g r e e o f MASTER OF ARTS. Member o f tfe-erDepartment' o f M a t h e m a t i c s Member o f t h e Departm^njj o f P h y s i c s THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1952 ABSTRACT A number o f p o t e n t i a l s have been suggested as a p p r o x i -mations t o t h e 't r u e ' p o t e n t i a l f u n c t i o n f o r t h e n u c l e i o f a diatomic molecule. The r e l a t i v e m e r i t s o f thes e poten-t i a l s are d i s c u s s e d . Whenever p o s s i b l e the e i g e n f u n c t i o n s and e i g e n v a l u e s c o r r e s p o n d i n g to t h e s e p o t e n t i a l s a re gi v e n . For the Morse p o t e n t i a l [2l] the c a l c u l a t i o n s o f the eigen-f u n c t i o n s and eigenvalues are reproduced i n d e t a i l . These e i g e n f u n e t i o n s a re used to d e r i v e g e n e r a l formulae f o r the r a d i a l p a r t s o f t h e d i p o l e and quadrupole m a t r i x elements. The e x p r e s s i o n f o r t h e d i p o l e m a t r i x element i s (-1) n+m+1 n. r'(n+b+l).b.b3 m! r(m+b'+l) a(n-m) (n-m+b) and .for t h e quadrupole ma t r i x element 1/2 m < n . ( - l ) n + m + 1 2 ml Hm+b'+l) b b f ni T(n+b+l) 1 / 2 ^ r (n-m+l) r (2d+l-n-m -1) j p b r u + i ) r ( 2 d - 2 m + * ) X [\Hm-jO - t(n-m+je-l) - (2d+J?-n-m-2)] + 2 m < n , The symbols are d e f i n e d i n s e c t i o n s 20, 21, and 22, The e x p r e s s i o n f o r 1% i s i n agreement w i t h t h e one d e r i v e d by I n f e l d and H u l l [l6] w h i l e the e x p r e s s i o n f o r MQ i s a r e -s u l t which, so f a r as t h e author i s aware, has not been pub-l i s h e d i n the l i t e r a t u r e . TABLE OF CONTENTS Page INTRODUCTION i CHAPTER I OUTLINE OF PHYSICAL BACKGROUND 1. I n t r o d u c t i o n 1 2 . Quantum M e c h a n i c s 1 3 . D i a t o m i c M o l e c u l e s 3 CHAPTER I I ENERGY LE V E L S AND POTENTIAL FUNCTIONS 4. I n t r o d u c t i o n 7 5 . E x p e r i m e n t a l D a t a 7 6 . Harmonic O s c i l l a t o r S 7 . Anharmonic O s c i l l a t o r 9 8. Morse P o t e n t i a l 9 9 . P S s c h l - T e l l e r P o t e n t i a l 1 2 1 0 . Manning-Rosen P o t e n t i a l 1 2 1 1 . M o d i f i e d Morse P o t e n t i a l I 1 3 1 2 . M o d i f i e d Morse P o t e n t i a l I I 1 3 1 3 . S e r i e s P o t e n t i a l 1 4 1 4 . L i n n e t t P o t e n t i a l 1 4 1 5 . P e r t u r b a t i o n C a l c u l a t i o n w i t h t h e Morse P o t e n t i a l 1 4 1 6 . N u m e r i c a l C u r v e P l o t t i n g 1 6 CHAPTER I I I MATRIX ELEMENTS 1 7 . I n t r o d u c t i o n 18 18. T h e o r y o f M a t r i x E l e m e n t s 18 1 9 . P o p u l a t i o n o f E n e r g y L e v e l s 1 9 2 0 . M a t r i x E l e m e n t s f o r t h e Harmonic O s c i l l a t o r 2 0 2 1 . M a t r i x E l e m e n t s f o r t h e Morse P o t e n t i a l 2 0 2 2 . The S o l u t i o n o f t h e R a d i a l Wave E q u a t i o n w i t h t h e Morse P o t e n t i a l 2 1 2 3 . D i p o l e M a t r i x E l e m e n t s f o r t h e Morse P o t e n t i a l 2 3 2 4 . Q u a d r u p o l e M a t r i x E l e m e n t s f o r t h e Morse P o t e n t i a l 2 5 2 5 . C a l c u l a t i o n s I n v o l v i n g M a t r i x E l e m e n t s 2 7 TABLE OF CONTENTS CHAPTER IV FUTURE INVESTIGATIONS 26. Topics of Further I n t e r e s t APPENDIX BIBLIOGRAPHY J ACKNOWLEDGEMENTS The a u t h o r w i s h e s t o t h a n k Dr. T.E. H u l l f o r s u g g e s t i n g t h e t o p i c o f t h i s t h e s i s and f o r h i s h e l p f u l d i s c u s s i o n d u r i n g i t s ; : p r e p a r a t i o n . H e w i s h e s t o t h a n k Dr. R.D. James f o r h i s a s s i s t a n c e w i t h m a t h e m a t i c a l p r o -b l e m s d u r i n g t h e summer o f 1951 and a l s o t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada f o r making i t p o s s i b l e t o c a r r y o u t t h i s r e s e a r c h u n d e r a Summer G r a n t . INTRODUCTION The purpose of t h i s t h e s i s i s twofold. I n f e l d and H u l l [ l 6 ] , u s i n g the f a c t o r i z a t i o n method., have c a l c u -l a t e d the r a d i a l matrix element f o r the d i p o l e t r a n s i t i o n with the Morse p o t e n t i a l * It was f e l t d e s i r a b l e t o con-f i r m t h i s c a l c u l a t i o n by standard methods of a n a l y s i s . These methods were then to be extended to the c a l c u l a t i o n of the r a d i a l matrix element f o r the quadrupole t r a n s i t i o n with the Morse p o t e n t i a l . The Morse p o t e n t i a l [ 2 l ] was chosen because,when the e f f e c t of r o t a t i o n is neglected, i t has been s u c c e s s f u l l y appl ied to the i n t e r p r e t a t i o n of spectroscopic data f o r diatomic molecules. Quantum mechanics leads to a successful t h e o r e t i c a l i n t e r p r e t a t i o n of spectroscopic data. In Chapter I the basic concepts of quantum mechanics are reviewed, e s p e c i a l l y those r e l e v a n t to the i n t e r p r e t a t i o n of spectroscopic data i n general and diatomic spectra i n p a r t i c u l a r . For diatomic molecules, i t i s necessary to make t e n t a t i v e approximations to the ' t r u e ' p o t e n t i a l f u n c t i o n and then to compare the corresponding c a l c u l a t e d energy l e v e l s and matrix elements with the observed frequencies and i n t e n s i t i e s . A number of these approximations are discussed i n Chapter II and t h e i r corresponding eigenfunotions and'eigenvalues are g i v e n when-ever p o s s i b l e . In Chapter I I I , the Morse p o t e n t i a l i s c o n s i d e r e d . F i r s t , the theory of matrix elements i s o u t l i n e d and a p p l i e d t o the harmonic o s c i l l a t o r . Then Morse's c a l c u l a t i o n s [2l] of the eigenfunctions and eigenvalues are reproduced, i n de-t a i l . F i n a l l y , the r a d i a l matrix elements for the d i p o l e 1 and quadrupole t r a n s i t i o n s with the Morse p o t e n t i a l are c a l -c u l a t e d . The dipole matrix element- agreed w i t h the r e s u l t obtained by I n f e l d and H u l l [ l 6 ] . The quadrupole matrix element i s a r e s u l t h i t h e r t o unpublished. Chapter IV b r i n g s the t h e s i s to a c l o s e w i t h a few suggestions about p o s s i b l e t h e o r e t i c a l further i n v e s t i g a t i o n s i n the f i e l d of d i a t o m i c s p e c t r a . CHAPTER I OUTLINE OF PHYSICAL BACKGROUND 1. I n t r o d u c t i o n In the a p p l i c a t i o n of quantum mechanics t o a p h y s i c a l system c o n s i s t i n g o f the two n u c l e i of a diato m i c molecule, the problem of determining a s u i t a b l e p o t e n t i a l f u n c t i o n a r i s e s . I t i s a d v i s a b l e to review some of t h e fundamental concepts of quantum mechanics before we c o n s i d e r t h i s pro-blem f u r t h e r . These concepts w i l l be examined i n t h i s chapter. Then t h e d e t a i l e d quantum mechanical a n a l y s i s of the motion of the n u c l e i o f a diatomic molecule w i l l be g i v e n . 2 . Quantum Mechanics The s t a r t i n g p o i n t i n the a n a l y s i s o f any p h y s i c a l system t o which a n o n - r e l a t i v i s t i c quantum mechanical treatment i s a p p l i c a b l e i s SchrSdinger's wave equation Hty = Eljf (2.1) where H i s the H a m i l t o n i a n of t h e system, E i s the t o t a l energy o f t h e system and ]]/ i s t h e wave f u n c t i o n of the sys-tem. F o r an n - p a r t i c l e system, (2.1) becomes 4 = o * where -n = h /2r r = 1 . 0 5 4 * IO""2? erg-sec (h - i s P l a n t s constant) m. i s the mass of t h e j t h p a r t i c l e •J A j i s the L a p l a c i a n which depends on t h e co o r d i n a t e of t h e j t h p a r t i c l e V i s the p o t e n t i a l energy which i s a f u n c t i o n o f a l l the c o o r d i n a t e s E i s the t o t a l energy o f t h e system. E q u a t i o n (2.2) can be c o n s i d e r e d s o l v e d when one knows the eige n v a l u e s f o r the energy E and the c o r r e s p o n d i n g e i g e n -f u n c t i o n s . The e x i s t e n c e of e i g e n v a l u e s a r i s e s from t h e requirement t h a t ]j/ s?be/square-integrable and s i n g l e - v a l u e d over the e n t i r e c o o r d i n a t e space. When i s normalized t o one,then i t c a n be i n t e r p r e t e d as t h e p r o b a b i l i t y d i s t r i -b u t i o n o f the p o s i t i o n s of t h e n p a r t i c l e s . E x p e r i m e n t a l l y , we can. observe n e i t h e r the energy l e v e l s nor the wave f u n c t i o n s . What we can observe i s t h e e f f e c t ^ o f a t r a n s i t i o n from one s t a t e of the system to another. These t r a n s i t i o n s a re u s u a l l y caused by an e x t e r n a l d i s t u r -bance, such as a r a d i a t i o n f i e l d . Such a t r a n s i t i o n causes a change i n the t o t a l energy of the system. The emitted o r absorbed energy, which, determines, t h e frequency v of the observed s p e c t r a l l i n e , i s given, by. |-E f - E± \ m hv (2.3) I f Ef < then energy i s emitted and i f E^ < E f t h e n energy i s absorbed. T h i s r e l a t i o n s h i p between t h e frequency and t h e energy i s a b a s i c assumption of quantum mechanics. The t y p e s of t r a n s i t i o n s t h a t occur can be c l a s s i f i e d e x p e r i m e n t a l l y a c c o r d i n g to t h e change i n J , t h e r o t a t i o n a l -quantum number.- The t y p e o f t r a n s i t i o n most f r e q u e n t l y ob-s e r v e d e x p e r i m e n t a l l y o c c u r s when A J = 0 o r l l and i s c a l l e d a d i p o l e t r a n s i t i o n . The n e x t commonest t y p e i s t h e q u a d r u p o l e t r a n s i t i o n f o r w h i c h O c c a s i o n a l l y , o c t u p o l e t r a n s i t i o n s c o r r e s p o n d i n g t o A J = - 3 a r e ob-a t o m i c s e r v e d . F o r a 2 - p a r t i c l e ^ s y s t e m , t h e r e l a t i v e p r o b a b i l i t y o f t h e s e t r a n s i t i o n s i s 1 : 1 0 " ^ : 1 0 " ^ a p p r o x i m a t e l y . The i n t e n s i t y o f a s p e c t r a l l i n e d e p e n d s on t h r e e con-d i t i o n s : (a) t h e f r e q u e n c y o f t h e l i n e ; (b) t h e p o p u l a t i o n o f t h e i n i t i a l s t a t e ; and ( c ) t h e t r a n s i t i o n p r o b a b i l i t y b e -t w e e n s t a t e s . The e x a c t f u n c t i o n a l d e p e n d e n c e o f t h e i n -t e n s i t y o n t h e s e c o n d i t i o n s d e pends o n t h e t y p e of. t r a n s i -t i o n . I t w i l l be c o n s i d e r e d i n g r e a t e r d e t a i l i n S e c t i o n 18. We have t h u s b r i e f l y s u r v e y e d t h e p h y s i c a l b a c k g r o u n d f o r o u r m a t h e m a t i c a l t r e a t m e n t o f t r a n s i t i o n p r o b a b i l i t i e s . F o r more d e t a i l e d a c c o u n t s t h e r e a d e r i s r e f e r r e d t o S c h i f f [ 3 2 ] o r Mott a n d Sneddon [ 2 3 ] . 3 . D i a t o m i c M o l e c u l e s L e t us now c o n s i d e r t h e m o t i o n o f t h e n u c l e i o f a d i a -t o m i c m o l e c u l e . We w i l l f o l l o w t h e quantum m e c h a n i c a l t r e a t -ment o f t h i s p r o b l e m due t o B o r n a n d Oppenheimer [ 3 ] • T h e y assume t h a t t h e wave f u n c t i o n f o r a s y s t e m c o n s i s t i n g o f two atoms f o r m i n g a m o l e c u l e c a n b e w r i t t e n i n t h e f o r m ^ e l e c t r o n ^ n u c l e i { r l » R 2 > ( 3 . ^ e l e c t r o n ^ s t n e n c a l c u l a t e d o n t h e a s s u m p t i o n t h a t t h e two n u c l e i a r e a t a f i x e d d i s t a n c e a p a r t R • T h i s a s s u m p t i o n i s j u s t i f i a b l e b e c a u s e t h e e l e c t r o n s move a r o u n d t h e n u c l e u s w i t h 4 a much high e r frequency of o s c i l l a t i o n t h a n the freque n c y with which the two n u c l e i o s c i l l a t e . I n o t h e r words, the e l e c t r o n s t r a v e l l a r g e d i s t a n c e s while the two .nuclei are r e l a t i v e l y f i x e d . Having s o l v e d f o r ^ e l e c t r o n > w e c a n w r i t e the d i f f e r e n t i a l e q u a t i o n f o r tynu%'i%lei» I t i n c l u d e s n a p o t e n t i a l energy t e r m i n v o l v i n g R as a parameter. S i n c e t h i s term i s d i f f i c u l t to c a l c u l a t e , we are f o r c e d t o a p p r o x i -mate to i t by a s u i t a b l e f u n c t i o n . The wave equation f o r the n u c l e i i s K V , 1 ^ + f l 5 ^ + ( E - V ^ = ° ...... O.*) where m-^, are the masses of t h e two n u c l e i Va* i s t h e L a p l a c i a n w i t h r e s p e c t to the co o r d i n a t e s of p a r t i b l e 1 • . V i s t h e L a p l a c i a n w i t h r e s p e c t t o t h e x c o o r d i n a t e s of p a r t i c l e 2 V(r.2, r 2 ) i s the p o t e n t i a l energy of the system when t h e p a r t i c l e s are at r ^ and r ^ r e s p e c t i v e l y . I f we change t o ce n t r e of mass and r e l a t i v e c o o r d i n a t e s , then the wave equ a t i o n s e p a r a t e s i n t o two p a r t s - i J V X = E^, ( 3 . 3 ) - J £ £ I J U ( E t - V ) l H ( 3 . 4 ) m-j_m2 where i]/ = tyntyo ; E = E-. + E^ ; M = m-, +m, ; = 11 <- , fr m l + m 2 E q u a t i o n ( 3 . 3 ) r e p r e s e n t s the motion of the c e n t r e of mass which behaves l i k e a f r e e p a r t i c l e of mass M . Eq u a t i o n ( 3 . 4 ) r e p r e s e n t s t h e e q u a t i o n of motion of a p a r t i c l e of mass u- i n a p o t e n t i a l V . R e w r i t i n g ( 3 . 4 ) i n s p h e r i c a l c o o r d i n a t e s and s e p a r a t i n g so t h a t V 2 ( x , y, z) =• <p(r, ©, $) = R ( r ) Y ( 9 , $ ) , . (3 .5) Then Y(©, .g?) i s t h e normalized s p h e r i c a l harmonic ' r -j V a . J , y ( e , » ) . "/ '""-'r i ' ' p:(co.e)e (3-6) '^TT « + | m . | ) j where Pg(cos 9) i s an a s s o c i a t e d Legendre f u n c t i o n . The r a d i a l wave equation becomes 1 To s o l v e t h i s e q u a t i o n , we must know the form of the poten-t i a l f u n c t i o n V ( r ) } as p r e v i o u s l y mentioned. S i n c e we,can-not d i r e c t l y observe i t i n a l a b o r a t o r y we must choose a f u n c t i o n such t h a t the above e q u a t i o n w i l l l e a d to eigen-v a l u e s and e i g e n f u n c t i o n s (hence to f r e q u e n c i e s and i n t e n -s i t i e s ) which agree with the experimental data f o r the v i b r a -t i o n r o t a t i o n s p e c t r a of d i a t o m i c molecules. U n f o r t u n a t e l y , i t can be shown that the,forms o f V(r) which enable us t o s o l v e t h e above equation i n terms o f known 1 1 o f u n c t i o n s c o n t a i n only the terms , ~ or r ^ and thexe forms do not l e a d t o s a t i s f a c t o r y e i g e n v a l u e s . Experimen-l(Jg+l) t a l l y , we know t h a t t h e e f f e c t of the r o t a t i o n a l term ^> i s s m a l l and so c a n be t r e a t e d as a p e r t u r b a t i o n . . I n what f o l l o w s , we w i l l n e g l e c t t h i s term and can t h e r e f o r e c o n s i d e r a much wider v a r i e t y of p o t e n t i a l f u n c t i o n s . 1 . .  The shape of t h e p o t e n t i a l curve i s r e s t r i c t e d by t h e f o l l o w i n g three c o n d i t i o n s . (a) 'V(r) h a s a s i n g l e minimum (b) V ( r ) h a s a f i n i t e v a l u e as r -*• °° ( c ) V ( r ) i s l a r g e f o r s m a l l v a l u e s o f r - . F i g u r e 1 E D i a g r a m s h o w i n g r t h e p o t e n t i a l c u r v e f o r t h e g r o u n d s t a t e o f H x • Above v "= 14 , t h e r e i s a c o n t i n u o u s s p e c t r u m i n d i -c c a t e d b y v e r t i c a l h a t c h i n g . ' E x a c t ' C u r v e Morse Curve CHAPTER I I ENERGY LEVELS AND POTENTIAL FUNCTIONS 4. I n t r o d u c t i o n I n t h i s c h a p t e r , we s h a l l examine energy l e v e l s from the experimental spectroscopist^s p o i n t of view. We w i l l t hen c o n s i d e r a number of f u n c t i o n s t h a t have been p r o -posed as p o s s i b l e approximations t o the ' t r u e ' p o t e n t i a l f u n c t i o n f o r the n u c l e i of a d i a t o m i c molecule. The eigen-f u n c t i o n s and e i g e n v a l u e s corresponding to these f u n c t i o n s are g i v e n whenever they a r e a v a i l a b l e . A p p r o p r i a t e com-ments are made about the r e l a t i v e m e r i t s of t h e s e f u n c t i o n s . 5. Experimental Data The experimental v i b r a t i o n - r o t a t i o n s p e c t r a o f diatomic molecules can be used to c a l c u l a t e c e r t a i n s p e c t r o s c o p i c c o n s t a n t s . These constants a r e r e l a t e d to t h e energy l e v e l s a c c o r d i n g to t h e f o l l o w i n g e m p i r i c a l r e l a t i o n E v = 2 Y l j ( v + i ) c . [ J ( J + i ) ] J where v i s t h e v i b r a t i o n a l quantum number and J i s the r o t a t i o n a l quantum number. I t should be noted t h a t the t h e o r e t i c a l p h y s i c i s t uses n and JI where the experimen-t a l s p e c t r o s c o p i s t uses v and J r e s p e c t i v e l y . (Herzberg [12] ). The more important of t h e s e constants are Y oo - D Heat o f D i s o c i a t i o n Y"lo _ w e C l a s s i c a l Frequency of O s c i l l a t o r Y 2 o - ~ w e x e Anharmonicity Constant Y Q2 - B g R o t a t i o n a l Constant ^11 ~ ~ a e Fine S t r u c t u r e Constant These f i v e c o nstants are u s u a l l y s u f f i c i e n t to guarantee good agreement w i t h s p e c t r o s c o p i c data. 6 . Harmonic O s c i l l a t o r The p o t e n t i a l f u n c t i o n i»s one of t h e s i m p l e s t f u n c t i o n s t h a t we can c o n s i d e r . A p a r t i c l e moving i n such a p o t e n t i a l i s s a i d to perform har-monic o s c i l l a t i o n s . I t would be t h e exact f u n c t i o n i f the f o r c e s h o l d i n g t h e n u c l e i t o g e t h e r obeyed Hooke's law. The wave equation f o r the harmonic o s c i l l a t o r i s V(x) = | px: .2 ( 6.1) ( 6 . 2 ) where m i s the mass of t h e p a r t i c l e . The s o l u t i o n o f ( 6 . 2 ) i s , a c c o r d i n g to Sommerfeld ( 6.3) 4Tr2mv where a = — ~ — and H n(Jc? x) i s the nth degree Hermite p o l y n o m i a l . The eigenvalues are given by . E n = (n + | ) h v Q (6 .4) where n = 0 , 1, 2 , . . . Hence the harmonic o s c i l l a t o r l e a d s to an eigenvalue ex-p r e s s i o n c o n t a i n i n g only the dominant term of the e m p i r i -c a l s e r i e s . 7. Anharmonic O s c i l l a t o r A p o t e n t i a l f u n c t i o n c l o s e l y r e l a t e d to the p o t e n t i a l f o r the harmonic o s c i l l a t o r i s V(x) = | p x 2 + qx3 (7.1) It introduces a s m a l l c o r r e c t i o n term (quadratic i n n ) to E n , when q i s s m a l l . The effect of the term i n x3 can be found by c o n s i d e r i n g i t as a p e r t u r b a t i o n of the harmonic o s c i l l a t o r . The eigenvalues are given by B i r t w i s t l e [2] as E = h V o ( n ^ ) - < l V.(30n a+30n+U) ( 7 . 2 ) The constant q i s a measure of the anharmonicity and the spectroscopic anharmonicity constant i s p r o p o r t i o n a l t o i t . Morse P o t e n t i a l Morse [ft] has suggested V(r) = D ( e - 2 a ( r " r e ) - 2 e - a ( r " r e ) ) (S.1) where r e i s the e q u i l i b r i u m distance between the n u c l e i D i s the heat of d i s s o c i a t i o n a i s a parameter JI {£ +1) I f the term ^ 2 i s n e g l e c t e d i n the r a d i a l wave equa-t i o n , t h e n a good approximation t o the s o l u t i o n i s g i v e n by where 3 = I d e ' * 0 ^ 4 = 2 d - l n - l H - b + l ) / The corresponding e i g e n v a l u e s are E n = -D + h [w e(n+l) - w e x e ( n + l ) 2 ] ( g # 3 ) ^ w .4.(22)1/2 . w x . hw| h £ The d e t a i l s of t h e c a l c u l a t i o n of these r e s u l t s w i l l be g i v e n i n Chapter I I I . Where t h e e f f e c t o f r o t a t i o n i s n e g l i g i b l e , the Morse p o t e n t i a l y i e l d s r e s u l t s which agree very w e l l with experimental data. Moreover, i t i s easy to handle mathema-t i c a l l y ; hence i t s g r e a t p o p u l a r i t y w i t h s p e c t r o s c o p i s t s . 11 F i g u r e 2. 15 2.0 2.5 R •—• 3.0 35 A 1 ~ ~ ~ • " -7 — — : — — ' Diagram showing a number of p o t e n t i a l c u rves f o r CdH Hulbert and H i r s c h f e l d e r S e c t i o n 12 Rydberg-Hyllerass S e c t i o n 16 C o o l i d g e , Vernon and James S e c t i o n 11 Morse S e c t i o n 6\ 12 9. PSschl-Teller Potential Pflschi and T e l l e r [2iJ have suggested * A V ( v ) = c o s ^ « ( r - v e ) (9.1) S£nk,«((v--rc^  where v , u. > 1 ; - H ~ Infeld and Hull [l6j give recurrence r e l a t i o n s f o r the eigenfunctions. The eigenvalues are E n = -a 2(-H + v + 2n) 2 &2) where n' -is a pos i t i v e integer and -\x + v + 2n < 0 . It should be noted that i f -u- + v = k then t h i s potential reduces to the Morse function. Because i t contains one more i parameter than the Morse function, the PSschl-Teller func-t i o n i s more f l e x i b l e and more accurate. However, the i n -creased accuracy does not warrant the increased labor of manipulation. 10. Manning-Rosen Potential The Manning-Rosen poten t i a l [l9] i s A e -o-e - * r Its eigenfunctions are given by Y M r ) - ( i - Z ) F(z) (10.1) (10.2) r where z = e~' ; k = ^ . ; m = ( - k p ^ ) 1 / 2 and F(z) i s a terminated hypergeometric series. The corresponding eigenvalues are ( A - / g ) - n ( n + * g ) (10.3) where n is a p o s i t i v e integer. Infeld and Hull [ l 6 ] give recurrence r e l a t i o n s f o r - 1 3 . the e i g e n f u n c t i o n s . Here, as w i t h the P 5 s c h l - T e l l e r poten-t i a l , the i n c r e a s e d accuracy does not warrant the i n c r e a s e d l a b o r of mani p u l a t i o n . 1 1 . M o d i f i e d Morse P o t e n t i a l I Coolidge, James and Vernon [4] have suggested which i s s i m i l a r to the Morse p o t e n t i a l . For n = 5 and 7 , t h e y comparied t h e i r p o t e n t i a l curve f o r H 2 with t h a t c a l c u l a t e d by H y l l e r a a s ' method (see s e c t i o n 1 6 ) and found b e t t e r agreement than when the Morse p o t e n t i a l was used. H o w e v e r t h i s p o t e n t i a l does not permit the e x p l i c i t s o l u -t i o n of the wave equa t i o n , nor can the eig e n v a l u e s be found e x a c t l y . 1 2 . M o d i f i e d Morse P o t e n t i a l I I Hulbert and H i r s c h f e l d e r [ 1 3 ] have suggested the f o l -lowing modified Morse f u n c t i o n . w h e r e c - 1 _ _L (1 + *e^e\ c I n * V l « T UB. 1 MB} W / l T h i s f u n c t i o n has the advantage of having f i v e parameters which can be r e l a t e d t o the f i v e s p e c t r o s c o p i c constants w , x ew e, D e, B e, and a e t h a t are the o n l y ones d e t e r -mined f o r most e l e c t r o n i c s t a t e s . However, i t does not l e a d to an exact s o l u t i o n of the wave e q u a t i o n nor can the 14. e i g e n v a l u e s be o b t a i n e d e x a c t l y . S i n c e i t consists o f t h e Morse f u n c t i o n p l u s a c o r r e c t i o n t e r m , i t w o u l d a p p e a r t h a t t h e e i g e n f u n c t i o n s and e i g e n v a l u e s c o u l d be o b t a i n e d u s i n g p e r t u r b a t i o n methods. U n f o r t u n a t e l y , t h e i n t e g r a l s en-c o u n t e r e d a p p e a r t o be t o o - c o m p l e x t o c a l c u l a t e i n t h e g e n e r a l c a s e . I n most c a s e s where t h i s f u n c t i o n h a s b e e n a p p l i e d , t h e f i t w i t h e x p e r i m e n t a l d a t a i s much b e t t e r t h a n t h a t o b t a i n e d w i t h t h e Morse f u n c t i o n . ( H e r z b e r g [ l 2 ] ) , 13. S e r i e s P o t e n t i a l Dunham [7] and o t h e r s have u s e d p o t e n t i a l f u n c t i o n s V ( 0 » ± + + c ( * - r e f (13.1) They have u s e d t h e W.K.B. method t o c a l c u l a t e t h e r e l a t i o n s b e t w e e n t h e p o t e n t i a l p a r a m e t e r s and t h e s p e c t r o s c o p i c c o n -s t a n t s . T h i s method i s v e r y l a b o r i o u s . The s e r i e s do n o t i n g e n e r a l c o n v e r g e and a r e o n l y a c c u r a t e f o r v a l u e s o f r n e a r r . e 14. L i n n e t t P o t e n t i a l L i n n e t t [ l S ] p r o p o s e d t h e p o t e n t i a l f u n c t i o n V(v) = A - b e " 0 ' (14.1) He h a s shown t h a t t h e r e l a t i o n s h i p b e t w e e n t h e p a r a m e t e r s i s more c l e a r l y d e f i n e d t h a n f o r t h e Morse f u n c t i o n . How-e v e r , i t c a n n o t be u s e d t o o b t a i n e i g e n f u n c t i o n s and e i g e n -v a l u e s e x p l i c i t l y and t h e p a r a m e t e r s have n ot been r e l a t e d t o t h e s p e c t r o s c o p i c c o n s t a n t s . 15. P e r t u r b a t i o n C a l c u l a t i o n w i t h t h e Morse P o t e n t i a l So f a r , we have had t o n e g l e c t t h e r o t a t i o n a l t e r m when , 15. we d e s i r e d exact s o l u t i o n s . Morse [21} has shown how t h i s term can be absorbed, to a f i r s t approximation, i n t o h i s p o t e n t i a l . He obtained a e « 2B ex e 0-5.D D = ^ t | e 3 15 .2) We which were ob t a i n e d e a r l i e r b y - K r a t z e r . (15.2) has been v e r i f i e d e x p e r i m e n t a l l y f o r a l a r g e number o f molecules. (15.1) i s too l a r g e by a f a c t o r 1 . 4 . Using p e r t u r b a t i o n method* P e r k e r i s [24] has been able to take the r o t a t i o n a l term i n t o account when the Morse p o t e n t i a l i s used. He 2 2 expanded £ e / r about rQ k av-e / where y = e " a ^ r " r e ^ and u s i n g the f i r s t t h r e e terms obtained an approximate s o l u t i o n of t h e r a d i a l wave e q u a t i o n W h S r e 1 = a d e " " 0 ^ l\-The c o r r e s p o n d i n g e i g e n v a l u e s a r e + Kc{j(3 + l ) B c L l - •(€]}+[J(3 * l ) ] \ c De H e r e n = - ^ Be (15.5) (15.6) (15.7) (15.7) y i e l d s v a l u e s f o r a e w h i c h a g r e e w i t h e x p e r i m e n t a l v a l u e s . P e r k e r i s shows t h a t t h e c o n t r i b u t i o n due t o t h e f i r s t n e g l e c t e d t e r m o f t h e s e r i e s i s s m a l l ; b u t he does n o t show t h a t t h e c o n t r i b u t i o n f r o m t h e e n t i r e n e g l e c t e d p a r t i s s m a l l . 1 6 . N u m e r i c a l C u r v e P l o t t i n g K l e i n [ 1 7 ] , R y d b e r g [2d], , [29] and H y l l e r a a s [ l 5 ] have d e v e l o p e d methods o f c o n s t r u c t i n g t h e p o t e n t i a l c u r v e f r o m t h e o b s e r v e d s p e c t r u m w i t h o u t a s s u m i n g a n a n a l y t i c a l e x p r e s s i o n f o r t h e p o t e n t i a l f u n c t i o n . T h e i r methods a r e now s t a n d a r d and i t i s c u s t o m a r y t o c a l l t h e r e s u l t i n g c u r v e s ' e x a c t ' p o t e n t i a l c u r v e s . O t h e r p o t e n t i a l s l i k e t h e Morse a r e t h e n compared w i t h t h e ' e x a c t ' c u r v e . The j u s t i f i c a t i o n of t h i s p r a c t i c e i s open t o d o u b t . Two ap-p r o x i m a t i o n s a r e i n v o l v e d . The W.K.B. p e r t u r b a t i o n method 17. i s u s e d and t h e e i g e n v a l u e s a r e assumed t o be o f a d e f i n i t e f o r m . Hence t h e c u r v e s a r e not e x a c t b u t r a t h e r a b e t t e r a p p r o x i m a t i o n t o t h e c o r r e c t c u r v e t h a n , s a y , t h e Morse f u n c t i o n . Rees [27] h a s f o r m u l a t e d an a n a l y t i c a l method o f p e r f o r m i n g t h e c u r v e c o n s t r u c t i o n b a s e d on t h e K l e i n -R y d b e r g p r o c e d u r e . A number o f a u t h o r s , i n c u l d i n g B a t e s [ l ] and P i l l o w [25] , have u s e d t h e s e c u r v e s t o p l o t t h e e i g e n f u n c t i o n s o f t h e v a r i o u s s t a t e s a n d have u s e d n u m e r i -c a l i n t e g r a t i o n t o c a l c u l a t e m a t r i x e l e m e n t s f r o m them. 20. M a t r i x E l e m e n t s f o r t h e H a rmonic O s c i l l a t o r We now t u r n to t h e p r o b l e m o f c a l c u l a t i n g m a t r i x e l e m e n t s . B e f o r e t h e s e c a l c u l a t i o n s a r e p e r f o r m e d f o r t h e Morse p o t e n t i a l , i t i s o f i n t e r e s t t o c a l c u l a t e t h e m a t r i x e l e m e n t s f o r t h e h a r m o n i c o s c i l l a t o r . T h i s i n -t e r e s t i s due t o t h e f a c t t h a t t h e h a r m o n i c o s c i l l a t o r h a s as e i g e n v a l u e s t h e p r i n c i p a l term o f t h e e m p e r i c a l e n e r g y l e v e l r e l a t i o n . T h e r e s u l t of t h e c a l c u l a t i o n o f t h e d i p o l e m a t r i x e l e m e n t f o r t h e h a r m o n i c o s c i l l a t o r i s • . (20 |£te±H m - n + i Thus we f i n d t h a t t h e o n l y a l l o w e d d i r e c t t r a n s i t i o n s f o r t h e h a r m o n i c o s c i l l a t o r a r e t o a d j a c e n t l e v e l s . The a b o v e r e s u l t i s c a l c u l a t e d i n d e t a i l i n B i r t w i s t l e [2] and Sommerfeld [33] . 21. M a t r i x E l e m e n t s f o r t h e Morse P o t e n t i a l B e f o r e c o m p u t i n g t h e r a d i a l m a t r i x e l e m e n t s w i t h t h e Morse p o t e n t i a l , we s h a l l r e p r o d u c e t h e c o m p l e t e s o l u t i o n o f t h e r a d i a l wave e q u a t i o n w i t h t h i s f u n c t i o n (Morse [21] ) From t h e method o f i t s s o l u t i o n and t h e c a l c u l a t i o n o f t h e n o r m a l i z a t i o n c o n s t a n t , we s h a l l o b t a i n t h e k e y t o t h e method o f c a l c u l a t i n g t h e v a r i o u s i n t e g r a l s w h i c h a r i s e i n f i n d i n g t h e d i p o l e and q u a d r u p o l e m a t r i x e l e m e n t s . I n t h e a p p e n d i x , t h e v a r i o u s i n d e n t i t i e s c o n c e r n i n g L a g u e r r e p o l y n o m i a l s , w h i c h a r e u s e d t h r o u g h o u t t h e f o l l o w i n g c a l c u -l a t i o n s ^ a r e enumerated., 21. R " 0 (22.1) 22* The S o l u t i o n of the.. R a d i a l Wave E q u a t i o n w i t h t h e  Morse P o t e n t i a l ^ The a p p r o x i m a t e s o l u t i o n o f t h e r a d i a l wave e q u a t i o n f o r t h e n u c l e a r m o t i o n o f a d i a t o m i c m o l e c u l e w i t h t h e Morse P o t e n t i a l has b e e n o b t a i n e d by Morse [2l] . I f t n = N n R n ( r ) Ij^ m( 8» 9) i s a s o l u t i o n o f t h e c o m p l e t e wave e q u a t i o n f o r a d i a t o m i c m o l e c u l e ( s e e s e c t i o n 3) where N n i s t h e n o r m a l i z a t i o n c o n s t a n t f o r R n ( r ) and Ijj m(©> <p) i s t h e n o r m a l i z e d s p h e r i c a l h a r m o n i c , t h e n t h e r a d i a l wave e q u a t i o n i s +[-$<E-V)-.4g> I f S(r) = r R ( r ) , t h e n (22.1) becomes it* 6yx \ **** I ' When V ( r ) h a s t h e f o r m o f t h e Morse p o t e n t i a l ( s e e s e c -t i o n 8) and i f we n e g l e c t t h e r o t a t i o n a l t e r m , t h e n ( 2 2 . 2 ) becomes I f u = r - r- and y = e ~ a u , t h e n (22.3) becomes L e t S(y) = e ~ z / 2 z b / 2 F ( z ) where z = 2dy T h e n - ^ £ F + ( k + 1 _ 3 ) i F + ( y - t - i ) F tt0 ( 2 2 - 5 ) The s o l u t i o n o f (22.5) i s a f i n i t e p o l y n o m i a l i f i s a p o s i t i v e i n t e g e r ; i n w h i c h c a s e , F ( z ) i s an a s s o c i a t e d L a g u e r r e p o l y n o m i a l (SzegB [34] ) . Hence b = 2 d - 2 n - l where n = 0 , 1, 2, .... I f b > 0 , t h e n S ( y ) i s f i n i t e 22. over (0, oo)' . The u s u a l boundary c o n d i t i o n s imposed on R(r)~ are t h a t i t be f i n i t e , s i n g l e - v a l u e d , and continuous i n the range 0 < r < °° . I f we assume t h a t i t must be continuous i n the range - 0 0 < r < °o , t h e n we c a n show t h a t the e r r o r i n t r o d u c e d i s , i n g e n e r a l , n e g l i g i b l e , (D. t e r Haar [10] ). Morse, F i s k and S c h i f f [22] have shown t h a t t h i s assumption i s not a p p l i c a b l e t o the case o f deu-te r i u m . Hence, i f t h e range of r i s (- 0 0 , °°) t h e n t h e range of y i s (0, °°) and a l l the c o n d i t i o n s imposed on R(r) are s a t i s f i e d by our s o l u t i o n . The s o l u t i o n o f (22.5) i s d e f i n e d by - ^ R — W U~V+t) T h i s can be shown t o l e a d t o the f o l l o w i n g two r e l a t i o n s (Appendix) ci) = e i T r t r ( n + b + i ) L b n v J ^ ( 2 2 - ? ) where L^^)= IZIT))^ ^22'^ or / nvb \ , , , ^ . . . (22.3b) (22.6) ( n-<.) " (n+V>X""»-t-l)... (ft+b-(n--M)) (22.3c) Hence r R„ (r) - ^ e " * ' \ W * ( 2 2-9) where; :z-==2de- a ( r- re) n- |- b E v a l u a t i n g the n o r m a l i z a t i o n c o n s t a n t N 0 i n v o l v e s the c a l c u l a t i o n o f the i n t e g r a l 00 S, .( R . W R . M - 1 ( 2 2 . 1 0 ) J- 00 . oo 23. S u b s t i t u t i n g (22.9) i n (22.10) and l e t t i n g z = 2 d e ~ a ^ r ~ r e ^ , we o b t a i n Making use o f (22.7) and 2 2 .gj and i n t e r c h a n g i n g t h e order o f summation and i n t e g r a t i o n , we o b t a i n ( t r U ( - ) ^ I ^ ( e " h " ) d v (22-12) Irt e g r a t i o n by p a r t s n times sho\vs that the o n l y t e r m f o r which we get a c o n t r i b u t i o n i s the one f o r which Ji = 0 . Notice t h a t a l l the i n t e g r a t e d terms vanish over the i n -t e r v a l ( 0 , °°)i Hence _a_ _ /n+b \ [ -s b-1 I, \Kf~ * n I I 5 1 ( 2 2 - 1 3 ) Then ^ / a b - o l Y*^ (22.14) L e t e £ T r b N o ( 2 2 - 1 5 ) Then r R ( r ) = e " 3 / 2 z b / 2 l £ + b ( 2 . ) where z = 2 d e - a ( r " r e ) (22.16) i s the normalized s o l u t i o n of the r a d i a l wave e q u a t i o n when the p o t e n t i a l f u n c t i o n has t h e Morse form. 23. D i p o l e M a t r i x Elements f o r the Morse P o t e n t i a l We s h a l l now c a l c u l a t e the r a d i a l m a t r i x element f o r the d i p o l e t r a n s i t i o n f o r the Morse p o t e n t i a l . The d e s i r e d m a t r i x element i s d e f i n e d by (23.1) 24. I n o r d e r t o o b t a i n a c l o s e d e x p r e s s i o n f o r M Q when t h e e i g e n f u n c t i o n s f o r Morse p o t e n t i a l a r e u s e d , i t i s n e c e s s a r y t o assume t h a t t h e r a n g e o f i n t e g r a t i o n i s (- 0 0 , 0 0 ) i n s t e a d o f (0, 0 0 ) . The v a l i d i t y of t h i s a s s u m p t i o n i s , , i n g e n e r a l , j u s t i f i a b l e (D. t e r Haar [ l o ] ) » S u b s t i t u t i n g t h e wave f u n c t i o n s o f (22.16) and making t h e change o f v a r i a b l e z = 2 d e ~ a ^ r ~ r e ^ , we o b t a i n M D - - m v . « r > ^ L > ( 2 3 . 2 ) ft J Q where b = 2d - 2n - 1, b' = 2d - 2 m - 1 . W r i t i n g t h e L a g u e r r e p o l y n o m i a l s as i n ( 2 2 . 3 ) , a s s u m i n g n >^ m , and i n t e r c h a n g i n g t h e o r d e r o f summation and i n t e -g r a t i o n , we o b t a i n where A „ f l y V ^ V V ^ S Jo ^ and P= n-m-1 + £ I n t e g r a t i n g A b y p a r t s p t i m e s g i v e s where g » f ^ i T " * ( e " ^ m t ) « ) s I n t e g r a t i n g B by p a r t s (m - S. + 1) t i m e s g i v e s B« r(<cd ,4/-n-r—'\) (23.5) By (23.4) and ( 2 3 . 5 ) , . (23.3) becomes ^ ° = ^ & I J ~ I i ( 2 3* 6 ) (23.6) c a n be aummed by i n d u c t i o n . Then s u b s t i t u t i n g f o r and TI^ , we o b t a i n 2 5 n l r . ( n v U l ) l . L ' m < n ( 2 3 . ( 2 3 . 7 ) can be shown to be i d e n t i c a l w i t h the r e s u l t obtained by I n f e l d and H u l l [ l 6 ] (the f a c t o r i z a t i o n method) and Heaps and Herzberg [ l l ] (a d i f f e r e n t method of standard a n a l y s i s ffom t h a t used above). M D had p r e v i o u s l y been c a l c u l a t e d by Wu [ 3 6 ] and Mizushima [ 2 0 ] f o r m = 0 ,, n = 1 , 2 , and 3 i n agreement wi t h ( 2 3 . I ) . 2 4 . Quadrupole M a t r i x Elements f o r the Morse P o t e n t i a l We s h a l l now c a l c u l a t e the r a d i a l matrix element f o r the quadrupole t r a n s i t i o n f o r the Morse p o t e n t i a l . The d e s i r e d m a t r i x element.-is d e f i n e d by /•CO 0 iEn order to o b t a i n a c l o s e d e x p r e s s i o n f o r M Q when the e i g e n f u n c t i o n s f o r t h e Morse p o t e n t i a l are used, i t i s necessary to assume t h a t the range of i n t e g r a t i o n i s ( _ 0 0 } 0 0 ) i n s t e a d o f (0, 0 9 ) . The v a l i d i t y of t h i s assump-t i o n : ^ , i n g e n e r a l , j u s t i f i a b l e (D. t e r Haar [ l o j ). S u b s t i t u t i n g the wave f u n c t i o n s of ( 2 2 . 1 6 ) and making the change of v a r i a b l e z = 2 d e ~ a ( r " " r e ) , we o b t a i n roo M Q = J I m T F T r 2 R n ( r ) r 2 d r ( 2 4 . 1 ) a"* where C ~ W r i t i n g t h e Laguerre "polynomials as i n (22.8), assuming n > m and i n t e r c h a n g i n g t h e order of summation and i n t e -g r a t i o n , we o b t a i n ( 2 4 . 3 ) 26. where . D - [ ( ^ ) V ^ U ' S ^ ) h and p = n m - 1 + JL I n t e g r a t i n g D by p a r t s p times g i v e s D - I [ E +- 2 {VO)**} F ] (24.4) where ^ ~ E u l e r ' s constant and W*)-ifr(™>r-V*%[$-h) I f z i s a p o s i t i v e i n t e g e r n , then f ( n ) = - X + 2 i i = l x (Whittaker and Watson [35] page 246) I n t e g r a t i n g . E by p a r t s (m -JL + 1) times g i v e s E = .2(m -|)« [-G + {^(m - 4 ) + *} H ] (24.5) r 0 0 where G = i n z e ~ z z 2 d + * - i ^ i i H 2 d x fOO and H = f e - z z 2 d + | - n - m - 2 d z JO By d e f i n i t i o n H = F ( 2 d + * - n - m - l ) (24.6) G = - A H = - P '(2d + 't- n - m - 1) (24.7) I n t e g r a t i n g F by p a r t s (m - X+ 1) times g i v e s - F * - ( m ' - j t ) 2 T (2d + n - m - 1) (24.8) By (24.3 - S ) , (24.2) becomes 2 7 . |VJ = ( - 1 ) a? n l Since t h e summation cannot be completely performed, ( 2 4 . 9 ) i s the most u s e f u l form i n which, f o r c a u l c u l a t i o n purposes, t h e e x p r e s s i o n f o r may be l e f t . S u b s t i t u t i n g the v a l u e s , we o b t a i n o f N£ and N m 4.0 ' i ^ * / P ( M - W ) * [q;(m-je)_ l//(n-m + *-:l) - If/fad**-( 2 4 . 1 0 ) where MQ" i s g i v e n by ( 2 3 . 7 ) and m < n ( 2 4 . 1 0 ) i s a r e s u l t which, as f a r as the author knows has not been reached elsewhere. 2 5 . Numerical C a l c u l a t i o n o f M a t r i x Elements Dunham [ 6 ] has c a l c u l a t e d , approximately, the d i p o l e matrix elements w i t h t h e Morse p o t e n t i a l . H i s r e s u l t s are compared with ours f o r HC1. The v a l u e s obtained w i t h t h e two ex p r e s s i o n s agree c l o s e l y . T r a n s i t i o n Dunham I n f e l d and H u l l n=l, m=0 7 . 6 7 i o " 1 0 7 . 6 6 4 8 8 I O " 1 0 •" n= 2 , . m=0 7 . 1 6 I O " 1 1 7 . 1 7 2 4 7 I O - 1 1 CHAPTER IV FUTURE INVESTIGATIONS 26. Topics of F u r t h e r I n t e r e s t As a r e s u l t o f the i n v e s t i g a t i o n s r e p o r t e d i n t h i s * study, a number of problems have a r i s e n , whose s o l u t i o n s are o f p h y s i c a l i n t e r e s t . The Morse, P 8 s c h l - T e l l e r , and Manning Rosen p o t e n t i a l s a l l have s i m i l a r e i g e n v a l u e s . They can be made t o f i t phy-s i c a l d ata e q u a l l y w e l l ; the Morse p o t e n t i a l b e i n g the e a s i e s t t o handle. I t would t h e r e f o r e be i n t e r e s t i n g t o compare the matrix elements o b t a i n e d f o r each o f these t h r e e p o t e n t i a l s . The t h e o r e t i c a l l y c a l c u l a t e d i n t e n s i t i e s of v a r i o u s t r a n s i t i o n s obtained from these m a t r i x elements c o u l d be compared w i t h t h e e x p e r i m e n t a l l y observed v a l u e s . T h i s procedure would provide another b a s i s f o r comparison o f p o t e n t i a l f u n c t i o n s . U n f o r t u n a t e l y , t h e p r e v i o u s l y mentioned problem of the d i s t r i b u t i o n of atoms i n t h e v a r i o u s s t a t e s makes comparison of experimental v a l u e s w i t h t h e o r e t i c a l v a l u e s d i f f i c u l t because o f t h e l a r g e e r r o r i n v o l v e d . Furthermore, t h e t e c h n i c a l problems i n -v o l v e d make i t d i f f i c u l t to measure the . i n t e n s i t i e s ac-c u r a t e l y . The c a l c u l a t i o n of most matrix elements i s based on 2 9 . the assumption t h a t t h e e l e c t r o n i c l e v e l remains unchanged d u r i n g the t r a n s i t i o n . E x p e r i m e n t a l l y , these l e v e l s u s u a l l y change, Schuler [31] and others are i n t e r e s t e d i n the c a l -c u l a t i o n of matrix elements i n which the e l e c t r o n i c l e v e l t r a n s i t i o n s are a l s o t a k e n i n t o account. When the Morse p o t e n t i a l i s assumed, the problem does not seem amenable to an exact s o l u t i o n , f o r the two wave f u n c t i o n s would i n v o l v e d i f f e r e n t parameters. Bates has t a b u l a t e d the v a l u e s of the Morse p o t e n t i a l parameters f o r the d i f f e r e n t e l e c t r o n i c l e v e l s i n the case of N a . These f i g u r e s show t h a t a l t h o u g h t h e changes are not l a r g e t h e y are s i g n i f i -c ant. The work of Rees [27] on t h e development of a n a l y t i c a l methods of c a l c u l a t i n g p o t e n t i a l f u n c t i o n s from spectroscopic ,data sheds l i g h t on the g e n e r a l problem o f p o t e n t i a l f u n c -t i o n s i n d i a t o m i c molecules. H i s o b s e r v a t i o n t h a t t h e r e are d i s c o n t i n u i t i e s i n the f o r c e f u n c t i o n at c e r t a i n v a l u e s of n i s of g r e a t i n t e r e s t . I t has been found t h a t i f the s p e c t r o s c o p i c constants are changed a f t e r c e r t a i n v a l u e s of n then the agreement w i t h experimental data i s c l o s e r . T h i s c e r t a i n l y supports h i s view. However, he says nothing about e l e c t r o n i c l e v e l s , a n d i t i s p o s s i b l e t h a t these may change and a p p a r e n t l y change the p o t e n t i a l f u n c t i o n . At, any r a t e , t h e r e i s v a l u a b l e work t o be done i n t h i s f i e l d which should broaden a n d _ c o n s o l i d a t e our knowledge of the f o r c e s between the two n u c l e i of a d i a t o m i c molecule. 3 0 . -APPENDIX LAGUERRE POLYNOMIALS The L a g u e r r e p o l y n o m i a l s a r e d e f i n e d b y where F ( x ) s a t i s f i e s t h e f o l l o w i n g d i f f e r e n t i a l e q u a t i o n x £f + ( U l - X ) JF + n F = O 2, The d e g r e e o f t h e p o l y n o m i a l s i s , u s i n g t h e above n o t a t i o n , g i v e n b y t h e s u b s c r i p t minus t h e . . s u p e r s c r i p t . The n o t a t i o n i s c o n v e n i e n t s i n c e b i s n o t a n i n t e g e r . I n o r d e r t o a p p l y d e f i n i t i o n 1. we must d e f i n e a f r a c -t i o n a l d e r i v a t i v e , and i t i s c o n v e n i e n t t o do so i n t e r m s o f a f r a c t i o n a l i n t e g r a l . The e q u a t i o n VM J o dt 4. d e f i n e s a f r a c t i o n a l i n t e g r a l where 0 < X < 1 ( C o u r a n t [5] page 3 4 0 ) . U s i n g 3. we c a n d e f i n e f r a c t i o n a l d e r i v a t i v e s b y d » - I r f f O J o J where u.-= m - p , 0 < p < 1 , and m v i s a posi t i v e i n -t e g e r . A p p l y i n g 4 . we c a n d e r i v e t h e f o l l o w i n g two u s e f u l i d e n t i t i e s <J*V> P(a-k +l) 31. and -L (e *) - e e Using 5. and 6., 1 . can now be written as where *—nvb £ j r 0 * j 8. The following useful i d e n t i t i e s can be shown to hold (Szego* [33] Ghapt. V). In a l l cases b = 2 d - 2 n - 1 , b T = 2 d - 2 m - l . LVi , x b+4 , W*i <ix ^ ' ft+b L h+W-i J 1 2 . 1 " r(n+b+0 ' 1 3 . I b n l 13. shows that e " " x / 2 x ^ b ~ l ^ / 2 L b + b ( x ) are orthogonal poly-nomials i n the i n t e r v a l ( 0 , 0 0 ) . An i n t e g r a l of considerable importance which a r i s e s i n perturbation calculations involving the Morse p o t e n t i a l , can. be evalulated using the same methods as were used i n Chapter I I . It has been calculated for b, b' integers by Schrodinger [ 3 0 ] . 3 2 . J o The case where p < 0 i s of no p h y s i c a l i n t e r e s t . Here,-b and b 1 are not n e c e s s a r i l y i n t e g e r s . BIBLIOGRAPHY 1. D.R, B a t e s , P r o c . R o y a l S o c . A. 1^6, 217 (1944). 2. G. B i r t w i s t l e ? The New Quantum M e c h a n i c s ( C a m b r i d g e U n i v e r s i t y P r e s s , London, 1928). 3. M. B o r n and J.R. Oppenheimer, Ann. d. P h y s i k 3A_, 457 (1927). 4. A.S. C o o l i d g e , H.M. James''and E . L . V e r n o n , P h y s . Rev. 726, (1938). 5. R. C o u r a n t , D i f f e r e n t i a l a n d I n t e g r a l C a l c u l u s , V o l . I I , E . J . .McShane t r a n s l a t o r (Nordeman Pub. Co. I n c . , New Y o r k , 1936). 6. J . L . Dunham, Phys. R e v . 2A_, 438 (1929). 7. P h y s . Rev. 41, 721 (1932). 3. A.G. Gaydon and R..W.B. P e a r s e , P r o c . R o y a l S o c . L o n d o n 173. 37 (1939). 9. G.E. G i b s o n , O.K. R i c e , and N.S. B o y l i s s , P h y s . Rev. LJ^ 193 (1933). 10. D. t e r Haar, Phys. Rev. 22, 2 2 2 (1946). 11. H.S. Heaps a n d G« H e r z b e r g , p r i v a t e c o m m u n i c a t i o n . 12. G. H e r z b e r g , S p e c t r a o f D i a t o m i c M o l e c u l e s (Van N o s t r a n d , New Y o r k , 1950) 2nd e d i t i o n . 1 3 . H.M. H u l b e r t and J.O, H i r s c h f e l d e r , J o u r n a l o f Chem. P h y s . 9 , 61 (1941). 14. E.A. H y l l e r a a s , Z. P h y s i k £ 6 , 643 (1935) 15. _ - . _ P h y s i k Z. J6, 599; ( I 9 3 6 ) 16. E J n f e l d and T.E. H u l l , Rev. Mod. P h y s . 2^, 21 (1951). 17. 0. K l e i n , Z. Physik• J6t 226 (1932). 18. J.W. L i n n e t , T r a n s . F a r a d a y S o c. l ^ ' , 1 (1942). 19. M.F. Manning and N. Rosen, P h y s . Rev. 59,, 341 (1941). 20. L. M i z u s h i m a , Phys. Rev. 76, 1263 (1949). i i 21. P.M. M o r s e , P h y s . Rev., 57 (1929). 22. P.M. M o r s e , J.B. F i s k , and L . I . S c h i f f , P h y s . Rev. j>0, 748 (1936). 23. N.F. Mott and I . Sneddon, Wave M e c h a n i c s and I t s A p p l i c a -t i o n s ( C l a r e n d o n P r e s s , O x f o r d , 1948]. 24. C L . P e k e r i s , P h y s . Rev. 98 (1934). 25. M.E. P i l l o w , P r o c . P h y s i c a l Soc. A. 64, 772 (1951) 26. G. P S s c h l and E . T e l l e r , Z. P h y s i k 143 (1933). 27. A.L.G. Rees , P r o c . Phys. S o c . Lo n d o n 998 (1947). 28. R. R y d b e r g , Z. P h y s i k , 21, 376 (1932). 29. Z. P h y s i k , 80, 514 (1933). 3 0 . E . S c h r o ' d i n g e r , Wave M e c h a n i c s ( B l a c k i e and Son, Lon d o n , 1928). 31. K.E. S c h u l e r , J o u r n a l o f Chem. Phys. 18, 1221(1950). 3 2 . L . I . S c h i f f , Quantum M e c h a n i c s ( M c G r a w - H i l l , New York,1949) 33. A. So m m e r f e l d , Wave M e c h a n i c s (Metheun and Co. L t d . ,London 1930) 1st e d i t i o n . 34. G. S z e g f l , O r t h o g o n a l P o l y n o m i a l s (Am. Math. Soc. , New Y o r k 1939) C o l l o q u i u m P u b l i c a t i o n s X X I I I . 35. E.T. W h i t t a k e r and G.N. Watson, Modern A n a l y s i s ( Cambridge U n i v e r s i t y P r e s s , London, 1946) 4th e d i t i o n . 3 6 . T.Y. Wu, V i b r a t i o n a l S p e c t r a and S t r u c t u r e o f P o l y a t o m i c M o l e c u l e s ( N a t i o n a l U n i v e r s i t y o f P e k i n g . Kun-Ming C h i n a , 1939). 

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