UBC Theses and Dissertations

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UBC Theses and Dissertations

Study of the BH molecule Gagnon, Paul Joseph 1970

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A STUDY OF THE BH MOLECULE by . ' • •• PAUL JOSEPH GAGNON B . S c , S t . F r a n c i s X a v i e r U n i v e r s i t y , 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF M.Sc. i n t h e Department o f . . C h e m i s t r y We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA J u l y , 1970 In presenting th i s thes i s in pa r t i a l f u l f i lment o f the requirements fo r an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make i t f r ee l y ava i l ab le for reference and study. I fu r ther agree that permission for extensive copying of th i s thes i s fo r scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It i s understood that copying or pub l i ca t ion of th i s thes i s f o r f i nanc i a l gain sha l l not be allowed without my wr i t ten permission. Department of CHEtN\\STRY The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada Date MX i A b s t r a c t Antisymmetrized gerainal product wavefunctions w i t h s e v e r a l l i m i t e d b a s i s s e t s were c a l c u l a t e d f o r the ground s t a t e of BH a t R=2.329 atomic u n i t s . Extending these wavefunctions t o i n c l u d e Kapuy's " o n e - e l e c t r o n t r a n s f e r " c o n f i g u r a t i o n s r e s u l t e d i n a lowering i n energy of 'V/O.Ol atomic u n i t s . In our case, t h i s improvement accounts f o r more than 90% o f t h a t achieved by a f u l l c o n f i g u r a t i o n -i n t e r a c t i o n wavefunction. A " c o n t r a c t e d " double-zeta b a s i s s e t y i e l d e d the b e s t o v e r a l l energy. A 13-term, " s p l i t c o r e " , c o n f i g u r a t i o n - i n t e r a c t i o n wavefunction was developed and y i e l d e d an energy o f -25.14769 atomic u n i t s . T h i s wavefunction was then made to s a t i s f y the v i r i a l theorem. P a r r and White's method f o r one-point f o r c e constant c a l c u l a t i o n was a p p l i e d t o the s c a l e d wavefunction w i t h n e g a t i v e r e s u l t s . A s i m i l a r wavefunction was p a r t i a l l y o p t i m i z e d a t three i n t e r n u c l e a r d i s t a n c e s f o l l o w e d by s c a l i n g w i t h f i x e d R. V a r i o u s p a r a b o l i c models were used t o f i t the v i r i a l f o r c e s and e n e r g i e s c o r r e s p o n d i n g t o each R v a l u e . The f o r c e c o n s t a n t s k e c a l c u l a t e d from these models were u s u a l l y very good and the e f f e c t o f s c a l i n g was shown t o be important. P a r a b o l i c expansions i n 1/R gave b e t t e r r e s u l t s than i i p a rabolas i n R, compared to a q u i n t i c model and to experimental v a l u e s . i i i TABLE OF CONTENTS Page ABSTRACT i LIST OF TABLES AND FIGURES V ACKNOWLEDGMENT v i CHAPTER I INTRODUCTION 1 CHAPTER I I GEMINALS 3 2-1 Concept 3 2-2 Computational Procedure 6 2- 3 A p p l i c a t i o n t o BH 9 CHAPTER I I I EXTENDED GEMINALS 12 3- 1 Kapuy's Theory 12 3-2 Kapuy's A p p l i c a t i o n 17 3- 3 A p p l i c a t i o n t o BH 18 CHAPTER IV ONE-POINT CALCULATION OF k e 22 4- 1 Theory 22 4-2 S c a l i n g Procedure Used t o Obtain f a t R g 24 4-3 S e l e c t i o n o f Wavefunction 26 4- 4 S c a l i n g and Force Constant R e s u l t s 31 CHAPTER V VIRIAL SCALING 38 5- 1 T h o r h a l l s s o n and Chong's Approach 38 5-2 Wavefunctions and Data 39 5-3 Numerical A n a l y s i s 42 BIBLIOGRAPHY APPENDIX I APPENDIX II V LIST OF TABLES AND FIGURES TABLE Page I Geminal C a l c u l a t i o n s 20 II 13-Term Wavefunction 27 I I I Data from Ohno 28 IV Optimized Wavefunction 29 V Wavefunction ¥ T 0 S C ^2 VI O r b i t a l Exponents f o r 13-Term Wavefunctions 33 VII S c a l i n g R e s u l t s 35 V I I I Force Constants 36 IX Unsealed Data 40 X S c a l e d Data 41 XI Q u i n t i c Polynomial F i t 43 XII P a r a b o l i c Model R e s u l t s 45 XIII Comparison of Data 47 XIV BH Wavefunctions 48 XV Data from B a s i s Set II 56 XVI C o e f f i c i e n t s of 13-Term Wavefunctions 57 FIGURE I E versus a 54 I I E versus £(H) 55 v i ACKNOWLEDGMENT The author wishes t o thank Dr. D. P. Chong f o r h i s encouragement and h e l p f u l a d v i c e i n the p r e p a r a t i o n of t h i s t h e s i s . The author a l s o acknowledges the f i n a n c i a l support r e c e i v e d from the N a t i o n a l Research C o u n c i l . CHAPTER I INTRODUCTION The general purpose of t h i s work i s to develop and, or, tes t several methods of seeking information on the e l e c t r o n i c representation of molecular systems. A l l calculations involve the ground state of BH only, and a Born-Oppenheimer spinless Hamiltonian. Three main projects were undertaken: to confirm the usefulness of Kapuy's extended geminal theory 11, 2]; to perform a one-point c a l c u l a t i o n of k e (quadratic force constant) according to Parr and White [3]; and to seek a better approach to three-point c a l c u l a t i o n of k e. Kapuy applied his extended geminal theory to the ir-electrons of trans-butadiene with good res u l t s [2] . Our application of extended geminal theory to BH at R=2.329 atomic units [4] treats a l l electrons and several limited basis sets are t r i e d . P u l l configuration-interaction calculations were within reach and were used for comparison. The re s u l t s i n every case support those of Kapuy. In Chapter II the basic geminal theory i s presented, while the extended geminal calculations are described i n Chapter I I I . Parr and White [3] developed a purely kinetic-energy perturbation at R e which i s used to describe the molecular -2-p o t e n t i a l energy function of diatomic molecules. Using t h e i r method and experimental data they calculate the force constants k e and l e for many diatomic molecules with excellent r e s u l t s . Since t h e i r method requires only the kinetic-energy matrix elements of a given wavefunction at Re, they suggest that actual calculations be carr i e d out using li m i t e d basis sets of atomic o r b i t a l s . In Chapter IV we begin by seeking a suitable wavefunction on which to apply Parr and White's theory. S p l i t t i n g the core o r b i t a l on boron into two o r b i t a l s for Ohno's 13-term configuration-interaction wavefunction [5] gives a large improvement i n energy. A perturbation s c a l i n g procedure i s then applied to t h i s " s p l i t core" wavefunction before force constants are calculated. The re s u l t s are very poor and i t i s concluded that a much larger basis set i s required. In Chapter V a v i r i a l scaling procedure, si m i l a r to that used on LiH by Thorhallsson and Chong [6], i s applied to a configuration-interaction wavefunction for BH at three internuclear distances. Various parabolic models i n v i r i a l force and energy are investigated and values for the force constant k e are found to be good. Parabolic f i t s to scaled data are found to give better r e s u l t s than parabolic f i t s to unsealed data. Expansion of the diatomic molecular p o t e n t i a l -energy function i n 1/R rather than R led to better r e s u l t s i n most cases. C H A P T E R I I G E M I N A L S 2 - 1 C o n c e p t T h e H a r t r e e - F o c k e n e r g y i s u s u a l l y w i t h i n a b o u t 1 p e r c e n t o f t h e e x p e r i m e n t a l v a l u e . H o w e v e r , t o t a l e n e r g i e s a r e n o t o f m u c h u s e i n t h e m s e l v e s , o n e i s u s u a l l y i n t e r e s t e d i n e n e r g y d i f f e r e n c e s , s u c h a s t h e e n e r g y d i f f e r e n c e b e t w e e n t w o s p e c t r o s c o p i c s t a t e s . U n f o r t u n a t e l y , t h e s e e n e r g y d i f f e r -e n c e s a r e o f t e n n o l a r g e r t h a n a b o u t 1 p e r c e n t o f t h e t o t a l e n e r g y o f e i t h e r s t a t e . T h u s , t h e r e i s a g r e a t i n t e r e s t i n q u a n t u m - m e c h a n i c a l c a l c u l a t i o n s w h i c h g i v e b e t t e r e n e r g i e s t h a n t h e H a r t r e e - F o c k m e t h o d . C o r r e l a t i o n e r r o r i n t h e H a r t r e e - F o c k m e t h o d i s d u e t o t h e f a c t t h a t c o u l o m b i c i n t e r -a c t i o n b e t w e e n p a i r s o f e l e c t r o n s , e s p e c i a l l y e l e c t r o n s w i t h a n t i p a r a l l e l s p i n s , i s n o t p r o p e r l y a c c o u n t e d f o r . E l e c t r o n s o f p a r a l l e l s p i n s a r e k e p t a p a r t b y t h e a n t i s y m m e t r y p r i n c i p l e a n d a r e t h u s d e s c r i b e d b e t t e r t h a n e l e c t r o n s o f a n t i p a r a l l e l s p i n . E l e c t r o n i c c o r r e l a t i o n i s f o u n d t o b e o f f u n d a m e n t a l i m p o r t a n c e i n c h e m i c a l b i n d i n g . A l l o n e - e l e c t r o n t r e a t m e n t s u s e d i n m o l e c u l a r q u a n t u m m e c h a n i c s a r e i n h e r e n t l y t o o i n a c c u r a t e . E l e c t r o n i c c o r r e l a t i o n i s u s u a l l y i n t r o d u c e d b y m e a n s o f a s u p e r p o s i t i o n o f c o n f i g u r a t i o n s . -4-Although t h i s method can lead to r e s u l t s of any desired accuracy, the wavefunctions become extremely complicated even for r e l a t i v e l y simple molecules. Also, as the complexity of the wavefunction increases i t becomes more and more d i f f i c u l t to assign any physical s i g n i f i c a n c e to the importance of any given configuration. The electron pair i s conceptually a t t r a c t i v e to chemists. Molecular e l e c t r o n i c structure i s usually described i n terms of inner s h e l l s , bond pairs and lone p a i r s . It has also been shown that the most important c o r r e l a t i o n e f f e c t s are those involving a p a i r of electrons at a time [7, 8 ] , There-fore i t seems reasonable to construct a method which uses two-electron functions instead of one-electron functions ( o r b i t a l s ) . Such electron-pair functions could emphasize i n t r a p a i r c o r r e l a t i o n and have l i t t l e i n t e r p a i r c o r r e l a t i o n . Correlation can e a s i l y be admitted within each pair function. The natural tendency to make the p a i r functions correspond to d i s t i n c t bond p a i r s , etc., allows the pair functions to be r e l a t i v e l y separated and highly l o c a l i z e d . For example two bonds at opposite ends of a long molecule or the sigma and p i bonds i n a double bond, are cases i n which, to a very good approximation, the s p a t i a l parts of the wavefunction describing d i f f e r e n t bonds do not overlap. The high degree of invariance of bond properties from molecule to molecule suggests that pair-functions w i l l be -5-t r a n s f e r a b l e f r o m o n e m o l e c u l e t o a n o t h e r t o a g o o d a p p r o x i m a t i o n [ 9 ] . S u c h t r a n s f e r a b i l i t y w o u l d g r e a t l y s i m p l i f y t h e c o n s t r u c t i o n o f g o o d w a v e f u n c t i o n s f o r l a r g e m o l e c u l e s , b e c a u s e o n e c o u l d u s e t h e r e s u l t s o b t a i n e d f r o m s i m p l e r s y s t e m s . D e v e l o p m e n t o f g o o d p a i r - f u n c t i o n s f o r o n l y a f e w o f t h e m o r e common b o n d s , s u c h a s C - C o r C - H , w o u l d a l l o w c a l c u l a t i o n s o n l a r g e s y s t e m s t o b e c a r r i e d o u t w i t h m u c h g r e a t e r a c c u r a c y a n d s i m p l i c i t y t h a n i s p o s s i b l e a t p r e s e n t . T h e i d e a o f p a i r f u n c t i o n s seems t o h a v e b e e n o r i g i n a t e d b y P a u l i n g [ 1 0 ] . U s u a l l y p a i r f u n c t i o n s a r e c a l l e d g e m i n a l s . A c t u a l l y , g e m i n a l t h e o r y ( o r " s e p a r a t e d p a i r t h e o r y " ) i s a p a r t i c u l a r c a s e o f t h e s e l f - c o n s i s t e n t g r o u p a p p r o a c h . ( S e e , f o r e x a m p l e , R e f s . [ 1 1 , 1 2 , 1 3 ] . ) A n e l e c t r o n g r o u p f u n c t i o n w o u l d d e s c r i b e a d i s t i n c t m a n y - e l e c t r o n g r o u p i n a m o l e c u l e , s u c h a s a i t - e l e c t r o n s y s t e m . I n t h e g e m i n a l a p p r o a c h a m o l e c u l a r w a v e f u n c t i o n i s w r i t t e n a s a n a n t i s y m m e t r i z e d p r o d u c t o f l o c a l i z e d g e m i n a l s . The s e p a r a t e d n a t u r e o f i n d i v i d u a l g e m i n a l s i s m a i n t a i n e d b y t h e f a c t t h a t i f a l l g e m i n a l s a r e l i n e a r i l y e x p r e s s e d i n t e r m s o f S l a t e r d e t e r m i n -a n t s b u i l t f r o m o r t h o g o n a l o n e - e l e c t r o n s p i n o r b i t a l s , n o s p i n o r b i t a l e n t e r s t h e d e s c r i p t i o n o f m o r e t h a n o n e g e m i n a l . T h e e f f e c t o f o n e g e m i n a l u p o n a n o t h e r i s t a k e n i n t o a c c o u n t i n a " s e l f - c o n s i s t e n t " m a n n e r , u s i n g a n ' . i t e r a t i v e m e t h o d a n a l o g o u s t o t h a t e m p l o y e d i n t h e s t a n d a r d SCF a p p r o a c h [ 1 4 ] . -6-2-2 Computational Procedure The following treatment i s according to Parks and Parr [12]. The molecular wavefunction for a 2n-electron system i s written (1,2)* (3,4)* (5,6)...* (2n-l,2n)], (2-1) a b c m where each geminal I|K i s an antisymmetric function of the space and spin coordinates of the two electrons involved. The square brackets represent the normalized p a r t i a l a n t i -symmetrization operator which generates a completely a n t i -symmetric Y from the simple products of the i n d i v i d u a l antisymmetric The geminals are well behaved and normalized to unity: 7/1^(1,2) | 2 d t 1 d x 2 = l for a l l i . (2-2) There i s a complete set of spin o r b i t a l s which can be par t i t i o n e d into subsets r a l , r a 2 , r a 3 , r b l / rb2' rb3' r i i * r i 2 ' *"* s u c ^ t n a t t n e geminal i|>^  may be expressed i n terms of Slater determinants b u i l t from the subset i o r b i t a l s only, for a l l i . That i s , V C i l * i l + C i 2 * i 2 + " - ' ( 2 - 3 ) where the C^j are constants and the <|>^j are Slater determinants b u i l t from the spin o r b i t a l s i n subset i . Geminals defined i n the above manner are mutually orthogonal, / * J ( l , 2 ) * j (1,4)^=0 for i ji j , (2-4) - 7 -a n d / / * * ( 1 , 2 ) ^ ( l f 2 ) d T 1 d x 2 = 0 f o r i ? j . ( 2 - 5 ) T h e s e o r t h o g o n a l i t y r e l a t i o n s g r e a t l y s i m p l i f y t h e c a l c u l a t i o n s . A l s o , f r o m t h e o r t h o g o n a l i t y r e l a t i o n s a n d t h e n o r m a l i z a t i o n o f i n d i v i d u a l g e m i n a l s , i t f o l l o w s t h a t t h e t o t a l w a v e f u n c t i o n ¥ i s n o r m a l i z e d . G e m i n a l s , a s d e f i n e d s o f a r , s a t i s f y t h e c o n d i t i o n s o f s e p a r a b i l i t y f o r e l e c t r o n p a i r s . The e l e c t r o n i c H a m i l t o n i a n o f a 2 n - e l e c t r o n s y s t e m o f f i x e d n u c l e i a w i t h c h a r g e s Z a e may b e w r i t t e n 2 n 2 n H ( l , 2 , . . . 2 n ) = 2 | H N ( 5 ) + l / 2 2 < e 2 / r P n ) , ( 2 - 6 ) 5^1 5,n=i ^ w h e r e H N ( S ) = T U ) + U N ( S ) , ( 2 - 7 ) U N ( e ) = - ^ a ( Z a e 2 / r ^ a ) . ( 2 - 8 ) U N ( C ) g i v e s t h e p o t e n t i a l e n e r g y o f a t t r a c t i o n b e t w e e n e l e c t r o n £ a n d t h e b a r e n u c l e i a n d T ( £ ) i s t h e k i n e t i c e n e r g y o p e r a t o r f o r e l e c t r o n £ i f t h e o t h e r e l e c t r o n s w e r e a b s e n t . F o r a " s y s t e m d e s c r i b e d b y t h e w a v e f u n c t i o n d i s c u s s e d e a r l i e r , t h e e x p e c t a t i o n v a l u e f o r t h e t o t a l e l e c t r o n i c e n e r g y w o u l d b e E=2?±i±+1/27?J~!$L ( J i j - K i j ) , ( 2 - 9 ) w h e r e t h e sums a r e o v e r t h e d i s t i n c t e l e c t r o n p a i r s a , b , . . . m . T h e q u a n t i t y 1^ i s t h e e l e c t r o n i c e n e r g y e l e c t r o n p a i r i w o u l d h a v e i f t h e o t h e r p a i r s w e r e a b s e n t . -8-I i = / / * * ( l , 2 ) H ° ( l , 2 ) * i ( l , 2 ) 0 1 x ^ 2 , (2-10) where H ° ( l , 2 ) E H N ( l ) + H N ( 2 ) + ( e 2 / r 1 2 ) • (2-11) The q u a n t i t y J ^ j i s t h e t o t a l Coulomb r e p u l s i o n between e l e c t r o n p a i r i and e l e c t r o n p a i r j : 3i±=JSft**(1,2)**(3,4) 1 3 1 3 X [ ( e 2 / r 1 3 ) + ( e 2 / r 1 4 ) + ( e 2 / r 2 3 ) + ( e 2 / r 2 4 ) ] X * i ( l , 2 ) * j ( 3 f 4 ) d T 1 d T 2 d T 3 d T 4 . (2-12) The K-LJ i s a c o r r e s p o n d i n g exchange r e p u l s i o n : K i j = / / / / * J ( l r 2 ) * j ( 3 , 4 ) .X t ( e 2 / r 1 3 ) * i ( 3 , 2 ) * j (1,4) + ( e 2 / r 1 4 ) * i ( 4 , 2 ) * . ( 3 , l ) + ( e 2 / r 2 3 ) * i ( l , 3 ) * j ( 2 , 4 ) + ( e 2 / r 2 4 ) * i ( l , 4 ) i | > ; . (3,2) ] d T 1 d T 2 d x 3 d T 4 . (2-13) T h e r e a r e two methods f o r m i n i m i z a t i o n o f t h e t o t a l e l e c t r o n i c e n e r g y o f a system o f g e m i n a l s . The f i r s t method s t a r t s w i t h some s p e c i f i c s e t o f o r t h o n o r m a l o n e - e l e c t r o n s p i n o r b i t a l s r p a r t i t i o n e d i n t o n o n o v e r l a p p i n g s u b s e t s r a , r b , . . . r m , i n some s p e c i f i c way, t h e n one seeks t h e b e s t f u n c t i o n [ * a * b . • • ip m l • To do t h i s , one f i x e s t h e form o f a l l g e m i n a l s e x c e p t o n e , say and t h e n m i n i m i z e s e n e r g y o f t h i s g e m i n a l . T h i s may be done b e c a u s e E q . (2-9) can be r e w r i t t e n E ^ E m - i + E i , (2-14) where Em_£ i s t h e t o t a l e l e c t r o n i c e n e r g y o f a l l t h e g e m i n a l p a i r s b u t i , i n c l u d i n g t h e i r i n t e r a c t i o n s . Then one s e l e c t s -9-another geminal, and v a r i e s the new geminal u n t i l i t s energy i s minimized. T h i s procedure i s repeated u n t i l no f u r t h e r improvements are o b t a i n e d . The second method i s more g e n e r a l and s t a r t s by seeking the b e s t s e t of o n e - e l e c t r o n f u n c t i o n s and the bes t p a r t i t i o n -i n g o f them. The f i r s t method i s used i n t h i s work but no i t e r a t i v e procedure i s r e q u i r e d s i n c e t h e r e i s o n l y one geminal, the bond geminal, which i s not of f i x e d form. The a c t u a l working equations f o r geminal c a l c u l a t i o n s , as presented by Parks and Pa r r [12], are i n an o p e r a t o r r e p r e s e n t a t i o n . K l e s s i n g e r and McWeeny [13] g i v e an e x a c t l y e q u i v a l e n t treatment of geminal theory; o n l y they use a d e n s i t y m a t r i x approach. Kapuy [1] d e s c r i b e s g e n e r a l i z e d geminal theory and a l s o uses p e r t u r b a t i o n techniques t o develop a method to extend geminal theory. 2-3 A p p l i c a t i o n t o BH Our treatment of BH i n v o l v e d f o u r b a s i s s e t s , each s e t c o n t a i n i n g f o u r o r b i t a l s . The o r b i t a l s a r e : k (Is on boron), s_ (2s on boron) , £(2pa on boron) , and h (Is on hydrogen) . In b a s i s s e t I, the o r b i t a l s on boron are the s i n g l e - z e t a S l a t e r - t y p e o r b i t a l s of Clementi and Raimondi [15], but the o r b i t a l exponent 5 of h ( a l s o a s i n g l e - z e t a S l a t e r - t y p e o r b i t a l ) i s o p t i m i z e d t o g i v e the lowest energy f o r the -10-antisymmetrized geminal product (AGP) wavefunction. B a s i s s e t I I i s l i k e s e t I, but s_ has been Schmidt o r t h o g o n a l i z e d to k. In s e t I I I , £(h)=l and the o r b i t a l s on boron are " c o n t r a c t i o n s " of the d o u b l e - z e t a s e t of Clementi [16, 171, t h a t i s , w i t h f i x e d l i n e a r c o e f f i c i e n t s [17]. B a s i s s e t IV i s i d e n t i c a l t o s e t I I I w i t h the e x c e p t i o n of h, the exponent of which i s o p t i m i z e d as i n s e t I. Two h y b r i d s were formed from s_ and a: 2 V 2 b = as + a , (2-15) n = ( 1 - a 2 ) 1 / 2 s - aa , (2-16) where the h y b r i d i z a t i o n parameter a i s a l s o o p t i m i z e d t o g i v e the b e s t energy f o r the AGP wavefunction*. The h y b r i d b i s bonding and n i s nonbonding. Next, the o r b i t a l s k, b, n, and h are symmetricly o r t h o n o r m a l i z e d i n t o K, B, N and H r e s p e c t i v e l y . The geminals f o r the core and lone p a i r s are simply ^!=(KK), (2-17) T|)2=(NN), (2-18) where the symbols r e p r e s e n t S l a t e r determinants. The bond p a i r i s d e s c r i b e d by: <J'3=C1(BB)+C2 [ (BH) + (HB) ]+C 3(HH). (2-19) Now the AGP wavefunction i s w r i t t e n f A G p=[i(» 1(l,2)T|> 2(3 f4)* 3(5,6)] . (2-20) * O p t i m i z a t i o n of a i s c a r r i e d out f o l l o w i n g any r e q u i r e d o p t i m i z a t i o n of the hydrogen Is exponent. -11-The form of geminals a n < ^ * 2 ^ s a l r e a d y f i x e d and o n l y geminal ty^ must be v a r i e d t o minimize the energy, and t h e r e f o r e no i t e r a t i o n procedure i s r e q u i r e d . T h i s p a r t i c u l a r AGP wavefunction i s e q u i v a l e n t t o a 3-term l i m i t e d c o n f i g u r a t i o n -i n t e r a c t i o n (CI) wavefunction with the c o n f i g u r a t i o n s (j)1= (KKNNBH) + (KKNNHB) , (2-21) <J>2= (KKNNBB) , (2-22) 4>3= (KKNNHH) . (2-23) In order to d e s c r i b e the methods g e n e r a l l y used i n the a c t u a l c a l c u l a t i o n s a sample geminal c a l c u l a t i o n i s presented i n Appendix I. The r e s u l t s of c a l c u l a t i o n s u s i n g V^Q-p are presented i n Chapter I I I f o r comparison w i t h the r e s u l t s of extended geminal and f u l l CI c a l c u l a t i o n s . CHAPTER I I I EXTENDED GEMINALS 3-1 Kapuy's Theory The best geminal wavefunction ¥ B G can be expressed as a l i n e a r combination of antisymmetrized geminal products Y.. <FR r = A., (3-1) B G i f j , . . . 1 1D...1 1 3 . . . 1 where the A,- -1 are numerical c o e f f i c i e n t s and ¥. . -, X J • • • X , X J • • • X c o n t a i n s a geminal from each subset, the geminal being d e s i g n a t e d by a s u b s c r i p t and the subset being d e s i g n a t e d by the p o s i t i o n of the s u b s c r i p t . Our BH ground s t a t e wave-f u n c t i o n V^GP C O U x d k e w r i t t e n as and i f a l l of the s u b s c r i p t s were not equal to 1 i t would r e p r e s e n t some e x c i t e d c o n f i g u r a t i o n of the system. The wavefunction *F B G and the corresponding energy E ( B G ) c o u l d be determined by s o l v i n g the s e c u l a r equation '"BG-^BG)1! - °' ( 3 ~ 2 ) where H B G i s the matrix of the t o t a l Hamiltonian o p e r a t o r i n the • -. r e p r e s e n t a t i o n . F o r t u n a t e l y , due t o the x j ... x p r o p e r t i e s of the geminals, nonvanishing matrix elements occur o n l y between geminal products * i j t # < x a n ^ 1 d i f f e r i n g i n not more than two s u b s c r i p t s . -13-Kapuy [1] i n t r o d u c e s p a r t i c l e number o p e r a t o r s N K i n each of the N subspaces (subsets) 2N N K = XL 21 /dcxr„Y (a) r * (a) , (3-3) 1£l X K X K X where r R ^ i s one of the o r b i t a l s used i n the Kth geminal. A l l such o p e r a t o r s commute w i t h each other and w i t h the t o t a l p a r t i c l e number operator«/C=SN k. I t can be shown t h a t a l l antisymmetrized geminal products and any a r b i t r a r y l i n e a r combination of them are e i g e n f u n c t i o n s of the N K w i t h e i g e n v a l u e s Nj=N 2=. . .=NN=2. Since none of the N K commutes wi t h the Hamiltonian H which c o n t a i n s one- and t w o - p a r t i c l e o p e r a t o r s , the N K and H cannot have simultaneous e i g e n s t a t e s . The r e s u l t i s t h a t cannot be the exact e i g e n s t a t e of the system. I t i s j u s t the b e s t p o s s i b l e approximate wave-f u n c t i o n w i t h the c o n s t r a i n t t h a t i t be an e i g e n f u n c t i o n of a l l of the N K which belong to the e i g e n v a l u e 2. I f the s e t {r} i s complete the t o t a l p a r t i c l e number operatort^/ 9 commutes wit h the Hamiltonian and they have simultaneous e i g e n s t a t e s . Any e i g e n s t a t e of H can be expanded i n terms of a l l l i n e a r l y independent e i g e n s t a t e s which belong t o the eigenvalue 2N. The s e t of the ^ i j . , . 1 comprises o n l y a p a r t o f these e i g e n s t a t e s o f « ^ a n d should be completed w i t h i t s "orthogonal complement". Since two e i g e n s t a t e s of ^ c o r r e s p o n d i n g t o d i f f e r e n t p a r t i t i o n s of the o c c u p a t i o n numbers N-^ , N 2, ... N N are a u t o m a t i c a l l y - 1 4 -o r t h o g o n a l , t h e o r t h o g o n a l c o m p l e m e n t c a n c o n s i s t o f a l l l i n e a r l y i n d e p e n d e n t e i g e n s t a t e s of</jf*which c o r r e s p o n d t o a l l p o s s i b l e p a r t i t i o n s o f t h e s e t N ^ , N 2 , . . . N n , 2 n K = 2 N , e x c e p t f o r w h i c h 1 ^ = ^ = . . . = N N = 2 . T h e p o s s i b l e e i g e n v a l u e s o f N K a r e 0 , 1, 2 , . . . 2 N , w h e n n R > 2 N ; o r 0 , 1, 2, . . . n R , w h e n n K £ 2 N , w h e r e n K i s t h e n u m b e r o f o n e - e l e c t r o n f u n c t i o n s i n s u b s p a c e K . T o a g i v e n p a r t i t i o n t h e r e e x i s t T T ^ N ^ l i n e a r l y i n d e p e n d e n t e i g e n s t a t e s o f e ^ ° w h i c h c a n b e o r t h o g o n a l i z e d . F o r s u i t a b l e b u i l d i n g b l o c k s t h e g r o u p f u n c t i o n s 4*Nvk K a r e i n t r o d u c e d a n d d e f i n e d a s . K k * N K k = < N K * > - l /22: ( " l ) P P x < ^ . > < v ^ . . v r K X ( 1 ) r K X < 2 ) - - - r K v ( N K ) ' ( 3 - 4 ) w h e r e N R = 0 r 1, 2, . . . n K ; k = l , 2 , .. '^N^ 7 X ' ^ ' *•* V = X ' 2 ' NK' G r o u p f u n c t i o n s b e l o n g i n g t o d i f f e r e n t s u b s p a c e s a r e o r t h o g o n a l i n t h e s t r o n g s e n s e a n d w i t h i n e v e r y s u b s p a c e n o r m a l i z e d a n d m u t u a l l y o r t h o g o n a l i n t h e u s u a l s e n s e : / ^ N K k i ' N K l d l d 2 - - - d N K = 6 k l - ( 3 - 5 ) T h e a n t i s y m m e t r i z e d p r o d u c t f u n c t i o n s i j . . . l s _ / N 1 ! N 2 ! ' - ' N N ! N 1 / 2 — P S l N 2 - ' . ' % ' \ (2N)1 / ^ P * N l i * N 2 j ' = 1 , 2 , . . . ( ^ , j = i , 2 , . . . ^ , . . . 1 = 1 , 2 , . . . ^ , 5V2N' ( 3 - 6 ) / n l \ A _ ^ n 0 \ J _ /n^v KT» i -c o n t a i n i n g o n e g r o u p f u n c t i o n f r o m e a c h o f t h e N s u b s p a c e s , f o r m a c o m p l e t e s e t s u c h t h a t t h e e x a c t w a v e f u n c t i o n o f t h e s y s t e m T c a n b e e x p a n d e d i n t e r m s o f t h e m . -15-I n o r d e r t h a t a l l t h e s t a t e s i n E q . (3-6) may b e d e r i v e d f r o m t h e g e m i n a l g r o u n d s t a t e K a p u y i n t r o d u c e s t h e i d e a s o f " s i m p l e e x c i t a t i o n " a n d " e l e c t r o n t r a n s f e r e x c i t a t i o n " . S i m p l e e x c i t a t i o n s u b s t i t u t e s o n e o f t h e g r o u p f u n c t i o n s i n ¥ b y a n o t h e r b e l o n g i n g t o t h e same s u b s p a c e : * -*• ijT , k ^ l . E l e c t r o n t r a n s f e r e x c i t a t i o n t r a n s f e r s o n e e l e c t r o n f r o m o n e o f t h e s u b s p a c e s t o a n o t h e r , c h a n g i n g t w o g r o u p f u n c t i o n s s i m u l t a n e o u s l y : 5jT . t"T . -*• , * . . TINK I ^ D N K - l ' k NL+1'X Now t h e e x a c t w a v e f u n c t i o n ¥ c a n b e w r i t t e n a s a s e r i e s o f a n t i s y m m e t r i z e d p r o d u c t f u n c t i o n s d e s c r i b e d i n E q . (3-6) w i t h t h e t e r m s g r o u p e d a c c o r d i n g t o t h e m i n i m u m n u m b e r o f e l e m e n t a r y e x c i t a t i o n s ( s i m p l e + e l e c t r o n t r a n s f e r ) n e e d e d t o r e a c h t h e a c t u a l t e r m f r o m t h e g r o u n d s t a t e ^ ¥ = c < ° > f 1 1 < - < 1 - g + 2CU>T<1> + Sc< 2>¥< 2> + ... +Z,c< 2 N- 2>y< 2 N- 2 ). (3-7) T h e K , N j r ^ r a n d N R>2 a r e s t i l l u n s p e c i f i e d s o f a r , t h e y may b e d e t e r m i n e d i n p r i n c i p l e , b u t f o r s i m p l i c i t y t h e y a r e i d e n t i f i e d w i t h t h e c o r r e s p o n d i n g r R ^ a n d * K k (1, 2, . . . N K ) r e s p e c t i v e l y , w h e r e * K k ( l » 2 , . . . . N K ) = - ( N K ! ) l / 2 2. ( - i ) P P r K X ( l ) r K X ( 2 ) . . . r R v ( N K ) , (3-8) a n d w h e r e X<A<I. ,<v a n d X , X , . . , v = l , 2 , . . . n K . I n E q . (3-7) i t i s t h e t e r m s o f t h e f i r s t a n d s e c o n d s u m m a t i o n s w h i c h may h a v e n o n v a n i s h i n g m a t r i x e l e m e n t s w i t h ' • ' j ^ 1« A l i s t o f t h e m i s g i v e n b e l o w i n w h i c h o n l y t h e -16-f a c t o r s b e i n g changed under e x c i t a t i o n are shown. A l l c o n f i g u r a t i o n s except those due to one simple e x c i t a t i o n may have nonvanishing m a t r i x elements w i t h ^ l i # > . l « F i r s t summation: one simple e x c i t a t i o n * K l ( l , 2 ) - 4>Kk(lr2) f o r a l l K and k ( k ^ l ) ; (3-9) one e l e c t r o n t r a n s f e r e x c i t a t i o n * K l ( i , 2 ) i | > L l ( 3 , 4 ) + r K X ( l ) i | > L l (2,3,4) , (3-10) f o r a l l K, L ( K ^ L ) , X and 1; second summation: two simple e x c i t a t i o n s iJ> K l<i,2)!ji L l(3,4) •* ^ ( 1 , 2 ) ^ ( 3 , 4 ) , f o r a l l K<L, k and l ( k , l ^ i ) ; (3-11) one simple + one e l e c t r o n t r a n s f e r e x c i t a t i o n s ^ K l ( l , 2 ) ^ L l ( 3 , 4 ) ^ M l ( 5 , 6 ) * r K X-(l)* L | l(2,3,4)i|» M i n(5,6), f o r a l l K, L, M(K^L^M), X, 1 and m(m^i); (3-12) two e l e c t r o n t r a n s f e r e x c i t a t i o n s i P K ± (1,2) tP L l (3,4) - i | i L l ( i , 2 , 3 , 4 ) , f o r a l l K, L(K^L) and 1; (3-13) ^ ( 1 , 2 ) ^ ( 3 , 4 ) ^ ( 5 , 6 ) + ^ ( 1 , 3 , 4 ) ^ ( 2 , 5 , 6 ) , f o r a l l K, L<M(K^L, M), 1 and m; (3-14) - 1 7 -* K l ( 1 . 2 ) * L l ( 3 , 4 ) * M l ( 5 , 6 ) * ^ ( 1 , ^ ( 3 1 * ^ ( 2 , 4 , 5 , 6 ) , f o r a l l K<L, M(M^K,L), X, X and m; (3-15) ^ ( 1 , 2 ) ^ ( 3 , 4 ) ^ ( 5 , 6 ) ^ ( 7 , 8 ) - ^ ( 1 ) 1 ^ ( 2 , 3 , 4 ) ^ ( 5 ) ^ ( 6 , 7 , 8 ) , f o r a l l J<L, K<M(J,L^K,M), X/ k, X and m. ( 3 - 1 6 ) In p r a c t i c a l c a l c u l a t i o n s the s e t {r} ob t a i n e d by determining the b e s t p o s s i b l e geminal product ^ i s a 11 • • • l f i n i t e one of course, but f o r systems w i t h N l a r g e i t i s necessary t o terminate the s e r i e s or use p e r t u r b a t i o n theory. Kapuy used Rayleigh-Schrfldinger p e r t u r b a t i o n theory upfeto second order i n h i s work. His r e s u l t s show t h a t the most important c o r r e c t i o n s t o the energy o f Y.. . come from one-•L X • • • J-e l e c t r o n t r a n s f e r c o n f i g u r a t i o n s (Eq. ( 3 - l 0 ) ) . 3 - 2 Kapuy's A p p l i c a t i o n Kapuy [2] a p p l i e d h i s extended geminal theory t o the T T-electrons o f t r a n s - b u t a d i e n e . The c a l c u l a t i o n s i n v o l v e d the f o l l o w i n g approximations: a - i r s e p a r a b i l i t y , second order p e r t u r b a t i o n theory and a Goeppert-Mayer and S k l a r H a m i l t o n i a n . Four SCF o r b i t a l s were used as a s t a r t i n g p o i n t and the same o r b i t a l s were used i n a complete CI c a l c u l a t i o n by Nesbet the r e s u l t of which i s taken as a standard (100%) to which a l l r e s u l t s are r e f e r r e d . The antisymmetrized geminal product wavefunction accounts f o r 9 3 . 1 % of the t o t a l c o r r e l a t i o n energy. The t o t a l c o n t r i b u t i o n of a l l con-f i g u r a t i o n s due t o two simple e x c i t a t i o n s i s 1 . 9 % . "One-electron -18-t r a n s f e r " c o n f i g u r a t i o n s make an important c o n t r i b u t i o n of 6.2% making the t o t a l c o r r e l a t i o n 101.3% of the standard v a l u e . The importance of " o n e - e l e c t r o n t r a n s f e r " c o n f i g u r a -t i o n s confirms Kapuy's p r e v i o u s estimate [1] of the magnitude of t h e i r c o n t r i b u t i o n i n the case of the Be atom. 3-3 A p p l i c a t i o n t o BH "One-electron t r a n s f e r " c o n f i g u r a t i o n s have th r e e s p i n o r b i t a l s from one subset and one s p i n o r b i t a l from another subset. In our case the subsets are K, K; N, N; B, H r B, H. These c o n f i g u r a t i o n s are i n c l u d e d i n the extended geminal c a l c u l a t i o n s l a b e l l e d G+l (geminals p l u s o n e - e l e c t r o n t r a n s f e r c o n f i g u r a t i o n s ) . F o r our simple example, ^Q+I i s e q u i v a l e n t t o a 7-term l i m i t e d CI wavefunction w i t h the c o n f i g u r a t i o n s fy^, <J>2 and <f>3 (see Eqs. (2-21) t o (2-23)) p l u s <j>4 = (KKHHNB) + (KKHHBN) , (3-17) <J>5 = (KKBBNH) + (KKBBHN), (3-18) $ c = (NNBBKH) + (NNBBHK), (3-19) D 4>7 = (NNHHKB) + (NNHHBK). (3-20) Because of the s m a l l b a s i s s e t s used, f u l l CI c a l c u l a -t i o n s can be e a s i l y performed. In a d d i t i o n t o <j>^  t o <f>^ , the on l y o t h e r c o n f i g u r a t i o n s i n are <j>8 = (KKBBHH) , (3-21) <t>9 = (NNBBHH) , (3-22) <J>10= (BBHHKN) + (BBHHNK) . (3-23) -19-These correspond to "two-electron t r a n s f e r " c o n f i g u r a t i o n s i n extended geminal terminology. In t h i s case Y C I i s e q u i v a l e n t t o ^ G + I + 2 ( 9 e m i n a l s p l u s one and t w o - e l e c t r o n t r a n s f e r c o n f i g u r a t i o n s ) . The r e s u l t s of AGP, G+l and f u l l CI c a l c u l a t i o n s w i t h the f o u r b a s i s s e t s are summarized i n Table I. The d i f f e r e n c e E (AGP) ~ E ( C I ) •"-s t ^ i e m a x i m u m improvement we o b t a i n f o r the p a r t i c u l a r b a s i s s e t ; and f r e p r e s e n t s the f r a c t i o n of t h i s maximum.we o b t a i n when we i n c l u d e o n l y " o n e - e l e c t r o n t r a n s f e r " c o n f i g u r a t i o n s . The e n e r g i e s are i n atomic u n i t s . Although s m a l l b a s i s s e t s and very l i t t l e o p t i m i z a t i o n have been used i n order to save computer time, the CI energy from b a s i s s e t IV i s q u i t e r e s p e c t a b l e compared to o t h e r c a l c u l a t i o n s on BH. (See compendia by Cade and Huo [18] and by Krauss [19].) However, the main purpose of t h i s work i s not to o b t a i n an e x c e l l e n t wavefunction and energy f o r the system but t o support Kapuy's extended geminal theory w i t h a b e t t e r example. T h i s i s achieved by the f a c t t h a t the f r a c t i o n f i n Table I i s over 0.9 w i t h a l l f o u r b a s i s s e t s used. T h i s seems to i n d i c a t e t h a t when one uses more extended b a s i s s e t s and f u l l CI becomes unmanageable the extended geminal wavefunction ^ G + i , i n c l u d i n g o n l y "one-e l e c t r o n t r a n s f e r " c o n f i g u r a t i o n s , may be s u f f i c i e n t . The r e s u l t s i n T a b l e I show t h a t the d ouble-zeta b a s i s s e t o f Clementi [16, 17], even w i t h c o n t r a c t i o n ( f i x e d l i n e a r -20-TABLE I GEMINAL CALCULATIONS A l l c a l c u l a t i o n s are c a r r i e d out a t an i n t e r n u c l e a r d i s t a n c e R of 2.329 a.u. E n e r g i e s are i n atomic u n i t s . B a s i s Set I I I I I I IV S ( H ) a 1. 26 1.31 (1.0) 1.362 ct a 0. 47 0.50 0.52 0.507 E(AGP) -25. 07331 -25.07601 -25.06673 -25.10324 E(G+1) -25. 08589 -25.08541 -25.08971 -25.11575 E ( C I ) -25. 08683 -25.08627 -25.09116 -25.11634 f 0. 930 0.916 0.941 0.955 E (Experimental) = -25.29 a.u. a Optimized f o r minimum E ( A G P ) , except the v a l u e i n parentheses. b Ref. [18]. -21-c o e f f i c i e n t s [17]) and without f u r t h e r exponent o p t i m i z a t i o n , i s capable of g i v i n g s i g n i f i c a n t improvement over a s i n g l e -z e t a b a s i s s e t . For example, a f u l l CI with an op t i m i z e d s i n g l e - z e t a s e t g i v e s an energy of -25.09034 a.u. f o r BH [20] compared t o our -25.11634 a.u. T h i s means t h a t , f o r some systems, c o n t r a c t e d d o u b l e - z e t a b a s i s s e t s may be more economical as w e l l as capable of g i v i n g lower e n e r g i e s . CHAPTER IV ONE-POINT CALCULATION OF k e 4-1 Theory The approach used i s t h a t o f Pa r r and White -[3] with some of the terms r e d e f i n e d . The Born-Oppenheimer e l e c t r o n i c Hamiltonian f o r a di a t o m i c molecule may be w r i t t e n i n the form , H = R~2 t + R" 1 v, (4-1) where R i s the i n t e r n u c l e a r d i s t a n c e , and t and v are the k i n e t i c - e n e r g y and p o t e n t i a l - e n e r g y o p e r a t o r s expressed i n c o n f o c a l e l l i p t i c c o o r d i n a t e s , f o r u n i t R. The terms t and v are independent of R and the eigenvalue W(R) of the equation H¥ = W(R)Y (4-2) i s the c o n v e n t i o n a l p o t e n t i a l - e n e r g y f u n c t i o n f o r the n u c l e a r motion o f the molecule as a f u n c t i o n of R. At the e q u i l i b r i u m i n t e r n u c l e a r s e p a r a t i o n the Hamiltonian w i l l be H e = T e + v e ' (4-3) where T— = t/Rz and V = v/R^. I f we i n t r o d u c e the e ' e e e parameter y = R e/R = 1 + 3 (4-4) we may w r i t e Eq. (4-1); as H = y 2 T e + u V e . (4-5) -23-D i v i d i n g Eq. (4-5) by y we o b t a i n H/y = u T e + V e , (4-6) and s u b s t i t u t i n g 1+3 f o r y g i v e s H/(l+ 3 ) = T e + 3T e + V e = H e + 3 T E . (4-7) A c c o r d i n g t o Eq. (4-7) we may w r i t e W(R)/(l+ 3 ) = W(R e) + 3T e , (4-8) where T g i s the m a t r i x of k i n e t i c - e n e r g y o p e r a t o r . Taking 3 as a p e r t u r b a t i o n parameter we expand ¥ and W(R)/(l+ 8 ) as a power s e r i e s i n 8 , Y = + 8 + 8 2 ¥ ( 2 ) . . . , (4-9) e i W(R)/(l+8')= oo0 + 0 ^ 8 + u 2 8 2 + ... . (4-10) Then 00 W(R) = w0 + S ( w k - 1 + w k ) 8 k , (4-11) and w0 = W(R e). (4-12) Taking the d e r i v a t i v e of W(R) wi t h r e s p e c t t o R, W(R) = dW(R)/dR = dW(R ) / d 8 . d 8/dR = ( - R e / R 2 ) 2 » k 8 ( " k . i + w k ) » k=l (4-13) we f i n d t h a t when R=R . / e W(RJ = (-R o/R 2 )(L0 N + oO = 0, (4-14) e v e e 0 1 f o r any wavefunction s a t i s f y i n g the v i r i a l theorem; t h e r e f o r e , u 0 = -w-^. The f o r c e constants p r e d i c t e d by t h i s p o t e n t i a l are k e = (d 2W(R)/dR 2) = 2.(o)1+w2)/R^ , (4-15) - 2 4 -l e = (d 3W(R)/dR 3) = [-6(o)2+u>3) -^(u^+o^) ]/R 3 , = [-6(o)2+w3) -6R 2 k e ] / R 3 , (4-16) m e = (d 4W(R)/dR 4) , = [24(o)3+a)4) -12R| l e -36R2 kJ/R 4 . (4-17) To c a l c u l a t e these f o r c e c onstants we r e q u i r e the k i n e t i c -energy m a t r i x a t R e o n l y . 4-2 S c a l i n g Procedure Used t o Obtain ¥ a t R e The s c a l i n g i s performed t o make the wavefunction s a t i s f y the v i r i a l theorem. In doing so we o b t a i n R £ and t h e r e f o r e can c a l c u l a t e T e . A l s o , when the wavefunction s a t i s f i e s the v i r i a l theorem, Eq. (4-14) i s s a t i s f i e d and i t i s then p o s s i b l e t o c a l c u l a t e the f o r c e c o n s t a n t s . L e t us r e p r e s e n t the unsealed and s c a l e d wavefunction of an N - e l e c t r o n d i a t o m i c molecule by, Y = Y ( £ l ' E 2 ' £N' R ) (4-18) and % = *( n£i» n E 2 ' ••• NEN' n R ) ' (4-19) where r ^ i s the p o s i t i o n v e c t o r of the >Ith e l e c t r o n , R i s the i n t e r n u c l e a r d i s t a n c e and n i s some a r b i t r a r y s c a l e f a c t o r . For d i a t o m i c i m o l e c u l e s i n the Born-Oppenheimer a p p r o x i -mation the t o t a l energy a s s o c i a t e d w i t h the s c a l e d wave-f u n c t i o n ¥ i s [21] E(n,R) = n 2 T(l,nR) + n V ( l , n R ) , (4-20) - 2 5 -w h e r e T ( l f n R ) = T f r ^ , r 2 , . . . r ^ , nR) , ( 4 - 2 1 ) a n d V ( l , n R ) = V ( r j y r , • • •" n R > • ( 4 - 2 2 ) I f we l e t o u r u n s e a l e d w a v e f u n c t i o n h a v e a n i n t e r n u c l e a r d i s t a n c e p=nR, t h e n t h e u n s e a l e d e n e r g y i s E = T ( l , p ) + V ( l , p ) , ( 4 - 2 3 ) w h i l e t h e s c a l e d e n e r g y i s E v = n 2 T ( l , p ) + n V ( l , p ) . ( 4 - 2 4 ) L e t n = 1 + X , t h e n E n = T ( l , p ) + 2 X T ( l , p ) + X 2 T ( l , p ) + V ( l f p ) + X V ( l , p ) = E + X [ 2 T ( l , p ) + V ( l , p ) ] + X 2 T ( l , p ) = E + X [ E + T ( l , p ) + X 2 T ( l , p ) ] = (1+X)E + X ( 1 + X ) T ( 1 , p ) . ( 4 - 2 5 ) T h e r e f o r e E n / ( 1 + X ) = E + X T ( l , p ) . ( 4 - 2 6 ) E x p a n d i n g E n / ( 1 + X ) a n d Y a s a p o w e r s e r i e s i n X , Y = y ( 0 ) + X T ( 1 ) + X 2 Y ( 2 ) . . . , ( 4 - 2 7 ) E n / ( l + \ ) = W Q + XW X + X 2 W 2 . . . , ( 4 - 2 8 ) t h e s c a l e d e n e r g y c a n b e w r i t t e n E ^ = WQ + X ( W Q + Wj_) + X 2 (W x + W 2 ) + . . . . ( 4 - 2 9 ) A t X=0 , WQ=E s o CO E n = E + 2 . Xk (w k - 1 + W R ) . ( 4 - 3 0 ) T o o b t a i n o p t i m u m s c a l i n g d E n / d X = 0 . ( 4 - 3 1 ) -26-Therefore co_ kX*' 1 (W. . + W. ) = 0 . (4-32) k=l K ~ l k Optimum X i s obtained by varying X u n t i l Eq. (4-32) i s s a t i s f i e d . Now we can generate the scaled wavefunction and from i t the kinetic-energy matrix at Re. 4-3 Selection of Wavefunction The extended basis set of Ohno [5] was selected as a st a r t i n g point. The Slater type o r b i t a l s used are: on boron ( l s B , 2s, 2 P Q * 2p + and 2p_) and on hydrogen ( l s H ) . A l l configurations, i n which two electrons are retained i n the ls_, o r b i t a l , are taken into account to obtain a th i r t e e n -term set of basic functions. (See Table I I . ) To check computer input a C I c a l c u l a t i o n i d e n t i c a l to Ohno's case (b) [5], except that our l s H o r b i t a l was not orthogonalized, was performed and the same ground state energy resulted. (See Table I I I . ) Using the t o t a l e l e c t r o n i c energy as a c r i t e r i o n , the o r b i t a l exponents were optimized through one cycle i n the following order: l s B , 2s, 2pQ, (2p +) and l s H . For r e s u l t -ing o r b i t a l exponents and energy see Table I V . The improve-ment i n energy on going from e ( Q H N 0 ) t o E(OPT) x s n o t t o ° s i g n i f i c a n t . Now the Is o r b i t a l on boron was s p l i t into two o r b i t a l s , 1st, and l s ^ . The number of Slater determinants i n each term - 2 7 -TABLE I I 13-TERM WAVEFUNCTION The l s B l s B c o r e , o c c u r i n g i n a l l t h e d e t e r m i n a n t s , o m i t t e d i n t h e f o l l o w i n g d e s c r i p t i o n . 1. (2s 2s* p Q p"^ ) 2. (2s 2s l s „ Is) H n 3. ( P 0 P~ i s R T F H ) 4. (2s Js p Q T s H ) + (2s "2s* l s H p"p 5. ( P 0 2 s I s H > + ( P 0 P "^ l s H 2s) 6. (2s V~ 1 S r Ts"H) + ( p 0 2s" l s H ls" H ) 7. (2s 2s" p + p~j + (2s 2s p_ p~) 8. (P 0 P^ " P + p j + (P 0 Py" P_ P7} 9. ( l s H T s H p + p^) + ( l s H T s H p_ p p 10. (2s PQ" p + p j + (2s p_ £T) + ( p Q 2? p + p i ) + (P 0 2"s p_ p"""") 11. (2s ls" H p + p j + (2s Ts*H p_ p"7) + ( l s R Is p + jT~) + ( l s H 2~s p_ p]"") 12. ( P 0 T s H p + p j + ( p Q T S h p_ p^ ") + ( l s H PQ" P+ P~D + < l s H Po" P - P+} 13. ( p + P^ P_ P J TABLE I I I DATA FROM OHNO O r b i t a l Exponents  Boron Hydrogen l s B 4.70136 l s H 1.0038 2s 1.30092 2p Q 1.30092 2p+ 2p_ •» I n t e r n u c l e a r D i s t a n c e R = 2.32911 a.u. T o t a l E l e c t r o n i c Energy E( 0HNO) = "25.11045 a.u. E (Experimental) = -25.29 a.u. a a Ref. [18]. TABLE IV OPTIMIZED WAVEFUNCTION O r b i t a l Exponents  Boron Hydrogen l s B 4.64915 l s H 1.14662 2s 1.33334 2p Q 1.44977 2p + 1.36771 2p_ I n t e r n u c l e a r D i s t a n c e R = 2.32911 a.u. T o t a l E l e c t r o n i c Energy E ( 0 p T ) = -25.11980 a.u. E (Experimental) = -25.29 a.u. a a Ref. [18]. -30-of the wavefunction i n Table I I i s doubled. In each o r i g i n a l determinant the s p i n o r b i t a l l s B i s r e p l a c e d by the s p i n o r b i t a l ls'g (or lSg) w h i l e the s p i n o r b i t a l Ti*B i s r e p l a c e d by the s p i n o r b i t a l TsBV (or • T h i s makes our w a v e f u n c t i o n an approximation t o a 39-term CI. S t a r t i n g v a l u e s of the o r b i t a l exponents of the s p l i t core o r b i t a l s l s f i and lSg were 5.5 and 3.9 r e s p e c t i v e l y . These v a l u e s are c l o s e t o those suggested by Silverman, P l a t a s and Matsen [22]. A c a l c u l a t i o n was c a r r i e d out u s i n g t h i s 13-term s p l i t core wavefunction w h i l e r e t a i n i n g the o r i g i n a l o r b i t a l exponents of Ohno (See Table I I I . ) f o r o r b i t a l s o t h e r than the s p l i t core o r b i t a l s . The t o t a l e l e c t r o n i c energy E ^ s c j of t h i s s p l i t core wavefunction ¥ g C was -25.13333 a.u. T h i s i s q u i t e good. Cade and Huo [18] used an i n t e r n u c l e a r d i s t a n c e of 2.305 a.u. and an extended b a s i s s e t of S l a t e r type f u n c t i o n s t o c a r r y out a SCF Hartree-Fock c a l c u l a t i o n r e s u l t i n g i n an energy o f -25.13147 a.u. The energy improved t o -2 5.134 07 8 a.u. when the o r b i t a l exponents of the l s B and l s ^ o r b i t a l s were op t i m i z e d t o 5.46404 and 3.87451 r e s p e c t i v e l y . Both exponents were op t i m i z e d t o g e t h e r , t h a t i s when optimum energy was reached both exponents were changed by the same f a c t o r . T h i s new wavefunction VQSQ i s q u i t e s u i t a b l e f o r our purpose but f o r g e n e r a l i t y we s t a r t a t the beginning again u s i n g the more -31-common i n t e r n u c l e a r d i s t a n c e of 2.329 a.u. and o r b i t a l exponents by S l a t e r ' s r u l e s [23], except f o r the s p l i t core o r b i t a l s and l s H , then o p t i m i z e a l l exponents. (See Table V.) The r e s u l t i n g energy, E(TOSC)' ^ s v e r v g°°d. H a r r i s o n and A l l e n [18] performed a 13-term VBCI c a l c u l a t i o n u s i n g Gaussian-lobe f u n c t i o n s and an i n t e r n u c l e a r d i s t a n c e of 2.336 a.u. to obtain, an energy of -25.1426 a.u. H a r r i s o n [19] a l s o performed a YB c a l c u l a t i o n u s i n g Gaussian-lobe f u n c t i o n s and an i n t e r n u c l e a r d i s t a n c e of 2.50 a.u. with the r e s u l t i n g energy equal to -25.1455 a.u. At p r e s e n t o n l y the c a l c u l a t i o n of Bender and Davidson [24] y i e l d s a ground s t a t e energy lower than E(TOSC)* B e n d e r a n d Davidson performed a 1123-term CI c a l c u l a t i o n u s i n g n a t u r a l o r b i t a l s and an i n t e r n u c l e a r d i s t a n c e of 2.336 a.u. to o b t a i n the value -25.26214 a.u. T h i s i s lower than the sum o f the ground s t a t e e n e r g i e s of the i n d i v i d u a l atoms i n v o l v e d , and thus i n c l u d e s some b i n d i n g energy of the molecule. A summary of o r b i t a l exponents and e n e r g i e s f o r the v a r i o u s 13-term wavefunctions i s g i v e n i n Table VI while the c o r r e s p o n d i n g term c o e f f i c i e n t s are g i v e n i n Appendix I I . 4-4 S c a l i n g and Force Constant R e s u l t s The eigenvalue matrix of the wavefunction ^ipQgQ was r e d i a g o n a l i z e d t o ensure t h a t the v a l u e of any o f f d i a g o n a l elements i s n e g l i g i b l e , then the new e i g e n v e c t o r matrix was generated. The o r i g i n a l ground s t a t e energy remained the -32-TABLE V WAVEFUNCTION 4* TOSC Order of exponent o p t i m i z a t i o n : ( 2 p + ) , 1 S H / 2 p Q , 2 S ' 1 S B and Is,' 'B-S t a r t i n g Values O r b i t a l Exponents Is B 2s 2P0 2p_ l s R 5.5 3.9 1.3 1.3 1.3 1.3 1.1 ( a r b i t r a r y s e l e c t i o n ) Optimized Values 5.47771 3.84914 1.30378 1.33424 1.32475 II 1.21170 I n t e r n u c l e a r D i s t a n c e R = 2.329 a.u. E ( S p l i t Core + T o t a l O p t i m i z a t i o n ) = E^QSC) E (Experimental) = -25.29 a.u. = -25.14769 a.u, -33-TABLE VI ORBITAL EXPONENTS FOR 13-TERM WAVEFUNCTIONS ^OHNO ''.OPT. ^OSC YTOSC 4.70136 4.64915 5.46406 5.47771 K 3.87451 3.84914 2s 1.30092 1.33334 1.30092 1.30378 2p 0 1.30092 1.44977 1.30092 1.33424 2p ± 1.30092 1.36771 1.30092 1.32475 1 S H 1.00038 1.14662 1.00038 1.21170 -E 25.11045 a.u. 25.11980 a.u. 25.134078 a.u. 25.14769 a.u. E (Experimental) = -25.29 a.u. -34-same however. Now the k i n e t i c - e n e r g y o p e r a t o r and new e i g e n v e c t o r s were used t o c a l c u l a t e the k i n e t i c - e n e r g y m a t r i x . The e i g e n v a l u e s a n d ' k i n e t i c - e n e r g y matrix were then used i n a p e r t u r b a t i o n computer program to c a l c u l a t e t o W^ . The r e s u l t s are g i v e n i n Table V I I . The v a l u e of n i s very c l o s e to t h a t obtained by a si m p l e r method [21] rf = -V/2T = 0.9993749 , (4-33) where V and T are the p o t e n t i a l and k i n e t i c - e n e r g y of the ground s t a t e wavefunction " ^ o s c * A new Hamiltonian matrix i s c o n s t r u c t e d a c c o r d i n g to Eq. (4-25). T h i s matrix i s d i a g o n a l i z e d to o b t a i n the e i g e n v e c t o r s of the s c a l e d wavefunction V^OSC' A check was c a r r i e d out to see t h a t dE n/d r j = £'(2 T + V ) c ' = 0 (4-34) where c ' i s the e i g e n v e c t o r of the s c a l e d wavefunction Y-posC c o r r e s p o n d i n g to the lowest energy, while T and V a r e , i n t h i s case, the u s u a l k i n e t i c and p o t e n t i a l - e n e r g y m a t r i c e s . Now the e i g e n v e c t o r s of * T0SC a r e u s e o - t o o b t a i n the s c a l e d k i n e t i c - e n e r g y matrix. Expanding the wavefunction i n terms of the parameter 8 we can c a l c u l a t e t o w^ . The f o r c e constants may be c a l c u l a t e d a c c o r d i n g to Eq. (4-15) to Eq. (4-17). The r e s u l t s are g i v e n i n Table V I I I . -35-TABLE V I I SCALING RESULTS k Wk k ( w k - l + Wk> 1 25.179196 0.0315011 2 -1.5153996 x 10" 1 50.0553121 3 1.7958395 x IO""1 0.0841320 4 -2.2055059 x IO" 1 -0.16386656 5 -2.6166944 x IO' 1 0.2055942 6 -2.9812845 x IO" 1 -0.2187541 7 3.1521269 x IO" 1 0.1195897 X Q p T (optimized X) = -0.0006293 n = 0.9993707 R g = R/n = 2.3304666 a.u. -36-TABLE V I I I FORCE CONSTANTS k u. k 1 25.147705 2 -1.5159298 x 10" 1 3 1.7968714 x 10" 1 4 -2.2068003 x 10" 1 5 2.6180703 x I O - 1 6 -2.9820331 x IO" 1 7 3.1509448 x IO" 1 C a l c u l a t e d Experimental k e 9.204849 0.1958 a l e -23.712046 - 0 . 5 3 l 9 a m e 61.049772 1.237.3a a From the unpublished r e s u l t s o f M u l l i k e n and Ramsy as r e p o r t e d by Cade and Huo [18]. -37-Using experimental data we see t h a t .2 io, + u)0 = 1/2 k , 1 2 e e = (0.5) (2.336 a . u . ) 2 (0.19590) , = 0.5345 . (4-35) The experimental v a l u e of O J 2 i s -24.756. I t seems t h a t the c a l c u l a t e d v a l u e s f o r and 0J3 are f a r too s m a l l . L i m i t e d b a s i s s e t c a l c u l a t i o n s are doomed to f a i l . Good r e s u l t s might be o b t a i n e d u s i n g a much l a r g e r b a s i s s e t , such as t h a t o f Bender and Davidson [24], CHAPTER V VIRIAL SCALING 5-1 T h o r h a l l s s o n and Chong's Approach T h o r h a l l s s o n and Chong [6] i n v e s t i g a t e d the p o s s i b i l i t y of expanding the v i r i a l f o r c e F i n powers of R. The v i r i a l f o r c e i s d e f i n e d as F(R) = dE(R)/dR = -(2T+V)/R, (5-1) where V and T are the p o t e n t i a l and k i n e t i c energy. T h i s equation holds o n l y f o r exact wavefunctions, but by a p p l y i n g the v i r i a l theorem t o an approximate wavefunction one can make dE(R)/dR = F ( R ) . T h o r h a l l s s o n and Chong used a 10-term valence bond c o n f i g u r a t i o n - i n t e r a c t i o n wavefunction f o r the ground s t a t e of L i H . At each of the th r e e R v a l u e s the o r b i t a l exponents were roughly o p t i m i z e d and s c a l i n g was performed. The method of f i n d i n g the c o r r e c t v a l u e of the s c a l i n g parameter n i s a c c o r d i n g t o LOwdin [21] n = - [ V ( l , p ) + p V / ( l , p ) ] / [ 2 T ( l , p ) + pT ( l , p ) ] , (5-2) ,P' P where p = nR, V = 3V(l,p)/3p and Tp = 3T ( l , p ) / 3 p . The P v- t> d e r i v a t i v e s V p and T p are the most d i f f i c u l t t o e v a l u a t e and were determined from the unsealed wavefunction a t R^-x and R k + x . -39-In t h i s p r e s e n t work we i n v e s t i g a t e expansions of E and F i n R and 1/R u s i n g s c a l e d and unsealed wavefunctions d e r i v e d from our wavefunction ¥ . Since Eq. (5-2) i s the e x p r e s s i o n f o r n d e r i v e d from the c o n d i t i o n t h a t 3E(n,R)/3ri = 0, we f i n d i t more convenient t o perform the s c a l i n g a t f i x e d R. A l l o r b i t a l exponents were m u l t i p l i e d by three d i f f e r e n t v a l u e s of n, and the r e s u l t i n g e n e r g i e s when f i t t e d t o a pa r a b o l a g i v e a minimum corresponding t o the c o r r e c t v a l u e of n. 5-2 Wavefunctions and Data Three i n t e r n u c l e a r d i s t a n c e s are s e l e c t e d such t h a t 1/R-^  - 1/R2 = 1/1*2 ~ 1 / / R 3 * T h e w a v e f u n c t i ° n ^TOSC' d e s c r i b e d i n T able V, i s used as a s t a r t i n g p o i n t , and the o r b i t a l exponents of the s p l i t core are op t i m i z e d once i n d i v i d u a l l y a t each i n t e r n u c l e a r d i s t a n c e . Two s e t s of c a l c u l a t i o n s were c a r r i e d out, one s e t i n v o l v e d data from the wavefunctions d e s c r i b e d above, the ot h e r i n v o l v e d t a k i n g the mentioned wavefunctions one step f u r t h e r , s c a l i n g them. The s c a l i n g was performed a t each i n t e r n u c l e a r d i s t a n c e by c a l c u l a t i n g 1 the energy o f the wavefunction as w e l l as the e n e r g i e s g i v e n when a l l of the o r b i t a l exponents were m u l t i p l i e d by a f a c t o r of 1±0.01. Then a p a r a b o l i c f i t on energy g i v e s the s c a l e d o r b i t a l exponents a t lowest energy and t h e r e f o r e the s c a l e d wavefunction. Data from the s c a l e d and unsealed wavefunctions i s g i v e n i n Tables IX and X. -40-TABLE IX UNSCALED DATA *1 *2 * 3 R 2.329 a.u. 2.433688 a.u. 2.548230 a.u. E -25.147692 a.u. -25.149526 a.u. -25.148820 a.u. V -50.344143 a.u. -50.373239 a.u. -50.360489 a.u. T 25.196451 a.u. 25.223713 a.u. 25.211669 a.u. y 0.252673 a.u. 0.224906 a.u. 0.190736 a.u. l s / e x p o n e n t 5.465763 5.479920 5.479919 ls"exponent 3.865785 3.860518 3.860635 F(R) -0.020936 a.u. -0.03483 a.u. -0.024664 a.u. E (Experimental) = -25.29 a.u. y = d i p o l e moment -41-TABLE X SCALED DATA f l *2 *3 R 2.329 a.u. 2.433688 a.u. 2.548230 a.u. Sc a l e F a c t o r 1.00043246 0.99877845 1.00000008 E -25.147697 a.u. -25.149563 a.u. -25.148928 a.u. V -50.365648 a.u. -50.312475 a.u. -50.258097 a.u. T 25.217952 a.u. 25.162912 a.u. 25.109169 a.u. y 0.252367 a.u. 0.225838 a.u. 0.192422 a.u. F -0.03016574 a.u. -0.00548509 a.u. +0.01560259 a.u. O r b i t a l Exponents lSg 5.468127 5.473226 5.468635 lSg 3.867456 3.855802 3.852686 2s 1.304343 1.302187 1.301095 2p Q 1.334822 1.3326151 1.331498 2 p + 1.325324 1.323133 1.3220237 2p_ H H n ls„ 1.212213 1.210209 1.209194 -42-5-3 Numerical A n a l y s i s The t h r e e e n e r g i e s and f o r c e s from the s c a l e d wavefunctions are f i t t e d t o a q u i n t i c p o l y n o m i a l . The f o r c e s c a l c u l a t e d from the unsealed wavefunctions are too i r r e g u l a r t o permit the unsealed data to be f i t t e d t o a q u i n t i c p o l y n o m i a l . Data from the q u i n t i c f i t i s g i v e n i n Table XI. S e v e r a l d i f f e r e n t p a r a b o l i c models are used to f i t v i r i a l f o r c e s or e n e r g i e s of both the s c a l e d and unsealed wavefunctions. The v a r i o u s p a r a b o l i c models are d e s c r i b e d below. Model A E(R) = A + BR + CR 2 (5-3) F(R) = dE/dR = B + 2CR (5-4) R e = -B/2C (5-5) k e = 2C (5-6) E(p) = A + B(p-p 0) + C ( p - p 0 ) 2 , (5-7) Model B where p = 1/R and p Q = 1/R 2« L e t X = E ( p 0 - h ) , Y = E ( p Q ) , and Z = E(p Q+h), where h = P^P-^ = P 3 ~ P 2 * Then, A = Y , (5-8) B = [Z-X]/2h , (5-9) C = [X+Z-2Y]/[2h 2] , (5-10) and F(R) = -[B+2C(p- P ( ))]/R 2. (5-11) Th e r e f o r e p -p = -B/2C and k = 2Cp 4 . e 0 e K e -43-TABLE XI QUINTIC POLYNOMIAL FIT C o e f f i c i e n t s -25.149634 -0.934849 x 10" 8 0.993244 x 10" 1 -0.131321 0.129005 0.255402 C a l c u l a t e d Experimental E e -25.149634 a.u. -25.29 a.u. R e 2.459919 a.u. 2.336 a.u. a k e 0.198649 a.u. 0.1958 a.u. a 1e' -0.787925 a.u. -0.5319 a.u. a a From the unpublished r e s u l t s o f M u l l i k e n and Ramsy as r e p o r t e d by Cade and Huo [18]. -44-Model C At F=0, F(R) = A + BR + CR 2 (5-12) R g = [-B±/B^ - 4ACJ/2C , (5-13) dF(R)/dR = B+2CR , (5-14) and k = + [B+2CR 1 = + [±/B^ - 4AC]. (5-15) T h e r e f o r e , take the p o s i t i v e r o o t . Model D F(p) = A + B(p-p 0) + C ( p - P ( ) ) 2 , (5-16) where p = 1/R and p Q = 1/R2 . L e t X = F ( p Q - h ) , Y = F ( p Q ) and Z = F(p Q+h) where h = p2~pi = p 3 ~ p 2 * T n e n A = Y, B = [Z-X]/2h and C = [X+Z-2Y]/2h 2. A t F(R) = 0, p - p = [-B±/B^ - 4AC]'/2C , (5-17) e 0 and ,2 At o e , dF(R)/dR = -[B+2C(p-p 0) ]/R'. (5-18) k = - p 2 [B+(-B±/B^ - 4AC)], = - p 2 [±/B^ - 4AC], (5-19) t h e r e f o r e , take n e g a t i v e r o o t . Table XII g i v e s the r e s u l t s o f each p a r a b o l i c model c a l c u l a t i o n f o r both s c a l e d and unsealed wavefunctions. The c u b i c f o r c e c o n s t a n t l e c a l c u l a t e d from the p a r a b o l i c models, where p o s s i b l e , was very poor except f o r s c a l e d Model C which gave the va l u e -0.471199. -45-TABLE XII PARABOLIC MODEL RESULTS Model k e E e (a.u.) R e (a.u.) A unsealed 0. 216050 -25.149615 2.462430 A s c a l e d 0. 213184 -25.149667 2.464954 B unsealed 0. 203903 -25.149589 2.458223 B s c a l e d 0. 199896 -25.149639 2.460911 C unsealed 0. 282013 2.669423 C s c a l e d 0. 198469 2.460473 D unsealed 0. 229515 2.689602 D s c a l e d 0. 196769 2.460718 Q u i n t i c 0. 198649 -25.149634 2.459919 Experimental 0.1958 -25.29 2.336 -46-The q u i n t i c f i t r e s u l t s are used as a standard f o r comparing the v a r i o u s p a r a b o l i c models. Table XIII g i v e s e n e r g i e s and v i r i a l f o r c e s c a l c u l a t e d from the q u i n t i c p o l y n o m i a l f o r the d i f f e r e n t RQ p r e d i c t e d by v a r i o u s models. Table XIII a l s o g i v e s the d i p o l e moments a t these d i f f e r e n t R e. The d i p o l e moments are c a l c u l a t e d from simple p a r a b o l i c f i t s i n R to the s c a l e d and unsealed data. V i r i a l s c a l i n g has r e s u l t e d i n an improved v a l u e f o r k e i n a l l p a r a b o l i c models. The importance of s c a l i n g confirms the r e s u l t s of T h o r h a l l s s o n and Chong [6]. P a r a b o l i c expansions of energy or v i r i a l f o r c e i n 1/R r a t h e r than R leads to an improved v a l u e f o r k Q , except i n the case of going from s c a l e d model C t o s c a l e d model D. Thus a Fues p o t e n t i a l appears t o be b e t t e r than an harmonic o s c i l l a t o r p o t e n t i a l . T h i s supports the approach of Parr and Borkmah [25]. Model C s c a l e d g i v e s the b e s t agreement with the q u i n t i c p o l ynomial r e s u l t s but the value f o r 1 seems o v e r l y f o r t u n a t e . The c a l c u l a t e d d i p o l e moments are too s m a l l by a f a c t o r g r e a t e r than two, the experimental value g i v e n by Thomson and Dalby [26] i s .4997 ± 0.08 a.u. A summary of e n e r g i e s c o r r e s p o n d i n g to wavefunctions used i n t h i s work and wavefunctions from o t h e r sources i s g i v e n i n Table XIV. TABLE XIII COMPARISON OF DATA Models R e p r e d i c t e d by v a r i o u s models Q u i n t i c Polynomial P a r a b o l i c F i t E (a.u.) F(R) (a.u.) y (u) (a.u.) y (s) (a.u.) A (u) A (s) 2.462430 2.464954 -25.149633 -25.149631 -0.000496 -0.000990 0.216710 0.217178 B (u) B (s) 2.458223 2.460911 -25.149634 -25.149634 +0.000338 -0.000197 0.217926 0.218318 Ci (u) C (s) 2.669423 2.460473 -25.149634 -25.149634 -0.053673 -0.000110 0.150276 0.218441 D (u) D (s) 2.689602 2.460718 -25.143994 -25.149634 -0.066635 -0.000158 0.143108 0.218372 Data P r e d i c t e d by Q u i n t i c Polynomial 2.459919 -25.149634 0.0 0.217430 0.218590 u ) = unsealed s) = s c a l e d y = d i p o l e moment -48-TABLE XIV BH WAVEFUNCTIONS Wavefunction R (a.u.) E (a.u.) CI (Fraga and R a n s i l [20]) 2. 329 a -25.09034 13-term CI (Ohno [5]) 2. 32911 -25.11018 10-term CI (Table I : i v ) 2. 329 -25.11634 13-term CI (Table IV) 2. 32911 -25.11980 Hartree-Fock (Cade and Huo [18]) 2. 305* -25.13147 VB '(Harrison [19]) 2. 50* -25.1455 13-term s p l i t core CI (Table V) 2. 329 -25.14769 13-term s p l i t core CI (Table XI) 2. 460* -25.14963 1123-term CI (Bender and Davidson [24]) 2. 336 -25.26214 Experimental 2. 336 b -25.29 c * C a l c u l a t e d R e a Experimental v a l u e r e p o r t e d by Herzberg [27]. b From the unpublished r e s u l t s of M u l l i k e n and Ramsy as r e p o r t e d by Cade and Huo [18]. c Cade and Huo [18]. i -49-BIBLIOGRAPHY 1. E. Kapuy, Theoret. chim. A c t a (Berl.) 6_, 281 (1966). 2. E. Kapuy, Theoret. chim. A c t a (Berl.) 12, 397 (1968). 3. R. G. Pa r r and R. J . White, J . Chem. Phys. 4_9, 1059 (1968) 4. P. J . Gagnon and D. P. Chong, Theoret. chim. A c t a ( B e r l . ) , i n p r e s s . 5. K. Ohno, J . Phys. Soc. of Japan 12, 938 (1957). 6. J . T h o r h a l l s s o n and D. P. Chong, Chem. Phys. L e t t . 4_, 405 (1969). 7. O. Sinano g l u , Phys. Rev. 122, 491, 493 (1961). 8. 0. Sinano g l u , Proc. Roy. Soc. (London) A260, 376 (1961). 9. T. L. A l l e n and H. S h u l l , J . Chem. Phys. 35_, 1644 (1961). 10. L. P a u l i n g , Proc. Roy. Soc. (London) A196, 343 (1949). 11. A. C. Hurley, J . E. Lennard-Jones, and J . A. Pople, Proc. Roy. Soc. (London) A220, 466 (1953). 12. J . M. Parks and R. G. P a r r , J . Chem. Phys. 2_8, 335 (1958). 13. M. K l e s s i n g e r and R. McWeeny, J . Chem. Phys. 4_2, 3343 (1965). 14. D. R. H a r t r e e , Proc. Cambridge P h i l . Soc. 2_4, 89 (1928). 15. E. Clementi and D. L. Raimondi, J . Chem. Phys. 3_8, 2686 (1963) . 16. E. Clementi, J . Chem. Phys. £0, 1944 (1964). 17. E. Clementi, IBM Tech. Report RJ-256 (1963). 18. P. E. Cade and W. M. Huo, J . Chem. Phys. £7, 614 (1967). 19. M. Krauss, Compendium of ab i n i t i o c a l c u l a t i o n s o f molecular e n e r g i e s and p r o p e r t i e s . Washington, D.C.: N a t i o n a l Bureau o f Standards, 1967. 20. S. Fraga and B. J . R a n s i l , J . Chem. Phys. 36, 1127 (1962). -50-21. P. O. Lowdin, J . Mol. S p e c t r y . 3, 46 (1959). 22. J . N. Silverman, 0. P l a t a s and F. A. Matsen, J . Chem. Phys. 32, 1402 (1960). 23. J . C. S l a t e r , Phys. Rev. 36^ , 57 (1930). 24. C . F . Bender and E. R. Davidson, Phys. Rev. 183, 23 (1969). 25. R. G. P a r r and R. F. Borkman, J . Chem. Phys. 49_, 1055 (1968) . 26. R. Thomson and F. W. Dalby, Can. J . Phys. 47, 1155 (1969). 27. G. Herzberg, S p e c t r a o f Diatomic M o l e c u l e s , (D. Van Nostrand Company, Inc., P r i n c e t o n , N.J., 1950). -51-APPENDIX I SAMPLE GEMINAL CALCULATION The f o l l o w i n g d e s c r i p t i o n d e a l s w i t h b a s i s s e t I I , (See Table I.) but most of the o p e r a t i o n s i n v o l v e d a l s o apply to the ot h e r b a s i s s e t s . The value of £(H) f o r * A G p was determined from a p a r a b o l i c f i t t o antisymmetrized geminal product e n e r g i e s c a l c u l a t e d a t t h r e e d i f f e r e n t v a l u e s of £(H). In t h i s case the v a l u e s were 1.26, 1.30 and 1.34. At each of the 5(H) the parameter a was o p t i m i z e d . The f o l l o w i n g procedure was c a r r i e d out a t each of the fo u r £(H) v a l u e s . F i r s t the fo u r s i n g l e - z e t a S l a t e r - t y p e o r b i t a l s , those on boron having the exponents of Clementi and Raimondi [15] w h i l e the Is o r b i t a l on hydrogen has some s e l e c t e d v a l u e f o r £ (H), were used i n a f o r t r a n language c o n f i g u r a t i o n -i n t e r a c t i o n computer program c a l l e d Diatom*. A f i c t i t i o u s wavefunction was used, the on l y purpose was to c a l c u l a t e a l l p o s s i b l e o v e r l a p , one and t w o - e l e c t r o n i n t e g r a l s . The 10 d i f f e r e n t o n e - e l e c t r o n and the 55 d i f f e r e n t t w o - e l e c t r o n i n t e g r a l s were then s t o r e d as two and f o u r dimensional a r r a y s r e s p e c t i v e l y . * Dr. D. P. Chong, Department of Chemistry, U n i v e r s i t y of B r i t i s h Columbia. -52-The next step was to Schmidt o r t h o g o n a l i z e s_ t o k, s = N[sf - Sk], (1-1) where N = l / ^ l - S 2 and S i s the o v e r l a p i n t e g r a l between £ and k: S = /k (1) s j ( l ) d T ^ . T h i s o r t h o g o n a l i z a t i o n was c a r r i e d p u t by m a t r i x m u l t i p l i c a t i o n , (k, s, a, h) = (k, £, o, h)A, (1-2) where A = 1 (-NS) 0 0 0 N 0 0 0 0 1 0 0 0 0 1 (1-3) N = 1.02242, and S = 0.20827. The r e q u i r e d h y d r i d i z a t i o n was performed with m a t r i x B, (k, b, n, h) = (k, s, a, h)B, (1-4) where r 0 0 0 fl 1 / 9 ( l - a 2 ) 1 / 2 o [ 1 - a 2 ) 1 / 2 -a 0 B = 1 0 0 0 (1-c^) 0 (1-5) 0 1 Symmetric o r t h o g o n o r m a l i z a t i o n was c a r r i e d out u s i n g matrix C, (K, B, N, H) = (k, b, n, h)C, (1-6) where C i s the ma t r i x A ~ l / 2 and A i s the matr i x o f o v e r l a p i n t e g r a l s . I f U i s the matr i x t h a t d i a g o n a l i z e s A, U+AU = X , (1-7) then U X " 1 / ^ = A"" 1/ 2. (1-8) -53-By these t h r e e t r a n s f o r m a t i o n s we have i n t r o d u c e d an a l t e r n a t i v e b a s i s , r± = 2 r ' T , (1-9) j where T j ^ i s a matrix element of the m a t r i x product ABC. In t h i s new b a s i s the corresponding one and t w o - e l e c t r o n i n t e g r a l s are r e l a t e d by: <r i|h|r j> = 2 T J i < r r l h l r s > T s j ' ( I " 1 0 ) < r . r . | g | r , r > = >^"\ T* . T* . <r r l I g| r ' r >T., T . , (1-11) i ^ k s 7 t r i s] r s'^« t u tk u l ' where the primes d e s i g n a t e the o r i g i n a l o r b i t a l s , and h and g are the one and t w o - e l e c t r o n o p e r a t o r s r e s p e c t i v e l y . A s h o r t computer program y i e l d e d the completely transformed i n t e g r a l s which were then used i n the geminal program to o b t a i n the energy. V a r i o u s s e t s of i n t e g r a l s , c orresponding to d i f f e r e n t v a l u e s of the parameter a, were used u n t i l optimum a was o b t a i n e d . F i g u r e I g i v e s a sample p l o t of E versus a and F i g u r e I I g i v e s a p l o t of E versus £(H). The completely transformed i n t e g r a l s c orresponding t o E(AGP) w e r e then used i n program DIATOM to o b t a i n E ( G + ] j and E ( C I ) * T ^ e c o e f f i c i e n t s of the terms i n the bonding geminal o f along w i t h the c o e f f i c i e n t s o f terms i n VG+± and are g i v e n i n Table XV. FIGURE I E VERSUS a H(?) = 1.34 E ( a . u . ) -25.0759 1 , • , , , 0.48 0.49 0.50 0.51 0.52 a FIGURE I I E VERSUS £(H) -25.0754 H -25.0755 E (a.u.) -25.0760 1.26 1.30 1.34 5(H) -56-TABLE XV DATA FROM BASIS SET I I Is 2s 2p 0 O r b i t a l Exponents Boron 4.6795 1.2881 1.2107 Is Hydrogen 1.3125 (optimized) AGP G + l CI C O E F F I C I E N T S C x 0.48473405 C 2 0.766514474 C 3 0.42129379 «(>! 0.54252394 <J)2 0.48625970 •3 0.41595987 <t>4 -0.13886659 cf>5 -0.14870197 <j> 0.15156648 cj>° -0.16986274 10-1 10 1 10-1 4>a <t>2 <j)3 <f>4 0.54239176 0.48537033 0.41656740 -0.12764533 x 10 -0.18499101 x 10"! -1 0.15135885 x 10 -0.16975390 x 10 -1 -1 <j/ -0.21502238 x 10*1 10 -0.30078238 x 0.10816345 x IO"3 IO"2 E -25.07601 a.u. -25.08541 a.u. -25.08627 a = 0.50 R = 2.329 a.u. APPENDIX I I TABLE XVI COEFFICIENTS OF 13-TERM WAVEFUNCTIONS TERM WAVEFUNCTION OHNO OPT *QSC YTOSC 1 0. 65160911 X 10" •1 0.97622451 X 10" •1 0.34391515 X 10" •1 0.71924579 X 10" 1 2 0. 21686080 0.23936148 0.1097724 0.76587498 X 10" 1 3 -0. 31982327 X 10" •1 -0.14220806 X 10" •1 -0.15799579 X 10" •1 -0.57602491 X 10" 2 4 0. 50163363 0.47944053 0.25833018 0.24332556 5 0. 11724651 0.11538376 0.60380664 X 10" 1 0.65463450 X 10" 1 6 -0. 59753748 X 10" •1 -0.65619440 X 10" •1 -0.30337938 X 10" 1 -0.26711443 X 10" 1 7 0. 13112438 X 10" 1 0.16147282 X 10" •1 0.68367720 X 10" 2 0.10012272 X 10" 1 8 0. 17893071 X 10" 1 0.20531238 X i o - 1 0.93060592 X 10" 2 0.14028880 X 10" 1 9 0. 34097752 X 10" 1 0.40209341 X 10" 1 0.17170934 X 10~ 1 0.12314110 X 10" 1 10 -0. 14793546 X i o - 1 -0.81108863 X 10" 2 -0.74646173 X 10" 2 -0.17865829 X 10" 2 11 0.28422092 X i o - 1 0.28109768 X 10" 1 0.14850164 X 10" 1 0.15692586 X 10" 1 12 0. 88044107 X i o - 1 0.82458844 X i o - 1 0.45148265 X 10" 1 0.41494663 X 10" 1 13 -0. 99948530 X i o - 2 -0.93721254 X i o - 2 -0.51528255 X 10" 2 -0.50561302 X 10" 2 

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