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+a^ . 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^ ^ 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 , 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

(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 ( 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 . 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