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An ab initio calculation of the potential energy curves of some excited electronic states of OH 1971

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AN AB INITIO CALCULATION OF THE POTENTIAL ENERGY CURVES OF SOME EXCITED ELECTRONIC STATES OF OH by IAN WHITMAN EASSON B . S c , U n i v e r s i t y o f B r i t i s h Columbia, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the department o f PHYSICS We a c c e p t t h i s t h e s i s as conforming t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA September, 1971 In presenting t h i s thesis in p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of t h i s thesis for f i n a n c i a l gain shall not be allowed without my written permission. Department of Physics The University of B r i t i s h Columbia Vancouver 8, Canada Date September 28. 1971 ABSTRACT A serie s of ab i n i t i o c a l c u l a t i o n s has been performed i n the Born-Oppenheimer approximation f o r some e l e c t r o n i c states of OH. Wavefunctions and energies are ca l c u l a t e d v a r i a t i o n a l l y . The form chosen f o r the wave- function i s a f i n i t e l i n e a r superposition of configurations. Molecular o r b i t a l s are formed by Schmidt-orthogonalizing the atomic o r b i t a l s , each of which i s represented by a si n g l e Slater-type o r b i t a l . The v a r i a t i o n a l parameters are the c o e f f i c i e n t s i n the l i n e a r expansion of the wave- function, and the non-linear parameters ^ of the S l a t e r - type o r b i t a l s . Wavefunctions and p o t e n t i a l energy curves are given f o r some of the lower-lying ^ and st a t e s . 2y - One r e s u l t of note i s that the lowest - state i s bound. This disagrees with an e a r l i e r c a l c u l a t i o n (Harris and Michels, 1969), but i t i s i n accord with a recent i n t e r - p r e t a t i o n of the spectrum (Pryce, 1971). i i TABLE OF CONTENTS Abstract i i Table of Contents i i i L i s t of Tables v L i s t of Figures v i Acknowledgement . v i i i Chapter I — Introduction 1 1. Motivation 1 2. The Nature of Potential Energy- Curves 2 Chapter II — Method of C a l c u l a t i o n 6 Chapter III — C l a s s i f i c a t i o n of the E l e c t r o n i c States 11 Chapter IV — Construction of T r i a l Wavefunctions of a Given Symmetry Type 15 1. Atomic O r b i t a l s 15 2. Molecular O r b i t a l s 17 3 .• Spin Eigenfunctions 21 4. S p a t i a l Part of the Configurations 26 5. S l a t e r Determinants 28" 6. Configurations f o r Non- States 30 7» Configurations f o r ^1 States. . 30 Chapter V — The Hamiltonian Matrix 33 Chapter VI — Implementation 37 i i i L I S T OF T A B L E S 4 . 1 M o l e c u l a r O r b i t a l s 20 4 . 2 S p i n E i g e n f u n c t i o n s 25 7.1 C o n f i g u r a t i o n s f o r T T c a l c u l a t i o n 51 2 r - 8 .1 C o n f i g u r a t i o n s f o r C a l c u l a t i o n 76 v i v C hapter VII — " ^ C a l c u l a t i o n 40 1. O u t l i n e o f C a l c u l a t i o n 40 2. The Ground S t a t e 43 3. The F i r s t E x c i t e d TTstate. . . . 47 4. The Second E x c i t e d TTstate . . . 46* Chapter V I I I — C a l c u l a t i o n 66 1. O u t l i n e o f C a l c u l a t i o n 66 2. The Lowest ^ S t a t e 68 3. The Second Lowest ^ ~ S t a t e . . 72 2 5 " " 4 . The T h i r d Lowest 4— S t a t e . . . 75 Chapter IX — C o n c l u d i n g Remarks 90 B i b l i o g r a p h y 91 Appendix 1 — M a t r i x Elements o f H Between S l a t e r Determinants 92 Appendix 2 -- I n t e g r a l s Over S l a t e r - T y p e O r b i t a l s 95 Appendix 3 — Comments on a Procedure Used i n the ^ " C a l c u l a t i o n 98 LIST OF FIGURES 4 . 1 Branching Diagrams 23 7 . 1 I s P a r a m e t e r 52 7 . 2 2s P a r a m e t e r 53 7 . 3 2 p P a r a m e t e r 54 7 . 4 3 s P a r a m e t e r 55 7 . 5 3 p P a r a m e t e r 56 7 . 6 3 d P a r a m e t e r 57 7 . 7 ( l s ) H P a r a m e t e r 53 7 . 3 ( 2 s ) H P a r a m e t e r 59 7 . 9 ( 2 p ) „ P a r a m e t e r 6 0 7 . 1 0 TTPotential E n e r g y C u r v e s 61 7 . 1 1 Ground S t a t e Wavefunction 62 7 . 1 2 Ground S t a t e Wavefunction 63 7 . 1 3 F i r s t E x c i t e d F T w a v e f u n c t i o n 64 7 . 1 4 S e c o n d E x c i t e d 2 T T w a v e f u n c t i o n 6 5 8 . 1 I s P a r a m e t e r 77 8 . 2 2s P a r a m e t e r 78 8 . 3 2 p P a r a m e t e r 79 8 . 4 3 s P a r a m e t e r 8 0 8 . 5 3p P a r a m e t e r 81 8 . 6 3 d P a r a m e t e r 82 8 . 7 ( 1 S ) H P a r a m e t e r 83 8 . 8 ( 2 s ) H P a r a m e t e r 84 8 . 9 ( 2 p ) H P a r a m e t e r 85 2 < r - 8 . 1 0 P o t e n t i a l E n e r g y C u r v e s 8 6 v i v i i 2 5 - - 8.11 Lowest Wavef u n c t i o n 87 2s - 8.12 Second Lowest *£— Wavefunction 88 8.13 T h i r d Lowest Wavefunction 89 ACKNOWLEDGEMENT I would l i k e t o thank Dr. M.H.L. Pryce, who suggested the t o p i c , f o r h i s s u p e r v i s i o n and g r e a t p a t i e n c e . I would a l s o l i k e t o thank the U.B.C. Computing Centre f o r t h e use o f t h e i r f a c i l i t i e s . F i n a l l y , I would l i k e t o thank my w i f e , who was as p a t i e n t as Dr. Pryce and who a l s o d i d the t y p i n g . v i i i CHAPTER I INTRODUCTION 1. M o t i v a t i o n The OH molecule i s one o f the s i m p l e r diatomic: h y d r i d e s , and has been s t u d i e d e x p e r i m e n t a l l y f o r many y e a r s . I t s spectrum has been d e t e c t e d i n the upper atmosphere, i n comets (Herzberg, 1971), and i n the i n t e r s t e l l a r medium ( T e r z i a n and Scharlemann, 1970). I t s presence i n the l a s t medium i s most p u z z l i n g . I t appears t h a t the OH i s forming a n a t u r a l maser. V a r i o u s pumping mechanisms have been p r o - posed, none o f which can s a t i s f a c t o r i l y e x p l a i n a l l the observed f e a t u r e s . A knowledge o f the e l e c t r o n i c a l l y e x c i t e d s t a t e s of OH might be h e l p f u l i n s e l e c t i n g some p o s s i b l e pumping mechanisms. Only a few e l e c t r o n i c a l l y e x c i t e d s t a t e s o f OH have been d e t e c t e d , however. The d i f f i c u l t y i s not i n the o b t a i n i n g o f l a b o r a t o r y s p e c t r a , but i n t h e i r i n t e r p r e t a t i o n . T h i s i n t e r p r e t a t i o n would be a i d e d g r e a t l y i f i t were even o n l y r o u g h l y known which bound e x c i t e d s t a t e s e x i s t , how d e e p l y bound t h e y a r e , and what t h e i r e q u i l i b r i u m s e p a r a t i o n s a r e . There i s a need, t h e r e f o r e , f o r a c a l c u l a t i o n from f i r s t p r i n c i p l e s o f the p r o p e r t i e s o f the e l e c t r o n i c a l l y e x c i t e d s t a t e s o f OH. 1 2 2. The Nature of Pote n t i a l Energy Curves The t o t a l n o n - r e l a t i v i s t i c ; Harailtonian f o r the OH molecule i s , neglecting spin, 2 9 " t o t ^ I ^ I ' P i ' ^ i 1 = P I / 2 M I + ^ p i / 2 m + 8 e 2 / l * l - ? 2 l 2 9 + i«EL e 2 /ft-?;l - 2- *EL Z j e 2 / ! ^ - ? . ! (1.1) i , j = l 1 J 1=1 i = l 1 1 1 In ( 1 . 1 ) , upper-case l e t t e r s r e f e r to the n u c l e i , and lower-case l e t t e r s to the electrons. Thus P j , M-j., Zj are the momentum operator, mass, and charge i n units of e th —^ of the I nucleus, and p^ and m are the momentum operator and mass of the i e l e c t r o n . The f i r s t term i n (1.1) represents the k i n e t i c energy of the 0 and H n u c l e i . The second term i s the k i n e t i c energy of the nine electrons. The t h i r d i s the repulsion between the oxygen nucleus, with charge +8e, and the hydrogen nucleus, with charge +e. The fourth i s the electron-electron repulsion, and the l a s t describes the ele c t r o n - n u c l e i a t t r a c t i o n s . I t i s desired to f i n d some of the eigenvalues E t o t a n d e i g e n f u n c t i o n s SP-tot o f Htot» t 0 a r e a s o n a D l e approximation: H t o t ^ t o t = E t o t ^ t o t { 1 ' 2 ) To a good approximation, . . can be written as the product of a part e r r i n g only to the electrons and to the r e l a t i v e positions of the n u c l e i , and a part ^ nuc r e ; E ' e r r i n S only to the nuc l e i (Born and Oppenheimer, 1 9 2 7 ) . This approximation can be made plausible by the following c l a s s i c a l argument: The n u c l e i are so much more massive than the electrons that the electrons move r e l a t i v e l y q u i c k l y . As the electrons t r a v e l around the n u c l e i , the nu c l e i hardly move at a l l . Thus to a good approximation, the electrons move i n the f i e l d of two f i x e d n u c l e i a distance R ~|^T_~^2^ apart. Let V(R) be the t o t a l energy of t h i s system. Then the e f f e c t of the electrons can be simulated by the e f f e c t i v e nucleus-nucleus p o t e n t i a l V(R). The problem of f i n d i n g the so l u t i o n of (1.2) has now been reduced i n t h i s approximation to fi n d i n g the solutions of H e l ( ^ i R ) ¥ e l - V(R) ¥ e l (1.3) and H n u c ( ? I ^ i ; R ) ^ n u c = E t o t ^ n u c > ^ where 4 9 9 H P l ( ? i ^ i i R ) = ^ P i / 2 m + * e 2 / l ? i " ? i l e i x l i = i 1 1 J + 8e 2/R - X Z Te 2/K-r. j (1.5) 1=1 i = l 1 1 1 and H. n u c ( P I ' R I > R ) P i / 2 M I + V ( R ) ' (1.6) H e^ i s the H a m i l t o n i a n f o r the motion o f the e l e c t r o n s i n the f i e l d o f two n u c l e i f i x e d a d i s t a n c e R a p a r t . S i n c e H e^ i s i n v a r i a n t under t r a n s l a t i o n s , i t depends on the p o s i t i o n s o f the n u c l e i o n l y through R, c o n s i d e r e d as a f i x e d parameter, d e s p i t e the e x p l i c i t appearance o f "Rj i n (1.5). H n u c i s the H a m i l t o n i a n f o r the two n u c l e i i n a p o t e n t i a l V ( R ) . A graph o f V v e r s u s R i s c a l l e d a p o t e n t i a l energy c u r v e . There w i l l i n g e n e r a l be a l a r g e number o f them, one f o r each s o l u t i o n o f (1.3), c o r r e s p o n d i n g to d i f f e r e n t e l e c t r o n i c s t a t e s . As R - » o Q , p o t e n t i a l energy curves f l a t t e n out, because the molecule s e p a r a t e s i n t o two n o n - i n t e r a c t i n g atoms or i o n s . As R -»0 , p o t e n t i a l energy curves r i s e r a p i d l y , because the term o c l/R becomes dominant. 5 At i n t e r m e d i a t e v a l u e s o f R, p o t e n t i a l energy- curves can have any shape. One p o s s i b i l i t y i s a curve which decreases m o n o t o n i c a l l y as R-^OO. T h i s corresponds t o a r e p u l s i v e s t a t e . Another p o s s i b i l i t y i s a s i n g l e minimum i n the cur v e , which, i f deep enough, corresponds t o a bound s t a t e . T h i s i s q u i t e common i n t h e lowest e l e c t r o n i c s t a t e . Not a l l curves have such a simple shape. The curve f o r a s t a t e o f a g i v e n symmetry type does not c r o s s a curve o f a s t a t e o f the same t y p e . Avoided curve c r o s s i n g can g i v e r i s e t o curves o f q u i t e c o m p l i c a t e d shapes. * The symmetry types are g i v e n i n Chapter I I I . CHAPTER I I METHOD OF CALCULATION I t i s n e c e s s a r y t o s o l v e e q u a t i o n (1.3) f o r the e n e r g i e s V ^ ( R ) and the a s s o c i a t e d wavefunctions f o r i =? 1,...,N, where N i s the number o f p o t e n t i a l energy curves which a r e to be computed. The s t a t e s w i l l be l a b e l e d so t h a t V ^ ^ ( R ) i s the lowest energy a t a g i v e n R, V^ 2^ i s the second l o w e s t , e t c . One o f the most s t r a i g h t f o r w a r d ways o f s o l v i n g (1.3) i s the V a r i a t i o n a l Method. Suppose a t r i a l wave- f u n c t i o n i s chosen* t h a t depends on a f i n i t e number of parameters 0( i ^ ( R ) . Then the v a l u e s o f 0{ [^^ which make X li^( OC i * ^ ) as good a s o l u t i o n o f (1.3) as p o s s i b l e a r e those which minimize the energy. I f V ( i ) ( R ; CX*. 1 *) i s d e f i n e d by J ^ ( i ) d V v ( i ) ( R . ^ ( i ) , . " 1 d V . (2-1) 3 where ^...dV means i n t e g r a t i o n over a l l space, t h e n the c o n d i t i o n f o r a minimum i s 3 v ( i , ( R i OL\±]) 9 2v ( i >(R;oci i }) 3 i o . __I i _ > 0 . ( 2 . 2 ) * H e n c e f o r t h , t h e s u b s c r i p t , on both H , and x i w i l l be o m i t t e d . - e x e x e l ( * \ ( * \ V (R; 0Q ) i s always an upper bound to the true energy V ( i ) ( R ) . It i s convenient to divide the parameters Q( \ into two classes: those upon which the wavefunctions depend linear l y , and those upon which they depend non-linearly. Let ^|3j^} > i ^ k e these two classes,respectively. Then the wavefunctions can be written ¥ < i > . ffh y*1') ( 2 .3) For convenience, the a r e taken to be orthonormal: For the next few paragraphs the internuclear distance i s considered fixed at a particular value R . I f , at this R , the ^^"^ are also fixed, i t is relatively easy to calculate the / ^ j using matrix algebra, Equation (2.2) i s equivalent to the problem of finding the lowest eigenvalue (R; <9( )̂ and the corresponding eigen- value of the matrix of the electron Hamiltonian H between the basis states ^ i 1 ^ . The results s t i l l depend upon the * ^ e £f should be varied in a systematic way so that the lowest energy eigenvalue becomes a minimum. In practice, this procedure i s tedious and time- consuming, even on a high-speed computer. This i s because 8 i t must be repeated for each wavefunction ^EM^ and for many values of lij^^ u n t i l the correct ^6are found. One possible resolution presents i t s e l f immed- iat e l y : use no non-linear parameters. If enough linear parameters are used, the set ^ i " ^ becomes almost complete, and no non-linear parameters are needed. A d i f f i c u l t y with this method i s that a very large number of linear para- meters are often needed to simulate the effect of a single well-chosen non-linear parameter. The approach taken i n this work is to use non-linear parameters, but only as few as necessary. Another way to speed up the calculation, although at the expense of accuracy, i s to choose the and V^. to be independent of ( i ) . Then the lowest eigen- value of the H-matrix i s an upper bound to V ^ ( R ) , the (2) second lowest eigenvalue i s an upper bound to V* (R), etc;. In this way, i f the basis states ty. = are well chosen, good approximations to a l l the desired V^'fR) can be calculated with a single dia&onalization of the H-matrix. The jQ ^ are s t i l l , calculated i n accordance with (2.2). There is now, however, no single prescription for calculating the ^ £ ^ since there is no longer just one eigenvalue corresponding to the set ̂ i c j > D u ^ N of them. One prescription often used is to choose the so that V^^RjOC^) i s a minimum. The lowest energy state 9 i s t h e n w e l l d e s c r i b e d . I t i s hoped t h a t these )f ̂  w i l l a l s o p r o v i d e a re a s o n a b l e d e s c r i p t i o n o f the h i g h e r s t a t e s . I f o n l y the lowest s t a t e i s t o be c a l c u l a t e d , t h i s p r e s c r i p t i o n i s the b e s t . On the o t h e r hand, i f one i s i n t e r - e s t e d i n the M l o w e s t - l y i n g s t a t e s , where 1<M^N, t h e r e i s no re a s o n t o suppose t h a t t h i s method w i l l a d e q u a t e l y d e s c r i b e s t a t e s 2,3,...,M u n l e s s N i s v e r y l a r g e . A d i f f e r e n t p r e s c r i p t i o n i s t h e r e f o r e used i n t h i s work. The a r e chosen so t h a t t h e average energy o f the lowest M s t a t e s i s a minimum. I f 7= 1/W J_Z V ( i ) (2.5) ' i = l i s the average energy, the n i t i s r e q u i r e d t h a t V s a t i s f y I t i s reas o n a b l e t o expect t h a t (2.6) w i l l p r o v i d e q u i t e good V^, because each one o f the M s t a t e s o f i n t e r e s t i s t h e r e b y t r e a t e d on an eq u a l f o o t i n g , i n c o n t r a s t w i t h the u s u a l p r e s c r i p t i o n , which d i s t i n g u i s h e s the lowest energy s t a t e . To produce good p o t e n t i a l energy c u r v e s , c a l c u l a t - i o n s must be done a t many v a l u e s o f R. I f the above p r o - cedure had t o be rep e a t e d a t each R, i t would be v e r y t i m e - consuming. I t i s not the c a l c u l a t i o n o f the f3 ̂  which i s so d i f f i c u l t , but the d e t e r m i n a t i o n o f the 2̂ .̂ The reason 10 i s t h a t the o n l y way i n g e n e r a l t o f i n d the ^ ^ which minimize V i s t o v a r y ^ s y s t e m a t i c a l l y . F o r each s e t o f t r i a l v a l u e s )j^, the H-matrix must be c a l c u l a t e d and d i a g o n a l i z e d . The s o l u t i o n t o t h i s d i f f i c u l t y adopted i n t h i s work i s t o c a l c u l a t e the ^ ( R ) a - t o n l y a few s e l e c t e d v a l u e s o f R, and t o es t i m a t e ^ o r o t h e r v a l u e s o f by p o l y n o m i a l e x t r a p o l a t i o n o r i n t e r p o l a t i o n . The ^ 3 J ^ ( R ) a r e s t i l l c a l c u l a t e d a t a l l R because t h e c a l c u l a t i o n does not ta k e v e r y much t i m e . T h i s approach i s r e a s o n a b l y s u c c e s s f u l because i t i s found t h a t i n p r a c t i c e the a r e r a t n e r smooth f u n c t i o n s o f R. R CHAPTER I I I CLASSIFICATION OF THE ELECTRONIC STATES The e l e c t r o n i c H a m i l t o n i a n H = H g^, d e f i n e d by- e q u a t i o n ( 1 . 5 ) , c o n t a i n s no s p i n terms. I t commutes w i t h a number o f o p e r a t o r s . These i n c l u d e S 2 , S Z , L Z , J Z , and . 2 S i s the square o f the t o t a l s p i n o p e r a t o r , S 2 i s the component o f s p i n a n g u l a r momentum a l o n g the i n t e r n u c l e a r a x i s (z a x i s ) , L_ i s the z-component o f the e l e c t r o n i c z o r b i t a l a n g u l a r momentum, J = L * S w i s the t o t a l e l e c t r o n i c z z z a n g u l a r momentum a l o n g the z - a x i s , and ^ i s the o p e r a t o r o f r e f l e c t i o n through a plane p a s s i n g through the z?-axis. A number o f t h e s e o p e r a t o r s commute w i t h each o t h e r ; o t h e r s do n o t . For example, ^ and L Z do not commute. To see t h i s , c o n s i d e r the case o f one e l e c t r o n , r a t h e r than n i n e . I n t h i s case, L„, i s g i v e n i n c o - o r d i n a t e space by z L = i 4- * (3-1) z df> where <p i s the a z i m u t h a l angle i n c y l i n d r i c a l p o l a r co- o r d i n a t e s , w i t h the i n t e r n u c l e a r a x i s as the a x i s o f symmetry. The r e f l e c t i o n ^ i n c o - o r d i n a t e space causes i\ : <f> -> - <f> ( 3 . 2 ) and t h e r e f o r e 11 12 IR: ^ L , ^ - 1 - ^ , . (3.3) so and L do not commute. I n f a c t , IR, takes an e i g e n -^ z s t a t e o f L w i t h e i g e n v a l u e m i n t o an e i g e n s t a t e o f L z z w i t h e i g e n v a l u e -m, as can be seen from (3.3). S and J z z a r e , l i k e L Z , z-components o f angular-momentum o p e r a t o r s , and s i m i l a r c o n c l u s i o n s f o l l o w f o r them. ^ can then be w r i t t e n as the product o f two o p e r a t o r s , ^ L and # s , which commute w i t h one another: ^ ^ i s the o p e r a t o r which causes # L : L Z ^ ^ ? L L Z - - L Z , (3.5) and 0r?g causes 0?S : S z ^ #S Sz ^ • " Sz • <3-6> 2? L and ( J ? s each commute wit h H . By (3.5), commutes w i t h 1 L \, and by (3.6), commutes w i t h |s | . The o p e r a t o r s H , S 2 , S Z , | L z l , | J z l and #? L then a l l commute w i t h one ano t h e r , and i t i s p o s s i b l e t o c l a s s i f y t he s t a t i o n a r y s t a t e s ^J^^) 0 f H i n t o d i f f e r e n t c l a s s e s l a b e l e d by the e i g e n v a l u e s o f S 2 , S Z , J L Z | , |J_\ and The m a t r i x elements o f H between s t a t e s b e l o n g i n g t o two d i f f e r e n t c l a s s e s v a n i s h . As a r e s u l t , i t i s o n l y n e c e s s a r y t o c o n s i d e r s t a t e s o f the same c l a s s when do i n g a c a l c u l a t i o n . 13 The eigenvalues of j L J can take on the values , where (\ = 0,1,2, etc. Eigenstates of ) L z| with these eigenvalues are called etc. states, respectively. The m u l t i p l i c i t y of a state Ĵ̂1* i s defined to be 2s+l, where ^ > - Vs(s+1) ^ U ) • (3.7) The mu l t i p l i c i t y i s indicated by a superscript, e.g. 2"|~[ i s a | \ state with s = |. Since ~ 1» can have eigenvalues -1. The appropriate eigenvalue is indicated by a superscript, The eigenvalue of |J I. t i l l , i s indicated by writing -TL. as a subscript, e.g. 2 / \ ^ Two states of different types can be degenerate. For example, the energy i s independent of S and J . Also, A z z states with A ^ 0 have energies which are independent of whether they have +1 or -1 as the eigenvalue of ^ t ^ . However, i f from a given ^L.state which is not an eigenstate of 0^L, both a 2L+ and a state can be formed, then these two w i l l not be degenerate. The explanation for this w i l l be given i n Chapter IV, Section 7, where the method of construction of ^L_* states is given. Because of th i s , i t does not matter i f the states with A ^ 0 are eigenvectors of . Such states can then be c l a s s i f i e d into groups labeled by the eigenvalues of 14 2 S ,S .L , and J _ . I n the s e groups, s t a t e s which d i f f e r o n l y Z Z Z i n t he s i g n o f S Z , L Z , o r J z are degenerate. S i n c e one i s i n t e r e s t e d i n c a l c u l a t i n g o n l y non- degenerate s t a t e s , i t w i l l be understood i n what f o l l o w s t h a t the maximum p o s s i b l e v a l u e s o f S_ and L_ a r e t o be z z t a k e n . For example, by a s t a t e w i l l be meant a s t a t e w i t h L = , S„ = z z I n the next c h a p t e r , the c o n s t r u c t i o n o f t r i a l w a vefunctions p o s s e s s i n g symmetries such as those mentioned 1 above w i l l be d e s c r i b e d . CHAPTER IV CONSTRUCTION OF TRIAL WAVEFUNCTIONS OF A GIVEN SYMMETRY TYPE 1. Atomic O r b i t a l s I t i s w e l l known t h a t , t o a l a r g e degree, the motions o f e l e c t r o n s i n atoms o r mol e c u l e s are u n c o r r e l - a t e d , so t h a t the e l e c t r o n w a v e f u n c t i o n a p p r o x i m a t e l y f a c t o r s i n t o a product o f o n e - e l e c t r o n w a v e f u n c t i o n s , o r o r b i t a l s . The OH molecule has c y l i n d r i c a l symmetry about the i n t e r n u c l e a r a x i s . I t i s re a s o n a b l e t o choose o r b i t a l s which a r e adapted t o t h i s symmetry. However, ano t h e r approach i s p o s s i b l e . Because the charge on the oxygen nucleus i s e i g h t times t h a t on the hydrogen n u c l e u s , t h e r e i s , a t s m a l l R, a p o i n t o f r o t a t i o n a l s ymmetry—the 0 n u c l e u s . At l a r g e R, t h e r e i s l i t t l e i n t e r - a c t i o n between the two n u c l e i , so t h a t t h e r e a r e two p o i n t s o f s p h e r i c a l symmetry—the n u c l e i . T h i s suggests an a l t e r n a t i v e approach, the one t a k e n here, namely, t h a t the o r b i t a l s s hould be c o n s t r u c t e d from t h i n g s which are c e n t r e d on the n u c l e i . These atomic o r b i t a l s a r e o f the form (4.1) 15 16 (r> Qi<p) form a s p h e r i c a l p o l a r c o - o r d i n a t e system, which may have i t s o r i g i n a t e i t h e r the G o r H n u c l e u s . Y^ m i s a s p h e r i c a l f u n c t i o n . T h i s i s convenient because the Y^ m a r e e i g e n f u n c t i o n s of L_: V l m = m I lm • «*-2> E q u a t i o n (4.2) i s t r u e i n d e p e n d e n t l y o f which nucleus >̂ i s c e n t r e d upon. The r a d i a l p a r t R ( r ) of the atomic o r b i t a l has been chosen i n t h i s work t o be o f the S l a t e r form R ( r ) = N(n,J ) r n - 1 e " ^ r . (4.3) N i s a n o r m a l i z a t i o n c o n s t a n t , n p l a y s a r o l e analogous t o t h a t of the principa - 1 quantum number i n the hydrogen atom w a v e f u n c t i o n . ^ i s a parameter which determines how q u i c k l y R ( r ) drops o f f w i t h i n c r e a s i n g r . The ^ of the d i f f e r e n t atomic o r b i t a l s are the n o n - l i n e a r parameters mentioned i n Chapter I I . n c o u l d a l s o have been chosen as a n o n - l i n e a r parameter, but t h i s was not done f o r two r e a s o n s . F i r s t , one n o n - l i n e a r parameter per atomic.1 o r b i t a l i s p r o b a b l y enough. Second, i f n i s f i x e d a t an i n t e g e r v a l u e d the i n t e g r a l s which have t o be e v a l u a t e d a r e e a s i e r t o c a l c u l a t e . I n o r d e r t o p r o v i d e a r e a s o n a b l e d e s c r i p t i o n of the l o w - l y i n g e l e c t r o n i c : s t a t e s o f OH, i t was d e c i d e d t o s e l e c t atomic o r b i t a l s o f two t y p e s . The f i r s t type c o n s i s t e d o f t h o s e atomic o r b i t a l s which a r e o c c u p i e d i n the ground 17 states of oxygen and hydrogen. These are the Is, 2 s , and 2p orbitals on oxygen and Is on hydrogen. The second type consisted of those atomic orbitals not included i n the f i r s t type which are occupied in the lower-lying excited states of oxygen and hydrogen. These are the 3 s , 3 p , and 3 d atomic orbitals on oxygen and the 2 s and 2p atomic orbitals on hydrogen. The number of (m=0) orbitals i s then nine: Is, 2s,2pc/ , 3 s , 3 p t f , 3 d c * on oxygen, and (Is ) H , ( 2 s ) H , (2pd) H on hydrogen. There are four TT (m=+l) orbitals: 2pK ,3prr, 3 d i r on oxygen, and ( 2 p i r ) H on hydrogen. There are four corresponding '^'(m=-l) orbitals. There i s one S(m=2) o r b i t a l : 3dS on oxygen, and a 3 d £ on oxygen. This i s a t o t a l of nineteen atomic orbitals. 2. Molecular Orbitals The atomic orbitals are not a l l orthogonal. For example, the overlap integral between the 2 p atomic orbital and the (1S)JJ does not vanish, unless R=0 or R=e°. The expressions for the matrix elements of H are simpler i f the electron orbitals are orthonormal. Orthonormal orbitals can be formed by taking appropriate linear combinations of atomic orbitals. Since these linear combinations can involve atomic orbitals centred on both * See Chapter V, Section 1 for these expressions. Also see Appendix 1. 18 n u c l e i , t he l i n e a r combinations may be a p p r o p r i a t e l y c a l l e d m o l e c u l a r o r b i t a l s . There a r e i n f i n i t e l y many ways o f formin g o r t h o - normal m o l e c u l a r o r b i t a l s from a g i v e n s e t o f atomic o r b i t a l s , i f t h e r e i s one way. T h i s can be seen as f o l l o w s : One s e t o f orthonormal m o l e c u l a r o r b i t a l s forms an orthonormal b a s i s o f the v e c t o r space o f e l e c t r o n o r b i t a l s spanned by the atomic o r b i t a l s . Such a b a s i s can be r o t a t e d a t w i l l t o form a n o t h e r orthonormal b a s i s , t h a t i s , a n o t h e r s e t o f orthonormal m o l e c u l a r o r b i t a l s . S i n c e t h e r e i s a h i g h degree o f a r b i t r a r i n e s s i n the m o l e c u l a r o r b i t a l s , t h e r e i s freedom t o choose them so t h a t t h e y a r e good approximations t o the o r b i t a l s i n OH. One o f the convenient p r o p e r t i e s o f OH i s t h a t the charge on the H nucleus i s e i g h t times s m a l l e r than t h a t on the 0 n u c l e u s . T h i s i m p l i e s t h a t , u n l e s s OH i s i n a h i g h l y e x c i t e d s t a t e , the i n n e r e l e c t r o n o r b i t a l s a r e oxygen atomic o r b i t a l s . At l a r g e R, anot h e r group o f m o l e c u l a r o r b i t a l s l o o k s l i k e atomic o r b i t a l s on hydrogen. The m o l e c u l a r o r b i t a l s formed s h o u l d then be chosen so t h a t t h e y l o o k as much as p o s s i b l e l i k e atomic o r b i t a l s . A way o f e n s u r i n g t h i s , which i s a l s o computat- i o n a l l y c o n v e n i e n t , i s Schmidt o r t h o g o n a l i z a t i o n . T h i s procedure w i l l be demonstrated u s i n g the ̂  m o l e c u l a r o r b i t a l s . 19 The innermost o r b i t a l l o o k s much l i k e a Is atomic o r b i t a l on oxygen, so the f i r s t d m o l e c u l a r o r b i t a l , i d , i s chosen t o be t h i s atomic o r b i t a l : i d = Is (4.4) The next innermost o r b i t a l l o o k s l i k e a 2s o r b i t a l c e n t r e d on oxygen. The second m o l e c u l a r o r b i t a l , 2 d , which i s chosen t o r e p r e s e n t t h i s , c o n s i s t s o f a 2s oxygen o r b i t a l minus enough o f a Is oxygen o r b i t a l t o make 2d and i d o r t h o g o n a l : 2 s - l d / i d *2sdV 2 d - J { 4 - 5 ) ( 2s-l(Sj id *2sdV^ * (^2s-ldf i d *2sdVJ dV The f a c t o r l/i'* * ensures t h a t 2 d i s p r o p e r l y n o r m a l i z e d . The next d m o l e c u l a r o r b i t a l , 3CS* , c o n s i s t s s o l e l y o f a 2 p d atomic: o r b i t a l on oxygen, because the p o r b i t a l i s a l r e a d y o r t h o g o n a l t o the i d and 2d m o l e c u l a r o r b i t a l s , which are s o r b i t a l s . The f i r s t d m o l e c u l a r o r b i t a l t o deserve the a d j e c t i v e 'molecular' because i t i s a l i n e a r combination o f an atomic o r b i t a l on hydrogen and atomic o r b i t a l s on oxygen i s t he 7 d m o l e c u l a r o r b i t a l . A l i s t o f the m o l e c u l a r o r b i t a l s and t h e i r con- s t i t u e n t s i s c o n t a i n e d i n Table 4.1 on page 20. 20 Table 4 .1 Molecular Orbitals Molecular Orbitals Main Other Constituent Constituents l t f Is 2 0' 2s Is 3C 2ptf 4 t f 3s Is,2s 5C 3 PCS' 2pd 6tf 3d<5 70' ( l s ) H ls,2s,2ptf,3s,3pd,3dc5. 8tf (2s) H ls,2s,2iW,3s,3pcr,3dd, (Is) 9cT (2pd) H ls,2s,2pd,3s,3pcS,3dO. Molecular Orbitals Main Other Constituent Constituents ITT 2pTf 2Tf 3ptT 2p-rr 3TT 3dTT 4-rr (2p7r) H 2pTr,3PTr,3dTT. Molecular Orbital l i » 3dS . 21 3. S p i n E i g e n f u n c t i o n s By the P a u l i P r i n c i p l e , a m o l e c u l a r o r b i t a l can be o c c u p i e d by a t most two e l e c t r o n s , and, i f the o r b i t a l i s doubly o c c u p i e d , the s p i n s o f the two e l e c t r o n s must be a n t i p a r a l l e l and coupled i n such a way as to form a system o f t o t a l s p i n z e r o . I n a d i s c u s s i o n o f the s p i n p r o p e r t i e s o f OH, then, any d o u b l y - o c c u p i e d m o l e c u l a r o r b i t a l can be n e g l e c t e d . S i n c e OH has nine e l e c t r o n s , t h e r e can be e i t h e r 1> 3, 5, 7/, or 9 s i n g l y - o c c u p i e d m o l e c u l a r o r b i t a l s i n the wa v e f u n c t i o n . The s p i n s o f the e l e c t r o n s i n the s e o r b i t a l s must be such t h a t the t o t a l w a v e f u n c t i o n i s an e i g e n s t a t e 2 o f S and S 2 . The t a s k a t hand i s then t o c o n s t r u c t s p i n 2 e i g e n f u n c t i o n s o f S and S f o r a g i v e n odd number o f e l e c t r o n s . A t t e n t i o n w i l l be r e s t r i c t e d t o s p i n d o u b l e t s (s = i ) . The reasons f o r t h i s c h o i c e are e x p l a i n e d i n Chapters V I I and V I I I . With one e l e c t r o n , t h e r e i s o n l y one way t o form a s t a t e w i t h s = ^, s z = t namely, s p i n 'up', denoted ct o r T . With t h r e e e l e c t r o n s , t h e r e a r e two ways t o form a s p i n d o u b l e t . The reason i s seen most e a s i l y u s i n g * Remember t h a t i t i s assumed t h a t the wav e f u n c t i o n f a c t o r s i n t o a product o f m o l e c u l a r o r b i t a l s . 22 the Branching Diagrams in Figure 4 . 1 . This type of diagram shows, graphically how the spins of a group of electrons can be coupled, one at a time. Two of the three electrons can be coupled to form either a singlet or a t r i p l e t . The singlet i s shown in I I I - l , the t r i p l e t i n III-2. To develop explicit expressions for these states, Clebsch-Gordon coefficients are used. Let |s,s z;N^ denote o an N-electron state which i s an eigenstate of S and S z with eigenvalues s(s+l) and "Ks , respectively. Then z <* = - l l , i ; l > and p= U,-4;l> . The singlet i s |o,0;2> = f ? U,i;i>U,-4;i>-fT|l,-l ; i>U,l ; i> = )/T(<*(i-(*<*). (4.6) The t r i p l e t with s = 1 i s z |i,i;2> = |4,4;i>| M;i> = ( X C X . (4.7) The t r i p l e t with s = 0 i s z |i,0;2> - f ? |i,i;i>|4,-4;i> \ i , - i;i> |i,i;i> = fT(o(|S+jJe(). (4.8) 23 One electron: •P I 5 i • H ' i"ZL"-3. •H I i i i 'i i 0 1 2 3 4 5 number of electrons 1-1 Three electrons: I l l - l Five electrons: V-l V-2 V-3 V-5 Figure 4.1. Branching Diagrams for Spin Doublets 24 To form a d o u b l e t w i t h t h r e e e l e c t r o n s , the t h i r d e l e c t r o n can be coupled w i t h e i t h e r the s i n g l e t o r the t r i p l e t . I f the e l e c t r o n i s coupled w i t h the s i n g l e t , then I 4,i;3> - I 0 , 0 ; 2 > | i,i ; i > » flWpcx - ). ( 4 . 9 ) I f t h e e l e c t r o n i s coupled w i t h the t r i p l e t , t h e n | i , i;3> - 1 ? | l , l ; 2 > | * f - 4 ; l > - t j|l,0;2>|i f i;l> = ( « c * )f3 - " j ? " ( * / * + ( * * ) o( = (2 o<cxp - ot|3c< - (3o(o( ) / fV . ( 4 . 1 0 ) The s p i n s t a t e s d e f i n e d i n ( 4 . 9 ) and ( 4 . 1 0 ) form 2 an orthonormal b a s i s o f the space of e i g e n f u n c t i o n s of S and S z c o r r e s p o n d i n g t o e i g e n v a l u e s -f "lr\ 2 and - H ^ , f o r t h r e e e l e c t r o n s . S i m i l i a r b a s i s s t a t e s can be formed f o r f i v e e l e c t r o n s . The b a s i s s t a t e s f o r one, t h r e e , and f i v e e l e c t r o n s c o u p l e d t o form d o u b l e t s a r e l i s t e d i n Table 4 . 2 . 25 Table 4.2 Spin Eigenfunctions One electron 1-1 : * Three electrons I I I - l : 'f7(<*(3°< - (3W) III-2 : (2ot«p_o((3c< - p*<*)/fJF Five electrons - oC^pdoC- pd^dtk- p|3curt©(} V-2 : 1 **P*(*-y<<p<>W|3-j|*<*e<c<p V-3 : <*p<**p - prtofe/p ) V-4 : f j t #<* + /*f3ctaW ) 26 4 . S p a t i a l Parts of the Configurations Since the electrons are to a large degree un- cor r e l a t e d , the s p a t i a l part, j., of each conf i g u r a t i o n * was chosen to be a product of nine of the molecular o r b i t a l s OCi constructed i n Section 2 of t h i s chapter. An example of a . i s 3 (Id )2(2d)2(lTr )(3d ) 2 ( 1 T T ) 2 (4.11) 2 In (4.11), the superscript indicates that the correspond- ing molecular o r b i t a l i s doubly occupied. The symmetry properties of the tyj place c e r t a i n r e s t r i c t i o n s on the possible forms Vj can have. Because the are antisymmetric, the^Pauli P r i n c i p l e r e s t r i c t s the molecular o r b i t a l s OCi±n ^ to be no more than doubly- occupied. The example i n (4.11) obeys t h i s r e s t r i c t i o n . The 9^ are eigenfunctions of S 2. Since there i s only one singly-occupied molecular o r b i t a l i n (4.11), (4.11) could be used only i n a c a l c u l a t i o n f o r spin doublets, The p r o j e c t i o n m ^ Q t $ \ of the t o t a l angular momentum of ty/ upon the inte r n u c l e a r axis i s the sum of «J the p r o j e c t i o n m of the angular momentum of each molecular o r b i t a l . Thus m t o t f o r (4.11) i s 1, so (4.11) can give r i s e only to a TT s t a t e . (4.11) i s , i n f a c t , the s p a t i a l * The basis states j. w i l l henceforth be c a l l e d con- f i g u r a t i o n s • J 27 p a r t o f t h a t c o n f i g u r a t i o n which i s the main c o n t r i b u t o r t o the J~[ ground s t a t e o f OH a t i t s e q u i l i b r i u m s e p a r a t i o n . Even w i t h t h e s e r e s t r i c t i o n s t h e r e a re s t i l l a l a r g e number o f which can be formed from the n i n e - t e e n m o l e c u l a r o r b i t a l s . From t h i s l a r g e number, a s m a l l s u bset must be chosen i n o r d e r t o make the c a l c u l a t i o n t r a c t a b l e . With a s i n g l e e x c e p t i o n , a l l c o n f i g u r a t i o n s chosen i n t h i s work have a ' f r o z e n c o r e ' ; t h a t i s , they a l l c o n t a i n 2 2 {id ) (2d) . The r e a s o n i n g behind t h i s r e s t r i c t i o n i\s t h a t these innermost f o u r e l e c t r o n s a r e well, s h i e l d e d by the o t h e r f i v e from the i n f l u e n c e o f the hydrogen n u c l e u s . T h i s a p p r o x i m a t i o n w i l l break down o n l y i n h i g h l y e x c i t e d s t a t e s , which a r e not c o n s i d e r e d h e r e . The o r b i t a l s i d ,2(3 t3d,7& ,1TT, , and llFare v e r y important i n the c o n s t r u c t i o n o f c o n f i g u r a t i o n s because t h e y are low i n energy, and are t h e most important o r b i t a l s i n the ground s t a t e o f OH (see (4.11), f o r example). I t i s r e a s o n a b l e , t h e n , t o i n c l u d e a l l c o n f i g u r a t i o n s formed from these o r b i t a l s i n t h e c a l c u l a t i o n . The o t h e r m o l e c u l a r o r b i t a l s a re h i g h e r i n energy. Those c o n f i g u r a t i o n s w i t h one s i n g l y - o c c u p i e d o r b i t a l o f t h i s type can be expected t o be the main c o n s t i t u e n t s o f the l o w e r - l y i n g e x c i t e d s t a t e s , and a l l such c o n f i g u r a t i o n s s h o u l d be i n c l u d e d i n the chosen s u b s e t . A _ * One c o n f i g u r a t i o n chosen i n t h i s work has a s i n g l y - o c c u p i e d 2 c * o r b i t a l . See S e c t i o n 1 of Chapter V I I I f o r a d i s c u s s i o n o f t h i s c o n f i g u r a t i o n . 28/; The configurations with two of the higher-energy o r b i t a l s w i l l probably be quite high i n energy, and thus w i l l not be major contributors to the states of i n t e r e s t , i n general. In the next three sections, the combination of with the spin eigenfunctions \JV constructed i n Section 3 of t h i s chapter to form configurations of a desired symmetry type w i l l be described. 5. S l a t e r Determinants The spin eigenfunctions 0^ constructed i n Section 3 of t h i s chapter are l i n e a r combinations of products s i n g l e electron spins. 9± = ^E- c.± Q. , ( 4 . 1 2 ) where the c.. are r e a l numbers, and (^). i s a product of an J J _.t t odd number of C\ s and p s. The problem of combining the 7 .̂ with the Q. w i l l be considered i n the next s e c t i o n . In t h i s section, the simpler problem of combining with one of (3)k w i l l be considered. A S l a t e r Determinant i s an antisymmetrized product of s p a t i a l and spin wavefunctions of the form ( 4 . 1 3 ) 29 ^ I n ( 4 . 1 3 ) ; each d o u b l y - o c c u p i e d o r b i t a l i n Sr . * 3 i s o c c u p i e d by one e l e c t r o n w i t h s p i n up, and one w i t h s p i n down. The s i n g l y - o c c u p i e d o r b i t a l s o f ^ have t h e i r s p i n s a s s i g n e d a c c o r d i n g t o (Ŝ > which has as many s p i n s as t h e r e a r e s i n g l y - o c c u p i e d o r b i t a l s i n ty.. Thus, f o r example, J the un-antisymmetrized product o f (4 .11) w i t h OC i s d d f )(icsi)(2<sT)(2cU)d?t )(3crt )(3dJ)(nrf M i n i ) . ( 4 . 1 4 ) The above i s a w a v e f u n c t i o n i n which e l e c t r o n number 1 i s i n a i d o r b i t a l w i t h s p i n up, number 2 i s i n a lc5 o r b i t a l w i t h s p i n down, e t c . I t t r e a t s the e l e c t r o n s , t h e r e f o r e , as d i s t i n g u i s h a b l e p a r t i c l e s . Because e l e c t r o n s are i n d i s t i n g u i s h a b l e f e r m i o n s , t h e i r w a v e f u n c t i o n should be t o t a l l y a n t i s y m m e t r i c . The a n t i s y m m e t r i z a t i o n o p e r a t o r used i n ( 4 . 1 3 ) accomplishes t h i s . The a p p l i c a t i o n of M- t o the w a v e f u n c t i o n (4.14) produces a w a v e f u n c t i o n which i s a sum o f 91 terms.. Each term i s l i k e ( 4.14), except t h a t the e l e c t r o n l a b e l s a r e permuted i n such a way t h a t the t o t a l w a v e f u n c t i o n i s a n t i - symmetric under i n t e r c h a n g e o f any two e l e c t r o n l a b e l s . The S l a t e r Determinants a r e e i g e n f u n c t i o n s o f j 4 * , as w e l l as e i g e n f u n c t i o n s of L and S . They a r e not, Z Z * i n g e n e r a l , e i g e n f u n c t i o n s o f S 2 , (R, , o r H. 30 6. Configurations for Noji-JsJL States The configuration may be written as V̂ J = ^ ( % • ( 4 , 1 5 ) where i s a spin eigenfunction compatible with j.. Using ( 4 . 1 2 ) and ( 4 . 1 3 ) , can be written as a linear combination of Slater Determinants: ¥3 = i r c j k s k • ^- 1 6> i s an eigenfunction of L Z , «̂ f", S 2, and S z. It i s not, i n general, an eigenfunction of or H. For non- WW states, i t i s not necessary in this approximation that they be eigenfunctions of as noted i n Chapter I I I 7. Configurations for States Suppose that i s a configuration which is an eigenstate of L , but not an eigenstate of ^PT. Define Z Li the reflected state ^ b v From ty. and i t i s possible to construct two states, and ^ T , which are eigenstates of % L with eigenvalues +1 and - 1 , respectively: 31 + N" a r e n o r m a l i z a t i o n c o n s t a n t s . where cTis the o v e r l a p i n t e g r a l (ĵ A and ^ R are degenerate. { H ^ R d V = | ^ * H ^.dV = E . (4.21) However. ^ do not n e c e s s a r i l y have the same energy: ~—i[f ¥TH VjdV - ̂ J- a-22) I f y . i s an e i g e n s t a t e o f L_ w i t h non-zero e i g e n v a l u e , then the r i g h t - h a n d s i d e o f (4.22) v a n i s h e s , and J and ty^" are degenerate, as noted i n Chapter I I I . However, i f ty. i s a^ELstate, then the r i g h t - h a n d s i d e o f (4«22) does not n e c e s s a r i l y v a n i s h , and the ^EL+ and s t a t e are not, i n g e n e r a l , degenerate. The W~ a r e , l i k e Sr., l i n e a r combinations o f S l a t e r Determinants. C a u t i o n must be used i n the c o n s t r u c t i o n o f e i g e n - s t a t e s o f ^ T , because a l i n e a r l y dependent s e t of b a s i s 2 5"" v e c t o r s can r e s u l t . F o r example, the two <£- s t a t e s c o n s t r u c t e d by combining ( l i t )(17T )(3tf ) w i t h c o u p l i n g s I I I - l and I I I - 2 form a l i n e a r l y dependent s e t o f s t a t e s . More p r e c i s e l y , the AC- s t a t e formed from (ITT )(17T. ) {3d ) and c o u p l i n g I I I - l v a n i s h e s i d e n t i c a l l y . CHAPTER V THE HAMILTONIAN MATRIX Once the configurations i n the case • • J J of <<d. states) have been formed, the matrix elements of the Hamiltonian can be calculated. By (4.16), H^j can be written as a linear combination of matrix elements of H between Slater Determinants. - .5 4-CIK ° J I J s* H  Sidv (5 ' z) The general form of H is a sum of zero-electron, one-electron, and two-electron operators. H ^ + ^ h j 1 ' + | Z h ' i ' j ) , (5.3) 0 i 1 i ^ j * where h Q = 8/R , (5,-4) 4 i } = "4 V i - */*± " l / | r . - f | , (5.5) and 33 34 I n (5.4) . t o ( 5 . 6 ) , atomic: u n i t s (e= t\2/m=l) have been u s e d . The u n i t o f l e n g t h i s the Bohr r a d i u s and the u n i t o f energy i s the H a r t r e e (1 H a r t r e e 27.2 e . v . ) . A s p h e r i c a l p o l a r c o - o r d i n a t e system ( r , S, <p ) c e n t r e d on the oxygen nucleus has been used. The hydrogen nucleus i s a t (R , 0 , 0 ) . The e x p r e s s i o n s f o r the m a t r i x elements o f sums of z e r o , one, and t w o - e l e c t r o n o p e r a t o r s between S l a t e r Determinants formed from orthonormal m o l e c u l a r o r b i t a l s a r e w e l l known ( S l a t e r , I 9 60 ) . These e x p r e s s i o n s a r e r e - produced i n Appendix 1. These e x p r e s s i o n s i n v o l v e o n e - e l e c t r o n , t h r e e - d i m e n s i o n a l i n t e g r a l s o f the form <Xi|h1|Xj> - j A * U ) h < 1 } J . ( l ) d v ( l ) , (5.7) and t w o - e l e c t r o n , s i x - d i m e n s i o n a l i n t e g r a l s o f the form JJXi(l)Xj(2)41,2)Xk(1) X 1 ( 2)dv(l)dv ( 2 ) , (5.8) where the volume element i s d v ( i ) = r 2 d r s±n6 d$ d<j) . (5.9) The m o l e c u l a r o r b i t a l s a p p e a r i n g i n these i n t e g r a l s a r e , as e x p l a i n e d i n Chapter IV, S e c t i o n 2, l i n e a r combinations o f atomic o r b i t a l s . ^ 1 " ^ " ^ i j (5.10) The i n (5.10) are r e a l c o e f f i c i e n t s . Thus OfJhJX]) - ££^aJb<Ahlft> , ( 5.n, and ' ^ . ^ l ^ l #c»0d> • '5.12) The i n t e g r a l s over Slater-type atomic o r b i t a l s i n (5.11) and (5.12) can be divided into two classes: one- centre and two-centre. One-centre i n t e g r a l s are those i n which a l l the atomic o r b i t a l s involved are centred on the same nucleus. There are numerically well-behaved, a n a l y t i c formulas f o r such i n t e g r a l s (Joy and Parr, 1958). These formulas are reproduced i n Appendix 2. The two-centre i n t e g r a l s are more d i f f i c u l t to c a l c u l a t e than the one-centre i n t e g r a l s . Although there are a n a l y t i c formulas f o r the former, they are often complicated, and are sometimes numerically i l l - c o n d i t i o n e d (Harris, 1969). The method of c a l c u l a t i o n of the two-centre i n t e g r a l s i n t h i s work i s to expand the o r b i t a l s on one centre as an i n f i n i t e sum of s p h e r i c a l harmonics on the other centre, and then to integrate numerically. Since t h i s method i s w e l l - 36 d e s c r i b e d elsewhere ( S w i t e n d i c k and Corbato, 1963), no f u r t h e r d e s c r i p t i o n w i l l be g i v e n h e r e . CHAPTER VI IMPLEMENTATION Three computer programs were w r i t t e n t o perform the work o u t l i n e d i n Chapters IV and V. The f i r s t program i s w r i t t e n i n FORTRAN and i n the assembly language f o r the IBM 360. The i n p u t t o t h i s program i n c l u d e s i n f o r m a t i o n about the number o f e l e c t r o n s , the m u l t i p l i c i t y , the component o f t o t a l o r b i t a l a n g u l a r momentum a l o n g the i n t e r n u c l e a r a x i s , the n, 1, and m v a l u e s and c e n t r e s f o r the atomic o r b i t a l s , the s p a t i a l p a r t s o f the c o n f i g u r a t i o n s , and, i f the s t a t e i s a s t a t e , whether i t i s IE[+ o r HEL~. The program does symbolic m a n i p u l a t i o n . I t d e t e r - mines which non-zero i n t e g r a l s over atomic o r b i t a l s have t o be c a l c u l a t e d , but does not a c t u a l l y c a l c u l a t e them i t - s e l f . R ather, i t a s s i g n s each o f t h e s e i n t e g r a l s a unique i d e n t i f y i n g l a b e l . The l a b e l s f o r the o v e r l a p i n t e g r a l s a r e used to form the symbolic e x p r e s s i o n s f o r the c o e f f i - c i e n t s A.. i n (5.10), each o f which i s then g i v e n i t s own l a b e l . The i n t e g r a l s over m o l e c u l a r o r b i t a l s a r e then expanded, as i n (5.11) and (5.12), as symbolic e x p r e s s i o n s i n v o l v i n g the Z^.. l a b e l s and the l a b e l s f o r the i n t e g r a l s o v e r atomic o r b i t a l s . Equal terms i n the expansion a r e a u t o m a t i c a l l y c o l l e c t e d t o g e t h e r . Each i n t e g r a l over m o l e c u l a r o r b i t a l s i s t h e n g i v e n i t s own l a b e l . 37 38 These l a b e l s a re used t o form the e x p r e s s i o n s g i v e n i n Appendix 1 f o r the m a t r i x elements o f H between each p a i r o f S l a t e r Determinants i n the c a l c u l a t i o n . E q u a l terms i n the e x p r e s s i o n s are c o l l e c t e d t o g e t h e r . Each m a t r i x element of H between S l a t e r Determinants i s g i v e n i t s own l a b e l . F i n a l l y , these l a b e l s a re used t o form the e x p r e s s i o n s (5.2) f o r the m a t r i x elements o f H between each p a i r o f c o n f i g u r a t i o n s i n the c a l c u l a t i o n . The o n l y n u m e r i c a l c a l c u l a t i o n i n the f i r s t program i s the m u l t i - p l i c a t i o n i n (5.2) o f c ^ k by c ^ , which a r e numbers. The output from t h i s program i s o f two k i n d s . The f i r s t k i n d i s a p r i n t o u t o f a l l the symbolic e x p r e s s - i o n s formed. T h i s p r i n t o u t i s v e r y u s e f u l as a debugging t o o l . The second k i n d o f output i s a s e r i e s o f numbers which are w r i t t e n on a d i s c f i l e . T h i s s e r i e s o f numbers c o n t a i n s e s s e n t i a l l y the same i n f o r m a t i o n as the p r i n t o u t . There i s a g r e a t advantage i n having a s e p a r a t e program t o perform the a l g e b r a . The advantage i s t h a t the a l g e b r a i s done once and f o r a l l , and the r e s u l t s may be used r e p e a t e d l y by the second program. The second program i s w r i t t e n i n FORTRAN a l o n e . I t reads i n the output on the d i s c f i l e from the f i r s t program. T h i s output t e l l s i t what t h i n g s t o c a l c u l a t e , and i n what o r d e r t o c a l c u l a t e them. In o r d e r t o perform t h i s c a l c u l a t i o n , s e v e r a l numbers must be s u p p l i e d . Three o f t h e s e , Z-, , Z 2 , and R, 39 a r e read i n . The n o n - l i n e a r parameters ^ must a l s o be s u p p l i e d . They can be read i n , i n t e r p o l a t e d o r e x t r a p o l - a t e d , or be s u p p l i e d a u t o m a t i c a l l y by a t h i r d program, about which more w i l l be s a i d p r e s e n t l y . G i v e n these numbers and the i n s t r u c t i o n s from the f i r s t program, the second program c a l c u l a t e s a l l i n t e - g r a l s and forms the H-matrix. The m a t r i x i s d i a g o n a l i z e d , and i t s e i g e n v a l u e s and e i g e n v e c t o r s are c a l c u l a t e d . The t h i r d program mentioned above i s a g e n e r a l program t o f i n d an u n c o n s t r a i n e d minimum of a f u n c t i o n of s e v e r a l v a r i a b l e s ( P o w e l l , 1964). I n t h i s case, the v a r - i a b l e s are the a n < 3 the f u n c t i o n i s u s u a l l y the V o f ( 2 . 5 ) . CHAPTER VII T~[ CALCULATION 1 . O u t l i n e o f C a l c u l a t i o n The f i r s t s e r i e s o f c a l c u l a t i o n s i n t h i s work was ; f o r s t a t e s , because the ground s t a t e o f OH i s a s t a t e (Herzberg, 1 9 7 1 ) , and l i t t l e i s known o f any- o t h e r 2T"T s t a t e s (Pryce, 1 9 7 1 ) . L i s t e d i n Table 7 . 1 are the s p a t i a l p a r t s o f the c o n f i g u r a t i o n s chosen f o r t h i s c a l c u l a t i o n , as w e l l as the s p i n c o u p l i n g s . An examination of t h i s t a b l e r e v e a l s t h a t the g u i d e l i n e s f o r s e l e c t i n g ty. o u t l i n e d i n Chapter IV, S e c t i o n 4 have not been c o m p l e t e l y f o l l o w e d . F o r example, the c o n f i g u r a t i o n (id )2{2d ) 2 ( l 7 C T) (IT* ) 2 ( 7 d ) 2 was not i n c l u d e d , as i t sho u l d have been. Furthermore, o f the f i v e p o s s i b l e f i v e - e l e c t r o n s p i n c o u p l i n g s o f Table 4.2, o n l y one, c o u p l i n g V - l , was u s e d . The r e a s o n f o r t h i s c h o i c e was Hund's r u l e , which suggests t h a t a s t a t e w i t h t h i s c o u p l i n g i s lower i n energy than a s t a t e w i t h e i t h e r o f the o t h e r f o u r c o u p l i n g s , p r o - v i d e d the s p a t i a l p a r t s o f the s t a t e s a r e the same. The e x p l a n a t i o n f o r t h i s f a i l u r e t o f o l l o w the g u i d e l i n e s i s i n e x p e r i e n c e w i t h the program.. In f a c t , the g u i d e l i n e s were a l t e r e d as more c a l c u l a t i o n s were performed, and a t t a i n e d t h e i r p r e s e n t form o n l y a f t e r the c a l c u l a t i o n s r e p o r t e d here had been completed. 40 41 The o r b i t a l parameters were c o n s t r a i n e d t o be equ a l f o r each group o f 2JL +1 atomic o r b i t a l s o f a g i v e n n and jC-value, f o r example, f o r t he group o f t h r e e 2p o r b i t a l s on oxygen. F o r the two s e p a r a t e d , n o n - i n t e r a c t i n g atoms, a l l the 2A+1 % s w i t h i n such a group a re e q u a l , by r o t a t i o n a l symmetry. Even i n the i n t e r a c t i n g case, a x i a l symmetry d i c t a t e s t h a t the ̂ 's a r e equal f o r two o r b i t a l s which d i f f e r o t h e r w i s e o n l y i n the s i g n o f t h e i r m-values. I t was expected, t h e r e f o r e , t h a t f o r c i n g a l l 2jt+l ^ s i n a group t o be equal would not be a bad approx- i m a t i o n , p a r t i c u l a r l y a t moderate and l a r g e i n t e r n u c l e a r s e p a r a t i o n s . A s h o r t d i s c u s s i o n o f the e f f e c t i v e n e s s o f t h i s a p p r o x i m a t i o n i s c o n t a i n e d i n S e c t i o n 2 of t h i s c h a p t e r . At a g i v e n R, the o r b i t a l parameters ^ were c a l c u l a t e d by m i n i m i z i n g t he t r a c e o f the H a m i l t o n i a n m a t r i x . T h i s i s e q u i v a l e n t t o m i n i m i z i n g the average energy V o f a l l twenty-nine c o n f i g u r a t i o n s . Only the d i a g o n a l elements o f H need be computed because the t r a c e i s i n v a r i a n t under d i a g o n a l i z a t i o n . There i s thus a l a r g e s a v i n g i n computer t i m e . I t was hoped t h a t the o r b i t a l parameters c a l c u l a t e d i n t h i s manner would be good enough f o r the l o w - l y i n g s t a t e s o f i n t e r e s t . A d i s c u s s i o n o f the e f f e c t i v e n e s s o f t h i s method i s i n S e c t i o n 2 of t h i s c h a p t e r . The t r a c e was minimized a t R=1.8342, 6, 10, and 15. R=s1.8342 i s the e q u i l i b r i u m s e p a r a t i o n o f the ground 42 s t a t e (Herzberg, 1971). R=6 and 10 were chosen because the y are moderate and l a r g e i n t e r n u c l e a r d i s t a n c e s , r e s p - e c t i v e l y . R=15 was chosen because i t was found t h a t not a l l the p o t e n t i a l energy curves had c o m p l e t e l y f l a t t e n e d out near R=10. To c a l c u l a t e a parameter v a l u e a t o t h e r than these f o u r p o i n t s , a p a r a b o l a was f i t t e d t o the c l o s e s t t h r e e p o i n t s , and the parameter v a l u e at the p o i n t o f i n t e r e s t was then i n t e r p o l a t e d or e x t r a p o l a t e d . Graphs of the parameter v a l u e s as f u n c t i o n s o f R a r e shown i n F i g u r e s 7.1 t o 7 . 9 . With s e v e r a l e x c e p t i o n s , t h e s e are smooth f u n c t i o n s o f R. Bach graph has a d i s c o n - t i n u i t y a t R=8, because the two p a r a b o l a s f i t t e d through (1.8342, 6, 10) and ( b , 10, 15) do not g i v e the same parameter v a l u e a t 4(6+10)=8. These d i s c o n t i n u i t i e s are p a r t i c u l a r l y a c u t e f o r the 3s, 3p, ( 2 s ) ^ , and (2p)^ atomic o r b i t a l parameters. The d i s c o n t i n u i t i e s c o u l d be e l i m i n a t e d by f i t t i n g a s i n g l e c u b i c r a t h e r than two p a r a b o l a s . However, another approach i s p o s s i b l e . T h i s a l t e r n a t i v e approach i s based on the o b s e r v a t i o n t h a t a parameter graph can be r o u g h l y d i v i d e d i n t o two r e g i o n s . The f i r s t r e g i o n , R < 6 , i s the r e g i o n where the 0 and H atoms i n t e r a c t c o n s i d e r a b l y , w i t h a r e - s u l t a n t l a r g e change i n o r b i t a l parameters w i t h R. I n the second r e g i o n , R">6, t h e r e i s l e s s i n t e r a c t i o n between the two atoms, and the curves are much smoother. 43 I f the best o r b i t a l parameters were calculated at a t h i r d point i n the f i r s t region, halfway between 1.8 and 6, f o r example, then not only would the r e s u l t s be better i n the f i r s t region, but the d i s c o n t i n u i t y i n the second region would be smaller. The calculated p o t e n t i a l energy curves f o r the lower-lying states are given i n Figure 7.10. A d i s - cussion of the lowest four curves follows. 2. The Ground State There are a c t u a l l y two calculated p o t e n t i a l energy curves f o r the ground state of OH shown i n Figure 7.10. The upper one, which w i l l be discussed f i r s t , i s the curve calcu l a t e d as explained i n the previous s e c t i o n . This f i r s t curve i s q u a l i t a t i v e l y c o r r e c t . I t predicts that the lowest state i s bound and separates o i n t o oxygen i n i t s ground ( P) state and hydrogen i n i t s ground ( S) s t a t e . The calculated equilibrium separation, about 2.2, i s approximately 25% l a r g e r than the experi- mentally observed separation. The calculated binding energy, about .035 Hartrees, i s considerably smaller than the exper- imental value of about .16 Hartrees (Carlone, 1969). The quantitative agreement with experiment i s therefore poor. The wavefunction f o r t h i s state as a function of R i s given i n Figure 7.11. The amplitudes of a l l major 44 contributors to the wavefunction, the yS 1^ of equation (2.3), are p l o t t e d . A 'major c o n t r i b u t o r 1 i s defined to be a configuration whose amplitude, i n absolute value, exceeds .1 f o r some value of R. For R<2.3, the l a r g e s t contributor i s configur- a t i o n #1, i n which a l l nine electrons are centred on the oxygen nucleus. The next l a r g e s t contributor i s #19, which i s s i m i l i a r to #1, except that a 3C$ electron on oxygen has been changed into a 7c£ e l e c t r o n on hydrogen. In #19, the ITT and 3d o r b i t a l s are coupled to form a spin t r i p l e t , which i s then coupled with the 7d to form a doublet. Con- f i g u r a t i o n #18, which has the same s p a t i a l part as #19, d i f f e r s i n that the lit" and 3d are coupled to form a s i n g l e t . Since the amplitude f o r #18 i s about -| times the amplitude f o r #19, the combined e f f e c t of #18 and #19 i s a configur- a t i o n i n which the 3cf and 7d are coupled to form a s i n g l e t , which i s then coupled with the ITT to form a doublet. This mixes very well with #1, i n which the two 3& s are coupled to form a s i n g l e t . The next most important configurations are those i n which a i n (or ItC ) o r b i t a l centred on oxygen i n #1 i s replaced by a 4TC (or 47* ) o r b i t a l centred on hydrogen. These are configurations #25 and #27. 45 I n the r e g i o n o f the e q u i l i b r i u m s e p a r a t i o n , then, the bond between 0 and H i s a ^ bond w i t h some T C -type c h a r a c t e r . As R->«0 , c o n f i g u r a t i o n #19 becomes predominant, and a l l o t h e r c o n f i g u r a t i o n s except #3, #5, and #7 become i n s i g n i f i c a n t . What i s happening i s t h a t an e l e c t r o n which at s m a l l R i s i n a 2pd o r b i t a l on oxygen a t t a c h e s i t s e l f t o the hydrogen and o c c u p i e s a pure o r b i t a l as R-*©6. Even a t R=12, though, t h i s e l e c t r o n has a p r o b a b i l i t y a m p litude o f about .1 o f b e i n g i n a d i f f u s e 4<^ , 5d , o r %d o r b i t a l c e n t r e d on oxygen. As can be seen, the amp- l i t u d e s f o r c o n f i g u r a t i o n s #3, #5, and #7 a r e d e c r e a s i n g w i t h i n c r e a s i n g R a t R=12, and s h o u l d go t o zero as R-*»6 . The form o f the w a v e f u n c t i o n seems q u i t e r e a s o n - a b l e . Why then i s the p o t e n t i a l energy curve so poor? I t c o u l d be t h a t the o r b i t a l parameters chosen by the method o f m i n i m i z i n g the t r a c e o f H are poor c h o i c e s f o r the ground s t a t e . To t e s t t h i s guess, a second c a l c u l a t i o n was performed f o r the ground s t a t e a l o n e . The o r b i t a l p a r a - meters f o r the 2 s , 2p, and ( l s ) j j o r b i t a l s were chosen t o minimize the lowest e i g e n v a l u e of H. The o r b i t a l p a r a - meter f o r the 2prf o r b i t a l was a l l o w e d t o v a r y independ- e n t l y o f the 2p i r , 2p?c parameter. The o t h e r o r b i t a l p a r a - meters were f i x e d a t the v a l u e s o b t a i n e d by m i n i m i z i n g the 46 t r a c e o f H. T h i s was done a t both R=1.8342 and 6. At R=6 t h e r e was no change i n the parameters. At R=l.6*342, how- ev e r , t h e r e was a l a r g e change i n the ( l s ) H o r b i t a l , and moder a t e l y l a r g e changes i n the 2pa ( T T ) and 2p^ o r b i t a l s (see F i g u r e s 7.3 and 7*7). The 2p o r b i t a l s became more d i f f u s e (^ d e c r e a s e d ) , and the ( l s ) H o r b i t a l c o n t r a c t e d c o n s i d e r a b l y . As can be seen from F i g u r e 7.12, though, the amplitudes o f the major c o n f i g u r a t i o n s changed v e r y l i t t l e . The r e s u l t i n g p o t e n t i a l energy curve i s the low- e s t curve o f F i g u r e 7.10. The p r e d i c t e d e q u i l i b r i u m sep- a r a t i o n i s about 1.8, i n v e r y good agreement wi t h exper- iment. The p r e d i c t e d b i n d i n g energy i s about .1 H a r t r e e s , i n f a i r agreement w i t h experiment. S e v e r a l c o n c l u s i o n s can be made a t t h i s p o i n t . The method o f m i n i m i z i n g the t r a c e o f the H-matrix i s d e f i n i t e l y not good enough f o r any more than a q u a l i t a t - i v e l y c o r r e c t c a l c u l a t i o n . The reason f o r t h i s i s p r o b a b l y as f o l l o w s . There a r e a l a r g e number of c o n f i g u r a t i o n s i n the c a l c u l a t i o n which are q u i t e u n l i k e the ground s t a t e w a v e f u n c t i o n . M i n i m i z i n g the t r a c e t r e a t s these c o n f i g - u r a t i o n s as i f t h e y were j u s t as important t o the ground s t a t e as those which a r e i t s major components. T h i s i s t r u e not j u s t f o r the ground s t a t e , but f o r a l l s t a t e s . The method, i n t r y i n g t o d e s c r i b e a l l s t a t e s e q u a l l y w e l l , d e s c r i b e s each s t a t e p o o r l y . 47 The constraint that a l l 2J£+1 parameters of a group be equal should be relaxed at small internuclear distances. Figure 7.3 shows that at R=1.8342, the d i f f - erence between the 2p7T, 2p9T parameter and the 2pd* para- meter, while not very large, i s not insignificant. At moderate and larger (R^-6) distances; as expected, the constraint should be retained. There are several things which could be done to increase the calculated binding energy further. One i s to include a configuration with a doubly-occupied 7^ o r b i t a l . Another is to find the best parameter values for the [2pi* (TC )] h and (2p^)^ atomic, orbitals, which are present i n some important configurations near the equilib- rium separation. If these things were done, the overall agreement with experiment would probably be quite good. 3 . The F i r s t Excited 2TT State According to i t s calculated potential energy curve, the f i r s t electronically excited 2TT state i s unbound. Beginning about R=5, the curve rises slowly and then more rapidly as R—* 0. At large R, the energy separation between i t and the ground state i s calculated to be .10 Hartrees, i n good agreement with the actual value of .08 Hartrees (Edien, 1943). 4 8 On the b a s i s o f ex p e r i e n c e w i t h the ground s t a t e r e s u l t s , i t can be r e a s o n a b l y expected t h a t the p o t e n t i a l , energy curve f o r the f i r s t e x c i t e d 2 T T s t a t e i s q u a l i t a t - i v e l y c o r r e c t ; i . e . , t h a t i t i s a c t u a l l y an unbound s t a t e . The w a v e f u n c t i o n i s g i v e n i n F i g u r e 7 . 1 3 . F o r R > 3 , i t i s v e r y much l i k e t he wa v e f u n c t i o n f o r the ground s t a t e . The important d i f f e r e n c e i s t h a t i n the f i r s t e x c i t e d 2 T 7 s t a t e , the 1TC and 3 d o r b i t a l s a re cou p l e d , a t l a r g e R, t o form a s i n g l e t , whereas i n the ground s t a t e t h e y a r e co u p l e d t o form a t r i p l e t . T h i s e x c i t e d s t a t e d i s s o c i a t e s 2 i n t o hydrogen i n i t s ground ( S) s t a t e and oxygen i n i t s f i r s t e x c i t e d (̂ "D) s t a t e . There i s an extremely abrupt change i n the wave- f u n c t i o n a t R=2 . 3 . F o r R < 2 . 3 , c o n f i g u r a t i o n s i n which a 2TT o r b i t a l i s prese n t predominate. 4 . The Second E x c i t e d 2TT S t a t e As can be seen from F i g u r e 7 . 1 4 , the wa v e f u n c t i o n f o r the second e x c i t e d 2TT s t a t e i s c o n s i d e r a b l y more comp- l i c a t e d than t h a t o f e i t h e r the ground s t a t e o r the f i r s t e x c i t e d 2TT s t a t e . The reason f o r t h i s c o m p l e x i t y i s the avoidance o f curve c r o s s i n g . F o r R ^ 2 . 3 , the dominant c o n f i g u r a t i o n s a r e those i n which a 3TT o r b i t a l i s s i n g l y - o c c u p i e d . F o r 2 . 3 < R < 7 . 6 , i t i s the 2~K o r b i t a l which i s s i n g l y - o c c u p i e d . I n the lower 49 p a r t o f t h i s range, 2.3 ̂ R ̂ 3 .6, the major c o n f i g u r a t i o n , #9, i s composed o f o r b i t a l s c e n t r e d on oxygen. I n the upper p a r t , 3 . 6 * R < 7 . 6 , the major c o n f i g u r a t i o n i s #28, which i s s i m i l i a r t o #9, except t h a t one o f the e l e c t r o n s i n a 3 ̂  o r b i t a l on oxygen has been changed t o an e l e c t r o n i n a 7 ^ o r b i t a l , c o n c e n t r a t e d near the hydrogen n u c l e u s . At R=7.6, t h e r e i s an abrupt change i n the wave- f u n c t i o n due to av o i d e d curve c r o s s i n g . Two c o n f i g u r a t i o n s , #23 and #21, predominate. They c o n s i s t o f oxygen i n i t s ground { P) s t a t e and an e l e c t r o n i n a 2 p ^ and 3s o r b i t a l on hydrogen, r e s p e c t i v e l y . An enumeration o f the combinations o f s t a t e s o f n o n - i n t e r a c t i n g oxygen and hydrogen atoms which can g i v e r i s e t o m o l e c u l a r s t a t e s shows i n f a c t t h a t the combination w i t h the t h i r d lowest t o t a l energy i s oxygen i n i t s ground s t a t e and hydrogen w i t h i t s e l e c t r o n i n a 2s or 2p o r b i t a l . I t i s not s u r p r i s i n g t h a t t h e r e i s a l a r g e mixture o f #23 and #21, s i n c e , a t l a r g e R, the ( 2 p ^ ) H and ( 2 s ) H o r b i t a l s a r e degenerate. A l s o , i t i s not s u r p r i s i n g t h a t #23 sho u l d be somewhat more important than #21, s i n c e t he ( 2 p d )^ o r b i t a l has, u n l i k e the ( 2 s ) ^ , a l o b e which l i e s a l o n g the i n t e r n u c l e a r a x i s . The p o t e n t i a l energy c u r v e , shown i n F i g u r e 7.10, has two minima, one a t R=2.6 and one a t R=9.5, w i t h a maximum a t R=6.5. The b i n d i n g energy f o r the minimum a t R=2.6 appears t o be r o u g h l y the same as f o r the ground s t a t e . 50 The v i b r a t i o n a l l e v e l s o f t h i s s t a t e w i l l be o v e r l a i d by the v i b r a t i o n a l l e v e l s o f the t h i r d e x c i t e d s t a t e , making s p e c t r o s c o p i c i d e n t i f i c a t i o n d i f f i c u l t . The broad, s h a l l o w minimum a t R=9.5 should be almost i m p o s s i b l e t o d e t e c t because t h e r e a r e no w e l l - known s t a t e s o f OH t o which a t r a n s i t i o n c o u l d be made from t h i s minimum. Of c o u r s e , these f e a t u r e s o f t h e p o t e n t i a l energy curve s h o u l d not be taken too s e r i o u s l y , because o f the poorness o f t h i s f i r s t c a l c u l a t i o n . Table 7.1 C o n f i g u r a t i o n s for 2TTcalculation C o n f i g u r a t i o n Number S p a t i a l Part S p i n C o u p l i n g 1 (1<5) 2 ( 2d)2( lrT)(30) 2 (nc) 2 1-1 2,3 2, 2d)2\ ll?)(3rf)(ll0 2(4tf) 111-1,2 4 , 5 (Id) 2, 2d)2i i*)(3d)(ir)2(5d) 111-1,2 6,7 (id) 2, 2d)2 kiw) (3d) (ITT)2(66) 111-1,2 3,9 (id) 2, ,2d)2 aTr ) ( 3^) 2(iTr ) (2 i r ) 111-1,2 10,11 (16) 2 ad)2 (lTf)(3d)2(lTT)(3xr) 111-1,2 12 (Id) 2 [2d)2 (lTr) 2(lTT) 2(27l) 1-1 13 (Id) 2 [2d)2 (iTf) 2(no 2 (3TO 1-1 14 (16) 2 [2d)2 (3rf) 2 (n0 2(27r) 1-1 15 (id) 2 [2d)2 (3rf)2(l7r)2(3TT) 1-1 16 ,17 (16) 2 [2d)2 (l?t)2(3<<)(l-ff)(lS) 111-1,2 18 ,19 (id) 2 (2d)2 (1F)(3<*)(1T02(7<*) 111-1,2 20,21 (id) 2 (2d)2 (ln)(3<*)(n02(84) 111-1,2 22 ,23 (id) 2 (2d)2 (liT)(3^)(iTr) 2(9d) 111-1,2 24,25 (id) 2 (2d)2 (lw ) ( 3^) 2(lTr)(47T) 111-1,2 26 (16) 2 (2d)2 (iT?) 2 ( iTD 2(4Tr) 1-1 27 (id) 2 (2<S)2 (3<*)2U?U2(4W) 1-1 28 (id) 2 (2d)2 (lTf)(3€f)(lTT)(2Tr)(7tf ) V - l 29 (id) 2 (2d)2 (lTT)(3^)(lw)(3Tr)(7rf) V - l 7 657—j 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 F i g u r e 7.1. Is Parameter ve r s u s I n t e r n u c l e a r Distance". • i n Bohr R a d i i . •• 2.26—i 2 .25 -H 2 2 4 H 1 j 1 — - r 1 1 1 1 1 \ 1 i i "I I 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Figure 7.2. 2s Parameter versus Internuclear Distance in Bohr Radii. 2.21 — y 2 .20 T 2 T 7 10 n 12 13 14 15 F i g u r e 7.3. ^Sp Parameter v e r s u s . I n t e r n u c l e a r ; D i s t a n c e , • i r i Bohr R a d i i . Upper. dottedV curve- i s 2po . : Lower d o t t e d curve i s 2p?r ,•' 2p7r . 0 45—1 F i g u r e .7.4. 3s Parameter v e r s u s I n t e r n u c l e a r Distance" i n Bohr R a d i i . 0.3 3 — r ; F i g u r e 7.5. 3p Parameter v e r s u s I n t e r n u c l e a r D i s t a n c e i n Bohr R a d i i . 0.30—r Q 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 F i g t ^ P T i ' S V 3,d Parameter v e r s u s I n t e r n u c l e a r D i s t a n c e • '-Vf^-..'^•:«***:-• -in Bohr Ra d i i . .    F i g u r e 7.10, 2-TT I [ P o t e n t i a l Energy C u r v e s . Dotted l i n e i s r e f i n e d ground s t a t e c a l c u l a t i o n . . 1 1 1 1 1 1 1 I I I I I 1 I I 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Ground S t a t e W avefunction. The a m p l i t u d e s of the major c o n f i g u r a t i o n s a r e p l o t t e d as a f u n c t i o n o f i n t e r n u c l e a r d i s t a n c e . S o l i d and dashed l i n e s r e p r e s e n t amplitudes o f o p p o s i t e s i g n . .• 0 s    CHAPTER V I I I 2 s - CALCULATION 1. O u t l i n e of C a l c u l a t i o n The second c l a s s o f s t a t e s f o r which c a l c u l a t i o n s were performed i s the c l a s s o f s t a t e s . T h i s c l a s s was chosen because t h e r e i s some i n d i c a t i o n t h a t the lowest s t a t e i s bound (Pryce, 1971). The c o n f i g u r a t i o n s chosen f o r the c a l c u l a t i o n s a r e l i s t e d i n T a b l e 8.1. S e v e r a l t h i n g s from t h i s t a b l e s h o u l d be n o ted. A l l c o n f i g u r a t i o n s which s h o u l d be p r e s e n t a c c o r d - i n g t o the g u i d e l i n e s l a i d down i n Chapter IV, S e c t i o n 4 a r e p r e s e n t . I n a d d i t i o n , t h e r e i s one c o n f i g u r a t i o n , #34, i n which the 2d o r b i t a l i s s i n g l y o c c u p i e d . T h i s c o n f i g u r a t i o n can be o b t a i n e d from c o n f i g u r a t i o n #9» an i mportant c o n s t i t u e n t o f the lowest — s t a t e , by moving one o f the two e l e c t r o n s i n the 2d o r b i t a l i n t o a 7c5* o r b i t a l . I t was hoped t h a t the e f f e c t i v e n e s s o f the 'frozen- c o r e ' a p p r o x i m a t i o n c o u l d be t e s t e d by d e t e r m i n i n g the importance of t h i s c o n f i g u r a t i o n t o the ground s t a t e . More w i l l be s a i d about t h i s i n S e c t i o n 2 o f t h i s c h a p t e r . A l l p e r m i s s i b l e s p i n c o u p l i n g s were used i n the c a l c u l a t i o n s , i n c o n t r a s t t o the 1 I c a l c u l a t i o n s . * See F i g u r e 8.11 f o r the w a v e f u n c t i o n o f t h i s s t a t e 66 67 Not a l l p o s s i b l e s p i n c o u p l i n g s a r e p e r m i s s i b l e , however. As noted i n Chapter IV, S e c t i o n 7 , some o f them l e a d t o i d e n t i c a l l y v a n i s h i n g ^ , s t a t e s . T h i s i s why, f o r example, s p i n c o u p l i n g s V - 4 and V - 5 a r e never used on- (ITT )(1"K ) ( 3 d ) ( 4 d ) ( 7 d ) . The average of the lowest f i v e e i g e n v a l u e s , V, was minimized a t R= 1 . 8 and 6 w i t h r e s p e c t t o the o r b i t a l parameters ^ . . A l l i t +1 parameters which belong t o the same group o f atomic o r b i t a l s , as e x p l a i n e d i n the l a s t c h a p t e r , were c o n s t r a i n e d t o be e q u a l . At R= 3 . 5 , a s e t of parameter v a l u e s was determined by l i n e a r i n t e r p o l a t i o n . Keeping the parameters f o r the I s , 2 s , and 2 p atomic o r b i t a l s f i x e d a t the s e v a l u e s , V was minimized by v a r y i n g the p a r a - meters f o r the 3 s , 3 p , 3 d , ( l s ) H , ( 2 s ) H , and ( 2 p ) H atomic o r b i t a l s . F i x i n g the I s , 2 s , and 2 p parameters a t t h e i r i n t e r p o l a t e d v a l u e s was f e l t t o be r e a s o n a b l e , s i n c e the f r a c t i o n a l changes i n these parameters as R v a r i e s s h o u l d be s m a l l e r than those o f the o t h e r atomic o r b i t a l s . T h i s i s t he case f o r the "TTca l c u l a t i o n . The r e s u l t i n g parameter v a l u e s as f u n c t i o n s o f R S c - are shown i n F i g u r e s 8 . 1 t o 8 . 9 . The f i v e lowest p o t e n t i a l energy curves a r e shown i n F i g u r e 8 . 1 0 . More r e f i n e d c a l c u l a t i o n s f o r the lowest and 2 5" " second lowest s t a t e s were performed. The lowest e i g e n v a l u e was minimized a t R= 2 . 1 , 3 . 5 , and 6 . The second 68 lowest eigenvalue was minimized at R=2 and 6. The values f o r those parameters which were varied are shown i n Figures 8.3, 8.4, and 8.7. The two p o t e n t i a l energy curves are shown i n Figure 8.10. 2 y - 2. The Lowest State The two calculated p o t e n t i a l energy curves f o r the lowest energy ' state are shown i n Figure 8.10. The more refined c a l c u l a t i o n produces the lower curve, of course. Both curves show the same q u a l i t a t i v e behavior. The state i s predicted to be bound, with an equilibrium separation of R ^ 3 « 5 . The calculated depth of the minimum, however, i s quite d i f f e r e n t i n the two c a l c u l a t i o n s . The cruder c a l c u l a t i o n gives about .09 Hartrees, and the more re f i n e d one .05 Hartrees. The spectrum of OH (Pryce, 1971) seems to ind i c a t e that the state i s bound by^.01 Hartrees. Only a lower l i m i t may be given with c e r t a i n t y because only one v i b - r a t i o n a l l e v e l has been detected, and i t may not be the lowest l e v e l . I f i t i s i n f a c t the lowest l e v e l , then the spacing of the observed r o t a t i o n a l l e v e l s indicates that the equilibrium separation i s at R=3.1. The present c a l c u l a t i o n s can be reconciled with observation i n at least two ways. In the f i r s t way, i t i s accepted that the observed v i b r a t i o n a l l e v e l i s the lowest * See Appendix 3 f o r further comments on t h i s . 69 one. T h i s i m p l i e s t h a t the more r e f i n e d c a l c u l a t i o n i s not yet a c c u r a t e enough. S i n c e the amount o f b i n d i n g i s lower i n the more r e f i n e d c a l c u l a t i o n , i t i s c o n c e i v a b l e t h a t a b e t t e r c a l c u l a t i o n would p r e d i c t an even lower b i n d i n g energy of ^ . 0 4 H a r t r e e s , c o n s i s t e n t w i t h a f i g u r e o f . 0 1 H a r t r e e s . I t i s a l s o c o n c e i v a b l e t h a t t h i s b e t t e r c a l c u l - a t i o n would s h i f t t he p r e d i c t e d e q u i l i b r i u m s e p a r a t i o n t o < 3 » 5 . I t sh o u l d be r e c a l l e d t h a t a s i m i l i a r s h i f t occured i n the TTground s t a t e c a l c u l a t i o n s . I n the second way, i t i s ac c e p t e d t h a t the c a l - c u l a t e d amount o f b i n d i n g and the c a l c u l a t e d e q u i l i b r i u m s e p a r a t i o n a r e r o u g h l y c o r r e c t . T h i s i m p l i e s t h a t the observed l e v e l i s a h i g h v i b r a t i o n a l l e v e l . An apparent e q u i l i b r i u m s e p a r a t i o n of 3 . 1 can be produced i f , c o n t r a r y t o t h e prese n t c a l c u l a t i o n s , the p o t e n t i a l energy curve r i s e s more r a p i d l y f o r R * > 3 » 5 than f o r R < 3 « 5 . I t may be o b j e c t e d t h a t a more u s u a l b e h a v i o r f o r the p o t e n t i a l energy curves i s f o r them t o r i s e more r a p i d l y f o r R < e q u i l i b r i u m s e p a r a t i o n than f o r R > e q u i l i b r i u m s e p a r a t i o n . T h i s b e h a v i o r i s produced by the 8/R term i n H. However, s i n c e an e q u i l i b r i u m s e p a r a t i o n o f 3 . 5 i s somewhat l a r g e r t h an the more u s u a l v a l u e o f RXZ2, t h i s o b j e c t i o n i s not a s t r o n g one. The w a v e f u n c t i o n f o r the lowest s t a t e i s shown i n F i g u r e 8 . 1 1 . T h i s i s the r e s u l t o f the f i r s t , l e s s r e f i n e d c a l c u l a t i o n . As i s the case f o r the TTground s t a t e , 70 the amplitudes a r e p r a c t i c a l l y the same i n both the f i r s t and more r e f i n e d c a l c u l a t i o n . That i s why the amplitudes f o r the more r e f i n e d c a l c u l a t i o n a r e not shown h e r e . F o r R<2.1, the dominant c o n f i g u r a t i o n s a re #2 and #1. The combined e f f e c t o f these two i s a w a v e f u n c t i o n c e n t r e d on the oxygen nucleus and c o n s i s t i n g o f a s i n g l e e l e c t r o n i n an e x c i t e d o r b i t a l , p l u s a core o f u n e x c i t e d 0 H +. The e x c i t e d o r b i t a l i s m o s t l y 3p<S w i t h some 3s c h a r a c t e r . The p d nature o f the o r b i t a l can be understood as due t o the l o b e p u l l e d out a l o n g the i n t e r n u c l e a r a x i s by the hydrogen n u c l e u s . At R=2.1, t h e r e i s an abrupt change i n the wave- f u n c t i o n . C o n f i g u r a t i o n #9, which d i s s o c i a t e s i n t o oxygen and hydrogen i n t h e i r ground s t a t e s , i s the predominant c o n f i g u r a t i o n f o r R>2.1. T h i s i s t o be expected, s i n c e the lowest energy s t a t e a t l a r g e i n t e r n u c l e a r d i s t a n c e s o i s formed from oxygen i n i t s ground ( P) s t a t e and hydrogen i n i t s ground ( S) s t a t e . What i s h i g h l y i n t e r e s t i n g i n the r e g i o n 2.1<" R <5 i s c o n f i g u r a t i o n #11, which d i s s o c i a t e s i n t o oxygen i n i t s ground s t a t e and hydrogen w i t h i t s e l e c t r o n i n a (2pd)^ o r b i t a l . I t i s the second most important c o n f i g - u r a t i o n i n t h i s r e g i o n , and i s c o m p l e t e l y unimportant e l s e - where. I t i s p r e c i s e l y i n t h i s r e g i o n where the s t a t e i s bound. Furthermore, i t i s i n the immediate r e g i o n o f the 71 equilibrium separation that #11 i s most important. It i s reasonable, then, to attribute the boundedness of the state to this configuration. For R">6, the only configurations of note besides #9 are #1, #2, and #3, xvhich are very similar to the three configurations i n the ITcalculations centred on oxygen with one very diffuse o r b i t a l . The lowest JET state and the ground IT state are degenerate at large R. It i s not surprising, then, that the calculated energies at i n f i n i t e separation are the same, as can be seen by comparing Figures 7.10 and 8.10. Configuration #34 i s not a major component of the wavefunction at any value of R of interest. It could be important only at such small R that the hydrogen nucleus affects the 2G* orbital s i g n i f i c a n t l y . Thus the 'frozen- core' approximation is a good one except possibly at very small R. Finally, i t should be mentioned that a previous 25-- calculation (Michels and Harris, 1969) on the lowest state predicted that the state is unbound. That calculation, i n the light of what has been said i n this section, i s most probably incorrect. 72 3. The Second Lowest - S t a t e The two c a l c u l a t e d p o t e n t i a l energy curves f o r the second lowest s t a t e a r e shown i n F i g u r e 8.10. Both c a l c u l a t i o n s p r e d i c t t h a t the s t a t e i s bound a t R s r 2 . 1 , and t h a t the p o t e n t i a l energy curve has a maximum a t R=4. One v i b r a t i o n a l l e v e l o f a s t a t e w i t h an e q u i l i b r i u m s e p a r a t i o n a t R^r2 has r e c e n t l y been observed (Douglas, 1 9 7 1 ) . The energy d i f f e r e n c e between t h i s l e v e l and the observed l e v e l i n the lowest *£— s t a t e ( P r y c e , 1971) i s v e r y n e a r l y .20 H a r t r e e s , the energy d i f f e r e n c e as c a l c u l a t e d from F i g u r e 8.10. I t seems q u i t e c e r t a i n , t h e n , t h a t the s t a t e observed by Douglas i s the f i r s t e x c i t e d s t a t e . 2 5 " " Another e x c i t e d s t a t e has been observed ( P r y c e , 1 9 7 1 ) . I t has an e q u i l i b r i u m s e p a r a t i o n o f R=3.7, and the minimum i n i t s p o t e n t i a l energy curve l i e s above 2S" t h e c a l c u l a t e d minimum of the f i r s t e x c i t e d s t a t e but below the c a l c u l a t e d maximum. The presen t c a l c u l a t i o n s are t h e r e f o r e c o m p l e t e l y unable t o account f o r the e x i s t a n c e o f t h i s o t h e r e x c i t e d s t a t e . A p o s s i b l e e x p l a n a t i o n , however, w i l l be o f f e r e d l a t e r i n t h i s s e c t i o n . The w a v e f u n c t i o n f o r the f i r s t e x c i t e d s t a t e as o b t a i n e d from the f i r s t c a l c u l a t i o n i s shown i n F i g u r e 8.12. As noted p r e v i o u s l y , the amplitudes change l i t t l e i n a more r e f i n e d c a l c u l a t i o n , and t h e r e f o r e the more r e f i n e d 73 wavefunction i s not shown. For R < 2 . 1 , the wavefunction i s effectively that of an electron in an excited o r b i t a l , plus an unexcited OH core. The orbital is mostly 3 s , with some 3 p d character . It had been guessed (Douglas, 1 9 7 1 ) that at the equilibrium separation, this orbital is probably 3 s , but that i t could possibly be 3p<^ • Figure 8 .12 shows that the orbital i s in fact an almost equal mixture of 3 s and 3 ^ at that distance. For 2 . 1<R< ^ 3 « 6 , this orbital i s almost purely 3 p d . As R - > 3 « 6 , an electron which at smaller separations occupies a 2 p d atomic orbital on oxygen begins to detach i t s e l f and occupies a ( l s ) ^ atomic o r b i t a l . This i s evid- enced by the increasing importance of configuration # 1 5 . At R=3.7, there i s a change i n the nature of the wavefunction. Configuration # 1 2 , which dissociates into oxygen i n an excited (^S°) state and hydrogen in i t s ground ( S) state, becomes predominant. It is near this point that the potential energy curve reaches i t s maximum. As R-»«ft, the wavefunction becomes pure # 1 2 , and the potential energy curve decreases monotohically. The existence of the other excited state (Pryce, 1 9 7 1 ) mentioned previously can be explained i f the maximum i n the potential energy curve were between R=2.1 and R= 3 . 7 , and i f there were a minimum near R=3.7; that i s , * Compare this with the description i n the previous section of the lowest 2«2E ~ wavefunction for R < 2 . 1 . 74 2 i f t he two observed e x c i t e d s t a t e s were the same s t a t e , but corresponded t o two minima i n the p o t e n t i a l energy c u r v e . 2 « r - I t s h o u l d be r e c a l l e d t h a t i n the lowest S . s t a t e c a l c u l a t i o n t h e r e was a c o n f i g u r a t i o n , # 1 1 , which i f i t had been o m i t t e d from the c a l c u l a t i o n would p r o b a b l y have caused the s t a t e t o appear t o be unbound. Furthermore, 2—- the minimum i n the lowest curve due t o #11 i s a t R = 3 . 5 , v e r y near the proposed minimum a t R= 3 « 7 i n the e x c i t e d 2 « S " <r - s t a t e . T h i s suggests t h a t a c o n f i g u r a t i o n s i m i l a r t o #11 has been o m i t t e d from the present c a l c u l a t i o n s , which, i f i t were i n c l u d e d , would produce a minimum a t the d e s i r e d 2 ^ " p l a c e i n the f i r s t e x c i t e d c u r v e . C o n f i g u r a t i o n #11 can be c o n s t r u c t e d by r e p l a c i n g the 7C? m o l e c u l a r o r b i t a l i n c o n f i g u r a t i o n # 9 , which i s the dominant c o n f i g u r a t i o n f o r the lowest <cL- s t a t e i n the r e g i o n of i t s e q u i l i b r i u m s e p a r a t i o n , by a 9 d m o l e c u l a r o r b i t a l . A s i m i l a r procedure f o r the f i r s t e x c i t e d s t a t e near R= 3 . 7 produces ( i d ) 2 ( 2 c * )2(1TT ) (ITT ) ( 3 c * ) ( 5 d ) ( 9 d * ) , w i t h a p p r o p r i a t e s p i n c o u p l i n g . A c a l c u l a t i o n s h o u l d be performed u s i n g t h i s c o n f i g u r a t i o n . The c a l c u l a t e d energy d i f f e r e n c e a t i n f i n i t e i n t e r - n u c l e a r d i s t a n c e between the lowest and the second lowest s t a t e i s ^ . 1 7 H a r t r e e s . The a c t u a l v a l u e , which i s the energy d i f f e r e n c e between the lowest and s t a t e s o f oxygen, o r e q u i v a l e n t l y , the d i f f e r e n c e i n i o n i z a t i o n 75 p o t e n t i a l s of a 2p and a 3s electron i n oxygen, i s c l o s e r to .42 Hartrees (Edlen, 1943) • The agreement i s poor. A d e t a i l e d analysis of the computer printout, which i s not reproduced here, has shown that the calculated 2p i o n i z a t i o n p o t e n t i a l i s low, and the 3s i s high. These two errors add to produce the large e r r o r . I t seems that the errors a r i s e because the program has been designed to describe molecules, but not separated atoms, w e l l . Even a s e l f - c o n s i s t e n t c a l c u l a t i o n , however, (Hartree et a l , 1940) f a i l s to get a good value f o r the i o n i z a t i o n p o t e n t i a l of a 2p electron i n oxygen. 4. The Third Lowest State The wavefunction f o r t h i s state i s shown i n Figure 8.13. I t appears to be s i m i l a r to the second lowest 4ttZ^ wavefunction. At R<"2.1, the wavefunction i s that of an electron i n a 3dtf atomic o r b i t a l on oxygen, plus an unexcited 0H + core centred on oxygen. At the equilibrium separation R = 2 . 2 , t h i s o r b i t a l i s a roughly equal mixture of 3dCf and 3 s , with some 3p<S character. Between 2 . 2 < " R < 3.7, the o r b i t a l i s 3 s . At R=3»7, the molecule begins to d i s s o c i a t e i n t o hydrogen i n i t s ground state and oxygen i n an excited s t a t e . 76 C o n f i g u r a t i o n S p a t i a l Part Spini Number C o u p l i n g 1 (l<*) 2(2tf) 2(lrr)(3d0 2(lTO(4d) III-2 2 (ltf) 2(2cT) 2(lTr}(3^) 2(nr)(5d) III-2 3 (10') 2(2^) 2(lff)(3^) 2 (lTT)(6a) ni-2 4,5 ( l d ) 2 ( 2 d ) 2 ( l T r ) 2 ( 3 d ) ( l * ) ( 2 n ) 111-1,2 6,7 (ltf ' ) 2 ( 2 ^ ) 2 ( l T f ) 2 ( 3 ^ ) ( l K ) ( 3 ^ ) 111-1,2 8 (l<J ' ) 2 (2^) 2(lTf ' ) 2 (3(5) 2(lS) 1-1 9 ( l d ) 2 ( 2 C) 2(in ) ( 3^) 2(iTc ) ( 7 d ) III-2 10 ( l r f ) 2 ( 2 d ) 2 ( l T r ) (3d) 2(lTT) (8d) ni-2 11 ( l^ ) 2 ( 2 d ) 2 ( l T f ) ( 3 d ) 2 ( l T V ) ( 9 d ) III-2 12,13,14 (lcr)2(2d) 2(lTf)(i-n:)(3d)(4<5)(7d) V-l.,2,3 15,16,17 ( l d ) 2 ( 2 t f) 2(l ^ ) ( l T T ) ( 3 d ) ( 5 d ) ( 7 d ) V-1,2,3 18,19,20 (l«?)2(2d)2(lTT)(rn)(3c<)(6c5)(7d) V-1,2,3 21,22 (l < * ) 2 ( 2 c f ) 2 ( l T r ) ( 3 c 0 2 ( 2 T T ) ( 7 d ) 111-1,2 23,24 ( l d ) 2 ( 2 d ) 2 ( l w ) (3d)2(3-n) (7d) III - 1 , 2 25,26 ( l d ) 2 ( 2 d ) 2 ( r r r ) 2 ( r n ) ( 2 7 r ) ( 7 d ) IH-1,2 27,28 (iar ) 2 ( 2 d ) 2 ( l T T ) 2 ( l T C)(3R ) ( 7 d ) 111-1,2 29,30 (l^) 2 (2c0 2(lTT ' ) 2 (3o ' ) ( l S ) (7d) 111-1,2 31,32 (ld ) 2 ( 2 d ) 2 ( l T f ) 2 ( 3 d)(l - K)(4Tc) 111-1,2 33 (l<3*) 2 (2d ) 2 ( l t r ) ( lTT ) (3d)(7d) 2 III-2 34 ( l c * ) 2 ( 1 7 r ) ( 3 d ) 2 ( l T T ) ( 7 d ) 2 < 2 d ) III-2 2 5~- Table 8.1. C o n f i g u r a t i o n s f o r ^— C a l c u l a t i o n   79 2 5- 2.4- 2.3- 2.2 — 2.V Figure 8.3 2p Parameters for ZL. States. Lowest dotted curve is for 2p?c , 2prt parameter i n lowest state. Second lowest dotted curve i s for 2pcf parameter in lowest state. Upper dotted curve i s for second lowest state. Solid line i s for lowest five states. o 8 — r 0 1 2 3 4 5 6 F i g u r e 8 . 4 . 3s Parameter, gutted l i n e i s o n l y 'for second lowest 2L~ s t a t e . 81 0A-\ 0.3-H 02—\ 0 1 2 3 4 5 6 Figure 8.5. 3p Parameter. 8-2 F i g u r e 8.6. 3d. Parameter .'$3 • 1.6- 1.4- 1.2- 1.0— 0.8' Figure 8.7. ( l s ) u Parameter. Lower dotted line is for lowest mZ~ state^ Upper dotted line i s for second lowest 21" state. Figure 8.8. (2s) u Parameter. 85 j 86 F i g u r e 8.10 Figure 8 . 1 1 . Lowest 2I^C ~ Wavefunction.  89 '! CHAPTER IX CONCLUDING REMARKS The computer programs developed here have been useful in the interpretation of the spectrum of OH, even though they have not yet been used as wisely as they could have been. I n the future, calculations of the , TT» and Es ta tes are planned. The TTstates wi l l also be recalculated, and possibly the second lowest m£— cal- culation wi l l be redone. 90 BIBLIOGRAPHY Born, M., and Oppenheimer, J.R., Ann. der Phys. 84, 457 (1927). C a r l o n e , C., Spectrum o f the Hy d r o x y l R a d i c a l , Ph.D. T h e s i s , U n i v e r s i t y o f B r i t i s h Columbia, 1969. Condon, E.U., and S h o r t l e y , G.H., The Theory o f Atomic S p e c t r a , Cambridge U n i v . P r e s s , 1951. o Douglas, A.E., " A b s o r p t i o n o f OH i n the 1200 A Region", t o be p u b l i s h e d (1971). E d l e n , B., Kungl. Svenska. Vetenskapsakad. Handl. 3 , 20, No. 10 (1943). H a r r i s , F.E., J . Chem. Phys. £1, 4770 (1969). H a r t r e e , D.R., H a r t r e e , W., and S w i r l e s , B., P h i l . T r a n s . Roy. Soc. A 23_8, 2 2 9 (1940). Herzberg, G., The S p e c t r a and S t r u c t u r e s o f Simple F r e e R a d i c a l s , C o r n e l l U n i v . P r e s s , 1971. Joy, H.W., and P a r r , R.G., J . Chem. Phys. 28, 444 (1958). M i c h e l s , H.H., and H a r r i s , F.E., Chem. Phys. L e t t . 3, 441 (1969). P o w e l l , M.J.D., Comp. J o u r . 2> !955 (1964). Pryce, M.H.L., "Steps i n A n a l y s i s o f OH Spectrum, 53,000- 58,000 cm"-1", u n p u b l i s h e d notes (1971). S l a t e r , J.C., Quantum Theory o f Atomic S t r u c t u r e . V . l , M c G r a w - H i l l , I960. S w i t e n d i c k , A.C., and Corbato, F . J . , i n Methods o f Comp- u t a t i o n a l P h y s i c s . I I , Academic P r e s s , 1963. T e r z i a n , Y., and Scharlemann, E., E a r t h and T e r r . S c i . 1, 103 (1970). 91 APPENDIX 1 MATRIX ELEMENTS OF H BETWEEN SLATER DETERMINANTS The d i a g o n a l m a t r i x elements a r e f 3 k h 0 V V " h 0 > ( A l .1) / s ^ Z h ^ h s ^ V = ZK^'fhJ , (A1 . 2 ) and i s t he m o l e c u l a r o r b i t a l , but i n c l u d i n g a s p i n f a c t o r o r |J . The S l a t e r Determinant S^ i s an a n t i - symmetrized product of nine D O s . ^ }C [h-jj ^ means v ( l ) . (A1.4) The i n t e g r a t i o n i n (A1.4) i n c l u d e s an i n t e g r a l over s p i n as w e l l as spac e . ^DC^,*Xj|h2( " ^ ' ^ l ^ I s d e f i n e d by x d v ( l ) d v ( 2 ) . (A1 . 5 ) The i n t e g r a t i o n i n (A1 . 5 ) i s over s p i n s as w e l l as space. The volume element d v ( i ) i n (Al.4) and (A1 . 5 ) i s 2 A r ^ d r ^ s i n 92 93 I f ^ S-̂ , permute the m o l e c u l a r o r b i t a l s w i t h s p i n , OQ, i n S k and S-̂  so t h a t t h e r e i s the maximum p o s s i b l e c o i n c i d e n c e between the two. I f , a f t e r t h i s p ermutation, the two S l a t e r Determinants d i f f e r o n l y i n t h a t S, has P K a i n t he p o s i t i o n where S^ has 3Cc, then fsl h Q S xdV = 0, (A1.6) / S k < ? h i i ) ) S l d V - "^VlI^O » { A 1 ' 7 ) and I- The + (-) s i g n i n (A1.7) and (A1.8) i s t o be taken i f an even (odd) number o f permutations are r e q u i r e d t o b r i n g S^ and S-̂  i n t o maximum c o i n c i d e n c e . I f , a f t e r the perm u t a t i o n , S k and S-̂  d i f f e r i n two p o s i t i o n s , 3£a a n d Df̂  f o r Sk> a n d c o r r e s p o n d i n g l y and y' d  f o r S-j^, t h e n S k h 0 S l d V = °» ( A 1 - 9 ) f 5T h J ^ j S ^ V = 0, (ALIO) and 94 -<V a 'Dfb'!h2IX 0̂'>]- ( A L U ) The i n t e r p r e t a t i o n o f the - s i g n i n ( A l . l l ) i s the same as i n (A1.7) and (A1.8). I f and S-̂  d i f f e r by more than two 3£ , then the m a t r i x elements of H between them v a n i s h . The i n t e g r a t i o n s over s p i n i n ( A l . l ) t o ( A l . l l ) a re e a s i l y done. F o r example, i n an obvious n o t a t i o n , < X ' * j ' M O £ > - £ S i > S k £ 8 i<&.3f,|h 2 i : K k,2;> . (A1.12) APPENDIX 2 INTEGRALS OVER SLATER-TYPE ORBITALS The simplest case i s an overlap i n t e g r a l < ^ l l * 2 > - Sg . ^ m A d ^ . ^ j A U g . ^ J / A ^ n . f ) . ^ ^ x» ^ (A2.1) In (A2.1), the Slater-type o r b i t a l ^ i s s p e c i f i e d by r ^ , Jt^t mi and (See (4.1) and (4.3)). 5 , "n, and A(n, £ ) are defined by f = + ^ 2 ) , (A2.2) "n = J d ^ + n 2 ) , (A2.3) and . 1 / ( 2 ^ ) 2 n + 1 A ( n , f ) = 1/ . (A2.4) P V (2n)l The three one-electron i n t e g r a l s are the kinetic, energy i n t e g r a l , x ^ 4n(2n - 1) [ ^ 2 n l ( n l " 1 } " 2 ^ 2 n l n 2 + ^ i n 2 ( n 2 - l ) - 4 ( ^ 1 + l ) ( 2 f ) 2 | , (A2,5) the oxygen a t t r a c t i o n i n t e g r a l , 1 ^ 2 ' -*1,^2 *«l,m2 A 2 ( n - | , ^ ) (A2.6) 95 96 and the hydrogen a t t r a c t i o n i n t e g r a l , m^L.U\= _ £ A(n 1,^ 1)A(n ^ ) • l'Jf* - K | I ' 2A m-pin2 ^ 2 V j 2 n x 2 1 C k ( 4,m 1^ 2,m 2)[ ( p ) " ( k + 1 ) y ( 2 n + k + l , p ) k =0,2 , • • • l. r—• __ +( p ) k P (2n - k,f> ) .(A2.7) The C k i n (A2.7) are Condon-Shortley c o e f f i c i e n t s (Condon and Shortley, 19$1). ^ ,K and Pare defined by = ̂  R , (A2.8) 7T(a,x) = e ^ t a ~ 1 d t , (A2.9) r 'X 0 and oo P ( a , x ) = / e"* t a - 1 d t . (A2.10) x The one-centre two-electron i n t e g r a l i s given by A(n 1,| 1)A(n 2,^ 2 )A(n 3 ,^ 3)A(n 4,|' 4) x r C k ( / 1 , r a 1 ; ^ , m 3 ) C k ( - ^ , r a 4 ; i 2 , m 2 ) x Ic—0 £ [A(Ej-ik-i,!"]_)A(n 2+ik.tg Q " 2 I i _ ^ ( 2n2+k+l ,25^-k) +{A (rT 2-|k-4 ,|2) A ( n ^ k , )J"2I>^( 2n1+k+l, 2n 2-k )J, (A2.ll) 97 where I i = i ( | i + t 3 } » (A2.12) f 2 = ^ t V ^ ) * (A2.13) \ =^1/(ii+^) » (A2.14) ~1 = *( nl + n3 )» (A2.15) "n2 = J(n 2+n^), (A2.16) and I Y(p,q) is the Incomplete Beta function I ( p , q ) = _ 0 . (A2.17) f t P " 1 ( l - t ^ d t ^0 APPENDIX 3 c u l a t e d by m i n i m i z i n g the second lowest e i g e n v a l u e o f the H a m i l t o n i a n m a t r i x . T h i s procedure can produce erroneous r e s u l t s . The t r u e w a v e f u n c t i o n f o r t h i s s t a t e i s o r t h o - g o n a l t o the t r u e w a v e f u n c t i o n f o r the lowest s t a t e . The t r i a l w a v e f u n c t i o n f o r the upper s t a t e s h o u l d t h e r e f o r e be c o n s t r a i n e d t o be o r t h o g o n a l t o the t r u e w a v e f u n c t i o n f o r the lowest s t a t e . S i n c e the l a t t e r i s not known, t h i s c o n s t r a i n t cannot be e x a c t l y r e a l i z e d . i s o r t h o g o n a l t o an approximate w a v e f u n c t i o n f o r the lowest s t a t e , s i n c e t h e y a r e non-degenerate e i g e n v e c t o r s o f the same H a m i l t o n i a n m a t r i x . I f the l a t t e r w a v e f u n c t i o n i s a good a p p r o x i m a t i o n t o the t r u e w a v e f u n c t i o n , the p r o - cedure can be expected t o produce r e a s o n a b l e r e s u l t s . has not been checked. There i s , however, reason t o expect i t t o be f a i r . The lowest s t a t e w a v e f u n c t i o n depends t o a l a r g e degree on the parameters f o r o n l y the I s , 2s, 2p, ( l s ) H , and ( 2 p ) H atomic o r b i t a l s (see F i g u r e 8.11). Of t h e s e parameters, o n l y those f o r the 2p and ( l s ) H o r b i t a l s 2 However, the c a l c u l a t e d upper s t a t e w a v e f u n c t i o n The a c c u r a c y of t h i s approximate w a v e f u n c t i o n 98 99 25~~ were v a r i e d i n the second lowest C— c a l c u l a t i o n . The d i f f e r e n c e s between the parameters c a l c u l a t e d i n t h i s manner and those c a l c u l a t e d by m i n i m i z i n g the lowest e i g e n - v a l u e , w h i l e not i n s i g n i f i c a n t , a re not g r e a t (see F i g u r e s 8.3 and 8 . 7 ) . I t i s q u i t e p r o b a b l e , t h e n , t h a t the r e s u l t s o b t a i n e d by m i n i m i z i n g the second lowest e i g e n v a l u e a r e not w i l d l y i n a c c u r a t e .

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