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Cusp conditions and properties at the nucleus of lithium atomic wave functions 1970

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CUSP CONDITIONS AND PROPERTIES AT THE NUCLEUS OF LITHIUM ATOMIC WAVE FUNCTIONS by JOHN ALVIN CHAPMAN ' B.Sc. (Honours), U n i v e r s i t y of V i c t o r i a , 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Chemistry We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard' THE UNIVERSITY OF BRITISH COLUMBIA March, 1970 In p resen t ing t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements f o r 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 i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r re ference and s tudy. I f u r t h e r agree tha permiss ion fo r ex tens i ve copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed wi thout my w r i t t e n pe rm iss i on . Department of The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada i ABSTRACT The dependence of the p o i n t p r o p e r t i e s at the nucleus, e l e c t r o n d e n s i t y ( Q e ( 0 ) ) and s p i n d e n s i t y ( Q S(0) ), on the n u c l e a r cusp i s examined f o r l i t h i u m atomic c o n f i g u r a t i o n i n t e r a c t i o n (CI) wave f u n c t i o n s . S e v e r a l s e r i e s of CI wave f u n c t i o n s w i t h 18 and fewer terms, are s t u d i e d . Importance of the t r i p l e t core s p i n f u n c t i o n to Q s(0) i s s u b s t a n t i a t e d . Necessary, "but not s u f f i c i e n t , s p i n and e l e c t r o n i n t e g r a l cusp c o n d i t i o n s are a p p l i e d as l i n e a r c o n s t r a i n t s . For the f u n c t i o n s s t u d i e d , Q s(0) improves on a p p l y i n g the s p i n cusp c o n s t r a i n t i f the f r e e v a r i a t i o n a l s p i n cusp i s g r e a t e r than -Z, but becomes worse otherwise. The e l e c t r o n cusp c o n s t r a i n t i n v a r i a b l y o v e r c o r r e c t s Q e ( 0 ) . The e f f e c t of necessary o f f - d i a g o n a l weighting c o n s t r a i n t s i s a l s o examined. No obvious trends c o u l d be found. I t i s concluded t h a t f o r c i n g CI f u n c t i o n s w i t h a s m a l l number of terms to s a t i s f y necessary d i a g o n a l or o f f - d i a g o n a l i n t e g r a l cusp c o n d i t i o n s has v e r y l i m i t e d u s e f u l n e s s . A good Q s(0) can be ob- t a i n e d without c o n s t r a i n i n g by ( l ) i n c l u d i n g t r i p l e t core s p i n terms. (2) o p t i m i z i n g o r b i t a l exponents. S u f f i c i e n t n u c l e a r cusp c o n s t r a i n t s are developed f o r CI wave f u n c t i o n s . The g e n e r a l i z e d c u s p - s a t i s f y i n g CI f u n c t i o n has m u l t i c o n f i g u r a t i o n a l SCF form w i t h the c o r r e c t cusp f o r each o r b i t a l . Sample c a l c u l a t i o n s w i t h a s m a l l b a s i s set are presented. These simple f u n c t i o n s g i v e extremely good Q s(0) e x p e c t a t i o n values but convergence of Q s(0) w i t h r e s p e c t t o b a s i s set s i z e i s yet t o be t e s t e d . The most i n t e r e s t i n g d i s - covery i s the appearance of D i r a c cf - l i k e c o r r e c t i o n b a s i s o r b i t a l s from energy m i n i m i z a t i o n of the o r b i t a l exponents. A scheme i s d e p i c t e d c l a s s i f y i n g p r e v i o u s and present work on cusp c o n s t r a i n t s i n terms of n e c e s s i t y and/or s u f f i c i e n c y . i i i TABLE OF CONTENTS ABSTRACT Page i LIST OF TABLES v i LIST OF FIGURES v i x i ACKNOWLEDGMENTS i x CHAPTER I . PRELIMINARIES 1.1 I n t r o d u c t i o n 1.2 Object of t h i s work 1 . 3 Survey of f o l l o w i n g chapters 1 5 6 CHAPTER I I . BACKGROUND 2.1 Review of S ground s t a t e l i t h i u m 8 atomic wave f u n c t i o n s and contact p r o p e r t i e s 2.2 What are cusp and coalescence 21 c o n d i t i o n s ? 2.3 Theory of coalescence c o n d i t i o n s 24 f o r exact wave f u n c t i o n s 2.4 Cusp c a l c u l a t i o n methods f o r 3 2 approximate wave f u n c t i o n s 2.5 Use of cusp and coalescence c o n d i - 40 t i o n s f o r improvement of approximate wave f u n c t i o n s i v Page CHAPTER I I I . INTEGRAL CUSP CONSTRAINTS AND APPLICATIONS TO LITHIUM 2 S GROUND STATE FUNCTIONS 3.1 Formation of c o n s t r a i n t s 4-5 3.2 E x p l o r a t o r y c a l c u l a t i o n s 50 3.3 Systematic study 63 3.4 O f f - d i a g o n a l cusp c o n s t r a i n t s 73 w i t h weighting f u n c t i o n s CHAPTER IV. SUFFICIENT CONDITIONS FOR CORRECT CUSP 4.1 Theory 87 4.2 A p p l i c a t i o n s to the l i t h i u m 2 S 103 ground s t a t e 4.3 A p p l i c a t i o n t o the lowest 119 l i t h i u m s t a t e CHAPTER V. SUMMARY AND CONCLUDING REMARKS 122 BIBLIOGRAPHY 128 APPENDIX A. ATOMIC UNITS 133 APPENDIX B. SOME IMPORTANT TYPES OF APPROX- 135 IMATE ATOMIC WAVE FUNCTIONS APPENDIX C. CONSTRAINED VARIATION C . l I n t r o d u c t i o n 142 C.2 S i n g l e c o n s t r a i n t s 143 C.3 M u l t i p l e c o n s t r a i n t s 146 C.4 O f f - d i a g o n a l c o n s t r a i n t s 148 V Page C.5 O f f - d i a g o n a l l i n e a r c o n s t r a i n e d 149 v a r i a t i o n method of Weber and Handy APPENDIX D. INTEGRAL CALCULATION D.l P r i m i t i v e i n t e g r a l s f o r S l a t e r - 1 5 2 type o r b i t a l s D .2 C o l l e c t i o n of p r i m i t i v e i n t e g r a l s 1 5 4 APPENDIX E. DESCRIPTIONS AND PROPERTIES OF 1 5 7 THE WAVE FUNCTIONS $ l 0 THROUGH l i s ' v i LIST OF TABLES Table I . I I . I I I . IV. V. VI. V I I . V I I I , IX. X. X I . R e p r e s e n t a t i v e wave f u n c t i o n s from the l i t e r a t u r e f o r the l i t h i u m 2 S ground s t a t e D e s c r i p t i o n and p r o p e r t i e s of j j j ^ Term-wise comparison of convergence f o r p e r t u r b a t i o n expansion of 4E, the energy- s a c r i f i c e from the J ? e = Y c o n s t r a i n t A n a l y t i c a l p a r a m e t r i z a t i o n s of c o n s t r a i n t on ^ D e s c r i p t i o n s of <j}/0 , <&/¥. ,$/Q Free v a r i a t i o n a l and c o n s t r a i n e d proper- t i e s of § t o , , Energy-weighted o f f - d i a g o n a l cusp c o n s t r a i n t s on § t ¥ Comparison of the d i a g o n a l and o f f - d i a g o n a l i t e r a t i v e methods f o r d i a g o n a l c o n s t r a i n t s . Example: <£>/v. , £e = Y c o n s t r a i n t Q S ( 0 ) , Q e (0 ) f o r the o f f - d i a g o n a l weights i n s p i n cusp c o n s t r a i n t s of M u l t i p l e w e i g h t i n g c o n s t r a i n t s on ̂ /y. The set of S l a t e r determinants, f < j > i ] z, and e i g e n f u n c t i o n s , \7/t'\ . of \d* , £. , 1 f o r the 2s wave f u n c t i o n s d e s c r i b e d i n case 2 of s e c t i o n 4 .1 Page 9 53 54 61 67 68 79 82 85 86 97 X I I . A d d i t i o n a l f<« and Wc] elements f o r Table XI when 0^ ^-type terms are i n c l u d e d 100 v i i Table X I I I , XIV. XV. XVI. XVII, I l l u s t r a t i v e c a l c u l a t i o n : 3 -term CI f u n c t i o n formed from {Xs-} b a s i s Simple DODS wave f u n c t i o n s w i t h CSO's ( 1 ) $£,a t r u e CI f u n c t i o n w i t h two 3 - o r b i t a l CSO's r e p r e s e n t i n g the core ( 2 ) , a s p i n - o p t i m i z e d CI func- t i o n w i t h -p c o r r e l a t i o n i n the core True CI f u n c t i o n s formed from STO bases l i s t e d i n Table XIV C a l c u l a t i o n s on the lowest l i t h i u m "P s t a t e of X V I I I . H i e r a r c h y of necessary and s u f f i c i e n t cusp c o n d i t i o n s f o r atoms XIX. Free v a r i a t i o n a l and c o n s t r a i n e d proper- t i e s of the wave f u n c t i o n s (£>/0 through <&/S Page 106 109 112 117 120 125 158 v i i i LIST OF FIGURES F i g u r e Page 1. C o r r e l a t i o n between e r r o r i n n u c l e a r cusp 43 and e r r o r i n Q e(0) f o r He wave f u n c t i o n s , found by Chong and Schrader. 2. (A) Graph of versus )\ f o r the 56 ground s t a t e of ^ i n e l e c t r o n i c cusp c o n s t r a i n t (B) Graphs of f i c t i t i o u s energy, Ef/ct ~ <#*A<?> , and t r u e energy ET =<(/// versus A , f o r the ground s t a t e of $ 7 i n e l e c t r o n i c cusp c o n s t r a i n t 3. (A) Q s(0) as a f u n c t i o n of the c o n s t r a i n t 72 X s = y f o r &D , §,+ , $ l 8 (B) Q e(0) as a f u n c t i o n of the c o n s t r a i n t r e = y f o r $ l 0 , , § / a 4. . (A) Q s(0) as a f u n c t i o n of the c o n s t r a i n t 73 JZ e =X' f o r , ^ , $,8 (B) Q e(0) as a f u n c t i o n of the c o n s t r a i n t r 5 - yy f o r §,0 , , $ t B 5. (A) Q s(0) p l o t t e d a g a i n s t (° and 84 i n w e i g h t i n g e l e c t r o n i c cusp const- r a i n t s on (B) Q e(0) p l o t t e d a g a i n s t f and j/f i n w e i g h t i n g e l e c t r o n i c cusp c o n s t - r a i n t s on (£>/lf. 6. Energy contour map of % (Table X I V ) : 114 Energy versus exponents Si's , Sis" of o -type cusp c o r r e c t i o n o r b i t a l s 7. P r o p e r t i e s of <g (Table XIV) correspond- 115 i n g t o energy contours i n f i g u r e 6. i x ACKNOWLEDGMENTS I thank Dr. D. P. Chong f o r h i s advice and en- couragement throughout my years at the U n i v e r s i t y of B r i t i s h Columbia and d u r i n g the p r e p a r a t i o n of t h i s t h e s i s . F i n a n c i a l w o r r i e s were removed by a MacMilla n F a m i l y F e l l o w s h i p (1965-1969) and a N a t i o n a l Research C o u n c i l F e l l o w s h i p (1968-1969). Work i n the l a s t few months has been made much e a s i e r by my w i f e , L e s l i e . 1 CHAPTER I PRELIMINARIES 1.1 I n t r o d u c t i o n The p o s t u l a t e s of quantum mechanics s t a t e t h a t to d e s c r i b e a system mathematically one needs t o decide on a H a m i l t o n i a n f o r t h a t system and then t o solve the corresponding Schrodinger equation. Instead of the t r u e H a m i l t o n i a n which can r a r e l y be deduced, one uses the n o n - r e l a t i v i s t i c , s p i n l e s s , time-independent H a m i l t o n i a n i n many problems of molecular p h y s i c s or quantum chemistry. Moreover, f o r molecules, the Born-Oppenheimer approximation i s u s u a l l y used t o parametrize the n u c l e a r c o o r d i n a t e s . This type of s i m p l i f i e d H a m i l t o n i a n w i l l be i m p l i e d throughout t h i s t h e s i s . S o l u t i o n s of the r e s u l t i n g ( s i m p l i f i e d ) SchrSdinger equation w i l l be c a l l e d exact. These exact wave f u n c t i o n s c o n t a i n a l l the i n f o r m a t i o n needed t o c a l c u l a t e any observable of the system by u s i n g the a p p r o p r i a t e H e r m i t i a n o p e r a t o r . Such a procedure w i l l y i e l d , what w i l l be c a l l e d , exact e x p e c t a t i o n v a l u e s . * Note t h a t e x p e r i m e n t a l l y d e r i v e d v a l u e s of observables are sometimes adjusted t o y i e l d an experimental estimate f o r the exact v a l u e . For example, the r e l a t i v i s t i c con- t r i b u t i o n t o the t r u e energy i s u s u a l l y s u b t r a c t e d from the t r u e energy g i v i n g the exact energy. 2 The complexity of many-body i n t e r a c t i o n s makes exact a n a l y t i c s o l u t i o n of the s i m p l i f i e d Schrb'dinger equation i m p o s s i b l e save f o r a s m a l l number of two and thre e p a r t i c l e systems. There are two gene r a l procedures t h a t can be f o l l o w e d at t h i s stage. E i t h e r the Ha m i l t o n i a n may be s i m p l i f i e d f u r t h e r i n a way t o a l l o w exact s o l u t i o n of the approximate Schrodinger e q u a t i o n — f o r example, the Hartree-Fock method, or approximate methods may be used t o solve the exact Schrb'dinger e q u a t i o n — f o r example, s e r i e s expansion of the s o l u t i o n . Often the two procedures are combined. The s o l u t i o n s from both r o u t e s are r e f e r r e d t o as approximate wave f u n c t i o n s . In e i t h e r case energy i s almost always used as the c r i t e r i o n f o r o b t a i n i n g the wave f u n c t i o n . This i s because there i s an e a s i l y a p p l i e d minimum energy;, p r i n c i p l e . The t r u e ground s t a t e i s determined u n i q u e l y by the lowest eigenvalue of the exact H a m i l t o n i a n . The energy, £ , of an approximate wave f u n c t i o n , ^ , approaches the exact energy as becomes more and more s i m i l a r t o the exact wave f u n c t i o n . This i s t r u e f o r other obser- v a b l e s a l s o . But they do not n e c e s s a r i l y approach t h e i r exact v a l u e s m o n o t o n i c a l l y . Here l i e s a problem: 3 the use of energy as a c r i t e r i o n f o r an approximate wave f u n c t i o n does not always ensure r e l i a b l e expec- t a t i o n v a l u e s of operators other than the Ha m i l t o n i a n . This i s e s p e c i a l l y t r u e f o r p o i n t - p r o p e r t i e s — t h o s e p r o p e r t i e s l i k e the h y p e r f i n e s p l i t t i n g , or e l e c t r o n d e n s i t y at the n u c l e u s — w h i c h depend on the value of the wave f u n c t i o n at a s i n g l e p o i n t i n space. Can r e l i a b l e e x p e c t a t i o n v a l u e s be c a l c u l a t e d ? There are two d i s t i n c t approaches t o t h i s problem. F i r s t , extremely accurate approximate wave f u n c t i o n s can be c a l c u l a t e d . These must be e s s e n t i a l l y exact to be sure of o b t a i n i n g accurate e x p e c t a t i o n v a l u e s . The e f f o r t needed t o o b t a i n such an accurate wave f u n c t i o n r a p i d l y i n c r e a s e s beyond f e a s i b i l i t y w i t h the number of p a r t i c l e s . Thus o n l y the hydrogen, helium and l i t h i u m atoms and the molecule have been d e s c r i b e d w e l l enough f o r the accurate p r e d i c t i o n of a l l ( n o n r e l a t i v i s t i c ) p r o p e r t i e s . To make r e l i a b l e estimates of p r o p e r t i e s of more i n t e r e s t i n g s p e c i e s , such as t r a n s i t i o n metal complexes, or even s m a l l - s i z e d organic compounds seems out of the q u e s t i o n at p r e s e n t . Now f o r a moment c o n t r a s t these accurate wave f u n c t i o n s w i t h s i m p l e r t y p e s . Simple wave f u n c t i o n s , though not n e c e s s a r i l y g i v i n g dependable p r o p e r t y 4- va l u e s when determined by the energy c r i t e r i o n are easy t o c o n s t r u c t and easy t o c a l c u l a t e ( c o m p a r a t i v e l y s p e a k i n g ) . Of course t h e i r c o m p l e x i t y a l s o grows tremendously w i t h the number of p a r t i c l e s , but can be lessened by c a r e f u l a p p l i c a t i o n of chemical i n - t u i t i o n — d i f f i c u l t t o do f o r the more complicated accurate wave f u n c t i o n s . The second a p p r o a c h — t h e one p a r t i a l l y explored i n t h i s t h e s i s — u t i l i z e s the s i m p l i c i t y of these s m a l l e r , l e s s accurate wave func- t i o n s , together w i t h the e x i s t e n c e of other c r i t e r i a as w e l l as energy f o r wave f u n c t i o n d e t e r m i n a t i o n , t o a r r i v e at a r e l i a b l e method f o r c a l c u l a t i n g atomic and molecular p r o p e r t i e s . These other c r i t e r i a i n c l u d e known c h a r a c t e r i s t i c s , t h e o r e t i c a l or experimental, which the exact wave f u n c t i o n must e x h i b i t . The quantum- mechanical v i r i a l theorem, the h y p e r v i r i a l theorems, the cusp c o n d i t i o n s , e x p e r i m e n t a l l y known e x p e c t a t i o n v a l u e s and the v a n i s h i n g of net f o r c e s are c o n d i t i o n s t h a t can a i d the c h a r a c t e r i z a t i o n of an approximate wave f u n c t i o n . N a t u r a l l y o n l y the exact wave f u n c t i o n w i l l s a t i s f y a l l p o s s i b l e c o n d i t i o n s . The u s u a l procedure i s t o f o r c e f u l f i l m e n t of those c o n d i t i o n s a f f e c t i n g the p r o p e r t y one wishes t o c a l c u l a t e . D i f f e r e n t p r o p e r t i e s w i l l have a d i f f e r e n t set of 5 * c o n d i t i o n s . The parameters i n the f u n c t i o n a l form chosen t o approximate the exact wavefunction are minimized w i t h r e s p e c t t o energy w h i l e being c o n s t r a i n e d t o s a t i s f y the d e s i r e d set of c o n d i t i o n s . This i s the i d e a behind the quantum-chemical theory of c o n s t r a i n t s . 1.2 Object of t h i s work The work r e p o r t e d i n t h i s t h e s i s w i l l t e s t the s e r v i c e a b i l i t y of nu c l e a r cusp c o n s t r a i n t s as a i d s t o make simple approximate wave f u n c t i o n s y i e l d good p o i n t p r o p e r t i e s at the nucleus. The l i t h i u m atom has been chosen as the system t o be i n v e s t i g a t e d f o r the f o l l o w i n g reasons: ( i ) S e v e r a l accurate treatments f o r l i t h i u m are available-?' ^' ^ ( i i ) The system i s a c c e s s i b l e e x p e r i m e n t a l l y . 8, 9, 10, 11, 12 ( i i i ) The l i t h i u m atom has c o r r e l a t i o n phen- omena c h a r a c t e r i s t i c of more complicated systems but i s simple enough t o r e v e a l the r e s u l t s of the method of c o n s t r a i n t s without undue computational problems. Thus, p r o p e r t i e s of constrained, simple l i t h i u m f u n c t i o n s can be compared w i t h r e s u l t s of both accurate c a l c u l a t i o n s For i n s t a n c e , c o n s t r a i n i n g the net f o r c e s to v a n i s h , s a t i s f a c t i o n of the h y p e r v i r i a l theorems; and f o r c i n g the c o r r e c t cusp behavior should improve c a l c u l a t e d f o r c e c o n s t a n t s , t r a n s i t i o n p r o b a b i l i t y c a l c u l a t i o n s and contact p r o p e r t i e s r e s p e c t i v e l y C l , 2, 3» 4- 3 . 6 and/or experiment a l l o w i n g a meaningful assessment of the u s e f u l n e s s of n u c l e a r cusp c o n s t r a i n t s . 1.3 iSurvey of f o l l o w i n g chapters Chapter I I c o n s i s t s of background m a t e r i a l nec- essary f o r understanding why the present work was undertaken. Cusp and coalesence c o n d i t i o n s are d e f i n e d and t h e i r a p p l i c a b i l i t y t o approximate wave f u n c t i o n s e x p l a i n e d . The e f f e c t s on s p i n and e l e c t r o n d e n s i t y at the nucleus, of f o r c i n g approximate c o n f i g u r a t i o n i n t e r - a c t i o n wave f u n c t i o n s t o s a t i s f y i n t e g r a l cusp cond- i t i o n s are presented i n Chapter I I I . S e v e r a l approaches are d e s c r i b e d . S u f f i c i e n t c o n d i t i o n s f o r ensuring a c o r r e c t cusp are developed and a p p l i e d t o approximate l i t h i u m wave f u n c t i o n s i n Chapter IV. Chapter V summarizes the work presented i n Chapters I I I . and IV. A scheme c l a s s i f y i n g cusp c o n d i t i o n s and c o n s t r a i n t s w i t h r e s p e c t t o n e c e s s i t y and s u f f i c i e n c y i s t a b l e d . Appendix A i s a l i s t of atomic u n i t s used i n t h i s work. D e f i n i t i o n s and forms of c e r t a i n b a s i c types of approximate wave f u n c t i o n s are presented 7 i n Appendix B. Knowledge of these types i s assumed i n the t e x t . Methods f o r a p p l y i n g l i n e a r c o n s t r a i n t s t o v a r i a t i o n a l l y determined approximate wave f u n c t i o n s are o u t l i n e d i n Appendix ,C. Appendix D d i s c u s s e s the i n t e g r a l s needed i n t h i s work and Appendix E c o n t a i n s the d e s c r i p t i o n and p r o p e r t i e s of the complete s e r i e s of f u n c t i o n s d e f i n e d i n S e c t i o n 3.3. 8 CHAPTER I I BACKGROUND 2 2.1 Review of S ground s t a t e l i t h i u m atomic wave f u n c t i o n s and contact p r o p e r t i e s To understand the reasons f o r development and a p p l i c a t i o n of s p e c i a l c o n s t r a i n t techniques to c a l c u l a t e n u c l e a r p o i n t p r o p e r t i e s , a look at some of the past work on l i t h i u m wave f u n c t i o n s i s necessary. No attempt to cover the v a s t l i t e r a t u r e i s made hut important aspects p e r t a i n i n g to the problem w i l l be b r i e f l y d i s c u s s e d . A c o l l e c t i o n n o f some of the more s i g n i - fic;ant c a l c u l a t i o n s i s presented i n Table I . E n t r i e s are energy o r d e r e d — t h e best at the bottom. A glance at the t a b l e r e v e a l s t h a t wave f u n c t i o n s so o r d e r e d — w i t h energy as a c r i t e r i o n of a c c u r a c y — a r e not i n the same sequence when the s p i n d e n s i t y at the nucleus, Q s ( 0 ) , i s a c r i t e r i o n . The d i f f i c u l t i e s i n c a l c u l a t i n g both Q s(0) and the corresponding e l e c t r o n d e n s i t y , Q e ( 0 ) , Other w o r k e r s — s e e [13, 14, 15, 16, 17l — h a v e a l s o t a b u l a t e d l i t h i u m groundstate c a l c u l a t i o n s from the l i t e r a t u r e . References [ 5 , 6, 18, 19, 20, 21, 22, 23, 24, 2 5 , 26j c o n t a i n c a l c u l a t i o n s t h a t have appeared i n the l i t e r - a t ure s i n c e L u n e l l ' s t a b u l a t i o n [17] i n 1968. Table I . E e p r e s e n t a t i v e wave f u n c t i o n s from the l i t e r a t u r e f o r the l i t h i u m S groundstate. Spin d e n s i t y E l e c t r o n d e n s i t y D e s c r i p t i o n of Reference Energy at the nucleus at the nucleus wave f u n c t i o n — T - Q s(0) % e r r o r a Refer- Q e(0) % e r r o r 0 R e f e r - ence ence 1 l i m i t e d CI, i n c l u d i n g f u n c t i o n s e-*'" where «T = 1 . 7 5 , 3.5, 7.0 27 -7.4-31849 0.2284 + 1.3 27 2 a n a l y t i c a l HIP 28 -7 . 4 3 2 7 2 7 0.1667 +2(7. 9 29 13 .8203 +0.1 29° 3 UHJ? 29 -7 . 4 3 2 7 5 1 0.2248 + 2.8 29 13 .8204 +0.1 29° 4 PUHJ? of S a c h s 2 9 17 -7.432768 0.1866 +19.3 17 5 EHF ( p r o j e c t e d ) 20 -7.4-32813 0.2412 - 4.3 20 6 open s h e l l , 2 determin- ants, e, s p i n f u n c t i o n 30 -7.4-436 0.3002 -29.8 31 13 . 5 1 9 3 +2.3 -e 7 open s h e l l , 3 determin- ants, (spin optimized) 32,33 -7.4-436 0.2417 d - 4.5 13 13 .5240 +2.2 8 open s h e l l , SEHJT, 2 determinants, 6, s p i n f u n c t i o n 17 -7.4-47529 0 . 2 0 5 5 +11.2 17 Table I. Representative wave:; functions from the l i t e r a t u r e f o r the lithium S groundstate. (continued) Spin density E l e c t r o n density Description of Reference Energy at the nucleus at the nucleus wave func t i o n — ~ ir QT(O) % error Refer- Q e(0) % e r r o r 0 Refer- enc e enc e C9 open s h e l l . SEHF, 3 det- erminants ( s p i n optim- ized) 10 Gl-EHF (projected) 11 EHF (spin-optimized) 12 s-type basis, 330 term CI (no r±.) 13 s,p-type b a s i s , 310 term CI (no r±.) 14 scaled 208-term CI (no r. .)with 2 non-linear .10 parameters 15 45 term CI (no r. .), STO basis 1 J 17 21 24 ,25 23 23 34 35 16 15 term c o r r e l a t e d func- t i o n , © i spin f u n c t i o n 37 -7.447536 0.2265 + 2.1 17 -7.447560 0.2095 +9.4 21 13.864 -0.2 21 -7.447565 0.2265 +2.1 24 ,25 13.8646 -0.2 25 -7.448520 0.2278 +11.5 23 -7.472680 0.2398 -.3.7 23 - 7 . W 6 9 •7.47710 0.2065 +10.7 13 -7.4771 13.8661 +0.2 36 Table I . R e p r e s e n t a t i v e wave f u n c t i o n s from the l i t e r a t u r e f o r the l i t h i u m S groundstate. (continued) D e s c r i p t i o n of Reference wave f u n c t i o n Energy Spin d e n s i t y at the nucleus E l e c t r o n d e n s i t y at the nucleus Q s(0) % e r r o r a R e f e r - ence Q e(0) % e r r o r b Refer ence 1 7 60 term c o r r e l a t e d f u n c- t i o n , ©i s p i n f u n c t i o n 6 -7.478010 0.2405 - 4.0 6 13.8327 0.0 38 18 100 term c o r r e l a t e d f u n c- t i o n (spin optimized) 6 -7.478025 0.2313 :0.0 6 13.8341 38 1 9 Bruckner-Goldstone Diagrammatic P e r t u r b a t i o n 5 -7.478 +0.002 0.230 +0.002 0.0 5 20 Exact (Experimental Q s(0) ) 39 -7.478069 0.2313 8 f °% e r r o r = Q s(0) (experimental) - Q s(0) ( c a l c u l a t e d ) x 100% Qs(0) (experimental) b % e r r o r = Q e(0) (La r s s o n . #18) - Q.e(0) ( c a l c u l a t e d ) x 100% Q e(0) (Larsson, #18) °Calculated from data i n r e f e r e n c e . C a l c u l a t e d i n t h i s work t o be Q s(0) = 0.2425. e C a l c u l a t e d i n the course of t h i s work. See r e f e r e n c e [40] . 12 are evident from the work of Jacobs and Larsson. Jacobs s t u d i e d the convergence p r o p e r t i e s of c o n f i g u r a t i o n i n t e r a c t i o n (CI) wave f u n c t i o n s and found e r r a t i c v a l u e s of Q s(0) f o r L i and Q e(0) f o r He f o r v a r i o u s expansions converging i n energy. * * Even Larsson's 100 term c o r r e l a t e d f u n c t i o n , the most accurate l i t h i u m groundstate f u n c t i o n a v a i l a b l e , has not converged i n these p a r t i c u l a r p r o p e r t i e s be- cause another 100 term c o r r e l a t e d f u n c t i o n , i d e n t i c a l i n energy, g i v e s a Q e ( 0 ) , Q s(0) d i f f e r e n t from the val u e s l i s t e d i n Table I , by 0.07% and 0.1% r e s p e c t i v e l y . For l a r g e r systems e r r o r s i n Q s(0) of 25-50% seem t o be common. See, f o r example, the c a l c u l a t i o n s on boron, carbon, n i t r o g e n , oxygen and f l u o r i n e by 41 Schaefer et a l . Some technique f o r s y s t e m a t i c - a l l y c a l c u l a t i n g such p o i n t p r o p e r t i e s i s c l e a r l y needed, e s p e c i a l l y f o r l a r g e r systems where fo r m a t i o n of accurate c o r r e l a t e d f u n c t i o n s becomes v i r t u a l l y i m p o s s i b l e . Cusp c o n s t r a i n t s may pr o v i d e a method. F i r s t of a l l , f o r what i s the s p i n d e n s i t y , Q s ( 0 ) , u s e f u l ? I t provides an important c o n t r i b u t i o n * From r e s u l t s communicated t o P r o f e s s o r D. P. Chong. A b r i e f d e s c r i p t i o n of v a r i o u s wave f u n c t i o n app- ro x i m a t i o n s i s g i v e n i n Appendix B. 13 t o the h y p e r f i n e i n t e r a c t i o n energy. This type of i n t e r a c t i o n a r i s e s from the c o u p l i n g of e l e c t r o n i c and n u c l e a r e l e c t r i c and magnetic f i e l d s . The r e s u l t a n t s p l i t t i n g of energy l e v e l s can "be a c c u r a t e l y measured f o r a l k a l i metal atoms i n atomic "beam magnetic r e s - onance experiments. Experimental r e s u l t s from the a l k a l i metal group can be e x p l o i t e d as a check i n developing t h e o r e t i c a l techniques of forming wave f u n c t i o n s . Improved techniques w i l l then enable t h e o r e t i c a l " a n a l y s e s of more complex systems where experiments are not so e a s i l y i n t e r p r e t e d . The h y p e r f i n e energy , A Eh-fe , has major c o n t r i - b u t i o n s from Fermi c o n t a c t , magnetic d i p o l e - d i p o l e , and e l e c t r i c quadrupole i n t e r a c t i o n s . For an S-state o n l y the Fermi contact term, d e s c r i b i n g e l e c t r o n i c s p i n i n t e r a c t i o n s at or w i t h i n the nucleus, i s non- zero and (2.1.1) where (2.1.2) See more comp l e t e v o u t l i n e s and f u r t h e r r e f e r e n c e s i n O l , 42 ~] . 1'4 jt/fi/ , y6(e are the magnitudes of the magnetic moments of the nucleus and an e l e c t r o n r e s p e c t i v e l y ; I i s the n u c l e a r s p i n and <ff£), the Di r a c d e l t a f u n c t i o n . For L i 7 4 7 " = 803.512 Mc/sec [ 8 ] . S u b s t i t u t i n g the accepted v a l u e s = 3 / 2 , = 3 - 2 5 6 3 1 0 n u c l e a r magnetons^, yZ/e = 1.00116 Bohr magnetons,into ( 2 . 1 . 1 ) one o b t a i n s the experimental s p i n d e n s i t y Q s ( 0 ) = 0 . 2 3 1 | «o 3 i n atomic u n i t s . The q u a n t i t y , Q s ( 0 ) , i s the g r e a t e s t source of e r r o r i n t h e o r e t i c a l h y p e r f i n e c a l c u l a t i o n s f o r l i g h t atoms. The reason f o r t h i s i s the inadequacy of approximate wave f u n c t i o n s t o d e s c r i b e i n d e t a i l c o r r e l a t i o n e f f e c t s and core p o l a r i z a t i o n . The c o r r e l a t i o n problem i s concerned w i t h the d e s c r i p t i o n of the instantaneous r e p u l s i o n s among e l e c t r o n s . Techniques of f o r m u l a t i n g approximate wave f u n c t i o n s must attempt t o d e a l w i t h t h i s t o o b t a i n h e l p f u l r e s u l t s , e s p e c i a l l y i n problems of i n t e r e s t t o chemists. The c o r r e l a t i o n energy, E , U U l 1 1 5 d e f i n e d as t h e d i f f e r e n c e between t h e H a r t r e e - F o c k (HF) e n e r g y , Eg-p, o f a system and t h e e x a c t e nergy, E c o r r = ^ F " E (2.1.3) p r o v i d e s a measurement o f t h e i n t e r a c t i o n . EJJJ, i s c h o s e n as a r e f e r e n c e because t h e HF method n e g l e c t s s h o r t - r a n g e i n t e r a c t i o n s c o m p l e t e l y ; each e l e c t r o n i s assumed t o move i n a p o t e n t i a l c r e a t e d by average movements o f a l l o t h e r e l e c t r o n s . C o n s e q u e n t l y an e l e c t r o n n e v e r e x p e r i e n c e s d i r e c t r e p u l s i v e f o r c e s i n a HF f u n c t i o n . ( A p p l i c a t i o n o f t h e P a u l i p r i n c i p l e by a n t i s y m m e t r i z i n g t h e f u n c t i o n h e l p s somewhat, t h o u g h , a u t o m a t i c a l l y i n c l u d i n g c o r r e l a t i o n between e l e c t r o n s o f t h e same s p i n ) . C o n s i d e r now t h e s p e c i f i c p c a s e , t h e l i t h i u m S ground s t a t e . I t s HF f u n c t i o n c a n be w r i t t e n i n t e rms o f f u n c t i o n s o f a t o m i c c o o r d i n a t e s , (Pc » t h e u s u a l s p i n f u n c t i o n s °^ , j$ and , t h e a n t i s y m m e t r i z a t i o n o p e r a t o r . S i n c e t h e I s c o n t r i - b u t i o n s c a n c e l e x a c t l y , o n l y t h e 2s o r b i t a l c o n t r i b u t e s 16 to the spin density ( r e a l o r b i t a l s assumed) Q'*,M = <P£<o), ( 2 - 1 - 5 ) and provides but 72% of the experimental value (Table I ) One might conclude that t h i s r e s u l t i s due to lack of correlation,but there i s another important e f f e c t — * core p o l a r i z a t i o n . Exchange forces are more a t t r a c t i v e between electrons with the same spin than electrons with d i f f e r e n t spins. The unpaired 2s e l e c t r o n thus exerts a d i f f e r e n t force on each core e l e c t r o n and so the K s h e l l o r b i t a l s should also be d i f f e r e n t — that i s there should be a s p l i t £ s h e l l . But the HP method forces the f u n c t i o n a l form of the cxL and ft spin core o r b i t a l s to be the same. In the u n r e s t r i c t e d Hartree-Fock (UHF) method t h i s p a r t - i c u l a r r e s t r i c t i o n i s relaxed, * Sometimes r e f e r r e d to as exchange, or spin, p o l a r i z a t i o n . * * The UHF method i s more c o r r e c t l y r e f e r r e d to as the spin-polarized Hartree-Fock method D-73 1? allowing p o l a r i z a t i o n of the core o r b i t a l s by the unpaired spin. Now the K s h e l l can contribute to the spin density; QlJO) = (Pnfo) + C2.1.7) and one can see the spectacular improvement from HF —> UHF i n Table I. But % ^ i s no longer an e i g e n f u n c t i o n 1 of the t o t a l spin. A small quartet c o n t r i b u t i o n e x i s t s . Since:.spin operators commute with the n o n - r e l a t i v i s t i c Hamiltonian, and since a spin dependent property i s being c a l c u l a t e d i t i s desirable from a p h y s i c a l viewpoint that a wave func- t i o n have sharp spin. Perhaps i t i s t h i s lack of sharp spin i n UHF functions that causes tremendous e r r o r s — e v e n the wrong s i g n — i n spin d e n s i t i e s of 4-0 c e r t a i n systems. , I f the quartet component i s a n n i h i l a t e d from a l i t h i u m ground state UHF function the spin projected u n r e s t r i c t e d Hartree-Fock (PUHF) f u n c t i o n — a pure doublet s t a t e — i s obtained ( f o r o L i S, s t i l l ) , b u t i t has a poorer spin density(Table I ) . The quartet component has a non-negligible con- t r i b u t i o n . An improvement to the PUHF procedure i s to project a UHF-type function and then minimize the 18 energy. Goddard , Kaldor, Schaefer and H a r r i s have obtained reasonable, but s t i l l e r r a t i c a l l y behaving s p i n d e n s i t i e s by a p p l y i n g t h i s method— c a l l e d the spin-extended Hartree-Fock (SEHF) t e c h - n i q u e — t o l i t h i u m . Q S ( 0 ) i s dependent upon the b a s i s set used i n these c a l c u l a t i o n s . Expansions such as c o n f i g u r a t i o n i n t e r a c t i o n (CI) or c o r r e l a t e d f u n c t i o n s can d e s c r i b e c o r r e l a t i o n i n p r i n c i p l e but sheer t e c h n i c a l problems have pre- vented accurate c a l c u l a t i o n s on systems l a r g e r than l i t h i u m by means of these approaches. Brueckner- Goldstone many-body p e r t u r b a t i o n t h e o r y does provide a w e l l - d e f i n e d procedure f o r c a l c u l a t i n g wave f u n c t i o n s and p r o p e r t i e s t o any d e s i r e d accuracy. However, i t a l s o becomes unwieldy f o r systems more complex than the f i r s t row elements. S e p a r a t i o n of core p o l a r i z a t i o n from c o r r e l a t i o n e f f e c t s i s d i f f i c u l t f o r approximate f u n c t i o n s . R a d i a l , or ' i n - o u t ' , c o r r e l a t i o n can appear t o s p l i t the core when sm a l l b a s i s s e t s are used, g i v i n g one core e l e c t r o n a s l i g h t l y d i f f e r e n t p r o b a b i l i t y d i s - 5 t r i b u t i o n from the o t h e r . Chang, Pu and Das e s t - imate by the many-body p e r t u r b a t i o n approach t h a t c o r r e l a t i o n and core p o l a r i z a t i o n c o n t r i b u t e 15% 19 and 80% r e s p e c t i v e l y of the d i f f e r e n c e between the HF and exact s p i n d e n s i t i e s . Core p o l a r i z a t i o n seems to be an important a t t r i b u t e t o b u i l d i n t o an approx- imate f u n c t i o n . The e x i s t e n c e of two degenerate s p i n f u n c t i o n s f o r doublet s p i n s t a t e s of t h r e e e l e c t r o n s f u r t h e r c omplicates the computational problem. These f u n c t i o n s are u s u a l l y designated &, , corresponding t o the c o u p l i n g of a s i n g l e t core w i t h the doublet valence s h e l l , and , a t r i p l e t core w i t h the doublet valence s h e l l . The most g e n e r a l t h r e e e l e c t r o n doublet f u n c t i o n can be w r i t t e n as the l i n e a r combination c9, would be expected t o d e s c r i b e a more s t a b l e core and indeed &^ has a s m a l l e f f e c t on energy. 14 30 44 ' ' ' - For a f u l l y o p t i mized f u n c t i o n , however, a 2 =̂  0. More important i n the present c o n t e x t , (2.1.8) S = a, e, +• a z . (2.1.9) 20 6 17 24 25 &2 h a s a profound e f f e c t on s p i n d e n s i t y ' ' "DT1 & ̂  n i ~ wo T*1C perhaps "by improving the d e s c r i p t i o n of core p o l a r i z a t i o n . The p r o j e c t i o n operator used i n PUHF or SEHF f u n c t i o n s f i x e s the r a t i o a- L/a 2 t o a value not n e c e s s a r i l y the best f o r energy or 25 other p r o p e r t i e s . R e c e n t l y "both Ladner and Goddard , 24 Kaldor and H a r r i s have overcome t h i s r e s t r i c t i o n i n t h e i r s p i n - o p t i m i z e d SEHF f u n c t i o n s . Dependence of Q s(0) on may "be seen i n Table I . The d i f f i c u l t y need not occur i n CI or c o r r e l a t e d expansions s i n c e a l / / a 2 """s i f f l p l i o i ' k l y o p t i mized i n the s e c u l a r equations, I f an accurate s p i n d e n s i t y i s d e s i r e d the ne g l e c t of r e l a t i v i s t i c e f f e c t s must be examined. These e f f e c t s should be g r e a t e s t near the nucleus where an e l e c t r o n has maximum k i n e t i c energy and hence might be important f o r Fermi cont a c t i n t e r a c t i o n s . A good d i s c u s s i o n of r e l a t i v i s t i c c o r r e c t i o n s i s presented by T t e r l i k k i s , Mahanti and D a s . ^ S o l v i n g the Dirac-Hartree-Fock (DHF) r e l a t i v i s t i c equations f o r the a l k a l i s e r i e s enabled the c o r r e c t i o n ^DHF*"^ ~ SEF^^ *° b e determined. T h e i r r e s u l t s i n d i c a t e t h a t r e l a t i v i s t i c c o r r e c t i o n s are sm a l l f o r l i t h i u m ( 0 . 2 % ) , sodium (0.7%) and potassium ( 2 % ) . 21 E l e c t r o n d e n s i t y Q%) = ( A &&)\ (2.1.10) i s needed t o express the isomer s h i f t i n Mossbauer 4-6 spectroscopy. Although core p o l a r i z a t i o n i s not so important, s i n c e o r b i t a l c o n t r i b u t i o n s are summed ( i n c o n t r a s t t o Q s(0) ), the b a s i c d i f f i c u l t i e s of c a l c u l a t i n g a p o i n t p r o p e r t y remain. D i s c u s s i o n of t h e o r e t i c a l c o n d i t i o n s t h a t , i f imposed on approx- imate f u n c t i o n s , might improve these p r o p e r t i e s are now presented. 2•2 What are cusp and coalescence c o n d i t i o n s ? Consider the u s u a l ( n o n - r e l a t i v i s t i c , time- independent) H a m i l t o n i a n i n atomic u n i t s f o r a system of N charged p a r t i c l e s : H = 7 " + V (2.2.1) where T, V, are the k i n e t i c and p o t e n t i a l energy o p e r a t o r s : , zEt- , the mass and charge of the i p a r t i c l e . The Coulomb p o t e n t i a l c o n t a i n s s i n g u l a r i t i e s at the set of p o i n t s {^=0^ > t h a t i s , at the coalescence * See Appendix A. .22 of any two ( o r more) p a r t i c l e s . The e i g e n f u n c t i o n s of H belong t o H i l b e r t space and must be continuous (save f o r a f i n i t e number of p o i n t s ) , square i n t e g - 47 r a b l e , and bounded everywhere. • Because an exact wave f u n c t i o n must be f i n i t e , even at the s i n g u l a r . p o i n t s of the 'Coulomb p o t e n t i a l , i t must f u l f i l the 2 c o n d i t i o n s known as coalescence c o n d i t i o n s . Coale- scence c o n d i t i o n s are r e f e r r e d t o as cusp c o n d i t i o n s f o r the case when the wave f u n c t i o n has no node at the p o i n t of coalescence. To i l l u m i n a t e the preceding remarks examine 48 an exact o n e - e l e c t r o n hydrogenic I s wave f u n c t i o n . y//5(£) = NGxp(-Zh) where N i s the n o r m a l i z a t i o n constant and Z, the atomic number, i s continuous everywhere but not d i f f e r e n t i a b l e s i n c e dW the d e r i v a t i v e s ^ — , , , do not e x i s t at r = 0, the coalescence of the e l e c t r o n w i t h the nucleus. However there i s a c u s p — a d i s c o n t i n u i t y i n the s l o p e — d e s c r i b e d by the cusp c o n d i t i o n , at the o n l y p o i n t of coalescence ( r = 0). A l l hyd- r o g e n i c wave f u n c t i o n s s a t i s f y t h i s r e l a t i o n s h i p but the coalescence c o n d i t i o n f o r non-s s t a t e s i s 23 t r i v i a l . Any well-behaved e i g e n f u n c t i o n of a Ham- i l t o n i a n must s a t i s f y the coalescence c o n d i t i o n s . The r a t i o HlP/Of/ i s then constant and does not c o n t a i n s i n g u l a r i t i e s when i s an exact eigen- 49 50 f u n c t i o n of H. " ^ The f o l l o w i n g d i s t i n c t i o n s are emphasized t o keep terminology c l e a r i n the r e - mainder of the t h e s i s : 1. A cusp of a f u n c t i o n , f ( x ) , i s the p o i n t , f ( x Q ) , at a d i s c o n t i n u i t y of the sl o p e , f ' ( x ) , where f ( x ) changes i t s d i r e c t i o n . I t i s a l s o a s s o c i a t e d w i t h a v a l u e : r < x 0 V f ( x 0 ) . 2. A coalescence c o n d i t i o n i s any mathemat- i c a l r e l a t i o n s h i p which an exact wave f u n c t i o n must s a t i s f y at one of i t s cusps. 3. A cusp c o n d i t i o n i s a case of a coalescence c o n d i t i o n when the wave f u n c t i o n has no node at the s i n g u l a r i t y . 4. Coalescence i s the s p a t i a l c o i n c i d e n c e of two or more p a r t i c l e s . Only the two p a r t i c l e case i s c o n s i d e r e d i n t h i s work. When the p a r t i c l e s are both e l e c t r o n s i t i s e l e c t r o n - e l e c t r o n coalescence; when one i s an e l e c t r o n , the other a 24 rmcleus, i t i s e l e c t r o n - n u c l e u s , or n u c l e a r coalescence. The cusp and coalescence c o n d i t i o n s f o r molecul-ar and atomic wave f u n c t i o n s w i l l now be reviewed. 2.3 Theory of coalescence c o n d i t i o n s f o r exact wave f u n c t i o n s 47 Kato d e r i v e d , f o r an N - e l e c t r o n , s p i n l e s s , atomic wave f u n c t i o n , the d i f f e r e n t i a l cusp c o n d i t i o n s lifter..*.-.*.)) = - -z.W°>&> i2'3'i:> 1 Jfj jh=o f o r e l e c t r o n - n u c l e u s coalescence .—\ f o r e l e c t r o n - e l e c t r o n coalescence. Here Z i s the -A atomic charge of the nucleus i n atomic u n i t s ; ^ i s the average of ft about a s m a l l sphere w i t h center at the c o a l e s c i n g p a r t i c l e s ; and r = (r-^ + A n o n r e l a t i v i s t i c H a m i l t o n i a n and the heavy-nucleus approximation were used i n Kato's d e r i v a t i o n as w e l l as the assumption t h a t o n l y two p a r t i c l e s were c o a l e s - c i n g . Steiner^"*" used the same assumptions and obtained cusp c o n d i t i o n s f o r the p r o b a b i l i t y o r , e l e c t r o n , 25 d e n s i t y — t h e d i a g o n a l element of the f i r s t order d e n s i t y m a t r i x . E q u i v a l e n t i n t e g r a t e d forms of Kato's cusp c o n d i t i o n s 48 g i v e n by Bingel,who extended them to molecules 52 and subsequently proved them r i g o r o u s l y , are: = W O * ! ? - ( 2 . 3 . l a ) and The c o n d i t i o n (2.3.1a) i s s a t i s f i e d f o r any nucleus, n u c l e a r charge iT^ , at the o r i g i n of the c o o r d i n a t e system. The v e c t o r a i s not determined by the Coulomb s i n g u l a r i t y but has magnitude depending on the c o o r d i - nates of the non-coalescing p a r t i c l e s and d i r e c t i o n p a r a l l e l t o the e l e c t r i c f i e l d produced by these 52 48 p a r t i c l e s < B i n g e l a l s o found cusp c o n d i t i o n s f o r the g e n e r a l f i r s t order d e n s i t y m a t r i x i n c l u d i n g 26 s p i n , extending S t e i n e r ' s d e r i v a t i o n t o i n c l u d e the s p i n d e n s i t y cusp. By Jr=o Note t h a t the s p h e r i c a l averaging operator needed to express the d i f f e r e n t i a l cusp c o n d i t i o n s (2.2.1) and (2.3.2) pre c l u d e s any p o s s i b i l i t y of o b t a i n i n g coalescence c o n d i t i o n s from these expressions t h a t are not t r i v i a l i n nature. Cusp c o n d i t i o n s f o r the s p e c i a l cases of the helium atom and hydrogen molecule have been found 4-9 50 and d i s c u s s e d by Roothaan and coworkers. ^' y T h e i r method was to e x p l i c i t l y c o n s i d e r the r a t i o , ^W/'y/ , f o r the exact ( s p i n l e s s ) wave f u n c t i o n . Among the necessary r e l a t i o n s needed to keep t h i s r a t i o constant are c o n d i t i o n s on s i m i l a r t o those of Kato * Higher order Coulomb s i n g u l a r i t i e s (coalescence of more than two p a r t i c l e s ) may be examined t h i s way. For more complicated cases, however, t h i s approach becomes v e r y i n v o l v e d . Pack p and Byers Brown were the f i r s t to d e r i v e r i g o r - o u s l y equations s i m i l a r t o (2.3.1a) and (2.3.2a) The c o n d i t i o n s of Roothaan et a l appear the same as Kato's,but a d i f f e r e n t set of independent v a r i a b l e s has been used. 27 allowing n o n - t r i v i a l coalescence conditions as well as cusp conditions to be found. They also removed the heavy-nucleus approximation. A b r i e f outline of t h e i r i n s t r u c t i v e method i s presented here: The general N - p a r t i c l e Schrodinger equation using the Hamiltonian (2.2.1) was solved i n the region of coalescence of two p a r t i c l e s ( l a b e l l e d '1' and '2' f o r convenience)—that i s , i n the manifold of points ^ — & , f^jy^& f o r a l l fij ̂ = /f^ and €. , some small p o s i t i v e constant. Transforming the space-fixed p o s i t i o n coordinates Jj , ^ / l to the center of mass and r e l a t i v e coordinates and S 7*?, -h ?r? of the two p a r t i c l e s , allowed the Schrodinger equation to be r e w r i t t e n i n the v i c i n i t y of the coalescence as Ij/^O (2.3.5) i s the LaPlacian operator f o r the v a r i a b l e £u , ytt/a. i s the reduced mass, the charges of the p a r t i c l e s , and contains a l l other terms of the Hamiltonian of order equal to or greater than 28 zero m constant. The g e n e r a l "bounded s o l u t i o n of ( 2 . 3 - 5 ) has form 2 / = £ 2 r'ftJHYttof*'** (2-3-6) where = (r-,&,<f>) and the W e s p h e r i c a l h a r - monics. For e l e c t r o n - n u c l e u s coalescence, w i t h the nucleus at the o r i g i n , Jrl2_ "becomes the r a d i u s v e c t o r Jj of the e l e c t r o n . S u b s t i t u t i n g ( 2 . 3 . 6 ) i n t o the d i f f e r e n t i a l equation ( 2 . 3 . 5 ) > expanding as a power s e r i e s i n r, and s o l v i n g , Pack and Byers Brown found a unique s o l u t i o n , t r u e f o r f/JL ^ S ̂ 1 * A (2 .5 .8) where Y = Z, 'z.y^/a. • They d e f i n e d ^ , a parameter r e l a t e d t o the nodal * s t r u c t u r e of the system, t o be the s m a l l e s t value The p h y s i c a l meaning of A as d e f i n e d here i s l o s t when the system does not have s p h e r i c a l symmetry about the coalescence.Examples are e l e c t r o n - e l e c t r o n coalescence i n atoms, or any type of coalescence i n molecules. 2 - 9 of z f o r which Tf/y, ^ 0 at the coalescence of the two p a r t i c l e s . The e q u i v a l e n t d i f f e r e n t i a l form of equation (2.3.8) i s ^ K ? $ ] _ « i i /n i (2.3.8a) ^ \ where the angular average o p e r a t o r , , i s m o d i f i e d to / c ( - ^ l j ^ r n • This equation g i v e s n o n - t r i v i a l coalescence c o n d i t i o n s f o r the case of a node at coalescence i n c o n t r a s t to e a r l i e r approaches. For the ( u s u a l ) nodeless case ^ i s zero and Kato's cusp c o n d i t i o n s can be recovered as ( ^ ) - & Y i s 1/2 f o r e l e c t r o n - e l e c t r o n coalescence and Y = f o r e l e c t r o n - n u c l e u s c o a l e s c e n c e — t h e same value t h a t Kato found, but w i t h a mass c o r r e c t i o n to the heavy-nucleus approximation."* Note again t h a t * * a l l exact s p i n l e s s wave f u n c t i o n s must have an ex- pansion l i k e (2.3.8) around a Coulomb s i n g u l a r i t y . yCC = 1 i f mass c o r r e c t e d atomic u n i t s are used. See Appendix A. * * Matsen's ' s p i n l e s s ' wave f u n c t i o n s are not r e a l l y s p i n l e s s . S p i n i s represented i m p l i c i t l y by a p p l y i n g a p p r o p r i a t e permutations of the symmetric group t o a s p a t i a l s o l u t i o n of the Schrb'dinger equation. See L~16J f o r f u r t h e r i n f o r m a t i o n . 30 But a s p i n l e s s wave f u n c t i o n i s not r e a l i s t i c . Any p h y s i c a l l y a c c e p t a b l e , quantum-mechanical wave f u n c t i o n must c o n t a i n s p i n c o o r d i n a t e s f o r i t s p a r t - i c l e s . Let us r e s t r i c t the d i s c u s s i o n now t o the s p e c i f i c case of N - e l e c t r o n atomic and molecular wave f u n c t i o n s . I n the Born-Oppenheimer, or heavy-nucleus, approximation the nuc l e a r c o o r d i n a t e s do not appear e x p l i c i t l y and only the e l e c t r o n i c c o o r d i n a t e s need be c o n s i d e r e d . ( A l s o the P a u l i p r i n c i p l e f o r a system of i d e n t i c a l fermions must be obeyed l e a d i n g t o a wave f u n c t i o n antisymmetric w i t h r e s p e c t to the i n t e r - change of any two set s of e l e c t r o n i c c o o r d i n a t e s ) . I n the n o n r e l a t i v i s t i c approximation both the t o t a l - s p i n operator and an a r b i t r a r y spin-component operator commute w i t h the H a m i l t o n i a n . Thus i t i s d e s i r a b l e t h a t a wave f u n c t i o n w i t h s p i n , (]5S)f1 , should have sharp t o t a l ( e l e c t r o n i c ) s p i n and a sharp s p i n component: (2.3.10) * 4 = n $S| *The e f f e c t s of n u c l e a r spins on the wave f u n c t i o n can be i n c l u d e d , i f necessary, as p e r t u r b a t i o n s . 31 where <|>s,M = ) JE*.?*S») i s a f u n c t i o n of space and s p i n c o o r d i n a t e s , Jj and Si , of each e l e c t r o n . Any wave f u n c t i o n c o n t a i n i n g s p i n 54 55 can be expanded^ 1 y y Here ^s,Mi s n o * normalized. The s p i n f u n c t i o n s |©s,(i;kj c o n s t i t u t e the complete set of l i n e a r l y i n d e - pendent s p i n f u n c t i o n s f o r N e l e c t r o n s . They are e i g e n f u n c t i o n s of W and jfc^z having eigenvalues S(S+1) and M r e s p e c t i v e l y . The f u n c t i o n s {%J are formed by symmetric group o p e r a t i o n s on some s p a t i a l s o l u t i o n of the Schrodinger equation and are a l l degenerate energy e i g e n f u n c t i o n s . Since an exact s p a t i a l s o l u t i o n s a t i s f i e s the coalescence c o n d i t i o n s ( 2 . 3 . 8 ) each % must a l s o , n e c e s s i t a t i n g the s a t i s - - f a c t i o n of ( 2 . 3 . 8 ) by an ( e x a c t ) s p i n c o n t a i n i n g f u n c t i o n , $s,/i . And as s p h e r i c a l l y averaging does not a f f e c t the argument, the d i f f e r e n t i a l cusp c o n d i t i o n s of Kato ( 2 . 3 . 1 ) , ( 2 . 3 . 2 ) a l s o apply. These preceding approaches a l l have the same gener a l l i m i t a t i o n s : 1. They o n l y t r e a t two p a r t i c l e coalescence. Higher order s i n g u l a r i t i e s , when tcj ^- ^ 32 f o r s e v e r a l c ,J , are assumed not t o occur. Thus the behaviour of exact wave f u n c t i o n s at 'Coulomb s i n g u l a r i t i e s has been i n v e s t i g a t e d f o r a l i m i t e d number of p o i n t s i n the m a n i f o l d { /jy = O ; i j = /, N j c ±j] . 2. They r e a l l y t r e a t o n l y the s p h e r i c a l l y symmetric p a r t of the cusp. A s p h e r i c a l average over the wave f u n c t i o n i n equations (2.3.1a), (2.3.2a), (2.3.8) must be taken i f r e l a t i o n s h i p s i n v o l v i n g completely determined q u a n t i t i e s are d e s i r e d . The angular dependence of the :Coulomb cusp p S P a r i s e s from the other (N-2) p a r t i c l e s . ' y I n a p h y s i c a l sense these l i m i t a t i o n s are not severe. The value of the approaches i s t h a t necessary c o n d i t i o n s f o r the behaviour of exact wave f u n c t i o n s ( w i t h s p i n ) at the most important Coulomb s i n g u l a r i t i e s have been d e r i v e d . 5 2.4- Cusp c a l c u l a t i o n methods f o r approximate wave f u n c t i o n s * Observe the f o l l o w i n g d i s t i n c t i o n s t o a v o i d l a t e r c o n f u s i o n . Cusp and coalescence c o n d i t i o n s g i v e n by equations (2.3.1), (2.3.2), (2.3.8) are the ones mentioned and a p p l i e d i n t h i s t h e s i s . An exact wave f u n c t i o n n e c e s s a r i l y s a t i s f i e s these c o n d i t i o n s , although they are not the o n l y ones. Approximate wave f u n c t i o n s can s a t i s f y them a l s o i n a necessary and/or s u f f i c i e n t way. These aspects of cusp c o n d i t i o n a p p l i c a t i o n s w i l l be d i s c u s s e d more f u l l y l a t e r i n the t e x t . Wave f u n c t i o n s ( w i t h t h e i r d e r i v a t i v e s ) obtained toy approximate methods do not n e c e s s a r i l y have the same types of d i s c o n t i n u i t i e s as the corresponding exact f u n c t i o n s . I f one i s s t r i v i n g to copy an exact f u n c t i o n , as i s u s u a l l y the case, the approximation c o u l d g i v e b e t t e r p o i n t p r o p e r t i e s i f i t has the c o r r e c t behaviour at the s i n g u l a r p o i n t s . P l u v i n a g e ^ 47 was among the f i r s t t o apply t h i s r e a s o n i n g ; Kato ' * was the f i r s t t o provide a ge n e r a l t o o l f o r d e s c r i b i n g Coulomb s i n g u l a r i t i e s . I n t h i s l i g h t i t i s of gene r a l i n t e r e s t t o analyse the importance of the cusp. How might the proper cusp be important f o r approximate wave f u n c t i o n s ? To answer t h i s l o o k at i t s e f f e c t on e x p e c t a t i o n v a l u e s . F i r s t , the energy. The e l e c t r o n - e l e c t r o n cusp seems, at f i r s t g lance, to be d i r e c t l y r e l a t e d t o the c o r r e l a t i o n problem. A proper d e s c r i p t i o n of c o r r e l a t i o n phenomena s u r e l y i n v o l v e s the behaviour of the wave f u n c t i o n at e l e c t r o n coalescence when two e l e c t r o n s approach the same p o i n t i n space. C o r r e l a t e d wave f u n c t i o n s ( c o n t a i n i n g i n t e r e l e c t r o n i c c o o r d i n a t e s , f'tj , e x p l i c i t l y ) converge more r a p i d l y than c o n f i g u r a t i o n i n t e r a c t i o n (CI) 35 57 expansions without r ^ ,^>-" ;rh e d i f f e r e n c e c o u l d * The absence of a treatment f o r the h i g h e r order s i n g - u l a r i t i e s should not be too s e r i o u s E49] . Three body e f f e c t s appear t o be much l e s s important than two body i n t e r a c t i o n s i n d e f i n i n g atomic and molecular p r o p e r t i e s . See, f o r example r e f e r e n c e [ 5J 34 be t h a t c o r r e l a t e d f u n c t i o n s can e a s i l y represent exact e l e c t r o n cusps. CI f u n c t i o n s f o r helium w i t h a f i n i t e number of terms cannot p o s s i b l y ^ ' s i n c e the occurrence of o n l y even powers of r ^ i n any expansion n e c e s s i - t a t e s f ^ J « ) = o . (Compare w i t h ( 2 . 3 . 2 ) ). Analyses by G i l b e r t - ^ and Gimarc, Cooney and P a r r , * however, su b s c r i b e t h a t adequate d e s c r i p t i o n of the Coulomb hole c o n t r i b u t e s more to c o r r e l a t i o n energy than does proper cusp behaviour. The cusp r e g i o n l i e s i n s i d e the energy-important p a r t of the Coulomb h o l e . Since e l e c t r o n - n u c l e u s c o n t r i b u t i o n to c o r r e l - a t i o n energy i s n e g l i g i b l e i t would seem t h a t the accuracy of cusps, both e l e c t r o n - e l e c t r o n and e l e c t r o n - nucleus has l i t t l e to do w i t h the accuracy of energy. There remains the q u e s t i o n w i t h r e s p e c t t o other e x p e c t a t i o n v a l u e s . R e c a l l t h a t t h e r e i s l i t t l e c o n n e c t i o n between the accuracy of approximate energy and accuracy of d i f f e r e n t , approximate p r o p e r t i e s . Consequently the c o n c l u s i o n s reached i n the previous paragraph f o r energy may not be v a l i d f o r other obser- v a b l e s . R e i t e r a t i n g s e c t i o n 1 . 2 , the o b j e c t of t h i s t h e s i s w i l l be to examine the r e l a t i o n s h i p of the cusp to c e r t a i n p r o p e r t i e s — t h e e l e c t r o n and s p i n d e n s i t i e s Quoted by Gimarc and P a r r • * This c o n c l u s i o n of Coulson and N e i l s o n f o r the case of helium was quoted by G i l b e r t [ 5 8 ] 35 at the nuc l e u s . Previous work done on t h i s s p e c i f i c 59 60 problenr ' w i l l be reviewed i n a l a t e r s e c t i o n . The next step i s t o decide how t o evaluate cusps f o r approximate wave f u n c t i o n s . Obviously i f a f u n c t i o n s a t i s f i e s equations (2.3.1), (2.3.2) or (2.3.8) i t has a proper cusp, but t h i s approach i s not p r a c t i c a l f o r almost a l l wave f u n c t i o n s , due t o the tediousness of the a l g e b r a , nor does i t g i v e an estimate of the close n e s s of the cusp t o the c o r r e c t value (r) . E a s i e r methods e x i s t . The e l e c t r o n - n u c l e u s cusp e v a l u a t i o n f o r s e l f - c o n s i s t e n t f i e l d (SCF) o r b i t a l s i s w e l l documented. (However the remarks concerning e l e c t r o n - e l e c t r o n cusps i n CI f u n c t i o n s made e a r l i e r i n t h i s s e c t i o n a l s o apply t o SCF f u n c t i o n s ; i t i s d i f f i c u l t f o r a wave f u n c t i o n without e x p l i c i t r . . c o r r e l a t i o n t o have the c o r r e c t e l e c t r o n - e l e c t r o n c u s p ) . An exact SCF o r b i t a l has the general form frm'*-*h£-fnA>Ym(e,<P) • (2.4.1) /7 , <£ , are the u s u a l o r b i t a l quantum numbers; j Yjlftj are the s p h e r i c a l harmonics. To s a t i s f y the gener a l coalescence c o n d i t i o n s (2.3.8a) f o r e l e c t r o n - nucleus coalescence i t i s s u f f i c i e n t t h a t the r a d i a l 3 6 p a r t of < P h $ > n , Y2jT^0r) , obey (2.4.2) Numerical s o l u t i o n s of the Hartree-Fock (HF) equations have t h i s c o n d i t i o n b u i l t i n t o them a u t o m a t i c a l l y and consequently should have good cusp v a l u e s . One i n d i c a t i o n of convergence of the non-exact, a n a l y t i c a l HF s o l u t i o n s i s the closeness of the r a t i o te±il{3fh< ) (2.4.2a) op. t o a = -Z. Roothaan, Sachs and Weiss have mentioned t h i s as an accuracy t e s t of t h e i r HF wave f u n c t i o n s i n the r e g i o n r —> 0. Clementi has evaluated the r a t i o f o r a n a l y t i c a l HF o r b i t a l s of helium through argon. Another method f o r cusp e v a l u a t i o n i s due t o Chong^ He changed the form of the coalescence c o n d i t i o n s ( 2 . 3 . 8 ) t o r e l a t i o n s h i p s between e x p e c t a t i o n values of SCF o r b i t a l s , c a l l e d i n t e g r a l coalescence cond- i t i o n s . I f an SCF o r b i t a l ,  (Png»,(^> &;&) , has a c o r r e c t cusp the r a d i a l f u n c t i o n -fhjtfH , (see (2.4.1) ), must n e c e s s a r i l y s a t i s f y (2.4.3) o 37 The formula <f(Jr) = -r~-, has been used here, c o r r e c t n r f o r the s p h e r i c a l l y symmetric r a d i a l f u n c t i o n . £(£) i s the D i r a c d e l t a f u n c t i o n . E quation (2.4 . 3 ) can be extended e a s i l y t o i n c l u d e the f u l l o r b i t a l exp- r e s s i o n . The s p i n dependence of SCF o r b i t a l s leads to no problems; arguments presented e a r l i e r can imme- d i a t e l y permit equations (2.4.2) and (2.4 . 3 ) t o be a p p l i e d t o any SCF s p i n o r b i t a l . I n Dirac n o t a t i o n , now, the i n t e g r a l coalescence c o n d i t i o n s f o r SCF s p i n o r b i t a l s appear l i k e The d e v i a t i o n from zero of t h i s i n t e g r a l w i l l g ive an estimate of how c l o s e the o n e - e l e c t r o n SCF o r b i t a l (Pft£frt comes t o having the proper cusp behaviour at the nucleus. CI wave f u n c t i o n s need a d i f f e r e n t approach.. The r a t i o , (2.4.2a), u s i n g d i f f e r e n t i a l coalescence c o n d i t i o n s may be a l l r i g h t f o r checking the cusp of SCF o r b i t a l s , but i t cannot be a p p l i e d t o many-electron CI f u n c t i o n s . Because CI f u n c t i o n s do not have the For the case of atoms, not molecules, equation (2.4.3) c o n t a i n s a 'pseudo-integration' which o n l y i n v o l v e s t a k i n g a l i m i t . Chong's i n t e g r a l c o n d i t i o n s (2.4.4) are thus necessary and s u f f i c i e n t f o r atoms. 38 simple independent p a r t i c l e i n t e r p r e t a t i o n of SCF wave f u n c t i o n s the i n t e g r a l coalescence c o n d i t i o n s (2.4.4) cannot be used d i r e c t l y e i t h e r . Chong has been able t o f i n d coalescence c o n d i t i o n s f o r CI f u n c t i o n s corresponding t o (2.4.4) by g e n e r a l i z i n g the cusp r e l a t i o n s of S t e i n e r (2.3.3) f o r e l e c t r o n d e n s i t y and of Bingel (2.3-4) f o r both e l e c t r o n and s p i n d e n s i t y . He obtained equations f o r i n t e g r a l s p i n , and i n t e g r a l e l e c t r o n coalescence c o n d i t i o n s which can be w r i t t e n compactly as a = s or e designates the s p i n or e l e c t r o n c o n d i t i o n s r e s p e c t i v e l y . Y , )\ have the same meaning as i n (2.3.8). , are the o n e - e l e c t r o n g r a d i e n t and d e n s i t y operators evaluated at the nucleus. N (2.4.6) 4 ^ *4- 39 rf~(r) i s the Dirac d e l t a f u n c t i o n . Note t h a t f o r s i n g l e t s p i n s t a t e s , (yS^*) - 6 and the s p i n c o a l - escence c o n d i t i o n s become t r i v i a l . For the nodeless case, /\ = 0, are the u s u a l d e n s i t y operators, and i n t e g r a l cusp c o n d i t i o n s are expressed. S p i n d e n s i t y at the nucleus i s g i v e n by Q S(°) - ( T - < Z > = o ) (2.4.8) and l i k e w i s e e l e c t r o n d e n s i t y i s Q e(0) = ^ > * o " ) (2.4.9) The expressions (2.4.5) are necessary r e l a t i o n s f o r exact wave f u n c t i o n s but they are extremely u s e f u l i n cusp e v a l u a t i o n f o r any approximate f u n c t i o n . Here, as b e f o r e , the c o r r e c t n e s s of the cusp i s i n d i - c ated by the va l u e s of the i n t e g r a l s i n (2.4.5) f o r the wave f u n c t i o n being examined. The value of the approximat e cusp, r = ( X + l ) < ^ * C T > . o = e . , , (2.4.10) i s t o be compared w i t h t h a t of the exact cusp Y = -Z . (2.4.11) 40 The approach i n t h i s work i s t o f o r c e t r i a l wave func t i o n s , i n v a r i o u s ways, to have Evidence t h a t t h i s procedure i s expected t o l e a d t o improved p r o p e r t i e s i s presented i n the next s e c t i o n . 2.5 Use of cusp and coalescence c o n d i t i o n s f o r im- provement of approximate wave f u n c t i o n s Touched upon i n the i n t r o d u c t i o n was the f a c t t h a t e x p e c t a t i o n v a l u e s of p o i n t p r o p e r t i e s are r a t h e r s p e c i a l compared w i t h the u s u a l type of observable. They depend on the value of a wave f u n c t i o n at a s i n g l e p o i n t and are not averaged out over the space surrounding the system. Thus a wave f u n c t i o n t h a t might be q u i t e good when con s i d e r e d throughout space c o u l d , indeed, be e x c e p t i o n a l l y poor at or near c e r t a i n p o i n t s . For few examples i s t h i s o b s e r v a t i o n more t r u e than the s p i n d e n s i t y at the nucleus. (See Table I ) . Any improvement of the approximate f u n c t i o n towards the exact i n the nu c l e a r r e g i o n should h o p e f u l l y im- prove such p o i n t p r o p e r t i e s as the e l e c t r o n and s p i n d e n s i t i e s t h e r e . N a t u r a l l y other f a c t o r s — c o r r e l a t i o n and core p o l a r i z a t i o n f o r e x a m p l e s — a l s o i n f l u e n c e the wave f u n c t i o n and these may o v e r r i d e any improvement, (2.4.12) 41 at the cusp. But s u r e l y a wave f u n c t i o n should have the c o r r e c t behaviour at a p o i n t of n o n - a h a l y t i c i t y , and s u r e l y t h e o r e t i c a l c o n d i t i o n s l i k e the cusp r e - l a t i o n s are j u s t as v a l i d as minimum energy f o r d e t e r - mining the f u n c t i o n . This r e a s o n i n g — t h a t a f u n c t i o n w i t h a c o r r e c t cusp i s a b e t t e r f u n c t i o n — h a s been s e i z e d on by many 40 50 workers. Hoothaan, Weiss and Kolos c o n s t r u c t e d c o r r e l a t e d f u n c t i o n s f o r helium and the hydrogen mol- ecule which have the c o r r e c t e l e c t r o n - e l e c t r o n and 63 e l e c t r o n - n u c l e u s cusps. Conroy ^ has used s p e c i a l c u s p - s a t i s f y i n g bases i n h i s unique c a l c u l a t i o n s and 64 P a r r , Weare and Weber have i n v e s t i g a t e d c u s p - s a t i s -/ 65 f y i n g Hulthen o r b i t a l s . K e l l y and Eoothaan ^ presented a treatment t h a t shows how to choose a S l a t e r - t y p e o r b i t a l (STO) b a s i s so t h a t atomic SCF o r b i t a l s w i l l s a t i s f y the coalescence c o n d i t i o n s at the nucleus; merely use the set of STO's I s , 3s, 4s,.;...; 2p, 4p, 5p,...; 3d, 5d, 6d, (2.5.1) where the f i r s t o r b i t a l of any angular symmetry ( Ji = n-1) has f i x e d o r b i t a l exponent n (2.5.2) 42 the members | 2s, 3p, 4 d , . . . j are not present, and a l l other exponents are f r e e t o be v a r i e d . Any atomic SCF o r b i t a l expressed as a l i n e a r combination of members from t h i s s p e c i a l set w i l l a u t o m a t i c a l l y have the c o r - r e c t behaviour at the nu c l e a r cusp.' This choice of b a s i s i s becoming q u i t e normal i n SCF-type atomic 21 65 66 67 c a l c u l a t i o n s . » » < Another procedure i s a v a i l a b l e f o r SCF c a l c u l a t i o n s — a c o n s t r a i n e d v a r i a t i o n a l 6ft approach of Handy, P a r r and Weber based on t h e i r elegant c o n s t r a i n t procedure ( [ 693 and Appendix C) — b u t i s d i s c u s s e d i n Chapter IV. The assumption t h a t a b e t t e r wave f u n c t i o n i s obtained i s the on l y apparent r a t i o n a l e behind t h i s f l u r r y of producing c u s p - s a t i s f y i n g approximate wave f u n c t i o n s . With a view towards c l a r i f y i n g the q u e s t i o n , 'Does a good cusp r e a l l y mean an i n t r i n s i c a l l y b e t t e r wave f u n c t i o n ? ' , Chong and Schrader examined the s t a t i s t i c a l c o r r e l a t i o n between e l e c t r o n d e n s i t y and cusp i n v a r i o u s helium wave f u n c t i o n s r a n g i n g from simple SCF t o h i g h l y c o r r e l a t e d ones. They d i s c o v e r e d a st r o n g c o r r e l a t i o n between the e r r o r i n the n u c l e a r cusp and the e r r o r i n the e l e c t r o n d e n s i t y at the nucleus. See f i g u r e 1. When compared w i t h the low degree of c o r r e l a t i o n between the e r r o r i n the cusp and the accuracy of energy, t h i s r e s u l t becomes important wave f u n c t i o n s , found by Chong and Schrader. 44 i m p l y i n g t h a t improvement of the cusp does improve at l e a s t the e l e c t r o n d e n s i t y , Q e ( 0 ) . A h i g h c o r r e - l a t i o n between the e l e c t r o n - e l e c t r o n cuspc'and the e x p e c t a t i o n value was a l s o found but here a l a r g e r correspondence between the cusp value and the energy e x i s t s making a s i m i l a r c o n c l u s i o n i n v a l i d f o r the p o i n t d e n s i t y 59 Having t h i s j u s t i f i c a t i o n Chong and Yue^ a p p l i e d the theory of l i n e a r c o n s t r a i n t s (Appendix C) to v a r i o u s helium CI (without r . .) f u n c t i o n s of 3-8 terms w i t h the i d e a of f o r c i n g simple, e a s i l y c a l c u l a t e d wave f u n c t i o n s t o have a good Q ( 0 ) . The c o n c l u s i o n s of Chong and S c h r a d e r ^ are q u a l i t a t i v e l y s u b s t a n t i a t e d i n t h e i r study. A f a i r l y f l e x i b l e f u n c t i o n was found to be necessary t o absorb the e f f e c t of the c o n s t r a i n t . I n the cases t e s t e d though, the a p p l i c a t i o n of the cusp c o n s t r a i n t , _ C = Y , (see the t e x t preceding (2.4.12) ), w h i l e improving Q e ( 0 ) , l e d t o a s l i g h t o v e r - c o r r e c t i o n . U n f o r t u n a t e l y , e r r o r s i n the values ^ r " ~ 2 ^ , ^ r ~ ^ d i d not decrease upon c o n s t r a i n t as might be expected from an improvement of the func- t i o n near the nucleus. This approach of Chong and Yue w i t h r e s p e c t t o cusp s a t i s f a c t i o n , however, i s unique and needs f u r t h e r i n v e s t i g a t i o n . I t w i l l be used i n the next chapter on l i t h i u m wave f u n c t i o n s to check p o s s i b l e improvements i n both Q e(0) and Q s ( 0 ) . 4-5 CHAPTER I I I INTEGRAL CUSP CONSTRAINTS AND APPLICATIONS TO LITHIUM 2 S GROUNDSTATE FUNCTIONS 3.1 Formation of c o n s t r a i n t s The c o r r e l a t i o n "between e r r o r i n n u c l e a r cusp and e r r o r i n Q e(0) f o r approximate helium wave f u n c t i o n s ^ , d i s c u s s e d i n S e c t i o n 2.5, l e d t o the d i s c o v e r y t h a t when a He CI wave f u n c t i o n was f o r c e d t o have a good re 59 = -Z, i t s e l e c t r o n d e n s i t y improved^ , p r o v i d i n g the f u n c t i o n had enough l i n e a r parameters t o absorb the e f f e c t of c o n s t r a i n i n g . Attempts t o p s u b s t a n t i a t e these r e s u l t s for.: L i S groundstate wave f u n c t i o n s are d e s c r i b e d i n t h i s chapter. Of prim- ary i n t e r e s t though i s the a d d i t i o n a l p o s s i b i l i t y of c o r r e l a t i o n s between Q s(0) and _C e or _CS , si n c e 59 60 the s t a t e s of helium examined-"' have no s p i n den- s i t y . Because JZ* need not equal JI f o r approximate wave f u n c t i o n s , the e f f e c t of a double c o n s t r a i n t , r . I s . -z i s t e s t e d . Various o f f - d i a g o n a l 'weighting' c o n s t r a i n t s are developed and a p p l i e d as w e l l . 4-6 Note the f o l l o w i n g p o i n t s : (1) I n t e g r a l cusp c o n s t r a i n t s are only necessary c o n d i t i o n s and the f a c t t h a t Ĵ *'5 = -Z does not mean the f u n c t i o n has the c o r r e c t cusp. (2) Since i s a r a t i o , c o n s t r a i n i n g i t does not d i c t a t e a value f o r Q e , s ( 0 ) . (3) There i s no immediate i n t e r e s t i n d e v e l - oping a c o n s t r a i n t f o r the e l e c t r o n - e l e c t r o n cusp. I n c o n s t r a i n e d v a r i a t i o n one wishes t o minimize the energy of an approximate wave f u n c t i o n subject t o a c e r t a i n number (k) of c o n s t r a i n t s . The b a s i c procedure i s t o d e f i n e c o n s t r a i n t o p e r a t o r s , &c' , d e s c r i b i n g the a t t r i b u t e s t o be c o n s t r a i n e d , so th a t when the c o n s t r a i n t s are s a t i s f i e d , ^Gt) = 0 f o r each / . The mo d i f i e d v a r i a t i o n a l p r i n c i p l e takes the form (Sy\\H+Z.*iGi-Ec}lp) = o ( j . i . i ) where H i s the Ha m i l t o n i a n f o r the system, E i s the energy of the c o n s t r a i n e d e i g e n f u n c t i o n and }\c' , 47 the Lagrange m u l t i p l i e r s , are t o be determined. The term s i n g l e c o n s t r a i n t means K = 1 . L i k e w i s e K = 2 i m p l i e s a double c o n s t r a i n t . Appendix C co n t a i n s a summary of methods t o solve ( 3 . 1 . 1 ) . F o l l o w i n g Chong^' 59 and Chong and Yue^ y the c o n s t r a i n t operators employed i n t h i s work f o r the n u c l e a r cusp c o n d i t i o n have the form ^-k(P" -Pat) , a^e.s ( 3 . 1 . 2 ) where y ? * = ^ £ * — Xc^=0 and J s ^ , are d e f i n e d i n equations ( 2 / * 4 . 7 ) . When such a cons- t r a i n t i s imposed, = 0 i m p l i e s as d e s i r e d . The c o n s t r a i n e d v a r i a t i o n a l s o l u t i o n of ( 3 - 1 . 1 ) i s the eigenvector having the lowest eigen- value of the m a t r i x r e p r e s e n t a t i o n of the f i c t i t i o u s H a m i l t o n i a n 4 f -H+±\& (5-I-« and s a t i s f y i n g ((^,;) = 0 . Thus (&i must be He r m i t i a n . The operator o<_ i s H e r m i t i a n but i s not; hence the form f o r i n ( 3 . 1 . 2 ) . This i s not a unique 48 c h o i c e . A whole h i e r a r c h y of H e r m i t i a n combinations of ^ and st£f w i l l a l s o l e a d t o ( 3 . 1 . 3 ) , f o r example. The r e s u l t s obtained from d i f f e r e n t are expected to be q u a l i t a t i v e l y the same. A d i a g o n a l c o n s t r a i n t r e s u l t s i n the c o n d i t i o n and i s best s o l v e d u s i n g the well-dev e l o p e d methods of Byers Brown, Chong and R a s i e l . 7 0 ' 7 1 ' ? 2 An o f f - d i a g o n a l c o n s t r a i n t , (0l<5lty> =0 , (3 . 1 .6) where $ may be an e x c i t e d s t a t e or even an (almost) a r b i t r a r y w e i g h t i n g f u n c t i o n , i s e a s i l y imposed by 69 the r e c e n t l y p u b l i s h e d method of Weber and Handy. A l l wave f u n c t i o n s i n t h i s work are i n i t i a l l y c h a r a c t e r i z e d by t h e i r f r e e v a r i a t i o n a l f o r m — t h a t i s , w i t h no c o n s t r a i n t imposed save n o r m a l i z a t i o n . The u s u a l S l a t e r - t y p e o r b i t a l (STO) on e - e l e c t r o n b a s i s , X.r , 8 r n0-,©,<M =A/»/~''-'e-Tf~)snfe)<l>) , ( 3 . 1 . 7 ) 4-9 i s always employed. The o r b i t a l exponents, S » i f v a r i e d , are s u c c e s s i v e l y o p t i m i z e d by p a r a b o l i c 30 i n t e r p o l a t i o n t o minimum energy. One i t e r a t i o n c y c l e i s completed when a l l exponents have been optimized once. U s u a l l y two or three c y c l e s w i l l ensure a m i n i - mized energy p r o v i d i n g the i n i t i a l estimates of the exponents are reasonable. Since the s p i n f u n c t i o n &^ i n (2.1.8) does not c o n t r i b u t e s i g n i f i c a n t l y towards energy, i t i s not i n c l u d e d u n t i l a f t e r ex- ponent o p t i m i z a t i o n . Terms c o n t a i n i n g 0*_ w i l l be designated ©i-type terms, or t r i p l e t core s p i n terms. To so l v e (3.1.1) a t r a n s f o r m a t i o n of a l l m a t r i c e s from c o n f i g u r a t i o n a l space t o the b a s i s of f r e e v a r - i a t i o n a l e i g e n f u n c t i o n s [ § { j i s advantageous f o r two reasons: (1) An orthonormal b a s i s s i m p l i f i e s c a l c u l a t i o n s . (2) This t r a n s f o r m a t i o n leads t o conceptual a d v a n t a g e s — a c o n s t r a i n e d f u n c t i o n appears as the f r e e v a r i a t i o n a l groundstate eigen- f u n c t i o n w i t h s m a l l ' c o r r e c t i v e ' terms. (3.1.8) \ai\«\ 50 The s e v e r i t y of the imposed c o n s t r a i n t can be estimated by e i t h e r the r a t e of convergence towards the c o r r e c t c o n s t r a i n e d f u n c t i o n , or the energy s a c r i f i c e ZlE = E - E„ ... 1 c f r e e v a r i a t i o n a l 3.2 E x p l o r a t o r y c a l c u l a t i o n s I t was hoped at the s t a r t of t h i s work t h a t r e a s - onable s p i n d e n s i t i e s c o u l d be obtained merely by con- s t r a i n i n g any simple f u n c t i o n to s a t i s f y the n u c l e a r cusp c o n d i t i o n s . Consequently a r a t h e r naive i n i t i a l approach was taken. The f i r s t waveJfunctions examined d i d not l e a d to unambiguous c o n c l u s i o n s . They d i d , however, i l l u s t r a t e the computational d i f f i c u l t i e s encountered and i n d i c a t e d a more r e f i n e d approach t o be d e s c r i b e d i n the next (3.3) s e c t i o n . Two types of f u n c t i o n s were developed f o r t h i s i n i t i a l study. The f i r s t type was comprised of a s e r i e s of f u n c t i o n s having 4 — 8 terms and p a r t i a l l y (not completely) optimized o r b i t a l exponents. The 7 and 8 term f u n c t i o n s , c o n t a i n i n g t r i p l e t core s p i n terms and some p-type angular c o r r e l a t i o n , are a c t u a l l y q u i t e good i n s p i t e of t h e i r s i m p l i c i t y but are not f l e x i b l e engugh f o r a meaningful study on the e f f e c t of c o n s t r a i n i n g . The second type, a s e r i e s of 1 0 — 1 5 5 1 terms, w i t h an i n c r e a s i n g number of t r i p l e t core s p i n terms, and w i t h p-type c o r r e l a t i o n , had o r b i t a l expo- 35 nents t r a n s p l a n t e d from the L i CI f u n c t i o n s of Weiss.' y This group i s poor indeed w i t h r e s p e c t t o energy, but s l i g h t l y more f l e x i b l e than' the f i r s t s e r i e s . S e v e r a l d i f f e r e n t attempts t o solve ( 3 . 1 . 1 ) f o r s i n g l e c o n s t r a i n t s are now d i s c u s s e d w i t h , the seven term f u n c t i o n from the f i r s t group, as an exam- p l e . I t became necessary t o i n v e s t i g a t e t h i s aspect 70 when the p e r t u r b a t i o n approach' f a i l e d t o give an i n i t i a l v a l u e t o \ , the Lagrange m u l t i p l i e r , f o r s e v e r a l f u n c t i o n s . S e r i e s divergences, exponent over- f l o w s , e t c . , are c h a r a c t e r i s t i c r e s u l t s of attempting to c o n s t r a i n i n f l e x i b l e f u n c t i o n s u s i n g a p e r t u r b a t i o n - type approach. Since p r e d i c t i o n of which f u n c t i o n s cause d i f f i c u l t i e s i s u n c e r t a i n , f i n d i n g a f o o l p r o o f method of s o l v i n g ( 3 . 1 . 1 ) i s h e l p f u l . The f r e e v a r i a t i o n a l d e s c r i p t i o n of 7^ , w i t h i t s p r o p e r t i e s f r e e and c o n s t r a i n e d i s l i s t e d i n Table I I . Although Q e(0) shows a 2% improvement, the energy s a c r i f i c e , A E, f o r the c o n s t r a i n t J T = If i s h i g h ; the strange value of Q s(0) a l s o i n d i c a t e s the s e v e r i t y of t h i s c o n s t r a i n t . Contrast w i t h the corresponding case The f u n c t i o n , (1/ , i s a t y p i c a l but i t pr o v i d e s a good t e s t case. These cusp c o n s t r a i n t s 5 2 p r o v i d e the f i r s t example of f a i l u r e of the p e r t u r b - a t i o n ^approach; the d e s i r e d behaviour i s i l l u s t r a t e d i n Table I I I by a f l e x i b l e ( f o r a 2 e l e c t r o n system) 7 term helium wave f u n c t i o n , (ft7 , of Yue and Chong^ 9 and compared w i t h the behaviour of ( L i ) . Only the p a r a m e t r i z a t i o n approach remains. Here f i c t i t i o n a l wave f u n c t i o n s are c a l c u l a t e d from ( 3 . 1 . 1 ) f o r d i f f e r e n t v a l u e s of A u n t i l , f o r some optimum, )iopt » a r o o t of CO) = (yl£/%y=<<S>=0 (2.2.!) where ffif/l) i s found from ( 3 . 1 . 1 ) . At )\ = \0p-t C()l+:)~0 ( 3 . 2 . 2 ) The problem i s t h a t u n l e s s one approximately knows ^ o p t > can be d i f f i c u l t t o l o c a t e . To understand b e t t e r what i s i n v o l v e d i n s o l v i n g ( 3 . 2 . 1 ) C.(\) was p l o t t e d a g a i n s t A f o r s e v e r a l f u n c t i o n s . The curve f o r ^ , £ = Q , i s shown i n f i g u r e 2 ( A ) . This ' t i t r a t i o n ' curve i s t y p i c a l f o r any c o n s t r a i n t as can e a s i l y be a s c e r t a i n e d by c o n s i d e r i n g cO) f o r )) > 0 and )\̂ < 0. When 0, A-f/ct^^C and i s the e i g e n f u n c t i o n having the lowest eigen- v a l u e £f/cJ)[) /\ C/ » C, being the lowest 53 Table I I . D e s c r i p t i o n and P r o p e r t i e s of ^ STO b a s i s : Xis X « 7C*p X35 Exponent: 3.298 2.068 0.433 0.639 3 . 9 9 2 1.090 C o n f i g u r a t i o n s : X / 5 Xis X/s ©1 > X\s X i / X i s > X,sX/s Xzs©,, L i n e a r c o e f f i c i e n t s : -0.108016, 0.004854, 0.540761, f o r the groundstate -0.004979, -0.030853, +0.007831, +0.092132 Prop- Pree e r t i e s : V a r i a t i o n a l S i n g l e C o n s t r a i n t s -£ AE >\ Q B ( 0 ) Q e ( 0 ) r 7.466022 0.22702 13.45413 -2.89513 -3.12497 7 . 3 4 0 2 9 0 0 . 1 2 5 7 3 2 0 . 8 0 6 8 3 3 3 5 x 1 0 2.40254 1 3 . 9 2 2 0 3 - 3 . 0 0 0 0 0 -3.81128 - 2 7.466004 0.000018 -0.99947345x10" 0.18609 13.45457 -2.89500 -3.00000 54 Table I I I . Term-wise comparison of convergence f o r p e r t u r b a t i o n expansion of A E, the energy s a c r i f i c e from the _L7e = Y c o n s t r a i n t 4?-,[59] f o r helium 2 f f o r l i t h i u m C o r r e c t A Order of c o n t r i b u t i o n E E E E E 5(1) (2) (3) (4) (5) (6) (7) E Sum of c o n t r i b u t i o n s t o . 7^ order Corr e c t 1.021239x10 3 . 7 4 9 9 5 0 x 1 0 " - 1 . 8 5 5 8 6 3 x 1 0 " -3 -1.26147x10 -7 - 9 . 4 3 1 9 x 1 0 " -10 -12 - 6 . 7 9 0 9 x 1 0 - 4 . 6 9 9 x l O ~ M - 3 . 1 0 9 x 1 0 -16 0.806833x10 -2 -1 1.43056x10 -2.56846x10 - 3 -1.91646x10 -1.68385x10 -1 . 4 7 3 0 5 x 1 0 " -1.28706x10 -1.12355x10 -3 -3 -3 - 3 Z E W = 1.881378x10-5 0.13300 1.881378x10 .-5 0.12573 5 5 eigenvalue of C . Thus icy^ (^)f^y ~~ c'' wnen>\« 0 , ^ k t ^ - \ \ l C and —\\\Cn where C> i s the hi g h e s t eigenvalue of & . Thus / y ~ w <m)lCtW> =cH . These r e l a t i o n s are compactly d e s c r i b e d C (+<*>) — C, ; (T^-o> <)=c A 7 , ( 3 . 2 . 3 ) and are i l l u s t r a t e d i n f i g u r e 2(A) a l s o . The f e a t u r e which appears to cause d i f f i c u l t y i s the extreme slope ( > 5 0 0 ) of Cfo) as i t crosses the A a x i s at Aopt Great s e n s i t i v i t y of t o A. might be a n t i c i p a t e d from the nature of the curve, and i s found; A oft must be computed t o 5 - 8 f i g u r e s t o ensure a s m a l l value (^ 1 0 " ^ ) for . The c o n s t r a i n e d energy i s not n e a r l y so s e n s i t i v e . The f i c t i t i o u s energy E-fleJ^ i s the lowest eigenvalue of the f i c t i t i o u s H a m i l t o n i a n /f^ = H ( 3 . 2 . 4 ) when ) = A opt where £~#*e i s the r e a l energy of the c o n s t r a i n e d func- t i o n . £--fictf)j) a n d / f y ^ p / ^ j are p l o t t e d i n f i g u r e 2(B) 56 F i g u r e 2. (A) Graph'of <£> versus A f o r the ground s t a t e of 2^ i n e l e c t r o n i c cusp c o n s t r a i n t A*fo c H Sxtr-etns Eigen- values of ((D) / n $7 last's (B) Graphs of f i c t i t i o u s energy,^>*t =(.H+)i<2)> , and t r u e energy, ET versus /\ , f o r ground s t a t e of ^ i n e l e c t r o n i c cusp c o n s t r a i n t . -7.0 -75* —8.0 1 1 1 1 £>v.r 1 1 1 1 -/ +3 ^ 7 57 f o r the c o n s t r a i n t of ^ . Because i s a m o n o t o n i c a l l y decreasing curve ( e a s i l y proved from r e s u l t s of the p e r t u r b a t i o n approach i n Appen- d i x C) i t has but one zero, opt , at which p o i n t E t r u e = E f i c t ' d e f i n i n S "the c o n s t r a i n e d ground s t a t e . As seen i n f i g u r e 2 ( B ) , ̂ t r u e = E f i c t w h e n A = 0 as w e l l . An a n a l y t i c a l approximation of the ' t i t r a t i o n ' curve c o u l d p r o v i d e what the p e r t u r b a t i o n approach f a i l e d t o — t h e i n i t i a l estimate of Aopt . S e v e r a l f u n c t i o n a l forms were i n v e s t i g a t e d , u t i l i z i n g as rea d - i l y o b t a i n a b l e parameters the extreme eigenvalues of the m a t r i x r e p r e s e n t a t i o n of , the f r e e v a r i a t i o n a l e x p e c t a t i o n value ( ^ r p and v a r i o u s d e r i v a t i v e s of Ch) at )i = 0. They are d e s c r i b e d here f o r the p o s s i b l e use and enjoyment of others doing c o n s t r a i n e d v a r i a t i o n s . The b a s i c parameters are d e f i n e d : v ' ( See (3.M) $ = C ( - c o ) j C = C ( o ) 0 =cYo) =£<Ers) 58 where CK(0) '"f-jyrj)i=o , and £ C k ) i s the * M order p e r t u r b a t i o n energy a s s o c i a t e d w i t h the f i c t - i t i o u s H a m i l t o n i a n ( 3 . 2 . 4 ) . Define R = (ot-fl/a. ; M = fe + P)/Z and the reduced q u a n t i t i e s M C - B _D m = R ; R = R ; b = "R ; d = _ R Then the g e n e r a l f u n c t i o n a l form has a zero at /\ (l+c-m)(l-c+m) ln(l+c-m)(l+m) //\\ R , O Q N y i o = 2b (l-c+m)(l-m) i / H i ] = ^ . ^ . / ; * I f the f i c t i t i o n a l energy i s expanded i n the p e r t u r - b a t i o n s e r i e s (see the p e r t u r b a t i o n approach i n App- endix C ) C"(0) =f9*EM) 3 '} = 0 5 9 I f the value of the second d e r i v a t i v e i s used ( C (b) i n s t e a d of <T'^ ), Now r e d e f i n e R, Then CM ~ M-Rarcta.n AfA-L) ( 3 . 2 . 8 ) has a zero at = 1 cos(c-m) si n c e ( B I ) = ( 3 . 2 . 9 ) ° b cos m v I f C f e j i s used i n s t e a d of C fc7 A . . . * ] / s i n 2 ( c - a ? | i g _ g (8^-(3.2.10) Cubic f a c t o r s as arguments of the tanh or a r c t a n func- t i o n s were t r i e d a l s o . The gene r a l form f o r these curves i s CM ^ M-Rtanh ^ [ f t - L ) 3 + a ] ; a,A,L,constants (C) = ( 3 . 2 . 1 1 ) 60 and C(fi M - R a r c t a n kW\-L? + *l ( D ) * ( 3 . 2 . 1 2 ) where R i s defined as i n (3.2.6) or (3.2.8) f o r tanh or arctan functions r e s p e c t i v e l y . Table IV gives the s t r a i n t . The parameters used f o r each evaluation are given i n the 'Parameter' column. An estimate should have at l e a s t the correct order of magnitude to be h e l p f u l . Remember that t h i s i l l u s t r a t i v e case i s p a t h o l o g i c a l ; l e s s severe constraint problems ( X " = Y f o r example) can be solved e a s i l y with these methods or any others. One other exotic a p p l i c a t i o n of the parametriz- a t i o n approach was t r i e d . Since the convergence of the perturbation series (see Table I I I ) depends on (/){ , the coordinate system f o r a t i t r a t i o n curve such as i n f i g u r e 2 was s h i f t e d , making use of the estimates just discussed. The transformation can be seen as follows: r e s u l t s obtained from f o r the con- where i s a new perturbation parameter. However 61 Table IV. A n a l y t i c a l p a r a m e t r i z a t i o n s of c o n s t r a i n t on ^ Equation Parameters employed ^ 0 a A l C(+oo ), C(0), C'(0), 0.05A- A 2 C(+cO ), C(0), C"(0) +0.0221 0 C(+oO ), C(0), C'(0), C"(0) 0.015 C C(+cx? ), C(0), C'(0) (a=0) 0.059 C C( + £>o ), C(0), C"(0) (a=0) 0.025 B l C( + <?o ), C(0), C'(0) 0.02 B 2 C( + Oo ), C(0), C"(0) 0.012 D C( + c?o ), C(0), C'(0), C"(0) 0.010 D G( + O 0 ) , C(0), C'(0) (a=0) 0 . 0 5 9 D C( + c?o ), C(0), C"(0) (a=0) 0 .017 aThe c o r r e c t v a l ue i s ^ o j > t = 0 . 0 0 8 0 6 8 3 3 3 5 . 62 the new p e r t u r b a t i o n s e r i e s d i d not always converge r a p i d l y enough. Having d i s c a r d e d more s o p h i s t i c a t e d approaches of s o l v i n g (3.1.1) f o r s i n g l e cusp c o n s t r a i n t s by p a r a m e t r i z a t i o n the method f i n a l l y s e t t l e d upon was a combination of r e g u l a f a l s i w i t h h a l f i n t e r v a l s . (Even r e g u l a f a l s i was not s u f f i c i e n t by i t s e l f because the steepness of the curve (3.2.1) at )iort sometimes l e d to i m p o s s i b l y slow convergences). Chong's f a s t 72 p e r t u r b a t i o n - i t e r a t i o n method' was employed f o r most J7 = y type c o n s t r a i n t s s i n c e these were u s u a l l y imposed e a s i l y . From t h i s e x p l o r a t o r y study came some p r e l i m i n a r y c o n c l u s i o n s noted now without f u r t h e r d e s c r i p t i o n of wave f u n c t i o n s or e x p e c t a t i o n v a l u e s . Except f o r Larsson's r e s u l t s the profound^ e f f e c t of £5*-type terms on Q s(0) was not w e l l documented at the time t h i s work was s t a r t e d . I t appears t h a t i n c l u s i o n of these terms improves Q s(0) o n l y f o r p a r t i a l l y or f u l l y o p t imized CI f u n c t i o n s . (See the e x c e l l e n t value of Q s(0) f o r ^ i n Table I I ) . £ € = Y c o n s t r a i n t s i n v a r i a b l y o v e r c o r r e c t e d Q e(0) o f t e n l e a v i n g a s i m i l a r e r r o r of opposite s i g n , w h i l e c o n s t r a i n t s y i e l d e d poorer s p i n d e n s i t i e s than the f r e e v a r i a t i o n a l v a l u e s . A p a r a d o x i c a l s i t u a t i o n e x i s t s . Optimized ( w i t h r e s p e c t 63 to energy) CI f u n c t i o n s having an a p p r e c i a b l e percen- tage of @a.-type terms g i v e good Q s(0) v a l u e s , but do not c o n s t r a i n e a s i l y , w h i l e l e s s a c c u r a t e , but more f l e x i b l e wave f u n c t i o n s do not g i v e a r e l i a b l e Q s(0) i n any case. The i d e a of c o n s t r a i n e d v a r i a t i o n does not appear to work. Optimum CI f u n c t i o n s f o r a many- e l e c t r o n case, however, are u n p r a c t i c a l to c o n s t r u c t and thus do not provide a good route t o accurate Q s(0) c a l c u l a t i o n s . So a f u r t h e r attempt was made to show Q s(0) improvement w i t h cusp c o n s t r a i n t s u s i n g method- i c a l l y c o n s t r u c t e d CI f u n c t i o n s . These f u n c t i o n s must be l o n g enough to absorb the c o n s t r a i n t , p a r t i a l l y o p t i m i z e d w i t h r e s p e c t to important exponents, and they must c o n t a i n -type terms. 3 . 3 Systematic study CI f u n c t i o n s now d i s c u s s e d provide a reasonable d e s c r i p t i o n of both core p o l a r i z a t i o n and c o r r e l a t i o n e f f e c t s . I t should not be too d i f f i c u l t t o o b t a i n s i m i l a r f u n c t i o n s capable of g i v i n g good s p i n d e n s i t i e s and other p r o p e r t i e s f o r at l e a s t the f i r s t row e l e - ments. The main purpose here, a g a i n , i s t o t r y t o f i n d a f a v o u r a b l e c o r r e l a t i o n between cusp c o n s t r a i n t s and s p i n d e n s i t i e s . The f o r m a t i o n of these CI f u n c t i o n s proceeded as f o l l o w s : 64- p A ,.:5 f u n c t i o n f o r groundstate l i t h i u m was rep- resented as (3.3.1) the a n t i s y m m e t r i z a t i o n o p e r a t o r , and the s , l i k e a n a l y t i c a l Hartree-Fock o r b i t a l s . These o r b i t a l s were l i n e a r l y expanded i n terms of STO's, <p„ = a, Xa 2.S 35 The n o t a t i o n i s s t r a i g h t forward; STO's w i t h the same o r b i t a l angular momentum quantum number ( £. ) and designated w i t h the same prime (') have i d e n t i c a l o r b i t a l exponents. $ was expanded as a CI f u n c t i o n ( w i t h the accompanying products of <%i Vand JJ-/S taken as independent l i n e a r c o e f f i c i e n t s ) i n terms of STO c o n f i g u r a t i o n s . Thus the key wave f u n c t i o n has ei g h t terms 65 ^ c A i + *z(X, sXu'Xzf) +Q3fciXs'Za) +4*(Xis X£ X&] + as fr,s Xts'&s) fafsXzi X3£) Because (Pn i s not equal t o core p o l a r i z a t i o n i s b u i l t i n t o the wave f u n c t i o n . There i s a minimum of n o n - l i n e a r parameters t o vary and absence of c o n f i g - u r a t i o n s m i x i n g the &c'savoids i n t e r f e r e n c e between terms. The three o r b i t a l exponents, 5s , Ss , 5S' were optimized f o r the f i r s t f o u r terms, ( ). With these v a l u e s as i n i t i a l estimates a s i n g l e optim- i z a t i o n c y c l e f o r ^ 8 p r o v i d e d the f i n a l v a l u e s of these n o n - l i n e a r parameters, p-type c o r r e l a t i o n was next i n c l u d e d . Three t r i a l a d d i t i o n s of two terms each were compared: 66 Here ( $si means the terms i n ^ 8 w i t h o n l y the l i n e a r c o e f f i c i e n t s t o be r e c a l c u l a t e d . The a d d i t i o n a l ex- ponents, Ss , S/? , were p a r t i a l l y o p t i m i z e d w i t h one c y c l e i n each 10-term f u n c t i o n . The s i m p l e s t , <J(0 , had the lowest energy and the other two were d i s c a r d e d . Next the e i g h t p o s s i b l e t r i p l e t core s p i n terms were added one at a time, i n no s p e c i a l order, changing no exponents, to <&/0 > y i e l d i n g a sequence of f u n c t i o n s from 10 to 18 terms i n l e n g t h , w i t h 0 t o 8 Ox -type terms ( i 8 ). Because the t r i p l e t core terms were added somewhat a r b i t r a r i l y , one other f u n c t i o n , ^ / f a , was computed, i n c o r p o r a t i n g the f o u r © a -type terms producing the l a r g e s t i n d i v - i d u a l f r a c t i o n a l energy decreases i n the above s e r i e s . Both s i n g l e ( JT = Y , AZ* = Y ) and double cusp c o n s t r a i n t s were a p p l i e d t o each f u n c t i o n . D e s c r i p t i o n s of , (3?n*. and <§IS are found i n Table V. Table VI c o n t a i n s t h e i r f r e e v a r i a t i o n a l and c o n s t r a i n e d p r o p e r t i e s . S i m i l a r t a b l e s f o r the complete s e r i e s are i n Appendix E. F i r s t examine the f r e e v a r i a t i o n a l p r o p e r t i e s (Table V I ) . Note t h a t ^ - t y p e terms do not con- t r i b u t e s i g n i f i c a n t l y t o the energy, the t o t a l increment, * Double d i a g o n a l c o n s t r a i n t s are e a s i l y imposed by the method of Chong and Benston [73j d e s c r i b e d i n Appendix C, making use of the r e s u l t s f o r s i n g l e c o n s t r a i n t s . 67 Table V. Descriptions of $ l 0 , $ I H. , STO Basis O r b i t a l s O r b i t a l Exponents: Configurations 14- Free V a r i a t i o n a l C o e f f i c i e n t s : 10 1? 3.168, 3.168, 2.840, 2.840, 0.765, 0.765, 4.974, 4.974 $ | t > : l s l s ^ s ' ^ , , ls2s'2s" e , , 2313*23"©, , 2s2s ,2s"©. , l s l s ' 3 s " e , , l s 2 s ' 3 s ' ^ , 2sls ' 3 s n e v , 2s2s ' 3 s " e , , 2s"(2p) 2©, , 2s"(3p) 2©. {<$,„}*, lsls^s"©* , ls2s'2s"6>2. , 2sls'2s"c9 2. , 2s2s'2s , l e i [ J^}*, l s l s ' 3 s " e t , ls2s'3s"© 2. , 2sls ,3s"© ?. , 2s2s'3s"© 2. +0.196099; +0.024179; -0.165463; -0.014436. +0.195868; 0.022274; -0.164641 -0.014436 -0.056793 +0.196595 +0.021249 -0.156697 -0.014440 -0.196107 -0.143961 +0.334607 +0.161653 +0.018404 +0.337217; +0.161696; +0.019763; -0.583381; +0.012474. +0.349087 +0.160845 +0.020425 -2 .035945 +0.036096 0.153566; - 0 . 2 3 0 8 3 1 +0.229143 -0 .010172 -0.230859; +0.226320; -0 .010174; +0.052004; -0.239527 +0.215740 -0.010176 0.181852; 1.605262; -0.026107. See the text following ( 3 . 3.4) f o r the meaning of the notation [<f>$ . Table VI. Free v a r i a t i o n a l and constrained properties of , J ^ - , $?/t (defined i n Table V) Function Constraint -Energy Q S ( 0 ) Q e ( 0 ) AE > None 7 . 4 6 7 3 8 9 0 . 2 6 7 7 1 3 . 7 5 2 2 - 2 . 9 7 3 2 - 3 . 2 7 9 5 — 0 . 0 7.467254 0 . 2 7 5 3 1 3 . 9 1 9 1 - 3 . 0 0 0 0 - 3 . 2 9 7 3 0 . 0 0 0 1 3 5 + 0 . 5 7 9 7 7 2 x l 0 " 4 7.465334 0 . 1 7 1 5 1 3 . 5 5 1 9 - 2 . 9 4 4 7 - 3 . 0 0 0 0 0 . 0 0 2 0 5 5 - 0 . 5 2 1 6 6 7 x l 0 ~ 2 7 . 4 6 4 7 3 6 0.1686 1 3 . 8 9 0 1 - 3 . 0 0 0 0 - 3 . 0 0 0 0 0 . 0 0 2 6 5 3 > e = + 0 . 1 2 5 2 7 7 x l 0 " 3 > s =-0.585347xl0" 2 None 7.467429 0 . 2 1 3 6 1 3 . 7 5 0 0 -2.9728 - 2 . 9 7 9 0 0 . 0 7 . 4 6 7 2 9 1 0 . 2 1 5 3 1 3 . 9 1 9 2 - 3 . 0 0 0 0 - 2 . 9 8 7 0 0.000138 0 . 5 8 2 3 2 9 x 1 0 " ^ 7.467429 0.2150 1 3 . 7 5 0 0 -2.9728 - 3 . 0 0 0 0 -10" 8 0.891716xl0" 6 7.467291 0.2163 13.9192 - 3 . 0 0 0 0 - 3 . 0 0 0 0 0.000138 - > e =0.582305xl0" 4 > s = 0 . 5 3 4 7 5 4 x l 0 " 6 None 7.46749^6) 0.2287 1 3 . 7 5 0 1 -2.9729 -3.0542 0 . 0 7.467360 0.2312 1 3 . 9 1 7 8 - 3 . 0 0 0 0 -3.0590 0.000135 +0.572685xl0" 4 7.467496 0.2246 1 3 . 7 5 0 1 -2.9729 - 3 . 0 0 0 0 ~ 1 0 " 7 - 0 . 2 39282x10 ~ 5 7.467360 0.2267 1 3 . 9 1 7 8 - 3 . 0 0 0 0 - 3 . 0 0 0 0 0 . 0 0 0 1 3 5 > e = + 0 . 5 7 2 7 4 5 x l 0 " 4 )f= - 0.252666xl0" 5 Larsson 1 s 7.478025 0.2313 13.8341 100 term correlated f u n c t i o n (see entry 18 Table i) 69 ^ $ , 8 - £ $ i o > being onl y 0.000106 h a r t r e e s . The e f f e c t on Q e(0) i s n e g l i g i b l e a l s o ; and $?l8 have e s s e n t i a l l y the same e l e c t r o n d e n s i t y . Q s ( 0 ) , however, improves tremendously. The h i g h s t a b i l i t y of the f i n a l v a l u e ( t h a t of ^? ( g ) cannot be seen i n Table VI but i s evident from the complete t a b l e i n Appendix E. From these complete r e s u l t s one can see c e r t a i n t r i p l e t core s p i n terms c o n t r i b u t e more than others to Q s ( 0 ) . Only about h a l f of the p o s s i b l e terms are necessary but because the important ones are d i f f i c u l t to p i c k out i n c l u s i o n of a l l of them seems a d v i s a b l e . Since most computer time i s spent on exponent o p t i m i z a t i o n the t r i p l e t core s p i n terms can be added p r a c t i c a l l y as a bonus. C o n s t r a i n i n g seems t o be a waste of time though. I n almost every case the wave f u n c t i o n d e t e r i o r a t e s . c o n s t r a i n t s apply c o r r e c t i o n s t o Q e(0) 59 as Yue and Chong found f o r He but the e r r o r 'improves' from «=*-0.6% t o «*+0.5% which r e a l l y i s not s u f f i c i e n t j u s t i f i c a t i o n f o r cusp c o n s t r a i n t s . The JZ e c o n s t r a i n t i s c o n s i s t e n t l y the best of the three c o n s t r a i n t s im- posed. F o r c i n g i s of l i t t l e use. When s e v e r a l Gx -type terms are present the c o n s t r a i n t s are easy t o apply and p r o p e r t i e s are not changed much. When th e r e are on l y a few, or none, the c o n s t r a i n t *See i . i f a . i n Appendix E f o r the demonstration. 70 becomes more severe i n d i c a t i n g t h a t f i n e adjustments i n the wave f u n c t i o n , a f f e c t i n g s p i n p r o p e r t i e s depend on t r i p l e t core s p i n terms. Wo s i g n i f i c a n c e i s attached to the double c o n s t r a i n t r e s u l t s . Because imposing the e l e c t r o n cusp c o n d i t i o n was more severe than f o r the s p i n cusp, f o r c i n g JZ = IZ = Y i n v a r i a b l y approximated the s i n g l e c o n s t r a i n t -E7 = . A f i n a l p o i n t i s t h a t the f r e e v a r i a t i o n a l Q (0) i s so good f o r t h a t c o n s t r a i n i n g would not be necessary even i f i t worked. No t r e n d i n improvements f o r Q s(0) r e s u l t i n g from f o r c i n g can be seen from t h i s study. I t i s p o s s i b l e t h a t f o r approximate wave f u n c t i o n s t h e r e i s an e m p i r i c a l ' e f f e c t i v e ' Y r a t h e r than the t h e o r e t i c a l iY = -Z. A f u r t h e r t h e o r e t i c a l c o n d i t i o n i s JZ = JZ w i t h Y u n s p e c i f i e d but one can con- f i d e n t l y p r e d i c t t h a t the minimum energy f o r such a c o n s t r a i n t w i l l occur at JZ e =-C ~-IZ^r^e va.v-io.-tUna.l f o r the s e r i e s — * <̂ ,g . To l o c a t e an e f f e c t i v e Y' the s i n g l e c o n s t r a i n t s JZC = Y' and JT = X were imposed on s e v e r a l f u n c t i o n s f o r d i f f e r e n t values of Y' . The a c t u a l dependence of Q e ( 0 ) , Q s(0) on the cusp c o n s t r a i n t s was more evident d u r i n g these c a l c u l a t i o n s than i n the preceding study. 7 1 Q (0) i s shown as a f u n c t i o n of the c o n s t r a i n t r S = i n f i g u r e 3(A) f o r , $lH. , . I t has a l i n e a r dependence on -17 One can see f o r these f u n c t i o n s ( a l s o f o r a l l others t e s t e d ) t h a t i f the f r e e v a r i a t i o n a l i s g r e a t e r than -Z = - 3 a s p i n cusp c o n s t r a i n t w i l l improve Q s ( 0 ) . ( I d e a l l y the l i n e s should pass through ( 0 . 2 3 1 , - 3.000) ). See the complete set of p r o p e r t i e s i n Appendix E to v e r i f y t h i s . I t i s evident a l s o t h a t t h e r e i s no ' e f f e c t i v e ' ^ v a l u e . The s c a t t e r of p o i n t s where the l i n e s c r o s s the experimental v a l u e ( 0 . 2 3 1 3 ) p r e - cludes t h i s . A s i m i l a r graph of Q e(0) versus X"6 = cV i n f i g u r e 3(B) a l s o shows a l i n e a r depen- dence. That the © A -type terms do not i n f l u e n c e Q e(0) i s obvious; the l i n e s f o r a l l f u n c t i o n s , , > $18 » a r e superimposed. Why _Ce = -Z over- c o r r e c t s Q e(0) i s c l e a r f o r these cases. An e f f e c t i v e y f o r Q e(0) cannot be s a i d t o e x i s t even though a l l l i n e s pass Q e(0) = 1 3 . 8 3 [ 3 8 ] at Y'= - 2 . 9 8 5 because the f u n c t i o n s <£lo through are a l l too s i m i l a r i n e l e c t r o n d e n s i t y . A comparison of f i g u r e s 3(A) and 3(B) w i l l a f f i r m t h a t c a l c u l a t i o n i s more d i f f i c u l t f o r Q s(0) than Q e(0) and w i l l r e f l e c t the s c a t t e r of Q s(0) values i n Table I . F i g u r e 4, (A) 7 2 F i g u r e 3. (A) O s(0) as a f u n c t i o n of the c o n s t r a i n t ~ x Z.8 -A.9 3.0 3.1 J.2. 3.3 Free v a r i a t i o n a l v a l u e s are marked e x p l i c i t l y f o r (B) Q e(0) as a f u n c t i o n of the c o n s t r a i n t £ * = f o r £,o , ^ , ^ 8 . Qfo) /J.o 1 1 1 i-ertica.1 /ue 1 1 J.8 l.f 3.o . J./ 3.% The common f r e e v a r i a t i o n a l v a l u e i s marked e x p l i c i t l y . 73 F i g u r e 4. (A) Q s(0) as a f u n c t i o n of the c o n s t r a i n t r= r' f o r J , 0 , ^ , $ | f . a <?.av — a F r e e v a r i a t i o n a l v a l u e s are marked f o r ^ , (8 (B) Q e(0) as a f u n c t i o n of the' c o n s t r a i n t 18 The £xactt38-j 10 t-ttica.( tue. 2.e 2.? 3.o 3./ 74 and ( B ) , shows Q s(0) as a f u n c t i o n of £ & = Y  / and Q e(0) as a f u n c t i o n of X " = Y ' . P r e v i o u s con- c l u s i o n s are borne out: l ) The s p i n cusp c o n s t r a i n t a d j u s t s o n l y minute d e t a i l s of a wave f u n c t i o n without a f f e c t i n g Q e(0) at a l l . 2) The e l e c t r o n cusp c o n s t r a i n t i s not l i k e l y t o improve s p i n d e n s i t i e s . F i n a l l y i t must be concluded t h a t these d i a g o n a l cusp c o n s t r a i n t s are not u s e f u l f o r improving Q s ( 0 ) . I f n o n - l i n e a r p a r a - meters are o p t i m i z e d and O*. -type s p i n terms are i n c l u d e d the wave f u n c t i o n should have a good s p i n d e n s i t y at the nucleus without any c o n s t r a i n t . 3.4 O f f - d i a g o n a l cusp c o n s t r a i n t s w i t h w e i g h t i n g f u n c t i o n s Since attempts thus f a r t o apply the p h ilosophy of c o n s t r a i n e d v a r i a t i o n towards c a l c u l a t i n g b e t t e r s p i n d e n s i t i e s have f a i l e d , the problem must be r e - examined. I f $ s , M s a t i s f i e s n u c l e a r cusp c o n d i t i o n s the i n t e g r a t i o n i n (2.4.5) can be seen as an a r t i f a c t , merely a l l o w i n g an e v a l u a t i o n of the e x p r e s s i o n (3.4.1) (Compare w i t h (2.3.1) ). Replacement of (J> S I M i n (2.4.5) by an a r b i t r a r y f u n c t i o n , T , seems reason- able i n t h i s l i g h t . To apply t h i s new concept, 75 r e s t r i c t i o n s on ~f must be examined. I f 1// i s a s p a t i a l e i g e n f u n c t i o n s a t i s f y i n g (2.3.8) f o r the case of n u c l e a r cusp, the c o n d i t i o n J-F*(^Co - ^ > J ^ d f > = O , * - * , s (3.4.2) i s v a l i d f o r any well-behaved f u n c t i o n ~P . Spin must be i n c l u d e d i n ~F when the s p i n c o n t a i n i n g f u n c t i o n $S,M i s considered because i n t e g r a t i o n over a s i n g l e s p i n v a r i a b l e i s undefined. To avoid t r i v i a l c o n d i t i o n s / should be an e i g e n f u n c t i o n of and w i t h the same s p i n f u n c t i o n s Oc as @SjM Then §SlH>-o (3.4.3) i s v a l i d . (Compare w i t h (2.4.5) ). This suggests the o f f - d i a g o n a l n u c l e a r cusp c o n s t r a i n t s , on an approximate ^ • T n e f u n c t i o n , . can be thought of as a weig h t i n g f u n c t i o n . I n p a r t i c u l a r ~f may be antisymmetric w i t h r e s p e c t to interchange of e l e c t r o n c o o r d i n a t e s . I f the dia g o n a l 76 cusp c o n s t r a i n t (3-2.1) i s recovered. The t r a d i t i o n a l approach t o c o n s t r a i n e d v a r i a t i o n , d e s c r i b e d i n the t e x t at equation (3.1.1), f o r c e s ^ ^ i ^ = 0. I n the working b a s i s s e t , (^K! , ( u s u a l l y the b a s i s of f r e e v a r i a t i o n a l e i g e n v e c t o r s ) t h i s r e l a - t i o n i s expressed i n m a t r i x n o t a t i o n e ^ C i O . =° • (3.4.5) <£2 i s the c o n s t r a i n e d (column) eige n v e c t o r sought, f = 2 , c and Cc = ( C k £ j £ ) , C^i^gjQIe^. (3.4.6) 69 In c o n t r a s t , the method of Weber and Handy y minimizes energy w h i l e c o n s t r a i n i n g (3.4.7) where i s a column v e c t o r c o n t a i n i n g necessary i n f o r m a t i o n about the i*h c o n s t r a i n t c o n d i t i o n . I n the o f f - d i a g o n a l case, (3-1.6) can be expanded i n a b a s i s t§<\ a s 7 7 jfCiCL = O (3.4 .8) f o r <p = ^ ^ ^ . Then ( 3 . 4 . 9 ) Note the important d i s t i n c t i o n between ( 3 . 4 . 9 ) w i t h ( 3.4.6), and ( 3 . 1 . 2 ) . The procedure of Weber and Handy does not need an H e r m i t i a n & o p e r a t o r . I f i s f i x e d , the c o n s t r a i n t s , ^-Y«-7 , are imposed i n a one-step m a t r i x d i a g o n a l i z a t i o n procedure. The d i a g o n a l c o n s t r a i n t case, ( 3 . 4 . 5 ) , can be handled, but an i t e r a t i o n procedure on i s necessary: Q - K = O , (3.4.10) S e v e r a l types of w e i g h t i n g f u n c t i o n s were t e s t e d t o determine t h e i r e f f e c t on Q s(0) and Q e ( 0 ) . The f i r s t t r i e d was ~P = i ^ - , f the set of f r e e v a r i a t i o n a l e i g e n v e c t o r s . Since the are energy ordered 78 the c o n s t r a i n t ( 3 . 4 . 4 ) amounts t o we i g h t i n g by energy. Computations are simple f o r t h i s choice because the m a t r i x elements ( 3 . 4 . 9 ) , ($k i f a - f ^ l ^ i ) = < ^ / / ^ - < # / / § ) had been evaluated f o r d i a g o n a l c o n s t r a i n t s . The r e s u l t s of w e i g h t i n g w i t h s e l e c t e d are presented i n Table V I I . Both e l e c t r o n and s p i n cusp s i n g l e c o n s t r a i n t s and the double c o n s t r a i n t were c a r r i e d out ( f o r Y = - Z ) . The values of J ^ 6 ' are i n c l u d e d so the consequences of ~f = , the groundstate e i g e n v e c t o r , can be seen. I n a l l thr e e cases, ( e ) , ( s ) , ( e , s ) , ~F = c l o s e l y dup- l i c a t e d the r e s u l t s of the d i a g o n a l c o n s t r a i n t s i n sec- t i o n . 3 . 3 , Table V I . The a p p r o p r i a t e _L equal -Z * * t o w i t h i n 3-4 decimal p l a c e s . Constrained energies are the same t o 10~ 6 h a r t r e e and Q e ( 0 ) , Q s ( 0 ) values a l s o o n l y d i f f e r i n the 3 r d , 4**1 decimal p l a c e s . This i s t o be expected s i n c e the overwhelming c o n t r i b u t i o n to the c o n s t r a i n e d i s ̂  . (See equation (3 .1 .8) ). The s i g n i f i c a n c e i s t h a t the o f f - d i a g o n a l , one-step method of Weber and Handy can be used t o R e a l i z e t h a t the groundstate of i_s ̂  . ^ V y . changes to some JF) when c o n s t r a i n e d . * * This shows t h a t the i n t e g r a l cusp c o n d i t i o n s are on l y necessary, f o r i f the c o r r e c t cusp existed,£"s Y always. Table V I I . Energy weighted o f f - d i a g o n a l cusp c o n s t r a i n t s on Co n s t r a i n t Weighting F u n c t i o n -E A E x l O 6 Q s ( 0 ) Q 6 ( 0 ) r I 5 None 7 . 4 - 6 7 4 2 9 0 . 2 1 3 6 13.7500 - 2 . 9 7 . 2 8 - 2 . 9 7 9 0 a=e f, 7 . 4 6 7 2 9 1 1 3 8 0 . 2 1 5 2 13.9211 - 3 . 0 0 0 0 - 2 . 9 8 6 0 II **. 7 . 4 6 7 4 2 9 0 0 . 2 1 3 7 13.7502 - 2 . 9 7 2 8 - 2 . 9 8 0 8 II $1 7 . 4 6 7 3 9 0 3 9 0 . 1 6 6 8 1 3 . 7 5 7 5 - 2 . 9 7 4 2 - 2 . 1 6 9 8 II 7 . 4 6 7 2 5 1 1 7 8 0 . 2 1 0 0 1 3 . 5 5 8 0 - 2 . 9 4 2 6 - 2 . 9 4 4 5 II f» 7 . 4 6 6 2 1 4 1215 0 . 1 8 3 9 1 4 . 2 0 1 0 - 3 . 0 4 6 8 - 2 . 4 2 9 9 a=s f, 7 . 4 6 7 4 2 9 0 0.2150 13.7500 - 2 . 9 7 2 8 - 2 . 9 9 9 9 ti & 7 . 4 6 7 1 9 9 230 0 . 0 8 1 0 1 3 - 7 2 8 7 - 2 . 9 6 9 8 + 1 . 9 1 4 6 II f, 7 . 4 6 7 2 4 8 1 8 1 0 . 1 3 8 7 1 3 . 6 7 1 9 - 2 . 9 6 1 1 - 1 . 4 7 6 1 II 7 . 4 6 7 4 2 9 0 0 . 2 1 5 4 13.7500 - 2 . 9 7 2 8 - 3 . 0 0 5 3 it 7 . 4 6 7 4 2 7 2 0 . 2 0 0 1 1 3 . 7 5 0 9 - 2 . 9 7 2 9 - 2 . 7 6 8 9 a=e,s 7 . 4 6 7 2 9 1 1 3 8 0 . 2 1 6 2 13.9211 - 3 . 0 0 0 0 - 2 . 9 9 9 9 u 7 . 4 6 7 1 6 7 2 6 2 0 . 0 6 6 0 13.7301 - 2 . 9 7 0 4 + 3 . 6 8 9 2 n 7 . 4 6 7 1 6 2 2 6 7 0 . 1 8 6 0 1 3 . 5 6 7 6 - 2 . 9 4 4 5 - 2 . 6 2 9 5 li 7 . 4 6 7 2 5 0 179 0 . 2 1 5 3 1 3 . 5 5 7 9 - 2 . 9 4 2 6 - 3 . 0 1 8 5 u 7 . 4 6 6 2 0 2 1227 0 . 2 1 4 4 1 4 . 2 0 5 5 - 3 . 0 4 7 5 - 2 . 9 5 7 2 80 c l o s e l y d u p l i c a t e the r e s u l t s of any d i a g o n a l c o n s t r a i n t p r o v i d i n g (3.1.8) h o l d s . Computational problems des- c r i b e d i n 3 .2 are bypassed and no i t e r a t i o n i s necessary. The w e i g h t i n g method w i t h •f - s , thus seems i d e a l f o r p r e - t e s t i n g the e f f e c t s of v a r i o u s c o n s t r a i n t s . As f a r as w e i g h t i n g w i t h other energy e i g e n f u n c t i o n s i s concerned, ~f = , K > 1, u n p r e d i c t a b l e , random, poor r e s u l t s occur. C o n f i g u r a t i o n w eighting was next attempted. Con- f i g u r a t i o n s , f ̂ - j , have an energy a s s o c i a t e d w i t h them but they are more s p a t i a l i n c h a r a c t e r than the energy e i g e n f u n c t i o n s , ( ^ K = 7^ ). P u t t i n g •f = ffl i s a k i n d of s p a t i a l w e i g h t i n g . Again, r e s u l t s are random and meaningless and w i l l not even be presented. A p p l i c a t i o n of Weber and Handy's method t o the d i a g o n a l cusp c o n s t r a i n t case w i t h (3.4.10) was i n v e s - t i g a t e d . I f convergence was r a p i d , computational problems of the s o r t mentioned i n 3 .2 would be avoided and the exact d i a g o n a l c o n s t r a i n t <&<:/¥> = 0 would be imposed. A f t e r convergence was achieved s l i g h t d i f f e r e n c e s from the pure d i a g o n a l approach (3.1.1) were found even though from both methods. The i n i t i a l guess <Z?0 = s, , i n (3.4.10) d i d not converge when an H e r m i t i a n G was used but o s c i l l a t e d back t o Q.XK - s~, • Groundstate energy and the over- l a p : 0^^X. were employed as convergence c r i t e r a . 81 A comparison of the two approaches to d i a g o n a l c o n s t r a i n t s f o r i s presented i n Table V I I I f o r academic i n t e r e s t . The reason f o r the anomalous r e s u l t s was not a program e r r o r but i m p o s i t i o n of a d i f f e r e n t c o n s t r a i n t . For the pure, o r i g i n a l d i a g o n a l technique the c o n s t r a i n t m a t r i x c = P + P was unambiguously employed. The H e r m i t i a n c o n s t r a i n t m a t r i x t h a t developed d u r i n g i t e r a t i o n had the form where $r i s the c o n s t r a i n e d e i g e n v e c t o r and need not equal <£L , the c o n s t r a i n e d e i g e n v e c t o r f o r the f i r s t case. Both these e i g e n v e c t o r s g i v e the c o n d i t i o n A f i n a l e x p l o r a t i o n of the w e i g h t i n g concept em- ployed , where the Xhs "*S are STO's having o r b i t a l exponents S/s = 2.7 (° , Jjj = 1-5 • C i s a v a r i a b l e parameter. The v a l u e s of S/S , Szs f o r C = 1 were chosen w i t h S l a t e r ' s r u l e s . Now the wave; f u n c t i o n t o be c o n s t r a i n e d can be weighted i n s e l e c t e d r e g i o n s . Of s p e c i a l i n t e r e s t i s the e x t r a p o l a t i o n XP - > o. 82 Table V I I I . Comparison of the d i a g o n a l and o f f - d i a g o n a l i t e r a t i v e methods f o r di a g o n a l c o n s t r a i n t s . Example: ^ ( ^ , JZ e = V c o n s t r a i n t Wave f u n c t i o n d i a g o n a l method i t e r a t i v e method a t t r i b u t e (see equation ( 3.4.5)) (see equation ( 3 .4.10)) Energy - 7.4672914 - 7-4672913 Q s(0) 0 .215387 .0.215193 Q e (0) 13.919166 13.920945 r - 3 . 0 0 0 0 0 0 - 3.000000 - 2 .986952 - 2 .985937 * a i + 0 .9999931 + 0 .9999933 a 2 - 0.1996x10~3 - 0.1757x10" •3 a 3 + 0.2666x10"*3 + 0.2446x10" •3 H + 0.2031xl0" 4 + 0.2099x10" -4 * a i = c o e f f i c i e n t i n c o n s t r a i n e d wave f u n c t i o n of the itt f r e e v a r i a t i o n a l e i g e n f u n c t i o n . S3 becomes a Di r a c o - l i k e f u n c t i o n , w eighting the n u c l e a r r e g i o n s t o an extreme degree. Q s ( 0 ) i s p l o t t e d a g a i n s t f and /ft i n f i g u r e 5 ( A ) . F i g u r e 5(B) shows the corresponding r e l a t i o n s of Q e ( 0 ) _ f o r e l e c t r o n i c cusp c o n s t r a i n t s on . Negative r e s u l t s are o b t a i n e d . A sampling of the corresponding r e s u l t s f o r s p i n cusp c o n s t r a i n t s i s shown i n Table IX. An i n t e r e s t i n g maximum e x i s t s around f = 1/4 but i s c e r t a i n l y c o i n c i d e n t a l . S i m i l a r s t u d i e s were c a r r i e d out on $>/0 and (j),7 of the same s e r i e s of f u n c t i o n s w i t h the same d i s c o u r a g i n g r e s u l t s . Energy s a c r i f i c e s , except f o r the o b v i o u s l y d i s t o r t e d cases, were n e g l i g i b l e . I n v e s t i g a t i o n of m u l t i p l e w e i g h t i n g c o n s t r a i n t s i s important because i f (3.4.4-) i s t r u e f o r a l l members of a complete s e t , an approximate wave f u n c t i o n must have the c o r r e c t cusp. Thus i f f o r c e d t o be zero f o r many "Pc f u n c t i o n s perhaps / ^ / w i l l more c l o s e l y s a t i s f y the n u c l e a r cusp con- d i t i o n . The method of Weber and Handy s i m p l i f i e s con- s t r a i n t c a l c u l a t i o n s . D i f f e r e n t weights were a p p l i e d s i m u l t a n e o u s l y t o $~/0 , and $ / 7 . Combinations of /W) , and f, , the f r e e v a r i a t i o n a l e i g e n f u n c t i o n , were used i n both e l e c t r o n i c and s p i n cusp c o n s t r a i n t s . R e p r e s e n t a t i v e r e s u l t s f o r $/tf- are presented i n Table X. Use of the di a g o n a l i t e r a t i o n technique to f o r c e Figure 5. (A) Q s(0) p l o t t e d against C and l/f i n o.zf Qs(o) O.ZZ o.zo weighting e l e c t r o n i c cusp constraint on 1 I 1 1 1 1 w - '[ °" z& in iff — r - * Q 0 ""' r 1 1 1 1 1 1 9- 8 0 / 2 . /6 20 2V 0.69- A/Z o.zo \y aze 03& 0.9? O-SC? 85 Table IX. Q s ( 0 ) , Q e(0) f o r the o f f - d i a g o n a l weights i n s p i n cusp c o n s t r a i n t s of <j>/tf e Q s(0) Q 6(0) 25 -0.1307 13.6474 10 -0.2005 13.7062 8 -0.1243• 13.7244 5 +0.0381 13.7443 2 +0.1817 13.7499 1 +0.2144 13.7500 1/2 +0.2260 13.7503 1/4 +0.2233 13.7505 1/8 +0.2148 13.7501 Free v a r i a t i o n a l r e s u l t s 0.2135 13.7500 86 Table X. M u l t i p l e weighting c o n s t r a i n t s on . Number of C o n s t r a i n t s AE(energy Q S(0) Q e(0) si m u l t a n - s a c r i f i c e eous due t o c o n s t r a i n t s c o n s t r a i n t s ) 3 lVS(k) >WstL)>Wsfr) 0.0135 0.0983 13.6157 3 §,*> WS(&)> WS&) 0.0140 0.0662 13.2261 2 IYW 0.0000 0.2181 13 .7517 2 gf, WSU) 0.0001 0 . 2 2 2 7 13 .7513 2 WVIL) >Wsf3.) 0.0001 0.2094 13.7519 3 Wita) ,14/%)^%) 0.0018 0.1968 13.9758 3 0 . 0 0 0 9 0.2052 13.8953 2 Jv% tt/fy) 0 . 0 0 0 9 0.2038 13.8998 2 0 . 0 0 0 9 0.2083 13.8952 2 J?*, 0.0001 0.2162 13.9211 4 /^k),^k) 0 . 0 0 0 9 0 .2205 13 . 8 9 9 0 6 f,e9f,,,^/ek)7^SHk), 0 .0177 0.0407 13 .7131 a = e,s gi v e s the type of c o n s t r a i n t a p p l i e d . •87 <^/(si iy> = ° a i ° * s w i t h (Mhieiy/y =... = 0 d i d not produce s i g n i f i c a n t l y d i f f e r e n t r e s u l t s from the case <?,/£/ I f > - <' WM/G/"V>'> = ... = 0 as expected. No d e f i n i t e t rends c o u l d he determined. The extreme number of p o s s i b l e c o n s t r a i n t s a l s o confuses the problem. I n c o n c l u s i o n , o f f - d i a g o n a l w e i g h t i n g c o n s t r a i n t s have o f f e r e d no sure method f o r improving s p i n and e l e c t r o n d e n s i t i e s . A more r e a l i s t i c form f o r , weighting o n l y one e l e c t r o n at a time but t h i s was not t r i e d . The g e n e r a l technique c o u l d perhaps be improved concep- t u a l l y and might be u s e f u l f o r other p r o p e r t i e s . •88 CHAPTER IV SUFFICIENT CONDITIONS FOR CORRECT CUSP 4 .1 Theory The f a i l u r e of necessary cusp c o n s t r a i n t s to provide accurate d e s c r i p t i o n s . o f wave f u n c t i o n s at nu c l e a r coalescence i s by now obvious. That these c o n s t r a i n t s need not f o r c e the c o r r e c t behaviour at the cusp i s re-emphasized. The weighting c o n s t r a i n t r e s u l t s i n Chapter I I I demonstrate t h i s f a c t . C o n s t r a i n i n g , <flii/-tel1f/)= 0 does not guarantee ̂ /^-U/W) = 0 as would be found i f ^ had the proper cusp. Are there p r a c t i c a l , s u f f i c i e n t r e s t r i c t i o n s on a f u n c t i o n t h a t w i l l p r o v i d e the c o r r e c t cusp? For an answer a r e t u r n to f i r s t p r i n c i p l e s i s i n d i c a t e d . The necessary and s u f f i c i e n t n u c l e a r coalescence c o n d i t i o n s i n d i f f e r e n t i a l form f o r any wave f u n c t i o n are embodied i n (2.3.8a), ( 4 . 1 . 1 X 2 . 3 . 8 a ) 89 and h o l d i n t u r n f o r each e l e c t r o n ( K = 1, N). Define an operator where Ifo^njrZL clHK)^m(eK)<pK) (4.1.3) T " rJVT -A J ^constant takes the s p h e r i c a l average about the p o i n t of c o a l e s - cence of the Kib e l e c t r o n i c c o o r d i n a t e s as i n (2.3.8a). -^^/^) i s a o n e - e l e c t r o n cusp e v a l u a t i o n o p e r a t o r . The n u c l e a r coalescence c o n d i t i o n s of a s p i n l e s s wave f u n c t i o n s a t i s f y i n g (2;.3.8) are w r i t t e n c o n c i s e l y , -jb/f) § —O (4.1.4) E x t e n s i o n t o spin-space i s s t r a i g h t f o r w a r d . The n u c l e a r coalescence c o n d i t i o n f o r an exact s p i n - c o n t a i n i n g func- t i o n (see (2.3.11) ) i s as each s a t i s f i e s (4.1.4). An approximate wave f u n c t i o n w i t h s p i n , ^ , has the c o r r e c t cusp i f and -90 o n l y i f The present d i s c u s s i o n w i l l "be l i m i t e d t o n u c l e a r cusp c o n d i t i o n s f o r CI wave f u n c t i o n s . That i s , X equals zero (and w i l l be suppressed). S p e c i a l i z e d c o n d i t i o n s d e r i v e d from (4-. 1.6) w i l l be o n l y s u f f i c i e n t because a c o n f i g u r a t i o n i n t e r a c t i o n expansion i s not a unique f u n c - t i o n a l form. An n-term CI f u n c t i o n w i t h sharp s p i n and o r b i t a l angular momentum and sharp Z components—an e i g e n f u n c t i o n of , JC.X »Wz •> oCz — i s u s u a l l y expanded i n the form fCI = 1^% (4.1.7) The {ci\ are v a r i a t i o n a l l y determined l i n e a r c o e f f i c i e n t s . The {%-] are eigenf u n c t i o n s of , '̂feŝz joZTg having the same eigenvalues as ^ r and are d e s c r i b e d by a predetermined, f i x e d l i n e a r combination of S l a t e r determinants f ^ x ] 5 = *> au <t>j (4.1.8) 9 1 Since -jbfk) operates o n l y on the e l e c t r o n each S l a t e r determinant i s c o n v e n i e n t l y expanded i n t o c o f a c t o r s of the o n e - e l e c t r o n f u n c t i o n s f o r the & e l e c t r o n : 4, = xl(os*(\) xicos/co (4-. 1 . 9 ) where S^fo) = oC or f? and i s the c o f a c t o r i n 0£ of the row and m^ column, (fy,^ i s thus a f u n c t i o n of a l l e l e c t r o n i c and s p i n c o o r d i n a t e s but those of the e l e c t r o n . The e x p l i c i t case f o r l i t h i u m has been d e p i c t e d t o avoid n o t a t i o n a l d i f f i c u l t i e s . The s u p e r s c r i p t ^ i s added when needed t o d i s t i n g u i s h the o n e - e l e c t r o n o r b i t a l s b elonging t o d i f f e r e n t d e t e r - minants. I n t h i s work only STO's,{^t] ( d e f i n e d i n ( 3 . 1 . 7 ) ), are considered although the approach can be developed f o r any b a s i s s e t . The P a u l i p r i n c i p l e makes the choi c e of K i r r e l e v a n t . Operating on (app- roximate) fflcl w i t h the cusp e v a l u a t i o n operator one o b t a i n s K 'jim (4-. 1 . 1 0 ) 9 2 which must equal zero i f the cusp i s c o r r e c t . I n g e n e r a l , S£,(K) ^ 0, u n l e s s there i s a l i n e a r dependence i n the set {Xi} • S u f f i c i e n t con- d i t i o n s f o r cusp s a t i s f a c t i o n are found by l e t t i n g the c o e f f i c i e n t of S»\ (K) fifrn equal zero f o r each J?> , (4.1.11) The system of equations, 2E.Cc<?t£ = 0 , ./= 1,..., # of determinants, leads t r i v i a l l y t o an i d e n t i c a l l y zero wave f u n c t i o n . This means t h a t ^ ^ J ^ = 0 when 0 f o r each member of the set,/><} The e x p r e s s i o n i s an o r b i t a l cusp c o n d i t i o n and must be evaluated e x p l i c i t l y t o f i n d the c o n d i t i o n s f o r STO's: A - v " « . vi/^' ( , . i . i 3 ) The d i s c u s s i o n can be r e s t r i c t e d to s-type o r b i t a l s s i n c e the s p h e r i c a l averaging operator ensures t h a t a l l STO's w i t h ./>0 a u t o m a t i c a l l y s a t i s f y (4.1.12). 93 Equation (4-. 1.13) reduces to Xr,s = f-Xjo)(f+lr) n = ns (9 h > 3 (4.1.14) For the o r b i t a l cusp c o n d i t i o n t o h o l d , (1) >JbKis^ 0 i m p l y i n g JT = ~Y = Z, (2) ^ 0 / t i s s 0 i s n o t P° s s i 1 : )le, (4.1.14a) (3) ^ ) X ^ ^ = 0 i s s a t i s f i e d f o r a l l o r b i t a l s except Vis , X z s . These are p r e c i s e l y the c o n d i t i o n s ( 2 . 5 . 1 ) , (2.5-2) g i v e n f o r STO's by Roothaan and K e l l y ^ w i t h a d i f f e r e n t d e r i v a t i o n when o n l y cusp c o n d i t i o n s ( \ = 0 ) are con s i d e r e d . The present treatment can e a s i l y be extended to i n c l u d e coalescence c o n d i t i o n s . Another approach e x i s t s . Examine (4.1.10). I f any c o f a c t o r s are equal, <̂>* = $/tr\y = there w i l l be other r e l a t i o n s h i p s t o i n v e s t i g a t e . A more gen e r a l e x p r e s s i o n of (4.1.10) i s S ^ J & . - ^ & r ' - a l l / ' , , * ' . 9 4 A l l equal c o f a c t o r s , and spins f o r the e l e c t r o n , have been factored out. Two e x p l i c i t cases are now presented to show what i s i n v o l v e d ; (4-. 1.15) i s too indeterminate to give recognizable cusp c o n d i t i o n s : Case 1 The s implest p o s s i b l e case has a bas is of three STO ' s — a , a 1 , b ( a ^ a 1 ) . The S l a t e r determinants are d e f i n e d The n o t a t i o n i s short f o r ( 4 . 1 . 9 ) , i d e n t i f y i n g a deter- minant by i t s p r i n c i p l e d i a g o n a l . X a n ^ "X w i t h i n the brackets i n (4 . 1.16) imply the s p i n - o r b i t a l s X oC a n d X ^ r e s p e c t i v e l y . The CI f u n c t i o n appears (The sum ( 4 . 1 . 8 ) has but one term). Now 4 and ( 4 . 1 . 1 5 ) reduces to ( 4 . 1 . 1 7 ) 95 (4.1.18) To f i n d the cusp c o n d i t i o n s f o r t h i s f u n c t i o n equate to zero c o e f f i c i e n t s of (fij^ f o r a l l £ , 4n- (1) pb = 0 (2) p [ c x a + c 2 a ' J = 0 (4.1.19) Equation (1) i m p l i e s t h a t i f ̂ > ^ = 0, o r b i t a l b must s a t i s f y the o r b i t a l cusp c o n d i t i o n , (4.1.12). That i s , b must f i t i n t o the scheme (4.1.14a). Such STO's w i l l be w r i t t e n Xj. . Thus pb == 0 and b i s chosen by the r u l e s (4.1.14a). Equation (2) f i x e s the r a t i o between c-̂  and Note t h a t ^ X i s a constant; a l l f u n c t i o n a l depend- ences have been removed. 9 6 f 2 = -pa . c 1 pa' (4-.1.20) a,a' must not s a t i s f y the o r b i t a l cusp c o n d i t i o n i f new r e l a t i o n s are t o be found. ( I f p a s . 0, but pa'^p; 0 the f u n c t i o n , T^ci > c o l l a p s e s t o a s i n g l e term). Of course t h e r e are no longer s e c u l a r equations f o r t h i s case: % x ^ C . f W - ^ p J ^.1.21) I f b = , JI > 0,the set of o r b i t a l s having sharp jC-z , ( On = , xJt~\ > "'' ° ) » m a y n e e d t o D e i n - cluded i n the b a s i s ensuring sharp t o t a l angular momen- nas 2 ( • CX example covers o n l y the cases turn f o r . I f ^X h g symmetry t h i s p a r t i c u l a r Jb Case '2 No b a s i s o r b i t a l needs t o s a t i s f y the o r b i t a l cusp c o n d i t i o n i n t h i s example. The S l a t e r determinants, £ , and eigenf u n c t i o n s of >£ , of , »o£°k f o r a 2 S f u n c t i o n , {tyf'} , are l i s t e d i n Table X I . Only 0/ -type terms are i n c l u d e d i n f ; 97 Table X I . The set of S l a t e r determinants, and e i g e n f u n c t i o n s , Szfz , o Z f z f o r the 2 S wave f u n c t i o n des- c r i b e d i n case 2 of s e c t i o n 4.1. (a,b,c) (b,a,c) ( V , a ' , c ) ( a 1 ,b,c) <#/ = ( a \ b , c ' ) (b,a',c) (b,a\',c') (a,b*,c) (a,B',c') *- (b',a,c) (b»,a\c') (a,b,c') <t>*= (a ,,B',c') 4>,= ( b , a , c 1 ) 4,1= ( V , a \ c ' ) + & + A %- + A K = 0/r 98 a d d i t i o n of t r i p l e t core s p i n terms w i l l be demonstrated s h o r t l y . Any d i f f e r e n t set w i l l cause a break- down i n the f o l l o w i n g equations. The CI f u n c t i o n i s < -/ I f j&M^ci = 0, e i t h e r each element of {\l\ must obey the o r b i t a l cusp c o n d i t i o n , or the f o l l o w i n g set of equations must be s a t i s f i e d : r where 1 0 0 0 0 0 0 0 1 0 0 0 0 0 <? 0 0 1 0 0 0 0 0 § 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 OC 0 1 0 0 0 0 0 Y 0 0 0 1 0 0 0 0 o< 0 1 .0 0 0 0 0 0 0 1 0 0 0 0 0 2r 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 oC = £a pa' 0 '3 '8 = 0 Y = E£ (4.1.22) There P C are seven l i n e a r l y independent equations and eigh t unknowns. The s o l u t i o n s t o a constant ( n o r m a l i z a t i o n ) f a c t o r a re: 99 c 1 = 1 = <X ft ' 3 _ p ^7 c 4 = - r c 8 = - c x ^ r ( 4 . 1 . 2 3 ) The l i n e a r c o e f f i c i e n t s have been completely f i x e d . From p r e v i o u s experience t r i p l e t core s p i n terms should he i n c l u d e d . Table X I I c o n t a i n s the a d d i t i o n a l elements needed i n Table XI f o r { f a ] and f Wc] • I f a l l p o s s i b l e 6?4 -type terms are not i n c l u d e d the equations w i l l breakdown. Now C-I The o n l y v a r i a t i o n a l l y determined l i n e a r parameter, k, couples the S\ and ©z. terms. The c o e f f i c i e n t s e t s ^ c i ~ " ~ * c 8 ^ ' a n d l c 9 ~ " ^ c 1 6 ^ a r e ^ o t h d e f i n e d by the equations (4-. 1.22), the s o l u t i o n s being c± , i = 1 , 8 as i n (4.1.23) c ^ o = kc. , i = 1,8 (4.1.24) l + O 1 Other cases, w i t h v a r i o u s combinations of the s e t s and [ji] , l e a d t o s i m i l a r r e s u l t s . Always 1Q0 Table X I I . A d d i t i o n a l \<$i\ and elements f o r Table XI when © z -type s p i n terms are i n c l u d e d . (j)n= (a,c,b) 4= (a',c,b) = (a,c",V) (j)zo= ( a , c \ b ) 0^ = (a',c,b') = (a',5',b) gjz3= ( a , c ' , V ) (fa ( a ' . c ' . V ) % - ^ - fa, + 2 < f ) n <Pt + 2<p/g fa + 2$,, fa - fa + 2<Pzo % = fa ~ fao fa, ~ fa*. ?£- <t>/3 ~ fat far ~ <t>/6 .101" the c o e f f i c i e n t s resemble (4.1.23). A tremendous con- c e p t u a l s i m p l i f i c a t i o n can be made when r e l a t i o n s h i p s between the c^'s are examined. Equations (4.1.23) can be r e w r i t t e n c2_ ° 5 c]_c2°3 c 2 = — < * c 6 = C 1 C 2 C 4 C j = ft Cry = C1 C3 C4- = — ̂  C g = c 2 c ^ c ^ (4.1.25) Now a c u s p - s a t i s f y i n g 'ffl-r becomes /^, z = (a+c 2a', b+c^b', c+c^c') + (b+c^b', a+c 2a', c+c^c') ( 4 . 1 . 2 5 a ) d e f i n i n g fflCI to a constant f a c t o r as a s p i n - p r o j e c t e d d i f f e r e n t o r b i t a l s f o r d i f f e r e n t s p i n s (PODS) wave f u n c t i o n . Each o r b i t a l of the f u n c t i o n ( 4 . 1 . 2 5 ) s a t - i s f i e s the o r b i t a l cusp c o n d i t i o n , f o r example p(k) ( a ( k ) + c 2 a ' ( k ) ) = 0 (4.1.26) Important g e n e r a l i z a t i o n s f o l l o w immediately. Let the b a s i s {~Ki\ be r e p l a c e d by a p p r o p r i a t e l i n e a r * UHF f u n c t i o n s are i n c l u d e d i n the group of DODS func- t i o n s . Reasons f o r the present c l a s s i f i c a t i o n are g i v e n i n Appendix B. 102 combinations, \ Cf{] , (PL = zLdCjXj (4.1.27) j the standard form f o r a n a l y t i c a l HE o r b i t a l s . A doub- l e t , t h r e e - e l e c t r o n CI f u n c t i o n can be formed from t h i s b a s i s : c where C: ty- (4.1.28) = ( (<P*,(P»><fi«)H<Pm,(Pjt,Cp») f o r ©,-type terms ((ft, fa,fa) Wn. f o r 6 > ^ - ~ t ^ > e t e r m s and the c^ are l i n e a r v a r i a t i o n a l c o e f f i c i e n t s . For jb(fc) = 0 each o r b i t a l must s a t i s f y the gen- e r a l i z e d o r b i t a l cusp c o n d i t i o n which holds e i t h e r i f ̂ bKi s 0 f o r the b a s i s or i f an a p p r o p r i a t e l i n e a r c o n s t r a i n t i s a p p l i e d t o 103 <Pi . At l e a s t two of the ~X] i n (4-.1 .27) should not s a t i s f y = 0 i f the l i n e a r c o n s t r a i n t i s t o be p o s s i b l e , showing why equations (4-. 1 . 1 9 ) , (4-. 1.22) can break down. F i n a l l y , the s p i n cusp c o n d i t i o n i s a t r i v i a l consequence of the e l e c t r o n cusp c o n d i t i o n . One can see from (4-. 1 . 1 5 ) t h a t a s p i n cusp e v a l u a t i o n operator, -jbikj^^fc) •> w i l l g ive e x a c t l y the same r e s t r i c t i o n s on ^pcz as p(k) si n c e w i l l operate only on the s p i n , SjT(k). p 4.2 A p p l i c a t i o n s t o the l i t h i u m S groundstate E x p r e s s i o n (4 . 1.28) poses d i f f i c u l t computational problems because the o r b i t a l s (fit must be v a r i a t i o n - a l l y determined ( w i t h or without c o n s t r a i n i n g ) as w e l l as • T r a c t a b l e procedures f o r c a l c u l a t i n g s p i n - o p t i m i z e d extended Hartree-Fock wave f u n c t i o n s have been 24 25 r e c e n t l y developed ' , making t h a t method an a t t r a c - t i v e scheme to t e s t the e f f e c t i v e n e s s of cusp c o n s t r a i n t s . The form of such f u n c t i o n s resembles the iyt i n ( 4 . 1 . 2 8 ) , where k i s v a r i a t i o n a l l y determined as i n (4.1.24). Nuclear cusp c o n s t r a i n t s by the method of Weber and 104 Handy .69 or of Chong 53* can "be a p p l i e d t o each Weber, Handy and P a r r have a l r e a d y done t h i s f o r Hartree-Fock f u n c t i o n s , the primary reason being to decrease the number of l i n e a r parameters by s a t i s f y i n g a t h e o r e t i c a l c o n d i t i o n . But HF f u n c t i o n s are not good enough, as d i s c u s s e d i n Chapter I , t o a c c u r a t e l y p r e - d i c t s p i n d e n s i t i e s , the o b j e c t of t h i s work. For t e c h - n i c a l reasons the s p i n - o p t i m i z e d EHF approach was d i s - carded and a l l f u n c t i o n s are c a s t i n a CI format. This d r a s t i c a l l y l i m i t s the number of terms i n the expansion (4.1.27) tout has the advantages: (1) No SCF i t e r a t i o n s on \CPij are necessary. (2) c91 , Oz. s p i n terms are e a s i l y i n c l u d e d . (3) Angular c o r r e l a t i o n can be added i f d e s i r e d Implementation of the l a s t A c o r r e s p o n d , to a gener- a l i z a t i o n of the EHF method which has not been accom- p l i s h e d y e t . Since simple wave f u n c t i o n s are d e s i r e d , expansion (4.1.26) w i l l be r e s t r i c t e d t o two or t h r e e terms. The approach i l l u s t r a t e d by cases 1 and 2 i n 4.1 i s u t i l i z e d i n t h i s s e c t i o n . O r b i t a l s l i k e the one i n (4.1.26) w i l l be c a l l e d c u s p - s a t i s f y i n g o r b i t a l s (CSO) and w i l l be designated a(k) - /pa(k) a'(k) = a a' == a a' (4.2.1) ( p a r ( k ) / — w h i l e m a i n t a i n i n g sharp oC , * For atoms. 105 where ^ X/s = - E ) , ^ = f S Z and = 0 f o r X 4 X,s , ~Xzs - T n u s (4.1.25a) becomes $z = lA[ a — a ' ( l ) b — b ' ( 2 ) c—c'(3 ) e , ] = ( a — a ' , b — b ' , c — c ' ) + ( b — b 1 , a=a', c — c 1 ) (4.2.2) As before f o r cases 1 and 2, n e i t h e r of the b a s i s o r b i t a l s "X or "X ' i n It ~x!_ must be i n d i v i d u a l l y cusp s a t i s f y i n g . For e x p l o r a t o r y purposes a simple CI wave f u n c t i o n formed from the b a s i s was c a l c u l a t e d . The func- t i o n and i t s p r o p e r t i e s appear i n Table X I I I . Although i t i s much l e s s e x t e n s i v e , t h i s i s the same type of b a s i s used by Roothaan and c o w o r k e r s ^ ' G o d d a r d 2 ^ and o t h e r s . A l l exponents were p a r t i a l l y optimized except J/S = z. Energy-wise the f u n c t i o n i s l i t t l e b e t t e r than HF-type f u n c t i o n s (Table I ) . Such a b a s i s makes cusp s a t i s f a c t i o n t r i v i a l but there are disadvan- tages i n t h i s approach as a p p l i e d t o simple CI func- t i o n s . F i r s t , e x p r e s s i o n of core p o l a r i z a t i o n i s d i f f i c u l t . The o r b i t a l making dominant c o n t r i b u t i o n s t o the core i s f i x e d . Second, Xzs STO's are not allowed. Thus the 2s e l e c t r o n i c f u n c t i o n i n l i t h i u m 106 Table X I I I . I l l u s t r a t i v e c a l c u l a t i o n : 3-term CI f u n c t i o n formed from b a s i s . STO b a s i s : o r b i t a l exponent 3 . 0 0 0 0 3.1126 X 3 5 0 . 3 0 0 0 0 . 9 4 - 3 5 CI f u n c t i o n : $ = 0 . 7 7 8 4 9 ( i s , l i ' , 3 s " ) + 0 . 1 6 3 7 5 [ ( I s , 5 i " , 3 s M ) + ( 3 s , T s , 3 s " ) ] - 0 . 0 0 0 3 3 9 [ ( l s , 3 s ' , 3 s " ) + ( 3 s • , l i " , 3 s " ) ] P r o p e r t i e s : -E Q S ( 0 ) ( % e r r o r ) 7.436183 0.1243(46.3) Q e ( 0 ) ( % e r r o r ) 1 3.6266 ( 1 . 5 ) r = i 5 - 3 . 0 0 0 0 107 must i n c l u d e s e v e r a l X3S o r b i t a l s i n c r e a s i n g the p o s s i b i l i t y of m u l t i p l e energy minima. C u s p - s a t i s f y i n g o r b i t a l s of the type K—X! are f a r more f l e x i b l e . Any Xzs or Xis w i t h the s t i p - u l a t i o n j°is 4 Z i s ac c e p t a b l e . The f i r s t f u n c t i o n s s t u d i e d had the form (4.2.2) 3) where <% s i s m r . , A- IS "XlS L =) k was e i t h e r p l a c e d equal to zero or v a r i a t i o n a l l y determined ( s p i n - o p t i m i z e d ) . The problem of exponent o p t i m i z a t i o n was s i m p l i f i e d by f i t t i n g each CSO i n t o the (DODS) f u n c t i o n = frS)^Jls)+feJs,^) (4.2.4) of Hurst et a l ^ i n the a p p r o p r i a t e p o s i t i o n and rough l y o p t i m i z i n g i t s p a i r of exponents. The expon- ents thus obtained were placed i n (4.2.3) and v a r i e d a g a i n t o be sure a roughly minimum energy r e s u l t e d . * The n o t a t i o n i s d i f f e r e n t from t h a t i n Chapter I I I . (1) Every o r b i t a l exponent can be d i f f e r e n t ; S\$ need not equal %zs (2) Wave f u n c t i o n s are not numbered according t o the number of terms. 108 Exponents and p r o p e r t i e s are shown i n Table XIV. There are s e v e r a l p o i n t s to note. S p i n - o p t i m i z a t i o n (k 4 0 cases) i s very important here as b e f o r e . The maximum e r r o r i n Q S(0) f o r the k =̂  0 cases i s 4%. Q e(0) v a l u e s are c o r r e c t w i t h i n 1%. These are very good numbers f o r such simple f u n c t i o n s . The best o r b i t a l exponents c a l c u l a t e d f o r the f u n c t i o n (4.2.4-) by Hurst et al 5° are f\s = 3.298, S\i = 2.068, $is = 0 . 6 3 9 . The best Q?/s CSO's are seen to be j u s t these o r b i t a l s w i t h Dirac & - l i k e f u n c t i o n s as c o r r e c t i o n s f o r the cusp, as a ~XiS o r b i t a l w i t h an exponent as l a r g e as 13.0 or 20.0 i s h i g h l y concentrated near the n u c l e u s . Y y / = = • - > - 6*-*)yL (4.2.5) f o r -^is = 3 . 3 , f\s =20.0. A corresponding £ - l i k e c o r r e c t i o n f o r the ^ s CSO c o u l d not be found. 27 This i s a v e r y i n t e r e s t i n g r e s u l t because Nesbet ' ( f i r s t e n t r y i n Table I ) i n c l u d e d ~X\s -type b a s i s elements i n h i s CI and HF f u n c t i o n s without any apparent j u s t i f i c a t i o n . As i s evident i n Table I he was not attempting to o b t a i n a good energy, only a good s p i n d e n s i t y and chose h i s b a s i s elements a c c o r d i n g l y . The l i n e a r c o e f f i c i e n t s and c o n f i g u r a t i o n s Table XIV. Simple DODS wave f u n c t i o n s w i t h CSO's. O r b i t a l Exponents STO "V" A-IS X,f F u n c t i o n 1P, % 3 - 3 3 . 3 3 . 3 20.0 20.0 20.0 2.065 2.065 2.065 1 3 . 0 1 3 . 0 1 3 . 0 0.60 0.95 0.80 0.68 0 . 7 0 1 .80 P r o p e r t i e s F u n c t i o n Energy Q S(Q)(% e r r o r ) Q e ( 0 ) ( % e r r o r ) J ] E , S ( a b s o l u t e ) ( a b s o l u t e ) k=0 k * 0 -7.445434 -7.445435 0 . 2 1 7 4 ( 6 . 0 ) 0.2232(3.;5) 13 .9626C.9) 13 .9625( .9) - 3 . 0 0 0 0 - 3 . 0 0 0 0 k=0 k^O -7.445419 - 7 . 4 4 5 4 2 O 0 . 2 1 6 0 ( 6 . 6 ) 0 . 2224 (3 . 8 ) 13 .9614( .9) 1 3 . 9 6 1 3 ( . 9 ) - 3 . 0 0 0 0 - 3 . 0 0 0 0 % k=0 k * 0 -7.445328 -7.445329 0.2265(2. ,1) 0 . 2 3 4 1 ( 1 . 2 ) 13 .9553( .9) 1 3 . 9 5 5 2 ( . 9 ) - 3 . 0 0 0 0 - 3 . 0 0 0 0 Hurst et a l a k=0 -7.4436 0 . 3 0 0 2 ( 2 9 . 8 ) 1 3 . 5 1 9 3 ( 2 . 3 ) Brigman and Matsen° k * 0 -7.4436 0.2417 ( 4 . , 5) 1 3 . 5 2 4 0 ( 2 . 2 ) Best -7 .478069° 0 . 2 3 1 3 d ( 0 . 0 ) 1 3 . 8 3 4 1 e ( 0 . 0 ) - 3 . 0 0 0 0 ; Functions are d e f i n e d i n equation ( 4 . 2 . 3 ) . aFrom Table I , e n t r y 6 . , ^Reference [ 8 ] . DFrom Table I , en t r y 7 . Reference [38] c R e f e r e n c e [39] . f T h e o r e t i c a l v a l u e . 110 of h i s f u n c t i o n s are not l i s t e d i n r e f e r e n c e ' so h i s cusp v a l u e s c o u l d not be c a l c u l a t e d . These CSO's resemble Hulthen o r b i t a l s , d e s c r i b e d by Weare, Weber and P a r r . Compare the Os Hulthen o r b i t a l w i t h a d/?/s CSO and a ^ 2 S CSO A p p a r e n t l y o n l y Os and l p Hulthen o r b i t a l s have been i n v e s t i g a t e d but they g i v e s u b s t a n t i a l improvements 64 i n energy over STO's. Perhaps t h i s e x p l a i n s the s l i g h t energy improvement of the CSO f u n c t i o n s over those 30 -52 of Matsen and coworkers^ ' ^ shown i n Table XIV f o r comparison. S l i g h t l y more complicated wave f u n c t i o n s were a l s o i n v e s t i g a t e d . An n o r b i t a l CSO, X'" , can be i n c o r p o r a t e d i n t o a CI f u n c t i o n without formal I l l c o n s t r a i n t procedures as f o l l o w s . The o r b i t a l , (jP , i n = j6 zLCc\( = 0 (4.2.6) can be expressed ( t o a n o r m a l i z a t i o n c o nstant) C P = . ^Lcdxc-^Xj) ' a s u m o f cso's- The choice of "Xj i s a r b i t r a r y . The case n = 3 f o r 0/5 CSO's i s demonstrated i n Table XV f o r the func- t i o n There are a c t u a l l y three independent l i n e a r c o e f f i c i e n t s (without the n o r m a l i z a t i o n f a c t o r ) i n the CI expansion i n s t e a d of two f o r a l e g i t i m a t e DODS f u n c t i o n because c 3 = °l c2 """s ̂ r e e ' t o v a r y « £ -type c o r r e c t i o n s a l s o e x i s t i n t h i s f u n c t i o n . Whether or not they would Table XV. ( l ) (J/^ , a t r u e CI f u n c t i o n with, two 3 - o r b i t a l CSO's r e p r e s e n t i n g the core . (2) fflr » a s p i n - o p t i m i z e d CI " f u n c t i o n w i t h p c o r r e l a t i o n i n the core O r b i t a l Exponents STO -y HI A-15 v » v *" v F u n c t i o n 3.5 2.2 20.0 15.0 0.7 1.82 3.0 2.5 % 3 . 3 24.0 2.065 14.0 0.7 1.8 5.0 P r o p e r t i e s F u n c t i o n Energy Q S ( 0 ) ( % e r r o r ) Q e ( 0 ) ( % e r r o r ) k=0 -7.446249 0.2298(0.6) k*0 -7.446276 0.2299(0.6) 13.8083(0.2) 13.8095(0.2) -3 -3 .0000 .0000 % k*0 -7.467491 0.2296(0.7) 13.9203(0.6) -3 .0000 * TP* i s d e f i n e d by equation (4.2.7). i s d e f i n e d by equation (4.2.8). 113 be found w i t h energy m i n i m i z a t i o n subject t o a con- s t r a i n t a p p l i e d by Weber and Handy's method i s unknown. A CI f u n c t i o n w i t h p c o r r e l a t i o n , , was c o n s t r u c t e d from j£f . I f a c o n f i g u r a t i o n i n v o l v i n g the product XxpX^f, had been i n c l u d e d , would be a t r u e DODS f u n c t i o n . % ^tAflLc^x-Xfs n—x*)i~c*.(Xzt>f (4.2.8) +C3 (X3PrJ ( XzT&X ©. + ke j ] For the Z 5 c o n f i g u r a t i o n s k = 0 a u t o m a t i c a l l y . The r e s t r i c t i o n Jzf> = s i m p l i f i e d exponent optim- i z a t i o n . Thus c a n toe expanded as a f o u r term CI f u n c t i o n . I t s exponents and p r o p e r t i e s are p r e - sented i n Table XV. When compared w i t h the e n t r i e s i n Table I i t i s seen to be a v e r y good f u n c t i o n indeed f o r i t s s i z e . With the e x c e p t i o n of ^ and a l l of these cusp- s a t i s f y i n g f u n c t i o n s are t r u e DODS f u n c t i o n s . S e l f c o n s i s t e n c y i s t r i v i a l f o r a t w o - o r b i t a l expansion w i t h a l i n e a r c o n s t r a i n t . The success w i t h CSO's may be f o r t u i t o u s . To t e s t t h i s d i s t i n c t p o s s i b i l i t y the s t a b i l i t y of prop- e r t i e s w i t h r e s p e c t to the S - l i k e c o r r e c t i o n s was examined f o r the s p i n - o p t i m i z e d (k = 0) ^ 3 F i g u r e 6 shows a rough energy contour map as a f u n c t i o n of J /5 and J /5 . F i g u r e 7 shows the same energy 114 F i g u r e 6. Energy contour map of £r 3 (Table XIV) Energy versus exponents Sfs , S/s' of & -type cusp c o r r e c t i o n o r b i t a l s . IS 13.0 14.0 15.0 16.0 S i 20.0 - 7 .445329\ -7.445339/ -7.445317 -7.445277 21.0 -7.445344 -\. 445327 22.0 -7.445326 77.445347\ -7.4453^4 -7.445302 2 3 . 0 -7.445324, - V . 4 4 5 3 3 B - 7 . 4 4 5 3 1 0 24.0 - 7 . 4 4 5 3 3 O -7-H5343 -7.445317 2 5 . 0 -7.445b 17 \7.445349 -7.445345 - V .445321 26.0 -7.4453JU -7.445348 -7.445346 -7.445525 115 F i g u r e 7- P r o p e r t i e s of (£3 (Table XIV) corresponding to energy contours i n f i g u r e 6. /// 13-0 J is 14.0 1 5 . 0 16.0 20.0 21.0 22.0 23.0 24.0 2 5 . 0 26.0 Qs= 0 . 2 3 3 4 Q e =13.8273 Q e = °" Q e=13. 2336 8093 K= 0. '=13. Q e = °̂  Q e=13 2343 8471 ,2345 £ 7 6 116 contours superimposed on the corresponding p r o p e r t i e s . The s e n s i t i v i t y of p r o p e r t i e s t o changes i n 5ts , Si's i s seen t o be small around the energy minimum. The main energy c o n t r i b u t i o n s come from X is , \ is' STO's. O p t i m i z a t i o n w i t h r e s p e c t t o S\s , £is i s more e a s i l y accomplished and a s e n s i t i v i t y study not as e s s e n t i a l . These f i g u r e s answer the q u e s t i o n , 'Can any £ -type exponent be employed i n a b a s i s ? 1 Another t e s t f o r CSO's l i e s i n i n c r e a s i n g the 41 s i z e of b a s i s . Schaefer et a l compare the b a s i s dependence of Q s(0) f o r p r e v i o u s l y c a l c u l a t e d SEHF f u n c t i o n s of the f i r s t row elements, boron through f l u o r i n e , and conclude t h a t a l a r g e b a s i s ensures s t a b i l i t y of p r o p e r t i e s . Perhaps the present successes do not a r i s e from s a t i s f a c t i o n of the n u c l e a r cusp c o n d i t i o n . R e c a l l t h a t Wesbet's 2^ r e s u l t s w i t h 6 -type f u n c t i o n s are e x c e l l e n t a l s o . A t r u e CI wave f u n c t i o n , (16 terms) was formed w i t h the same b a s i s as the c u s p - s a t i s - f y i n g f u n c t i o n , <^ » Du"k w i t h f r e e v a r i a t i o n of a l l l i n e a r c o e f f i c i e n t s . A second t r u e CI f u n c t i o n , fflj" (4 terms) d i d not use the & -type ~K\s cusp c o r r e c t i o n o r b i t a l s . The e f f e c t s of these & o r b i t a l s can be seen i n Table XVI. (Table X I V ) , of course, u t i l i z e s the <f o r b i t a l s and has the c o r r e c t cusp. Table XVI. True CI f u n c t i o n s formed from STO b a s i s l i s t e d i n Table XIV. CI F u n c t i o n Corresponding CSO F u n c t i o n C o n f i g u r a t i o n s S p i n Functions ( w i t h * i * 1 o r b i t a l s ) XisXisXzs JXIŜ /SXzs fXisXity^s Xis ?Cis Xzs j X/s X/s Xzs) Xs X/s Xzs "V ' V* V "V ' ~VF" V ' *W5 A/5 AZS 9 A/5 X/S A2-S T } w - (without 1 b ' o r b i t a l s ) XisX15 X2S , X/s~X>ts Xzs ©, ,©i P r o p e r t i e s F u n c t i o n Energy Q S ( 0 ) ( % e r r o r ) Q e ( 0 ) ( % e r r o r ) I t I 5 %' -7.445560 0 . 2 2 3 9 ( 3 . 2 ) 13.9086(0.5) -2.9249 -1.9217 -7.444890 0.2416(4.4) 13.4671(2.7) -2.8956 - 3 . 1 2 5 3 118 One must keep i n mind t h a t the f u n c t i o n of Brigman and * Matsen has a reasonable s p i n d e n s i t y ( e n t r y 7 i n Table I ) i n the f i r s t p l a c e . c l o s e l y d u p l i c a t e s t h e i r r e s u l t s as expected. A d d i t i o n of £ -type o r - b i t a l s t o the b a s i s improves Q s(0) c o n s i d e r a b l y and i n f a c t , , w i t h a very poor cusp, has p r o p e r t i e s s i m i l a r to . The p r o g r e s s i o n from (16 term CI) to (2 term c u s p - s a t i s f y i n g f u n c t i o n ) , however, shows a 2.2% improvement of the e r r o r i n Q s(0) at the , _ Z L . expense of a t i n y s a c r i f i c e i n energy (2 x 10 a.u.J. The b e t t e r s p i n d e n s i t y of i s a p p a r e n t l y due t o cusp s a t i s f a c t i o n . U n f o r t u n a t e l y i t c o u l d be concluded from t h i s study, t a k i n g i n t o account the r a t h e r good s p i n d e n s i t i e s of and the f u n c t i o n of Brigman and Matsen, t h a t S -type o r b i t a l s r a t h e r than c o r r e c t cusp c o n d i t i o n s might be r e s p o n s i b l e f o r the e x c e l l e n t r e s u l t s . Authors 5 14- 15 74- 27 ' ' ^' have c r i t i c i z e d Wesbet ' i n v a r y i n g degrees f o r i n c l u d i n g S -type terms i n h i s bases. This work does demonstrate t h a t n u c l e a r cusp c o n d i t i o n s provide a t h e o r e t i c a l avenue to Nesbet's approach even though they cannot yet be s a i d to a f f e c t , to any great e x t e n t , the accuracy of c a l c u l a t e d s p i n d e n s i t y . F u r t h e r i n v e s t i g a t i o n w i l l determine ,the g e n e r a l i t y of CSO's. The s p i n - o p t i m i z e d EHF methods of Goddard The f u n c t i o n of Brigman and Matsen [32J i s e s s e n t i a l l y the f u n c t i o n of Hurst et a l [30] w i t h a t r i p l e t v ' c o r e ; s p i n term. 119 25 24 and Ladner , Kaldor and H a r r i s are i d e a l f o r such a study. E f f e c t s of exponent o p t i m i z a t i o n , s i z e of STO b a s i s , i n c l u s i o n of & -type terms w i t h and without n u c l e a r cusp c o n s t r a i n t s , and e x t e n s i o n of CSO's to l a r g e r systems should be examined. 2 4 . 3 A p p l i c a t i o n t o the lowest l i t h i u m P s t a t e The method of t w o - o r b i t a l CSO's i s now u t i l i z e d t o c a l c u l a t e Q s(0) f o r the l i t h i u m 2 2 P s t a t e . A f u n c t i o n s i m i l a r t o Tp'i > i = 1 > 2 , 3 f o r l i t h i u m 2 S w i t h CSO's, was p a r t i a l l y o p t i m i z e d . S~ -type cusp c o r r e c t i o n s were a l s o found here, at minimum energy. Since t r i v i a l l y s a t i s f i e s the nu c l e a r cusp c o n d i t i o n no con- s t r a i n t i s needed to be a p p l i e d t o the o r b i t a l , (pzp = Xzp+ C\ /v3p . The more ge n e r a l coalescence c o n d i t i o n s (2 . 3.8a) c o u l d have determined c-^. The f u n c t i o n p r o p e r t i e s are compared w i t h p r e v i o u s l y p u b l i s h e d r e s u l t s i n Table XVII , ordered by energy, whereas p CSO's f o r l i t h i u m S groundstate g i v e s i m i l a r r e s u l t s to the s p i n - o p t i m i z e d c a l c u l a t i o n s (compare the eleventh e n t r y i n Table I w i t h p r o p e r t i e s l i s t e d i n Table XIV), Table X V I I . C a l c u l a t i o n s on the lowest P s t a t e of l i t h i u m . Spin d e n s i t y E l e c t r o n d e n s i t y -n . , . „ „ -[-, at the nucleus at the nucleus D e s c r i p t i o n of Reference Energy wave f u n c t i o n s Q s(0) Reference Q'e(0) Reference HF 4-2 -7.365069 0.00000 42 13.6534- 42 UHF 4-2 -7.365076 -0.01747 42 13.6535 42 PUHF. 4-2 -7.365080 -0.00582 42 13.6535 42 GF 4-2 -7.365091 -0.02304 42 13.6534- 42 Spin-optimized DODS w i t h CSO's . a,b present ' -7.377569 -0.02234 p r e s e n t 5 13.5501 pres Spin-optimized EHF 24 -7.380087 -0.0169 24 Spin-optimized EHF 25 -7.380116 -0.0172 2-5 13.7065 208 -term CI 23 -7.4-0366 4-5-term CI 67 -7.4-0838 -0.02222 75 Experimental 67 -7.41016 -0.0181 10 a *is= 3 . 2 7 , 5S =2.08,&#">>S\~ =17.0, Sp =0.526 f o r the f u n c t i o n d e f i n e d i n (4.3.1). £ E= £ S =-3.0000 M u l t i p l i c a t i o n of exponents by the s c a l e f a c t o r 1.00144 gi v e s a f u n c t i o n w i t h prop- e r t i e s E=-7.377584, Qfe(O)=-O.02226, Q e(0)=13-6031. 121 f o r l i t h i u m 2P there i s a s u b s t a n t i a l d i f f e r e n c e . 42 Goddard has cas t doubt on the r e l i a b i l i t y of the ex- p e r i m e n t a l s p i n d e n s i t y . I f the p r e s e n t l y accepted experimental value i s a c t u a l l y too low then the s p i n - o p t i m i z e d DODS f u n c t i o n may be b e t t e r than the s p i n - o p t i m i z e d EHF f u n c t i o n s . On the other hand i n c l u s i o n of c o r r e l a t i o n i n the K s h e l l o r b i t a l s w i l l decrease the magnitude of the c a l c u l a t e d (DODS) va l u e somewhat. More work w i t h a l a r g e r b a s i s i s d e f i n i t e l y needed t o assess the value of CSO's. 122 CHAPTER V SUMMARY AND CONCLUDING REMARKS The hyp o t h e s i s t h a t s a t i s f a c t i o n of n u c l e a r cusp c o n d i t i o n s should l e a d to good p o i n t p r o p e r t i e s at the nucle u s , was i n v e s t i g a t e d f o r approximate wave f u n c t i o n s by employing s e v e r a l d i f f e r e n t cusp c o n s t r a i n t s . F o r c i n g necessary i n t e g r a l cusp c o n d i t i o n s , although c o r r e c t i n g the f r e e v a r i a t i o n a l e l e c t r o n d e n s i t y at the nucleus (as found by Chong and Yue f o r helium CI f u n c t i o n s ) i n v a r i a b l y o v e r c o r r e c t e d t o an extent t h a t the magnitudes of e r r o r before and a f t e r con- s t r a i n t were s i m i l a r . No g e n e r a l l y a p p l i c a b l e r e l a - t i o n s h i p s between s p i n d e n s i t y at the nucleus and cusp c o n s t r a i n t s were found. The CI f u n c t i o n s w i t h ®2.- type s p i n terms s t u d i e d had the p r o p e r t y t h a t i f the f r e e v a r i a t i o n a l value of was g r e a t e r than -Z some improvement occured upon f o r c i n g JT = -Z w h i l e the opposite was t r u e when _L7 was l e s s than -Z. Necessary w e i g h t i n g c o n s t r a i n t s d i d not appear to be u s e f u l f o r c a l c u l a t i n g Q e ' s ( 0 ) . But we i g h t i n g w i t h the f r e e v a r i a t i o n a l groundstate eig e n v e c t o r c l o s e l y 123 approximated t r a d i t i o n a l d i a g o n a l c o n s t r a i n t r e s u l t s 69 w h i l e u t i l i z i n g the method of Weber and Handy y to a v o i d computational problems. For c o n s t r a i n t s t h a t are not too severe, (the c o n s t r a i n e d f u n c t i o n i s almost equal t o the f r e e v a r i a t i o n a l f u n c t i o n ) , t h i s weighting procedure should provide a good estimate of a t r u e d i a g o n a l c o n s t r a i n t w i t h l e s s e f f o r t . S u f f i c i e n t n u c l e a r cusp c o n s t r a i n t s were a p p l i e d t o CI wave f u n c t i o n s . The r e s u l t i n g form resembled a n a l y t i c a l , s p i n - o p t i m i z e d , extended Hartree-Fock f u n c t i o n s . Only a minuscule b a s i s was employed, because the c o n s t r a i n t f u n c t i o n s were evaluated i n c o n f i g u r - a t i o n i n t e r a c t i o n form. The most important r e s u l t was the appearance of Dirac & - l i k e o r b i t a l s c o r - r e c t i n g the cusp when energy was r o u g h l y minimized. p Very good s p i n d e n s i t i e s f o r the l i t h i u m S ground- s t a t e were c a l c u l a t e d but f u r t h e r t e s t s are necessary to determine i f cusp s a t i s f a c t i o n i s r e s p o n s i b l e . Questions to be answered, i n c l u d e (1) W i l l a l a r g e r b a s i s a d v e r s e l y e f f e c t Q s(0)? (2) W i l l & - l i k e c o r r e c t i o n s , demonstrated 27 by Nesbet ' and by t h i s work to improve Q s ( 0 ) , appear when a l a r g e r b a s i s i s used? Only s t u d i e s w i t h a l a r g e r b a s i s i n c o n j u n c t i o n w i t h the c o n s t r a i n e d v a r i a t i o n methods of Weber and Handy 124 or Chong w i l l i n d i c a t e whether s a t i s f a c t i o n of n u c l e a r cusp c o n d i t i o n s t r u l y a f f e c t p o i n t p r o p e r t i e s at the nucleus. A scheme f o r nu c l e a r cusp c o n d i t i o n s found i n Table X V I I I u n i f i e s the v a r i o u s approaches. The nec- essary and s u f f i c i e n t cusp c o n d i t i o n i s w r i t t e n f o r any e l e c t r o n . The o n e - e l e c t r o n form of t h i s con- d i t i o n d i c t a t e s t h a t i n d i v i d u a l e l e c t r o n o r b i t a l s i n o r i g i n a l l y d e r i v e d i n a d i f f e r e n t manner by Roothaan and K e l l y f ^ Very r e c e n t l y , G o d d a r d ^ u t i l i z e d t h i s type of b a s i s f o r a n a l y t i c a l EHF atomic wave f u n c t i o n s of boron through f l u o r i n e w i t h the expressed purpose of improving Qf(0) v a l u e s . He concluded th a t Q s(0) converged much f a s t e r ( w i t h r e s p e c t t o the s i z e of the ( 5 . 1 . 1 ) must s a t i s f y the cusp c o n d i t i o n . Table X V I I I . H i e r a r c h y of necessary and s u f f i c i e n t cusp c o n d i t i o n s f o r atoms. Wave f u n c t i o n S u f f i c i e n t cc Necessary and. Necessary cc S u f f i c i e n t cc Exact, <| CI f u n c t i o n without SCF o n e - e l e c t r o n o r b i t a l , STO c u s p - s a t i s - f y i n g b a s i s STO c u s p - s a t i s - f y i n g b a s i s of Roothaan and K a t o 4 7 Pack and Byers Brown 2 3) />§ =0° 1 ) 2 ) CSO's d-I n t e g r a l cusp c i t i o n s of Chong' lj) I n t e g r a l cusp • c o n s t r a i n t s of Chong 6 2 c 2 ) Weighted cusp c o n s t r a i n t s 0 1 ) I n t e g r a l cusp c o n s t r a i n t s of C h o n g 5 3 2 ) Cusp c o n s t r a i n t s of Weber. Handy and P a r r 6 8 3) CSO's { ( % } c a N e c e s s i t y and s u f f i c i e n c y r e f e r to cusp c o n d i t i o n s a f t e r wave f u n c t i o n type i s s p e c i f i e d . Cusp c o n d i t i o n s ( c c ) . c T h i s work. 126 two o r b i t a l s should be important both f o r energy and d e s c r i p t i o n of the r e g i o n about the nucleus. Goddard's c o n c l u s i o n s do r e i n f o r c e the i d e a behind t h i s work. The more general o n e - e l e c t r o n o r b i t a l s , {CPi} , can be l i n e a r l y c o n s t r a i n e d to s a t i s f y s u f f i c i e n t and necessary cusp c o n d i t i o n s . The r e s u l t i n g c u s p - s a t i s f y i n g o r b i t a l s , , are f l e x i b l e and do not show the d i s - advantages of the s e t , {X_i} The P a u l i p r i n c i p l e ensures t h a t i s a l s o a necessary and s u f f i c i e n t c o n d i t i o n . Thus the weighted cusp c o n s t r a i n t s are d e r i v e d immediately by i n t e g r a t i o n : ( • f l * 2 . / ? f c ) \ % j y ( 5 . 1 . 3 ) These c o n d i t i o n s are s u f f i c i e n t i f they h o l d f o r each member of a complete s e t . The necessary cusp c o n d i t i o n s are the s p e c i a l case, f = r I n t e g r a t i o n d e s t r o y s s u f f i c i e n c y because o r b i t a l over- l a p s c o n t r i b u t e t o the q u a n t i t y </7^/*/^rather than j u s t ( ^>(k) ). 127 There i s a f i n a l comment on cusp and coalescence c o n d i t i o n s . Experiments are being performed on p o s i t r o n - e l e c t r o n a n n i h i l a t i o n s i n molecules. One would expect e f f e c t s from a n n i h i l a t i o n to be extremely dependent 77 on the wave f u n c t i o n at coalescence. The u s e f u l n e s s and a p p l i c a t i o n of cusp c o n s t r a i n t s should s t i l l be examined w i t h r e s p e c t t o t h i s important new development. 128 BIBLIOGRAPHY 1. D. P. Chong and W. Byers Brown, J . Chem. Phys. 45_, 392 (1966). 2. R. CD. Pack and W. Byers Brown, J . Chem. Phys. 45_, 556 (1966). 3 . D. P. Chong, Theoret. Chim. A c t a ( B e r l . ) 11, 205 (1968). 4. C P . Yue and D. P. 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Q u a n t i t y a.u. mass charge l e n g t h angular momentum energy magnetic moment For atomic (as opposed to m o l e c u l a r ) c a l c u l a t i o n s a d i f f e r e n t d e f i n i t i o n of mass i s used: 1 = #k = 9.1091x10 g. 1 = e = 4.80298X10"10 e.s.u. 1 bohr = Q» = J j Z = 5.29167xl0_9cm. -* L ^ £ i -27 1 = 7? = _h_ = 1.0544x10 ^ e r g sec. 1 h a r t r e e = = 4.3594x10 erg Q° p-i 1 bohr magneton = Cn = 9.2732xlQ-^- Ierg 3.?neC Gauss where M i s the mass of the 134 nucleus. The a.u. f o r l e n g t h , energy, e t c . are re- d e f i n e d Ct = — i - — = mass c o r r e c t e d bohr € \ = mass c o r r e c t e d h a r t r e e . 1 3 5 APPENDIX B SOME IMPORTANT TYPES OF APPROXIMATE ATOMIC WAVE FUNCTIONS F i r s t a remark on the symmetry of atomic wave f u n c t i o n s . The n o n - r e l a t i v i s t i c H a m i l t o n i a n f o r an atom commutes w i t h j^f , , %fz , Jlz , the t o t a l s p i n and o r b i t a l angular momenta and t h e i r Z components r e s p e c t i v e l y . An exact wave f u n c t i o n , t h e r e f o r e , has sharp values f o r these observables and i t seems proper t h a t approximate wave f u n c t i o n s should t o o . Because e l e c t r o n s are fermions, the P a u l i p r i n c i p l e holds a l s o . The a n t i s y m m e t r i z a t i o n o p e r a t o r , w i l l d esignate t h i s symmetry. The most w i d e l y used approximation i n atomic and molecular p h y s i c s i s the Hartree-Fock (HF) method a r r i v e d at by c o n s i d e r i n g each e l e c t r o n to be i n an average p o t e n t i a l f i e l d c r e a t e d by a l l other e l e c t r o n s . This r e s u l t s d i r e c t l y i n a s i n g l e p a r t i c l e i n t e r p r e t a - t i o n f o r the approximate f u n c t i o n where an e l e c t r o n cannot experience d i r e c t i n t e r a c t i o n s w i t h the o t h e r s . The HF f u n c t i o n i s the ve r y b e s t , w i t h energy as a c r i t e r i o n , antisymmetrized product of o n e - e l e c t r o n 136 s p i n o r b i t a l s : The (/{• may be expanded i n a complete set of on e - e l e c t r o n f u n c t i o n s . I f t h i s expansion i s t r u n c a t e d f o r pr a c - t i c a l a p p l i c a t i o n s , an a n a l y t i c a l HF approximation t o the t r u e HF f u n c t i o n r e s u l t s . 17 There are a number of r e s t r i c t i o n s ' t h a t must be made on the most gen e r a l HF o r b i t a l s so t h a t the f u n c t i o n w i l l be an e i g e n f u n c t i o n of , , ( l ) The s p i n o r b i t a l , should be separable i n t o s p i n and o r b i t a l components. The more general ( u n r e s t r i c t e d ) case i s (2) The o r b i t a l should be separable i n t o r a d i a l and angular components. (pft,e,<p) - U;0lYi(e,4>) * The present d i s c u s s i o n a p p l i e s t o atoms o n l y . 137 ( 3 ) I f ( 2 ) i s t r u e , U{(H should be independent of ^ . (4) I f (1) i s t r u e , UtfH should be independent of % * C l o s e d - s h e l l systems present no problems. A l l the above r e s t r i c t i o n s a u t o m a t i c a l l y h o l d and the r e s u l t i n g best f u n c t i o n i s c a l l e d the r e s t r i c t e d Hartree-Fock (EHF) or ( u s u a l l y ) j u s t the HF f u n c t i o n . Open-shell systems having an unpaired s p i n e x h i b i t core p o l a r i z - a t i o n o n l y i f (4) i s l i f t e d . This i s c a l l e d the un- r e s t r i c t e d Hartree-Fock (UHF) approximation, ambiguously s i n c e o n l y one r e s t r i c t i o n has been l i f t e d . I t i s a l s o r e f e r r e d to as a s p i n - p o l a r i z e d Hartree-Fock. E e l e a s i n g of the r e s t r i c t i o n s ( 2 ) , ( 3 ) has not been i n v e s t i g a t e d t o any extent. Sharpness of , , c v ^ . » oCz can be r e s t o r e d t o u n r e s t r i c t e d f u n c t i o n s by the a p p r o p r i a t e p r o j e c t i o n o p e r a t o r ( s ) . The p r o j e c t e d u n r e s t r i c t e d Hartree-Fock (PUHF) i s thus obtained by p r o j e c t i n g a (minimized) UHF f u n c t i o n to have sharp fe/*2" . Spin p r o j e c t i o n before m i n i m i z a t i o n i s p h y s i c a l l y more r e a l - i s t i c , r e s u l t i n g i n the extended Hartree-Fock (EHF), a l s o r e f e r r e d t o as s p i n - p o l a r i z e d p r o j e c t e d H a r t r e e - Fock or s p i n extended Hartree-Fock (SEHF). *The present d i s c u s s i o n a p p l i e s t o atoms o n l y . 138 Such, p r o j e c t e d f u n c t i o n s are no longer t r u e HF f u n c t i o n s s i n c e they c o n t a i n more than one determinant, hut they have an independent p a r t i c l e i n t e r p r e t a t i o n i f they are c a l c u l a t e d w i t h averaged Coulomb p o t e n t i a l s . The independent p a r t i c l e i n t e r p r e t a t i o n of the HF method, although l e a d i n g t o conceptual advantages, a l s o p r o v i d e s s e r i o u s shortcomings. E l e c t r o n s are per- m i t t e d by the f u n c t i o n a l form of the HF f u n c t i o n to come too c l o s e t o g e t h e r . The P a u l i p r i n c i p l e auto- m a t i c a l l y p r o v i d e s a 'Fermi h o l e ' f o r c o r r e l a t i n g the movement of two e l e c t r o n s w i t h p a r a l l e l s p i n s — t h e ( d e t e r m i n a n t a l ) wave f u n c t i o n can v a n i s h i d e n t i c a l l y . The ' c o r r e l a t i o n h o l e ' or 'Coulomb h o l e ' d e s c r i b i n g the instantaneous i n t e r a c t i o n s between two e l e c t r o n s of d i f f e r e n t s p i n s does not e x i s t i n HF f u n c t i o n s . The method of d i f f e r e n t o r b i t a l s f o r d i f f e r e n t s p i n s (DODS) improves on the HF procedure by a l l o w i n g e l e c t r o n s having d i f f e r e n t spins (and thus not a f f e c t e d by the P a u l i p r i n c i p l e ) t o occupy d i f f e r e n t s p a t i a l p o s i t i o n s . To be an e i g e n f u n c t i o n of a DODS func- t i o n i s u s u a l l y s p i n p r o j e c t e d . The term, DODS, i s c o l l e c t i v e , i n c l u d i n g a l l UHF, EHF, e t c . (except EHF) f u n c t i o n s but i m p l y i n g a degree of approximation below t h a t of a n a l y t i c a l HF d e r i v e d f u n c t i o n s . For 139 example i f where jP i s a s p i n p r o j e c t i o n o p e r a t o r , the o r b i t a l s are expanded i n the same o n e - e l e c t r o n b a s i s , "usually r e l a t i v e l y l a r g e , P,s= l^QiYi , <pfs= *Z£r(Xi , e t c . whereas f o r the o r b i t a l s <P are expanded i n a s m a l l b a s i s — s o t h a t t h e i r resemblence t o accurate a n a l y t i c a l HF-type o r b i t a l s i s i n n o t a t i o n o n l y — t h a t can be d i f f e r e n t f o r each o r b i t a l . 0/5 = X, i 0% =XL , etc. A remark on open and c l o s e d s h e l l s can be made here. 'Open s h e l l ' can r e f e r t o an atom l i k e l i t h i u m w i t h an u n f i l l e d outer s h e l l or i t can r e f e r t o a s p l i t s h e l l d e s c r i b e d by the DODS method. Helium f o r example 140 lias a c l o s e d \( s h e l l hut a DODS d e s c r i p t i o n i s an open s h e l l d e s c r i p t i o n . E n t r i e s 6, 7, 8, 9 i n Table I c o u l d be more a c c u r a t e l y labelled, as f u n c t i o n s having s p l i t K s h e l l s . There are two main types of f u n c t i o n s c o n t a i n i n g c o r r e l a t i o n : a c o n f i g u r a t i o n i n t e r a c t i o n (CI) f u n c t i o n (a k i n d of g e n e r a l i z a t i o n of the DODS method), and a c o r r e l a t e d f u n c t i o n (sometimes r e f e r r e d to as a H y l l e r a a s - t y p e f u n c t i o n ) . Both, of n e c e s s i t y , depart from the independent p a r t i c l e p i c t u r e . A CI f u n c t i o n c o n s i s t s of a sum of antisymmetrized products of one- e l e c t r o n s p i n o r b i t a l s . A complete o n e - e l e c t r o n b a s i s w i t h a l l p o s s i b l e pro- ducts can d e s c r i b e an exact f u n c t i o n . I n p r a c t i c e / ^ C I i s t r u n c a t e d and the determined by the u s u a l s e c u l a r equations. I f a f u l l b a s i s ( i n the sense t h a t a l l combinations of any p a r t i c u l a r t r u n c a t e d set of (fy 9s » w i t h a l l p o s s i b l e ^ , , «S" quantum numbers, are p r e s e n t ) i s used, i s a u t o m a t i c a l l y an e i g e n f u n c t i o n of sJ , c£X , Wz , • Or determinants may be grouped to i n d i v i d u a l l y be eigen- f u n c t i o n s of these operators (the examples i n the t e x t ) . Any complete set of on e - e l e c t r o n f u n c t i o n s w i l l generate 141 a CI f u n c t i o n , Gaussian o r b i t a l s , S l a t e r - t y p e o r b i t a l s (STO's), Laguerre po l y n o m i a l s , e t c . STO's provide a p h y s i c a l l y r e a l i s t i c b a s i s . The CI approach s u f f e r s from slow convergence and the growing number of impor- t a n t c o n f i g u r a t i o n s w i t h the number of p a r t i c l e s . C o r r e l a t e d wave f u n c t i o n s c o n t a i n i n t e r e l e c t r o n i c c o o r d i n a t e s , r . ., e x p l i c i t l y and so are not expanded i «J p i n a o n e - e l e c t r o n b a s i s s e t . Such a f u n c t i o n f o r S l i t h i u m atom c o u l d appearf' ^® The Si terms are the two l i n e a r l y independent s p i n doublet f u n c t i o n s f o r t h r e e e l e c t r o n s , and the c's are v a r i a t i o n a l l y determined l i n e a r c o e f f i c i e n t s . A form f o r the £7J-is <p<j«**h = K£rJif h£ t/Texp(-«/7-fa-rr3) F a c t o r s l i k e r l , j , r i ~ r , i c o u l d be i n c l u d e d as r. + r . r . . w e l l . Disadvantages of t h i s method are: (1) The .number of p o s s i b l e (ps i n c r e a s e s trem- endously w i t h the number of e l e c t r o n s . (2) The necessary i n t e g r a l s can be complicated to e v a l u a t e . 142 APPENDIX C CONSTEAINED VARIATION C . l I n t r o d u c t i o n Constrained v a r i a t i o n i s a technique t h a t "builds s e l e c t e d i n f o r m a t i o n , e i t h e r of a t h e o r e t i c a l or em- p i r i c a l n a t u r e , i n t o a v a r i a t i o n a l l y determined approx- imate wave f u n c t i o n . The purpose i s the a n t i c i p a t e d improvement over the f r e e v a r i a t i o n a l f u n c t i o n i n r e l a t e d e x p e c t a t i o n v a l u e s . B a s i c a l l y the problem i s the m i n i - m i z a t i o n of energy of a t r i a l f u n c t i o n w h i l e f o r c i n g i t t o have predetermined p r o p e r t i e s . Since the procedure removes degrees of freedom i n the v a r i a t i o n a l c o e f f i c - i e n t s of the t r i a l f u n c t i o n i t r e s u l t s i n an energy s a c r i f i c e , Z l E , from the energy of a f r e e v a r i a t i o n a l f u n c t i o n . The i d e a of c o n s t r a i n e d v a r i a t i o n was f i r s t i n t r o - o-i duced by M u k h e r j i and Karplus and b a s i c theory was op nn developed by R a s i e l and Whitman and Byers Brown.' O - l O p Q 7 I n i t i a l successes i n a p p l i c a t i o n s ' ' D have l e d t o r e f i n e m e n t s 6 ^ ' ^ 1' ? 2 ' 73 and f u r t h e r a p p l i c a t i o n s . 3 4 59 84 85 ' * ' ' Methods of s o l v i n g c o n s t r a i n e d s e c u l a r equations are now presented. 143 C.2 S i n g l e c o n s t r a i n t s The energy of a t r i a l f u n c t i o n , E = < I H 1 7 ? ( C 2 . 1 ) i s to be minimized subject to a c o n s t r a i n t (lFMilTP>/<y//W> = A< (0-2-2) c o n v e n i e n t l y expressed ( C 2 . 3 ) can be any observable or a t t r i b u t e not commuting w i t h // . The m o d i f i e d v a r i a t i o n a l p r i n c i p l e becomes o or more e x p l i c i t l y (C.2.4) The d e t e r m i n a t i o n of the Lagrange m u l t i p l i e r , X , c o n s t i t u t e s the major problem i n a p p l y i n g c o n s t r a i n t s . 144 The form of (C.2.4) r e s t r i c t s C t o "be an H e r m i t i a n 70 o p e r a t o r . Byers Brown' has developed the most exten- s i v e treatment of c o n s t r a i n e d v a r i a t i o n by c o n s i d e r i n g z\ as a p e r t u r b a t i o n parameter. In t h i s p e r t u r b a t i o n approach the energy i n (C.2.4) i s designated as Efj_ct» a f i c t i t i o u s energy and i s expanded i n a power s e r i e s Z- n /*> ( C 2 . 5 ) E f i c t - E \n E where i s the i& order p e r t u r b a t i o n energy. Since E f l o t ( A ) . { m c y . ^ = y — Cf/l) toy the Hellmann-Feynman theorem. When the c o n s t r a i n t (C.2.3) i s s a t i s f i e d , )[ — A opt » the optimum value of X . Because / o p t r\ £_ = o . (c.2.7) The s e r i e s (C.2.5) must converge r a p i d l y i f a p e r t u r - b a t i o n approach i s to be of value and can be t r u n c a t e d a f t e r 'k' terms. The E ^ ^ are r e a d i l y evaluated f o r the E a y l e i g h - S c h r o d i n g e r p e r t u r b a t i o n expansion and a value 145 f o r s^opt c a n D e obtained by i n v e r t i n g the power s e r i e s (C.2.7) A opt 'Ao Z ^ ^ Z ^ - J ^ ( C 2 . 8 ) P r o p e r t i e s can be evaluated by a double p e r t u r b a t i o n approach, or d i r e c t l y from the wave f u n c t i o n s a t i s f y i n g The c l o s e n e s s of the approximation, "X opt A 0 , depends on the convergence of the i n v e r t e d s e r i e s and the t r u n c a t i o n e r r o r s i n v o l v e d . 71 T n e p a r a m e t r i z a t i o n approach' avoids the q u e s t i o n of convergence o c c u r i n g i n the p e r t u r b a t i o n approach. I t i s simple t o apply; (C.2.4) i s r e p e a t e d l y s o l v e d f o r d i f f e r e n t v a l u e s of A u n t i l C( Aopt ) = 0 i s found. The problem here i s t h a t a good i n i t i a l guess f o r \opt i s d i f f i c u l t . I f one i s f o r t u n a t e enough t o c l o s e l y estimate Xopt , i t may be obtained to h i g h accuracy by su c c e s s i v e l i n e a r i n t e r p o l a t i o n s or e x t r a - p o l a t i o n s of C( ^ ). 72 Chong' has developed a p e r t u r b a t i o n - i t e r a t i o n approach by i n c o r p o r a t i n g p a r a m e t r i z a t i o n i n t o the per- t u r b a t i o n approach. C( )\ ) i s expanded i n a Taylor 146 s e r i e s about an estimate /\n . For )\h+r = + X. Cft„+I) = CQJ + i V W C U ( C 2 . 1 0 ) where Dv( (>») = 2 / JTV ^7 + ' ) E rv\ = n (ri'] are the b i n o m i a l c o e f f i c i e n t s . Equation (C.2.8) p r o v i d e s an i n i t i a l guess, ^ co.) = <wM/^M>/r^j/mj>i-o evaluated w i t h the s o l u t i o n t o (C.2.9) f o r ^ ̂  . Since = \0pt i s d e s i r e d C ( ) \ h + , ) i s set t o zero and i n v e r s i o n of the t r u n c a t e d s e r i e s (C.2.10) gi v e s an estimate f o r "X . U s u a l l y a v e r y few i t e r a t i o n s s u f f i c e t o g i v e /\o],t to d e s i r e d accuracy. C.7> M u l t i p l e c o n s t r a i n t s 70 Byers Brown' extended h i s p e r t u r b a t i o n approach to i n c l u d e m u l t i p l e c o n s t r a i n t s . The v a r i a t i o n p r i n c i p l e i s SE +- Z Ai£CC =0 ( C 3 . 1 ) R e s u l t i n g s e r i e s expansions and i n v e r s i o n s are not e a s i l y worked out and thus h i s e x t e n s i o n i s not too p r a c t i c a l . 147 73 Chong and Benston'^ observed approximate l i n e a r r e l a t i o n s h i p s i n double c o n s t r a i n t s . The c o n s t r a i n t c o n d i t i o n s = C, O.pt , ^.pt) = o A , 7j the two Lagrange m u l t i p l i e r s , are c l o s e l y estimated by s o l v i n g C i O ^ ) = ^ + ^ » ^ * / 9 ~ ^ ( C 3 . 3 ) f o r C i = Cx - 0 a t A = A opt , ?l = ^lopt . The c o e f f i c i e n t s { flc'j] can be i n i t i a l l y determined from the e x p e c t a t i o n values of . and f o r the f r e e v a r i a t i o n a l and s i n g l y c o n s t r a i n e d e i g e n f u n c t i o n s , Cs(o^o) = R i o = y ( f r e e v a r i a t i o n a l ) C-L(O}O) — flz_o = ( £ ? _ y " c i ( A ^ t , o) = ft* + fiuXort =o 148 where A opt, tyopt are optimum s i n g l e c o n s t r a i n t v a l u e s . The s e c u l a r equation i s s o l v e d u s i n g Xopt » rf°t>t from ( C . 3 . 3 ) . Accurate e v a l u a t i o n s of C, O o p t , ??°pt), C ^ O i o p t , ??opt) are made en a b l i n g the set of p o i n t s (C . 3.4) to be improved. I t e r a t i o n proceeds u n t i l ( C . 3 . 2 ) h o l d s . U s u a l l y o n l y one or two c y c l e s are needed. An a l t e r n a t i v e to t h i s approach i s successive p a r a m e t r i z a t i o n . The best )\ f o r some value of 9j i s found so t h a t C\{Y\?i) = 0. A i s f i x e d and a new 7f i s found g i v i n g Cz. (}\ , = 0. The process i s repeated u n t i l s e l f - c o n s i s t e n c y i s 86 achieved. Loeb and R a s i e l have employed t h i s method. The disadvantage i s t h a t m a t r i x e i g e n f u n c t i o n s must be found at each step i n the p a r a m e t r i z a t i o n w h i l e e q u a t i o n s ( C . 3 . 2 ) are simple, l i n e a r , a l g e b r a i c equations. C.4 O f f - d i a g o n a l c o n s t r a i n t s A l l methods d i s c u s s e d so f a r have been d i a g o n a l , 1 4 9 An o f f - d i a g o n a l c o n s t r a i n t can be imposed by d e f i n i n g a new pseudo-diagonal con- s t r a i n t o p e r a t o r ^ £' = eii?xw/ + \iyxwi<2 . c c . 4 . 2 ) The c o n s t r a i n t c o n d i t i o n (C.4.1) becomes C =<tyltS'\W>=o co .4 .3) where ffi' i s f i x e d and i s v a r i a t i o n a l l y d e t e r - mined. S e l f - c o n s i s t e n c y i s achieved by reforming ' a f t e r every c a l c u l a t i o n of . G.5 O f f - d i a g o n a l l i n e a r c o n s t r a i n e d v a r i a t i o n of Weber and Handy No i t e r a t i o n or p a r a m e t r i z a t i o n i s necessary i n t h i s one-step approach. M u l t i p l e c o n s t r a i n t s are handled e a s i l y . The p r e s e n t a t i o n of Weber and Handy J i s f o l l o w e d . Define the c o n s t r a i n e d wave f u n c t i o n 150 represented "by the column v e c t o r , (C , i n some orthonormal b a s i s set (% J . Weber and Handy consid- ered the c o n s t r a i n t c o n d i t i o n s f€ = o , c = /,..»m ( c . 5 . 1 ) where each d e f i n e s a c o n s t r a i n t . The set {-{p^} can be o r t h o g o n a l i z e d to f^p)cJ where f t p , =- & « ( c - 5 - 2 ) The r e s u l t i n g c o n s t r a i n e d s e c u l a r equations are f/H-el)c = *LXifie (c-"> I i s the u n i t m a t r i x , i s the H a m i l t o n i a n m a t r i x i n the b a s i s , and <~ i s the ( c o n s t r a i n e d ) energy of 7p . The beauty of the method l i e s i n the e l i m i n a t i o n of the Lagrange m u l t i p l i e r s , /^« . M u l t - i p l y i n g (C.5.5) "by J/Df< one o b t a i n s ~~2 = j & t ' M * ( C 5 . 4 ) These v a l u e s f o r the V s are s u b s t i t u t e d i n t o (C.5.3) which i s then manipulated i n t o the set of s e c u l a r equations 151 OB -ez)<£ = <? ( c . 5 . 5 ) where /£> = ) M l ~/l ) m a t r i x and = ^ £̂>*- ^2*- an H e r m i t i a n When i s d i a g o n a l i z e d , 7?i extraneous r o o t s , € i , appear because 977 degrees of freedom are absorbed i n the c o n s t r a i n t s . Weber and Handy show t h a t these r o o t s a l l have value zero. O f f - d i a g o n a l c o n s t r a i n e d v a r i a t i o n i s thus reduced to o r t h o g o n a l i z a t i o n of the set f ^ t ] and s o l u t i o n of (C . 5 . 5 ) . 152 APPENDIX D INTEGRAL CALCULATION D.1 P r i m i t i v e i n t e g r a l s f o r S l a t e r - t y p e o r b i t a l s (STO 1 s ) A g e n e r a l STO has the form The Yurr^s are the u s u a l s p h e r i c a l harmonics. P r i m i t i v e i n t e g r a l s are those a r i s i n g between o n e - e l e c t r o n b a s i s f u n c t i o n s . I n energy c a l c u l a t i o n s the f o l l o w i n g prim- i t i v e i n t e g r a l s occur f o r atoms: o v e r l a p i n t e g r a l s , S(-j fx^h) yjfc) dfi k i n e t i c energy i n t e g r a l s , ~^f)(iCc)V**jfc) n u c l e a r a t t r a c t i o n i n t e g r a l s , ?f~X*(r)_L-~)Cj(h) Coulomb and exchange /" „ ^ i n t e g r a l s /*• fc)XjfhJ_/_ K j & t / ^ A n a l y t i c a l formulae are g i v e n by Roothaan?7 153 For cusp c a l c u l a t i o n s the f o l l o w i n g p r i m i t i v e i n t e g r a l s a r i s e : e l e c t r o n d e n s i t y ,. A***s^.\ X~M ^/ at the nucleus, Q S ) 7 ^ * , ^ ) ^ Formulae are e a s i l y d e r i v e d f o r these i n t e g r a l s from the d e f i n i t i o n of STO's ( D . l . l ) : Q , . _ f y ^ , 3 > f o r ' Xj both ~X\S o r b i t a l s , i f X i , X j are not both X.\s ^ o r b i t a l s = ( -JlVSifs . ^ . , £ ; are Xts o r b i t a l s -4- ?• l/r/ ?'"J , i f Xc i s a XZJJ o r b i t a l and t ^ L - ' X; i s a * 1 S o r b i t a l vrn \̂  o 1 f ° r a l l other combinations of o r b i t a l s The s p h e r i c a l averaging operator d e f i n e d i n the t e x t has not been shown, but i t ensures t h a t a l l -fi , c( , -f ,... o r b i t a l s ( £ ^> 0) g i v e a z e r o ' c o n t r i b u t i o n to the cusp. 154 D.2 C o l l e c t i o n of p r i m i t i v e i n t e g r a l s The i n t e g r a l s d i r e c t l y employed i n atomic c a l - c u l a t i o n s are those between S l a t e r determinants (<Pi] where l< </( i s the a n t i s y m m e t r i z a t i o n o p e r a t o r , {X} are STO's and 'V~< are s p i n f u n c t i o n s oc , ^ The 88 ge n e r a l method of h a n d l i n g these i s g i v e n by S l a t e r . For three e l e c t r o n f u n c t i o n s w i t h S = 1/2 = S^ a sim- p l i f i e d approach can be used. I f g e n e r a l p r i m i t i v e i n t e g r a l s are designated the S l a t e r determinants (D.2.1) may be compactly w r i t t e n Then 155 - pf/MM ffafite <?&)o43) } >. I f T^C) i s s p i n l e s s the spins may be i n t e g r a t e d out immediately. F t i = < a O ) £ t i c ® lil'H)] e(\) fa its) -~Q(\)&)e(i)} = fae Sb-f Scj + S * e ft-fScj + S * e Sb* fog - f « j S l { S c e - 5 « . j f 6 - F S c e S t r i k e . i n terms of p r i m i t i v e overlap and J~ i n t e g r a l s . ^1E£(K} c o n t a i n s s p i n — f o r example, the s p i n d e n s i t y o p e r a t o r , i p j p ^ f c ^ k ^ — t h e s p i n f u n c t i o n s may be operated on and then i n t e g r a t e d out i n an analogous way. 156 F u r t h e r i n t e g r a l c o l l e c t i o n s , between e i g e n f u n c t i o n s of , e t c . , as used i n the t e x t , are t r i v i a l , con- s i s t i n g of l i n e a r combinations of the i n t e g r a l s be- tween S l a t e r determinants. 157 APPENDIX E DESCRIPTIONS AND PROPERTIES OP THE WAVE FUNCTIONS $ / 0 THROUGH §/ g The S l a t e r - t y p e o r b i t a l b a s i s f o r t h i s s e r i e s i s de f i n e d i n Table V, as are the c o n f i g u r a t i o n s i n $/o , the key wave f u n c t i o n . The s i g n i f i c a n c e of ex p l a i n e d i n the t e x t f o l l o w i n g equation C3.3.4-). The n o t a t i o n , f & 1} , used below, means the e n t i r e c o l l e c t i o n of c o n f i g u r a t i o n s making up the CI f u n c t i o n , <§( . P r o p e r t i e s of $ / 0 through (ft/8 are l i s t e d i n Table XIX. The f u n c t i o n s through have the c o n f i g u r a t i o n s $ : $,1.: > \sis'2s"e*_ £»: 2S\S2S"&?. f&3) , 2S2S /J.s"oz_ /<W > /sts3s"ez. /S2s'SS"&2_ 25 /s'3s "&z_ 2S25'js" &z Table XIX. Free v a r i a t i o n a l and c o n s t r a i n e d p r o p e r t i e s of the wave f u n c t i o n s through <j5/9 F u n c t i o n C o n s t r a i n t -Energy Q S(0) Q e(0) r A E J-/o 7z a. v3 7¥ None j / e = 2r None None None None 7.467389 7.467254 7.465334 7.464736 7.467408 7.467278 7.467394 7.467264 7.467418 7.467283 7.467418 7.467282 7.467424 7.467289 7.467420 7.467287 7.467429 7.467291 7.467429 7.467291 0 . 2 6 7 7 0 . 2 7 5 3 0 . 1 7 1 5 0.1686 0.2244 0.2265 0.1867 0.1879 0.2148 0 . 2 1 9 7 0 . 2 2 0 1 0 . 2 1 7 7 0.2048 0.2088 0 . 2 2 2 9 0 . 2 2 1 9 0 . 2 1 3 5 0 . 2 1 5 3 0 . 2 1 5 0 0 . 2 1 6 3 1 3 . 7 5 2 2 1 3 . 9 1 9 1 1 3 . 5 5 1 9 1 3 . 8 9 0 1 13.7538 1 3 . 9 1 7 9 13.7524 13 .9168 1 3 . 7 5 1 2 1 3 . 9 1 8 9 1 3 . 7 5 1 8 1 3 . 9 1 9 0 1 3 . 7 5 1 5 1 3 . 9 1 8 9 1 3 . 7 5 2 7 1 3 . 9 1 8 7 1 3 . 7 5 0 0 1 3 . 9 1 9 2 1 3 . 7 5 0 0 1 3 . 9 1 9 2 - 2 . 9 7 3 2 - 3 . 0 0 0 0 - 2 . 9 4 4 7 - 3 . 0 0 0 0 - 2 . 9 7 3 7 - 3 . 0 0 0 0 - 2 . 9 7 3 6 - 3 . 0 0 0 0 - 2 . 9 7 3 0 - 3 . 0 0 0 0 - 2 . 9 7 3 1 - 3 . 0 0 0 0 - 2 . 9 7 3 1 - 3 . 0 0 0 0 - 2 . 9 7 3 3 - 3 . 0 0 0 0 - 2 . 9 7 2 8 -3.0000 - 2 . 9 7 2 8 -3.0000 -3 .2795 -3 .2973 - 3 . 0 0 0 0 - 3 . 0 0 0 0 -3 .1545 -3 .1587 - 3 . 0 0 0 0 - 3 . 0 0 0 0 - 2 . 9 4 8 7 -3 .0193 - 3 . 0 0 0 0 -3 . 0000 -2.7800 -2.8446 -3.0000 -3.0000 -2.9790 -2.9870 -3.0000 -3.0000 0 . 0 0 0 1 3 5 0 . 0 0 2 0 5 5 0 . 0 0 2 6 5 3 0 . 0 0 0 1 3 0 .0.000014 0.000144 0.000136 0 . 0 0 . 0 0 0 1 3 7 0 . 0 0 0 1 3 5 0.000004 0 . 0 0 0 1 3 7 0.000138 0.0 0.000138 0 . 0 + 0 . 5 7 9 7 7 2 x 1 0 -4 -2 -0.521667x10 , >e =+0.125277x10-^ >s =-0.585347x10 ^ +0.569242x10 7 -0.621749x10 ^ j, >\ =+0.569979x10 7 >s =-0.615931x10 ^ 0.0 , +0.5776325x10. +0.144304x10 ^ . >e =+0.574015xl0"7; > =+0.100169x10"^ 1-I—1 VJ1 00 Table XIX (continued) F u n c t i o n C o n s t r a i n t -Energy Q S(0) Q e(0) r A 0n None None None None None 7.467483 7.467354 7.467483 7.467354 7.467490 7.467356 7.467490 7.467356 7.467492 7.467359 7.467491 7.467358 7-467495 7.467360 7.467495 7.467360 7.467479 7 . 4 6 7 3 4 8 7.467479 0.2269 0.2297 0.2248 0.2268 0.2289 0.2310 0.2241 0.2264 0.2298 0.2320 0.2240 0.2263 0.2287 0 .2312 0.2246 0.2267 0.2304 0.2360 0.2260 13.7529 13.9168 13.7529 13.9168 13.7510 13.9176 13.7509 13.9176 13.7511 13.9176 13.7511 13.9176 13.7501 13.9178 13.7501 13.9178 13.7521 13.9172 13.7517 -2.9736 - 3 . 0 0 0 0 -2.9736 - 3 . 0 0 0 0 -2.9730 - 3 . 0 0 0 0 -2.9731 - 3 . 0 0 0 0 -2.9731 - 3 . 0 0 0 0 -2.9731 - 3 . 0 0 0 0 -2.9729 - 3 . 0 0 0 0 -2.9729 - 3 . 0 0 0 0 -2.9733 - 3 . 0 0 0 0 -2.9732 -3.0271 -3.0380 - 3 . 0 0 0 0 - 3 . 0 0 0 0 -3.0622 -3.0596 -3.0000 -3.0000 -3.0754 -3.0741 -3.0000 -3.0000 -3.0542 -3.0590 -3.0000 -3.0000 -3.0394 -3.0988 -3.0000 0.000129 0 . 0 0.000129 0.000134 0.0 0.000134 0.000133 0.000001 0.000134 0.000135 0 . 0 0.000135 0.000131 0.0 0.0 +0.563295x10 £ -0.121657x10 y . >* =+0.563403x10 Z > 5=-0.166013xl0 _ : : > 0.0 _ 4 +0.570254x10 J : -0.278766x10 y . >e=+0.570195x10 £ >s=-0.258561x10 y 0.0 +0.569446x10 -0.337136x10 -4 - 5 -4 X=+0.569374x10"£ > =-0.320967x10 ? 0.0 +0.572685x10 -0.239282x10 -4 - 5 >* =+0.572745x10" J > 5 =-0.252666xl0"' ? 0.0 +0.566694x10 -0.487691x10" -4 Lars son's 100 term c o r r e l a t e d f u n c t i o n , see entr y 18 i n Table I 7.478025 0.2313 13.8341

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China 3 19
United States 2 2
City Views Downloads
Tokyo 5 0
Beijing 3 0
Unknown 3 0
Mountain View 1 2
Ashburn 1 0

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