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Cusp conditions and properties at the nucleus of lithium atomic wave functions Chapman, John Alvin 1970

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CUSP CONDITIONS AND PROPERTIES AT THE NUCLEUS OF LITHIUM ATOMIC WAVE FUNCTIONS  by  JOHN ALVIN CHAPMAN B.Sc.  '  ( H o n o u r s ) , U n i v e r s i t y o f V i c t o r i a , 1965  A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  i n t h e Department of Chemistry  We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e required standard'  THE UNIVERSITY OF BRITISH COLUMBIA March, 1970  In p r e s e n t i n g 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  for  an advanced degree at the U n i v e r s i t y of B r i t i s h C o l u m b i a , I agree t h a t the L i b r a r y s h a l l make i t I f u r t h e r agree tha  freely available for  r e f e r e n c e and s t u d y .  p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s  thesis  f o r s c h o l a r l y purposes may be g r a n t e d by the Head of my Department by h i s r e p r e s e n t a t i v e s . of  this thesis for  It  financial  i s understood t h a t c o p y i n g o r  Department  publication  g a i n s h a l l not be a l l o w e d w i t h o u t my  written permission.  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  or  i  ABSTRACT The dependence o f t h e p o i n t p r o p e r t i e s a t t h e n u c l e u s , e l e c t r o n d e n s i t y ( Q ( 0 ) ) and s p i n d e n s i t y e  ( Q ( 0 ) ), on t h e n u c l e a r cusp i s examined f o r l i t h i u m S  a t o m i c c o n f i g u r a t i o n i n t e r a c t i o n ( C I ) wave f u n c t i o n s . S e v e r a l s e r i e s o f C I wave f u n c t i o n s w i t h 18 and fewer terms, a r e s t u d i e d .  Importance o f t h e t r i p l e t  core  spin function to Q (0) i s substantiated. s  N e c e s s a r y , "but n o t 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 a r e 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 . F o r t h e f u n c t i o n s s t u d i e d , Q ( 0 ) i m p r o v e s on a p p l y i n g s  the s p i n cusp c o n s t r a i n t i f t h e f r e e v a r i a t i o n a l  spin  cusp i s g r e a t e r t h a n -Z, b u t becomes worse o t h e r w i s e . 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 Q (0). e  overcorrects  The e f f e c t o f n e c e s s a r y 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 i s a l s o examined. be f o u n d .  No o b v i o u s t r e n d s  could  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 o f terms t o s a t i s f y n e c e s s a r y d i a g o n a l o r o f f - d i a g o n a l i n t e g r a l cusp has  very l i m i t e d usefulness.  conditions  A good Q ( 0 ) c a n be obs  t a i n e d w i t h o u t 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) optimizing 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 a r e d e v e l o p e d 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 f o r m w i t h t h e c o r r e c t cusp f o r each o r b i t a l . with a  small  Sample c a l c u l a t i o n s  b a s i s set are presented.  These  f u n c t i o n s g i v e e x t r e m e l y good Q ( 0 ) e x p e c t a t i o n s  but c o n v e r g e n c e o f Q ( 0 ) w i t h r e s p e c t s  s i z e i s y e t t o be t e s t e d .  simple values  to basis set  The most i n t e r e s t i n g d i s -  c o v e r y i s t h e appearance o f 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 f r o m 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 previous  and  p r e s e n t work on cusp c o n s t r a i n t s i n t e r m s o f n e c e s s i t y and/or s u f f i c i e n c y .  iii  TABLE OF CONTENTS  Page ABSTRACT  i  L I S T OF TABLES  vi v i x i  L I S T OF FIGURES  i  ACKNOWLEDGMENTS CHAPTER I .  x  PRELIMINARIES  1.1  Introduction  1  1.2  O b j e c t o f t h i s work  5  1.3  Survey o f f o l l o w i n g  chapters  6  CHAPTER I I . BACKGROUND 2.1  Review o f S g r o u n d s t a t e l i t h i u m atomic wave f u n c t i o n s and c o n t a c t properties  8  2.2  What a r e cusp and c o a l e s c e n c e conditions?  21  2.3  Theory o f c o a l e s c e n c e c o n d i t i o n s f o r e x a c t wave f u n c t i o n s  24  2.4  Cusp c a l c u l a t i o n methods f o r a p p r o x i m a t e wave f u n c t i o n s  32  2.5  Use o f cusp and c o a l e s c e n c e c o n d i 40 t i o n s f o r improvement o f a p p r o x i m a t e wave f u n c t i o n s  iv Page CHAPTER I I I . INTEGRAL CUSP CONSTRAINTS AND APPLICATIONS TO LITHIUM S GROUND STATE FUNCTIONS 2  3.1  Formation of c o n s t r a i n t s  4-5  3.2  Exploratory  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 with weighting functions  73  CHAPTER I V .  calculations  SUFFICIENT CONDITIONS FOR CORRECT CUSP  4.1  Theory  4.2  Applications to the lithium S ground s t a t e  4.3  A p p l i c a t i o n t o t h e lowest lithium state  119  SUMMARY AND CONCLUDING REMARKS  122  CHAPTER V.  87 2  BIBLIOGRAPHY  103  128  APPENDIX A.  ATOMIC UNITS  133  APPENDIX B.  SOME IMPORTANT TYPES OF APPROXIMATE ATOMIC WAVE FUNCTIONS  135  APPENDIX C.  CONSTRAINED VARIATION  C.l  Introduction  142  C.2  Single constraints  143  C.3  Multiple constraints  146  C.4  Off-diagonal constraints  148  V  Page C.5  APPENDIX D.  Off-diagonal l i n e a r constrained v a r i a t i o n method o f Weber and Handy  149  INTEGRAL CALCULATION  D.l  Primitive integrals f o r Slatertype o r b i t a l s  152  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  154  DESCRIPTIONS AND PROPERTIES OF THE WAVE FUNCTIONS $ THROUGH  157  APPENDIX E.  l0  lis  '  vi  L I S T OF TABLES  Page  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 f r o m t h e l i t e r a t u r e f o r t h e l i t h i u m 2 ground state  9  S  II.  D e s c r i p t i o n and p r o p e r t i e s o f j j j ^  53  III.  Term-wise c o m p a r i s o n o f c o n v e r g e n c e f o r p e r t u r b a t i o n e x p a n s i o n o f 4 E , t h e energys a c r i f i c e from the J ? = Y constraint  54  IV.  A n a l y t i c a l parametrizations c o n s t r a i n t on ^  of  61  V.  Descriptions  o f <j} , <& . ,$/Q  67  VI.  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 t i e s of § , ,  e  /0  /¥  proper-  68  to  VII.  E n e r g y - w e i g h t e d o f f - d i a g o n a l cusp c o n s t r a i n t s on § 79 t ¥  C o m p a r i s o n o f t h e 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: <£>. , £ = Y constraint  82  IX.  Q ( 0 ) , Q (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 o f  85  X.  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.  86  XI.  The s e t o f S l a t e r d e t e r m i n a n t s , f < j > i ] , and e i g e n f u n c t i o n s , \7/t'\ . o f \d* , £. , 1 f o r t h e 2 s wave f u n c t i o n s d e s c r i b e d i n case 2 o f s e c t i o n 4 . 1  97  XII.  A d d i t i o n a l f<« T a b l e X I when included  VIII,  e  / v  S  e  z  and Wc] e l e m e n t s f o r 0^ ^-type terms a r e  100  vii  Table  Page  XIII,  Illustrative calculation: 3 - t e r m CI f u n c t i o n formed from {Xs-} b a s i s  106  XIV.  S i m p l e DODS wave f u n c t i o n s w i t h CSO's  109  XV.  ( 1 ) $£,a t r u e C I 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 t h e c o r e (2) , 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 t h e c o r e  112  XVI.  True C I f u n c t i o n s formed f r o m STO b a s e s l i s t e d i n Table XIV  117  XVII,  C a l c u l a t i o n s on t h e l o w e s t "P s t a t e o f lithium  120  X V I I I . H i e r a r c h y o f n e c e s s a r y 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.  125  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 o f t h e wave f u n c t i o n s (£> t h r o u g h <&/ /0  S  158  viii  L I S T OF FIGURES Figure  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 and e r r o r i n Q ( 0 ) f o r He wave f u n c t i o n s , f o u n d by Chong and S c h r a d e r .  43  2.  ( A ) Graph o f v e r s u s )\ for the ground s t a t e o f ^ i n electronic cusp c o n s t r a i n t  56  e  (B) Graphs o f f i c t i t i o u s e n e r g y , Ef/ct ~ <#*A<?> , and t r u e energy E =<(/// versus A , for the ground s t a t e o f $ 7 i n electronic cusp c o n s t r a i n t T  3.  ( A ) Q ( 0 ) as a f u n c t i o n o f t h e c o n s t r a i n t X = y f o r & , §,+ , $ s  72  s  D  l 8  (B) Q ( 0 ) as a f u n c t i o n o f t h e c o n s t r a i n t e  r  4.  e  =y  for$ , ,§ . ( A ) Q ( 0 ) as a f u n c t i o n o f t h e c o n s t r a i n t JZ e =X' for , ^ , $, l0  / a  s  73  8  (B) Q ( 0 ) as a f u n c t i o n o f t h e c o n s t r a i n t e  r  5.  5  -  f o r §, , ,$ (A) Q ( 0 ) p l o t t e d against (° and 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 y  y  0  t B  s  84  (B) Q ( 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 (£>. e  / l f  6.  E n e r g y c o n t o u r map o f % (Table XIV): E n e r g y v e r s u s exponents Si's , Sis" o f o - t y p e cusp c o r r e c t i o n o r b i t a l s  114  7.  P r o p e r t i e s o f <g ( T a b l e XIV) c o r r e s p o n d - 115 i n g t o e n e r g y c o n t o u r s i n f i g u r e 6.  ix  ACKNOWLEDGMENTS I t h a n k Dr. D. P. Chong f o r h i s a d v i c e and encouragement t h r o u g h o u t my y e a r s a t t h e U n i v e r s i t y of B r i t i s h C o l u m b i a and d u r i n g t h e p r e p a r a t i o n o f this  thesis. F i n a n c i a l w o r r i e s were removed b y a M a c M i l l a 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 R e s e a r c h Council Fellowship  (1968-1969).  Work i n t h e l a s t few months h a s been made much e a s i e r b y my w i f e ,  Leslie.  1  CHAPTER I PRELIMINARIES 1.1  Introduction The p o s t u l a t e s o f quantum m e c h a n i c s s t a t e t h a t  to  d e s c r i b e a system m a t h e m a t i c a l l y one needs t o  d e c i d e on a H a m i l t o n i a n f o r t h a t system  and t h e n t o  solve the corresponding Schrodinger equation. of  Instead  t h e t r u e H a m i l t o n i a n w h i c h c a n r a r e l y be deduced,  one u s e s t h e 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 o f m o l e c u l a r p h y s i c s or  quantum c h e m i s t r y .  Moreover, f o r molecules, t h e  Born-Oppenheimer a p p r o x i m a t i o n i s u s u a l l y used t o parametrize the nuclear coordinates.  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 t h r o u g h o u t this thesis.  Solutions of the r e s u l t i n g  (simplified)  S c h r S d i n g e r e q u a t i o n w i l l be c a l l e d e x a c t .  These  e x a c t wave f u n c t i o n s c o n t a i n a l l t h e i n f o r m a t i o n needed t o c a l c u l a t e any o b s e r v a b l e o f t h e system by u s i n g t h e 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 . a procedure  w i l l y i e l d , what w i l l be c a l l e d ,  Such exact  expectation values. * 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 o f o b s e r v a b l e s a r e sometimes a d j u s t e d t o y i e l d an e x p e r i m e n t a l e s t i m a t e f o r t h e e x a c t v a l u e . F o r example, t h e r e l a t i v i s t i c cont r i b u t i o n t o t h e 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 t h e t r u e energy g i v i n g t h e e x a c t energy.  2  The  c o m p l e x i t y o f 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 o f t h e s i m p l i f i e d  Schrb'dinger  e q u a t i o n i m p o s s i b l e save f o r a s m a l l number o f two and t h r e e p a r t i c l e s y s t e m s . procedures  There a r e two g e n e r a l  t h a t c a n be f o l l o w e d a t t h i s  stage.  E i t h e r t h e H a 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 e x a c t s o l u t i o n o f t h e a p p r o x i m a t e Schrodinger  e q u a t i o n — f o r example, t h e H a r t r e e - F o c k  method, o r a p p r o x i m a t e methods may be u s e d t o s o l v e t h e e x a c t Schrb'dinger e q u a t i o n — f o r example, s e r i e s expansion  of the solution.  a r e combined.  O f t e n t h e two  procedures  The s o l u t i o n s f r o m b o t h r o u t e s a r e  r e f e r r e d t o as a p p r o x i m a t e wave f u n c t i o n s . I n e i t h e r c a s e energy i s a l m o s t  a l w a y s u s e d as  t h e c r i t e r i o n f o r o b t a i n i n g t h e wave f u n c t i o n .  This  i s because t h e r e i s an e a s i l y a p p l i e d minimum energy;, principle.  The t r u e ground s t a t e i s d e t e r m i n e d  u n i q u e l y by t h e l o w e s t e i g e n v a l u e o f t h e e x a c t Hamiltonian.  The energy,  wave f u n c t i o n , ^ as  £  , o f an a p p r o x i m a t e  , a p p r o a c h e s t h e e x a c t energy  becomes more and more s i m i l a r t o t h e e x a c t  wave f u n c t i o n vables also.  .  This i s true f o r other  obser-  But t h e y do n o t n e c e s s a r i l y a p p r o a c h  t h e i r exact values m o n o t o n i c a l l y .  Here l i e s a p r o b l e m :  3  t h e u s e o f energy as a c r i t e r i o n f o r a n a p p r o x i m a t e wave f u n c t i o n does n o t a l w a y s e n s u r e r e l i a b l e  expec-  t a t i o n values of operators other than the Hamiltonian. This i s e s p e c i a l l y true f o r p o i n t - p r o p e r t i e s — t h o s e properties l i k e the hyperfine s p l i t t i n g , density at the nucleus—which  or electron  depend on t h e v a l u e o f  t h e wave f u n c t i o n a t a s i n g l e p o i n t i n s p a c e . 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 a r e two d i s t i n c t a p p r o a c h e s t o t h i s p r o b l e m . First,  e x t r e m e l y a c c u r a t e a p p r o x i m a t e wave f u n c t i o n s  c a n 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  t o be s u r e o f o b t a i n i n g a c c u r a t e 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 a c c u r a t e 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 t h e number o f p a r t i c l e s .  with  Thus o n l y t h e h y d r o g e n ,  h e l i u m and l i t h i u m atoms and t h e  molecule  been d e s c r i b e d w e l l enough f o r t h e a c c u r a t e 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 .  have  prediction  To make r e l i a b l e  e s t i m a t e s o f p r o p e r t i e s o f 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 m e t a l c o m p l e x e s , o r even s m a l l s i z e d o r g a n i c compounds seems o u t o f t h e q u e s t i o n at  present. Now f o r a moment c o n t r a s t t h e s e a c c u r a t e wave  functions with simpler types.  S i m p l e 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-  v a l u e s when d e t e r m i n e d  by t h e energy c r i t e r i o n a r e  e a s y t o c o n s t r u c t and easy t o c a l c u l a t e speaking).  (comparatively  Of c o u r s e 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 t h e number o f p a r t i c l e s , b u t c a n  be l e s s e n e d by c a r e f u l a p p l i c a t i o n o f c h e m i c a l i n tuition—difficult  t o do f o r t h e more c o m p l i c a t e d  a c c u r a t e 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 e x p l o r e d i n t h i s t h e s i s — u t i l i z e s t h e s i m p l i c i t y o f t h e s e s m a l l e r , l e s s a c c u r a t e wave f u n c t i o n s , together with the existence of other  criteria  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 , to  a r r i v e a t a r e l i a b l e method f o r c a l c u l a t i n g  and m o l e c u l a r p r o p e r t i e s .  atomic  These o t h e r 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 o r e x p e r i m e n t a l , w h i c h t h e e x a c t wave f u n c t i o n must e x h i b i t . mechanical  The quantum-  v i r i a l theorem, t h e h y p e r v i r i a l theorems,  t h e c u s p 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 t h e v a n i s h i n g o f n e t f o r c e s a r e c o n d i t i o n s t h a t c a n a i d t h e c h a r a c t e r i z a t i o n o f an a p p r o x i m a t e wave f u n c t i o n . will  N a t u r a l l y o n l y t h e e x a c t wave f u n c t i o n  satisfy a l l possible conditions.  procedure  The u s u a l  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 t h e p r o p e r t y one w i s h e s 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 s e t o f  5 *  conditions  .  The p a r a m e t e r s i n t h e f u n c t i o n a l f o r m  chosen t o approximate the exact wavefunction are minimized with respect  t o energy w h i l e being  to s a t i s f y the desired set of c o n d i t i o n s .  This i s  t h e i d e a b e h i n d t h e quantum-chemical t h e o r y 1.2  constrained  of c o n s t r a i n t s .  O b j e c t o f 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 o f n u 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 s i m p l e  a p p r o x i m a t e wave f u n c t i o n s y i e l d good  point properties at the nucleus.  The l i t h i u m atom  has been c h o s e n as t h e 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)  Several accurate  treatments f o r l i t h i u m  a r e a v a i l a b l e - ? ' ^' ^ (ii)  The system i s a c c e s s i b l e  experimentally.  8, 9, 10, 11, 12 (iii)  The l i t h i u m atom has c o r r e l a t i o n phenomena c h a r a c t e r i s t i c o f more systems b u t i s s i m p l e  complicated  enough t o r e v e a l  t h e r e s u l t s o f t h e method o f c o n s t r a i n t s w i t h o u t undue c o m p u t a t i o n a l p r o b l e m s . Thus, p r o p e r t i e s o f c o n s t r a i n e d , s i m p l e  lithium functions  can be compared w i t h r e s u l t s o f b o t h a c c u r a t e  calculations  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 t o vanish, s a t i s f a c t i o n o f t h e h y p e r v i r i a l theorems; and f o r c i n g t h e c o r r e c t c u s p b e h a v i o r s h o u l d improve c a l c u l a t e d force constants, 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 c o n t a c t 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 e x p e r i m e n t a l l o w i n g a m e a n i n g f u l assessment of t h e u s e f u l n e s s  1.3  of n u c l e a r cusp c o n s t r a i n t s .  iSurvey o f f o l l o w i n g c h a p t e r s Chapter I I c o n s i s t s o f background m a t e r i a l nec-  e s s a r y f o r u n d e r s t a n d i n g why t h e p r e s e n t work was undertaken.  Cusp and c o a l e s e n c e c o n d i t i o n s a r e  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 a p p r o x i m a t e wave functions  explained.  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 a t t h e 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 condi t i o n s a r e presented i n Chapter I I I . are  Several  approaches  described. S u f f i c i e n t conditions f o r ensuring  a correct  cusp a r e d e v e l o p e d and a p p l i e d t o a p p r o x i m a t e l i t h i u m wave f u n c t i o n s i n C h a p t e r I V . C h a p t e r V summarizes t h e work p r e s e n t e d i n C h a p t e r s I I I . a n d I V . 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 and  to necessity  sufficiency i s tabled. A p p e n d i x A i s a l i s t o f a t o m i c 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 o f c e r t a i n b a s i c  t y p e s o f a p p r o x i m a t e wave f u n c t i o n s a r e p r e s e n t e d  7  i n A p p e n d i x B. i n the t e x t .  Knowledge  o f t h e s e t y p e s i s assumed  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 d e t e r m i n e d a p p r o x i m a t e wave f u n c t i o n s a r e o u t l i n e d i n Appendix ,C.  Appendix D  discusses  t h e i n t e g r a l s needed i n t h i s work and A p p e n d i x E c o n t a i n s t h e d e s c r i p t i o n and p r o p e r t i e s o f t h e c o m p l e t e s e r i e s o f 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 o f  S ground  s t a t e l i t h i u m a t o m i c wave  f u n c t i o n s and c o n t a c t p r o p e r t i e s To u n d e r s t a n d t h e r e a s o n s f o r development and a p p l i c a t i o n of s p e c i a l constraint 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 l o o k a t some o f t h e p a s t work on l i t h i u m wave f u n c t i o n s i s n e c e s s a r y .  No attempt  t o c o v e r t h e v a s t l i t e r a t u r e i s made h u t i m p o r t a n t a s p e c t s p e r t a i n i n g t o t h e p r o b l e m w i l l be b r i e f l y discussed.  A c o l l e c t i o n n o f some o f t h e 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 . a r e energy o r d e r e d — t h e b e s t a t t h e b o t t o m . at t h e 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  Entries A glance  so o r d e r e d —  w i t h energy as a c r i t e r i o n o f a c c u r a c y — a r e n o t i n t h e same sequence when t h e s p i n d e n s i t y a t t h e n u c l e u s , Q (0),is a criterion. s  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  b o t h Q ( 0 ) and t h e c o r r e s p o n d i n g e l e c t r o n d e n s i t y , Q ( 0 ) , s  e  Other w o r k e r s — s e e [ 1 3 , 14, 1 5 , 16, 17l — h a v e also tabulated l i t h i u m groundstate c a l c u l a t i o n s from t h e l i t e r a t u r e . References [ 5 , 6, 18, 1 9 , 20, 2 1 , 2 2 , 2 3 , 24, 2 5 , 2 6 j 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 t h e l i t e r ature since Lunell's t a b u l a t i o n [ 1 7 ] i n 1968.  Table I .  Eepresentative  wave f u n c t i o n s from t h e l i t e r a t u r e f o r t h e l i t h i u m  D e s c r i p t i o n of wave f u n c t i o n  Reference  Energy  Spin density at the nucleus  — Q (0) % e r r o r s  1  a  Reference  S groundstate.  Electron density at the nucleus T Q (0) % e r r o r Reference e  0  limited CI, including 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  -7.432727  0.1667  +2(7. 9  29  13 .8203  +0.1  29°  3  UHJ?  29  -7.432751  0.2248  + 2.8  29  13 .8204  +0.1  29°  4  PUHJ? o f S a c h s  17  -7.432768  0.1866  +19.3  17  5  EHF  20  -7.4-32813  0.2412  - 4.3  20  6  open s h e l l , 2 d e t e r m i n ants, e, s p i n function  30  -7.4-436  0.3002  -29.8  31  13 . 5 1 9 3  +2.3  -e  open s h e l l , 3 d e t e r m i n ants, (spin optimized)  32,33  -7.4-436  0.2417  - 4.5  13  13 .5240  +2.2  open s h e l l , SEHJT, 2 d e t e r m i n a n t s , 6, spin function  17  -7.4-47529  0.2055  +11.2  17  -  7  8  2 9  (projected)  d  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 t h e l i t h i u m  S groundstate.  (continued)  Description of wave f u n c t i o n  C9  Reference  e  0  + 2.1  17  21  -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  0.2065  +10.7  13  13.8661  +0.2  36  EHF ( s p i n - o p t i m i z e d )  12  s-type b a s i s , 330 term CI (no r .) ±  s,p-type b a s i s , 310 term CI (no r .) ±  24,25 23 23  s c a l e d 208-term CI (no r . .)with 2 n o n - l i n e a r .10 parameters  34  45 term CI (no r . . ) , STO b a s i s  35  1  16  Referenc e  0.2265  11  15  — QT(O) % e r r o r  Electron density a t the n u c l e u s ~ ir Q (0) % e r r o r Referenc e  -7.447536  Gl-EHF ( p r o j e c t e d )  14  Spin density a t the nucleus  open s h e l l . SEHF, 3 d e t erminants ( s p i n optimized) 17  10  13  Energy  J  15 term c o r r e l a t e d f u n c tion, © i s p i n f u n c t i o n 37  -7.W69  •7.47710 -7.4771  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 t h e l i t e r a t u r e f o r t h e l i t h i u m  S groundstate.  (continued)  Description of wave f u n c t i o n  Spin density at t h e n u c l e u s  Energy  Reference  Q (0) % e r r o r s  a  Reference  Electron density at the nucleus Q (0) % e r r o r e  17  60 t e r m c o r r e l a t e d f u n c tion, ©i s p i n f u n c t i o n  6  -7.478010  0.2405  - 4.0  6  13.8327  18  100 t e r m 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  19  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  39  -7.478069  0.2313  20  Exact (Experimental Q (0) ) s  8  °% e r r o r = Q ( 0 ) ( e x p e r i m e n t a l ) - Q ( 0 ) ( c a l c u l a t e d ) Qs(0) ( e x p e r i m e n t a l )  x  100%  % e r r o r = Q ( 0 ) ( L a r s s o n . #18) - Q. (0) ( c a l c u l a t e d ) Q ( 0 ) ( L a r s s o n , #18)  x  100%  s  b  s  e  e  e  °Calculated from d a t a i n r e f e r e n c e . Calculated e  Calculated  i n t h i s work t o be Q ( 0 ) = 0.2425. s  i n t h e c o u r s e o f t h i s work.  See r e f e r e n c e  [40]  .  f  0.0  b  Refer ence  38 38  12  are evident  f r o m t h e work o f J a c o b s  and L a r s s o n .  J a c o b s s t u d i e d t h e convergence p r o p e r t i e s o f c o n f i g u r a t i o n i n t e r a c t i o n ( C I ) wave f u n c t i o n s and f o u n d e r r a t i c v a l u e s o f Q ( 0 ) f o r L i and Q ( 0 ) f o r s  He f o r v a r i o u s  e  expansions converging i n energy. **  Even L a r s s o n ' s 100 t e r m c o r r e l a t e d f u n c t i o n  , the  most a c c u r a t e l i t h i u m g r o u n d s t a t e 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 a n o t h e r 100 t e r m c o r r e l a t e d f u n c t i o n ,  identical  i n energy, g i v e s a Q ( 0 ) , Q ( 0 ) d i f f e r e n t from t h e e  s  v a l u e s l i s t e d i n T a b l e I , by 0 . 0 7 % and 0.1% r e s p e c t i v e l y . F o r l a r g e r systems e r r o r s i n Q ( 0 ) o f 2 5 - 5 0 % seem s  t o be common.  See, f o r example, t h e c a l c u l a t i o n s  on b o r o n , c a r b o n , 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 t e c h n i q u e f o r  systematic-  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  formation  of a c c u r a t e 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 impossible.  Cusp c o n s t r a i n t s may p r o v i d e  F i r s t of a l l , Q (0), useful? s  f o r what i s t h e s p i n  a method. density,  I t p r o v i d e s an i m p o r t a n t 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 o f v a r i o u s wave f u n c t i o n appr o x i m a t i o n s i s g i v e n i n A p p e n d i x B.  13  to t h e 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 t h e c o u p l i n g o f 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 m a g n e t i c f i e l d s . splitting for  of  The r e s u l t a n t  e n e r g y l e v e l s c a n "be a c c u r a t e l y  a l k a l i m e t a l atoms i n a t o m i c "beam m a g n e t i c  onance e x p e r i m e n t s .  measured res-  E x p e r i m e n t a l r e s u l t s from t h e  a l k a l i m e t a l group c a n be e x p l o i t e d as a check i n d e v e l o p i n g t h e o r e t i c a l t e c h n i q u e s o f f o r m i n g wave functions.  Improved  theoretical"analyses  techniques w i l l then enable o f more complex systems where  e x p e r i m e n t s a r e n o t 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 b u t i o n s from Fermi c o n t a c t , magnetic  contri-  dipole-dipole,  and e l e c t r i c q u a d r u p o l e i n t e r a c t i o n s . o n l y the Fermi c o n t a c t term, d e s c r i b i n g  F o r an S - s t a t e electronic  s p i n i n t e r a c t i o n s a t o r w i t h i n t h e n u c l e u s , i s nonz e r o and  (2.1.1)  where  (2.1.2) See more c o m p 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  a r e t h e magnitudes o f t h e m a g n e t i c moments  o f t h e n u c l e u s 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 t h e n u c l e a r s p i n and <ff£), t h e D i r a c d e l t a  function.  For L i  7  4 7"  =  Substituting =  803.512 Mc/sec  [8].  the accepted values  3-256310  =  nuclear magnetons^,  3/2, yZ/e  =  1.00116  Bohr m a g n e t o n s , i n t o ( 2 . 1 . 1 ) one o b t a i n s t h e e x p e r i m e n t a l spin  density  Q (0)  =  s  The  quantity,  0.231|  Q (0), s  «o  3  i n atomic  i s t h e g r e a t e s t source of e r r o r  i n theoretical hyperfine calculations The  units.  f o r l i g h t atoms.  reason f o r t h i s i s t h e inadequacy o f approximate  wave f u n c t i o n s  to describe i n d e t a i l  correlation  e f f e c t s and c o r e 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 t h e  description  of the instantaneous repulsions  electrons.  Techniques o f f o r m u l a t i n g approximate  wave f u n c t i o n s  among  must a t t e m p t 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 e n e r g y , E  , UUl  1  15 defined (HF)  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  e n e r g y , Eg-p, o f a s y s t e m a n d t h e e x a c t  E  corr  =  ^ F  "  (2.1.3)  E  EJJJ, i s  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 . chosen as a r e f e r e n c e  b e c a u s e t h e HF m e t h o d  short-range i n t e r a c t i o n s completely; each is  a s s u m e d t o move i n a p o t e n t i a l c r e a t e d  movements o f a l l o t h e r e l e c t r o n s .  by  by average  Consequently an forces  (Application of the P a u l i p r i n c i p l e  antisymmetrizing  the function helps  though, a u t o m a t i c a l l y electrons  neglects  electron  e l e c t r o n never experiences d i r e c t r e p u l s i v e i n a HF f u n c t i o n .  energy,  somewhat,  i n c l u d i n g c o r r e l a t i o n between  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 case, the l i t h i u m can  S ground s t a t e .  function  be w r i t t e n  i n terms of f u n c t i o n s the  I t s HF  usual  spin functions  antisymmetrization butions  o f atomic coordinates,  cancel  °^  operator.  exactly,only  , j$  and  (Pc » ,the  Since the I s c o n t r i t h e 2s o r b i t a l  contributes  16  t o the  spin density  Q'*,M  ( r e a l o r b i t a l s assumed)  =  <P£<o),  72%  of the  ( 2  and  p r o v i d e s but  One  might conclude t h a t t h i s r e s u l t i s due  of c o r r e l a t i o n , b u t  - 1  5 )  experimental value (Table to  I)  lack  t h e r e i s another important  effect—  *  core p o l a r i z a t i o n . Exchange f o r c e s electrons  w i t h the  different  spins.  are more a t t r a c t i v e between  same s p i n t h a n e l e c t r o n s The  u n p a i r e d 2s  electron  with thus  exerts  a d i f f e r e n t f o r c e on each c o r e e l e c t r o n  so the  K s h e l l o r b i t a l s s h o u l d a l s o be  that HP  i s t h e r e s h o u l d be  method f o r c e s the ft  a split  But  f u n c t i o n a l form of the  u n r e s t r i c t e d H a r t r e e - F o c k (UHF) icular restriction is  different—  £ shell.  s p i n c o r e o r b i t a l s t o be the  same.  method  and  In  the cxL  and  the  this part-  relaxed,  *  Sometimes r e f e r r e d t o as exchange, or s p i n , 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 t o as the s p i n - p o l a r i z e d H a r t r e e - F o c k method D-73  1?  allowing  p o l a r i z a t i o n of the  unpaired spin. spin  Now  the K s h e l l can  the  contribute  to  the  density;  QlJO) = (Pnfo) and  c o r e o r b i t a l s by  one  HF —>  can  UHF  see  contribution  C2.1.7)  spectacular  i n Table I.  eigenfunction  w i t h the  the  +  improvement from  But % ^  i s no  of the t o t a l s p i n .  1  exists.  Since:.spin  longer  A small  an  quartet  o p e r a t o r s commute  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  since  a  s p i n dependent p r o p e r t y i s b e i n g 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 v i e w p o i n t t h a t  t i o n have sharp s p i n . sharp s p i n i n UHF  a wave f u n c -  Perhaps i t i s t h i s l a c k  functions  that  of  causes tremendous  e r r o r s — e v e n the wrong s i g n — i n s p i n 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 the  spin projected  function—a  function  u n r e s t r i c t e d H a r t r e e - F o c k (PUHF)  pure d o u b l e t s t a t e — i s  obtained ( f o r  o  Li I).  S,  still),but  The  quartet  tribution. to project  An  i t has  a poorer s p i n  component has  density(Table  a non-negligible  con-  improvement t o the PUHF procedure i s  a UHF-type f u n c t i o n  and  t h e n minimize  the  18  energy.  Goddard  , K a l d o r , S c h a e f e r and  have o b t a i n e d r e a s o n a b l e , b u t s t i l l  Harris  erratically  b e h a v i n g 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 m e t h o d — c a l l e d t h e s p i n - e x t e n d e d H a r t r e e - F o c k (SEHF) t e c h Q (0)  nique—to lithium. s e t used i n t h e s e  S  i s dependent upon t h e  calculations.  E x p a n s i o n s s u c h as c o n f i g u r a t i o n (CI) or c o r r e l a t e d  basis  functions  interaction  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 s h e e r t e c h n i c a l p r o b l e m s have p r e vented accurate c a l c u l a t i o n s  on systems l a r g e r  l i t h i u m by means o f t h e s e a p p r o a c h e s . G o l d s t o n e many-body p e r t u r b a t i o n  than  Brueckner-  t h e o r y does p r o v i d e  a well-defined  p r o c e d u r e f o r c a l c u l a t i n g wave  and p r o p e r t i e s  t o any d e s i r e d  accuracy.  functions  However, i t  a l s o becomes u n w i e l d y f o r systems more complex t h a n t h e 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 e f f e c t s i s d i f f i c u l t f o r approximate Radial,  or 'in-out',  functions.  c o r r e l a t i o n c a n appear t o s p l i t  t h e c o r e when s m a l l b a s i s core e l e c t r o n  correlation  sets are used, g i v i n g  one  a slightly different probability dis5  t r i b u t i o n from the o t h e r .  Chang, Pu and Das  i m a t e by t h e many-body p e r t u r b a t i o n  approach  c o r r e l a t i o n and c o r e p o l a r i z a t i o n c o n t r i b u t e  estthat 15%  19 and 80% r e s p e c t i v e l y o f t h e d i f f e r e n c e between t h e HF and e x a c t 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  t o be an i m p o r t a n t a t t r i b u t e t o b u i l d i n t o an imate  function. The  for  approx-  e x i s t e n c e o f two d e g e n e r a t e  spin functions  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  complicates the computational problem.  These f u n c t i o n s  are u s u a l l y designated  (2.1.8)  &,  , corresponding to 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 , triplet  and  core w i t h the doublet valence s h e l l .  , a The  most g e n e r a l t h r e e e l e c t r o n d o u b l e t f u n c t i o n can w r i t t e n as t h e l i n e a r  S = a, e, +• c9,  ' ' ' 2  combination  a  z  .  (2.1.9)  would be e x p e c t e d t o d e s c r i b e a more s t a b l e  c o r e and i n d e e d 14 30 44  a  be  =^ 0.  -  &^  has a s m a l l e f f e c t on  energy.  F o r a f u l l y o p t i m i z e d f u n c t i o n , however,  More i m p o r t a n t i n t h e p r e s e n t c o n t e x t ,  20  &2 h  a  "DT & ^  s  n i~  1  p r o f o u n d e f f e c t on s p i n d e n s i t y  a  6 '  17  24 '  25  wo T*1C  p e r h a p s "by i m p r o v i n g t h e 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 o p e r a t o r  used  i n PUHF o r SEHF f u n c t i o n s f i x e s t h e r a t i o a- /a L  2  t o a value n o t n e c e s s a r i l y t h e b e s t f o r energy o r 25 other p r o p e r t i e s .  Recently  "both L a d n e r 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 spin-optimized Q (0) s  on  SEHF f u n c t i o n s .  may "be seen i n T a b l e I .  Dependence o f The d i f f i c u l t y  need n o t o c c u r i n C I o r c o r r e l a t e d e x p a n s i o n s s i n c e a  l  / / a  2 """ i f f l p l i o i ' k l y o p t i m i z e d s  I f an a c c u r a t e neglect  i n the secular  equations,  spin density i s desired the  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 s h o u l d be g r e a t e s t n e a r t h e n u c l e u s where a n e l e c t r o n has maximum k i n e t i c e n e r g y and hence m i g h t be i m p o r t a n t f o r F e r m i c o n t 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 o f r e l a t i v i s t i c  corrections i s  p r e s e n t e d by T t e r l i k k i s , M a h a n t i and D a s . ^ the Dirac-Hartree-Fock for  the a l k a l i  ^DHF*"^ ~ SEF^^  (DHF) r e l a t i v i s t i c  Solving  equations  s e r i e s enabled the c o r r e c t i o n *°  b  e  determined.  indicate that r e l a t i v i s t i c  Their r e s u l t s  c o r r e c t i o n s are small f o r  l i t h i u m ( 0 . 2 % ) , sodium ( 0 . 7 % ) and p o t a s s i u m ( 2 % ) .  21  Electron density  Q%)  =  ( A  &&)\  (2.1.10)  i s needed t o e x p r e s s t h e i s o m e r 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 i m p o r t a n t ,  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 a r e summed  ( i n c o n t r a s t t o Q ( 0 ) ), t h e b a s i c d i f f i c u l t i e s o f s  c a l c u l a t i n g a point property remain.  Discussion 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 a p p r o x imate f u n c t i o n s , might improve t h e s e p r o p e r t i e s a r e now  presented.  2•2  What a r e cusp and c o a l e s c e n c e Consider  conditions?  the usual ( n o n - r e l a t i v i s t i c ,  independent) Hamiltonian  i n atomic u n i t s  timefora  system o f N c h a r g e d p a r t i c l e s :  H  =7"  +V (2.2.1)  where  T, V, a r e t h e k i n e t i c and p o t e n t i a l e n e r g y , The  zE - , t h e mass and c h a r g e o f t h e i t  Coulomb p o t e n t i a l c o n t a i n s  s e t o f p o i n t s {^=0^ * See A p p e n d i x A.  operators: particle.  s i n g u l a r i t i e s at the  > that i s , at the coalescence  .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 b e l o n g t o H i l b e r t space and must be c o n t i n u o u s ( s a v e f o r a f i n i t e number o f p o i n t s ) , square  integ-  47  r a b l e , and bounded e v e r y w h e r e . • wave f u n c t i o n must be f i n i t e ,  Because an e x a c t  even a t t h e s i n g u l a r .  p o i n t s o f t h e 'Coulomb p o t e n t i a l , i t must f u l f i l t h e 2 c o n d i t i o n s known as c o a l e s c e n c e c o n d i t i o n s .  Coale-  scence c o n d i t i o n s a r e r e f e r r e d t o as cusp c o n d i t i o n s for  t h e case when t h e wave f u n c t i o n has no node a t  the p o i n t of coalescence. To i l l u m i n a t e t h e p r e c e d i n g remarks examine 48  an e x a c t o n e - e l e c t r o n h y d r o g e n i c  y/ (£) /5  =  NGxp(-Zh)  I s wave f u n c t i o n . where N i s t h e  n o r m a l i z a t i o n c o n s t a n t and Z, t h e atomic  number,  i s c o n t i n u o u s everywhere b u t n o t 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 n o t e x i s t a t  r = 0, t h e c o a l e s c e n c e o f t h e e l e c t r o n w i t h t h e nucleus.  However t h e r e i s a c u s p — a  discontinuity  i n t h e s l o p e — d e s c r i b e d by t h e c u s p c o n d i t i o n ,  at  the only point of coalescence ( r = 0).  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  A l l hyd-  relationship  but t h e c o a l e s c e n c e c o n d i t i o n f o r non-s s t a t e s i s  23  trivial.  Any w e l l - b e h a v e d  eigenfunction  i l t o n i a n must s a t i s f y t h e c o a l e s c e n c e The r a t i o  o f a Ham-  conditions.  HlP/Of/ i s t h e n c o n s t a n t and does n o t  c o n t a i n s i n g u l a r i t i e s when 49 50 f u n c t i o n o f H. "  ^  i s an exact eigen-  The f o l l o w i n g d i s t i n c t i o n s  a r e emphasized t o keep t e r m i n o l o g y c l e a r i n t h e r e mainder o f t h e t h e s i s : 1.  A cusp o f a f u n c t i o n , f ( x ) , i s t h e p o i n t , f ( x ) , at a d i s c o n t i n u i t y of the slope, Q  f ' ( x ) , where f ( x ) changes i t s d i r e c t i o n . It i s also associated with a value: r<x Vf(x ). 0  2.  0  A c o a l e s c e n c e c o n d i t i o n i s any mathemati c a l r e l a t i o n s h i p w h i c h an e x a c t wave f u n c t i o n must s a t i s f y a t one o f i t s cusps.  3.  A cusp c o n d i t i o n i s a c a s e o f a c o a l e s c e n c e c o n d i t i o n when t h e wave f u n c t i o n h a s no node a t t h e s i n g u l a r i t y .  4.  Coalescence  i s the s p a t i a l coincidence  o f two o r more p a r t i c l e s .  O n l y t h e 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 t h e p a r t i c l e s a r e b o t h  electrons  i t i s electron-electron coalescence; when one i s an e l e c t r o n , t h e o t h e r a  24  rmcleus, i t i s electron-nucleus, or nuclear  coalescence.  The c u s p and c o a l e s c e n c e and  c o n d i t i o n s f o r molecul-ar  a t o m i c wave f u n c t i o n s w i l l now be r e v i e w e d .  2.3  Theory o f c o a l e s c e n c e  c o n d i t i o n s f o r exact  wave  functions 47 Kato  d e r i v e d , f o r an N - e l e c t r o n ,  spinless,  a t o m i c wave f u n c t i o n , t h e d i f f e r e n t i a l c u s p c o n d i t i o n s  lifter..*.-.*.))  1 Jfj  =  jh=o  for electron-nucleus  -  -z.W°>&>  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 t h e -A  atomic charge of t h e nucleus i n atomic u n i t s ; i s t h e average of  ft  ^  about a s m a l l sphere w i t h  c e n t e r a t t h e 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 Hamiltonian approximation  and t h e h e a v y - n u c l e u s  were used i n K a t o ' s d e r i v a t i o n as w e l l  as t h e a s s u m p t i o n 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 cing.  Steiner^"*" u s e d t h e same a s s u m p t i o n s and o b t a i n e d  cusp c o n d i t i o n s f o r t h e p r o b a b i l i t y o r , e l e c t r o n ,  ''  i2 3 i  25  d e n s i t y — t h e d i a g o n a l element o f t h e f i r s t o r d e r density matrix.  E q u i v a l e n t i n t e g r a t e d forms o f K a t o ' s cusp c o n d i t i o n s 48 g i v e n by B i n g e l , w h o extended them t o m o l e c u l e s 52 and s u b s e q u e n t l y p r o v e d 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 n u c l e u s , n u c l e a r charge system.  iT^  , at the o r i g i n of the coordinate  The v e c t o r a i s not d e t e r m i n e d by t h e Coulomb  s i n g u l a r i t y but has magnitude d e p e n d i n g on t h e c o o r d i n a t e s o f t h e n o n - c o a l e s c i n g p a r t i c l e s and  direction  p a r a l l e l t o t h e e l e c t r i c f i e l d p r o d u c e d by t h e s e p a r t i c l e s <52 B i n g e l 48 a l s o f o u n d cusp c o n d i t i o n s f o r the general f i r s t order density matrix 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 t h e s p h e r i c a l a v e r a g i n g o p e r a t o r needed to  e x p r e s s t h e 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) p r e c l u d e s any p o s s i b i l i t y o f o b t a i n i n g c o a l e s c e n c e c o n d i t i o n s from t h e s e e x p r e s s i o n s 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 t h e s p e c i a l c a s e s o f t h e h e l i u m atom and h y d r o g e n m o l e c u l e have been f o u n d 4-9 50 and d i s c u s s e d by Roothaan and c o w o r k e r s . T h e i r method was  ^W/'y/  ^'  to e x p l i c i t l y c o n s i d e r the  y  ratio,  , f o r t h e e x a c t ( s p i n l e s s ) wave f u n c t i o n .  Among t h e n e c e s s a r y r e l a t i o n s needed t o keep t h i s r a t i o c o n s t a n t a r e c o n d i t i o n s on t h o s e o f Kato  *  s i m i l a r to  H i g h e r o r d e r Coulomb s i n g u l a r i t i e s  ( c o a l e s c e n c e o f more t h a n two p a r t i c l e s ) may  be  examined t h i s way. F o r more c o m p l i c a t e d c a s e s , however, t h i s a p p r o a c h becomes v e r y i n v o l v e d . P a c k p and B y e r s Brown  were t h e f i r s t  to derive r i g o r -  o u s l y e q u a t i o n s 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 o f Roothaan e t a l appear t h e same as K a t o ' s , b u t a d i f f e r e n t s e t o f i n d e p e n d e n t v a r i a b l e s has been u s e d .  27  a l l o w i n g n o n - t r i v i a l c o a l e s c e n c e c o n d i t i o n s as w e l l as cusp c o n d i t i o n s t o be f o u n d .  They a l s o removed  the heavy-nucleus a p p r o x i m a t i o n .  A brief  outline  of t h e i r i n s t r u c t i v e method i s p r e s e n t e d h e r e : The g e n e r a l N - p a r t i c l e S c h r o d i n g e r u s i n g the H a m i l t o n i a n (2.2.1) was o f c o a l e s c e n c e o f two  equation  s o l v e d i n the r e g i o n  particles (labelled  '1' and  '2' f o r c o n v e n i e n c e ) — t h a t i s , i n the m a n i f o l d of  points  ^  and  , some s m a l l p o s i t i v e c o n s t a n t .  €.  —  &  ,  f^jy^&  f o r a l l fij ^= /f^  the s p a c e - f i x e d p o s i t i o n c o o r d i n a t e s Jj  Transforming , ^/l  to  the c e n t e r of mass and r e l a t i v e c o o r d i n a t e s  and  of the two  S  7*?, -h ?r?  p a r t i c l e s , a l l o w e d the S c h r o d i n g e r  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 c o a l e s c e n c e as  Ij/^O  (2.3.5)  i s the L a P l a c i a n o p e r a t o r f o r the v a r i a b l e  £u  ,  ytt/a.  i s the reduced mass,  of the p a r t i c l e s , and  the  charges  c o n t a i n s a l l o t h e r terms  of the H a m i l t o n i a n of o r d e r equal t o or g r e a t e r than  28  zero m  constant.  The g e n e r a l "bounded s o l u t i o n o f  2/= £ 2 where  W  (2  -3  6)  spherical har-  e  For electron-nucleus coalescence, with the  n u c l e u s a t t h e o r i g i n , Jr _ l2  Jj  has form  r'ftJHYttof*'**  = (r-,&,<f>) and t h e  monics.  (2.3-5)  of the electron.  "becomes t h e r a d i u s v e c t o r  Substituting  the d i f f e r e n t i a l e q u a t i o n  (2.3.5)>  (2.3.6)  into  expanding  as a power s e r i e s i n r,  and s o l v i n g , Pack and B y e r s Brown f o u n d a u n i q u e solution, true f o r  1  where  Y  *  =  They d e f i n e d  f  /JL  ^ S ^  (2.5.8)  A  Z,  ^  'z.y^/a.  •  , a parameter r e l a t e d t o t h e nodal *  structure  o f t h e system,  t o be t h e s m a l l e s t v a l u e  The p h y s i c a l meaning o f A as d e f i n e d h e r e i s l o s t when t h e system does n o t have s p h e r i c a l symmetry about t h e c o a l e s c e n c e . E x a m p l e s a r e e l e c t r o n - e l e c t r o n c o a l e s c e n c e i n atoms, o r any t y p e o f c o a l e s c e n c e i n molecules.  2-9 of  z  f o r w h i c h Tf/y,  two p a r t i c l e s .  ^ 0 at the coalescence of the  The e q u i v a l e n t d i f f e r e n t i a l  form  o f e q u a t i o n (2.3.8) i s ^  K? $  ]  _  «  i i  /n i  (2.3.8a)  ^ \  where t h e a n g u l a r average o p e r a t o r , to  /c(-^l  •  j^rn  , i s modified  This equation gives  n o n - t r i v i a l c o a l e s c e n c e c o n d i t i o n s f o r t h e case o f a node a t c o a l e s c e n c e i n c o n t r a s t t o e a r l i e r F o r t h e ( u s u a l ) n o d e l e s s case  ^  approaches.  i s z e r o and K a t o ' s  c u s p c o n d i t i o n s c a n be r e c o v e r e d 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 c o a l e s c e n c e and  Y =  f o r electron-nucleus coalescence—the  same v a l u e t h a t Kato f o u n d , b u t w i t h a mass c o r r e c t i o n to the heavy-nucleus  approximation."* Note a g a i n t h a t **  all  e x a c t s p i n l e s s wave f u n c t i o n s  must have an e x -  p a n s i o n 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 a r e u s e d . See A p p e n d i x A. ** Matsen's ' s p i n l e s s ' wave f u n c t i o n s a r e n o t r e a l l y spinless. S p i n i s r e p r e s e n t e d 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 p e r m u t a t i o n s o f t h e symmetric group t o a s p a t i a l s o l u t i o n o f t h e Schrb'dinger e q u a t i o n . 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 n o t 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 icles.  L e t us r e s t r i c t t h e d i s c u s s i o n now t o t h e  s p e c i f i c case o f N - e l e c t r o n atomic functions.  and m o l e c u l a r wave  I n t h e Born-Oppenheimer, o r  heavy-nucleus,  a p p r o x i m a t i o n t h e n u c l e a r c o o r d i n a t e s do n o t appear e x p l i c i t l y and o n l y t h e 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 t h e P a u l i p r i n c i p l e f o r a system  o f i d e n t i c a l f e r m i o n s must be obeyed l e a d i n g t o a wave f u n c t i o n a n t i s y m m e t r i c w i t h r e s p e c t t o t h e i n t e r change o f any two s e t s o f e l e c t r o n i c c o o r d i n a t e s ) . In 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 o p e r a t o r and an a r b i t r a r y spin-component o p e r a t o r 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 ,  (]5  S)f1  , s h o u l d have s h a r p t o t a l ( e l e c t r o n i c ) s p i n  and a s h a r p s p i n component:  *4  = n$  (2.3.10)  S|  *The e f f e c t s o f n u c l e a r s p i n s on t h e wave f u n c t i o n c a n be i n c l u d e d , i f n e c e s s a r y , as p e r t u r b a t i o n s .  31  <|>s,M =  where  of  ) JE*.?*S»)  i s a function  space and s p i n c o o r d i n a t e s , Jj  each e l e c t r o n .  1  ^s,Mi  s  n o  Si  , of  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 c a n be expanded^  Here  and  55 y  y  * 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 s e t of l i n e a r l y  pendent  spin functions for N electrons.  eigenfunctions of  W  and jfc^z h a v i n g  S(S+1) and M r e s p e c t i v e l y .  inde-  They a r e eigenvalues  The f u n c t i o n s {%J  formed b y symmetric group o p e r a t i o n s on some  are spatial  s o l u t i o n o f t h e S c h r o d i n g e r e q u a t i o n and a r e a l l degenerate energy e i g e n f u n c t i o n s .  S i n c e an exact  spatial  s o l u t i o n s a t i s f i e s the coalescence conditions  (2.3.8)  each %  must a l s o , n e c e s s i t a t i n g t h e s a t i s - -  f a c t i o n o f ( 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  does n o t a f f e c t t h e argument, c o n d i t i o n s o f Kato ( 2 . 3 . 1 ) ,  averaging  the d i f f e r e n t i a l  ( 2 . 3 . 2 ) also  cusp  apply.  These p r e c e d i n g approaches a l l have t h e same general 1.  limitations: They o n l y t r e a t two p a r t i c l e Higher order s i n g u l a r i t i e s ,  coalescence. when  tcj  ^- ^  32  f o r s e v e r a l c ,J occur.  , a r e assumed not t o  Thus t h e b e h a v i o u r o f e x a c t wave  f u n c t i o n s a t '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 2.  j  c ±j]  They r e a l l y t r e a t o n l y t h e symmetric p a r t o f t h e c u s p .  .  spherically A spherical  a v e r a g e o v e r t h e wave f u n c t i o n i n e q u a t i o n s ( 2 . 3 . 1 a ) , ( 2 . 3 . 2 a ) , ( 2 . 3 . 8 ) must be i f relationships involving determined  taken  completely  q u a n t i t i e s are d e s i r e d .  The  a n g u l a r dependence o f t h e :Coulomb cusp p  a r i s e s from t h e o t h e r (N-2) p a r t i c l e s .  SP  '  y  I n a p h y s i c a l sense t h e s e l i m i t a t i o n s a r e n o t severe.  The v a l u e o f t h e a p p r o a c h e s i s t h a t n e c e s s a r y  c o n d i t i o n s f o r t h e b e h a v i o u r o f e x a c t wave f u n c t i o n s ( w i t h s p i n ) a t t h e most i m p o r t a n t 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 a p p r o x i m a t e  wave f u n c t i o n s  Observe t h e 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 c o a l e s c e n c e c o n d i t i o n s g i v e n by e q u a t i o n s ( 2 . 3 . 1 ) , ( 2 . 3 . 2 ) , ( 2 . 3 . 8 ) a r e t h e ones m e n t i o n e d and a p p l i e d i n t h i s t h e s i s . An e x a c t 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 t h e s e c o n d i t i o n s , a l t h o u g h t h e y a r e not t h e o n l y ones. A p p r o x i m a t e wave f u n c t i o n s c a n s a t i s f y them a l s o i n a n e c e s s a r y and/or s u f f i c i e n t way. These a s p e c t s o f 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 t h e t ext.  Wave f u n c t i o n s  (with t h e i r derivatives)  obtained  toy a p p r o x i m a t e methods do n o t n e c e s s a r i l y have t h e same t y p e s o f d i s c o n t i n u i t i e s as t h e c o r r e s p o n d i n g exact f u n c t i o n s .  I f one i s s t r i v i n g t o copy an e x a c t  f u n c t i o n , as i s u s u a l l y t h e c a s e , t h e a p p r o x i m a t i o n 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 t h e correct behaviour at the s i n g u l a r p o i n t s .  Pluvinage^ 47 was among t h e f i r s t t o a p p l y t h i s r e a s o n i n g ; Kato ' *  was t h e f i r s t t o p r o v i d e Coulomb s i n g u l a r i t i e s .  a general  tool  f o r describing  I n t h i s l i g h t i ti s of general  i n t e r e s t t o analyse the importance of the cusp. How might t h e p r o p e r cusp be i m p o r t a n t f o r a p p r o x i m a t e wave f u n c t i o n s ? on e x p e c t a t i o n The  To answer t h i s l o o k a t i t s e f f e c t values.  F i r s t , t h e energy.  e l e c t r o n - e l e c t r o n cusp seems, a t f i r s t  glance,  t o be d i r e c t l y r e l a t e d t o t h e c o r r e l a t i o n p r o b l e m . A p r o p e r d e s c r i p t i o n o f 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 t h e b e h a v i o u r o f t h e wave f u n c t i o n a t e l e c t r o n c o a l e s c e n c e when two e l e c t r o n s a p p r o a c h t h e 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  i n t e r e l e c t r o n i c coordinates,  f'tj  (containing  , e x p l i c i t l y ) converge  more r a p i d l y t h a 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 ( C I ) 35 57 e x p a n s i o n s w i t h o u t r ^ ,^>-" ;rh d i f f e r e n c e could e  *  The absence o f a t r e a t m e n t f o r t h e h i g h e r o r d e r s i n g u l a r i t i e s s h o u l d n o t be t o o s e r i o u s E 4 9 ] . Three body e f f e c t s appear t o be much l e s s i m p o r t a n t t h a n two body i n t e r a c t i o n s i n d e f i n i n g a t o m i c and m o l e c u l a r properties. 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 r e p r e s e n t e l e c t r o n cusps.  exact  CI f u n c t i o n s f o r h e l i u m w i t h a f i n i t e  number o f terms cannot p o s s i b l y ^ ' s i n c e t h e of o n l y even powers of r ^ = o.  tates  f^J«)  Analyses  by G i l b e r t - ^ and  i n any  occurrence  expansion n e c e s s i -  (Compare w i t h ( 2 . 3 . 2 ) Gimarc, Cooney and  ).  Parr,*  however, s u b s c r i b e t h a t adequate d e s c r i p t i o n o f  the  Coulomb h o l e c o n t r i b u t e s more t o c o r r e l a t i o n energy t h a n does p r o p e r cusp b e h a v i o u r .  The  cusp r e g i o n  l i e s i n s i d e t h e e n e r g y - i m p o r t a n t p a r t o f t h e Coulomb hole.  Since electron-nucleus c o n t r i b u t i o n to  a t i o n energy i s n e g l i g i b l e accuracy nucleus  correl-  i t would seem t h a t  of c u s p s , b o t h 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 has  little  t o do w i t h t h e a c c u r a c y  of energy.  There r e m a i n s t h e q u e s t i o n w i t h r e s p e c t t o expectation values.  and a c c u r a c y  other  Recall that there i s l i t t l e  c o n n e c t i o n between t h e a c c u r a c y  of a p p r o x i m a t e energy  of d i f f e r e n t , a p p r o x i m a t e 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 p a r a g r a p h f o r energy may vables.  the  previous  not be v a l i d f o r o t h e r  R e i t e r a t i n g s e c t i o n 1.2,  obser-  t h e o b j e c t of t h i s  t h e s i s w i l l be t o examine t h e r e l a t i o n s h i p of t h e 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  spin densities  Quoted by Gimarc and P a r r • * T h i s c o n c l u s i o n o f C o u l s o n and N e i l s o n f o r t h e c a s e o f h e l i u m was quoted by G i l b e r t [ 5 8 ]  35  at t h e n u c l e u s . 59 problenr  P r e v i o u s work done on t h i s  specific  60 '  w i l l be r e v i e w e d  i n a later  section.  The n e x t s t e p i s t o d e c i d e how t o e v a l u a t e c u s p s f o r a p p r o x i m a t e wave f u n c t i o n s . s a t i s f i e s equations  Obviously i f a f u n c t i o n  ( 2 . 3 . 1 ) , ( 2 . 3 . 2 ) o r (2.3.8) i t  has a p r o p e r c u s p , b u t t h i s a p p r o a c h i s n o t 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 t h e t e d i o u s n e s s  o f t h e a l g e b r a , n o r does i t g i v e an e s t i m a t e o f t h e c l o s e n e s s o f t h e cusp t o t h e c o r r e c t v a l u e  (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 -  consistent f i e l d  (SCF) o r b i t a l s i s w e l l documented.  (However t h e remarks c o n c e r n i n g e l e c t r o n - e l e c t r o n c u s p s i n C I 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 a p p l y t o SCF f u n c t i o n s ; i t i s wave f u n c t i o n w i t h o u t e x p l i c i t  d i f f i c u l t  r.. correlation to  have t h e 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 ) . SCF  An e x a c t  o r b i t a l has t h e g e n e r a l f o r m frm'*-*h -fnA>Ym(e,<P) £  /7  fora  , <£  ,  •  (2.4.1)  a r e t h e u s u a l o r b i t a l quantum numbers;  j Yjlftj a r e t h e s p h e r i c a l h a r m o n i c s .  To s a t i s f y t h e  g e n e r a l c o a l e s c e n c e 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 ti s s u f f i c i e n t that the r a d i a l  36  part of  < P  h  $ > n ,  Y T^0r) , obey 2j  (2.4.2)  Numerical s o l u t i o n s of the Hartree-Fock  (HF) e q u a t i o n s  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 c o n s e q u e n t l y s h o u l d have good cusp v a l u e s . One indication  o f convergence  o f t h e non-exact,  analytical  HF s o l u t i o n s i s t h e c l o s e n e s s o f t h e r a t i o  te±il{3f < h  )  (2.4.2a) op.  a = -Z.  to  Roothaan, Sachs and Weiss  have m e n t i o n e d  t h i s as a n a c c u r a c y t e s t o f t h e i r HF wave f u n c t i o n s i n t h e 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 o f h e l i u m  through  argon. A n o t h e r method f o r cusp e v a l u a t i o n i s due t o Chong^  He changed t h e f o r m o f t h e c o a l e s c e n c e 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 v a l u e s o f SCF o r b i t a l s , c a l l e d i n t e g r a l c o a l e s c e n c e c o n d itions.  I f a n SCF o r b i t a l ,  Png»,(^> ;&)  (  c o r r e c t cusp t h e r a d i a l f u n c t i o n ( 2 . 4 . 1 ) ), must n e c e s s a r i l y  &  -fhjtfH  ,  has a  , (see  satisfy (2.4.3)  o  37  The f o r m u l a  <f(Jr)  =  -r~-,  has been used h e r e , c o r r e c t  n r  f o r t h e s p h e r i c a l l y symmetric  radial function.  i s the Dirac delta function.  Equation (2.4.3) can  be extended ression.  £(£)  e a s i l y t o i n c l u d e t h e f u l l o r b i t a l exp-  The s p i n dependence o f SCF o r b i t a l s l e a d s  t o no p r o b l e m s ;  arguments p r e s e n t e d e a r l i e r c a n imme-  d i a t e l y p e r m i t e q u a t i o n s (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 notation,  now, t h e i n t e g r a l c o a l e s c e n c e 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 z e r o o f t h i s i n t e g r a l w i l l g i v e an e s t i m a t e o f how c l o s e t h e o n e - e l e c t r o n SCF o r b i t a l (Pft£ comes t o h a v i n g t h e p r o p e r cusp b e h a v i o u r a t frt  the nucleus. CI wave f u n c t i o n s need a d i f f e r e n t The r a t i o , ( 2 . 4 . 2 a ) , u s i n g d i f f e r e n t i a l  approach.. coalescence  c o n d i t i o n s may be a l l r i g h t f o r c h e c k i n g t h e cusp o f SCF o r b i t a l s , b u t i t cannot be a p p l i e d t o m a n y - e l e c t r o n CI f u n c t i o n s . Because CI f u n c t i o n s do n o t have t h e For t h e contains taking a are thus  c a s e o f atoms, n o t m o l e c u l e s , e q u a t i o n (2.4.3) a ' p s e u d o - i n t e g r a t i o n ' which o n l y i n v o l v e s limit. Chong's i n t e g r a l c o n d i t i o n s (2.4.4) n e c e s s a r y and s u f f i c i e n t f o r atoms.  38  simple  i n d e p e n d e n t p a r t i c l e i n t e r p r e t a t i o n o f SCF  wave f u n c t i o n s t h e i n t e g r a l c o a l e s c e n c e  conditions  ( 2 . 4 . 4 ) cannot be used d i r e c t l y e i t h e r . has  Chong  been a b l e t o f i n d c o a l e s c e n c e c o n d i t i o n s f o r CI  f u n c t i o n s c o r r e s p o n d i n g t o ( 2 . 4 . 4 ) by g e n e r a l i z i n g t h e cusp r e l a t i o n s o f 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 o f Bingel spin density.  (2.3-4) f o r b o t h e l e c t r o n and  He o b t a i n e d  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  conditions  w h i c h c a n be w r i t t e n c o m p a c t l y as  a = s or e designates respectively.  Y  (2.3.8).  ,  and  , )\  the spin or electron  have t h e same meaning as i n are the one-electron  density operators  conditions  evaluated  gradient  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 .  singlet spin states,  (yS^*)  -  6  escence c o n d i t i o n s become t r i v i a l .  /\  case, and  = 0,  Note t h a t f o r  and t h e s p i n c o a l For the nodeless  are the usual density  i n t e g r a l cusp c o n d i t i o n s a r e e x p r e s s e d .  operators, Spin  density  at t h e nucleus i s g i v e n by  Q (°)  -  S  and  ( T - < Z > = o  (2.4.8)  )  likewise electron density i s  Q (0)  =  e  The  ^>*o")  expressions  (2.4.9)  (2.4.5) a r e n e c e s s a r y r e l a t i o n s  f o r e x a c t wave f u n c t i o n s b u t t h e y a r e e x t r e m e l y u s e f u l i n cusp e v a l u a t i o n f o r any a p p r o x i m a t e f u n c t i o n . H e r e , as b e f o r e , t h e c o r r e c t n e s s o f t h e cusp i s i n d i c a t e d b y t h e v a l u e s o f t h e i n t e g r a l s i n (2.4.5) f o r t h e wave f u n c t i o n b e i n g  examined.  The v a l u e o f t h e  approximat e cusp,  r=(X l) +  < ^ * C T >  .  o = e . ,  i s t o be compared w i t h t h a t o f t h e e x a c t  Y  =  -Z .  (2.4.10)  ,  cusp  (2.4.11)  40  The  a p p r o a c h i n t h i s work i s t o f o r c e t r i a l wave f u n c  t i o n s , i n v a r i o u s ways, t o have  (2.4.12)  Evidence that 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  c o n d i t i o n s f o r im-  Use o f cusp and c o a l e s c e n c e  provement o f a p p r o x i m a t e wave f u n c t i o n s Touched upon i n t h e i n t r o d u c t i o n was t h e f a c t that expectation values of point properties are rather s p e c i a l compared w i t h t h e u s u a l t y p e o f  observable.  They depend on t h e v a l u e o f a wave f u n c t i o n a t a s i n g l e p o i n t and a r e n o t a v e r a g e d o u t o v e r t h e space the system.  surrounding  Thus a wave f u n c t i o n t h a t m i g h t be q u i t e  good when c o n s i d e r e d  t h r o u g h o u t space c o u l d ,  indeed,  be e x c e p t i o n a l l y p o o r a t o r n e a r c e r t a i n p o i n t s . F o r few examples i s t h i s o b s e r v a t i o n more t r u e t h a n the s p i n d e n s i t y at the nucleus. Any  (See T a b l e I ) .  improvement o f t h e a p p r o x i m a t e f u n c t i o n towards  the exact  i n the nuclear r e g i o n should h o p e f u l l y im-  p r o v e s u c h p o i n t p r o p e r t i e s as t h e e l e c t r o n and s p i n densities there.  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 c o r e 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 t h e wave f u n c t i o n and t h e s e may o v e r r i d e any improvement,  41  at the cusp.  But s u r e l y a wave f u n c t i o n s h o u l d have  the c o r r e c t behaviour a t a p o i n t o f n o n - a h a l y t i c i t y , and  surely theoretical conditions  l i k e t h e cusp r e -  l a t i o n s a r e j u s t as v a l i d as minimum e n e r g y 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 function with a correct  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  workers.  Hoothaan, Weiss and K o l o s  50  constructed  c o r r e l a t e d f u n c t i o n s f o r h e l i u m and t h e h y d r o g e n m o l e c u l e w h i c h have t h e 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 c u s p s . Conroy ^ has used s p e c i a l c u s p - s a t i s f y i n g b a s e s i n h i s u n i q u e 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 H u l t h e n o r b i t a l s . K e l l y and E o o t h a a n ^ p r e s e n t e d a t r e a t m e n t t h a t shows how t o 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 a t o m i c SCF o r b i t a l s  will  s a t i s f y the coalescence conditions at the nucleus; m e r e l y u s e t h e s e t o f STO's  Is,  3 s , 4s,.;...; 2p, 4p, 5p,...; 3 d , 5d, 6 d ,  where t h e f i r s t  (2.5.1)  o r b i t a l o f any a n g u l a r symmetry  ( Ji = n-1) has f i x e d o r b i t a l exponent  n  (2.5.2)  42  t h e members  | 2 s , 3p, 4 d , . . . j  a r e n o t p r e s e n t , and  a l l o t h e r exponents a r e f r e e t o be v a r i e d .  Any a t o m i c  SCF o r b i t a l e x p r e s s e d as a l i n e a r c o m b i n a t i o n o f members f r o m t h i s s p e c i a l s e t w i l l a u t o m a t i c a l l y have t h e c o r r e c t behaviour at the nuclear  cusp.' T h i s c h o i c e o f  b a s i s i s becoming q u i t e n o r m a l i n SCF-type a t o m i c c a l c u l a t i o n s . 21 » 65 66 » 67 < 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  a p p r o a c h o f Handy, P a r r and Weber  b a s e d on t h e i r  e l e g a n t c o n s t r a i n t p r o c e d u r e ( [ 693 — b u t i s discussed  i n Chapter IV.  and A p p e n d i x C) The a s s u m p t i o n  t h a t a b e t t e r wave f u n c t i o n i s o b t a i n e d  i s the only  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 a p p r o x i m a t e wave f u n c t i o n s . W i t h a v i e w towards c l a r i f y i n g t h e 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 wave f u n c t i o n ? ' , Chong and S c h r a d e r  better  examined t h e  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 h e l i u m wave f u n c t i o n s r a n g i n g f r o m s i m p l e SCF t o h i g h l y c o r r e l a t e d ones.  They  discovered  a s t r o n g c o r r e l a t i o n between t h e e r r o r i n t h e n u c l e a r cusp and t h e e r r o r i n t h e e l e c t r o n d e n s i t y a t t h e nucleus.  See f i g u r e 1.  When compared w i t h t h e low  degree o f c o r r e l a t i o n between t h e e r r o r i n t h e cusp and t h e a c c u r a c y o f energy, t h i s r e s u l t becomes i m p o r t a n t  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 o f t h e 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 ( 0 ) .  A high  e  corre-  l a t i o n between t h e e l e c t r o n - e l e c t r o n cuspc'and t h e expectation value  was  a l s o f o u n d but  here  a l a r g e r c o r r e s p o n d e n c e between t h e cusp v a l u e  and  t h e e n e r g y 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 for  the p o i n t  invalid  density 59  H a v i n g t h i s j u s t i f i c a t i o n Chong and the theory  applied  o f l i n e a r c o n s t r a i n t s ( A p p e n d i x C) t o  h e l i u m CI ( w i t h o u t  r . .) f u n c t i o n s o f 3-8  the i d e a of f o r c i n g simple,  The  A fairly  terms w i t h  conclusions  S c h r a d e r ^ are q u a l i t a t i v e l y  i n t h e i r study.  various  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 ) . Chong and  Yue^  of  substantiated  f l e x i b l e f u n c t i o n was  found  t o be n e c e s s a r y t o a b s o r b t h e e f f e c t o f t h e 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 _C  cusp c o n s t r a i n t ,  =  Y  , (see the t e x t  the preceding  (2.4.12) ), w h i l e i m p r o v i n g Q ( 0 ) , l e d t o a s l i g h t e  over-correction. ^r"~ ^ 2  Unfortunately,  , ^ r ~ ^  as might be  e x p e c t e d f r o m an improvement of t h e  with respect  values  d i d not d e c r e a s e upon c o n s t r a i n t  t i o n near the nucleus. Yue  e r r o r s i n the  T h i s a p p r o a c h of Chong  funcand  t o cusp s a t i s f a c t i o n , however, i s  u n i q u e and needs f u r t h e r i n v e s t i g a t i o n .  It will  be  u s e d i n t h e n e x t c h a p t e r on l i t h i u m wave f u n c t i o n s t o c h e c k p o s s i b l e improvements i n b o t h Q ( 0 ) e  and  Q (0). s  4-5  CHAPTER I I I  INTEGRAL CUSP CONSTRAINTS AND APPLICATIONS TO LITHIUM S GROUNDSTATE FUNCTIONS 2  3.1  Formation of constraints The  and  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  e r r o r i n Q ( 0 ) f o r a p p r o x i m a t e h e l i u m wave f u n c t i o n s ^ , e  discussed  i n S e c t i o n 2.5, l e d t o t h e d i s c o v e r y  that  when a He C I 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 i m p r o v e d ^ ,  p r o v i d i n g t h e f u n c t i o n had enough l i n e a r p a r a m e t e r s t o absorb t h e e f f e c t o f c o n s t r a i n i n g .  Attempts t o  p  substantiate  t h e s e r e s u l t s for.: L i  wave f u n c t i o n s  are described  S groundstate  i n t h i s chapter.  Of p r i m -  ary i n t e r e s t though i s t h e a d d i t i o n a l p o s s i b i l i t y o f c o r r e l a t i o n s between Q ( 0 ) and _C o r _C 59 60 s  the  e  s t a t e s o f h e l i u m examined-"' Because JZ*  sity.  S  , since  have no s p i n den-  need n o t e q u a l JI  f o r approximate  wave f u n c t i o n s , t h e e f f e c t o f a d o u b l e c o n s t r a i n t , r  .  I  s  'weighting' well.  . -z i s t e s t e d .  Various  off-diagonal  c o n s t r a i n t s a r e d e v e l o p e d and a p p l i e d as  4-6  Note t h e 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 a r e o n l y conditions  necessary  and t h e f a c t t h a t J^*'  = -Z  5  does n o t mean t h e f u n c t i o n has t h e c o r r e c t cusp. (2)  Since  i s a r a t i o , constraining i t  does n o t d i c t a t e a v a l u e f o r Q (3)  e , s  (0).  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.  In constrained the  v a r i a t i o n one w i s h e s t o m i n i m i z e  e n e r g y o f a n a p p r o x i m a t e wave f u n c t i o n  t o a c e r t a i n number ( k ) o f c o n s t r a i n t s .  subject  The b a s i c  procedure i s t o define c o n s t r a i n t operators,  &'  d e s c r i b i n g t h e 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 t h a t  when t h e c o n s t r a i n t s a r e s a t i s f i e d , ^Gt) each / . the  The m o d i f i e d  c  ,  = 0 for  v a r i a t i o n a l p r i n c i p l e takes  form  (Sy\\H+Z.*iGi-Ec}lp)  =o  (j.i.i)  where H i s t h e H a m i l t o n i a n f o r t h e s y s t e m , E energy o f t h e c o n s t r a i n e d  eigenfunction  i s the  and }\' , c  47  t h e Lagrange m u l t i p l i e r s , a r e t o be d e t e r m i n e d . t e r m s i n g l e c o n s t r a i n t means K = 1 . i m p l i e s a double  constraint.  Likewise K = 2  Appendix C c o n t a i n s a  summary o f methods t o s o l v e ( 3 . 1 . 1 ) . 59 and Chong and Y u e ^  y  The  Following Chong^'  t h e c o n s t r a i n t o p e r a t o r s employed  i n t h i s work f o r t h e n u c l e a r c u s p c o n d i t i o n have t h e form  ^-k(P" where  -P ) , at  y ? * = ^ £ *  — Xc^  (3.1.2)  and J s ^ ,  =0  are d e f i n e d i n equations ( 2 * 4 . 7 ) . /  When such a c o n s -  = 0 implies  t r a i n t i s imposed,  as d e s i r e d .  a^e.s  The c o n s t r a i n e d v a r i a t i o n a l  solution  of ( 3 - 1 . 1 ) i s the e i g e n v e c t o r having the lowest value o f the m a t r i x r e p r e s e n t a t i o n o f the  eigen-  fictitious  Hamiltonian  4  f  -H+±\&  and s a t i s f y i n g ((^,;) = 0 . The  o p e r a t o r o<_  the form f o r  - -«  (5 I  Thus (&i  i s H e r m i t i a n but in (3.1.2).  must be H e r m i t i a n . i s n o t ; hence  This i s not a unique  48  choice. of  A whole h i e r a r c h y  ^  of Hermitian combinations  and st£f w i l l a l s o l e a d t o ( 3 . 1 . 3 ) ,  for  example.  The r e s u l t s o b t a i n e d  are  e x p e c t e d t o be q u a l i t a t i v e l y t h e same. A diagonal  and  from d i f f e r e n t  constraint results i nthe condition  i s b e s t s o l v e d u s i n g t h e w e l l - d e v e l o p e d methods  of B y e r s Brown, Chong and R a s i e l . diagonal  $  '  7 1  ' ?  2  An o f f -  constraint,  (0l<5lty> where  7 0  =0  (3.1.6)  ,  may be a n e x c i t e d s t a t e o r even an ( a l m o s t )  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 b y 69 the r e c e n t l y p u b l i s h e d  method o f Weber and Handy.  A l l wave f u n c t i o n s i n t h i s work a r e 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  normalization.  The u s u a l S l a t e r - t y p e o r b i t a l (STO) o n e - e l e c t r o n  X.r, 0-,©,<M 8rn  =A/»/~''-'e- ~)snfe <l>) Tf  )  ,  basis,  (3.1.7)  4-9  i s a l w a y s employed.  The o r b i t a l e x p o n e n t s , S  »  i f v a r i e d , are s u c c e s s i v e l y optimized 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 e n e r g y .  One i t e r a t i o n c y c l e  i s c o m p l e t e d when a l l exponents have been once.  optimized  U s u a l l y two o r t h r e e 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 t h e i n i t i a l exponents a r e r e a s o n a b l e . &^  estimates  of the  Since the s p i n f u n c t i o n  i n (2.1.8) does n o t c o n t r i b u t e  significantly  t o w a r d s e n e r g y , i t i s n o t i n c l u d e d u n t i l a f t e r exponent o p t i m i z a t i o n . designated  Terms c o n t a i n i n g 0*_  ©i-type t e r m s , o r t r i p l e t c o r e  w i l l be spin  terms. To s o l v e (3.1.1) a t r a n s f o r m a t i o n o f a l l m a t r i c e s f r o m c o n f i g u r a t i o n a l space t o t h e b a s i s o f f r e e v a r i a t i o n a l eigenfunctions [§{j two  i s advantageous f o r  reasons: (1)  An o r t h o n o r m a l 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 transformation  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 t h e f r e e v a r i a t i o n a l g r o u n d s t a t e function with small  eigen-  ' 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 o f t h e imposed c o n s t r a i n t c a n be e s t i m a t e d by e i t h e r t h e r a t e o f c o n v e r g e n c e t o w a r d s t h e c o r r e c t constrained  ZlE  3.2  =  f u n c t i o n , o r t h e energy  E  c  Exploratory  -  sacrifice  E„ ... free v a r i a t i o n a l 1  calculations  I t was hoped a t t h e s t a r t o f t h i s work t h a t o n a b l e s p i n d e n s i t i e s c o u l d be o b t a i n e d  reas-  m e r e l y by c o n -  s t r a i n i n g any s i m p l e f u n c t i o n t o s a t i s f y t h e 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  a p p r o a c h was t a k e n . did  The f i r s t  initial  w a v e J f u n c t i o n s examined  n o t l e a d t o unambiguous c o n c l u s i o n s .  however, i l l u s t r a t e t h e c o m p u t a t i o n a l  They d i d ,  difficulties  e n c o u n t e r e d and i n d i c a t e d a more r e f i n e d a p p r o a c h 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 t y p e s o f f u n c t i o n s were d e v e l o p e d f o r t h i s initial  study.  The f i r s t  t y p e was c o m p r i s e d o f a  s e r i e s o f f u n c t i o n s h a v i n g 4 — 8 terms and p a r t i a l l y (not completely) optimized  o r b i t a l exponents.  7 and 8 t e r m 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 c o r e  The spin  terms and some p - t y p e a n g u l a r c o r r e l a t i o n , a r e a c t u a l l y q u i t e good i n s p i t e o f t h e i r s i m p l i c i t y b u t a r e n o t f l e x i b l e engugh f o r a m e a n i n g f u l s t u d y on t h e e f f e c t of c o n s t r a i n i n g .  The second t y p e ,  a series of 10—15  51  t e r m s , w i t h a n i n c r e a s i n g number o f t r i p l e t c o r e s p i n t e r m s , and w i t h p - t y p e c o r r e l a t i o n , h a d o r b i t a l expo35 from t h e L i C I f u n c t i o n s o f W e i s s . '  nents transplanted  T h i s group i s p o o r i n d e e d w i t h r e s p e c t  y  t o energy, b u t  s l i g h t l y more f l e x i b l e than' t h e f i r s t s e r i e s . Several d i f f e r e n t attempts t o solve  (3.1.1)  s i n g l e c o n s t r a i n t s a r e now d i s c u s s e d w i t h  for  , the  s e v e n t e r m f u n c t i o n from t h e f i r s t g r o u p , as an example.  I t became n e c e s s a r y t o i n v e s t i g a t e t h i s a s p e c t 70  when t h e p e r t u r b a t i o n a p p r o a c h ' i n i t i a l value t o \ several functions.  f a i l e d t o give an  , t h e Lagrange m u l t i p l i e r , f o r Series divergences,  exponent o v e r -  flows, 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 o f attempting to c o n s t r a i n i n f l e x i b l e functions using 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 o f which functions  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 o f s o l v i n g The  (3.1.1)  i s helpful.  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 o f 7^  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 II.  i s l i s t e d i n Table  A l t h o u g h Q ( 0 ) shows a 2% improvement, t h e energy e  sacrifice,  A E, f o r t h e c o n s t r a i n t  the strange  value  JT  i tprovides  i s high;  s  Contrast  with the corresponding  The f u n c t i o n , (1/  case  = If  of Q (0) also indicates the s e v e r i t y  of t h i s c o n s t r a i n t .  but  , with  a good t e s t c a s e .  , i satypical  These cusp c o n s t r a i n t s  52  provide the f i r s t  example of f a i l u r e of t h e  perturb-  a t i o n ^approach; t h e d e s i r e d b e h a v i o u r i s i l l u s t r a t e d i n T a b l e 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 t e r m h e l i u m wave f u n c t i o n , and  (ft  7  , of Yue  compared w i t h t h e b e h a v i o u r o f  and  Chong^  (Li).  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 a r e c a l c u l a t e d from f o r d i f f e r e n t v a l u e s of » a root  )iopt  A  (3.1.1)  u n t i l , f o r some optimum,  of  CO) = (yl£/%y=<<S>=0 where ffif/l) i s f o u n d f r o m  (3.1.1).  At  ( . .!) 2  )\  =  0  (3.2.2)  p r o b l e m i s t h a t u n l e s s one  ^opt  >  approximately  c a n be d i f f i c u l t t o l o c a t e .  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 plotted against for ^ This  ,  £  2  \ p-t  C()l+:)~0  The  A = Q  To  (3.2.1)  knows understand  C.(\)  for several functions.  The  was  curve  , i s shown i n f i g u r e 2 ( A ) .  ' t i t r a t i o n ' c u r v e i s t y p i c a l f o r any c o n s t r a i n t  as c a n 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)  for  A-f/ct^^C  and  )) >  0 and  )\^<  0.  When  0,  i s the e i g e n f u n c t i o n having v a l u e £f J)[) /c  9  /\ C/  » C,  the lowest  eigen-  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 o f  STO b a s i s : Exponent:  3.298  Configurations:  X  2.068  / 5  Linear coefficients: f o r the groundstate  PropPree erties: Variational  7.466022  -£ AE  Xis  X «  7C*p  X  0.433  0.639  3.992  1.090  Xis X/s ©1 > X\s X i / X i s  > X,sX/s Xzs©,,  Single  Constraints  7.340290  7.466004  0.125732  0.000018 -2  -0.99947345x10"  Q (0)  0.22702  2.40254  0.18609  Q (0)  13.45413  13.92203  13.45457  -2.89513  -3.00000  -2.89500  -3.12497  -3.81128  -3.00000  B  r e  35  -0.108016, 0.004854, 0.540761, -0.004979, -0.030853, +0.007831, +0.092132  0.80683335x10  >\  ^  54  Table I I I .  Term-wise c o m p a r i s o n o f c o n v e r g e n c e f o r p e r t u r b a t i o n e x p a n s i o n o f A E, t h e energy s a c r i f i c e f r o m t h e _L7 = Y constraint e  4?-,[59] f o r h e l i u m  2ffor lithium  1.021239x10 -3  0.806833x10 -2  (1)  3.749950x10"  1.43056x10  E (2)  -1.855863x10"  -2.56846x10  E (3) E (4)  -1.26147x10 -7 -10 -9.4319x10"  -1.91646x10 -3  E (5) E (6)  -6.7909x10  Correct  A  Order o f contribution 5  E (7) Sum o f contributions to .7^ order Correct  -12  -4.699xlO~ -3.109x10  ZEW=  M  -16  -3  -1.68385x10 -3 -1.47305x10"  -1.28706x10 -3 -1.12355x10 - 3  1.881378x10-5  1.881378x10 .-5  -1  0.13300  0.12573  55  eigenvalue wnen>\«  of 0,  where C>  C ^  k  icy^ (^)f^y ~~ '' c  . t  Thus  and  ^ - \ \ l C  —\\\C  n  i s the highest eigenvalue  of &  .  Thus  <m)lCtW> =c .  /y~w  H  These r e l a t i o n s a r e c o m p a c t l y d e s c r i b e d  — C,  C (+<*>)  (T^-o> )=c  ;  <  A7  i n figure 2(A) also.  and a r e i l l u s t r a t e d  w h i c h a p p e a r s t o cause d i f f i c u l t y (>500)  Great  o f Cfo)  t o A.  of  The f e a t u r e  i s t h e extreme s l o p e  as i t crosses t h e A  sensitivity  (3.2.3)  ,  a x i s a t Aopt  m i g h t be a n t i c i p a t e d  from t h e nature o f t h e curve, and i s found;  A oft  must be computed t o 5 - 8 f i g u r e s t o e n s u r e a s m a l l v a l u e (^ 1 0 " ^ ) for  .  not n e a r l y so s e n s i t i v e . i s the lowest  eigenvalue  /f^  when  The f i c t i t i o u s  energy  of the f i c t i t i o u s  H  E-f J^ le  Hamiltonian (3.2.4)  ) = A opt  where £~#* tion.  =  The c o n s t r a i n e d energy i s  e  i s t h e r e a l energy o f t h e c o n s t r a i n e d f u n c -  £--fi tf)j) c  and/fy^p/^j  are p l o t t e d i n figure 2(B)  56 F i g u r e 2. (A)  Graph'of <£> s t a t e of 2^  versus A f o r t h e ground i n e l e c t r o n i c cusp c o n s t r a i n t  cH Sxtr-etns Eigenvalues of ((D) $7 last's  A*fo  (B)  / n  Graphs o f f i c t i t i o u s energy,^>*t =(.H+)i<2)> and t r u e e n e r g y , E v e r s u s /\ , f o r ground s t a t e o f ^ i n electronic cusp c o n s t r a i n t . T  -7.0  1  1  1  1  1 +3  1  -75* £>v.r  —8.0  1 -/  1 ^7  ,  57  for  the  constraint of ^  i s a monotonically  .  Because  decreasing curve ( e a s i l y proved  f r o m r e s u l t s o f t h e p e r t u r b a t i o n a p p r o a c h i n Appendix E  opt  C) i t has b u t one z e r o ,  true  =  E  fict'  d  e  fi  n  i  n  S  , at which point  "the c o n s t r a i n e d ground s t a t e .  As s e e n i n f i g u r e 2 ( B ) , ^  = fi t E  t r u e  w  h  e  n  c  A  =  0  as w e l l . An a n a l y t i c a l a p p r o x i m a t i o n  of the ' t i t r a t i o n '  c u r v e c o u l d p r o v i d e what t h e p e r t u r b a t i o n a p p r o a c h failed to—the  i n i t i a l estimate of  Aopt  .  Several  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 r e a d i l y o b t a i n a b l e p a r a m e t e r s t h e extreme e i g e n v a l u e s o f the m a t r i x r e p r e s e n t a t i o n of  , the free v a r i a t i o n a l  e x p e c t a t i o n v a l u e ( ^ r p and v a r i o u s d e r i v a t i v e s o f Ch)  at  )i = 0.  They a r e d e s c r i b e d h e r e f o r t h e  p o s s i b l e u s e and enjoyment o f o t h e r s d o i n g variations. The b a s i c p a r a m e t e r s a r e d e f i n e d :  ' ( See C(-co) j v  $  =  C  =  0  = Yo)  (3.M)  C ( o )  c  =£<E  rs)  constrained  58  C (0) '"f-jyrj K  where  , and £  )i=o  i s the  C k )  *  M  o r d e r 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 t h e f i c t (3.2.4).  i t i o u s Hamiltonian  R  =  (ot-fl/a.  Define  ;  M  = fe +  P)/Z  and t h e r e d u c e d q u a n t i t i e s C  M  m  R  R ;  =  =  R  -B  ;  b  =  "R  _D  ;  d  =  _  R  Then t h e g e n e r a l f u n c t i o n a l f o r m  has a z e r o a t  y  /\ i  o  (l+c-m)(l-c+m) l n ( l + c - m ) ( l + m ) 2b (l-c+m)(l-m)  =  *  //\\ , N i/Hi]=^.^./; R  O Q  I f t h e f i c t i t i o n a l energy i s expanded i n t h e p e r t u r b a t i o n s e r i e s ( s e e t h e p e r t u r b a t i o n a p p r o a c h i n Appendix C)  "(0)  C  =f9*E ) M  3  '} = 0  59  I f t h e v a l u e o f t h e 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  ~  AfA-L)  M-Rarcta.n  (3.2.8)  has a z e r o a t  =  °  1  b  cos(c-m) s i n c e cos m  v  (BI)=(3.2.9)  I f C f e j i s u s e d i n s t e a d o f C fc7  A..  .  * ]/sin 2(c-a?  (8^-(3.2.10)  | i g _ g  C u b i c f a c t o r s as arguments o f t h e t a n h o r a r c t a n f u n c t i o n s were t r i e d a l s o .  The g e n e r a l form f o r t h e s e  curves i s CM  ^  M-Rtanh  ^[ft-L) +a] 3  ;  a,A,L,constants (C) =  (3.2.11)  60  and  M  C(fi  -  R a r c t a n kW\-L?  (D)*(3.2.12)  + *l  where R i s d e f i n e d as i n (3.2.6) o r (3.2.8) f o r t a n h or a r c t a n f u n c t i o n s r e s p e c t i v e l y . r e s u l t s obtained straint.  Table  IV g i v e s t h e  f o r the  from  con-  The parameters used f o r each e v a l u a t i o n a r e  g i v e n i n t h e 'Parameter' column.  An  estimate  should have a t l e a s t t h e c o r r e c t o r d e r o f magnitude t o be h e l p f u l .  Remember t h a t t h i s i l l u s t r a t i v e  i s p a t h o l o g i c a l ; l e s s severe (  X"  = Y  case  c o n s t r a i n t problems  f o r example) c a n be s o l v e d e a s i l y  with  t h e s e methods or any o t h e r s . One o t h e r e x o t i c a p p l i c a t i o n o f t h e p a r a m e t r i z a t i o n approach was t r i e d .  S i n c e t h e convergence o f  the p e r t u r b a t i o n s e r i e s ( s e e Table (/){  , the coordinate  I I I ) depends on  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 o f the estimates  just discussed.  The t r a n s f o r m a t i o n  can be seen as f o l l o w s :  where  i s a new p e r t u r b a t i o n parameter.  However  61  Table IV.  Equation l A A  2  A n a l y t i c a l parametrizations of c o n s t r a i n t on ^  P a r a m e t e r s employed  ^  a 0  C(+oo ) , C ( 0 ) , C ' ( 0 ) ,  0.05A-  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  C( + <?o ), C ( 0 ) , C'(0)  0.02  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)  0.059  D  C( + c?o ), C ( 0 ) , C"(0)  (a=0)  0.017  B  l  B  2  a  ),  The correct value i s  ^oj>t =  0.0080683335.  62  t h e new  perturbation  s e r i e s d i d n o t always  converge  r a p i d l y enough. Having discarded  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 parametrization  t h e method f i n a l l y  by  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 . ( E v e n r e g u l a f a l s i was n o t s u f f i c i e n t by i t s e l f t h e s t e e p n e s s o f t h e c u r v e ( 3 . 2 . 1 ) a t )io t r  led  to impossibly  slow c o n v e r g e n c e s ) . 72  p e r t u r b a t i o n - i t e r a t i o n method' J7  = y  was  because  sometimes  Chong's f a s t  employed f o r most  t y p e c o n s t r a i n t s s i n c e t h e s e 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 conclusions  n o t e d now  wave f u n c t i o n s results Q (0) s  was  was  or expectation  values.  Except f o r Larsson's  £5*-type terms  on  n o t w e l l documented a t t h e t i m e t h i s work  Q (0) s  CI f u n c t i o n s . ^  without f u r t h e r d e s c r i p t i o n of  t h e profound^ e f f e c t o f  started.  improves  s t u d y came some p r e l i m i n a r y  I t appears t h a t i n c l u s i o n o f t h e s e only f o r p a r t i a l l y or f u l l y  overcorrected  optimized  (See t h e e x c e l l e n t v a l u e o f Q ( 0 ) s  i n Table I I ) . Q (0) e  £  €  =  Y  terms  for  constraints invariably  often leaving a s i m i l a r error of  opposite sign, while  constraints  yielded  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 values. A paradoxical  situation exists.  Optimized (with  respect  63  to  e n e r g y ) CI f u n c t i o n s h a v i n g  an a p p r e c i a b l e  @a.-type terms g i v e good Q ( 0 )  tage of  percen-  values,  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 i n any c a s e .  The  Q (0) s  i d e a o f c o n s t r a i n e d v a r i a t i o n does  not appear t o 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 c a s e , however, a r e u n p r a c t i c a l t o c o n s t r u c t and t h u s do n o t p r o v i d e a good r o u t e t o a c c u r a t e calculations. Q (0)  So a f u r t h e r a t t e m p t was  Q (0) s  made t o show  improvement w i t h c u s p c o n s t r a i n t s u s i n g method-  s  ically  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 t o a b s o r b t h e c o n s t r a i n t , p a r t i a l l y optimized w i t h respect to important t h e y must c o n t a i n  3.3  Systematic  -type  exponents,  terms.  study  CI f u n c t i o n s now  discussed provide a  d e s c r i p t i o n o f b o t h c o r e p o l a r i z a t i o n and effects.  I t s h o u l d not be t o o d i f f i c u l t  s i m i l a r f u n c t i o n s capable and  reasonable correlation to obtain  o f g i v i n g good s p i n d e n s i t i e s  o t h e r p r o p e r t i e s f o r a t l e a s t t h e f i r s t row  ments.  The  and  main purpose here,  ele-  again, i s to t r y to  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 c u s p c o n s t r a i n t s and  spin densities.  p r o c e e d e d as f o l l o w s :  The  f o r m a t i o n of these  CI f u n c t i o n s  64-  p  A ,.:5 f u n c t i o n f o r g r o u n d s t a t e  l i t h i u m was r e p -  r e s e n t e d as  (3.3.1)  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 , and t h e l i k e a n a l y t i c a l Hartree-Fock  orbitals.  s ,  These o r b i t a l s  were l i n e a r l y expanded i n terms o f STO's,  <p„ = a, X  2.S  a  35  The n o t a t i o n i s s t r a i g h t f o r w a r d ; STO's w i t h t h e same o r b i t a l a n g u l a r momentum quantum number ( £. ) and d e s i g n a t e d w i t h t h e same p r i m e (') have i d e n t i c a l o r b i t a l exponents.  $  was expanded as a C I f u n c t i o n  ( w i t h t h e accompanying p r o d u c t s o f  <%i Vand JJ-/S t a k e n  as i n d e p e n d e n t l i n e a r c o e f f i c i e n t s )  i n terms o f STO  configurations. terms  Thus t h e k e y wave f u n c t i o n h a s e i g h t  65  ^  c  +4*(Xis  Because  A  i  +  *z(X, Xu'Xzf) s  s  5' S  3  There i s a minimum  t o v a r y and absence o f c o n f i g -  u r a t i o n s m i x i n g t h e &c'savoids terms.  X £)  core p o l a r i z a t i o n  i s b u i l t i n t o t h e wave f u n c t i o n . of n o n - l i n e a r parameters  3  fafsXzi  X£ X&] + as fr, Xts'&s)  (Pn i s n o t e q u a l t o  +Q fciXs'Za)  i n t e r f e r e n c e between  The t h r e e o r b i t a l e x p o n e n t s ,  5s  , Ss  ,  were o p t i m i z e d f o r t h e f i r s t f o u r t e r m s , (  W i t h t h e s e v a l u e s as i n i t i a l i z a t i o n cycle f o r ^  8  next i n c l u d e d .  estimates a s i n g l e optim-  provided the f i n a l values of  these n o n - l i n e a r parameters,  p - t y p e c o r r e l a t i o n was  Three t r i a l a d d i t i o n s o f two terms  each were compared:  ).  66  Here  ( $si  means t h e terms i n  ^  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 . ponents, one  S  ,  s  S/?  w i t h only the l i n e a r  8  The  , were p a r t i a l l y o p t i m i z e d  c y c l e i n each 10-term f u n c t i o n .  <J  (0  , had  discarded.  t h e l o w e s t e n e r g y and  The  exwith  simplest,  t h e o t h e r two  were  Next t h e e i g h t p o s s i b l e t r i p l e t c o r e  terms were added one c h a n g i n g no functions  additional  at a t i m e , i n no  exponents, to  <&  special  spin  order,  > y i e l d i n g a sequence of  /0  f r o m 10 t o 18 terms i n l e n g t h , w i t h 0 t o  Ox - t y p e terms (  i  ).  8  Because  8  the  t r i p l e t c o r e terms were added somewhat a r b i t r a r i l y , one  o t h e r f u n c t i o n , ^ / f a , was  the f o u r  ©  a  computed,  incorporating  - t y p e terms p r o d u c i n g t h e 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 d e c r e a s e s i n t h e above s e r i e s . JT  Both s i n g l e (  =  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 .  Descriptions  a r e f o u n d i n T a b l e V. v a r i a t i o n a l and  of  T a b l e VI c o n t a i n s  constrained  properties.  f o r the complete s e r i e s are i n Appendix  , (3?n*. and <§  IS  t h e i r free Similar tables 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  t r i b u t e s i g n i f i c a n t l y t o the e n e r g y , t h e t o t a l  conincrement,  * Double d i a g o n a l c o n s t r a i n t s a r e e a s i l y imposed by t h e method of Chong and B e n s t o n [73j described i n A p p e n d i x C, making use of the r e s u l t s f o r s i n g l e constraints.  67  T a b l e V.  D e s c r i p t i o n s of  $  ,  l 0  $ . , IH  STO B a s i s O r b i t a l s O r b i t a l Exponents: Configurations $  |t>  :  3.168, 3.168, 2.840, 2.840, 0.765, 0.765, 4.974, 4.974 lsls^s'^,  , ls2s'2s"e,  ,  2313*23"©,  , 2s2s 2s"©.  ,  lsls'3s"e,  , ls2s'3s'^  ,  2sls'3s ev  , 2s2s'3s"e,  ,  2s"(2p) ©,  , 2s"(3p) ©.  n  2  14-  ,  2  {<$,„}*, l s l s ^ s " © * 2sls'2s"c9 . 2  [ J^}*,  2sls 3s"© . ?  2  , 2s2s'2s e , l  lsls'3s"e  ,  , ls2s'2s"6> . ,  t  i  , ls2s'3s"© . , 2  , 2s2s'3s"© . 2  Free V a r i a t i o n a l Coefficients:  10  +0.196099;  +0.334607  -0.230831  +0.024179;  +0.161653  +0.229143  -0.165463; +0.018404  -0.010172  -0.014436. +0.195868;  +0.337217; -0.230859;  0.022274; +0.161696;  1?  +0.226320;  -0.164641  +0.019763; - 0 . 0 1 0 1 7 4 ;  -0.014436  -0.583381; +0.052004;  -0.056793  +0.012474.  +0.196595  +0.349087  -0.239527  +0.021249  +0.160845  +0.215740  -0.156697  +0.020425  -0.010176  -0.014440  -2.035945  0.181852;  -0.196107  +0.036096  1.605262;  -0.143961  0.153566; -0.026107.  See t h e t e x t f o l l o w i n g ( 3 . 3 . 4 ) f o r t h e meaning o f t h e n o t a t i o n [<f>$ .  T a b l e V I . 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 o f Function  , J ^ - , $?  (defined i n Table  /t  AE  C o n s t r a i n t -Energy  Q (0)  None  7.467389  0.2677  13.7522  -2.9732  -3.2795  7.467254  0.2753  13.9191  -3.0000  -3.2973  0.000135  +0.579772xl0"  4  7.465334  0.1715  13.5519  -2.9447  -3.0000  0.002055  -0.521667xl0~  2  7.464736  0.1686  13.8901  -3.0000  -3.0000  0.002653  S  Q (0)  V)  e  > 0.0  —  > =+0.125277xl0" e  3  > =-0.585347xl0" s  None  7.467429  0.2136  13.7500  -2.9728  -2.9790  7.467291  0.2153  13.9192  -3.0000  -2.9870  7.467429  0.2150  13.7500  -2.9728  -3.0000  7.467291  0.2163  13.9192  -3.0000  -3.0000  0.0 0.000138 -10"  8  0.000138 -  0.582329x10"^ 0.891716xl0"  6  > =0.582305xl0" e  > =0.534754xl0" s  None  7.46749^6)  0.2287  13.7501  -2.9729  -3.0542  7.467360  0.2312  13.9178  -3.0000  -3.0590  0.000135  +0.572685xl0"  7.467496  0.2246  13.7501  -2.9729  -3.0000  ~10"  -0.239282x10~  7.467360  0.2267  13.9178  -3.0000  -3.0000  0.000135  4  6  0.0  7  4  5  > =+0.572745xl0" e  )f=-0.252666xl0" 7.478025 Larsson s 100 term correlated function (see e n t r y 18 T a b l e i ) 1  0.2313  13.8341  2  4  5  69  ^ $ ,  -  8  > b e i n g o n l y 0.000106 h a r t r e e s .  £ $ i o  e f f e c t on Q ( 0 ) i s n e g l i g i b l e a l s o ;  and $?  e  e s s e n t i a l l y t h e same e l e c t r o n d e n s i t y . improves tremendously. value (that of ^?  (g  The have  l8  Q ( 0 ) , however, s  The h i g h s t a b i l i t y o f t h e f i n a l  ) cannot be seen i n T a b l e V I b u t  i s e v i d e n t f r o m t h e complete t a b l e i n A p p e n d i x E. From t h e s e c o m p l e t e r e s u l t s one c a n see c e r t a i n t r i p l e t c o r e s p i n terms c o n t r i b u t e more t h a n o t h e r s t o Q ( 0 ) . s  O n l y about h a l f o f t h e p o s s i b l e  terms a r e n e c e s s a r y  b u t because t h e i m p o r t a n t ones a r e d i f f i c u l t t o p i c k out i n c l u s i o n o f a l l o f them seems a d v i s a b l e . most computer  t i m e i s spent on exponent  Since  optimization  t h e t r i p l e t c o r e s p i n terms c a n be added p r a c t i c a l l y as a bonus. Constraining  seems t o be a waste o f t i m e though.  I n a l m o s t e v e r y c a s e t h e wave f u n c t i o n d e t e r i o r a t e s . constraints apply corrections t o Q (0) 59 e  as Yue and Chong  found f o r He b u t t h e e r r o r  'improves'  from «=*-0.6% t o «*+0.5% w h i c h r e a l l y i s n o t 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 J Z  e  constraint  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 imposed.  Forcing  several  i s o f l i t t l e u s e . When  G - t y p e terms a r e p r e s e n t t h e c o n s t r a i n t s x  a r e easy t o a p p l y and p r o p e r t i e s a r e n o t changed much. When t h e r e a r e o n l y a few, o r none, t h e c o n s t r a i n t *See  i.ifa.  i n Appendix E f o r t h e d e m o n s t r a t i o n .  70  becomes more s e v e r e i n d i c a t i n g t h a t f i n e  adjustments  i n t h e 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 on t r i p l e t  core s p i n terms.  depend  Wo s i g n i f i c a n c e i s a t t a c h e d  to t h e double c o n s t r a i n t r e s u l t s .  Because i m p o s i n g  t h e e l e c t r o n cusp c o n d i t i o n was more s e v e r e t h a n f o r JZ  t h e s p i n cusp, f o r c i n g  =  IZ  =  approximated the s i n g l e c o n s t r a i n t  Y -E  7  invariably =  . A  f i n a l p o i n t i s t h a t t h e 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 n o t be n e c e s s a r y  even i f i t worked.  No t r e n d i n improvements f o r Q ( 0 ) s  r e s u l t i n g from f o r c i n g  c a n be seen f r o m  t h i s study. I t i s p o s s i b l e t h a t f o r a p p r o x i m a t e wave t h e r e i s an e m p i r i c a l iY = -Z.  theoretical JZ  is  'effective'  = JZ  with  Y  functions  rather than the  A further theoretical condition  Y  u n s p e c i f i e d b u t one c a n c o n -  f i d e n t l y p r e d i c t t h a t t h e minimum energy f o r such a c o n s t r a i n t w i l l occur at for Y'  the series  —*  JZ <^,  g  the single constraints  e  =-C . JZ  ~-IZ^r^e  va.v-io.-tUna.l  To l o c a t e an e f f e c t i v e C  = Y'  JT  and  = 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 v a l u e s of  Y'  .  The a c t u a l dependence o f Q ( 0 ) , Q ( 0 ) on e  s  t h e cusp c o n s t r a i n t s was more e v i d e n t d u r i n g t h e s e c a l c u l a t i o n s than i n the preceding study.  71  Q ( 0 ) i s shown as a f u n c t i o n o f t h e r  i n f i g u r e 3(A)  =  S  I t has  for  a l i n e a r dependence on  these functions  , -17  constraint $ .  ,  lH  One  can  t h a n -Z  i s greater  a s p i n cusp c o n s t r a i n t w i l l improve Q ( 0 ) . the  l i n e s should pass through ( 0 . 2 3 1 ,  See  t h e c o m p l e t e s e t of p r o p e r t i e s  the  I t i s evident ^  X"  dence. e  experimental value ( 0 . 2 3 1 3 )  i n f i g u r e 3(B)  That t h e  ©  A  pre-  versus  a l s o shows a l i n e a r depen-  - t y p e t e r m s do not  influence  i s obvious; the l i n e s f o r a l l f u n c t i o n s , »  superimposed.  a r e  c o r r e c t s Q (0) f o r Q (0)  cannot be  e  a l l l i n e s pass Q (0) because the f u n c t i o n s  <£  lo  [38]  through  similar i n electron density. 3(B)  = -Z  over-  An e f f e c t i v e  at  Y'=  -2.985  are a l l too  A comparison of  figures  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 (0) s  s c a t t e r of Q ( 0 ) s  e  ,  s a i d t o e x i s t even though  = 13.83  e  and  _C  Why  i s c l e a r f o r these cases.  e  3(A)  i s no  s c a t t e r of p o i n t s where  e  > $18  y  i n Appendix E to  also that there  The  -3  =  - 3 . 0 0 0 ) ).  A s i m i l a r g r a p h of Q ( 0 )  = cV  6  Q (0)  value.  l i n e s c r o s s the  cludes t h i s .  for  (Ideally  s  'effective'  see  (also f o r a l l others tested) that i f  the f r e e v a r i a t i o n a l  verify this.  .  than Q (0) e  and w i l l r e f l e c t  values i n Table I .  F i g u r e 4,  the (A)  72  F i g u r e 3. ( A )  O ( 0 ) as a f u n c t i o n o f t h e c o n s t r a i n t ~ s  Z.8  -A.9  3.0  3.1  J.2.  x  3.3  F r e e v a r i a t i o n a l v a l u e s a r e marked e x p l i c i t l y f o r (B)  Q ( 0 ) as a f u n c t i o n o f t h e c o n s t r a i n t £* = for £, , ^ , ^ . e  o  1  8  1  1 i-ertica.1 /ue  Qfo)  /J.o  1 J.8  l.f  1 3.o  . J./  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  3.%  explicitly.  73  F i g u r e 4. ( A ) Q ( 0 ) s  as a f u n c t i o n o f t h e c o n s t r a i n t a r' f o r J,0 , ^ , $|f .  r=  <?.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  (B)  Q (0) e  ^  ,  (8  as a f u n c t i o n o f the' c o n s t r a i n t 18  £xactt38-j  10  Thet-ttica.( tue. 2.e  2.?  3.o  3./  74  and  ( B ) , shows Q ( 0 ) s  Q (0)  as a f u n c t i o n of  e  c l u s i o n s a r e borne o u t : adjusts  X"  = Y '  l)  The  .  e  Y  and  /  P r e v i o u s con-  s p i n cusp c o n s t r a i n t  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  l i k e l y t o improve s p i n d e n s i t i e s .  must be c o n c l u d e d t h a t t h e s e d i a g o n a l are not u s e f u l f o r i m p r o v i n g Q ( 0 ) . meters are o p t i m i z e d included  and  Finally i t  cusp c o n s t r a i n t s I f non-linear  s  the wave f u n c t i o n s h o u l d have a good s p i n  Off-diagonal  constraint.  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  Since attempts thus f a r to a p p l y the of c o n s t r a i n e d  functions  philosophy  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 , t h e p r o b l e m must be examined.  para-  O*. - t y p e s p i n terms are  d e n s i t y a t t h e n u c l e u s w i t h o u t any  3.4  =  o n l y m i n u t e d e t a i l s o f a wave f u n c t i o n w i t h o u t  a f f e c t i n g Q (0) i s not  £&  as a f u n c t i o n o f  I f $s,M s a t i s f i e s nuclear  t h e i n t e g r a t i o n i n (2.4.5) c a n be  cusp  conditions  seen as an  m e r e l y a l l o w i n g an e v a l u a t i o n o f t h e  re-  artifact,  expression  (3.4.1)  (Compare w i t h  (2.3.1) ).  Replacement o f  (2.4.5) by an a r b i t r a r y f u n c t i o n , able i n t h i s l i g h t .  T  To a p p l y t h i s new  (J>  SIM  in  , seems r e a s o n concept,  75  restrictions If  1//  on  ~f  must be examined.  i s a spatial eigenfunction  (2.3.8) f o r t h e case o f n u c l e a r  J-F*(^Co  - ^ > J ^ d f >  satisfying  cusp, t h e c o n d i t i o n  =  O ,*-*,s ~P  i s v a l i d f o r any w e l l - b e h a v e d f u n c t i o n must be i n c l u d e d i n ~F  when t h e s p i n  $S,M i s c o n s i d e r e d  function  and  /  .  Spin  containing  because i n t e g r a t i o n o v e r  a single spin v a r i a b l e i s undefined. conditions  (3.4.2)  To a v o i d  trivial  s h o u l d be an e i g e n f u n c t i o n o f  w i t h t h e same s p i n f u n c t i o n s  as  Oc  @SjM  Then  § >-o  (3.4.3)  SlH  is valid.  (Compare w i t h (2.4.5) ) .  the o f f - d i a g o n a l nuclear  on an a p p r o x i m a t e  ^  •  T  n  e  function,  If  . c a n be  In particular  may be a n t i s y m m e t r i c w i t h r e s p e c t  of e l e c t r o n coordinates.  suggests  cusp c o n s t r a i n t s ,  t h o u g h t o f as a w e i g h t i n g f u n c t i o n . ~f  This  to interchange the diagonal  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 a p p r o a c h t o c o n s t r a i n e d described  i n the t e x t a t equation  (3.1.1),  forces  I n t h e w o r k i n g b a s i s s e t , (^K!  ^ ^ i ^ = 0.  the b a s i s o f f r e e v a r i a t i o n a l t i o n i s expressed i n matrix  variation,  , (usually  eigenvectors)  this rela-  notation  e ^ C i O . =° • <£2  i s the constrained  f  =  (3.4.5)  (column) e i g e n v e c t o r  2  sought,  ,  c  and  Cc  =  ( C  k  £  j  £  )  ,  C^i^gjQIe^.  (3.4.6) 69  I n c o n t r a s t , t h e method o f Weber and Handy energy w h i l e where  constraining  y  minimizes (3.4.7)  i s a column v e c t o r c o n t a i n i n g n e c e s s a r y  i n f o r m a t i o n about t h e  i*  h  constraint condition.  I n t h e o f f - d i a g o n a l c a s e , ( 3 - 1 . 6 ) c a n be expanded i n a basis  t§<\  a  s  77  jfCiCL = O for  <p =  ^  ^  ^  (3.4.8)  .  Then  (3.4.9)  Note t h e i m p o r t a n t d i s t i n c t i o n between ( 3 . 4 . 9 ) ( 3 . 4 . 6 ) , and  (3.1.2).  The p r o c e d u r e  Handy does n o t need an H e r m i t i a n  &  with  o f Weber and operator.  If  i s f i x e d , t h e c o n s t r a i n t s , ^-Y«-7 , a r e 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  diagonal c o n s t r a i n t case, ( 3 . 4 . 5 ) ,  c a n be  but an i t e r a t i o n p r o c e d u r e  i s necessary:  Q-K =  O  on  handled,  ,  (3.4.10)  S e v e r a l t y p e s o f 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 to determine t r i e d was  t h e i r e f f e c t on Q ( 0 ) s  ~P = i ^ -  eigenvectors.  , f  Since the  and Q ( 0 ) . e  the set of f r e e are energy  The  first  variational  ordered  78 t h e c o n s t r a i n t ( 3 . 4 . 4 ) amounts t o w e i g h t i n g by energy. C o m p u t a t i o n s a r e s i m p l e f o r t h i s c h o i c e because t h e m a t r i x elements ( 3 . 4 . 9 ) ,  i f a - f ^ l ^ i )  ($k  =  < ^ / / ^ - < # / / § )  had been e v a l u a t e d 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  are p r e s e n t e d  with selected  i n Table V I I .  B o t h 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 t h e d o u b l e c o n s t r a i n t were c a r r i e d o u t ( f o r J^ ' 6  Y  = -Z).  The v a l u e s o f  a r e i n c l u d e d so t h e consequences o f  the groundstate  e i g e n v e c t o r , c a n be s e e n .  =  ,  In a l l  ~F =  three 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 diagonal c o n s t r a i n t s i n section.3.3,  T a b l e V I . The a p p r o p r i a t e _L  e q u a l -Z  **  t o w i t h i n 3-4 d e c i m a l p l a c e s . a r e t h e same t o 10~  6  h a r t r e e and Q ( 0 ) , Q ( 0 ) e  also only d i f f e r i n the 3 i s t o be e x p e c t e d to the constrained  Constrained  r d  s  energies values  , 4** d e c i m a l p l a c e s .  This  1  s i n c e t h e overwhelming c o n t r i b u t i o n 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 t h e o f f - d i a g o n a l , o n e - s t e p method o f Weber and Handy c a n be used t o R e a l i z e t h a t t h e groundstate of i_s ^ changes t o some JF) when c o n s t r a i n e d . **  .  ^Vy.  T h i s shows t h a t t h e i n t e g r a l cusp c o n d i t i o n s a r e o n l y n e c e s s a r y , f o r i f t h e c o r r e c t cusp existed,£"s Y always.  Table V I I .  Energy w e i g h t e d o f f - d i a g o n a l cusp c o n s t r a i n t s on  Constraint Weighting Function None a=e  -E  AExlO  7.4-67429  6  Q ( 0 ) s  Q ( 0 ) 6  r  I  5  0.2136  13.7500  -2.97.28  -2.9790  f,  7.467291  138  0.2152  13.9211  - 3 . 0 0 0 0  - 2 . 9 8 6 0  **.  7.467429  0  0.2137  13.7502  -2.9728  - 2 . 9 8 0 8  7.467390  39  0.1668  13.7575  - 2 . 9 7 4 2  - 2 . 1 6 9 8  7.467251  178  0.2100  13.5580  - 2 . 9 4 2 6  - 2 . 9 4 4 5  7.466214  1215  0.1839  14.2010  - 3 . 0 4 6 8  - 2 . 4 2 9 9  7.467429  0  0.2150  13.7500  - 2 . 9 7 2 8  - 2 . 9 9 9 9  7.467199  230  0.0810  13-7287  - 2 . 9 6 9 8  +1.9146  7.467248  181  0.1387  13.6719  - 2 . 9 6 1 1  - 1 . 4 7 6 1  7.467429  0  0.2154  13.7500  - 2 . 9 7 2 8  -3.0053  7.467427  2  0.2001  13.7509  -2.9729  - 2 . 7 6 8 9  7.467291  138  0.2162  13.9211  - 3 . 0 0 0 0  -2.9999  7.467167  262  0.0660  13.7301  - 2 . 9 7 0 4  +3.6892  n  7.467162  267  0.1860  13.5676  - 2 . 9 4 4 5  - 2 . 6 2 9 5  li  7.467250  179  0.2153  13.5579  - 2 . 9 4 2 6  -3.0185  u  7.466202  1227  0.2144  14.2055  - 3 . 0 4 7 5  -2.9572  II  II  $1 II  II  a=s ti  f»  f,  &  II  f, II  it  a=e,s u  80  c l o s e l y d u p l i c a t e t h e r e s u l t s o f 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 . c r i b e d i n 3.2  Computational problems des-  a r e b y p a s s e d and no i t e r a t i o n i s n e c e s s a r y . •f  The w e i g h t i n g method w i t h  - s,  t h u s 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 various  constraints.  As f a r as w e i g h t i n g w i t h o t h e r energy e i g e n f u n c t i o n s ~f =  i s concerned,  K > 1, u n p r e d i c t a b l e ,  ,  random, p o o r r e s u l t s o c c u r . C o n f i g u r a t i o n w e i g h t i n g was n e x t a t t e m p t e d . figurations, f^ - j  Con-  , have an energy a s s o c i a t e d w i t h them  but t h e y a r e more s p a t i a l i n c h a r a c t e r t h a n t h e energy eigenfunctions •f  =  ffl  , ( ^  K  =  7^  ).  i s a kind of s p a t i a l weighting.  Putting Again,  r e s u l t s a r e random and m e a n i n g l e s s and w i l l n o t even be p r e s e n t e d . A p p l i c a t i o n o f Weber and Handy's method t o t h e d i a g o n a l c u s p c o n s t r a i n t c a s e w i t h (3.4.10) was tigated.  I f convergence  was r a p i d , c o m p u t a t i o n a l  p r o b l e m s o f t h e s o r t mentioned i n 3.2 and t h e e x a c t d i a g o n a l c o n s t r a i n t be imposed.  inves-  would be a v o i d e d  <&<:/¥>  = 0 would  A f t e r convergence was a c h i e v e d s l i g h t  d i f f e r e n c e s f r o m t h e pure d i a g o n a l a p p r o a c h (3.1.1) were f o u n d even though  f r o m b o t h methods.  The i n i t i a l guess <Z? = s, , i n (3.4.10) d i d n o t c o n v e r g e when an H e r m i t i a n G was u s e d b u t o s c i l l a t e d back t o Q.XK - s~, • G r o u n d s t a t e e n e r g y and t h e o v e r l a p : 0^^X. were employed as c o n v e r g e n c e c r i t e r a . 0  81  A c o m p a r i s o n o f t h e two a p p r o a c h e s t o d i a g o n a l c o n s t r a i n t s for  i s p r e s e n t e d i n T a b l e V I I I f o r academic i n t e r e s t .  The r e a s o n f o r t h e anomalous r e s u l t s was  n o t a program  e r r o r but i m p o s i t i o n o f 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 diagonal technique the c o n s t r a i n t matrix  c  =  P  + P  was  u n a m b i g u o u s l y 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 d e v e l o p e d  during  i t e r a t i o n had t h e f o r m  $r  where  not e q u a l  i s t h e c o n s t r a i n e d e i g e n v e c t o r and need <£L  f i r s t case.  , the c o n s t r a i n e d eigenvector f o r the Both these eigenvectors 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 o f t h e w e i g h t i n g concept ployed  em-  , where  t h e Xhs  "*S  S/s  = 2.7 (°  parameter.  a r e STO's h a v i n g o r b i t a l ,  Jjj  = 1-5  The v a l u e s o f  chosen w i t h S l a t e r ' s r u l e s .  •  S/  , Szs  S  Now  be c o n s t r a i n e d can be w e i g h t e d  exponents C for  i s a variable C  = 1 were  t h e wave; f u n c t i o n t o  i n selected regions.  Of s p e c i a l i n t e r e s t i s t h e e x t r a p o l a t i o n XP  ->  o.  82  Table V I I I .  Comparison o f t h e 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 Example:  ^ ^,  JZ = V e  (  constraints.  constraint  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 attribute (see equation (3.4.5)) (see equation(3.4.10)) Energy  - 7.4672914  Q (0) s  Q  (0)  e  r *  -  0.215387  .0.215193  13.919166  13.920945  -  3.000000  -  3.000000  -  2.986952  -  2.985937  0.9999931  + 0.9999933  a  i  +  a  2  - 0.1996x10~  a  3  3  H *  7-4672913  - 0 . 1 7 5 7 x 1 0 "•3  + 0.2666x10"*  3  + 0.2446x10"•3  + 0.2031xl0"  4  -4 + 0.2099x10"  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 o f i 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 . a  =  S3  o - l i k e function, weighting  becomes a D i r a c the nuclear  regions f  p l o t t e d against 5(B)  Q (0) i s  t o an extreme d e g r e e .  i n f i g u r e 5(A).  /ft  and  s  Figure  shows t h e c o r r e s p o n d i n g r e l a t i o n s o f Q ( 0 ) _ f o r e  e l e c t r o n i c cusp c o n s t r a i n t s on are o b t a i n e d .  .  Negative r e s u l t s  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 T a b l e I X . i n t e r e s t i n g maximum e x i s t s around certainly coincidental. out  on  $>  /0  (j),  and  f  An  = 1/4 b u t i s  S i m i l a r s t u d i e s were c a r r i e d  o f t h e same s e r i e s o f f u n c t i o n s  7  w i t h t h e same d i s c o u r a g i n g except f o r the o b v i o u s l y  results.  Energy  sacrifices,  d i s t o r t e d c a s e s , 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 weighting  constraints  i s i m p o r t a n t because i f (3.4.4-) i s t r u e f o r a l l members o f a c o m p l e t e s e t , an a p p r o x i m a t e wave f u n c t i o n must have t h e c o r r e c t c u s p .  Thus i f  f o r c e d t o be z e r o f o r many "Pc /  ^  functions  perhaps  w i l l more c l o s e l y s a t i s f y t h e n u c l e a r  /  dition.  cusp c o n -  The method o f Weber and Handy s i m p l i f i e s c o n -  straint calculations.  D i f f e r e n t w e i g h t s were a p p l i e d  simultaneously  ,  o f /W)  to  , and f,  $~  /0  and $  / 7  .  , the free v a r i a t i o n a l  Combinations eigenfunction,  were u s e d i n b o t h 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- a r e p r e s e n t e d i n T a b l e X.  Use o f t h e d i a g o n a l  i t e r a t i o n technique to force  F i g u r e 5.  o.zf  (A)  Q (0) p l o t t e d against C and /f in 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 on s  l  I  1  1  1  1  1  Q (o) s  w  O.ZZ  o.zo  r  0.69-  '[ °" z&  in  iff  — r - *  0 Q  1  1  9-  8 0 / 2 .  A/Z  o.zo  1 \y  -  ""'  aze  1  1  1  /6  20  2V  03&  0.9?  O-SC?  85  Table I X . Q ( 0 ) , Q ( 0 ) f o r t h e 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 o f <j>/tf s  e  e  Q (0) s  Q (0) 6  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  0.2135  13.7500  Free v a r i a t i o n a l r e s u l t s  86  T a b l e X. Number o f simultaneous constraints  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 Constraints  .  AE(energy Q (0) Q (0) sacrifice due t o constraints) S  e  3  lV (k) >W tL)>W fr)  0.0135  0.0983  13.6157  3  §,*> W (&)> W &)  0.0140  0.0662  13.2261  IYW  0.0000  0.2181  13.7517  W U)  0.0001  0.2227  13.7513  S  s  s  S  2  S  2  gf,  2  WVIL) >W f3.)  3  Wita) ,14/%)^%)  S  s  3 2  Jv% tt/fy)  2 2  J?*,  0.0001  0.2094  13.7519  0.0018  0.1968  13.9758  0.0009  0.2052  13.8953  0.0009  0.2038  13.8998  0.0009  0.2083  13.8952  0.0001  0.2162  13.9211  4  /^k),^k)  0.0009  0.2205  13.8990  6  f, 9f, ,^ k)7^ Hk),  0.0177  0.0407  13.7131  e  ,  /e  S  a = e,s g i v e s t h e t y p e o f c o n s t r a i n t applied.  •87  <^/(si  iy>  = °  a i  °*s  with  (Mhieiy/y =... =  0  d i d n o t p r o d u c e 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  <?,/£/  as e x p e c t e d .  If  >  -  = ... = 0  <' WM/G/"V>'>  No d e f i n i t e t r e n d s c o u l d he d e t e r m i n e d .  The  extreme number o f p o s s i b l e c o n s t r a i n t s a l s o c o n f u s e s  the  problem. In conclusion,  off-diagonal weighting  constraints  have o f f e r e d no s u r e method f o r i m p r o v i n g s p i n and electron densities.  A more r e a l i s t i c f o r m f o r , weighting  o n l y one e l e c t r o n a t a t i m e b u t t h i s was n o t t r i e d . The  general  t e c h n i q u e c o u l d p e r h a p s be i m p r o v e d concep-  t u a l l y and might be u s e f u l f o r o t h e r p r o p e r t i e s .  •88  CHAPTER I V  SUFFICIENT CONDITIONS FOR CORRECT CUSP 4.1  Theory The  f a i l u r e o f n e c e s s a r y cusp c o n s t r a i n t s t o p r o v i d e  a c c u r a t e d e s c r i p t i o n s . o f wave f u n c t i o n s c o a l e s c e n c e i s b y now o b v i o u s .  at nuclear  That t h e s e c o n s t r a i n t s  need n o t f o r c e t h e c o r r e c t b e h a v i o u r a t t h e cusp i s re-emphasized.  The w e i g h t i n g 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 .  <flii/-tel1f/)=  0 does n o t g u a r a n t e e  as w o u l d be f o u n d i f ^ practical,  t h e c o r r e c t cusp?  to f i r s t p r i n c i p l e s  ^/^-U/W)  had the proper cusp.  sufficient restrictions  w i l l provide  The  Constraining  on a f u n c t i o n  , = 0 Are there that  F o r a n answer a r e t u r n  i s indicated.  n e c e s s a r y 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 f o r m f o r any wave f u n c t i o n are embodied i n ( 2 . 3 . 8 a ) ,  (4.1.1X2.3.8a)  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 ) . D e f i n e an o p e r a t o r  where  clH )^ (e <p )  Ifo^njrZL T  "  K  rJVT-A  m  K)  (4.1.3)  K  J  ^constant  t a k e s t h e s p h e r i c a l average about t h e p o i n t o f c o a l e s cence o f t h e Kib e l e c t r o n i c c o o r d i n a t e s -^^/^) i s a o n e - e l e c t r o n The n u c l e a r  as i n ( 2 . 3 . 8 a ) .  cusp e v a l u a t i o n  operator.  c o a l e s c e n c e c o n d i t i o n s o f 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) a r e w r i t t e n c o n c i s e l y ,  -jb/f) § Extension  —O  (4.1.4)  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  c o a l e s c e n c e c o n d i t i o n f o r an e x a c t s p i n - c o n t a i n i n g  func-  t i o n ( s e e (2.3.11) ) i s  as each  s a t i s f i e s (4.1.4).  function with spin, ^  An a p p r o x i m a t e wave  , has t h e c o r r e c t cusp i f and  -90  only i f  The p r e s e n t 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 . z e r o (and w i l l be s u p p r e s s e d ) . d e r i v e d f r o m (4-. 1.6)  Specialized  equals  conditions  w i l l be o n l y s u f f i c i e n t because a  configuration interaction tional  That i s , X  expansion i s not a unique f u n c -  form.  An n-term CI f u n c t i o n w i t h s h a r p s p i n and o r b i t a l a n g u l a r momentum and s h a r p Z c o m p o n e n t s — a n e i g e n f u n c t i o n , JC.  of the  •> oCz  »Wz  X  —i  s  u s u a l l y expanded i n  form  f  =  CI  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  The {%-]  are eigenf unctions of  h a v i n g t h e same e i g e n v a l u e s as ^  , r  coefficients.  '^fes^z  joZTg  and a r e d e s c r i b e d  by a p r e d e t e r m i n e d , f i x e d l i n e a r c o m b i n a t i o n o f S l a t e r determinants  f^x]  5  = *> a <t>j u  (4.1.8)  91  Since  -jbfk)  determinant  operates  o n l y on t h e  e l e c t r o n each S l a t e r  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 &  electron:  xl(os*(\) xicos/co  4, =  (4-.1.9)  where in  =  S^fo)  0£  oC  of the  o r f?  and  i s the cofactor  row and m^ column,  (fy,^  i s thus  a f u n c t i o n o f 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 b u t those of the  electron.  The e x p l i c i t c a s e f o r l i t h i u m  has been d e p i c t e d t o a v o i d n o t a t i o n a l d i f f i c u l t i e s . The  superscript  ^  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 belonging t o d i f f e r e n t minants. (3.1.7)  I n t h i s work o n l y S T O ' s , { ^ t ]  (defined i n  ), are considered although t h e approach can  be d e v e l o p e d  f o r any b a s i s s e t .  makes t h e c h o i c e o f K i r r e l e v a n t . roximate) one  deter-  ffl  cl  The P a u l i  principle  O p e r a t i n g on (app-  w i t h t h e cusp e v a l u a t i o n o p e r a t o r  obtains K 'jim  (4-.1.10)  92 w h i c h must e q u a l z e r o i f t h e c u s p i s c o r r e c t . In general,  S£,(K)  ^  a l i n e a r dependence i n t h e s e t {Xi}  0, u n l e s s t h e r e i s •  Sufficient  con-  d i t i o n s f o r cusp s a t i s f a c t i o n a r e f o u n d by l e t t i n g t h e c o e f f i c i e n t of  S»\ (K)  fifrn  e q u a l z e r o f o r each  J?>  ,  (4.1.11)  The system o f e q u a t i o n s , # of determinants,  2E.C <?t£ c  =0,  ./=  1,...,  l e a d s t r i v i a l l y t o an i d e n t i c a l l y  z e r o wave f u n c t i o n .  T h i s means t h a t ^ ^ J ^ = 0 when  0 f o r each member o f t h e 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 e v a l u a t e d e x p l i c i t l y t o f i n d t h e c o n d i t i o n s f o r STO's:  A- "«. v  vi/^'  The d i s c u s s i o n c a n be r e s t r i c t e d t o s-type  (,.i.i3)  orbitals  s i n c e t h e s p h e r i c a l a v e r a g i n g o p e r a t o r ensures 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  that  (4.1.12).  93  Equation  (4-. 1.13)  Xr,sns =  reduces t o  f-Xjo)(f+lr)  n  (9  =  (4.1.14)  h>3  For t h e 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^  (2)  ^0/tis  (3)  ^)X^^=0 i s satisfied for a l l orbitals except  implying  0 s  0  i  Vis  s  n  o  t  P°  , Xz  s  =  JT  ssi1:)  ~Y  =  le,  Z,  (4.1.14a)  .  These a r e p r e c i s e l y t h e 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 ( considered. to  The p r e s e n t t r e a t m e n t  include coalescence  \  = 0 ) are  c a n e a s i l y be e x t e n d e d  conditions.  Another approach e x i s t s . any c o f a c t o r s a r e e q u a l ,  Examine ( 4 . 1 . 1 0 ) .  <^>* =  $/tr\  y  =  If there  w i l l be o t h e r 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 g e n e r a l e x p r e s s i o n o f (4.1.10) i s  S ^ J & . - ^ & r  ' - a l l / ' , , * '  .  94  A l l equal c o f a c t o r s ,  and spins f o r the  have been f a c t o r e d out.  electron,  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) indeterminate to give r e c o g n i z a b l e  i s too  cusp c o n d i t i o n s :  Case 1  The s i m p l e s t p o s s i b l e case has a b a s i s of STO's—a, a , b ( a ^ a ) . 1  The S l a t e r d e t e r m i n a n t s  1  are  defined  The  n o t a t i o n i s short f o r  (4.1.9),  minant by i t s p r i n c i p l e d i a g o n a l .  identifying a deterX  a n  ^  w i t h i n the brackets i n ( 4 . 1 . 1 6 ) imply the X oC  andX^  three  respectively.  "X  spin-orbitals  The CI f u n c t i o n appears  (4.1.17)  (The  and  sum  (4.1.8)  (4.1.15)  has but one t e r m ) .  reduces to  Now  4  95  (4.1.18)  To f i n d t h e c u s p c o n d i t i o n s to zero c o e f f i c i e n t s of  f o r t h i s f u n c t i o n equate  (fij^ f o r a l l  (1)  pb  (2)  p [ c a + c a'J  £  , 4n-  = 0 x  =  2  0  E q u a t i o n ( 1 ) i m p l i e s t h a t i f ^> ^  (4.1.19)  = 0, o r b i t a l b  must s a t i s f y t h e o r b i t a l cusp c o n d i t i o n , ( 4 . 1 . 1 2 ) . That i s , b must f i t i n t o t h e scheme ( 4 . 1 . 1 4 a ) . STO's w i l l be w r i t t e n  Xj.  .  Thus pb == 0 and b i s  c h o s e n by t h e r u l e s ( 4 . 1 . 1 4 a ) . ratio  Such  Equation (2) f i x e s the  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 ences have been removed.  depend-  96  f2 c  =  -pa . pa'  1  (4-.1.20)  a,a' must n o t s a t i s f y t h e 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 a r e t o be f o u n d . t h e f u n c t i o n , T^ci  ( I f p a s . 0, b u t pa'^p; 0  > c o l l a p s e s t o a s i n g l e term).  Of c o u r s e t h e r e a r e no l o n g e r s e c u l a r e q u a t i o n s f o r this  case:  %  ^ C . f W - ^ p J  x  If b =  , JI > 0,the s e t o f o r b i t a l s h a v i n g , ( On =  jC-z  ^.1.21)  , Jt~\ > "'' °  )»  x  m  a  y  n  e  e  d  t  o  D e  sharp  in-  c l u d e d i n t h e b a s i s e n s u r i n g s h a r p t o t a l a n g u l a r momenturn f o r  .  I f ^X • CX  hag nas  symmetry t h i s  2 (  particular  example c o v e r s o n l y t h e c a s e s  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 t h e o r b i t a l cusp c o n d i t i o n i n t h i s example. £  , and e i g e n f u n c t i o n s o f f o r a S f u n c t i o n , {tyf'} 2  XI.  Only  The S l a t e r >£  ,  of  determinants,  ,  »o£°k  , are l i s t e d i n Table  0/ - t y p e terms a r e i n c l u d e d i n f  ;  97  T a b l e X I . The s e t o f S l a t e r d e t e r m i n a n t s , and  eigenfunctions,  , oZfz  Szfz  f o r t h e S wave f u n c t i o n 2  des-  c r i b e d i n case 2 o f s e c t i o n 4.1. (a,b,c) (b,a,c) ( a ,b,c)  <#/ = ( a \ b , c ' )  (b,a',c)  (b,a\',c')  (a,b*,c)  (a,B',c')  (b',a,c)  (b»,a\c')  1  *-  (V,a',c)  (a,b,c')  4>,=( b , a , c ) 1  <t>*= ( a  4,1=  ,  ,B',c')  (V,a\c')  + & + +  A A  %-  K = 0/r  98  addition of t r i p l e t shortly.  c o r e s p i n terms w i l l be d e m o n s t r a t e d  Any d i f f e r e n t s e t  w i l l cause a b r e a k -  down i n t h e f o l l o w i n g e q u a t i o n s .  The CI f u n c t i o n i s  < -/  If  j&M^ci  = 0, e i t h e r each element o f  obey t h e o r b i t a l cusp  {\l\  condition,  o r t h e f o l l o w i n g s e t o f e q u a t i o n s must be  r  1 0 0  <?  0 0 0 0 0  §  0 0  0 0 0 0  where are  0 1 0 0 0 OC  Y  0 0 1 0 0 0 0  0 0 0 1 0 1 0  0 0  0 0  o<  0  0  0  2r  0 0  0 0  0 0  0 0  oC = £a  pa'  must  0 0  0 0 0 0 1 0 0 1  0 0 0 0  0 0 0 0  0 0 1  0 0 0  .0  0 0  1 0  0 0  0 0  1  '3  satisfied:  = 0  '8  1 1 1  (4.1.22)  Y 0  = E£  There  PC  s e v e n l i n e a r l y i n d e p e n d e n t e q u a t i o n s and e i g h t  unknowns. factor are:  The s o l u t i o n s t o a c o n s t a n t  (normalization)  99  c  =  1  '3 c  The  = <X  1  ^7  p  _  =-r  4  ft  c  (4.1.23)  =-cx^r  8  l i n e a r c o e f f i c i e n t s have been c o m p l e t e l y f i x e d . From p r e v i o u s e x p e r i e n c e t r i p l e t c o r e s p i n terms  s h o u l d he i n c l u d e d .  Table X I I c o n t a i n s the  needed i n T a b l e X I f o r { f a ]  elements  If a l l possible  additional  and f Wc]  •  6? - t y p e terms a r e not i n c l u d e d 4  t h e e q u a t i o n s 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 S\  k, c o u p l e s t h e sets  ^ i ~ " ~ * 8^ c  '  c  linear  and  ©z.  a n d  l 9 ~ " ^ 16^ c  terms. c  parameter,  The a r e  coefficient  ^oth defined  by t h e e q u a t i o n s (4-. 1.22), t h e s o l u t i o n s b e i n g  c  , i = 1,8  ±  c^o  = kc.  1  l+O  as i n (4.1.23) , i = 1,8  Other c a s e s , w i t h v a r i o u s c o m b i n a t i o n s sets  and  [ji]  (4.1.24)  of t h e  , lead to 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 T a b l e X I when  ©  z  elements f o r - t y p e s p i n terms a r e  included. (j) =  (a,c,b)  4=  (a',c,b)  n  0^ = ( a ' , c , b ' ) =  = (a,c",V) zo  % - ^ - fa,+ 2 < f ) <Pt  fa  %  =  ?£-  fa -  fa  fa  fao  ~  fa, ~  fa*.  <t>/  ~  fat  ~  <t>/6  far  3  gj =  (a,c',V)  (fa  (a'.c'.V)  z3  (j) = ( a , c \ b )  +  2<p  n  /g  + 2$,, +  2<Pzo  (a',5',b)  .101"  the c o e f f i c i e n t s resemble (4.1.23).  A tremendous c o n -  c e p t u a l s i m p l i f i c a t i o n c a n be made when r e l a t i o n s h i p s between t h e c^'s a r e examined.  E q u a t i o n s (4.1.23)  c a n be r e w r i t t e n  °5  2_  c  c  = — <*  2  Cj  c  =  ft  Cry  = —^  ^,  =  z  =  C  =  'ffl-r  ]_ 2°3 c  1 2 4 C  C  Cg =  Now a c u s p - s a t i s f y i n g  /  6  c  C  1 3 4C  C  c c^c^ 2  becomes  ( a + c a ' , b+c^b', c+c^c') 2  + (b+c^b', a + c a ' , c+c^c') 2  defining  ffl  CI  (4.1.25)  (4.1.25a)  t o a c o n s t a n t 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 function.  Each o r b i t a l o f t h e f u n c t i o n ( 4 . 1 . 2 5 ) s a t -  i s f i e s t h e 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 a'(k) ) 2  =  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. L e t t h e 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  linear  UHF f u n c t i o n s a r e i n c l u d e d i n t h e group o f DODS f u n c t i o n s . Reasons f o r t h e p r e s e n t c l a s s i f i c a t i o n a r e g i v e n i n A p p e n d i x B.  102  c o m b i n a t i o n s , \ Cf{]  (PL  ,  zLdCjXj  =  (4.1.27)  j the  s t a n d a r d f o r m f o r a n a l y t i c a l HE o r b i t a l s .  l e t , three-electron this  A doub-  CI f u n c t i o n c a n be formed f r o m  basis:  :  ty-  C  (4.1.28)  c where  =  (  (<P*,(P»><fi«)H<Pm,(Pjt,Cp»)  for  ((ft,  fa,fa) Wn.  f  and t h e c^ a r e l i n e a r v a r i a t i o n a l jb(fc) eralized  =  ©,-type terms  0  o  r  6  >  ^ - ~ ^ t  >  e  coefficients.  t  e  r  m  s  For  each o r b i t a l must s a t i s f y t h e gen-  o r b i t a l cusp  condition  w h i c h h o l d s e i t h e r i f ^bKi o r i f an a p p r o p r i a t e  s 0 f o r the basis  linear constraint i s applied to  103  <Pi not  .  A t l e a s t two o f t h e  satisfy  ~X]  i n (4-.1.27) s h o u l d  = 0 i f t h e 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 e q u a t i o n s (4-. 1 . 1 9 ) ,  (4-. 1.22)  c a n b r e a k down. F i n a l l y , t h e 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  o f t h e e l e c t r o n cusp c o n d i t i o n .  see f r o m (4-. 1 . 1 5 )  -jbikj^^fc) on  ^p  •> w i l l  t h a t a s p i n cusp e v a l u a t i o n  on t h e s p i n ,  can  operator,  g i v e e x a c t l y t h e same r e s t r i c t i o n s  as p ( k ) s i n c e  cz  One  w i l l operate only  SjT(k). p  4.2  A p p l i c a t i o n s to the l i t h i u m  S groundstate  E x p r e s s i o n ( 4 . 1 . 2 8 ) poses d i f f i c u l t c o m p u t a t i o n a l p r o b l e m s because t h e 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 (with or without c o n s t r a i n i n g ) as  •  Tractable  as w e l l  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 e x t e n d e d H a r t r e e - F o c k 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 t o t e s t t h e e f f e c t i v e n e s s o f cusp c o n s t r a i n t s . The f o r m o f s u c h f u n c t i o n s r e s e m b l e s t h e  iyt  i n (4.1.28),  where k i s v a r i a t i o n a l l y d e t e r m i n e d as i n ( 4 . 1 . 2 4 ) . N u c l e a r cusp c o n s t r a i n t s by t h e method o f Weber and  104  Handy.69 o r o f Chong 53* Weber, Handy and P a r r Hartree-Fock  can "be a p p l i e d t o each have a l r e a d y done t h i s f o r  f u n c t i o n s , the primary r e a s o n being  d e c r e a s e t h e number of l i n e a r p a r a m e t e r s by a theoretical condition.  to  satisfying  But HF f u n c t i o n s a r e not good  enough, as d i s c u s s e d i n C h a p t e r 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 , t h e o b j e c t o f t h i s work. n i c a l r e a s o n s t h e s p i n - o p t i m i z e d EHF carded  and  For  a p p r o a c h was  a l l f u n c t i o n s a r e c a s t i n a CI  techdis-  format.  T h i s d r a s t i c a l l y l i m i t s t h e number o f terms i n t h e e x p a n s i o n (4.1.27) tout has t h e  advantages:  (1)  No SCF  i t e r a t i o n s on  (2)  c91  (3)  A n g u l a r c o r r e l a t i o n c a n be added i f d e s i r e d  ,  Oz.  \CPij  oC  ,  of the l a s t c o r r e s p o n d , to a generA  a l i z a t i o n o f t h e EHF plished yet.  necessary.  s p i n terms a r e e a s i l y i n c l u d e d .  w h i l e m a i n t a i n i n g sharp Implementation  are  method w h i c h has not been accom-  S i n c e s i m p l e wave f u n c t i o n s a r e d e s i r e d ,  e x p a n s i o n (4.1.26) w i l l be r e s t r i c t e d t o two terms.  The  or three  a p p r o a c h i l l u s t r a t e d by c a s e s 1 and 2 i n  4.1  i s u t i l i z e d i n this section.  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  a(k) *  For  atoms.  the  designated  /pa(k) a'(k) = (pa (k)/ r  Orbitals like  a  a' == a  a' —  (4.2.1)  105  X  where ^  =  /s  = 0 for  -E), ^  X  4  X,s  f S Z  =  , ~Xzs  -  T  n  u  and  (4.1.25a)  s  becomes  $  lA[  =  z  =  a — a ' ( l ) b—b'(2)  c—c'(3)e,]  ( a — a ' , b—b', c — c ' ) + ( b — b , a=a', c — c ) 1  1  (4.2.2)  As b e f o r e f o r c a s e s 1 and 2, n e i t h e r o f t h e b a s i s "X  orbitals  o r "X '  in  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 . F o r e x p l o r a t o r y p u r p o s e s a s i m p l e C I wave f u n c t i o n formed f r o m t h e b a s i s  was c a l c u l a t e d .  t i o n and i t s p r o p e r t i e s appear i n T a b l e X I I I .  The f u n c 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 t h e same t y p e o f b a s i s u s e d 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  except  J/  S  = z.  optimized  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 t h a n HF-type f u n c t i o n s ( T a b l e 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 b u t t h e r e a r e d i s a d v a n t a g e s i n t h i s approach as a p p l i e d t o s i m p l e CI f u n c tions.  F i r s t , expression of core p o l a r i z a t i o n i s  difficult.  The o r b i t a l making dominant c o n t r i b u t i o n s  to the core i s f i x e d . allowed.  Second,  Xzs  STO's a r e n o t  Thus t h e 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 .  Illustrative calculation:  3 - t e r m CI  f u n c t i o n formed f r o m  basis.  STO b a s i s :  orbital  exponent 3.0000  3.1126 0.3000  X35  0.94-35  CI f u n c t i o n : $  =  0.77849(is,li',3s") + 0.16375 [(Is,5i",3s ) M  -  +  (3s,Ts,3s")]  0.000339[ ( l s , 3 s ' , 3 s " ) + (3s • ,li",3s")]  Properties: -E  Q (0)(% error)  7.436183  0.1243(46.3)  S  Q (0)(% e  error)  13.6266(1.5)  r = i  5  -3.0000  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 increasing the  p o s s i b i l i t y o f m u l t i p l e e n e r g y minima. K—X!  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 a r e f a r more f l e x i b l e .  Any Xzs  o r Xis  j°is 4 Z i s a c c e p t a b l e .  ulation  s t u d i e d had t h e f o r m  with the s t i p -  The f i r s t  functions  (4.2.2)  3) where  <%  s  is  m A- IS  k was e i t h e r p l a c e d  r  . ,  "XlS  L  =)  equal t o zero o r 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 p r o b l e m o f 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  = of H u r s t e t a l ^ roughly  fr ^J )+feJ ,^) S)  ls  i n the appropriate  p o s i t i o n and  o p t i m i z i n g i t s p a i r of exponents.  ents thus obtained  were p l a c e d  (4.2.4)  s  The expon-  i n (4.2.3) and v a r i e d  a g a i n t o be s u r e a r o u g h l y minimum e n e r g y 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 ) E v e r y o r b i t a l exponent c a n be d i f f e r e n t ; S\$ need n o t e q u a l %zs ( 2 ) Wave f u n c t i o n s a r e n o t numbered a c c o r d i n g t o t h e number o f t e r m s .  108  E x p o n e n t s and p r o p e r t i e s a r e shown i n T a b l e There are s e v e r a l p o i n t s t o n o t e .  XIV.  Spin-optimization  (k 4 0 c a s e s ) i s v e r y i m p o r t a n t h e r e as b e f o r e . maximum e r r o r i n Q ( 0 )  f o r t h e k =^ 0 c a s e s i s  S  Q (0)  values  e  are c o r r e c t w i t h i n 1%.  good numbers f o r such s i m p l e The  best  4%.  These a r e  o r b i t a l exponents c a l c u l a t e d f o r t h e  $is  = 2.068,  a r e s e e n t o be  = 0.639.  The  = 3.298,  f\s  5  Q?/s  best  CSO's  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 t h e c u s p , as ~Xi 20.0  very  functions.  f u n c t i o n (4.2.4-) by H u r s t e t a l ° a r e S\i  The  a  o r b i t a l w i t h an exponent as l a r g e as 13.0  S  i s h i g h l y concentrated  Y  for  y  near the  f\s  =20.0.  l i k e c o r r e c t i o n f o r the  nucleus.  6*-*)yL  / = = • - > -  -^is = 3 . 3 ,  or  ^  s  (4.2.5)  A corresponding CSO  c o u l d not be  £  -  found. 27  T h i s 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 entry i n Table I ) i n c l u d e d b a s i s e l e m e n t s i n h i s CI and HF apparent j u s t i f i c a t i o n . he was  not  attempting  The  -type  f u n c t i o n s without  As i s e v i d e n t  any  i n Table I  t o o b t a i n a good e n e r g y , o n l y  a good s p i n d e n s i t y and accordingly.  ~X\s  chose h i s b a s i s  elements  l i n e a r c o e f f i c i e n t s and  configurations  Table XIV.  S i m p l e 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  "V"  STO Function  X,f  A-IS  1P,  3-3 3.3 3.3  %  20.0 20.0 20.0  2.065 2.065 2.065  13.0 13.0 13.0  0.60 0.95  0.80  0.68 0.70  1.80  Properties Function  Energy  % Hurst e t a l Brigman and Matsen° Best Functions a  a  Q (Q)(% error) (absolute) S  J]  e  E , S  k=0 k*0  -7.445434 -7.445435  0.2174(6.0) 0.2232(3.;5)  13.9626C.9) 13.9625(.9)  -3.0000 -3.0000  k=0 k^O  -7.445419 -7.44542O  0.2160(6.6) 0.2224(3.8)  13.9614(.9) 13.9613(.9)  -3.0000 -3.0000  k=0 k*0  -7.445328 -7.445329  0.2265(2.,1) 0.2341(1.2)  13.9553(.9) 13.9552(.9)  -3.0000 -3.0000  k=0  -7.4436  0.3002(29.8)  13.5193(2.3)  k*0  -7.4436  0.2417 ( 4 . , 5)  13.5240(2.2)  -7.478069°  0.2313 (0.0)  13.8341 (0.0)  are defined i n equation  From Table I , entry 6 . , From Table I , entry 7 . Reference [39] .  d  (4.2.3).  D  c  Q (0)(% error) (absolute)  f  ^Reference [ 8 ] . Reference [38] T h e o r e t i c a l value.  e  -3.0000  ;  110  o f h i s f u n c t i o n s a r e n o t 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 n o t be  calculated.  These CSO's r e s e m b l e H u l t h e n 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 t h e Os H u l t h e n  orbital  with a  and a ^  d/?/s  2 S  CSO  CSO  A p p a r e n t l y o n l y Os and l p H u l t h e n o r b i t a l s have been i n v e s t i g a t e d but t h e y g i v e s u b s t a n t i a l improvements 64 i n energy o v e r STO's. Perhaps t h i s e x p l a i n s t h e s l i g h t e n e r g y improvement o f t h e CSO f u n c t i o n s o v e r t h o s e 30 -52 of Matsen and c o w o r k e r s ^ ' ^  shown i n T a b l e X I V f o r  comparison. S l i g h t l y more c o m p l i c a t e d wave f u n c t i o n s were also investigated.  An n o r b i t a l CSO, X'"  c a n 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 w i t h o u t f o r m a l  ,  Ill  c o n s t r a i n t p r o c e d u r e s as f o l l o w s .  The o r b i t a l ,  (jP  ,  in  =  j6 zLCc\(  = 0  (4.2.6)  c a n be e x p r e s s e d ( t o a n o r m a l i z a t i o n  C P = .  ^Lcdxc-^Xj)  '  The c h o i c e o f "Xj i s a r b i t r a r y . 0  /5  CSO's i s d e m o n s t r a t e d  a  s  constant)  u  m  o  f  cso  's  The c a s e n = 3 f o r  i n T a b l e XV f o r t h e f u n c -  tion  There a r e a c t u a l l y t h r e e i n d e p e n d e n t (without the normalization  linear coefficients  f a c t o r ) i n t h e CI expansion  i n s t e a d o f 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 2 """ ^ c  s  r e e  '  to v a r  y«  also exist i n t h i s function.  £ -type  corrections  Whether o r n o t t h e y would  T a b l e XV.  (J/^  (l)  , a t r u e C I 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  spin-optimized  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 t h e  core O r b i t a l Exponents -y 3.5  %  HI  A-15  STO Function 3.3  2.2 24.0  15.0  20.0 2.065  14.0  0.7 0.7  1.82 1.8  v»  v *"  3.0  2.5  v  5.0  Properties Function  % *  Energy  Q (0)(% S  error)  Q (0)(% e  error)  k=0 k*0  -7.446249 -7.446276  0.2298(0.6) 0.2299(0.6)  13.8083(0.2) 13.8095(0.2)  -3 .0000 -3 .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 e q u a t i o n i s d e f i n e d by e q u a t i o n  (4.2.7). (4.2.8).  113  be f o u n d w i t h e n e r g y m i n i m i z a t i o n s u b j e c t t o a c o n 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 , c o n s t r u c t e d f r o m j£f the product  XxpX^f,  .  ,  was  If a configuration involving  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 +C  For the The  r e s t r i c t i o n Jzf> = Thus  CI f u n c t i o n . sented  (X rJ ( XzT&X 3P  ©. ke j] +  configurations k = 0 automatically.  Z 5  ization.  3  (4.2.8)  s i m p l i f i e d exponent  c a n  optim-  toe expanded as a f o u r term  I t s exponents and p r o p e r t i e s a r e  i n T a b l e XV.  pre-  When compared w i t h t h e e n t r i e s  i n T a b l e I i t i s seen t o be a v e r y good f u n c t i o n i n d e e d for i t s size. W i t h t h e e x c e p t i o n of ^  and  a l l of these  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 .  cusp-  Self  consistency i s t r i v i a l f o r a t w o - o r b i t a l expansion with a linear constraint. The  s u c c e s s 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 t h e s t a b i l i t y of p r o p e r t i e s w i t h r e s p e c t to the S was  -like corrections  examined f o r t h e s p i n - o p t i m i z e d (k = 0)  F i g u r e 6 shows a r o u g h energy c o n t o u r map o f J/5  and  J/5  .  ^  3  as a f u n c t i o n  F i g u r e 7 shows t h e same energy  114  F i g u r e 6.  E n e r g y c o n t o u r map o f £r ( T a b l e X I V ) E n e r g y v e r s u s exponents Sf , S/ ' of 3  s  &  - t y p e cusp c o r r e c t i o n  s  orbitals.  IS  14.0  13.0  Si  15.0  20.0  - 7 . 4 4 5 3 2 9 \ -7.445339/ -7.445317  21.0  -7.445344  16.0 -7.445277  -\. 445327  22.0  -7.445326  23.0  -7.445324,  -V.44533B  -7.445310  24.0  -7.44533O  -7-H5343  -7.445317  25.0  -7.445b 17  \7.445349  -7.445345  -V.445321  26.0  -7.4453JU  -7.445348  -7.445346  -7.445525  7  7 . 4 4 5 3 4 7 \ -7.4453^4  -7.445302  115  F i g u r e 7- P r o p e r t i e s o f (£3 ( T a b l e X I V ) c o r r e s p o n d i n g t o energy c o n t o u r s i n f i g u r e 6. J  13-0 20.0  ///  is  14.0  16.0  15.0 Q = 0.2334 Q =13.8273 s  e  21.0  22.0  Q  e  =  °"  2336  Q =13. 8093 e  23.0  24.0  25.0  26.0  K=  0. 2343 '=13. 8471  Q  e  =  °^ ,2345  Q =13 e  £76  116  c o n t o u r s superimposed on t h e c o r r e s p o n d i n g p r o p e r t i e s . The s e n s i t i v i t y o f p r o p e r t i e s t o changes Si's  in  5ts  ,  i s seen t o be s m a l l around t h e e n e r g y minimum.  The main energy c o n t r i b u t i o n s come f r o m STO's.  Optimization  with respect  to  X is , \ is'  S\s , £is  is  more e a s i l y a c c o m p l i s h e d and a s e n s i t i v i t y s t u d y n o t as e s s e n t i a l . any  These f i g u r e s answer t h e q u e s t i o n ,  £ - t y p e exponent be employed  i n a basis?  'Can  1  A n o t h e r 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 t h e 41 s i z e of b a s i s .  Schaefer et a l  compare t h e b a s i s  dependence o f Q ( 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 s  functions  o f t h e f i r s t row e l e m e n t s , b o r o n t h r o u g h  f l u o r i n e , and c o n c l u d e 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 . P e r h a p s t h e p r e s e n t s u c c e s s e s do n o t a r i s e from s a t i s f a c t i o n of the nuclear that Wesbet's ^ 2  excellent also.  cusp c o n d i t i o n .  Recall  6 -type functions are  r e s u l t s with  A t r u e CI wave f u n c t i o n ,  (16  t e r m s ) was formed w i t h t h e same b a s i s as t h e c u s p - s a t i s fying function,  <^  » "k w i t h f r e e v a r i a t i o n o f a l l Du  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 , (4 t e r m s ) d i d n o t use t h e orbitals.  & -type  The e f f e c t s o f t h e s e  seen i n T a b l e X V I . u t i l i z e s t h e <f  &  fflj"  ~K\s cusp c o r r e c t i o n o r b i t a l s c a n be  (Table XIV), of course,  o r b i t a l s and has t h e c o r r e c t c u s p .  Table XVI. CI Function  True C I f u n c t i o n s formed from STO b a s i s  Configurations  Corresponding CSO F u n c t i o n (with orbitals)  * i*  l i s t e d i n T a b l e XIV.  1  XisXisXzs JXIS^/SXzs  Spin Functions  fXisXity^s  Xis ?Cis Xzs j X/s X/s FXzs) Xs X/s Xzs}  T  "*W5 V ' V* A/5 V AZS"V 'A/5 ~V "X/S V 'A2-S 9  w  1  (without b ' orbitals)  XisX15  X2S , X/s~X>ts Xzs  ©, ,©i  Properties Function  Energy  Q (0)(% error)  %'  -7.445560  0.2239(3.2)  -7.444890  0.2416(4.4)  S  Q (0)(% e  error)  It  I  13.9086(0.5)  -2.9249  -1.9217  13.4671(2.7)  -2.8956  -3.1253  5  118  One must keep i n mind t h a t t h e f u n c t i o n o f Brigman and * Matsen  has a r e a s o n a b l e 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 . t h e i r r e s u l t s as e x p e c t e d .  closely duplicates A d d i t i o n of  b i t a l s t o t h e b a s i s improves Q ( 0 ) s  fact,  CI) to  .  The p r o g r e s s i o n  -type o r -  considerably  , w i t h a v e r y p o o r c u s p , has  s i m i l a r to  £  properties  from  (16 t e r m  (2 t e r m c u s p - s a t i s f y i n g f u n c t i o n ) ,  shows a 2.2% improvement  and i n  however,  of the e r r o r i n Q (0) at the s  ,  .  _ZL  expense o f 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  satisfaction. Unfortunately  i t c o u l d be c o n c l u d e d f r o m t h i s  s t u d y , t a k i n g i n t o a c c o u n t t h e r a t h e r good s p i n d e n s i t i e s of  and t h e f u n c t i o n o f Brigman and M a t s e n , S  - t y p e o r b i t a l s r a t h e r than c o r r e c t cusp  that  conditions  might be r e s p o n s i b l e f o r t h e e x c e l l e n t r e s u l t s . A u t h o r s 5 14- 15 7427 ' ' ^' 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 - t y p e terms i n h i s b a s e s .  T h i s work does d e m o n s t r a t e t h a t n u c l e a r provide  cusp  conditions  a t h e o r e t i c a l avenue t o N e s b e t ' s a p p r o a c h even  though t h e y cannot y e t be s a i d t o a f f e c t , t o any  great  extent, the accuracy of c a l c u l a t e d s p i n d e n s i t y . Further  i n v e s t i g a t i o n w i l l d e t e r m i n e ,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 o f Goddard  The f u n c t i o n o f Brigman and M a t s e n [32J i s essentially t h e f u n c t i o n o f H u r s t et a l [30] with a triplet 'core ; s p i n term. v  119  and  Ladner  a study.  25  , K a l d o r and H a r r i s  24  a r e i d e a l f o r such  E f f e c t s o f exponent o p t i m i z a t i o n , s i z e o f  STO b a s i s , i n c l u s i o n o f without nuclear  &  - t y p e t e r m s w i t h and  cusp c o n s t r a i n t s , and e x t e n s i o n o f  CSO's t o l a r g e r systems should be examined.  4.3  2 A p p l i c a t i o n t o the lowest l i t h i u m The  P state  method o f t w o - o r b i t a l CSO's i s now u t i l i z e d  to c a l c u l a t e Q (0) f o r the l i t h i u m 2 P s t a t e . s  2  function similar to  A  Tp'i > i = 1 > 2 , 3 f o r l i t h i u m S 2  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~ - t y p e c u s p  were a l s o f o u n d h e r e , a t minimum e n e r g y . t r i v i a l l y s a t i s f i e s the nuclear  corrections Since  c u s p c o n d i t i o n no c o n -  s t r a i n t i s needed t o be a p p l i e d t o t h e o r b i t a l ,  (pzp = conditions  Xzp+ C\ /v3p  .  The more g e n e r a l  coalescence  ( 2 . 3 . 8 a ) c o u l d have d e t e r m i n e d c-^. The  f u n c t i o n p r o p e r t i e s a r e compared w i t h p r e v i o u s l y r e s u l t s i n Table XVII,  o r d e r e d by e n e r g y ,  published  whereas  p  CSO's f o r l i t h i u m  S groundstate give  to the spin-optimized  similar results  c a l c u l a t i o n s (compare t h e e l e v e n t h  entry 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 t h e l o w e s t  -n . ,. „ Description of wave f u n c t i o n s  „ Reference  P state of lithium.  -[-, Energy  Spin density a t the nucleus  Electron density at the nucleus  Q (0)  Q' (0)  s  Reference  e  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  13.5501  pres  S p i n - o p t i m i z e d DODS w i t h CSO's  . a,b present ' -7.377569 -0.02234 p r e s e n t  S p i n - o p t i m i z e d EHF  24  -7.380087 -0.0169  24  S p i n - o p t i m i z e d EHF  25  -7.380116 -0.0172  2-5  208  23  -7.4-0366  4-5-term C I  67  -7.4-0838  -0.02222  75  Experimental  67  -7.41016  -0.0181  10  a  -term CI  *is=3.27, £ = E  £  S  5S =2.08,&#">>S\~ =17.0,  =-3.0000  5  13.7065  Sp =0.526 f o r t h e f u n c t i o n d e f i n e d i n ( 4 . 3 . 1 ) .  M u l t i p l i c a t i o n o f exponents b y t h e s c a l e f a c t o r 1.00144 g i v e s a f u n c t i o n w i t h p r o p e r t i e s E=-7.377584, Q (O)=-O.02226, Q (0)=13-6031. fe  e  121  for  lithium 42  Goddard  2P t h e r e i s a s u b s t a n t i a l  difference.  has c a s t doubt on t h e r e l i a b i l i t y  perimental spin density.  I f the presently  o f t h e exaccepted  e x p e r i m e n t a l v a l u e i s a c t u a l l y t o o low t h e n t h e 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 t h a n t h e s p i n o p t i m i z e d EHF f u n c t i o n s .  On t h e o t h e r hand i n c l u s i o n  of c o r r e l a t i o n i n t h e K s h e l l o r b i t a l s w i l l  decrease  t h e magnitude o f t h e c a l c u l a t e d (DODS) v a 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 a s s e s s t h e v a l u e o f CSO's.  122  CHAPTER V  SUMMARY AND CONCLUDING REMARKS The  hypothesis  c o n d i t i o n s should nucleus,  that s a t i s f a c t i o n of nuclear  cusp  l e a d t o good p o i n t p r o p e r t i e s a t t h e  was i n v e s t i g a t e d f o r a p p r o x i m a t e wave f u n c t i o n s  by e m p l o y i n g 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 n e c e s s a r y i n t e g r a l cusp c o n d i t i o n s , a l t h o u g h correcting the free v a r i a t i o n a l electron density at t h e n u c l e u s ( a s f o u n d by Chong and Yue f o r h e l i u m 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 h a t t h e magnitudes o f e r r o r b e f o r e s t r a i n t were s i m i l a r .  t o an e x t e n t  and a f t e r c o n -  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 a t t h e n u c l e u s and cusp c o n s t r a i n t s were f o u n d .  The C I f u n c t i o n s w i t h  t y p e s p i n terms s t u d i e d had t h e p r o p e r t y free v a r i a t i o n a l value  of  was t r u e when _L7  Necessary weighting  that i f the  was g r e a t e r t h a n -Z  some improvement o c c u r e d upon f o r c i n g the opposite  ®2.-  JT  = -Z w h i l e  was l e s s t h a n -Z.  c o n s t r a i n t s d i d n o t appear t o be  useful f o rcalculating Q ' (0). e  s  But w e i g h t i n g  the f r e e v a r i a t i o n a l groundstate e i g e n v e c t o r  with  closely  123  approximated t r a d i t i o n a l diagonal 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 t h e method o f Weber and Handy  to  y  avoid computational  problems.  For c o n s t r a i n t s t h a t  a r e not t o o s e v e r e ,  (the c o n s t r a i n e d f u n c t i o n i s almost  equal to 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 procedure should provide  a good e s t i m a t e  weighting  of a t r u e  diagonal constraint with less 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 , spin-optimized, functions.  extended Hartree-Fock  O n l y a m i n u s c u l e b a s i s was  employed, because  t h e c o n s t r a i n t f u n c t i o n s were e v a l u a t e d a t i o n i n t e r a c t i o n form. was  The  t h e appearance of D i r a c  i n configur-  most i m p o r t a n t &  r e c t i n g t h e cusp when e n e r g y was  result  -like orbitals roughly  cor-  minimized. p  V e r y good s p i n d e n s i t i e s f o r t h e 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 n e c e s s a r y t o d e t e r m i n e 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 . Q u e s t i o n s t o be answered, i n c l u d e (1)  W i l l a larger basis adversely  (2)  Will  &  effect Q (0)? s  - 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 t o improve Q ( 0 ) , appear when a l a r g e r b a s i s i s used? s  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  with  t h e 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  o r 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 o f 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 a t t h e nucleus. A scheme f o r n u c l e a r  cusp c o n d i t i o n s f o u n d i n  Table X V I I I u n i f i e s the v a r i o u s approaches. essary  The nec-  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  (5.1.1)  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 dictates that i n d i v i d u a l electron o r b i t a l s i n must s a t i s f y t h e cusp c o n d i t i o 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 y p e o f b a s i s f o r a n a l y t i c a l EHF a t o m i c wave  this  functions  of b o r o n t h r o u g h f l u o r i n e w i t h t h e e x p r e s s e d purpose of improving Q f ( 0 ) v a l u e s .  He c o n c l u d e d t h a t  c o n v e r g e d much f a s t e r ( w i t h r e s p e c t  Q (0) s  t o the s i z e of the  Table X V I I I .  Hierarchy  Wave f u n c t i o n  o f n e c e s s a r y 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.  S u f f i c i e n t cc  E x a c t , <|  N e c e s s a r y and. S u f f i c i e n t cc 1) 2)  Kato Pack and B y e r s Brown / > § =0° 4 7  N e c e s s a r y cc  I n t e g r a l cusp c di t i o n s o f Chong'  2  3) CI f u n c t i o n w i t h o u t  STO c u s p - s a t i s fying basis  CSO's  lj) I n t e g r a l cusp • c o n s t r a i n t s of Chong c 2) Weighted cusp constraints 6 2  0  SCF one-electron orbital,  STO c u s p - s a t i s f y i n g basis of Roothaan and  1)  I n t e g r a l cusp c o n s t r a i n t s of  2)  Cusp c o n s t r a i n t s of Weber. Handy and P a r r CSO's { ( % }  Chong53  6 8  3)  c  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 t o 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 t y p e i s specified.  a  Cusp c o n d i t i o n s ( c c ) . c  This  work.  126  two o r b i t a l s s h o u l d be i m p o r t a n t b o t h f o r energy and d e s c r i p t i o n o f t h e r e g i o n about t h e n u c l e u s .  Goddard's  c o n c l u s i o n s do r e i n f o r c e t h e i d e a b e h i n d t h i s work. The more g e n e r a l o n e - e l e c t r o n o r b i t a l s , {CPi} , c a n be l i n e a r l y c o n s t r a i n e d t o s a t i s f y s u f f i c i e n t and n e c e s s a r y cusp c o n d i t i o n s . orbitals, advantages  The r e s u l t i n g  cusp-satisfying  , a r e f l e x i b l e and do n o t show t h e d i s o f t h e s e t , {X_i}  The P a u l i p r i n c i p l e e n s u r e s t h a t  i s a l s o a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n .  Thus  t h e w e i g h t e d cusp c o n s t r a i n t s a r e d e r i v e d i m m e d i a t e l y by  integration:  ( • f l  (5.1.3)  *2./?fc)\%jy  These c o n d i t i o n s a r e s u f f i c i e n t i f t h e y h o l d f o r each member o f a complete are t h e s p e c i a l  f  set.  The n e c e s s a r y cusp c o n d i t i o n s  case,  = 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 laps contribute t o the quantity t h a n j u s t ( ^>(k)  ).  over-  </7^/*/^rather  127  There i s a f i n a l comment on cusp and c o a l e s c e n c e conditions.  Experiments  a r e b e i n g performed  electron a n n i h i l a t i o n s i n molecules.  on p o s i t r o n -  One would  expect  e f f e c t s from a n n i h i l a t i o n t o be e x t r e m e l y dependent 77  on t h e wave f u n c t i o n a t c o a l e s c e n c e .  The u s e f u l n e s s  and a p p l i c a t i o n o f cusp c o n s t r a i n t s s h o u l d s t i l l be examined w i t h r e s p e c t t o t h i s i m p o r t a n t new development.  128  BIBLIOGRAPHY 1.  D. P. Chong and W. B y e r s Brown, J . Chem. Phys. 45_, 392 ( 1 9 6 6 ) .  2.  R. CD. P a c k and W. B y e r s Brown, J . Chem. Phys. 45_, 556 ( 1 9 6 6 ) .  3.  D. P. Chong, T h e o r e t . Chim. 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J o n e s , Can. J . Chem. 41 255,  84.  (1965). 586 ( 1 9 6 3 ) .  D. P. Chong and M. L. B e n s t o n , J . Chem. P h y s . 42,  1302  (1968).  85.  J . T h o r h a l l s s o n , J . L a r c h e r and D. P. Chong, T h e o r e t . Chim. A c t a ( B e r l . ) , i n p r e s s .  86.  R. J . Loeb and Y. R a s i e l , t o be p u b l i s h e d .  87.  C. C. J . Roothaan, J . Chem. P h y s . 1^, 1445 ( 1 9 5 1 ) ; E r r a t a , J . Chem. P h y s . 2 2 , 765 ( 1 9 5 4 ) .  88.  J . C. S l a t e r , Quantum Theory o f M o l e c u l e s and S o l i d s , V o l . 1, ( M c G r a w - H i l l T o r o n t o , 1 9 6 3 ) , Appendix 9 .  133  APPENDIX A  ATOMIC UNITS A d i s c u s s i o n o f t h e s i g n i f i c a n c e and u s e f u l n e s s of atomic u n i t s (a.u.) i s found i n r e f e r e n c e  [78^  A r e c e n t t a b u l a t i o n o f v a l u e s of t h e fundamental cons t a n t s a p p e a r s i n [79]  .  The b a s i c a.u. a r e d e f i n e d  here i n c.g.s. u n i t s . Quantity  a.u.  mass  1 = #k = 9.1091x10  charge  1 = e  length  1 bohr = Q» =  a n g u l a r momentum  L 1 = -* 7? = _h_ = ^ 1.0544x10-27 ^ e r g sec.  energy  1 hartree =  m a g n e t i c moment  g.  = 4.80298X10" J j £  10  Z  e.s.u. =  5.29167xl0 cm. _9  i  = 4.3594x10 Q°  1 bohr magneton = Cn  3.?n C e  =  erg  p-i 9.2732xlQ-^- erg Gauss  F o r a t o m i c ( a s opposed t o 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 o f mass i s u s e d :  where M  i s t h e mass o f t h e  I  134  nucleus.  The a.u. f o r l e n g t h , e n e r g y , e t c . a r e r e -  defined  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 .  135  APPENDIX B  SOME IMPORTANT TYPES OF APPROXIMATE ATOMIC WAVE FUNCTIONS F i r s t a remark on t h e symmetry o f atomic functions.  wave  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  , %f  z  , Jl  , the  z  t o t a l s p i n and o r b i t a l a n g u l a r momenta and t h e i r Z components r e s p e c t i v e l y .  An e x a c t wave f u n c t i o n , t h e r e f o r e ,  has s h a r p v a l u e s f o r t h e s e o b s e r v a b l e s and i t seems proper t h a t approximate  wave f u n c t i o n s s h o u l d t o o .  Because e l e c t r o n s a r e f e r m i o n s , t h e P a u l i holds also.  principle  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 ,  will  d e s i g n a t e t h i s symmetry. The most w i d e l y u s e d a p p r o x i m a t i o n i n atomic and molecular physics i s the Hartree-Fock  (HF) method  a r r i v e d a t by c o n s i d e r i n g each e l e c t r o n t o 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 o t h e r This r e s u l t s d i r e c t l y i n a single p a r t i c l e t i o n f o r t h e approximate cannot  electrons.  interpreta-  f u n c t i o n where an e l e c t r o n  experience d i r e c t i n t e r a c t i o n s w i t h the others.  The HF f u n c t i o n i s t h e v e 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 one-electron  136  spin orbitals:  The  (/{• may be expanded i n a c o m p l e t e s e t o f  functions.  I f t h i s expansion i s truncated  one-electron  f o r prac-  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 a p p r o x i m a t i o n t o the t r u e HF f u n c t i o n  results. 17  There a r e a number o f r e s t r i c t i o n s be made on t h e most g e n e r a l  HF o r b i t a l s  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 (l)  The s p i n o r b i t a l , into  so t h a t t h e ,  s h o u l d be s e p a r a b l e  ( u n r e s t r i c t e d ) case i s  The o r b i t a l s h o u l d be s e p a r a b l e i n t o and a n g u l a r components.  (pft,e,<p) *  ,  s p i n and o r b i t a l components.  The more g e n e r a l  (2)  of  ' t h a t must  The p r e s e n t d i s c u s s i o n  U;0lYi(e,4>)  a p p l i e s t o atoms  only.  radial  137  (3)  I f (2) i s true, of  (4)  ^  s h o u l d be i n d e p e n d e n t  UtfH  s h o u l d be i n d e p e n d e n t  .  I f (1) i s t r u e , of %  U{(H  *  C l o s e d - s h e l l systems p r e s e n t no p r o b l e m s .  A l l t h e 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 t h e 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) o r ( u s u a l l y ) j u s t t h e HF f u n c t i o n .  Open-shell  systems h a v i n g an u n p a i r e d s p i n e x h i b i t c o r e p o l a r i z a t i o n only 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 H a r t r e e - F o c k (UHF) a p p r o x i m a t i o n , a m b i g u o u s l y 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 .  Itis  a l s o r e f e r r e d t o as a s p i n - p o l a r i z e d H a r t r e e - F o c k . Eeleasing  o f t h e r e s t r i c t i o n s ( 2 ) , ( 3 ) has n o t been  i n v e s t i g a t e d t o any  extent.  Sharpness of  , c v ^ . » oCz  ,  c a n 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 t h e projection operator(s).  The p r o j e c t e d  H a r t r e e - F o c k (PUHF) i s t h u s o b t a i n e d  appropriate  unrestricted  by p r o j e c t i n g a  ( m i n i m i z e d ) UHF f u n c t i o n t o have s h a r p fe/* " .  Spin  2  p r o j e c t i o n before minimization  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 t h e extended H a r t r e e - F o c k ( E H F ) , 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  Hartree-  Fock o r s p i n e x t e n d e d H a r t r e e - F o c k (SEHF). *The p r e s e n t d i s c u s s i o n a p p l i e s t o atoms  only.  138  Such, p r o j e c t e d f u n c t i o n s a r e no l o n g e r t r u e HF f u n c t i o n s s i n c e t h e y c o n t a i n more t h a n one d e t e r m i n a n t , hut t h e y have an i n d e p e n d e n t p a r t i c l e i n t e r p r e t a t i o n i f t h e y a r e c a l c u l a t e d w i t h a v e r a g e d 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 o f the  HF method, a l t h o u g h also provides  leading t o conceptual  serious shortcomings.  advantages,  Electrons are per-  m i t t e d by t h e f u n c t i o n a l f o r m o f t h e HF f u n c t i o n t o come t o o c l o s e t o g e t h e r . m a t i c a l l y provides  The P a u l i p r i n c i p l e  a 'Fermi h o l e ' f o r c o r r e l a t i n g t h e  movement o f two e l e c t r o n s w i t h p a r a l l e l (determinantal) The  auto-  wave f u n c t i o n c a n v a n i s h  spins—the identically.  ' c o r r e l a t i o n h o l e ' o r '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 n o t e x i s t i n HF f u n c t i o n s . The  method o f 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) i m p r o v e s on t h e HF p r o c e d u r e by a l l o w i n g electrons having  d i f f e r e n t spins (and thus not a f f e c t e d  by t h e 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 positions.  To be an e i g e n f u n c t i o n o f  tion i s usually spin projected.  a DODS f u n c -  The t e r m , 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 . ( e x c e p t f u n c t i o n s b u t i m p l y i n g a degree o f  EHF)  approximation  below t h a t o f 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 spin p r o j e c t i o n operator, the o r b i t a l s  are  expanded i n t h e 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  P, = s  l^QiYi  large, ,  <pf = s  *Z£r Xi (  ,  etc.  whereas f o r  the  orbitals  <P  a r e expanded i n a s m a l l  basis—so  t h a t t h e i r resemblence t o accurate a n a l y t i c a l 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 c a n be  HF-type  different  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 c a n be made h e r e . 'Open s h e l l ' c a n 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 d e s c r i b e d by t h e DODS method.  to a s p l i t  shell  H e l i u m f o r example  140  lias a c l o s e d \( s h e l l h u t a DODS d e s c r i p t i o n i s an open shell description.  E n t r i e s 6, 7, 8, 9 i n T a b l e 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 h a v i n g K  split  shells. There a r e two m a i n t y p e s o f f u n c t i o n s  correlation:  containing  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)  function  ( a k i n d o f g e n e r a l i z a t i o n o f t h e 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 t o as a Hylleraas-type  function).  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 o f a sum o f a n t i s y m m e t r i z e d p r o d u c t s o f oneelectron spin orbitals.  A complete o n e - e l e c t r o n ducts can describe /  ^  i s truncated  C I  secular equations.  basis with a l l possible  an e x a c t f u n c t i o n . and t h e  In practice  determined by the  usual  I f a f u l l b a s i s ( i n t h e sense t h a t  a l l c o m b i n a t i o n s o f any p a r t i c u l a r t r u n c a t e d (fy s » w i t h a l l p o s s i b l e 9  ^  ,  , «S"  numbers, a r e p r e s e n t ) i s u s e d , an e i g e n f u n c t i o n  pro-  o f sJ  , c£  X  set o f quantum  i s automatically , Wz  ,  • Or  d e t e r m i n a n t s may be grouped t o i n d i v i d u a l l y be e i g e n f u n c t i o n s o f these operators Any  ( t h e examples i n t h e  complete set o f one-electron  functions w i l l  text).  generate  141  a CI f u n c t i o n , G a u s s i a n o r b i t a l s , S l a t e r - t y p e (STO's), Laguerre p o l y n o m i a l s ,  etc.  a physically r e a l i s t i c basis.  The  f r o m s l o w c o n v e r g e n c e and  STO's  t h e g r o w i n g number of i m p o r particles.  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 r . ., e x p l i c i t l y and  lithium  The  basis set.  terms are the two  e l e c t r o n s , and  £  well.  like  coefficients.  l,j , r. + r . r  r  3  i ~ ,i r. . r  c o u l d be i n c l u d e d  The .number of p o s s i b l e  The  necessary i n t e g r a l s  to  evaluate.  as  are:  (ps  e n d o u s l y w i t h t h e number o f (2)  c's  t/Texp(-«/7-fa-rr )  D i s a d v a n t a g e s o f t h i s method (1)  the  £7J-is  = K rJif h£ Factors  S  l i n e a r l y independent s p i n  are v a r i a t i o n a l l y determined l i n e a r  <p<j«**h  p  ^®  doublet functions f o r three  A form f o r the  expanded  Such a f u n c t i o n f o r  atom c o u l d a p p e a r f '  Si  interelectronic  so a r e not  i «J  i n a one-electron  provide  CI a p p r o a c h s u f f e r s  t a n t c o n f i g u r a t i o n s w i t h t h e number o f  coordinates,  orbitals  increases  trem-  electrons.  can be  complicated  142  APPENDIX C  CONSTEAINED VARIATION C.l  Introduction Constrained  v a r i a t i o n i s a t e c h n i q u e t h a t "builds  selected information,  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 approxi m a t e wave f u n c t i o n .  The  purpose i s the  anticipated  improvement o v e r t h e 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 expectation mization  values.  B a s i c a l l y t h e p r o b l e m i s the  o f energy of a t r i a l f u n c t i o n w h i l e  i t t o have p r e d e t e r m i n e d p r o p e r t i e s .  mini-  forcing  Since the  procedure  removes d e g r e e s o f freedom i n t h e v a r i a t i o n a l c o e f f i c i e n t s o f t h e 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 ZlE,  sacrifice,  from the  energy  energy o f a f r e e v a r i a t i o n a l  function. 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  first  intro-  o-i  duced by M u k h e r j i and  Karplus  and b a s i c t h e o r y op  d e v e l o p e d by R a s i e l and  Whitman  I n i t i a l successes i n a p p l i c a t i o n s to refinements ^' ^ ' 6  3  '  4  *  59  '  84  '  1  85  e q u a t i o n s a r e now  ? ' 2  73 and  nn  and O-l  '  B y e r s Brown.' O p  '  Q7 D  have l e d  further applications.  Methods o f s o l v i n g c o n s t r a i n e d presented.  was  secular  143  C.2  Single The  E  constraints  energy o f a t r i a l f u n c t i o n  =  <  I H  1  7  ,  (  ?  C  i s t o be m i n i m i z e d s u b j e c t  to a constraint  (lFMilTP>/<y /W>  = A<  /  conveniently  2  .  1  )  --  (0  2  2)  expressed  ( C 2 . 3 )  can be any o b s e r v a b l e o r a t t r i b u t e n o t commuting with  //  .  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 o r 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 o f t h e Lagrange m u l t i p l i e r , X  c o n s t i t u t e s t h e major p r o b l e m i n a p p l y i n g  ,  constraints.  144  The  f o r m of (C.2.4) r e s t r i c t s C 70  operator.  B y e r s Brown'  has  t o "be an  Hermitian  d e v e l o p e d t h e most  s i v e t r e a t m e n t of c o n s t r a i n e d v a r i a t i o n by z\  as a p e r t u r b a t i o n p a r a m e t e r .  In t h i s  exten-  considering perturbation  a p p r o a c h t h e energy i n (C.2.4) i s d e s i g n a t e d  as Efj_ t» c  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 Zn /*> (C2.5) E  f ict  where E  f l o t  E \ E n  -  i s the i& o r d e r p e r t u r b a t i o n energy. (A) .  ^  =  theorem. )[ — A opt  {  m  c  y —  y. Cf/l)  toy  t h e Hellmann-Feynman  When t h e c o n s t r a i n t (C.2.3) i s s a t i s f i e d , » t h e optimum v a l u e o f  /opt  The  Since  X  r\ £_  .  Because  =o.  (c.2.7)  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 a p p r o a c h i s t o be of v a l u e and after  'k' t e r m s .  The  Eayleigh-Schrodinger  E^^  can be  truncated  are r e a d i l y e v a l u a t e d  p e r t u r b a t i o n e x p a n s i o n and  f o r the a  value  145  for  s^opt  c  a  n  D e  o b t a i n e d by i n v e r t i n g t h e 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 c a n be e v a l u a t e d by a d o u b l e p e r t u r b a t i o n a p p r o a c h , o r d i r e c t l y f r o m t h e 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 o f t h e a p p r o x i m a t i o n , "X opt  A  ,  0  depends on t h e convergence o f t h e 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 question  of convergence o c c u r i n g i n t h e p e r t u r b a t i o n approach. I t i s s i m p l e t o a p p l y ; (C.2.4) i s r e p e a t e d l y s o l v e d  A  f o r d i f f e r e n t values of found. f o r \opt  u n t i l C( A t op  ) = 0is  The p r o b l e m h e r e i s t h a t a good i n i t i a l i sdifficult.  c l o s e l y e s t i m a t e Xopt  guess  I f one i s f o r t u n a t e enough t o , i t may be o b t a i n e d t o h i g h  a c c u r a c y by s u 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 o r e x t r a p o l a t i o n s o f C( ^  ).  72 Chong'  has developed  a perturbation-iteration  a p p r o a c h 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 t h e p e r t u r b a t i o n approach.  C(  )\  ) i s expanded i n a T a y l o r  146  s e r i e s about a n e s t i m a t e  Cft„) = CQJ + +I  Dv( (>»)  where (ri']  =  +  coefficients.  and  E  E q u a t i o n (C.2.8)  ^  i sdesired C ( ) \ , h +  i n v e r s i o n of the truncated  s u f f i c e t o give  Multiple  X.  (C2.10)  w i t h t h e s o l u t i o n t o (C.2.9) f o r  "X  an e s t i m a t e f o r  C.7>  +  <wM/^M>/r^j/mj>i-o  = = \0pt  Since  =  )\h+r  JTV ^7 ' )  p r o v i d e s an i n i t i a l g u e s s ,  evaluated  For  i V W C U  2 / rv\ = n  are t h e b i n o m i a l  co.)  .  /\n  .  ^ ^ .  ) i s s e t t o zero  s e r i e s (C.2.10) g i v e s  U s u a l l y a v e r y few i t e r a t i o n s  /\o],t t o d e s i r e d  accuracy.  constraints 70  B y e r s Brown'  extended h i s p e r t u r b a t i o n  to include m u l t i p l e c o n s t r a i n t s .  approach  The v a r i a t i o n p r i n c i p l e  is SE  Resulting  +-  Z  Ai£C  C  =0  s e r i e s e x p a n s i o n s and i n v e r s i o n s a r e n o t  e a s i l y worked o u t and t h u s h i s e x t e n s i o n practical.  (C3.1)  i s not too  147  73 Chong and B e n s t o n ' ^ o b s e r v e d a p p r o x i m a t e 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  conditions  = C, O.pt , ^.pt) = o  , 7j  A  t h e two Lagrange m u l t i p l i e r s , a r e c l o s e l y  estimated by s o l v i n g  C i O ^ )  for  Ci  =  C  = ^  -  x  0  ^ » ^  +  a  A  t  (C3.3)  */9~^  =  A opt  ,  ?l  =  ^lopt  .  The c o e f f i c i e n t s { flc'j] c a n be i n i t i a l l y d e t e r m i n e d from the  expectation values of  . and  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  =  C (o^o) s  Rio =  "  }  i  (A^t  , o)  =  ft*  +  fiuXort  eigenfunctions,  (free variational)  y  C-L(O O) — flz_o = ( £ ? _ y  c  f o r the  =o  148  where The  A opt, tyopt a r e optimum s i n g l e c o n s t r a i n t v a l u e s .  secular  equation  i s solved using  Xopt  evaluations of  C,  »  rf°t>t  from ( C . 3 . 3 ) .  O o p t , ??°pt),  Accurate  C ^ O i o p t , ??opt)  are  made e n a b l i n g t h e s e t o f p o i n t s ( C . 3 . 4 ) t o be i m p r o v e d . I t e r a t i o n proceeds u n t i l one  ( C . 3 . 2 ) holds.  Usually only  o r two c y c l e s are needed. An a l t e r n a t i v e t o t h i s a p p r o a c h i s s u c c e s s i v e  parametrization. 9j and  )\  f o r some v a l u e o f  i s f o u n d so t h a t C\{Y\?i) = 0. a new 7f  The p r o c e s s achieved. The  The b e s t  i s found g i v i n g  Cz. (}\ ,  A  i s fixed = 0.  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 Loeb and R a s i e l  disadvantage  have employed t h i s method.  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 f o u n d a t each s t e p i n t h e p a r a m e t r i z a t i o n w h i l e equations(C.3.2) C.4  are simple, l i n e a r , algebraic equations.  Off-diagonal constraints 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 ,  149  An o f f - d i a g o n a l c o n s t r a i n t  c a n be imposed by d e f i n i n g a new p s e u d o - d i a g o n a l c o n straint  operator^  £' = eii?xw/  + \iyxwi<2 .  cc.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 ffi'  where mined. '  G.5  i s f i x e d and  co.4.3)  i svariationally  S e l f - c o n s i s t e n c y i s achieved a f t e r every c a l c u l a t i o n o f  deter-  by r e f o r m i n g .  Off-diagonal l i n e a r constrained v a r i a t i o n of Weber and Handy No i t e r a t i o n o r p a r a m e t r i z a t i o n i s n e c e s s a r y i n  t h i s one-step a p p r o a c h . handled e a s i l y . is  M u l t i p l e constraints are  The p r e s e n t a t i o n o f Weber and Handy  followed. D e f i n e t h e c o n s t r a i n e d wave f u n c t i o n  J  150  represented  "by t h e column v e c t o r ,  o r t h o n o r m a l b a s i s s e t (%  J .  (C  , i n some  Weber and Handy c o n s i d -  ered the c o n s t r a i n t c o n d i t i o n s  f€  ,  = o  where each  to f^p) J c  f t p ,  =-  The  s e t {-{p^}  where  & «  (  r e s u l t i n g constrained secular equations  f/H-el)c I  (c.5.1)  /,..»m  defines a constraint.  can be o r t h o g o n a l i z e d  The  c =  =  m a t r i x i n the b a s i s energy of  7p  .  The  <~  These v a l u e s f o r t h e  Hamiltonian  i s the  (constrained)  j & t ' M *  V s  .  Mult-  obtains  (C5.4)  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 equations  )  b e a u t y of t h e method l i e s i n t h e  i p l y i n g (C.5.5) "by J/Df< one  =  2  -">  e l i m i n a t i o n o f t h e Lagrange m u l t i p l i e r s , /^«  ~~2  -  5  (c  i s the , and  -  are  *LXifie  i s the u n i t m a t r i x ,  c  secular  151  OB -ez)<£ = <? where  /£> =  m a t r i x and  ) M =  i s d i a g o n a l i z e d , 7?i because  977  constraints.  l ^  ~/l  )  (c.5.5)  an  ^£>*- ^2*-  Hermitian  When  extraneous r o o t s , € i  , appear  d e g r e e s o f freedom a r e a b s o r b e d i n t h e Weber and Handy show t h a t  r o o t s a l l have v a l u e z e r o .  Off-diagonal  these constrained  v a r i a t i o n i s thus reduced t o o r t h o g o n a l i z a t i o n of the s e t f ^ t ] and s o l u t i o n o f ( C . 5 . 5 ) .  152  APPENDIX D  INTEGRAL CALCULATION D.1  Primitive integrals f o r Slater-type orbitals (STO s ) 1  A g e n e r a l STO has t h e f o r m  The  Yurr^s a r e  the usual s p h e r i c a l harmonics.  Primitive  i n t e g r a l s a r e t h o s e a r i s i n g between o n e - e l e c t r o n b a s i s functions.  I n energy c a l c u l a t i o n s t h e 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 o c c u r f o r atoms:  overlap i n t e g r a l s ,  S -j (  fx^h)  yjfc)  ~^f)(iCc)V**jfc)  k i n e t i c energy i n t e g r a l s ,  nuclear a t t r a c t i o n i n t e g r a l s ,  Coulomb and exchange integrals  dfi  /" „ /*•  ?f~X*(r)_L-~)Cj(h)  ^ fc)XjfhJ_/_  A n a l y t i c a l f o r m u l a e a r e g i v e n by Roothaan?7  K  j  &  t  /  ^  153  For  cusp c a l c u l a t i o n s  integrals  the following  primitive  arise:  electron density at t h e n u c l e u s ,  Q  ,. S)  A***s^.\  X~M  7 ^ * , ^ ) ^  Formulae a r e 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 the  ^/  from  d e f i n i t i o n o f STO's ( D . l . l ) :  Q, .  _  fy ^ ,  3  >  , i f Xi , Xj orbitals  ^  = ( -JlVSifs -4-  ?• l/r/  t^L- '  vrn  \^  o  The s p h e r i c a l  ' Xj  f o r  . ^  ?'" J  . , £;  b o t h ~X\  S  orbitals  a r e n o t b o t h X.\s  are  Xts  orbitals  , i f Xc i s a XZJJ o r b i t a l and X; i s a * orbital 1 S  1 f ° a l l other combinations of o r b i t a l s r  averaging operator defined i n the text  has n o t been shown, b u t i t e n s u r e s 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  integrals  The i n t e g r a l s d i r e c t l y employed i n atomic culations  (<Pi]  a r e t h o s e between S l a t e r  cal-  determinants  where  l<  </(  i s 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 , {X}  STO's and  'V~< a r e s p i n f u n c t i o n s  oc  are  , ^  The 88  g e n e r a l method o f h a n d l i n g t h e s e 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 s i m -  p l i f i e d a p p r o a c h c a n be u s e d .  I f general  primitive  i n t e g r a l s are designated  the S l a t e r determinants  Then  (D.2.1) may be c o m p a c t l y  written  155  - pf/MM ffafite  <?&)o43)  }  >.  I f T^C) i s s p i n l e s s t h e s p i n s may be i n t e g r a t e d o u t immediately.  = <aO)£tic®  F t i  =  fae  Sb-f Scj  - f « j S l { S c e  lil'H)] + S * e  contains  operator,  e  Sb*  - 5 « . j f 6 - F S c e  i n terms o f p r i m i t i v e o v e r l a p ^1E£(K}  + S *  ft-fScj  e(\) fa its)  and J~  S t r i k e .  integrals.  s p i n — f o r example, t h e s p i n  ipjp^fc^k^  —the  fog  density  s p i n f u n c t i o n s may  be o p e r a t e d o n and t h e n i n t e g r a t e d o u t i n a n a n a l o g o u s way.  -~Q(\)&)e(i)}  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 t h e t e x t , a r e t r i v i a l ,  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 tween S l a t e r d e t e r m i n a n t s .  con-  be-  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 d e f i n e d i n T a b l e V, as a r e t h e c o n f i g u r a t i o n s i n $/o , t h e k e y wave f u n c t i o n .  The s i g n i f i c a n c e o f  e x p l a i n e d i n t h e t e x t f o l l o w i n g e q u a t i o n C3.3.4-). The n o t a t i o n , f & 1} , used below, means t h e e n t i r e c o l l e c t i o n o f c o n f i g u r a t i o n s making up t h e C I f u n c t i o n , <§( .  Properties of  i n Table XIX. the  $  / 0  The f u n c t i o n s  configurations  $  :  $,1.:  >  £»:  \sis'2s"e*_ 2S\S2S"&?.  f& ) 3  , 2S2S J.s"o _ /  z  /<W > /sts3s"ez.  /S2s'SS"&2_ 25 /s'3s 2S25'js"  "&z_ &  z  through  (ft a r e l i s t e d /8  through  have  Table XIX.  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 o f t h e wave f u n c t i o n s  Function Constraint J-/o  None j / = 2r e  None  7z  a. v3  -Energy  Q (0)  Q (0)  7.467389 7.467254 7.465334 7.464736  0.2677 0.2753 0.1715 0.1686  13.7522 13.9191 13.5519 13.8901  7.467408 7.467278 7.467394 7.467264  S  r  e  0.2244 0.2265 0.1867 0.1879  13.7538 13.9179 13.7524 13.9168  -2.9732 -3.0000 -2.9447 -3.0000 -2.9737 -3.0000 -2.9736 -3.0000  -3.2795 -3.2973 -3.0000 -3.0000 -3.1545 -3.1587 -3.0000 -3.0000  t h r o u g h <j5  /9  AE  0.000135 0.002055 0.002653  0.000130 .0.000014 0.000144  0.0 -4 +0.579772x10 -2 -0.521667x10 , > =+0.125277x10-^ > =-0.585347x10 ^ e  s  +0.569242x10 7 -0.621749x10 ^ j, >\ =+0.569979x10 7 > =-0.615931x10 ^ s  None  7.467418 7.467283 7.467418 7.467282  0.2148 0.2197 0.2201 0.2177  13.7512 13.9189 13.7518 13.9190  -2.9730 -3.0000 -2.9731 -3.0000  -2.9487 -3.0193 -3.0000 -3 . 0000  0.000136 0.0 0.000137  None  7.467424 7.467289 7.467420 7.467287  0.2048 0.2088 0.2229 0.2219  13.7515 13.9189 13.7527 13.9187  -2.9731 -3.0000 -2.9733 -3.0000  -2.7800 -2.8446 -3.0000 -3.0000  0.000135 0.000004 0.000137  0.2135 0.2153 0.2150 0.2163  13.7500 13.9192 13.7500 13.9192  0.0 , +0.5776325x10. +0.144304x10 ^ . > =+0.574015xl0"7; > =+0.100169x10"^ e  I— 1VJ1 1  7¥  None  7.467429 7.467291 7.467429 7.467291  -2.9728 -3.0000 -2.9728 -3.0000  -2.9790 -2.9870 -3.0000 -3.0000  0.000138 0.0 0.000138  00  Table XIX (continued) Function Constraint None  -Energy 7.467483 7.467354 7.467483 7.467354  r  Q (0) Q (0) S  0.2269 0.2297 0.2248 0.2268  e  13.7529 13.9168 13.7529 13.9168  -2.9736 -3.0000 -2.9736 -3.0000  -3.0271 -3.0380 -3.0000 -3.0000  A 0.000129 0.0 0.000129  0.0 +0.563295x10 £ -0.121657x10 . >* =+0.563403x10 Z > =-0.166013xl0 y  5  None  0n  None  None  7.467490 7.467356 7.467490 7.467356  0.2289 0.2310 0.2241 0.2264  13.7510 13.9176 13.7509 13.9176  -2.9730 -3.0000 -2.9731 -3.0000  -3.0622 -3.0596 -3.0000 -3.0000  _  4  -0.278766x10  0.0 -4 +0.569446x10 -0.337136x10- 5 -4 X=+0.569374x10"£ > =-0.320967x10 ?  0.2298 0.2320 0.2240 0.2263  13.7511 13.9176 13.7511 13.9176  -2.9731 -3.0000 -2.9731 -3.0000  -3.0754  -3.0741 -3.0000 -3.0000  0.000133 0.000001 0.000134  7-467495 7.467360 7.467495 7.467360  0.2287  13.7501 13.9178 13.7501 13.9178  -2.9729 -3.0000 -2.9729 -3.0000  -3.0542 -3.0590 -3.0000 -3.0000  0.000135 0.0 0.000135  0.2246 0.2267  0.0  0.000134 0.0 0.000134  7.467492 7.467359 7.467491 7.467358  0.2312  _::>  +0.570254x10 J : y  > =-0.258561x10 s  7.467479 7.467348 7.467479  0.2304 0.2360 0.2260  13.7521 13.9172 13.7517  L a r s son's 7.478025 100 t e r m correlated f u n c t i o n , see e n t r y 18 i n T a b l e I  0.2313  13.8341  -2.9733 -3.0000 -2.9732  -3.0394 -3.0988 -3.0000  0.000131 0.0  y  0.0 -4 +0.572685x10 -0.239282x10 5  >* =+0.572745x10" J  > =-0.252666xl0"' 5  None  .  > =+0.570195x10 £ e  0.0 +0.566694x10 -4 -0.487691x10"  ?  

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