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UBC Theses and Dissertations

Quantum chemical calculations on hf and some related molecules 1972

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QUANTUM CHEMICAL CALCULATIONS ON HF AND SOKE RELATED MOLECULES by ROBERT EMERSON BRUCE B . S c . , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1970 A T H E S I S SUBMITTED I N P A R T I A L FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE l n t h e D e p a r t m e n t 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 a s 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 B R I T I S H COLUMBIA J u n e 1972 In present ing th is thes is in p a r t i a l f u l f i l m e n t o f the requirements for an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L ib ra ry sha l l make i t f r e e l y a v a i l a b l e for reference and study. I fu r ther agree that permission for extensive copying of t h i s thes is for s c h o l a r l y purposes may be granted by the Head of my Department or by h is representa t ives . It i s understood that copying or p u b l i c a t i o n of t h i s thes is f o r f i n a n c i a l gain sha l l not be allowed without my wr i t ten permiss ion. Department of The Un ivers i ty of B r i t i s h Columbia Vancouver 8, Canada Date - i i - A b s t r a c t T h i s t h e s i s r e p o r t s some qu a n t u m c h e m i c a l c a l c u - l a t i o n s d i r e c t e d a t e l u c i d a t i n g p r i n c i p l e s u s e f u l f o r r e f i n i n g c a l c u l a t i o n s o f e l e c t r o n d i s t r i b u t i o n and o t h e r p r o p e r t i e s f o r c o m p l e x m o l e c u l e s . I n t h i s w o r k c a l c u - l a t i o n s h a v e b e e n made w i t h t h e v a l e n c e b o n d and m o l e - c u l a r o r b i t a l m e t h o d s u s i n g minimum b a s i s s e t s o f S l a t e r - t y p e o r b i t a l s o n t h e g r o u n d s t a t e s o f t h e m o l e c u l e s HP and HO, a n d o n s t a t e s o f H F + c o r r e s p o n d i n g t o t h e i o n i - z a t i o n o f e i t h e r a I s e l e c t r o n o r a ZpfC e l e c t r o n f r o m f l u o r i n e i n HF. C a l c u l a t i o n s h a v e b e e n made f o r m o l e - c u l a r e n e r g i e s , b o n d l e n g t h s , f o r c e c o n s t a n t s , d i p o l e moments, a n d e l e c t r o n d i s t r i b u t i o n s a s g i v e n by M u l l l - k e n p o p u l a t i o n a n a l y s i s . F o r HF, t h e p e r f e c t p a i r i n g m o d e l w i t h m o l e c u l e - o p t i m i z e d e x p o n e n t s y i e l d s m o l e c u l a r e n e r g i e s a b o u t 6 k c a l . / m o l e l o w e r t h a n t h e c o m p a r a b l e m o l e c u l a r o r b i t a l c a l c u l a t i o n s ; t h e d i p o l e moment c a l c u l a t e d b y t h e p e r - f e c t p a i r i n g m e t h o d i s 0.3 D. c l o s e r t o t h e e x p e r i m e n - t a l v a l u e (1.82 D.) t h a n t h a t c a l c u l a t e d b y t h e m o l e c u - l a r o r b i t a l m e t h o d . The HF e q u i l i b r i u m b o n d l e n g t h a n d f o r c e c o n s t a n t s a r e c a l c u l a t e d t o a r e a s o n a b l e d e g r e e o f a c c u r a c y w i t h t h e two m e t h o d s , a l t h o u g h t h e f i r s t i o n i z a t i o n p o t e n t i a l s seem t o be b e t t e r c a l c u l a t e d b y t h e m o l e c u l a r o r b i t a l m e t h o d e i t h e r by Koopman's T h e o r e m - I i i - or by taking the difference between the energies of the two states. The calculations reported in this thesis show clearly that in general free atom exponents are not re- liable for calculating molecular properties, and this is important for calculations on larger molecules which most frequently use basis functions appropriate to free atoms. As part of a programme for finding ways of op- timizing exponents relatively inexpensively, for use with more complex molecules, an approximation due to Lowdin, for overlap charge distributions ln electron repulsion integrals, was tested. The results reported In this thesis show that the method has promise in pro- viding a way of i n i t i a l l y optimizing exponents prior to the actual calculation wherein a l l integrals are evalu- ated exactly. - i v - T a b l e o f C o n t e n t s page A b s t r a c t i i T a b l e o f C o n t e n t s i v L i s t o f T a b l e s v A c k n o w l e d g e m e n t v i i C h a p t e r One - I n t r o d u c t i o n 1 The M o l e c u l a r O r b i t a l M e t h o d % 6 The V a l e n c e B o n d M e t h o d 12 A i m s o f t h e T h e s i s 21 C h a p t e r Two - C a l c u l a t i o n s o n HF, H F + a n d HO 2^ B a s i s F u n c t i o n s 27 V a l e n c e B o n d a n d M o l e c u l a r O r b i t a l Wave F u n c t i o n s 31 C o m p u t a t i o n a l D e t a i l s 36 C h a p t e r T h r e e - R e s u l t s a n d D i s c u s s i o n UrZ A t o m i c O r b i t a l E x p o n e n t s 51 M o l e c u l a r E n e r g i e s 55 Bond L e n g t h s a n d F o r c e C o n s t a n t s 6l E l e c t r o n D i s t r i b u t i o n s 6*4- C o n c l u d i n g R e m a r k s 71 B i b l i o g r a p h y 76 - V - L l s t o f T a b l e s T a b l e s page 1 V a l e n c e bond c o n f i g u r a t i o n s f o r t h e s t a t e o f HP lb 2 R e s u l t s o f some p r e v i o u s c a l c u l a t i o n s o f mo- l e c u l a r p r o p e r t i e s f o r HF t^ 25 3 R e s u l t s o f some p r e v i o u s c a l c u l a t i o n s o f mo- l e c u l a r p r o p e r t i e s o f HF+.J,. a n d H047J. 26 k Z e r o - o r d e r wave f u n c t i o n s i n e q u a t i o n (36) f o r t h e 1 I s t a t e o f HF 33 5 O r b i t a l e x p o n e n t s and m o l e c u l a r p r o p e r t i e s f o r d i f f e r e n t wave f u n c t i o n s o f HF a t t h e e x p e r i - m e n t a l bond d i s t a n c e (1.733 a.u.) 4 3 6 V a r i a t i o n p a r a m e t e r s and M u l l i k e n p o p u l a t i o n s f o r d i f f e r e n t wave f u n c t i o n s o f HF a t t h e e x - p e r i m e n t a l bond d i s t a n c e (1.733 a.u.) 7 O r b i t a l e x p o n e n t s and m o l e c u l a r p r o p e r t i e s f o r d i f f e r e n t wave f u n c t i o n s o f HF a t c a l c u l a t e d e q u i l i b r i u m bond d i s t a n c e s ^5 8 V a r i a t i o n p a r a m e t e r s a n d M u l l i k e n p o p u l a t i o n s f o r d i f f e r e n t wave f u n c t i o n s o f HF a t c a l c u l a - t e d e q u i l i b r i u m b o nd d i s t a n c e s 46 9 O r b i t a l e x p o n e n t s and m o l e c u l a r p r o p e r t i e s f o r d i f f e r e n t wave f u n c t i o n s o f H F + 2 ^ a t HF e x p e - r i m e n t a l bond d i s t a n c e (1.733 a.u.) k7 - v l - Tables page 10 Variation parameters and Kulliken populations for different wave functions of HF+â . at HF ex- perimental bond distance (1.733 a.u.) 4-8 11 Orbital exponents and molecular properties for a series of wave functions for HF*^ and H0aff» ^9 12 Variation parameters and Kulliken populations for a series of wave functions for HF+^«- and H0a 50 A c k n o w l e d g e m e n t I w o u l d l i k e t o g r a t e f u l l y a c k n o w l e d g e t h e a s s i s - t a n c e and e n c o u r a g e m e n t w h i c h I r e c e i v e d f r o m D r . K. A. R. M i t c h e l l t h r o u g h o u t t h e c o u r s e o f t h i s s t u d y . H i s g u i d a n c e , a d v i c e a n d many h e l p f u l d i s c u s s i o n s w e r e i n - v a l u a b l e b o t h f o r t h e r e s e a r c h a n d t h e p r e p a r a t i o n o f t h i s t h e s i s . I a l s o w i s h t o t h a n k J . K. Wannop f o r t h e p r e p a r a t i o n o f t h e m a n u s c r i p t , a n d my p a r e n t s f o r t h e i r c o n s t a n t s u p p o r t . Chapter One Introduction Quantum mechanics is important in chemistry for several reasons. In the most fundamental sense, i t provides, in principle, the means of determining the- oretically a l l the properties of molecules, either by the time-dependent or the time-independent Schrodinger equation,* and, given the properties of individual molecules and the interaction energies between them, s t a t i s t i c a l mechanics allows predictions to be made for macroscopic collections of molecules. That the pos s i b i l i t i e s for exact quantum mechanical calcula- tions on individual molecules are somewhat limited, can be assessed by noting that agreement between the- ory and experiment for the binding energy of the sim- plest neutral molecule, H?, has only recently been -2- reached. 2 Thus, f o r molecular systems of general i n - tere s t to the chemist, t h e o r e t i c a l treatments must be based on some degree of approximation. Molecular properties i n organic and inorganic chemistry are often discussed i n terms of electron d i s t r i b u t i o n s , ^ 1 ' ' and l n t h i s v e i n P l a t t ^ has argued that a theory of chemistry i s primarily a theory of e- l e c t r o n density. Early quantum mechanical c a l c u l a t i o n s on atoms and molecules, and experimental studies, espe- c i a l l y i n s t r u c t u r a l chemistry, have le d to quantum chemical concepts such as o r b i t a l s , i o n i c character, h y b r i d i z a t i o n , and electron pair bonds. These con- cepts are f r e e l y used i n discussing electron density i n molecules,^'^ although density d i s t r i b u t i o n s can r a r e l y be obtained d i r e c t l y by experiment. Another use of quantum mechanics i n chemistry has evolved with the development, during the l a s t two or three decades, of experimental techniques, such as nu- c l e a r magnetic resonance, electron spin resonance, nu- c l e a r quadrupole resonance, Mossbauer spectroscopy and photoelectron spectroscopy, which are now widely used by chemists l n attempting to gain an improved under- standing of chemical bonding. Quantum mechanics has been employed i n t h i s context, both f o r e l u c i d a t i n g the basic physics of these experiments, and f o r developing approximate computational schemes from which calculated -3- m o l e c u l a r p r o p e r t i e s c a n be c o m p a r e d w i t h e x p e r i m e n t a l v a l u e s . T h i s p r o v i d e s i m p o r t a n t I n f o r m a t i o n f o r a s s e s - s i n g t h e v a l i d i t y o f t h e m o d e l s o f e l e c t r o n d e n s i t y a n d c h e m i c a l b o n d i n g u s e d by c h e m i s t s . Two m a j o r a p p r o a c h e s h a v e b e e n d e v e l o p e d f o r a p - p r o x i m a t e c a l c u l a t i o n s o n m o l e c u l e s , a n d t h e s e a r e t h e m o l e c u l a r o r b i t a l m e t h o d a n d t h e v a l e n c e bond m e t h o d . The f o r m e r h a s b e e n more g e n e r a l l y u s e d , m a i n l y b e c a u s e i t h a s b e e n c o n s i d e r e d t o be c o m p u t a t i o n a l l y s i m p l e r . N e v e r t h e l e s s , r e c e n t d e v e l o p m e n t s h a v e l e d t o e f f i c i e n t c o m p u t a t i o n a l schemes f o r v a l e n c e b o n d c a l c u l a t i o n s , a n d , m o r e o v e r , a t t e m p t s a r e now b e i n g made t o d e v e l o p o s e m i - e m p i r i c a l schemes w i t h t h i s m e t h o d . A l s o i t h a s b e e n known f o r some t i m e t h a t c a l c u l a t i o n s u s i n g t h e p e r f e c t p a i r i n g m o d e l , s u c h a s t h a t p r o p o s e d b y H u r l e y , L e n n a r d - J o n e s and P o p l e , ^ w h i c h r e p r e s e n t s a n e x t e n s i o n t o p o l y a t o m i c m o l e c u l e s o f t h e H e i t i e r - L o n d o n c a l c u l a - t i o n o n Hg,*^ c a n g i v e b e t t e r m o l e c u l a r e n e r g i e s t h a n t h e c o r r e s p o n d i n g m o l e c u l a r o r b i t a l c a l c u l a t i o n s . 1 * T h i s i m p r o v e m e n t o c c u r s b e c a u s e e l e c t r o n m o t i o n s a r e b e t t e r c o r r e l a t e d i n H e i t l e r - L o n d o n t y p e wave f u n c t i o n s t h a n i n m o l e c u l a r o r b i t a l wave f u n c t i o n s . The u s e - f u l n e s s o f p e r f e c t p a i r i n g wave f u n c t i o n s i n p o l y a t o m i c s i s c l o s e l y r e l a t e d t o t h e u s e f u l n e s s o f t h e c o n c e p t o f h y b r i d i z a t i o n , w h i c h i s i t s e l f d e p e n d e n t o n t h e p r o - p e r t i e s o f a t o m i c o r b i t a l s i n m o l e c u l e s . The b e h a v i o u r of atomic orbitals in molecules i s of general interest, but i t i s also of particular Importance for studying molecules containing the heavier atoms (such as those of the second row of the periodic table and beyond, i n - cluding transition metals) for which the details of chemical bonding have not yet been established unam- biguously i n a number of Important c a s e s . ^ " ^ Large basis set calculations on these molecules would seem to be impractical in the near future, and the alterna- tive is to attempt to make reasonable calculations of molecular properties by using well chosen restricted basis sets of atomic orbitals. In any event, large basis set calculations are d i f f i c u l t to interpret i n terms of quantum chemical c o n c e p t s , a n example being 1 ft K u l l l k e ^ s suggestion that the increase in bond length observed on ionizing a TT electron ln many dia- tomic hydrides indicates a degree of Tt bonding in these molecules, and therefore the Involvement of 2prr atomic orbitals on hydrogen. Although large basis set calculations have been performed for diatomic hydrides, including up to 3 d orbitals on hydrogen ln the basis 19 set, 7 the chemical significance of hydrogen 2p7r orbi- tals i n bonding has not been determined. In discussing the valence bond and molecular or- b i t a l methods of molecular calculations, one starts with the time-independent Schrodinger equation -5- w h e r e H i s t h e H a m i l t o n i a n o p e r a t o r , E i s t h e e n e r g y o f t h e s y s t e m , and V i s t h e s t a t e f u n c t i o n . I n t h e n o n r e - l a t i v i s t i c a p p r o x i m a t i o n , t h e H a m i l t o n i a n o p e r a t o r c a n be w r i t t e n a s f o r a c o l l e c t i o n o f N e l e c t r o n s and S n u c l e i , w h e r e t h e f i r s t t e r m r e p r e s e n t s t h e summed k i n e t i c e n e r g i e s o f t h e n u c l e i , t h e s e c o n d t e r m r e p r e s e n t s t h e summed k i n e t i c e - n e r g i e s o f t h e e l e c t r o n s , t h e t h i r d t e r m r e p r e s e n t s t h e a t t r a c t i o n e n e r g y b e t w e e n t h e e l e c t r o n s and t h e n u c l e i , and t h e f o u r t h and f i f t h t e r m s r e p r e s e n t r e s p e c t i v e l y t h e n u c l e a r - n u c l e a r r e p u l s i o n s and t h e e l e c t r o n - e l e c t r o n r e - p u l s i o n s . I n m o l e c u l a r c a l c u l a t i o n s t h e B o r n - O p p e n h e i m e r a p p r o x i m a t i o n 2 0 i s f r e q u e n t l y made. P h y s i c a l l y t h i s a p - p r o x i m a t i o n c o n s i s t s o f r e g a r d i n g t h e m o t i o n s o f t h e n u c - l e i i n a m o l e c u l e a s i n s i g n i f i c a n t l y s m a l l i n c o m p a r i s o n t o t h e m o t i o n s o f t h e e l e c t r o n s , a n d t h i s i s d e p e n d e n t o n t h e m a s s e s o f t h e n u c l e i b e i n g v e r y much g r e a t e r t h a n t h e m a s s e s o f t h e e l e c t r o n s . T h u s one r e g a r d s t h e n u c l e i a s r e m a i n i n g e s s e n t i a l l y a t r e s t r e l a t i v e t o t h e m o t i o n s o f t h e e l e c t r o n s . U s i n g t h e B o r n - O p p e n h e i m e r a p p r o x i m a t i o n , t h e r e f o r e , t h e wave f u n c t i o n i s a p p r o x i m a t e d a s a f u n c t i o n o f t h e e l e c t r o n c o - o r d i n a t e s o n l y , t h e n u c l e i b e i n g r e g a r d - ed a s s t a t i o n a r y . T h e n t h e e l e c t r o n m o t i o n s a r e c o n t a i n e d - 6 - i n t h e e l e c t r o n i c wave f u n c t i o n , V , w h i c h i s o b t a i n e d i n p r i n c i p l e by s o l v i n g t h e e q u a t i o n K% = , (3) w h e r e ^ . i ^ A»l >* A<« A V I n t h e B o r n - O p p e n h e i m e r a p p r o x i m a t i o n , E i s E g p l u s t h e n u c l e a r - n u c l e a r r e p u l s i o n e n e r g y . The m o l e c u l a r o r b i t a l a n d v a l e n c e bond m e t h o d s p r o - v i d e s c h emes f o r w r i t i n g down a p p r o x i m a t e f o r m s o f t h e e- l e c t r o n i c wave f u n c t i o n , and f o r c a l c u l a t i n g t h e a p p r o x c o r r e s p o n d i n g e l e c t r o n i c e n e r g i e s a c c o r d i n g t o ( i n t h e D i r a c n o t a t i o n ) t e * (5) The a p p r o x i m a t e e l e c t r o n i c wave f u n c t i o n s a r e o b t a i n e d a c - 21 c o r d i n g t o t h e v a r i a t i o n p r i n c i p l e ; by w h i c h t h e b e s t : wave f u n c t i o n i s s e l e c t e d a c c o r d i n g t o t h e c r i t e r i o n o f minimum e n e r g y . The M o l e c u l a r O r b i t a l M e thod The m o l e c u l a r o r b i t a l method o r i g i n a t e d f r o m s t u d i e s 22 2^ by Hund a n d M u l l i k e n made w i t h i n a f e w y e a r s o f t h e f o r m u l a t i o n o f q u a n t u m m e c h a n i c s , a n d t h i s method r e p r e - s e n t s t h e d i r e c t e x t e n s i o n t o m o l e c u l e s o f t h e a t o m i c -7- 2^ 25 orbital method for atoms. ' For singlet states of a molecule containing 2N electrons, the electronic wave function in molecular orbital theory is approximated by a single determinant as in where only the diagonal elements of the determinant are defined e x p l i c i t l y . The determinantal form of equa- tion (6) is convenient for ensuring consistency with 26 _i. the antisymmetry principle? (2N)~2 i s the normaliza- tion factor. Each molecular orbital is doubly occupied by electrons of opposite spin, fi spin being indicated in equation (6) by a bar over the molecular o r b i t a l . The molecular orbitals are one-electron functions which extend over the whole molecule and they can be defined to be that set of orthonorraal functions, satisfying the conditions which minimize the electronic energy of the system according to F Ofc. fHe f %.>  H'~ (tJt> ' ( 8 ) where H is the electronic Hamiltonian defined in e equation (M . In earlier work on atoms, orbitals were given in -8- numerical formj in practical applications to molecules, however, they are usually expanded following the pro- cedure reviewed by Roothaan2? over a set of basis func- tions as i n *. • (9) Eoothaan's procedure consists of determining, by the variation theorem, the coefficients i n equation (9) In order to specify the molecular orbitals. Often the basis functions ^ i n equation (9) may be identified as atomic orbitals. In practice, a linear combination of atomic orbitals represents an approximation to a mo- lecular orbital wave function because only a restricted number of atomic orbitals are included in the basis set, although, in principle, one may approach as close to the limit as desired. The atomic orbitals in equation (9) may be centred on only one atom in a molecule, how- ever, the convergence to minimum energy is then slow, and, with modern computing f a c i l i t i e s , this approxima- tion seems to be of only limited value. The energy of the determinantal wave function i n equation (6), where the one-electron orbitals satisfy the conditions in equation (?), can be expressed 2^ as i j where -9- H i i - C M - W - z i l i O ( I D g i v e s t h e c o n t r i b u t i o n o f one e l e c t r o n i n ̂  t o t h e t o t a l e l e c t r o n i c e n e r g y . The t e r m i n e q u a t i o n (11) i n - v o l v i n g t h e L a p l a c l a n o p e r a t o r r e p r e s e n t s t h e k i n e t i c e n e r g y o f one e l e c t r o n i n Y^i t h e s e c o n d t e r m r e p r e s e n t s t h e a t t r a c t i o n b e t w e e n a n e l e c t r o n i n a n d t h e n u c l e i . I n e q u a t i o n (10), J 1 j a n d r e p r e s e n t , r e s p e c t i v e l y , C o u l o m b a n d e x c h a n g e e l e c t r o n r e p u l s i o n i n t e g r a l s d e - f i n e d a s a n d * - > r [ [ V i ( 1 ) * i < V i f c * j M * i ( i )  d r i J e i < ( 1 3 ) The i n t e g r a l s J±y a n d K i j » c a n r e a d i l y be e x - p a n d e d i n t e r m s o f t h e b a s i s o r b i t a l s i n e q u a t i o n (9)» a n d , f o l l o w i n g E o o t h a a n ' s p r o c e d u r e , a s e l f - c o n s i s t e n t f i e l d c a l c u l a t i o n a l l o w s t h e d e t e r m i n a t i o n o f t h e c o - e f f i c i e n t s , c ^ , i n e q u a t i o n (9) g i v e n t h e m o l e c u l a r I n t e g r a l s o v e r t h e b a s i s o r b i t a l s . F o r o p e n s h e l l s y s t e m s more t h a n one d e t e r m i n a n t c a n be w r i t t e n f o r a g i v e n c o n f i g u r a t i o n , a n d t h e d e - t e r m i n a n t s m u s t be c o m b i n e d a c c o r d i n g t o t h e a p p r o p - r i a t e e l e c t r o n i c s t a t e i n o r d e r t o o b t a i n a p p r o x i m a t i o n s -10- t o t h e t o t a l wave f u n c t i o n o f t h e s y s t e m . A s i m p l e e x - a m p l e i s t h e t r i p l e t s t a t e o f a t w o - e l e c t r o n s y s t e m f o r w h i c h t h e t o t a l e l e c t r o n i c wave f u n c t i o n i s s e t u p i n t e r m s o f t h e o r b i t a l s a n d ^ 2 a s , (1*0 s z . - l 1 A d e t a i l e d d i s c u s s i o n o f t h e m o l e c u l a r o r b i t a l m e t h o d f o r o p e n s h e l l s y s t e m s h a s b e e n g i v e n by R o o t h a a n . - ^ M o l e c u l a r o r b i t a l c a l c u l a t i o n s u s i n g t h e R o o t h a a n p r o c e d u r e a n d e v a l u a t i n g a l l m o l e c u l a r i n t e g r a l s w i t h - o u t a p p r o x i m a t i o n become e x c e s s i v e l y e x p e n s i v e a s t h e number o f e l e c t r o n s i n t h e m o l e c u l e a n d t h e s i z e o f t h e b a s i s s e t i n c r e a s e . The g r e a t e r c o m p u t a t i o n a l e f - f o r t a n d e x p e n s e i s due i n p a r t t o t h e number o f e l e c - t r o n - e l e c t r o n r e p u l s i o n i n t e g r a l s t o be e v a l u a t e d , w h i c h i n c r e a s e s a s a p p r o x i m a t e l y t h e f o u r t h power o f t h e b a s i s s e t . - ^ * A l s o , f o r l a r g e b a s i s s e t s , i t i s o f t e n f o u n d t h a t more t i m e i s r e q u i r e d t o e v a l u a t e i n t e g r a l s i n v o l v i n g h i g h e r members o f t h e b a s i s s e t t h a n t o e - v a l u a t e i n t e g r a l s i n v o l v i n g t h e l o w e r members o f t h e s e t . T h e s e f a c t o r s h a v e l e d t o t h e d e v e l o p m e n t o f a number o f a p p r o x i m a t e m o l e c u l a r o r b i t a l m e t h o d s s u i t a b l e - 1 1 - f o r a p p l i c a t i o n o n a r o u t i n e b a s i s t o m o l e c u l e s w h i c h a r e t o o c o m p l e x t o be r e a d i l y t r e a t e d u s i n g t h e more c o m p l e t e m e t h o d s . I n t h e s e a p p r o x i m a t e m o l e c u l a r o r b i t a l m e t h o d s one a t t e m p t s t o make j u d i c i o u s a p p r o x i m a t i o n s w h i c h w i l l s i m p l i f y t h e c o m p u t a t i o n s s o t h a t p r o p e r t i e s o f f a i r l y l a r g e m o l e c u l e s c a n be c a l c u l a t e d w i t h o u t e i t h e r i m p o s i n g c o n c e p t s s u c h a s p r e c o n c e i v e d b o n d i n g s c h e m e s , o r e l i m i n a t i n g e s t a b l i s h e d p h y s i c a l f e a t u r e s s u c h a s t h e r e l a t i v e e n e r g y l e v e l s o f a t o m i c o r b i t a l s . One d e v e l o p m e n t h a s b e e n t o i n c o r p o r a t e e m p i r i c a l d a t e I n t o a m o d e l s u c h a s i s done i n t h e H u c k e l method-^*33 d e v e l o p e d f o r jr e l e c t r o n s i n o r g a n i c s y s t e m s and e x - t e n d e d t o i n c l u d e a l l t h e v a l e n c e e l e c t r o n s T h i s m e t h o d d o e s n o t e x p l i c i t l y I n c l u d e e l e c t r o n - e l e c t r o n r e p u l s i o n s , b u t b y r e l a t i n g H u c k e l ' s C o u l o m b I n t e g r a l s t o v a l e n c e i o n i z a t i o n p o t e n t i a l s , a n d e x p r e s s i n g t h e r e s o n a n c e i n t e g r a l s i n t e r m s o f t h e C o u l o m b a n d o v e r l a p i n t e g r a l s , H o f f m a n n - ^ h a s d i s c u s s e d c h a r g e d i s t r i b u t i o n s a n d c o n f o r m a t i o n e n e r g i e s o f a l a r g e number o f h y d r o - c a r b o n s , a n d s i m i l a r m e t h o d s h a v e b e e n a p p l i e d t o many I n o r g a n i c m o l e c u l e s . 36,37 L e s s d r a s t i c a p p r o x i m a t i o n s a r e made i n t h e Com- p l e t e N e g l e c t o f D i f f e r e n t i a l O v e r l a p a nd r e l a t e d me- t h o d s w h i c h a r e d i s c u s s e d i n a r e c e n t b o o k by P o p l e a n d B e v e r i d g e ^ a n d a l s o i n a b o o k e d i t e d by S i n a n o g l u -12- and Wiberg .39 In these methods, emphasis i s placed on the v a l e n c e e l e c t r o n s , and e l e c t r o n r e p u l s i o n I n - t e g r a l s are Incl u d e d , but approximations are made such as Atomic s p e c t r a l d a t a are a g a i n i n c o r p o r a t e d i n these methods, but a g u i d i n g p r i n c i p l e i s t h a t they a re f o r - mulated so t h a t the c a l c u l a t e d r e s u l t s are i n v a r i a n t t o the r o t a t i o n of axes. T h i s p r o p e r t y i s r e q u i r e d phy- s i c a l l y , but i s not shown by the extended Huckel method. Many a p p l i c a t i o n s have been made to the c a l c u l a t i o n o f mo l e c u l a r e n e r g i e s , m o l e c u l a r geometries, charge d i s t r i - b u t i o n s , i o n i z a t i o n p o t e n t i a l s , and n u c l e a r magnetic resonance p a r a m e t e r s a n d these methods have been e s t a b l i s h e d as p r o v i d i n g a reasonable balance be- tween computational expense and worthwhile c a l c u l a t i o n s of mo l e c u l a r p r o p e r t i e s . The Valence Bond Method (15) and (16) H i s t o r i c a l l y , the v a l e n c e bond theory p r o v i d e d -13- the f i r s t method for molecular calculations, and this theory originated from the work of Heitler, London, Slater, and Pauling.^ In this method one assumes a set of basis functions for a molecule, and these func- tions are most frequently identified as atomic orbitals. In the most complete form of the valence bond method, combinations of determinantal functions are written down for a l l possible ways of accommodating the elecfc trons in the various atomic orbital functions in ac- cordance with both the Pauli principle, and with the symmetry of the particular electronic state for which the wave function i s being expressed. The determinan- t a l functions are defined by the various valence bond configurations for a given electronic state. As an il l u s t r a t i v e example, a l l the valence bond configu- rations are li s t e d i n Table 1 for the state of HP using a basis set of the Is atomic orbital at hydrogen, and the Is, 2s, 2pO% and 2p*r atomic orbitals at fluorine. The ground state wave function Is then obtained by a free mixing of the zero-order wave functions corres- ponding to a l l the configurations as ln where c^ Is the linear mixing coefficient, and i s the appropriate combination of determinantal functions for the i ^ n valence bond configuration. As examples, - 1 4 - Table 1. Valence bond c o n f i g u r a t i o n s f o r the ^ s t a t e of HF 1. 2. 3 . 4 . 5. 6. 7. 8. 9. 10. 11. I s 2 2 s 2 77£ 2 73^2 cr h I s 2 2 s 2 7 7 ^ 2 7 T 2 2 CT2 I s 2 2 s 2 i r t 2 i r z 2 h 2 2 2 2 2 I s 2s 77^ ^ <rr2 & h I s 2 2s TT2 Tf2 <r h 2 2 2 9 9 I s 2s^ -77̂  7r-2 <r^ h^ 2 2 2 2 Is 2s 7T1 7T 2 cr- h 2 2 ? 2 i s 2s 7r± ir2 * h ., „ 2 2 2 .2 I s 2s ir. <TC7 cr h 2 2 1 2 2 9 2s 7^ TT <r h The symbols I s , 2s, ^ , 7r^, *r, h r e f e r r e s p e c t i v e l y to I s , 2s, 2p-Tlt 2pir 2, 2pcr f u n c t i o n s a t F and the I s f u n c - t i o n a t H, -15- the specific forms of the unnormalized zero-order wave functions for the f i r s t two configurations in Table 1 are t and f z | is U2*ZZ Tr^tr^ir-rl m (19) The linear coefficients in equation ( 1 7 ) and the corres- ponding energies are obtained by application of the 12 variation principle. The technique is well known, and involves solving the secular equation D E T JM-.J - S i i £ {-- o ( 2 0 ) for the energies, and Z I ci (Hi3 - ES.-j)-* (21) j » for the coefficients. The matrix elements In equations ( 2 0 ) and ( 2 1 ) are defined as H;j * $y*Ht Vi U ( 2 2 ) and ) •* ( 2 3 ) When a l l the configurations formed from a given basis set are mixed, as in equation ( 1 7 ) , the valence b o n d m e t h o d i s e q u i v a l e n t t o a c o m p l e t 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 c a l c u l a t i o n i n t e r m s o f m o l e c u l a r o r b i t a l s e x p a n d e d o v e r t h e same b a s i s . ^ C l e a r l y , a s t h e s i z e o f a b a s i s s e t i s i n c r e a s e d , more c o n f i g u r a t i o n s c a n be f o r m e d w i t h t h e a p p r o p r i a t e s y m m e t r y , a n d , i n t h e l i m i t , t h e e n e r g i e s o b t a i n e d by t h e v a l e n c e bond method c o n - v e r g e t o t h e e i g e n v a l u e s f o r a c o m p l e t e s o l u t i o n o f t h e S c h r o d i n g e r e q u a t i o n i n t h e B o r n - O p p e n h e i m e r a p p r o x i - m a t i o n . I n l e s s a c c u r a t e a p p l i c a t i o n s o f t h e m e t h o d , h o w e v e r , i t t u r n s o u t t h a t some c o n f i g u r a t i o n s c a n be d i s r e g a r d e d . A v a l e n c e b o n d s t u d y by H a r r i s a n d M i c h e l s ^ o n HF f o r a r a n g e o f b o n d l e n g t h s , h a s shown t h a t m i x i n g t h e s i x m o s t i m p o r t a n t c o n f i g u r a t i o n s l e a d s t o a c a l - c u l a t e d m o l e c u l a r e n e r g y o n l y 0.35 k c a l . / m o l e a b o v e t h a t o b t a i n e d f r o m t h e m i x i n g o f a l l e l e v e n c o n f i g u r a t i o n s . The f i v e c o n f i g u r a t i o n s w h i c h may be n e g l e c t e d , w i t h o n l y s m a l l e r r o r , e i t h e r c o r r e s p o n d t o c h a r g e d i s t r i - b u t i o n s i n t h e s e n s e H""-F+, w h i c h i s c o n t r a r y t o c h e - m i c a l e x p e r i e n c e b a s e d o n t h e c o n c e p t o f e l e c t r o n e g a - t i v i t y , o r c o r r e s p o n d t o c o n f i g u r a t i o n s i n v o l v i n g e x - c i t a t i o n o f e l e c t r o n s f r o m t h e f l u o r i n e I s c o r e . S i - m i l a r l y , c a l c u l a t i o n s by M a g l a g a n a n d S c h n u e l l e ^ f o r BeHg show, f o r t h e p a r t i c u l a r c a s e o f u s i n g f r e e a t o m e x p o n e n t s a n d a Be-H d i s t a n c e o f 1.3^ A, t h a t t h e e f f e c t o f n e g l e c t i n g t h o s e v a l e n c e bond c o n f i g u r a t i o n s , w h e r e Be i s more n e g a t i v e l y c h a r g e d t h a n H,- a n d n e g l e c t i n g -17- also those configurations where Be is non-bonding, raises the total energy by only 1,2 kcal./mole. These consi- derations indicate ways of selecting for approximate c a l - culations those valence bond configurations which are most significant to a given basis set. A consequence of such a selection Is an appreciable saving In computational effort and expense, and this becomes more important for larger molecules. A more restricted form of the valence bond method is that which involves perfect pairing,^ and this method is usually based on hybrid rather than natural atomic orbitals. In this approach, electron pair bonds are constructed between orbitals in a molecule, and the to- t a l electronic wave function i s given in terms of the determinantal functions appropriate to the various spin couplings for the electron pair bonds. As an ex- ample of this approach, one can consider BeHg, for which two electron pair bonds are assumed to be formed from the overlap of the Is orbitals (designated 1 1 and 1 2 ) on each hydrogen with the appropriate directed hybrid functions h^ and h 2 at the Be atom. For this example, the unnormalized perfect pairing wave function is written as - l ^ x ^ ' - ^ M ^ J - l A j . / . A I W M . V * 1,(24) where one electron pair bond is formed by the overlap -18- of 1 1 and h l t and the other bond is formed by the overlap of 1 2 and hg. Two interpretations can be given to V p p . ^ In the f i r s t , ^pp involves only neutral configurations at Be, and h^ and h 2 are the digonal hybrids formed from the 2s and 2p«" atomic orbitals at Be. The second interpretation allows for ionic character in the BeH bonds by expressing h^ and h 2 as suitable combinations of the Is orbitals at H and the digonal hybrids as ln h« = N(d. + kl«) 1 1 1 (25) h 2 = N(d 2 + k l 2 ) , where k i s a measure of the ionic character of the bond and may be taken as a variation parameter} N i s the nor- malization factor. In general, within the perfect pairing model, for a closed-shell molecule with n electron-pair bonds, there w i l l be 2 n determinants in the electronic wave function. Hurley, Lennard-Jones, and Pople have showr? that, provided that the orbitals involved i n any ele- tron pair bond are orthogonal to a l l other orbitals in a molecule, a comparatively simple expression can be written in closed form for the electronic energy cor- responding to the perfect pairing wave function of the molecule. The evaluation of this expression is related to the methods which are discussed ln chapter two for evaluating the Hamiltonian and overlap matrix elements -19- in equations (22) and (23). A useful criterion for different methods for mo- lecular calculations i s provided by the agreement between calculated and experimental molecular properties. Cal- culations performed using the valence bond and molecular orbital methods with comparable basis sets, enable com- parisons to be made between the two methods. Detailed comparisons of the valence bond and mole- cular orbital methods have been made for Hg. ' Using a minimum basis set of Slater-type orbitals with energy- 48 optimized exponents, the valence bond method predicts an equilibrium bond length of 0.743 A and a binding energy of 85.94 kcal./mole. The experimental values are 0.741 A and 109.98 kcal./mole respectively, while the corresponding quantities from the molecular orbital me- t h o d * * are 0.732 A and 80.27 kcal./mole with a minimum basis set. The molecular orbital method becomes much less reliable as the bond distance Is increased, and this i s associated with the overemphasis of ionic con- tributions in the molecular orbital method 5̂ ' Karo and Olsen^ 0 have compared the molecular orbital and valence bond methods for the ground state (*£ ) of LiH using a basis set of numerical Is, 2s, and 2p orbitals at L i and a Slater Is orbital at H. At the equilibrium bond length of 1,56 A , both methods predict a dipole moment of 6.05 which may be compared with the experimental -20- val\ie of 5.88 D.;-3 and a t t h i s bond d i s t a n c e , the v a - le n c e bond method g i v e s a m o l e c u l a r energy which Is o . 4 kcal./mole lower than t h a t g i v e n by the m o l e c u l a r o r b i t a l method. Again, as the i n t e r n u c l e a r d i s t a n c e i n c r e a s e s , the m o l e c u l a r o r b i t a l method becomes r e l a t i v e l y l e s s Q r e l i a b l e . Maglagan and S c h n u e l l e have noted t h a t the va l e n c e bond method g e n e r a l l y g i v e s lower m o l e c u l a r e n e r g i e s than the mo l e c u l a r o r b i t a l method f o r mole- c u l e s l n which the model of e l e c t r o n p a i r bonds i s f r e q u e n t l y used. By c o n t r a s t , the m o l e c u l a r o r b i t a l me- thod i s c o m p a r a t i v e l y b e t t e r f o r d e l o c a l i z e d systems such as benzene, a l t h o u g h , as noted p r e v i o u s l y , w i t h s u f f i c i e n t refinement the two methods merge. A de- t a i l e d comparison has r e c e n t l y been made by M i t c h e l l and Thi r u n a m a c h a n d r a n ^ f o r BeHg employing a b a s i s s e t of S l a t e r - t y p e o r b i t a l s w i t h energy-optimized exponents. The m o l e c u l a r energy c a l c u l a t e d w i t h the p e r f e c t p a i r - i n g model i s 15 kcal./mole below t h a t f o r the molecu- l a r o r b i t a l method, and the c a l c u l a t e d Be-H d i s t a n c e s are 1.35 A and 1.37 A f o r the p e r f e c t p a i r i n g and mo- l e c u l a r o r b i t a l methods r e s p e c t i v e l y . A f u r t h e r i n - t e r e s t i n g comparison between the mo l e c u l a r o r b i t a l and va l e n c e bond methods has been made by H a r r i s o n and Allen-* f o r the ground s t a t e and two low e x c i t e d s t a t e s *A^ and *B^ of CH^. U s i n g a b a s i s s e t of Gaussian lobe f u n c t i o n s , these authors c a l c u l a t e the bond a n g l e s o f - 2 1 - the ^3±, 1 A 1 , and 1 B 1 s t a t e s to be 1 3 1 ° . 1 1 1 ° , and 15^° a c c o r d i n g to the m o l e c u l a r o r b i t a l method, and u s i n g the v a l e n c e bond approach the c o r r e s p o n d i n g a n g l e s are 1 3 8 ° , 1 0 8 ° , and 1 4 8 ° which are to be compared w i t h ex- p e r i m e n t a l e q u i l i b r i u m bond an g l e s of ~ 1 3 5 ° » 1 0 3 °. and 1 4 0 ° f o r the three s t a t e s . ^ 3 F i n a l l y , , the 1 A 1 - 1 B 1 v e r - t i c a l t r a n s i t i o n i s 0.96 kcal./mole c l o s e r to the exper i m e n t a l v a l u e u s i n g the v a l e n c e bond method. Aims of the T h e s i s T h i s study f o l l o w s r e c e n t c a l c u l a t i o n s by M i t c h e l l and T h i r u n a m a c h a n d r a n ^ » ^ on BeH^, i n which the mole- c u l a r o r b i t a l and v a l e n c e bond methods have been compared u s i n g a minimum b a s i s s e t of energy-optimized S l a t e r - type o r b i t a l s . The work on BeRg i n d i c a t e s t h a t the per- f e c t p a i r i n g model p r o v i d e s a v e r y good approximation f o r t h i s molecule, and f u r t h e r comparative c a l c u l a t i o n s are necessary i n order to provide b a s i c i n f o r m a t i o n w i t h which to a s s e s s these methods on a wider b a s i s . HF was chosen as a s u i t a b l e molecule f o r c o n t i n u i n g t h i s work because the computational e f f o r t i s reduced f o r d i - atomic h y d r i d e s , and the l a r g e e l e c t r o n e g a t i v i t y d i f - -22- ference between H and P contrasts with the much smal- le r difference in electronegativities between neigh- bouring atoms in BeH2. Although many calculations have already been made on HF using large basis sets, 7*-J i t s t i l l seems worthwhile to make a direct comparison of the different models for molecular calculations asing an energy-optimized minimum basis set of Slater- type orbitals, ln part because the large basis set calculations are d i f f i c u l t to interpret in terms of quantum chemical concepts. Connected with the perfect pairing model is the concept of hybridization, and ln this work, attempts are made to compare the atomic orbital hybridization at P in HF and HF+, and also to compare with the hybridization at 0 in the diatomic KO which Is isoelectronic with HF+, Calculations on HF + were made for states in which a Is or a 2ptf elec- tron has been ionized from fluorine in the neutral mo- lecule . Since minimum basis set calculations are performed in the various semi-empirical molecular orbital methods currently being used,-^'39 and because these methods have been designed especially for evaluating properties of related series of molecules, i t seems necessary, in order to gain a better understanding of these semi-em- p i r i c a l methods, to know Just how well minimum basis set calculations are able, In principle, to give useful -23- calculated values of properties such as bond lengths, ionization potentials, dipole moments, and molecular energies. This is especially so for more complicated molecules, such as those containing an atom of the se- cond row of the Periodic Table or beyond, for which computational expense most often requires calculations to be carried out using restricted basis sets. It is hoped that the calculations on HF, HF+, and HO reported in this thesis can contribute both to a better understanding of the r e l i a b i l i t y of minimum basis set computations for calculating molecular properties of basic interest to the chemist, and to a better a- wareness of the value and limitations of the perfect pairing model and the use of hybridization In molecular wave function calculations. Finally, a preliminary attempt i s made to assess the value of an approximation, proposed by Lowdin,^^ for simplifying molecular integrals for the purpose of optimizing atomic orbital exponents. The need for suitable integral approximations i s great- est for more complex molecules, but i t i s necessary to assess the applicability of any approximations by comparing results obtained using the approximations with those from the more complete calculations and also with experimental results, and this indicates ln part the reason for this study on HF. - 2 4 - C h a p t e r Two C a l c u l a t i o n s o n HF, H F + , and HO I n p r i n c i p l e , q u a n t u m c h e m i c a l c a l c u l a t i o n s o n m o l e c u l e s c a n g i v e d i r e c t i n f o r m a t i o n a b o u t e l e c t r o n i c d i s t r i b u t i o n by s q u a r i n g t h e e l e c t r o n i c wave f u n c t i o n ? some a s s e s s m e n t o f t h e r e l i a b i l i t y o f a c t u a l c a l c u l a - t i o n s may be o b t a i n e d by c o m p a r i n g c a l c u l a t e d m o l e c u l a r p r o p e r t i e s w i t h t h o s e m e a s u r e d e x p e r i m e n t a l l y . I n T a b l e s 2 and 3 r e s u l t s a r e c o l l e c t e d f r o m a number o f r e c e n t c a l c u l a t i o n s o n HF £^ , HF"^. , and H 0 i ; r , w h i c h a r e o f s p e c i a l i n t e r e s t t o t h i s t h e s i s . The m a i n p a r t o f t h i s c h a p t e r d e s c r i b e s t h e d e t a i l s o f t h e c a l c u l a t i o n s w h i c h h a v e b e e n c a r r i e d o u t d u r i n g t h i s w o r k , and t h e r e s u l t s i n T a b l e s 2 and 3 p r o v i d e a r e f e r e n c e f o r a s s e s - s i n g t h e r e s u l t s w h i c h a r e d i s c u s s e d i n c h a p t e r t h r e e . Table 2. Results of some previous calculations of molecular properties for HFj Wave function Molecular properties Model Basis set Ref. Energies ( a - u - L H-F Distance (a.u.) Dipole Moments (D.) Ionization P o t l . ( l tr) (eV) Force Constants •10^(dynes/cm) mo min. STO free atom exp. 57 -99.4915 1.733 1.12 mo min. STO opt. exp. 58 -99.5361 1.733 1.44 vb min. STO opt. exp. 59 -99.563* 1.860 mo + c i min. STO opt. exp. 58 -99.5640 1.733 1.30 mo ext. GTO opt. atom exp. 60 -99.8873 1.7^3 14.00 9.60 mo ext. GTO opt. exp. 55 -100.0622 1.733 mo ext. STO opt. exp. 19 -100.0708 1.696 14.6* 11.22 Experimental -100.5271 1.7238 1 . 8 1 9 5 16.06 9.657 * The value i s taken from a graph **1 a.u.= 27.205e.V. Table 3. Results of some previous calculations of molecular properties of EF+ifr and H04fl. Wave function H0'-a Molecular properties Model Basis' set Ref. Energies (a.u.) Bond dis- tance (a.u.) Dipole Moments (D.) Ionization P o t l . U )(eV) Force Constants •lo5(dynes/cm) mo ext. GTO opt. atom exp. 60 -75.2872 1.813 11.7 vb H.-Fock a.o. 61 -75.325 1.80 mo H.-Fock a.o. 61 -75.327 2.00 mo ext. STO opt.exp. 19 -75.^208 1.80 11.3* 7.79 Experimental -75.778 1.834 1.66 13.2 9.216 E F l r i t mo ext. GTO opt. atom exp. 60 -99.373** 1.96 mo ext. STO opt. exp 62 -99.53* 1.85 14.45 Experimental -27- B a s l s F u n c t i o n s A s s e e n i n T a b l e 2, a number o f d i f f e r e n t t y p e s o f f u n c t i o n s h a v e b e e n u s e d a s b a s e s f o r m o l e c u l a r c a l - c u l a t i o n s . I n p r i n c i p l e , a n y s e t o f f u n c t i o n s w h i c h i s c o m p l e t e may be u s e d t o e x p a n d a m o l e c u l a r wave f u n c t i o n } i n p r a c t i c e t h e c h o i c e o f f u n c t i o n s i s d e t e r m i n e d by t h e c o m p u t a t i o n a l e f f o r t r e q u i r e d t o o b t a i n a d e s i r e d l e v e l o f c o n v e r g e n c e . M o s t o f t e n t h e b a s i s f u n c t i o n s u s e d i n m o l e c u l a r c a l c u l a t i o n s a r e r e l a t e d i n some way t o a t o m i c o r b i t a l f u n c t i o n s . T r a d i t i o n a l l y , t h e H a r t r e e - F o c k method f o r a t o m s l e a d s t o a t o m i c o r b i t a l s w h i c h a r e i n n u m e r i c a l f o r m , h o w e v e r , t h e s e t a b u l a t e d f u n c t i o n s a r e t o o u n w i e l d y t o be u s e f u l l n m o l e c u l a r c o m p u t a t i o n s . I n more r e c e n t w o r k a n a l y t i c e x p r e s s i o n s a r e u s e d t o r e p r e - s e n t a t o m i c o r b i t a l s and t h e s e a n a l y t i c a t o m i c o r b i t a l s a r e e x p r e s s e d a s w h e r e r e p r e s e n t s t h e a p p r o p r i a t e s p h e r i c a l h a r m o n i c s ( l i s t e d f o r e x a m p l e by P a u l i n g and W i l s o n * ^ )t a r K 3 t h e R t ( r ) a r e some t y p e o f r a d i a l f u n c t i o n . One commonly u s e d r e p r e s e n t a t i o n o f t h e r a d i a l f u n c t i o n s I s t h a t g i - v e n by S l a t e r ^ a n d e x p r e s s e d a s % - r"'"1- t x F ( - < i r ) 9 (27) -28- w h e r e n ^ a n d a r e t h e p r i n c i p a l q u a ntum number and o r b i t a l e x p o n e n t r e s p e c t i v e l y ; i s a n o r m a l i z a t i o n f a c t o r . T h i s f o r m was p r o p o s e d b y S l a t e r i n o r d e r t o a p p r o x i m a t e , i n a s i m p l e way, t h e r a d i a l f u n c t i o n s f r o m t h e H a r t r e e - F o c k c a l c u l a t i o n s f o r a t o m i c o r b i t a l s . The r a d i a l f u n c t i o n i n e q u a t i o n (2?), when c o m b i n e d w i t h a s p h e r i c a l h a r m o n i c d e f i n e s a S l a t e r - t y p e o r b i t a l . S l a - t e r - t y p e o r b i t a l s h a v e no r a d i a l n o d e s , h o w e v e r , t h e y do c o n v e r g e e f f i c i e n t l y i n a t o m i c a n d m o l e c u l a r c a l c u l - a t i o n s . A n a l t e r n a t i v e r a d i a l f u n c t i o n i s t h e G a u s s i a n f u n c t i o n , c o n v e n i e n t l y e x p r e s s e d a s Rn£ - N - r ^ * . <up (-*.r*) . ( 2 8 ) The u s e o f G a u s s i a n f u n c t i o n s was f i r s t p r o p o s e d b y 64 B o y s , b e c a u s e i n t e g r a l s r e q u i r e d i n m o l e c u l a r c a l c u l - a t i o n s a r e more e a s i l y o b t a i n e d u s i n g G a u s s i a n f u n c t i o n s t h a n S l a t e r f u n c t i o n s . T h i s a d v a n t a g e m u s t be w e i g h e d a g a i n s t t h e f a c t t h a t t h e s e f u n c t i o n s p r o v i d e much s l o w - e r c o n v e r g e n c e f o r m o l e c u l a r e n e r g i e s t h a n do S l a t e r - t y p e o r b i t a l s ; i n f a c t i t r e q u i r e s a b o u t h a l f t h e number o f S l a t e r b a s i s f u n c t i o n s a s G a u s s i a n f u n c t i o n s t o o b - t a i n a g i v e n m o l e c u l a r e n e r g y . I n t h e p r e s e n t s t u d y , t h e i n t e r e s t c e n t r e s m a i n l y o n t h e u s e o f minimum b a s i s s e t c a l c u l a t i o n s and a s s e s - s i n g t h e p o s s i b i l i t i e s f o r c a l c u l a t i n g p r o p e r t i e s o f l a r g e m o l e c u l e s ; t h e r e f o r e S l a t e r - t y p e r a d i a l f u n c t i o n s -29- combined with the appropriate s p h e r i c a l harmonics are used as bas is se t s . To overcome the lack of orthogo- n a l i t y which occurs f o r pure S la ter - type o r b i t a l s of the same symmetry on the same centre , orthogonal ized S la ter - type o r b i t a l s are constructed according to the Schmidt or thogona l i za t ion p r o c e d u r e . ^ p o r o r b i t a l s on the same centre t h i s procedure cons i s t s of r e i n s t a - t i n g the r a d i a l nodes which were neglected i n the ap- proximation i m p l i c i t In equation (27). In the general case, to Schmidt orthogonal ize a func t ion ^ to another func t ion o n e "takes * i . H f c . , = *«• " 5 , (29) where S ' • (30) When normalized, ^ o r ^ n 0 g defines the funct ion P^ which has been Schmidt orthogonal ized to The S l a t e r funct ions used as a basis set f o r the study of HF cons i s t of a Is funct ion at H and I s , 2s, 2p<r and the degenerate p a i r of 2p;r funct ions at F . The formulat ion of the energy expressions i s s i m p l i f i e d i f a l l valence o r b i t a l funct ions are orthogonalized to the F l s core , and t h i s i s achieved by Schmidt or thogona l i z ing both His and F2s i n d i v i d u a l l y to F l s . In the perfect p a i r i n g model of KF there i s one -30- electron pair bond constructed from the overlap of the Is atomic orbital at H with a hybrid formed from the 2s and 2p«r atomic orbitals at F, The second hybrid formed from these two atomic orbitals at F i s regarded as a non- bonding orbital, and is doubly occupied. These two hyb- rids may be expressed as d 1 = s i n a 2s + cos <* 2per- (31) and d 2 = sln/3 2s» + cos ,3 2pa~/ (32) where d^ i s the bonding hybrid and d 2 is the non-bond- ing hybrid. The primes in the expression for d 2 allow for the possibility that the radial functions used for the bonding hybrid could be different from those used for the non-bonding hybrid» ^ Is a variation parameter which determines the mixing of the atomic orbitals ln d^, and is chosen so as to ensure that d^ and d«> are orthogonal by taking tan/9 = -S p p / (S s s»tan<* ), (33) where S g g is the overlap integral between the 2s and 2s' functions and S ^ i s the overlap between 2p and PP 2p«. -31- Valence Bond and Molecular Orbital Wave Functions In the perfect pairing model the ground state wave function of HF can be written in unnormalized form as a combination of two determinants as in where there is a single electron pair bond between the hybrid d^ and the orbital combination designated h in equation (3*0s Is, ff^t and rr^ refer to doubly occupied non-bonding orbitals at F. Allowance Is made in equation (34) for the possibility of ionic character in the H-F <r bond, expected since F is more electronegative than H, by forming the electron pair bond between the hybrid d^ and a linear combination of the Is function at H with d^ defined as = ( s i n Y . a + cos Y - <J t ) , (35) In equation (35). a Is the H^g atomic orbital orthogo- nalized to the F core, and y is a variation parameter X s which gives a measure of the ionic character of the H-F a- bond. This formulation corresponds to the second i n - terpretation given to on page 18. When d^, dg» and h, as given by equations (3l)» (32) , and (35) are substituted into equation (3k), Y p p can be ex- panded as -32- ^P-JL (36) w h e r e e a c h V c 6 r r e s p o n d s t o a v a l e n c e b o n d c o n f i g u - r a t i o n e x p r e s s e d I n t e r m s o f t h e o r t h o g o n a l i z e d S l a - t e r - t y p e o r b i t a l s . The f i f t e e n w h i c h c a n be f o r m e d f r o m t h e e x p a n s i o n o f V , a r e l i s t e d i n T a b l e 4. The PP c o e f f i c i e n t s c ^ i n e q u a t i o n (36) a r e t h e n d e t e r m i n e d b y t h e v a l u e s o f t h e p a r a m e t e r s , /3 , a n d Y . F o r t h e s p e c i a l c a s e t h a t 2 s == 2 s 1 a n d 2 p s 2 p ' , c o r r e s p o n d i n g t o t h e minimum b a s i s s e t s i t u a t i o n , o n l y t h r e e o f t h e f i f t e e n c o n f i g u r a t i o n s l n T a b l e 4 a r e d i f f e r e n t , a n d i n t h i s c a s e t h e wave f u n c t i o n o b t a i n e d b y s o l v i n g t h e s e c u l a r e q u a t i o n f o r t h e s e t h r e e s t r u c t u r e s i s e q u i - v a l e n t t o t h e p e r f e c t p a i r i n g wave f u n c t i o n i n e q u a - t i o n (34). T h e s e p e r f e c t p a i r i n g a n d v a l e n c e b o n d wave f u n c t i o n s f o r t h e g r o u n d s t a t e o f HF a r e t o be c o m p a r e d w i t h t h e c o r r e s p o n d i n g m o l e c u l a r o r b i t a l wave f u n c t i o n s d i s c u s s e d b e l o w ; h o w v e r , c o n s i d e r a t i o n i s f i r s t g i v e n , w i t h i n t h e p e r f e c t p a i r i n g f r a m e w o r k , t o some d o u b l e t s t a t e s o f H F + w h i c h a r e o f i n t e r e s t , i n i n t e r p r e t i n g r e s u l t s f r o m p h o t o e l e c t r o n s p e c t r o s c o p y ^ a n d e l e c t r o n s p e c t r o s c o p y f o r c h e m i c a l a n a l y s i s ^ o n t h e n e u t r a l HF m o l e c u l e . Two s t a t e s o f H F + a r e c o n s i d e r e d i n a n a p p r o x i m a t e way. The f i r s t i s a 25i s t a t e - 3 3 - Table If. Zero-order wave functions in equation ( 3 6 ) for the 1X. state of HF 1. V |lsfsn*Jr*ff^g-s 2s'2s2s| 2. *2= | l s l S jr^n^/Tgff^S '2T'2p2pl 3 . Jlslsfr^TTg^ 2? 2p»2s2s| 4. 11 s l i s / r 2 2 p1 fp^pSipl 5 . |lsls'7r1fr^'n'2^22s "2s"'2sa| + J1 s I s T T 2 ^ 2 s ' 2s1 a2s | 6. V | l s i s n^fr^ t^n^s 1 2s~'2paf + |lsls/T 1F 1 ^ 2 ^2 2 s ' 2"s* a 2 Pl 7. Jl s 1 s ̂ 7?^ ̂2^2 2 s »2s"»2s2p] + | lslsP'177jfr2J7^2s'2s12p2s| 8. V 11 s 1 s n2*~2 2 p 2p12sa| + [lsrsTr1fr^7r2?fj2p,2p'a2s | 9. V llslsT^ff^trg^2? 2p»2pa| + (lsls 7T1lf^7r2?f22pl2p,a2p| 1 0 . * i o - 11 s i s ? JV ' : 2 2 p 2^282^1 + |lsls7r 1fr^7r 2^2p«2p -'2p2sj 1 1 . llsrs/r.;F" ôjf~2s 1 1 1 2 2 2p~'2s2s| +11 s Is r t TT^ tr2"F2 2 p 12s 12 s 2s | 12.t 1 2= 11 s i s •7r̂ 7r̂ /77'2"̂ 22s '2p»2p2p| + |lsrs^^rr2^2p ,2s" ,2p2p| 1 3.V 1 3= 11 s Is ̂ ^ i ^ ^ 2 s' 2p»2s2p| +1 ls ls ^ ^ 7 r 2 ^ 2 P ' 2 s T 2 s 2 p | + 11 S1 S ff^TT^ J T g ^ 2 S 2p'2p2si + |lsrsV1rr^r2^2p,'2s12p2s| Islsi^rF^r^7^?23 2p»2sa|+ \ lsisr 17r^7r 2 7^2p , 2F l 2sa | + (lsls^/F^r^fl^s' ^ a ^ l + | l s i s ^ ^ ^ 2 ^ 2 ? 1 2s' a2s| 1 5 . * 1 5 = l l s l S T T ^ ^ n ^ T T ^ S 1 2p~'2pa| + 11 s Is ̂  7^ ̂ 2^2 2 p1 2~sT 2 pa/ + |lslsr^ff^rr2^2s 1 2p~'a2p| + [ l s f s ^ t r ^ /r2l7^2pl2s",a2p| Symbols are a$ in the text. -34- o b t a i n e d by the i o n i z a t i o n of an e l e c t r o n from the P 1 o r b i t a l i n the n e u t r a l HFj the second i s a " u s t a t e ob- t a i n e d by removing an e l e c t r o n from a 2pTr atomic o r b i t a l a t F. I n the p e r f e c t p a i r i n g approximation, the appro- p r i a t e wave f u n c t i o n s f o r these two s t a t e s of HF T may be expressed i n unnormalized form as % r * I i s i r ^ r ^ i ^ X l r i ^ U | j s n - j T , . i r , F ^ X ^ T I ( 3 7 ) and = I is is B i ^ K l t | + I is is ff, ff* ff* J JI + 11» fi ffi ̂ J x U j + I I s H jr,^ 7TAd **XK I , ( 3 8 ) where S i n both c a s e s . The wave f u n c t i o n i n e q u a t i o n (38) a l s o r e p r e s e n t s an approximation w i t h i n the p e r f e c t p a i r i n g model f o r the ground s t a t e of the di a t o m i c spe<* c i e s HO which i s i s o e l e c t r o n i c w i t h HF*. Wave f u n c t i o n s i n the m o l e c u l a r o r b i t a l approach have been c o n s t r u c t e d so as to enable as c l o s e a compa- r i s o n as p o s s i b l e w i t h the v a l e n c e bond and p e r f e c t p a i r i n g wave f u n c t i o n s d i s c u s s e d above. Consequently, the m o l e c u l a r o r b i t a l s were c o n s t r u c t e d from o n l y the v a l e n c e b a s i s o r b i t a l s . The m o l e c u l a r o r b i t a l s which would i n v o l v e mainly the i n n e r b a s i s o r b i t a l s , were sim- pl y taken to be core atomic o r b i t a l s as was done i n the p e r f e c t p a i r i n g c a l c u l a t i o n s . On t h i s b a s i s , the e l e c t r o n i c wave f u n c t i o n f o r the ground s t a t e of HF i n - 3 5 - the molecular orbital approximation can be expressed as ( 3 9 ) where and are respectively the bonding and es- sentially non-bonding molecular orbitals, defined as irt = Ns ( s«* S • a + coi & • <k 4) (24.0) and a- = t4, (i'^e-a + cos 6 (s/'» £ ^ t t e * » £ 0 ) , , , 6 (41) in such a way as to be useful for discussing hybri- dization in this model. In equations (40) and (41) Nj and N t are normalization factors; h and £ (along with <*) are variation parameters, and £ i s fixed byi the condition that <r̂  and <r^ are mutually orthogonal. Since the F ^ g core orbital has the same symmetry as and t T g , a free mixing of this core orbital must lead to a slight lowering of energy by the variation prin- ciple. The corresponding approximate wave functions in the molecular orbital approximation for the 2- and ^TT electronic states of HF formed by removing either an F ^ g or an F2prr electron from HF are expressed as and -36- Comrjutatlonal Details Given the electronic wave function of a molecule and the electronic Hamiltonian, the electronic energy of the system is obtained by equation (5). When the determinantal wave functions defined in the previous section are substituted into equation (5). the expression for the electronic energy involves functions of the type and where ^ and <Kj are many-electron determinantal func- tions. The general procedures for the evaluation of these many-electron matrix elements In terms of coef- ficients and one and. two-electron molecular integrals 68 have been given by Lowdin and are f u l l y discussed by 69 Slater. 7 The latter treatment was followed in this work with a l l expressions for the overlap and Hamilto- -37- n i a n m a t r i x e l e m e n t s b e i n g e v a l u a t e d by h and by d e t e r - m i n i n g t h e a p p r o p r i a t e c o e f f i c i e n t s , i n t e r m s o f t h e o v e r l a p I n t e g r a l s , f o r a l l t h e m o l e c u l a r i n t e g r a l s o c - c u r r i n g f o r t h e p a r t i c u l a r b a s i s s e t . C o m p u t e r p r o - grammes w e r e w r i t t e n t o sum a l l t h e s e c o n t r i b u t i o n s a n d c o n s i d e r a t i o n i s now g i v e n t o t h e m e t h o d s e m p l o y e d t o o b t a i n t h e v a r i o u s m o l e c u l a r i n t e g r a l s . The b a s i s s e t , a s d e s c r i b e d a b o v e , becomes c o n - t a m i n a t e d by t h e o r t h o g o n a l i z a t i o n p r o c e d u r e s , h o w e v e r , t h e one and t w o - e l e c t r o n m o l e c u l a r i n t e g r a l s o v e r t h e b a s i s o r b i t a l s a r e r e a d i l y e x p a n d e d i n t e r m s o f one a n d t w o - e l e c t r o n i n t e g r a l s i n v o l v i n g o n l y S l a t e r - t y p e o r b i t a l s . A p r o c e d u r e due t o M a g n u s s o n a n d Z a u H ? 0 p r o - v i d e s a c o n v e n i e n t way o f o b t a i n i n g t h o s e e l e c t r o n - e l e c - t r o n r e p u l s i o n i n t e g r a l s w h i c h I n v o l v e a c h a r g e d i s t r i - b u t i o n o n a s i n g l e c e n t r e s u c h a s w h e r e a a n d b a r e t h e two n u c l e i and ^ t o a r e S l a t e r - t y p e o r b i t a l s o n t h e i n d i c a t e d c e n t r e s . T h i s p r o c e d u r e i n v o l v e s e x p r e s s i n g t h e i n t e g r a l i n (46) a s w h e r e ^ ( i f ^ f g ) * S t J i e P ° t e n t i a l d u e t o * n e c h a r g e d i s - t r i b u t i o n t/ 1 S |/ /2 a c e n t r e d o n a . P o t e n t i a l s o f t h i s t y p e f o r S l a t e r - t y p e o r b i t a l s h a v e b e e n t a b u l a t e d by Magnus- -38- 70 s o n a n d Z a u l i / and some e x t e n s i o n s and c o r r e c t i o n s 71 h a v e b e e n r e p o r t e d by M i t c h e l l . ' When u s i n g p o l a r c o - o r d i n a t e s ( r , 9 , <P ) a t e a c h c e n t r e , t h e i n t e g r a l i n (47) i n v o l v e s s i x v a r i a b l e s ? h o w e v e r , by i n t r o - d u c i n g t h e e l l i p t i c a l c o - o r d i n a t e s ^ a n d v , d e f i n e d a s a n d y = jr ( - r b ) , (49) w h e r e R i s t h e i n t e r - n u c l e a r d i s t a n c e , t h e i n t e g r a l i n (47) c a n r e a d i l y be e x p r e s s e d i n t e r m s o f t h e t h r e e v a r i a b l e s ju. t v , a n d <f> , w h e r e ^ m e a s u r e s t h e a n g l e o f r o t a t i o n a b o u t t h e i n t e r - n u c l e a r a x i s a - b . A l s o f o r t h e known f o r m o f t h e S l a t e r - t y p e o r b i t a l s and t h e p o t e n t i a l V, t h e I n t e g r a t i o n o v e r <p i s t r i v i a l a n d c a n be d o n e a n - a l y t i c a l l y . T h u s t h e e v a l u a t i o n o f t h e i n t e g r a l i n (47) r e q u i r e s I n t e g r a t i o n o v e r t h e two c o - o r d i n a t e s ^ a n d v , and t o c o v e r a l l s p a c e t h e r e s p e c t i v e r a n g e s a r e 1 t o °° and -1 t o +1. The e v a l u a t i o n o f i n t e g r a l s o f t h e t y p e i n (46) was p e r f o r m e d u s i n g t w o - d i m e n s i o n a l G a u s s i a n q u a d r a t u r e ? 2 w i t h L e g e n d r e p o l y n o m i a l s o f o r d e r 16; a n d a s a n i l l u s t r a t i o n o f t h i s t e c h n i q u e , a n i n t e g r a l i n v o l v i n g a s i n g l e v a r i a b l e w i t h l i m i t s p a n d q i s g i v e n i n t h i s a p p r o x i m a t i o n by - 3 9 - ffth)l..l£ I f ( x , ?-' • VU) , (50) where x^ is the i ^ n root of the Legendre polynomial of order n, and a^ are tabulated constants associated with each x^. The form of equation (50) can be directly ex- tended to any number of variables. One advantage of this approach for the evaluation of the two-electron integrals of the type in (46) is that a l l the one-electron integrals can be obtained by the same methods at the same time. The overlap and nuclear attrac- tion integrals respectively, written in general as < fi I (51) and < * i l $ l * i > (52) represent special cases of the Integral in (47). Fur- thermore, the kinetic energy integrals Ul-X^K-) (53) can be expressed in terms of overlap integrals as shown by Roothaan''7^ Who gave the expression - 4 0 - f o r t h e e f f e c t o f t h e k i n e t i c e n e r g y o p e r a t o r o n a S l a t e r - t y p e o r b i t a l r e p r e s e n t e d by ( n l m ) w i t h e x p o - n e n t cH • The method u s e d i n g o i n g f r o m (46) t o (47) i s n o t a p p l i c a b l e i n a c o n v e n i e n t way f o r e v a l u a t i n g t h e e l e c t r o n - e l e c t r o n r e p u l s i o n e x c h a n g e i n t e g r a l s o f t h e t y p e < W I V . 1 ) , ( 5 5 ) E x a c t n u m e r i c a l v a l u e s o f t h e s e i n t e g r a l s w e r e o b t a i n e d by u s i n g a c o m p u t e r programme w r i t t e n by P i t z e r , W r i g h t 74 a n d B a r n e t t ' a n d t r a n s l a t e d i n t o F o r t r a n I V by M i t c h e l l . S i n c e t h e s e I n t e g r a l s w e r e much t h e m o s t t i m e c o n s u m i n g , a n a p p r o x i m a t i o n p r o p o s e d by L o w d i n - ^ was a l s o u s e d t o o b t a i n v a l u e s o f t h e i n t e g r a l s . L o w d i n ' s a p p r o x i m a t i o n c o n s i s t s o f e x p r e s s i n g t h e c h a r g e d i s t r i b u t i o n V^Y^ a s - s ^ x w r ^ K C W ) ] , (56) w h e r e S ^ i s t h e o v e r l a p i n t e g r a l b e t w e e n ^ a n d W^, a n d and A 2 a r e d e t e r m i n e d by t h e c o n d i t i o n t h a t t h e d i p o l e moments o f t h e c h a r g e d i s t r i b u t i o n s o n t h e r i g h t a n d l e f t h a n d s i d e s o f (56) a r e e q u a l . S u b s t i t u t i o n o f (56) i n t o (55) y i e l d s -41- and the right hand side now involves integrals which can be evaluated by the numerical method discussed a- bove. Secular equations for wave functions of the type in equation (36) were solved with computer programmes from Quantum Chemistry Programme Exchange. ?5»76 For the molecular orbital and perfect pairing calculations, the molecular energies were minimized by varying the relevant mixing parameters by making successive five point per variable grid searches u n t i l the energy con- verged to the f i f t h decimal place (energies in atomic units). The optimum orbital exponents were obtained by varying the individual exponents in turn u n t i l self-con- sistency was achieved in the exponent values to two de- cimal places. The bond distances corresponding to mi- nimum energies for the various wave functions were ob- tained by determining the orbital exponents for minimum energy for a series of bond distances, and then inter- polating exponents linearly and calculating energies for the intermediate lengths, thereby allowing estima- tion of the equilibrium distance. -42- C h a p t e r T h r e e R e s u l t s a n d D i s c u s s i o n U s i n g t h e wave f u n c t i o n s and p r o c e d u r e s d e s c r i b e d i n c h a p t e r t w o , a s e r i e s o f c a l c u l a t i o n s h a v e b e e n made f o r HF, H F + , a n d HO i n t h e i r g r o u n d s t a t e s , and a l s o f o r H F + i n t h e 2 £ s t a t e o b t a i n e d o n i o n i z i n g a f l u o r i n e c o r e I s e l e c t r o n f r o m HF, C o m p u t a t i o n s h a v e b e e n made u s i n g m o l e c u l e - o p t i m i z e d e x p o n e n t s f o r t h e S l a t e r - t y p e f u n c t i o n s , and t h e r e s u l t i n g wave f u n c t i o n s , m o l e c u l a r e n e r g i e s , o n e - e l e c t r o n e n e r g i e s , M u l l i k e n p o p u l a t i o n s , d i p o l e moments, H-F bond d i s t a n c e s and f o r c e c o n s t a n t s a r e r e p o r t e d i n T a b l e s 5 - 1 2 , I n c l u d e d i n t h e s e t a b l e s a r e c o m p a r a t i v e r e s u l t s o b t a i n e d f r o m c a l c u l a t i o n s u s i n g f r e e atom e x p o n e n t s . 7 ? Table 5. Orbital exponents and molecular properties for different wave functions of HF at the experimental bond distance (1.733 a.u.) Orbital exponents Molecular properties ^ Wave function His F2s F2pcr F2p7T Energy (a.u.) Dipole Moment (D) Ionization potentials (l<r)(eV) (lir) (eV) pp a 1.34 2.55 2.60 2.49 -99.5450 1.73 713.29 12.39 PP b 1.38 2.56 2.59 2.49 -99.5449 1.77 PP c 1.00 2.56 2.55 2.55 -99.4956 1.44 712.32 12.45 mo a 1.32 2.56 2.67 2.50 -99.5355 1.44 714.07 12.66 mo b 1.36 2.56 2.63 2.49 -99.5346 1.55 mo c 1.00 2.56 2.55 2.55 -99.4908 1.12 714.12 13.32 A l l Fls exponents have been optimized at 8,65. * A l l properties have been calculated exactly, a - A l l orbital exponents have been optimized completely. b - Orbital exponents have been optimized with the Lowdin approximation in (56). c - Free atom exponents have been used. Table 6 . Variation parameters and Mulliken populations for different wave functions of HF at the experimental bond distance (1.733 a.u.) Variation parameters Mulliken populations Wave function sin <* sin JT sin 5 s i n ^ sin 6 His F2s F2p<r pp a 0.1203 0 . 9 3 ^ 0.685 1.999 1.316 PP b 0.1281 0.9313 0.679 1.999 1.322 pp c 0.0875 0.9^38 0.702 2.000 1.298 mo a 0.4000 0.6000 0.0344 -0.1970 0.773 1.9^4 1.284 mo b 0.2797 0.5922 -0.0422 -0.1488 0 . 7 ^ 1.9^5 1.312 mo c 0.309^ 0.6250 -0.0219 -0.2042 0.781 1.959 1.260 a, b, c f - are as in Table 5 . T a b l e 7. O r b i t a l e x p o n e n t s and m o l e c u l a r p r o p e r t i e s f o r d i f f e r e n t wave f u n c t i o n s o f HF a t c a l c u l a t e d e q u i l i b r i u m b o n d d i s t a n c e s O r b i t a l e x p o n e n t s M o l e c u l a r p r o p e r t i e s Wave f u n c t i o n H i s F2s F2po- F2pjr E n e r g y ( a . u . ) B o nd l e n g t h ( a . u . ) D i p o l e Moment(D.) F o r c e c o n s t a n t pp a 1.33 2.55 2.60 2.49 -99.5456 1.77 1.72 a. 3 PP b 1.35 2.55 2.58 2.49 -99.5410 1.79 I.83 7.1 pp c 1.00 2.56 2.55 2.55 -99.5088 1.93 1.31 7.1 mG a 1.31 2.56 2.66 2.50 -99.5356 1.75 1.41 8.5 mo b 1.35 2.56 2.62 2.49 -99.5263 1.76 1.71 7.6 mo c 1.00 2.56 2.55 2.55 -99.5021 1.93k 0.89 7.2 A l l F l s e x p o n e n t s have b e e n o p t i m i z e d a t 8.65. a - A l l o r b i t a l e x p o n e n t s have b e e n o p t i m i z e d e x a c t l y . b - The L o w d l n a p p r o x i m a t i o n i n (56) h a s b e e n u s e d f o r c a l c u l a t i n g e x p o n e n t s and m o l e c u l a r p r o p e r t i e s , c - F r e e atom e x p o n e n t s have b e e n u s e d and m o l e c u l a r p r o p e r t i e s h a v e b e e n c a l - c u l a t e d e x a c t l y . Table 8. Variation parameters and Mulliken populations for different wave functions of HP at calculated equilibrium bond distances Variation parameters Mulliken ] populations Wave function sin <* sin if sin b sin ̂  sin 6 His F2s F2p<r PP a 0.1188 0.9378 0.694 1.999 1.306 PP b 0.1376 0.9313 0.682 1.999 1.319 PP c 0.0700 0.9613 0.769 2.000 1.232 mo a 0.4938 0.5875 0.1000 -0.2301 0.777 1,9^8 1.275 mo b 0.3125 0.5750 -0.0313 -0.1428 0.721 1.936 1.3^3 mo c 0.3375 0.6563 0.0187 -0.2185 0.860 1.965 1.175 a, b, c, - are as in Table 7 . Table 9 . Orbital exponents and molecular properties for different wave functions of HF* at HF experimental bond distance (1 , 7 3 3 a.u.) Orbital exponents Molecular properties Wave function His Fls F2s F2p<r- F2pfl" Energy (a.u.) Dipole Moment (D.) PP a 1.49 8.97 2.77 3 . 0 3 2.97 -74.1245 2.70 pp b 1.50 8.97 2.77 3.02 2.97 -74.1245 2.73 PP c 1.00 8.65 2.56 2.55 2.55 -73 . 5 3 0 3 3.80 mo a 1.49 8.97 2.77 3 . 0 3 2 . 9 6 -74 . 1 1 0 3 2.81 mo b 1.50 8.97 2.77 3.02 2 . 9 6 -74.1102 2.80 mo c 1.00 8.65 2.56 2.55 2.55 -73.5316 3.83 a, b, cr are as in Table 7 . Table 10. Variation parameters and Mulliken populations for different wave functions of HF \ at HF experimental bond distance ( 1 . 7 3 3 a, ,u.) Variation parameters Mulliken populations Wave function sin o< sin * sin & sin % sin e. His F2s F2po- PP a 0 . 1 6 2 5 0 . 7 9 3 8 0.466 1 . 9 9 9 1 . 5 3 6 PP b 0 . 1 6 2 5 0 . 7 8 8 8 0 . 4 5 9 1 . 9 9 9 1 . 5 ^ 2 PP c 0 . 0 0 0 0 0 . 3 7 1 9 0.146 2 . 0 0 0 1 . 8 5 4 mo a 0 . 4 1 2 5 0.4000 0 . 0 0 0 0 - 0 . 1 1 5 7 0 . 4 5 2 1 . 9 5 1 1 . 5 9 7 mo b 0 . 3 6 2 5 0 . 4 0 3 1 -0.0187 -0.1024 0.451 1 . 9 5 ^ 1 . 5 9 5 mo c 0.0469 0.1719 - 0 . 0 5 0 0 - 0 . 0 5 5 5 0.108 2 . 0 3 3 1.859 a, b, c,- are as in Table 7 . T a b l e 11. O r b i t a l e x p o n e n t s and m o l e c u l a r p r o p e r t i e s f o r a s e r i e s o f wave f u n c t i o n s f o r HF l t and H0_ O r b i t a l e x p o n e n t s M o l e c u l a r p r o p e r t i e s Wave f u n c t i o n H i s F l s F2s F2p<r F 2 p T E n e r g y ( a . u . ) Bond l e n g t h ( a . u , ) D i p o l e Moment(D.) HF pp a 1.48 8.64 2 . 6 3 2.74 2.67 -99.1983 1.73 2.44 b 1.48 8.64 2 . 6 3 2.73 2.67 -99.1983 1.73 2.44 c 1.00 8.65 2.56 2 . 5 5 2 . 5 5 -99.0989 1.73 2.79 d 1.43 8.64 2.63 2.71 2.68 -99.2031 1.84 2 . 5 5 HF mo a 1.47 8.64 2 . 6 3 2.74 2.67 -99.1835 1.73 2.46 b 1.47 8.64 2 . 6 3 2.74 2.67 -99.1835 1.73 2.46 c 1.00 8.65 2.56 2 . 5 5 2 . 5 5 -99.0900 1.73 2.87 H0*pp b 1.28 7 . 6 6 2.24 2.27 2.18 -75.1154 I.83 1.16 * C a l c u l a t i o n f o r HO ** 1 . 7 3 i s a f i x e d v a l u e f o r t h e bond, l e n g t h , a , b, c , - a r e a s i n T a b l e 7 . d. - The L o w d i n a p p r o x i m a t i o n h a s b e e n used, f o r c a l c u l a t i n g e x p o n e n t s a n d m o l e c u l a r p r o p e r t i e s . Table 12. Variation parameters and Mulliken populations for a series of wave functions for and HO Variation parameters Mulliken populations Wave function sin <* sin r sin b sin ̂  sin £ His F2s F2p<r- PP a 0.1560 0.8391 0.525 1.999 1.476 PP b 0.1563 0.8375 0.523 1.999 1.478 PP c O.0656 0.7313 0.384 2.000 1.616 pp d 0.1594 0.8438 0.53^ 1.999 1.467 mo a 0.0734 0.4563 -0.1391 -0.0311 0.530 1.948 1.522 mo b 0.0734 0.4563 -0.1391 -0.0311 0.530 1.948 1.522 mo c 0 . 2 9 H 0.3625 -0.0438 -0.1115 0.373 1.982 1.645 PP b* 0.1500 0.9881 0.863 1.999 1.139 * Calculation for HO. a, h, c, - are as in Table 7. d - is as in Table 11. -51- A t o m l c O r b i t a l E x t x m e n t s The f i r s t c h o i c e o f v a r i a b l e s f o r m o l e c u l a r c a l - c u l a t i o n s w i t h a b a s i s s e t o f S l a t e r - t y p e o r b i t a l s c o n - c e r n s t h e s e l e c t i o n o f a p p r o p r i a t e o r b i t a l e x p o n e n t s . I n s e m i - e m p i r i c a l s c h e m e s , f r e e a t om e x p o n e n t s a r e u s u - a l l y used,3^»39 h o w e v e r , e x a m i n a t i o n o f t h e o p t i m i z e d e x p o n e n t v a l u e s i n T a b l e s 5» 7» 9» a n d 11 shows t h a t i n c e r t a i n c a s e s t h e e x p o n e n t v a l u e s a r e c o n s i d e r a b l y mo- d i f i e d f r o m a t o m i c v a l u e s ; a n d t h i s i n d i c a t e s t h a t i n g e n e r a l t h e c h o i c e o f s u i t a b l e e x p o n e n t v a l u e s i s n o t a t r i v i a l o n e . L o o k i n g f i r s t a t t h e e x p o n e n t v a l u e s f o r HF i n T a b l e s 5 and 7, a s i g n i f i c a n t c h a n g e h a s o c c u r r e d i n t h e H i s e x p o n e n t , f r o m 1.00 f o r t h e f r e e atom v a l u e t o t h e o p t i m i z e d v a l u e o f a b o u t 1.32, d e p e n d i n g o n t h e p a r t i c u l a r wave f u n c t i o n . As t h e d i s t a n c e o f maximum p r o b a b i l i t y f o r a S l a t e r - t y p e o r b i t a l i s g i v e n by • , (58) w h e r e n i s t h e p r i n c i p a l q u a n t u m number and °< i s t h e o r b i t a l e x p o n e n t , a n i n c r e a s e i n a n e x p o n e n t v a l u e c o r - r e s p o n d s t o a c o n t r a c t i o n o f t h e S l a t e r - t y p e o r b i t a l . The H i s o r b i t a l a p p e a r s t o be c o n t r a c t e d l n HF com- p a r e d w i t h t h e f r e e H a t o m , a n d t h e c o n t r a c t i o n c a n be r e l a t e d , I n p a r t a t l e a s t , t o a t r a n s f e r o f c h a r g e f r o m -52- H t o F e x p e c t e d by e l e c t r o n e g a t i v i t y a r g u m e n t s a n d shown by t h e K u l l l k e n p o p u l a t i o n s i n T a b l e s 6 and 8 . T h i s c h a r g e t r a n s f e r r e s u l t s i n H b e c o m i n g p o s i t i v e l y c h a r g e d and t h e e l e c t r o n i c d e n s i t y a t H i s i n c o n s e - q u e n c e h e l d more t i g h t l y . The o p t i m i z e d e x p o n e n t s f o r t h e F l s , F2s, a n d F2pfl" o r b i t a l s e x p e r i e n c e o n l y s m a l l c h a n g e s f r o m t h e f r e e a t o m v a l u e s , a l t h o u g h a n i n c r e a s e o f a r o u n d 0.1 i s shown by t h e F2p<?~ o r b i t a l . P r e v i o u s e x p e r i e n c e ^ 6 , 7 8 , 7 9 j i a g ±nftic8Lte& t h a t e x p o n e n t v a l u e s o f t e n t e n d t o i n c r e a s e by t h i s amount f o r o r b i t a l s i n - v o l v e d i n b o n d i n g , a n d t h i s c a n be r e l a t e d t o t h e V i - r i a l T h e o r e m . ^ 0 When h i g h l y p o l a r i z a b l e e x c i t e d o r b i - t a l s a r e i n v o l v e d i n b o n d i n g h o w e v e r , t h e c h a n g e s i n 81 e x p o n e n t v a l u e s may be l a r g e . x S i m i l a r e x p o n e n t v a - l u e s a r e f o u n d f r o m b o t h t h e p e r f e c t p a i r i n g and m o l e - c u l a r o r b i t a l c a l c u l a t i o n s f o r HF, t h e g r e a t e s t d i f - f e r e n c e f o r t h e c a l c u l a t e d e q u i l i b r i u m b o n d l e n g t h i s 0.06 f o r F2p*r. The r e s u l t s f o r HF and H F + J r show t h a t t h e F l s e x p o n e n t i s n o t s e n s i t i v e t o c h a n g e s i n t h e v a l e n c e s h e l l e l e c t r o n i c s t r u c t u r e , a n d t h i s i s e x p e c t e d f o r a c o r e o r b i t a l w h i c h h a s a v e r y l o w p o l a r i z a b l l i t y , A s shown i n T a b l e 9 h o w e v e r , o n i o n i z i n g a n F l s e l e c - t r o n f r o m HF t h i s e x p o n e n t v a l u e i s i n c r e a s e d v e r y s i g n i f i c a n t l y . I n g e n e r a l , f o r t h i s i o n i z a t i o n a l l t h e e x p o n e n t s a r e i n c r e a s e d a n d t h i s c o r r e s p o n d s t o a -53- c o n t r a c t i o n o f t h e a t o m i c o r b i t a l s w h i c h i s e x p e c t e d s i n c e t h e r e m a i n i n g e l e c t r o n s w i l l be h e l d more t i g h t l y i n t h e p o s i t i v e l y c h a r g e d s p e c i e s , a s h a s b e e n n o t e d i n p r e v i o u s c a l c u l a t i o n s o n C 2 H2 a n d C 2 H 2 + b y Goodman a n d 8? G r i f f i t h , a l t h o u g h t h e s e w o r k e r s d i d n o t o p t i m i z e t h e H i s e x p o n e n t w h i c h t h e y f i x e d a t 1.20. I n HF"t t t h e F l s e x p o n e n t now h a s a v a l u e o f 8.97 w h i c h i s v e r y c l o s e t o t h e v a l u e (9.00) o b t a i n e d b y S l a t e r ' s r u l e s . ^ The H i s e x p o n e n t i n H F + a ^ i s 0.5 l a r g e r t h a n t h e f r e e a t o m v a l u e , a n d 0,2 l a r g e r t h a n t h a t f o r HF. A g a i n , t h i s i n c r e a s e f o r HF+j^ c a n be a s s o c i a t e d w i t h t h e l a r g e t r a n s f e r o f e l e c t r o n i c c h a r g e f r o m H t o F. I t may be n o t e d t h a t t h e e f f e c t o f t h i s c h a r g e t r a n s f e r i s t h a t H F + a ^ a p p r o x i m a t e s t o t h e s i t u a t i o n r e p r e s e n t e d b y H+-F w h e r e t h e e l e c t r o n d i s t r i b u t i o n a t F t e n d s t o w a r d s s p h e r i c a l s y m m e t r y . T h i s i s r e f l e c t e d i n t h e F2p(j-and F2pn- e x p o n e n t s b e i n g more n e a r l y e q u a l t h a n , f o r e x a m p l e , i n HF, The o p t i m u m e x p o n e n t v a l u e s i n t h e p e r f e c t p a i r - i n g a n d m o l e c u l a r o r b i t a l m o d e l s a r e n e a r l y i d e n t i c a l , t h e g r e a t e s t d i f f e r e n c e b e i n g 0,01, and a g a i n t h i s c a n by r a t i o n a l i z e d b y t h e t e n d e n c y t o a p p r o a c h H + - F , The o p t i m u m e x p o n e n t v a l u e s f o r HF i f t., l i s t e d i n T a b l e 11, a r e i n t e r m e d i a t e b e t w e e n t h e e x p o n e n t v a l u e s f o r HF and t h o s e f o r K F I ^ , W i t h t h e same d o u b l y o c c u - p i e d F l s c o r e , t h e F l s e x p o n e n t f o r HF"^^ h a s t h e v a l u e o f 8,65 e q u a l b o t h t o t h a t f o r t h e f r e e a t o m , and t h a t - 5 4 - f o r HF. The c o n t r a c t i o n o f t h e H i s o r b i t a l i n H F + r i s l e s s t h a n t h a t i n HF*^ b u t g r e a t e r t h a n t h e H i s c o n - t r a c t i o n i n HF. The r e s u l t s l n T a b l e s 5 - 1 2 g i v e e v i d e n c e t h a t c a l c u l a t i o n s o f m o l e c u l a r p r o p e r t i e s s u c h a s b o n d l e n g t h s and d i p o l e moments w i t h minimum b a s i s s e t s a r e much i m - p r o v e d i f m o l e c u l e - o p t i m i z e d e x p o n e n t s a r e u s e d r a t h e r t h a n f r e e a t o m e x p o n e n t s . I n a p p l y i n g t h i s r e s u l t t o more c o m p l e x m o l e c u l e s , i t w i l l be n e c e s s a r y t o h a v e c o n - v e n i e n t a n d r e l a t i v e l y i n e x p e n s i v e m e t h o d s f o r o p t i m i z i n g S l a t e r o r b i t a l e x p o n e n t s , a nd w i t h t h i s i n m i n d c o n s i d e r - a t i o n h a s b e e n g i v e n t o o b t a i n i n g o p t i m a l e x p o n e n t s when t h e t i m e - c o n s u m i n g e l e c t r o n r e p u l s i o n i n t e g r a l s i n v o l v i n g two t w o - c e n t r e c h a r g e d i s t r i b u t i o n s a r e e v a l u a t e d u s i n g t h e a p p r o x i m a t i o n due t o L o w d i n i n e q u a t i o n ( 5 6 ) . The f i r s t p o i n t t o n o t e i s t h a t i n a l l c a s e s i n T a b l e s 5t 7» 9» a n d 1 1 , t h e o p t i m u m e x p o n e n t s o b t a i n e d u s i n g t h e Low- d i n a p p r o x i m a t i o n a r e q u i t e s i m i l a r t o t h e v a l u e s f r o m e - x a c t c a l c u l a t i o n s a n d t h e r e f o r e a r e r a t h e r d i f f e r e n t f r o m f r e e a t o m v a l u e s ; i n t h e c a s e s o f HF"^ and t h e a - g r e e m e n t i s v e r y c l o s e . F o r t h e s e c a s e s t h e c o n t r a c t i o n i n t h e H i s o r b i t a l s , a s r e f l e c t e d i n t h e l a r g e H i s e x p o - n e n t v a l u e s , r e d u c e s t h e n u m e r i c a l v a l u e s o f t h e t w o - c e n t r e e x c h a n g e I n t e g r a l s w i t h t h e c o n s e q u e n c e t h a t t h e e r r o r s i n - t r o d u c e d by t h e a p p r o x i m a t i o n a r e r e d u c e d a l s o . L i k e w i s e , f o r t h e s e two s t a t e s o f H F + , t h e m o l e c u l a r p r o p e r t i e s c a l - - 5 5 - culated with the Lowdin approximation are very similar to those from the exact calculations, and even for neu- t r a l HF the errors introduced are not large considering the saving in computation time. This suggests i t could be advantageous to investigate further in this context. Molecular Energies As noted in Tables 2 and 3, (with the exception of HF + 3£ ), molecular energies lower than the values repor- ted in Tables 5 t 7 t and 11 have been given previously for the molecular species of interest here. However, the intention in the present work is to rest r i c t the ba- sis sets to forms which have applications to more com- plex molecules, and consequently the results obtained w i l l be discussed more in relation to similar calcula- tions, rather than to those with the extended basis sets noted in Tables 2 and 3. The molecular energies for HF at the calculated e- qullibrium bond distances l i s t e d in Table 7» show that the perfect pairing model gives a molecular energy ( - 9 9 . 5 ^ 5 6 a.u.) which is 6.27 kcal./mole lower than that from the molecular orbital model ( - 9 9 . 5 3 5 6 a.u.) for a minimum basis set of Slater-type orbitals with energy-optimized exponents. Using the Lowdin approx- -56- i m a t i o n f o r t h e t w o - c e n t r e e x c h a n g e i n t e g r a l s i n t h e way d e s c r i b e d a b o v e , t h e p e r f e c t p a i r i n g wave f u n c t i o n g i v e s a m o l e c u l a r e n e r g y (-99.5410 a.u.) 9.22 k c a l . / m o l e l o w e r t h a n t h e m o l e c u l a r o r b i t a l c a l c u l a t i o n (-99.5263 a . u . ) , and t h i s p e r f e c t p a i r i n g e n e r g y i s 2.88 k c a l . / m o l e h i g h e r t h a n t h a t o b t a i n e d when a l l I n t e g r a l s a r e e v a l u a t e d e x - a c t l y . U s i n g f r e e a t o m e x p o n e n t s ? ? and e v a l u a t i n g a l l i n t e g r a l s e x a c t l y , t h e m o l e c u l a r e n e r g y f o r HF f o r t h e p e r f e c t p a i r i n g wave f u n c t i o n (-99.5088 a.u.) i s 3.20 k c a l . / m o l e l o w e r t h a n t h a t f o r t h e m o l e c u l a r o r b i t a l wave f u n c t i o n (-99. 5021 a.u.) b u t i t i s 23.07 k c a l . / m o l e a b o v e t h a t o b t a i n e d w i t h e n e r g y - o p t i m i z e d e x p o n e n t s . S i m i l a r l y , t h e e n e r g y f o r HF c a l c u l a t e d w i t h t h e m o l e - c u l a r o r b i t a l m o d e l i s 21.00 k c a l . / m o l e h i g h e r when f r e e a t o m e x p o n e n t s a r e u s e d I n s t e a d o f m o l e c u l e - o p t i m i z e d e x p o n e n t s . T h u s f o r t h e t h r e e d i f f e r e n t s e t s o f c a l c u - l a t i o n s i n T a b l e 7, t h e p e r f e c t p a i r i n g m o d e l y i e l d s l o w e r e n e r g i e s t h a n t h e c o r r e s p o n d i n g m o l e c u l a r o r b i t a l c a l c u l a t i o n s , and t h e u s e o f f r e e a t o m e x p o n e n t s g i v e s e n e r g i e s more t h a n 20 k c a l . / m o l e h i g h e r t h a n t h e e n e r - g i e s o b t a i n e d w i t h m o l e c u l e - o p t i m i z e d e x p o n e n t s . A s e x p e c t e d , r e s u l t s i n T a b l e 5 show s i m i l a r t r e n d s f o r c a l c u l a t i o n s o n HF w i t h t h e bond l e n g t h f i x e d a t t h e e x p e r i m e n t a l v a l u e (1.733 a . u . ) . P r e v i o u s l y , H a n s i l ^ ? h a s u s e d a minimum b a s i s s e t o f S l a t e r - t y p e o r b i t a l s f o r a m o l e c u l a r o r b i t a l c a l c u l a - - 5 7 - t l o n o n KP a t t h e e x p e r i m e n t a l bond d i s t a n c e and r e p o r - t e d a n e n e r g y o f - 9 9 . 4 7 8 5 a » u » u s i n g o r b i t a l e x p o n e n t s o b t a i n e d f r o m S l a t e r ' s r u l e s ( K l s = 1 . 0 0 , F l s = 8 . 7 0 , F 2 s = 2 . 6 0 , F 2 p = 2 . 6 0 ) . A n e n e r g y 0 . 0 0 0 6 a . u . ( O . 3 8 k c a l . / m o l e ) h i g h e r t h a n R a n s i l ' s e n e r g y vras o b t a i n e d w i t h t h e c o m p u t e r programme u s e d i n t h i s s t u d y f o r t h e same v a l u e s o f b o n d l e n g t h a n d o r b i t a l e x p o n e n t s . T h i s d i f f e r e n c e i s a t t r i b u t e d t o t h e u s e i n t h i s w o r k o f t h e p u r e I s a t o m i c o r b i t a l a t F i n t h e m o l e c u l a r o r b i t a l wave f u n c t i o n i n e q u a t i o n ( 3 9 ) . By t h e v a r i a t i o n p r i n - c i p l e t h i s c o n s t r a i n t must r a i s e t h e e n e r g y c o m p a r e d w i t h t h e c a s e when a l l a t o m i c o r b i t a l s o f t h e same sym- m e t r y t y p e a r e f r e e l y m i x e d t o f o r m t h e m o l e c u l a r o r b i - t a l s . When t h e same v a l u e s a r e g i v e n s e p a r a t e l y t o t h e F 2 s and t h e F2pff" e x p o n e n t s i n t h e h y b r i d s d^ a n d d g t t h e p e r f e c t p a i r i n g wave f u n c t i o n i n e q u a t i o n ( 3 ^ ) c o r r e s - p o n d s t o a f r e e m i x i n g o f t h e c o n f i g u r a t i o n s ( F 2 s ) ( H l s ) 1 ( F 2 p < r ) 1 t ( F 2 s ) 2 ( F 2 p < r ) 2 , a n d ( F 2 p < r ) 2 ( H i s ) 1 ( F 2 S ) 1 , ( o m i t t i n g t h e common c o r e ( F l s ) 2 ( F 2 p 7 T ' 1 ) 2 ( F 2 p i T 1 ) 2 ) . P r e - v i o u s l y , S i l k and M u r r e l l , 5 9 u s i n g a minimum b a s i s s e t o f S l a t e r - t y p e o r b i t a l s , m i x e d t h e s e t h r e e c o n f i g u r a - t i o n s I n c o m b i n a t i o n w i t h t h r e e more c o n f i g u r a t i o n s c o r - r e s p o n d i n g t o E _ F + . S i l k a n d K u r r e l l ' g a v e t h e same e x - p o n e n t v a l u e s f o r t h e F2pc" and F2pfl" o r b i t a l s and c a l c u - l a t e d a n e q u i l i b r i u m b o n d l e n g t h o f 1 . 8 6 a . u . i n o n l y -58- f a i r a g r e e m e n t w i t h t h e e x p e r i m e n t a l v a l u e (1.733 a . u . ) . M o l e c u l a r e n e r g i e s f o r t h e p e r f e c t p a i r i n g wave f u n c t i o n i n e q u a t i o n (34) c a n n o t be c o m p a r e d d i r e c t l y w i t h S i l k a n d M u r r e l l ' s b e c a u s e t h e y m i x e d more c o n f i g u r a t i o n s , b u t a t 1.4 a . u . w here t h e y f i n d t h e H ~ P + c o n f i g u r a t i o n s t o c o n t r i b u t e o n l y s l i g h t l y , t h e programmes u s e d i n t h i s w o r k g i v e a m o l e c u l a r e n e r g y (-99.472 5 a.u.) 1.25 k c a l . / m o l e h i g h e r t h a n t h e i r p u b l i s h e d v a l u e (-99.4745 a . u . ) , u s i n g t h e i r e x p o n e n t s . As f o r HF, c a l c u l a t i o n s o n HF",^ u s i n g o p t i m i z e d o r b i t a l e x p o n e n t s g i v e l o w e r m o l e c u l a r e n e r g i e s f o r t h e p e r f e c t p a i r i n g wave f u n c t i o n t h a n f o r t h e m o l e c u l a r o r - b i t a l m o d e l . When e x p o n e n t s a r e o p t i m i z e d f o r e x a c t c a l c u l a t i o n o f a l l m o l e c u l a r I n t e g r a l s , t h e p e r f e c t p a i r i n g e n e r g y i s 8.91 k c a l . / m o l e l o w e r t h a n t h e m o l e - c u l a r o r b i t a l e n e r g y . A s n o t e d a b o v e , t h e u s e o f t h e L o w d i n a p p r o x i m a t i o n y i e l d s e x p o n e n t s i n c l o s e a g r e e m e n t w i t h t h o s e f r o m t h e e x a c t c a l c u l a t i o n , and t h e i n c r e a s e i n e n e r g y i n u s i n g L o w d i n e x p o n e n t s i s o n l y 0.07 k c a l . / m o l e . A v e r y g r e a t d i f f e r e n c e o c c u r s when f r e e a t o m e x p o n e n t s a r e u s e d ; i n t h e p e r f e c t p a i r i n g m o d e l t h e m o l e c u l a r e n e r g y i s t h e n 373 k c a l . / m o l e a b o v e t h e v a l u e o b t a i n e d w i t h m o l e c u l e - o p t i m i z e d e x p o n e n t s a n d t h e c o r r e s p o n d i n g v a l u e f o r t h e m o l e c u l a r o r b i t a l m o d e l i s 363 k c a l . / m o l e . T h i s l a r g e d i f f e r e n c e b e t w e e n e n e r g i e s o b t a i n e d u s i n g f r e e a t om e x p o n e n t s and t h o s e o b t a i n e d -59- w i t h o p t i m i z e d e x p o n e n t s , e m p h a s i z e s t h a t e l e c t r o n i c r e - l a x a t i o n m ust be i n c l u d e d i n c a l c u l a t i o n s o f E.S.C.A. e n e r g i e s . A s n o t e d i n t h e s e c t i o n o n o r b i t a l e x p o n e n t s , h o w e v e r , t h i s s t u d y d o e s I n d i c a t e t h a t i t may be p o s s i b l e t o u s e S l a t e r ' s r u l e s ^ o r s o m e t h i n g s i m i l a r ? ? f o r e s t i - m a t i n g e x p o n e n t s when t h e p o s i t i v e i o n i s f o r m e d by t h e r e m o v a l o f a c o r e e l e c t r o n . The c o m p a r i s o n o f p e r f e c t p a i r i n g a n d m o l e c u l a r o r b i t a l e n e r g i e s f o r HP + J f f. i s s i m i l a r t o t h a t r e p o r t e d a b o v e f o r HF and HF 4"^ , and t h e d e t a i l s a r e t o be f o u n d i n T a b l e 1 1 . The i n t e r e s t i n t h e e n e r g y o f H F i n t h i s w o r k i s m a i n l y i n r e l a t i o n t o c a l c u l a t i n g t h e f i r s t i o n i z a t i o n p o t e n t i a l o f HF w i t h a minimum b a s i s s e t o f S l a t e r - t y p e o r b i t a l s . The e n e r g i e s a r e p r e s e n t e d i n T a b l e 5 f o r t h e i o - n i z i n g o f a n F l s e l e c t r o n ( l t r ) o r a n F2p7T e l e c t r o n ( i f f ) a s c a l c u l a t e d a s s u m i n g no r e o r g a n i z a t i o n o f t h e r e m a i n - i n g e l e c t r o n s ( t h i s i s u s u a l l y r e f e r r e d t o a s Koopman' T h e o r e m ^ ) . A v a l u e o f 13.32 e.V, i s c a l c u l a t e d f o r t h e 1 vr i o n i z a t i o n p o t e n t i a l w i t h f r e e a t o m e x p o n e n t s , and a bond l e n g t h o f 1.733 a . u , , and t h i s i s t o be c o m p a r e d 84 w i t h t h e e x p e r i m e n t a l v a l u e o f 16,06 e.V, by p h o t o e - l e c t r o n s p e c t r o s c o p y . U s i n g K o o p m a n s 1 T h e o r e m and e x - p o n e n t s g i v e n by S l a t e r ' s r u l e s , P o p l e and B e v e r i d g e ^ r e p o r t a v a l u e o f 12.65 e.V. w i t h t h e m o l e c u l a r o r b i t a l t h e o r y , and w i t h S l a t e r - t y p e o r b i t a l s o p t i m i z e d f o r HF -60- ( T a b l e 5) c a l c u l a t e d v a l u e s o f 12.66 a n d 12.39 e.V. a r e o b t a i n e d f o r t h e m o l e c u l a r o r b i t a l and p e r f e c t p a i r i n g m o d e l s r e s p e c t i v e l y . I n p r i n c i p l e , a n i m p r o v e d c a l c u l a - t i o n o f t h e v e r t i c a l i o n i z a t i o n p o t e n t i a l i s made by t a k i n g t h e d i f f e r e n c e s b e t w e e n t h e m o l e c u l a r e n e r g i e s o f HF and H F + J / r f o r t h e H-F bond l e n g t h , b u t s u r p r i s i n g - l y t h e f i r s t i o n i z a t i o n p o t e n t i a l c a l c u l a t e d t h i s way h a s a v a l u e i n l e s s g o o d a g r e e m e n t f o r b o t h t h e p e r f e c t p a i r i n g a n d t h e m o l e c u l a r o r b i t a l m o d e l u s i n g e i t h e r o p t i m i z e d o r f r e e a t o m e x p o n e n t s , t h a n t h e v a l u e o b - t a i n e d w i t h Koopmans' T h e o r e m . T h u s a t t h e e x p e r i m e n t a l bond l e n g t h (1.733 a.u.) e n e r g y d i f f e r e n c e s b e t w e e n T a - b l e s 5 and 11 g i v e a v a l u e o f 10.90 e.V. f r o m t h e m o l e - c u l a r o r b i t a l m e t h o d u s i n g f r e e a t o m e x p o n e n t s . The r e a s o n s f o r t h e l e s s good a g r e e m e n t i n t a k i n g t h e d i f - f e r e n c e s b e t w e e n t h e s t a t e e n e r g i e s i s n o t c l e a r , a l - t h o u g h p r e s u m a b l y I t i s r e l a t e d i n p a r t t o t h e r e s t r i c t e d f o r m o f t h e b a s i s s e t . A s n o t e d a l r e a d y , t h e d i f f e r e n c e s b e t w e e n u s i n g f r e e a t om e x p o n e n t s and m o l e c u l e - o p t i m i z e d e x p o n e n t s a r e much l a r g e r f o r c a l c u l a t i n g t h e e n e r g y o f i o n i z i n g a n F l s e l e c t r o n . U s i n g Koopmans* T h e o r e m , t h i s i o n i z a t i o n p o t e n t i a l i s c a l c u l a t e d t o be 71^.12 e.V. when u s i n g f r e e a t o m e x p o n e n t s and t h e m o l e c u l a r o r b i t a l m e t h o d . T h i s v a l u e c a n be c o m p a r e d w i t h t h e v a l u e s o f 706.26 e.V, a n d 691.57 e.V, o b t a i n e d r e s p e c t i v e l y w i t h f r e e a t om -61- e x p o n e n t s and m o l e c u l e - o p t i m i z e d e x p o n e n t s when d i f - f e r e n c e s i n t h e m o l e c u l a r o r b i t a l e n e r g i e s f o r HF and HF+j£ i n T a b l e s 5 and 9 a r e t a k e n . U n f o r t u n a t e l y , t h e s e numbers c a n n o t be c o m p a r e d w i t h e x p e r i m e n t s i n c e HF d o e s n o t seem t o h a v e b e e n s t u d i e d by E.S.C.A. y e t . Bond L e n g t h s and F o r c e C o n s t a n t s E q u i l i b r i u m b o n d l e n g t h s h a v e b e e n c a l c u l a t e d u - s i n g t h e p e r f e c t p a i r i n g a n d m o l e c u l a r o r b i t a l wave f u n c t i o n s f o r v a r i o u s s e t s o f o r b i t a l e x p o n e n t s , a n d t h e m e t h o d f o r o b t a i n i n g t h e e q u i l i b r i u m d i s t a n c e h a s b e e n d e s c r i b e d o n page M. The r e s u l t s i n T a b l e 7 show t h a t t h e m o l e c u l a r o r b i t a l m e t h o d u s i n g m o l e c u l e - o p t i m i z e d e x p o n e n t s g i v e s a n e q u i l i b r i u m b o n d d i s t a n c e o f 1.75 a . u . w h i c h i s o n l y 0.02 a . u . (0.01 A) l o n g e r t h a n t h e e x p e r i m e n t a l v a l u e o f 1.733 a.u.$3 The p e r f e c t p a i r i n g m e t h o d g i v e s a c a l c u l a t e d v a l u e o f 1.77 a . u . . U s i n g e x p o n e n t v a l u e s o b t a i n e d w i t h t h e L o w d i n a p p r o x i m a t i o n , t h e c a l c u l a t e d e q u i l i b r i u m b o n d l e n g t h s a r e w i t h i n 0.02 a . u . (0.01 A) o f t h e b e s t c a l c u l a t e d v a l u e s , and t h e r e - f o r e a r e i n r e a s o n a b l e a g r e e m e n t w i t h e x p e r i m e n t . I t i s s i g n i f i c a n t t h a t t h e c a l c u l a t i o n s w i t h f r e e a t o m e x p o - n e n t s g i v e bond l e n g t h s i n p o o r a g r e e m e n t w i t h e x p e r i - -62- m e n t a l v a l u e s . Thus u s i n g f r e e atom e x p o n e n t s , the p e r f e c t p a i r i n g model p r e d i c t s an e q u i l i b r i u m bond l e n g t h 0.20 a.u. g r e a t e r t h a n t h a t o b t a i n e d u s i n g op- t i m i z e d e x p o n e n t s . L i k e w i s e , the m o l e c u l a r o r b i t a l c a l - c u l a t i o n g i v e s a bond d i s t a n c e 0.18 a.u. l a r g e r w i t h f r e e atom exponents t h a n t h a t o b t a i n e d w i t h m o l e c u l e - o p - t i m i z e d e x p o n e n t s . Thus f o r t h e s e c a l c u l a t i o n s , t h e e r r o r i n c a l c u l a t e d bond l e n g t h s u s i n g f r e e atom expo- n e n t s i s an o r d e r o f magnitude g r e a t e r t h a n the e r r o r i n t r o d u c e d u s i n g exponent v a l u e s o p t i m i z e d w i t h Low- d i n ' s a p p r o x i m a t i o n . As d i s c u s s e d above, and shown i n T a b l e 11, the optimum o r b i t a l exponent v a l u e s f o r HF + X^. c a l c u l a t e d w i t h and w i t h o u t Lowdin's a p p r o x i m a t i o n a r e v e r y n e a r l y e q u a l . T h e r e f o r e i t seems r e a s o n a b l e t o c a l c u l a t e the e q u i l i b r i u m bond l e n g t h o f HF+, by u s i n g the Lowdin a p p r o x i m a t i o n f o r o p t i m i z i n g exponents a t d i f f e r e n t bond d i s t a n c e s . The c a l c u l a t e d e q u i l i b r i u m v a l u e o f 1.84 a.u, i s c l o s e t o the v a l u e (1.85 a.u.) c a l c u l a t e d 6? r e c e n t l y by R i c h a r d s and R a f t e r y , a l t h o u g h a p p r e c i a b l y l o w e r t h a n the v a l u e o f 1.96 a.u, quoted by F o p l e ^ 0 u- s i n g a b a s i s s e t o f G a u s s i a n f u n c t i o n s d e r i v e d from a- t o m l c wave f u n c t i o n s . No e x p e r i m e n t a l v a l u e f o r t h e e q u i l i b r i u m d i s t a n c e i n HF"^ i s p r e s e n t l y a v a i l a b l e . The Lowdin a p p r o x i m a t i o n has a l s o been used f o r a p r e - l i m i n a r y c a l c u l a t i o n on the e q u i l i b r i u m d i s t a n c e i n HO -63- f o r w h i c h t h e e x p e r i m e n t a l v a l u e i s I.83 a . u . ^ i n t h e 2/T s t a t e . U s i n g t h e L o w d i n a p p r o x i m a t i o n , t h e o p t i - m i z e d e x p o n e n t v a l u e s a t 1.83 a . u . a r e H l s = 1 . 2 8 , 01s=7 . 66 , 0 2 s = 2 . 2 4 , 02p^-=2.2?, and 0 2 p t f = 2 . l 8 and w i t h t h e s e e x - p o n e n t s t h e c a l c u l a t e d e q u i l i b r i u m d i s t a n c e i s 1.91 a . u , , 0.08 a . u , l o n g e r t h a n t h e e x p e r i m e n t a l v a l u e . As t h e r e i s l e s s c h a r g e t r a n s f e r f r o m H i n EO c o m p a r e d w i t h EF"£j. t h e L o w d i n a p p r o x i m a t i o n may be l e s s r e l i a b l e f o r HO, . I t i s p o s s i b l e , t h e r e f o r e , t h a t a c a l c u l a t i o n w i t h a l l i n t e g r a l s e v a l u a t e d e x a c t l y w o u l d g i v e a n i m p r o v e d v a - l u e f o r t h e e q u i l i b r i u m b o n d d i s t a n c e i n HO, L i k e b o n d l e n g t h s , v a l u e s o f f o r c e c o n s t a n t s a r e o f t e n u s e d t o g i v e i n f o r m a t i o n a b o u t t h e n a t u r e o f t h e b o n d i n g . The c a l c u l a t i o n o f a f o r c e c o n s t a n t d e p e n d s o n c a l c u l a t i n g m o l e c u l a r e n e r g i e s a s a f u n c t i o n o f t h e d i s p l a c e m e n t f r o m e q u i l i b r i u m , a n d i n t h e h a r m o n i c a p - 2Q / \ p r o x i m a t i o n , 7 t h e s t r e t c h i n g f o r c e c o n s t a n t ( k ) f o r a d i a t o m i c m o l e c u l e i s o b t a i n e d f r o m £ = T (r ' r * ) X (59) w h e r e E i s t h e m o l e c u l a r e n e r g y c a l c u l a t e d a t a b o n d l e n g t h r , a n d r e i s t h e c a l c u l a t e d e q u i l i b r i u m b o n d d i s t a n c e . The f o r c e c o n s t a n t , t h e r e f o r e , i s r e a d i l y d e t e r m i n e d f r o m a p l o t o f E v e r s u s ( r - r e ) . U s i n g m o l e c u l e - o p t i m i z e d e x p o n e n t s , HF s t r e t c h i n g f o r c e c o n s t a n t s e q u a l t o 8,3 x 1 0 ^ d y n e s / c m and 8 ,5 x 1 0 ^ - 6 4 - d y n e s / c m a r e o b t a i n e d f r o m t h e p e r f e c t p a i r i n g a n d mo- l e c u l a r o r b i t a l m o d e l s r e s p e c t i v e l y , a n d t h e s e v a l u e s a r e t o be com p a r e d w i t h a n e x p e r i m e n t a l v a l u e o f 9 . 6 6 x 1CK d y n e s / c m . The c o r r e s p o n d e n c e t o t h e e x p e r i m e n - t a l v a l u e i s l e s s g o o d by a b o u t 1 x 1 0 ^ d y n e s / c m when f r e e a t o m e x p o n e n t s a r e u s e d t o c a l c u l a t e t h e f o r c e c o n s t a n t . E l e c t r o n D i s t r i b u t i o n s The c a l c u l a t e d c h a r g e d i s t r i b u t i o n s i n t h e m o l e - c u l e s o f i n t e r e s t i n t h i s w o r k a r e d e t e r m i n e d b y t h e v a l u e s o f t h e v a r i a t i o n p a r a m e t e r s <*• , * , 5 a n d ̂  i n e- q u a t i o n s (31)» (35)»(40), and ( 4 l ) . E x p e r i m e n t a l l y , i n - f o r m a t i o n r e l a t i n g t o c h a r g e d i s t r i b u t i o n s i s o b t a i n e d by m e a s u r e m e n t s o f d i p o l e moments a n d o f t h e h i g h e r mo- 8 5 8 6 m e n t s , ' and t h e s e moments may be c a l c u l a t e d f r o m m o l e c u l a r wave f u n c t i o n s . T h u s , f o r a m o l e c u l e , t h e d i p o l e moment, w h i c h i s a v e c t o r q u a n t i t y , i s g i v e n b y ^ , - * < f / f c | r < > ( 6 o ) f o r a s t a t e f u n c t i o n f ; r i s a sum o f t h e e l e c t r o n p o - s i t i o n v e c t o r s . F o r d i a t o m i c h y d r i d e s w i t h c y l i n d r i c a l s y m m e t r y a b o u t t h e i n t e r n u c l e a r a x i s , t h e d i p o l e moment i s d i r e c t e d a l o n g t h i s a x i s w i t h m a g n i t u d e -65- (61) where ^ is the electronic wave function, z i s the sum of components along the internuclear axis of electron positions, r^ and are respectively the position and charge of the i ^ n nucleus. Another convenient measure of electron distribu- tions which is used frequently for molecular wave func- tions expressed as a basis of atomic orbital functions O n is provided by the population analysis due to Mulliken. In the molecular orbital model, when the l ^ n molecular orbital is expressed as % •• 1 C« ^ . (62) the total electron population of in the linear com- bination of atomic orbitals - molecular orbital method, is given by PJ*° * 1 " i {CiJ 4- £ C,-u C;YSUV ] (63) where The summation over i is over a l l occupied molecular or- bitals and n^ is the occupation number. Implicit in equation (63) is that the overlap charge distribution has been partitioned equally between the two centres -66- i n v o l v e d . An e q u i v a l e n t p o p u l a t i o n a n a l y s i s f o r v a - l e n c e bond wave f u n c t i o n s i s o b t a i n e d a c c o r d i n g : t o t h e o o f o l l o w i n g p r o c e d u r e : 0 0 - f , <65) w h e r e t h e z e r - o r d e r wave f u n c t i o n V'; c o r r e s p o n d s t o a c o n f i g u r a t i o n w i t h o c c u p a n c y n ( i ) f o r t h e a t o m i c o r - b i t a l fiu. T h e n t h e t o t a l e l e c t r o n p o p u l a t i o n i n fiu i n t h e v a l e n c e bond method i s g i v e n b y i ( i*i ) w h e r e V itiWi). (67) Some o f t h e v a r i a t i o n p a r a m e t e r s i n e q u a t i o n s (35) a n d ( 4 0 ) p r o v i d e m e a s u r e s o f e l e c t r o n d i s t r i b u t i o n s . T h u s f o r t h e p e r f e c t p a i r i n g m o d e l a n I n c r e a s e i n s i n * i n d i c a t e s a n i n c r e a s e i n t h e c h a r g e a t H; s i n 5 e q u a l t o 1.00 i m p l y i n g no c h a r g e t r a n s f e r w h i l e s i n * e q u a l t o 0.0 c o r r e s p o n d s t o t r a n s f e r o f one e l e c t r o n f r o m H. S i m i - l a r l y i n t h e m o l e c u l a r o r b i t a l m o d e l s i n S i s a m e a s u r e o f t h e c h a r g e a t H i n t h e b o n d i n g m o l e c u l a r o r b i t a l . Q u a n t i t a t i v e l y , a s t h e v a l u e s o f e i t h e r s i n * o r s i n $ d e c r e a s e , one may e x p e c t t h e H i s o r b i t a l p o p u l a t i o n t o d e c r e a s e a n d c o r r e s p o n d i n g l y t h e d i p o l e moment t o i n - c r e a s e . I n b o t h t h e m o l e c u l a r o r b i t a l and p e r f e c t p a i r - -67- i n g models, as used i n t h i s work, s i n <* i s a measure o f the sp h y b r i d i z a t i o n a t F. As s i n o<" i n c r e a s e s , the hy- b r i d d e s i g n a t e d d^ has more F2s c h a r a c t e r , and c o r r e s - p o n d i n g l y the h y b r i d d e s i g n a t e d dg has l e s s F2s c h a r a c - t e r . The t r e n d s i n t h e s e v a r i o u s measures o f e l e c t r o n d i s t r i b u t i o n w i l l now be examined f o r the d i f f e r e n t mo- l e c u l a r wave f u n c t i o n s . L o o k i n g f i r s t a t t h e r e s u l t s i n T a b l e s 7 and 8 f o r t h e c a l c u l a t e d HF e q u i l i b r i u m d i s t a n c e s , the a g r e e - ment t o 0.01 i n the v a l u e s o f s i n Jf o r s i n 5 u s i n g the e x a c t l y - o p t i m i z e d and the L o w d i n - o p t i m i z e d exponents i s r e f l e c t e d i n t h e H i s p o p u l a t i o n s b e i n g s i m i l a r f o r e i t h e r s e t o f e x p o n e n t s . The H i s p o p u l a t i o n s a r e , how- e v e r , s l i g h t l y h i g h e r f o r the m o l e c u l a r o r b i t a l model (0.78) t h a n f o r the p e r f e c t p a i r i n g model (0.69); and t h i s i s c o n s i s t e n t w i t h t h e c a l c u l a t e d d i p o l e moment b e i n g h i g h e r f o r t h e p e r f e c t p a i r i n g c a l c u l a t i o n (1.72 D.) t h a n f o r the m o l e c u l a r o r b i t a l model ( l . 4 l D.), The e x p e r i m e n t a l l y - m e a s u r e d d i p o l e moment o f HF i s 1.82 D . .^9 W i t h f r e e atom e x p o n e n t s , t h e charge r e d i s t r i - b u t i o n on f o r m a t i o n o f HF i s c a l c u l a t e d t o be l e s s , and t h i s i s r e f l e c t e d i n t h e l o w e r c a l c u l a t e d v a l u e s o f the d i p o l e moment, b e i n g 1.31 D » and 0.89 D. f o r the p e r - f e c t p a i r i n g and m o l e c u l a r o r b i t a l c a l c u l a t i o n s r e s p e c - t i v e l y . S i m i l a r t r e n d s i n r e s u l t s a r e found f o r the c a l c u l a t i o n s a t e x p e r i m e n t a l d i s t a n c e o f 1.733 a.u.. -68- The v a l u e s o f s i n <*. i n T a b l e s 6 and 8 i n d i c a t e t h a t t h e p e r f e c t p a i r i n g m o d e l i s c o n s i s t e n t w i t h some- what l e s s s p h y b r i d i z a t i o n t h a n t h e m o l e c u l a r o r b i t a l m e t h o d , a n d a l t h o u g h i t i s w e l l known t h a t t h e c o n c e p t o f h y b r i d i z a t i o n i s n o t n e c e s s a r y i n t h e m o l e c u l a r o r - 12 b i t a l t h e o r y , r e s u l t s o f c o m p a r i n g v a l u e s o f s i n «* a r e c o n s i s t e n t w i t h t h e M u l l i k e n p o p u l a t i o n s o n t h e P 2 s o r b i t a l b e i n g 0.05 g r e a t e r i n t h e p e r f e c t p a i r i n g c a l - c u l a t i o n t h a n i n t h e m o l e c u l a r o r b i t a l c a l c u l a t i o n . N e v e r t h e l e s s , t h e h y b r i d i z a t i o n a t F i n HF i s s m a l l i n b o t h m o d e l s , a s e x p e c t e d f r o m t h e a v e r a g e 2 s t o 2p p r o - m o t i o n e n e r g y , w h i c h f r o m a t o m i c s p e c t r a l d a t a i s e s - 12 t i m a t e d t o be 20.8 e.V. The r e s u l t s o b t a i n e d a r e c o n s i s t e n t w i t h t h e b o n d i n g h y b r i d d^ b e i n g e s s e n t i a l l y F2p<r, and t h e r e f o r e , t h e n o n - b o n d i n g h y b r i d d e s i g n a t e d d 2 b e i n g m a i n l y F 2 s i n c h a r a c t e r . E v e n t h o u g h t h e s p h y b r i d i z a t i o n a t F seems t o be s m a l l , t h e h y b r i d s h a v e b e e n l o o k e d a t i n a d i f f e r e n t way f o r t h e p u r p o s e o f m o l e c u l a r c a l c u l a t i o n s . T h i s e x - t e n s i o n i n v o l v e d a s s i g n i n g one o r b i t a l e x p o n e n t t o d^ a n d a d i f f e r e n t e x p o n e n t t o d 2 w i t h o u t r e g a r d t o t h e b a s i s S l a t e r - t y p e o r b i t a l s ; t h a t i s t h e S l a t e r e x p o n e n t s a r e s e l e c t e d s u c h t h a t c< is = (68) -69- and \ w i t h a. 5 ^ <=< -?s' . ( 7 0 ) With these b a s i s f u n c t i o n s , the energy was completely minimized f o r the p e r f e c t p a i r i n g wave f u n c t i o n . The o p t i m i z e d exponents were found to be Kls=1.35» Fls=8,65» F2pjr=2.49, d 1 =2.62 and d 2=2.55. The optimum exponents f o r d^ and d 2 are w i t h i n 0.02 of the optimum v a l u e s of F2p<r and F2s i n T a b l e ?, however, t h i s approach r e s u l t s i n a p e r f e c t p a i r i n g energy of -99.5^59 a.u. which i s 0.19 kcal./mole lower than the p r e v i o u s b e s t p e r f e c t p a i r i n g energy, and a c a l c u l a t e d e q u i l i b r i u m H-F d i s t a n c e of 1.75 a.u. which i s 0.02a.u. b e t t e r than the p e r f e c t p a i r i n g c a l c u l a t i o n u s i n g the more c o n v e n t i o n a l b a s i s of atomic o r b i t a l s . A s s i g n i n g exponents to h y b r i d s r a - t h e r than n a t u r a l atomic o r b i t a l s might be expected to g i v e g r e a t e r improvements i n the c a l c u l a t i o n s of p r o - p e r t i e s of molecules l n which the h y b r i d i z a t i o n of a- tomic o r b i t a l s i s suggested to occur to a g r e a t e r ex- t e n t than a t F i n HF, An i n t e r e s t i n g o b s e r v a t i o n from a l l the r e s u l t s i s t h a t with the s i n g l e e x c e p t i o n of the m o l e c u l a r o r - b i t a l c a l c u l a t i o n f o r HF* atomic o r b i t a l h y b r i d i z a - -70- t i o n a t P i s l e s s when f r e e atom exponents are used i n - st e a d of m o l e c u l e - o p t i m i z e d exponents. T h i s suggests t h a t one should perhaps be c a u t i o u s i n deducing c o n c l u - s i o n s r e g a r d i n g p o s s i b i l i t i e s o f h y b r i d i z a t i o n p u r e l y from c o n s i d e r a t i o n s r e l a t e d to da t a f o r f r e e atoms, a l - though i t may be noted t h a t i n the l i m i t of complete t r a n s f e r of e l e c t r o n i c charge from H to F, the h y b r i - d i z a t i o n p i c t u r e becomes i r r e l e v a n t . I t i s not too c l e a r a t present, but the f a c t t h a t the m o l e c u l a r o r b i - t a l c a l c u l a t i o n of HF + a j r i s out of l i n e may be r e l a t e d to t h i s c o n s i d e r a t i o n . Another odd f e a t u r e of t h i s c a l c u l a t i o n i s t h a t the M u l l i k e n p o p u l a t i o n f o r F2s i s indic a t e d , t o be g r e a t e r than 2. T h i s r e s u l t i s a s s o - c i a t e d w i t h the eq u a l p a r t i t i o n i n g of o v e r l a p charge between the two c e n t r e s i n the M u l l i k e n a n a l y s i s . T h i s i s not r e a l i s t i c and i s w e l l known to y i e l d n e g a t i v e 90 p o p u l a t i o n s i n some c a s e s . The r e s u l t s i n Tables 10 and 12 show t h a t l a r g e m o d i f i c a t i o n s occur i n the c a l c u l a t e d e l e c t r o n d i s t r i - b u t i o n s f o r HF*^ and H F + a 7 r when f r e e atom exponents are used i n s t e a d of m o l e c u l e - o p t i m i z e d exponents. How- ever, when the Lowdin approximation i s used f o r d e t e r - mining o r b i t a l exponents f o r these m o l e c u l a r s p e c i e s , the M u l l i k e n p o p u l a t i o n s and d i p o l e moments are i n c l o s e agreement w i t h those from the exact c a l c u l a t i o n s . As expected, the charge t r a n s f e r from H to F i n c r e a s e s i n - 7 1 - the s e r i e s HF, K ?+ r , HF*^ , and t h i s i s r e f l e c t e d i n the values f o r the K u l l i k e n populations and the c a l c u - l a t e d d i p o l e moments. A l s o , as noted above, the His ex- ponent tends to increase w i t h t h i s charge t r a n s f e r . A f u r t h e r comparison wi t h HF + l l r i s provided by a p r e l i m i n a r y c a l c u l a t i o n on H0ijr w i t h the p e r f e c t p a i r - i n g model and u t i l i z i n g the Lowdin approximation. In t h i s case i t turns out t h a t s i n ^ i s c l o s e to u n i t y and, c o r r e s p o n d i n g l y , the His e l e c t r o n p o p u l a t i o n i s 0.86. This i s to be expected because of the much lower e l e c - t r o n e g a t i v i t y of 0 compared w i t h F i n the species be- i n g considered. The s m a l l e r charge t r a n s f e r compared w i t h F r e s u l t s i n a lower d i p o l e moment; our c a l c u l a t e d value of 1 . 1 6 D, i s to be compared w i t h the experimental value of 1 . 6 6 D.91 f o r HO. Concluding Remarks The i n t e r e s t i n f i n d i n g u s e f u l methods f o r d e r i v i n g wave f u n c t i o n s w i t h a p p l i c a t i o n s to complex molecules stems from many c o n s i d e r a t i o n s , but without doubt there i s c u r r e n t l y much i n t e r e s t i n developing quantum chemical methods wit h a p p l i c a t i o n s to molecular systems as d i - verse as those of b i o l o g i c a l I n t e r e s t ^ a n d those present i n s t r u c t u r e s of the s o l i d s t a t e . 9 - 2 I n e v i t a b l y , methods -72- w i t h a p p l i c a t i o n to complex systems must f i r s t be t e s t e d on s i m p l e r molecules. On the whole, the molecular o r b i - t a l method has proved most u s e f u l i n a p p l i c a t i o n s to com- ple x systems so f a r , 38,39 but recent advances i n compu- t a t i o n a l techniques have i n d i c a t e d the f e a s i b i l i t y of ma- k i n g valence bond c a l c u l a t i o n s on a more r o u t i n e b a s i s to polyatomic molecules ,8»̂ 5 and i t has been known f o r some time t h a t c a l c u l a t i o n s i n the p e r f e c t p a i r i n g mo- d e l can be formulated r e a d i l y w i t h comparatively simple expressions f o r molecular e n e r g i e s . 9»69 Probably i n the f u t u r e , wave f u n c t i o n s f o r complex molecules w i l l be w r i t t e n so as to represent some h y b r i d of the molecular o r b i t a l and p e r f e c t p a i r i n g schemes, such as i s b u i l t i n t o many co n v e n t i o n a l bonding models (eg. the l o c a l i z e d and d e l o c a l i z e d components of the e l e c t r o n i c s t r u c t u r e of benzene). In p a r t , t h i s t h e s i s has been d i r e c t e d at comparing the molecular o r b i t a l and p e r f e c t p a i r i n g mo- d e l s f o r some simple diatomic h y d r i d e s , and t h i s r e p r e - sents an e x t e n s i o n of the recent work by K i t c h e l l and 46 Thirunamac hand ran on BerL,. We can conclude t h a t f o r minimum b a s i s set c a l c u l a t i o n s , both the p e r f e c t p a i r i n g and molecular o r b i t a l models provide reasonable accounts of a number of b a s i c p r o p e r t i e s of HP, HF*^ and H F + j ^ . More s p e c i f i c a l l y , c a l c u l a t i o n s of molecular energies u s i n g wave f u n c t i o n s i n the p e r f e c t p a i r i n g framework are between 6 and 10 kcal/mole lower than those from -73- c o r r e s p o n d i n g c a l c u l a t i o n s u s i n g m o l e c u l a r o r b i t a l wave f u n c t i o n s . E q u i l i b r i u m bond l e n g t h s c a l c u l a t e d w i t h b o t h t h e p e r f e c t p a i r i n g and m o l e c u l a r o r b i t a l mo- d e l s a r e w i t h i n 0.04 a . u . (0.02 A) o f t h e e x p e r i m e n t a l v a l u e f o r HF; w h e r e c o m p a r i s o n w i t h e x p e r i m e n t a l d a t a i s p o s s i b l e d i p o l e moments seem t o be b e t t e r p r e d i c t e d w i t h t h e p e r f e c t p a i r i n g m o d e l , a l t h o u g h , f o r c e c o n - s t a n t s and i o n i z a t i o n p o t e n t i a l s a r e c a l c u l a t e d w i t h s i - m i l a r r e l i a b i l i t i e s w i t h t h e p e r f e c t p a i r i n g and m o l e - c u l a r o r b i t a l m e t h o d s . To some e x t e n t t h i s c o n t r a s t s w i t h t h e s i t u a t i o n f o r BeHg, w h e r e t h e p e r f e c t p a i r i n g m o d e l seems t o be p r e f e r a b l e t o t h e m o l e c u l a r o r b i t a l m o d e l ; i n p a r t t h i s c o n t r a s t c a n be a s s o c i a t e d w i t h t h e g r e a t e r e l e c t r o n e g a t i v i t y d i f f e r e n c e b e t w e e n H a n d F? t h u s i n t h e l i m i t o f H+F*" b o t h t h e p e r f e c t p a i r i n g and t h e m o l e c u l a r o r b i t a l m o d e l s w o u l d merge w i t h t h e i o n i c m o d e l . The c o m p a r i s o n s made h e r e b e t w e e n c a l c u l a t i o n s w i t h f r e e a t o m e x p o n e n t s and t h o s e w i t h m o l e c u l e - o p t i - m i z e d e x p o n e n t s show t h a t f r e e a t o m e x p o n e n t v a l u e s c a n n o t i n g e n e r a l be c o n s i d e r e d a p p r o p r i a t e f o r c a l c u l a t i o n s w h i c h a t t e m p t t o e v a l u a t e p r o p e r t i e s s u c h a s bond l e n g t h s and d i p o l e moments. O f t e n s e m i - e m p i r i c a l m e t h o d s 3 ^ g i v e t h e same e x p o n e n t v a l u e s t o t h e d i f f e r e n t 2p o r b i t a l s , w h i c h , i n a m o l e c u l a r e n v i r o n m e n t , a r e o f t e n i n e q u l v a - l e n t . T h i s r e s t r i c t i o n i s made i n p a r t f o r s i m p l i c i t y -74- and i n part to maintain r o t a t i o n a l l n v a r i a n c e . Ne- v e r t h e l e s s , i t should be noted that t h i s c o n s t r a i n t does introduce e r r o r compared with the s i t u a t i o n where the symmetrically d i f f e r e n t 2p o r b i t a l s have d i f f e r e n t exponents. I t may be noted that i n t h i s work, the p e r - f e c t p a i r i n g c a l c u l a t i o n s f o r HF gave much b e t t e r c a l - c u l a t i o n s of the H-F e q u i l i b r i u m distance than that r e - ported by S i l k and M u r r e l l , 7 even though these workers used a l a r g e r set of b a s i s c o n f i g u r a t i o n s f o r t h e i r v a - lence bond wave f u n c t i o n s . The d i f f e r e n c e i n bond l e n g t h seems to be due to the allowance i n t h i s work of d i f f e - rent 2pr and 2pJT exponents, whereas S i l k and M u r r e l l c o n s t r a i n e d t h e i r s to have a s i n g l e v a l u e . For t h i s b a - s i s set with S l a t e r - t y p e o r b i t a l s i t seems therefore that the choice of o r b i t a l exponents i s q u i t e c r u c i a l f o r e s t i m a t i n g molecular p r o p e r t i e s . Further s t u d i e s of the changes i n exponents from free atom values are neces- sary to enable reasonable p r e d i c t i o n s of s u i t a b l e ex- ponents f o r l a r g e r molecules. The determination of molecule-optimized exponent values by means of exact c a l c u l a t i o n s does, however, become very expensive f o r large molecules, and f o r t h i s reason i t i s important that there are approximate schemes a v a i l a b l e f o r l i m i t i n g the computational e f f o r t . One way Is to approximate the molecular i n t e g r a l s by d r a s - t i c a l l y l i m i t i n g the number that need to be c a l c u l a t e d -75- exactly. This study has shown that an approximation proposed by Lowdin,5^ when applied to c a l c u l a t i n g some two-centre electron-repulsion exchange i n t e g r a l s , pre- d i c t s optimum exponent values which are close (0.04 f o r a l l cases considered here) to the values obtained from c a l c u l a t i o n s i n which a l l i n t e g r a l s are evaluated exactly. The approximation as used here does make a sub s t a n t i a l saving i n computational expense f o r the diatomic hydrides considered i n t h i s t h e s i s , but to be r e a l l y u s e f u l f o r more complex molecules i t would be necessary f o r the Lowdin approximation to be made f o r a l l overlap charge d i s t r i b u t i o n s occurring i n ele c t r o n - repulsion i n t e g r a l s . 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