UBC Theses and Dissertations

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UBC Theses and Dissertations

Studies in constrained variation Yue, Tony Chee Ping 1969

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STUDIES IN' CONSTRAINED VARIATIONS by TONY CHEE PING YUE B . S c . ( H o n o u r s ) , C h i n e s e U n i v e r s i t y o f Hong Kong, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e Department o f C h e m i s t r y We a c c e p t t h i s t h e s i s -as c o n f o r m i n g to t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF J u l y , BRITISH 1969 COLUMBIA In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may b e g r a n t e d b y t h e Head o f my D e p a r t m e n t o r b y h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r pub 1 i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f C h e m i s t r y  The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8 , C a n a d a P a t e J u l y 16f iq60  ABSTRACT Three d i f f e r e n t c o n s t r a i n t s a r e c o n s i d e r e d i n t h i s t h e s i s , namely: t h e i n t e g r a l e l e c t r o n c u s p c o n d i t i o n as a c o n s t r a i n t ; t h e o f f - d i a g o n a l h y p e r -v i r i a l theorems as c o n s t r a i n t s ; and p e r t u r b a t i o n -i n d u c e d c o n s t r a i n t s . E l e v e n a p p r o x i m a t e c o n f i g u r a t i o n - i n t e r a c t i o n w a v e f u n c t i o n s f o r t h e ground s t a t e o f h e l i u m a r e used t o t e s t t h e a p p l i c a t i o n o f t h e i n t e g r a l e l e c t r o n c u s p c o n d i t i o n as a c o n s t r a i n t . The r e s u l t s i n d i c a t e t h a t , i f t h e a p p r o x i m a t e w a v e f u n c t i o n i s f l e x i b l e enough, t h e c a l c u l a t e d e l e c t r o n d e n s i t y a t t h e n u c l e u s i s improved when t h e cusp c o n s t r a i n t i s . i m p o s e d . -1 -? However, t h e e x p e c t a t i o n v a l u e s o f r and r " do. n o t change s i g n i f i c a n t l y . . F u r t h e r i n v e s t i g a t i o n i s made on t h e use o f o f f - d i a g o n a l h y p e r v i r i a l theorems as c o n s t r a i n t s . 1 1 The t r a n s i t i o n f r o m t h e 1 S s t a t e t o t h e 2 P s t a t e o f h e l i u m i s chosen as an example. F i r s t l y , i t i s f o u n d t h a t an e n e r g y - i n d e p e n d e n t o f f - d i a g o n a l c o n s t r a i n t i s n o t u s e f u l i n i m p r o v i n g c a l c u l a t i o n o f t r a n s i t i o n p r o b a b i l i t i e s . S e c o n d l y , when t h e i i approximate wavefunction i s f l e x i b l e enough, i t e r a t i o n on the t r a n s i t i o n energy converges very r a p i d l y . F i n a l l y , the study i s extended to the i s o e l e c t r o n i c species L i + and B e + + . A five-term 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 wavefunction f o r the ground state of helium i s used to t e s t the v a l i d i t y of the perturbation-induced constraints scheme. D i f f e r e n t c o n s t r a i n t operators are constructed f o r d i f f e r e n t properties. The properties studied are - 2 - 1 +2 expectation values of r , r , r and r . Two methods of the perturbation-induced constraints are tested: in one, the f i r s t - o r d e r wavefunction i s fix e d and constant f o r a l l constrained properties; i n the other, the f i r s t -order wavefunction v a r i e s with the constrained properties. I t i s disappointing to f i n d that for t h i s p a r t i c u l a r wavefunction chosen, both methods f a i l to improve the properties studied when one imposes the c o n s t r a i n t s . TABLE OF CONTENTS ABSTRACT LIST OF TABLES LIST OF FIGURES ACKNOWLEDGMENTS CHAPTER I . INTRODUCTION 1.1 G e n e r a l cons i d e r a t i o n s 1.2 E a r l y work, on t h e c o n s t r a i n e d  v a r i a t i o n method 1.3 Theory o f t h e c o n s t r a i n e d v a r i a t i o n method 1.4 Methods o f s o l v i n g c o n s t r a i n e d  s e c u l a r e q u a t i o n 1.5 Ot h e r d e v e l o p m e n t s i n t h e con- s t r a i n e d v a r i a t i o n method 1.6 O b i e c t o f t h e t h e s i s CHAPTER I I . INTEGRAL ELECTRON CUSP CONDITION AS A CONSTRAINT 2.1 I n t r o d u c t i o n 2.2 On t h e use o f t h e i n t e g r a l e l e c t r o n c u s p c o n d i t i o n as a c o n s t r a i n t i v Page 2.3 C a l c u l a t i o n s and r e s u l t s 30 2.4 D i s c u s s i o n 39 CHAPTER I I I . OFF-DIAGONAL HYPERVIRIAL THEOREMS AS CONSTRAINTS 3.1 I n t r o d u c t i o n 57 3.2 On t h e use o f t h e o f f - d i a g o n a l 62 h y p e r v i r i a l theorems as c o n s t r a i n t s 3.3 A t o m i c t r a n s i t i o n p r o b a b i l i t i e s 66 3.4 C a l c u l a t i o n s and r e s u l t s 69 3.5 D i s c u s s i o n 80 CHAPTER I V . PERTURBATION-INDUCED CONSTRAINTS 4.1 I n t r o d u c t i o n 88 4.2 Theory ' 90 4.3 C a l c u l a t i o n s and r e s u l t s 109 4.4 D i s c u s s i o n 117 CHAPTER V. CONCLUDING REMARKS 124 BIBLIOGRAPHY 126 APPENDIX A. CALCULATIONS AND RESULTS FOR THE 130 ILLUSTRATIVE EXAMPLE IN CHAPTER I APPENDIX B. LINEAR COEFFICIENTS FOR THE FREE 134 VARIATION APPROXIMATE WAVEFUNCTIONS APPENDIX C. PRIMITIVE INTEGRALS APPENDIX D. A COMPUTING PROGRAM FOR THE OFF-DIAGONAL HYPERVIRIAL THEOREMS DOUBLE CONSTRAINT WITH ITERATION IN THE TRANSITION ENERGY v i L I ST OF TABLES TABLE Page I . C o n f i g u r a t i o n - i n t e r a c t i o n a p p r o x i m a t e w a v e f u n c t i o n s f o r t h e ground s t a t e o f h e l i u m . Each c o n f i g u r a t i o n b e l o n g s t o a I s s t a t e . 32 I I . O r b i t a l exponents o f S l a t e r - t y p e , a t o m i c o r b i t a l s used i n t h e c o n f i g u r a t i o n -i n t e r a c t i o n a p p r o x i m a t e w a v e f u n c t i o n s shown i n T a b l e I . 33 I I I . R e s u l t s o f f r e e v a r i a t i o n s ( i n a t o m i c u n i t s ) . The % e r r o r s o f t h e cusp v a l u e F and A , t h e e l e c t r o n d e n s i t y a t t h e n u c l e u s , f rom t h e i r e x a c t v a l u e s a r e g i v e n i n p a r e n t h e s e s . 34 IV. R e s u l t s o f c o n s t r a i n e d v a r i a t i o n s ( i n a t o m i c u n i t s ) . The % e r r o r s o f /S. , t h e e l e c t r o n d e n s i t y a t t h e n u c l e u s , f r o m t h e e x a c t v a l u e i s g i v e n i n p a r e n -t h e s e s . V. C o e f f i c i e n t s o f c o n s t r a i n e d w a v e f u n c t i o n s i n terms o f f r e e v a r i a t i o n f u n c t i o n s . • 41 'VI. T y p i c a l b e h a v i o r o f v a r i o u s q u a n t i t i e s as a f u n c t i o n o f >^ (as i l l u s t r a t e d by t h e c a s e o f 4* 7) • T n e % e r r o r s o f t h e e l e c t r o n c u s p v a l u e F and A , th e e l e c t r o n , d e n s i t y a t the n u c l e u s from t h e i r e x a c t v a l u e s a r e g i v e n i n p a r e n -t h e s e s . 45 -2 -] V I I . E x p e c t a t i o n v a l u e s o f r and r f o r th e more f l e x i b l e w a v e f u n c t i o n s i n t h e c a s e o f cusp c o n s t r a i n t . ' 50 V I I I . R e s u l t s o f c o n s t r a i n e d v a r i a t i o n s ( i n a t o m i c u n i t s ) f o r t h e a p p r o x i m a t e wave-f u n c t i o n s <p 4 , V4» a n < 3 ^ 4 * The % e r r o r o f t h e cusp v a l u e JT from t h e e x a c t v a l u e i s g i v e n i n p a r e n -t h e s e s . 51 TABLE IX . C o e f f i c i e n t s o f c o n s t r a i n e d w a v e f u n c t i o n s i n terms o f f r e e v a r i a t i o n f u n c t i o n s . -2 -1 X. E x p e c t a t i o n v a l u e s o f r and r i n the c a s e o f u s i n g A , t h e e l e c t r o n d e n s i t y a t t h e n u c l e u s as a c o n s t r a i n t . X I . E c k a r t - t y p e f u n c t i o n s f o r t h e 2"^ P s t a t e o f He, L i + and B e + + . X I I . O r b i t a l exponents o f S l a t e r - t y p e a t o m i c o r b i t a l s and energy ( i n a t o m i c u n i t s ) f o r t h e a p p r o x i m a t e w a v e f u n c t i o n s T 6 and V 7 f ° r t h e 1^ -S ground s t a t e o f He, Li+ and Be++. X I I I . R e s u l t s o f u s i n g the e n e r g y - i n d e p e n d e n t o f f - d i a g o n a l h y p e r v i r i a l theorem as a c o n s t r a i n t on t h e f u n c t i o n ^ 5 f o r th e l i s ground s t a t e of He. The % e r r o r o f M l , M 2 ' a n d M 3 from t h e e x a c t v a l u e s a r e g i v e n i n p a r e n t h e s e s . XIV. R e s u l t s o f f r e e v a r i a t i o n ( i n a t o m i c u n i t s ) f o r t h e l i s ground s t a t e f u n c -t i o n s and ^r, o f He, L i + and B e + + . The A E i s t h e 1 1 S - 2 1 P t r a n s i t i o n e n e r g y . XV. R e s u l t s o f c o n s t r a i n e d v a r i a t i o n s ( i n a t o m i c u n i t s ) on ^f 7 6 A N C * ' 7 f o r t h e l i s ground s t a t e o f He. The r e s u l t s f o r t h e a s s u m p t i o n o f c o n s t a n t A P a r e g i v e n i n p a r e n t h e s e s . X V I . R e s u l t s o f c o n s t r a i n e d v a r i a t i o n s ( i n a t o m i c u n i t s ) on and ^ 7 . f o r t h e l i s ground s t a t e o f L i + . t h e r e s u l t s f o r t h e a s s u m p t i o n o f c o n -s t a n t A E a r e g i v e n i n p a r e n t h e s e s . TABLE X V I I . R e s u l t s o f c o n s t r a i n e d v a r i a t i o n s ( i n a t o m i c u n i t s ) on g and ' ^¥ 7 f o r t h e 1-^ S ground s t a t e o f B e + + . The r e s u l t s f o r t h e a s s u m p t i o n o f c o n s t a n t A E a r e g i v e n i n p a r e n t h e s e s . X V I I I . C o e f f i c i e n t s f o r t h e s e v e n - t e r m d o u b l e c o n s t r a i n e d w a v e f u n c t i o n s i n terms o f f r e e v a r i a t i o n f u n c t i o n s . X I X . R e s u l t s o f d i a g o n a l p e r t u r b a t i o n -i n d u c e d c o n s t r a i n t s ( i n a t o m i c u n i t s ) . XX. R e s u l t s o f i t e r a t i v e method o f o f f -d i a g o n a l p e r t u r b a t i o n - i n d u c e d c o n s t r a i n t s ( i n a t o m i c u n i t s ) . X X I . Comparison o f t h e e f f e c t s on e x p e c t a t i o n v a l u e s o f r - 2 , r _ 1 , r and r + 2 o b t a i n e d f r o m t h e d i a g o n a l p e r t u r b a t i o n - i n d u c e d c o n s t r a i n t s (DPIC) and t h e f r e e v a r i a t i o n c o r r e c t i o n method (FVC). X X I I . V a r i a t i o n s o f t h e energy £ , A , t h e e l e c t r o n d e n s i t y a t t h e n u c l e u s and e x p e c t a t i o n v a l u e <r~2 > o f r~2 ( i n a t o m i c u n i t s ) as a f u n c t i o n o f t h e o r b i t a l e xponent f o r t h e f u n c -t i o n SB . The a b s o l u t e % e r r o r o f E , % e r r o r o f ^ and < r ~ 2 > f r o m t h e i r e x a c t v a l u e s a r e g i v e n i n . p a r e n t h e s e s . X X I I I . F r e e v a r i a t i o n a p p r o x i m a t e c o n f i g u r a t i o n -i n t e r a c t i o n w a v e f u n c t i o n s f o r t h e ground s t a t e o f h e l i u m (see C h a p t e r I I ) . * XXIV. F r e e v a r i a t i o n a p p r o x i m a t e c o n f i g u r a t i o n -i n t e r a c t i o n w a v e f u n c t i o n s f o r t h e ground s t a t e o f He, L i + and Be r l" +(see C h a p t e r I I I ) . TABLE XXV. I n p u t d a t a f o r t h e o f f - d i a g o n a l d o u b l e c o n s t r a i n t i n t h e c a s e o f ^ 7 f o r t h e ground s t a t e o f h e l i u m . XXVI. R e s u l t s from t h e o f f - d i a g o n a l d o u b l e c o n s t r a i n t c a l c u l a t i o n i n t h e c a s e o f j f o r t h e ground s t a t e o f h e l i u m . L I S T OF FIGURES FIGURE 1. V a r i a t i o n s o f (a) t h e % a b s o l u t e e r r o r o f t h e energy E , (b) % e r r o r o f •A , t h e e l e c t r o n d e n s i t y a t t h e n u c l e u s and (c) % e r r o r o f <* r - 2 > th e e x p e c t a t i o n v a l u e o f r ~ 2 a s a f u n c t i o n o f t h e o r b i t a l exponent . 2. C o r r e l a t i o n between t h e a c c u r a c i e s o f ^ and IT f o r f r e e v a r i a t i o n wave-f u n c t i o n s . The dashed l i n e i s t h e p e r f e c t c o r r e l a t i o n c u r v e . 3. B e h a v i o r o f C , % e r r o r o f r and % e r r o r o f <£± as a f u n c t i o n o f >v i n th e c a s e o f 4> y. 4. C o r r e l a t i o n between t h e a c c u r a c i e s o f /\ and IT as a f u n c t i o n o f ^ i n th e c a s e o f <p 7 . The dashed l i n e i s t h e p e r f e c t c o r r e l a t i o n c u r v e . 5. A s i m p l i f i e d f l o w c h a r t f o r t h e o f f - d i a g o n a d o u b l e c o n s t r a i n t program. ACKNOWLEDGMENTS The a u t h o r w i s h e s t o e x p r e s s h i s g r a t i t u d e f o r th e a d v i c e , h e l p and encouragement g i v e n by Dr. D. P. Chong throxaghout t h e c o u r s e o f t h e r e s e a r c h and p r e p a r a t i o n o f t h i s t h e s i s . The a u t h o r a l s o w i s h e s t o thank t h e U n i v e r s i t y . o f B r i t i s h C o l u m b i a f o r t h e award o f a U n i v e r s i t y G r a d u a t e F e l l o w s h i p (1967-1969). 1 CHAPTER I INTRODUCTION 1.1 General considerations Since i t i s very d i f f i c u l t to obtain accurate solutions f o r the Schrodinger equation f o r atoms or molecules other than simple systems such as the hydrogen atom, hydrogen molecular ion and helium atom, approximate wavefunctions must be used. The usual c r i t e r i o n f o r the determination of the best approximate wavefunction i s the minimum energy p r i n c i p l e of the v a r i a t i o n method. I t has been recognised that t h i s minimum energy p r i n c i p l e which leads to approximate wavefunctions best in terms of t o t a l energy, does not n e c e s s a r i l y give r e l i a b l e p r e d i c t i o n of other properties. For example, l e t us approximate the ground state wavefunction f o r helium by the product of two normalised screened hydrogenic Is wavefunctions, ii ( l . i . D 2 where <ptl) = [ ^ ] (1.1.2) and s*7 i s the o r b i t a l exponent. The v a r i a t i o n of (a) the % absolute error i n the energy E, (b) the % error i n Z\ , the electron density at the nucleus, 2 * and (c) the % error i n the expectation value <^~^> as a function of the o r b i t a l exponent ^ are shown in F i g . 1. The d e t a i l s of c a l c u l a t i o n s are given i n Appendix A. The minimum energy of the helium atom obtainable with t h i s approximate wavefunction i s -2.847656 a.u. when ^ i s 1.6875, at which the % absolute error i n the energy E i s 1.93 while the % errors i n J-S. , the electron density at the nucleus, -2 and the expectation value of r are -15.5 and -5.3 r e s p e c t i v e l y . I t i s quite c l e a r from F i g . 1 that with the best energy does not give the best Z\ nor <r > = < r i > + < r 2 > the best <r~ ^ > . In general, approximate wavefunctions neither y i e l d exact expectation values of operators nor s a t i s f y conditions obeyed by exact wavefunctions. But i t i s possible to improve our approximate wavefunctions i f we constrain them to s a t i s f y some of these conditions or to give expectation values * of some operators i n agreement with known values . This constraint method to improve approximate wave-functions was f i r s t suggested by Oppenheim"'" and 2 l a t e r by Coulson". Oppenheim considered constraints on t r a n s i t i o n moments while Coulson considered constraints on h y p e r v i r i a l r e l a t i o n s h i p s . The approximate wavefunction obtained from the pure co n s t r a i n t method of Oppenheim and Coulson would probably give a very high energy unless many constrain are used. In order f o r an approximate wavefunction to s a t i s f y some constraints and, at the same time, give a reasonably low energy, the con s t r a i n t method can be * By known values, we mean exact t h e o r e t i c a l values or experimental values. % Absolute error i n E o ro co • ' cn i. I i i O + -r + . + + >ft» c o r o i - i i-i r o w ^ c n o o o o o o . o o o % E r r o r s i n A , <r > 5 used s i m u l t a n e o u s l y w i t h t h e minimum energy p r i n c i p l e o f t h e v a r i a t i o n method. T h i s l e a d s t o t h e method o f c o n s t r a i n e d v a r i a t i o n s . 1.2 E a r l y work on t h e c o n s t r a i n e d v a r i a t i o n method The c o n s t r a i n e d v a r i a t i o n method was i n t r o d u c e d 3 by M u k h e r j i and K a r p l u s i n 1962. They a p p l i e d t h i s method t o t h e hydrogen f l u o r i d e m o l e c u l e . The c a l c u l a t e d v a l u e s o f d i p o l e moment and q u a d r u p o l e c o u p l i n g c o n s t a n t were c o n s t r a i n e d t o agree w i t h e x p e r i m e n t a l r e s u l t s . The same method was l a t e r 4 a p p l i e d t o t h e h e l i u m atom by Whitman and C a r p e n t e r , and t o t h e l i t h i u m h y d r i d e m o l e c u l e by R a s i e l and 5 Whitman . The r e s u l t s o f t h e s e c a l c u l a t i o n s i n d i c a t e t h a t t h e s a c r i f i c e i n energy i s s m a l l when c o n s t r a i n t s a r e imposed. F u r t h e r m o r e , t h e c o n s t r a i n e d w a v e f u n c t i o n s a l s o y i e l d e d improved e x p e c t a t i o n v a l u e s f o r some o t h e r p r o p e r t i e s . 1.3 Theory o f t h e c o n s t r a i n e d v a r i a t i o n method p r o c e d u r e a c o m p l e t e A n a l y s i s o f t h e c o n s t r a i n e d v a r i a t i o n f i r s t g i v e n by R a s i e l and Whitman**, b u t was 6 formulation of the theory was l a t e r made by Byers 6 Brown based on a perturbation approach. The idea of the constrained v a r i a t i o n method i s to constrain the v a r i a t i o n a l wavefunction to give the known t h e o r e t i c a l or experimental value f o r the expectation value of some operator. This c o n s t r a i n t w i l l lead to a s a c r i f i c e i n energy. Byers Brown^ stated that i f the d i f f e r e n c e A between the fre e v a r i a t i o n expectation value and the known value i s small, then the s a c r i f i c e i n energy w i l l only be of order A when the constraint i s imposed. I t i s hoped that, fo r a n e g l i g i b l e cost i n energy, the constrained wavefunction w i l l y i e l d more r e l i a b l e p r e d i c t i o n of other properties. Let us consider an operator 97V which does not commute with the Hamiltonian ~fl> of the system. [ 771 , -k ] t o (1.3.1) The constrained v a r i a t i o n p r i n c i p l e can be written as ( 6 i | { f t - £ ) I £ > = o (1.3.2) 7 where £ i s a constrained wavefunction to be determined, 6 £ i s the v a r i a t i o n i n , h, i s the energy and ft i s a (f i c t i t i o u s ) constrained Hamiltonian. i t = £ + X - 6 d.3.3) In equation (1.3.3), ^ i s a Lagrangian m u l t i p l i e r and - 6 i s a constraint operator "L? = - y f c (1.3.4) where i s an experimental or t h e o r e t i c a l constant within the bounds of M , the expectation value of , i . e . M = ( £ | ^ / | 5 > / ( £ | £ > d-3.5) The constraint condition i s (i|-6 | £>/ (J | £_> = 0 (1.3.6) 8 The method of constrained v a r i a t i o n s i s a procedure to determine a constrained wavefunction ^ such that the energy is. = < £ | l | £ ) / ( i l i > ( 1 . 3 . 7 ) i s a minimum, subject to the constraint condition stated i n equation (1.3.6). In p r a c t i c e , i t i s convenient to use the free . v a r i a t i o n solutions ^ ^ ^ j as the basis with corresponding energies £ 6.^  j- . Then the constrained wavefunction can be written as £ =ZI' % (1.3.8) k where the c o e f f i c i e n t s are to be determined. The constrained v a r i a t i o n p r i n c i p l e , equation (1.3.2), leads to the constrained secular equation [ 4 + ^ £ " £ 5] - = 0 (1.3.9) and the constraint condition, equation (1.3.6) becomes 9 ct c a = o (1.3.10) where "L and C are the matrix representations of the Hamiltonian and the co n s t r a i n t operator r e s p e c t i v l e l y . The column vector consists of the c o e f f i c i e n t s GLk i s simply diagonal and the overlap matrix =- i s a un i t matrix i n the basis chosen. The extension of the theory to the imposition of more than one co n s t r a i n t i s straightforward and has 6 also been included i n the work of Byers Brown .' The double constrained v a r i a t i o n method was f i r s t applied by Mukherji and Karplus^ i n t h e i r pioneering work and 7 l a t e r by Chong and Byers Brown. 1.4 Methods of solving constrained secular equation In the method of constrained v a r i a t i o n s , f o r the *In t h i s t h e s i s , a l l matrices are double underlined + and a l l column vectors are s i n g l e underlined, CU w i l l then be a row vector. as elements I t should be noted 10 determination of the constrained wavefunction £ and the energy £ the system of equations one has to solve are [ 4 * \£ (1.4.1) o (1.4.2) The c o r r e c t value of the Lagrangian m u l t i p l i e r X.. i s f i x e d by the constraint condition (1.4.2), and the corresponding values f o r the matrix elements &-fc of Several methods have been proposed f o r solving the system of equations (1.4.1) and (1.4.2), namely, 6 8 perturbation , parametrisation and perturbation-9 i t e r a t i o n . A method in v o l v i n g the solutions of two coupled l i n e a r equations has also been suggested"~^ i n the case of double c o n s t r a i n t s . Perturbation perturbation parameter and the constraint operator matrix Q. as a perturbation matrix. The constraint CC w i l l give the s o l u t i o n of £ In the perturbation approach by Byers Brown , the Lagrangian m u l t i p l i e r i s treated as a 11 condition becomes £ Y\ X « - ' E ( M = O (1.4.3) and the value of ^\ i s given by the inverted power series of equation (1,4.3) X - e - A 3 + [ - A 4 - + * A j ] 6>3 + Of©4-) (1.4.4) where e -~ - E t o A £ t 4 ) (1.4.5) /\ n = n E t n / 2 E t t > ( 1 . 4 . 6 ) and are the nth-order perturbation energies. The change i n energy due to co n s t r a i n t i s 0 0 A E ~ H ^ (1.4.7) 12 In p r a c t i c e , a truncated power s e r i e s i s used for both equations (1.4.3) and (1.4.7). Parametrisation 8 In the parametrisation method , we solve the constrained secular equation (1.4.1) v/ith t r i a l values of , the eigenvector ± s then substituted into equation (1.4.2). The correct value of i s obtained when the constraint condition i s s a t i s f i e d to the desired accuracy and the corresponding vector 2- f o r the constrained secular equation i s the correct representation of the constrained wavefunction Ht- ' I n p r a c t i c e , the value of obtained from the perturbation method i s used as an i n i t i a l guess fo r the present method. Per turba t i o n - i ter a t ion The p e r t u r b a t i o n - i t e r a t i o n method proposed by 9 Chong i s in f a c t a combination of the perturbation and parametrisation methods. The i n i t i a l value of X. i s obtained from a truncated power seri e s expansion (1.4.4). For a p a r t i c u l a r value of /V the constrained secular equation (1.4.1) i s solved and 13 t h e c o r r e s p o n d i n g e x p e c t a t i o n v a l u e o f t h e c o n s t r a i n t o p e r a t o r i s o b t a i n e d f r o m t h e c o n s t r a i n t c o n d i t i o n (1.4.2). A new e x p e c t a t i o n v a l u e o f t h e c o n s t r a i n t o p e r a t o r i s t h e n g i v e n by a t r u n c a t e d T a y l o r ' s s e r i e s e x p a n s i o n around t h e p r e c e e d i n g . A new v a l u e o f X i s th e n o b t a i n e d by s e t t i n g t h i s new e x p e c t a t i o n v a l u e o f t h e c o n s t r a i n t o p e r a t o r t o z e r o , and i s a g a i n o b t a i n e d by i n v e r t i n g t h e T a y l o r ' s s e r i e s . The s t e p s a r e r e p e a t e d u n t i l t h e d e s i r e d a c c u r a c y on t h e c o n s t r a i n t c o n d i t i o n i s a c h i e v e d . P a r a m e t e r s f o r d o u b l e c o n s t r a i n t s F o r s i n g l e c o n s t r a i n t , t h e i n i t i a l guess o f />v f o r t h e p a r a m e t r i s a t i o n method i s u s u a l l y o b t a i n e d f r o m t h e p e r t u r b a t i o n a p p r o a c h . However, t h i s p r o c e d u r e becomes t e d i o u s f o r d o u b l e c o n s t r a i n t s and f o r m i d a b l e f o r m u l t i p l e c o n s t r a i n t s . R e c e n t l y , Chong and Benston"*^ s u g g e s t e d a new method t o o b t a i n good i n i t i a l g u e s s e s f o r t h e p a r a m e t e r s f o r d o u b l e c o n s t r a i n t s and t o a r r i v e a t t h e c o r r e c t v a l u e s f o r t h e p a r a m e t e r s . The method d i s c o v e r e d by Chong and B e n s t o n i s t o s o l v e t h e f o l l o w i n g two c o u p l e d l i n e a r e q u a t i o n s 14 C,CA, , A i ) = Aio + A / A , + /4^AJ_ (1.4.8) C T C A L ; Ao.) - Azo + /U. A, + / t a A2. (1.4.9) where the ^ 's are the expectation values of the constraint operators. I n i t i a l l y , the Aio and A^ o c o e f f i c i e n t s are simply the expectation values of the constraint operators obtained with the free v a r i a t i o n function; the An and c o e f f i c i e n t s are given A„~~Ato//\u ) Ax*. - - A** I (1.4.10) where A M and X i i are the values of A s a t i s f y i n g the corresponding sing l e c o n s t r a i n t C, = 0 and C i = 0 r e s p e c t i v e l y . The Aix and A^( c o e f f i c i e n t s are obtained from -A,ol (1.4.11) A i l = [ ^ ( > n , o ) In subsequent i t e r a t i o n s we use values of and C*. at three sets of ( A, , A.2_ ) around the previous estimates, we solve for the s i x A 's and then f o r better X, and X x. u n t i l the c o n s t r a i n t condition i s s a t i s f i e d to the desired accuracy. 1.5 Other developments in the constrained v a r i a t i o n  method A modified c l o s e d - s h e l l s e l f - c o n s i s t e n t f i e l d formalism to include constraints for one electron 11 properties has been given by Fraga and B i r s s . A formalism f o r the off-diagonal constrained v a r i a t i o n method fo r open-shell s e l f - c o n s i s t e n t f i e l d theory (1.4.12) 16 has also been developed by Benston and Chong Off-diagonal h y p e r v i r i a l theorems, coalescence conditions and diatomic forces have been proposed as constraints i n order to c a l c u l a t e better o s c i l l a t o r 13 14 15 strengths , e l e c t r o n i c properties at the n u c l e i ' , 16 and force constants r e s p e c t i v e l y . More rec e n t l y , 17 Chong and Benston proposed a perturbation-induced constraints scheme for f i r s t - o r d e r properties i n which d i f f e r e n t c o n s t r a i n t operators are constructed f o r d i f f e r e n t properties. 1.6 Object of the thesis The object of t h i s thesis i s t h r e e f o l d . F i r s t l y , in Chapter I I , the a p p l i c a t i o n of the i n t e g r a l electron cusp condition as a c o n s t r a i n t i s tested. Secondly, i n Chapter I I I , the use of off-diagonal h y p e r v i r i a l theorems as constraints i s further investigated. F i n a l l y , i n Chapter IV, the v a l i d i t y of the perturbation-induced constraints scheme i s tested. In a l l the c a l c u l a t i o n s presented i n t h i s t h e s i s , we do not aim f o r high accuracies, our object i s to t e s t various constraints with r e l a t i v e l y simple wavefunctions and to look f o r improvements of expectation values. In many cases, we study the 1 1 S ground state properties of the helium atom which have been computed very accurately by 29 Pekeris and provide a convenient basis of comparison f o r the r e s u l t s i n the present work. 18 CHAPTER II INTEGRAL ELECTRON CUSP CONDITION AS A CONSTRAINT 2.1 Introduction Let us consider the co n f i g u r a t i o n a l ( s p a t i a l wavefunction j£_ s a t i s f y i n g the n o n r e l a t i v i s t i c ti Schrodinger equation H i - E t ( 2 ' - 1 f o r i N charged p a r t i c l e s system with the spinless Hamiltonian H = ~ £ > ^ V Z A Z J ^ ' ( 2 . 1 . 2 ) where WL^  and Zj^ are the mass and charge of the itb p a r t i c l e , f/ij i s the distance between the itb and the jtfc p a r t i c l e s . The Coubmb p o t e n t i a l that appears in the Hamiltonian becomes singular when any two p a r t i c l e i and j coalesce, i . e . * A J =0. For the coalescence of one electron with a nucleus i n an atom or a molecule, 19 the eigenfunction <J> has a cusp due to the sing-i) u l a r i t y that appears i n the Schrodinger equation where the corresponding Coulomb a t t r a c t i o n p o t e n t i a l i s i n f i n i t e . There i s a s i m i l a r cusp f o r the eigen-function <£, when two electrons coalesce. The conditions which must be s a t i s f i e d i n order that _ II remains a solut i o n of the Schrodinger equation at these s i n g u l a r i t i e s are c a l l e d the coalescence conditions. When does not vanish when p a r t i c l e s coalesce, the conditions are known as the cusp cond-i t i o n s . For an atom i n the heavy-nucleus approximation, 18 Kato r i g o r o u s l y derived the following cusp condition =• «y §L<rijso) (2.1.3) where <|_ i s the average value of over a small sphere about the s i n g u l a r i t y with T/j = constant, The constant Y i s equal to - t£ , the atomic number, f o r an electron-nucleus coalescence and i s equal to % f o r an electron-electron coalescence. 19 Roothaan and Weiss also obtained a s i m i l a r ex-pression f o r an exact wavefunction f o r the ground 20 state of helium. The el e c t r o n - e l e c t r o n cusp cond-i t i o n , f o r the hydrogen molecule has been given 20 by Kolos and Roothaan . Integral forms which are completely equivalent to d i f f e r e n t i a l forms of the 21 cusp conditions have been derived by Bingel The sp h e r i c a l average r e s t r i c t i o n i s removed from equation (2.1.3) by adding an angle dependent term and equation (2.1.3) then becomes J =• §L<Sj-o) [ |. + i f£j + £cj -U/j + OCfxj)] (2.1.4) The vector U aj i s not determined by the Cotilomb s i n g u l a r i t y and depends on the other p a r t i c l e s . The symbol O(Jkj) means terms of order tjcj and higher. Bingel also indicated that the cusp cond-i t i o n s are s t i l l v a l i d f o r molecular wavefunctions around each n u c l e i i n a polyatomic molecule. He has also obtained the analogue of equation 21 (2.1.4) f o r the f i r s t - o r d e r density matrix In order to set up a properly antisymmetrized eigenfunction & of the Hamiltonian, i t i s necessary to include spin. The proper wavefunction 5. 21 must also be an eigenfunction of the antisymmetrizer ^ , the spin operators /4 and /6z , with eigenvalues +1, s(s+l) and M r e s p e c t i v e l y . Hence, i t i s not easy to apply equation (2.1.3) i n general. In order to apply equation (2.1.3), one can construct the electron and spin d e n s i t i e s . Using equation 22 (2.1.3), Steiner was able to show that [af%L.. =*rP<.> (2.1.5, which holds f o r any eigenfunction of the spin independent Hamiltonian of an atom;pC£) i s the t o t a l electron density at the point JC i n space pCr) =<£|£S<r-Xc,|£> (2.1.6) -1=1 where £cr-£c) i s the Dirac d e l t a function, and by $ti~ - f/0 we mean (^ -"fl'X'i"") 6 ^ r " " * * ) . .Bingel also obtained the cusp conditions f o r both the electron 21 and the spin d e n s i t i e s . The cusp condition f o r the t o t a l electron density i s the same as equation (2.1.5) whereas that f o r the spin density, i t can be written as [ d p S'%r] r = 0 = ^Psc°> (2.1.7) where pSC£) i s the t o t a l spin density at the point r i n space M and ydg i s the usual spin operator on the z-comp-onent. A general treatment of coalescence conditions 2 3 was given by Pack and Byers Brown i n 1966 The new features of t h e i r d e r i v a t i o n are: (a) the f i x e d nucleus approximation which was used by pre-vious authors i s not required; (b) the wavefunction i s not s p h e r i c a l l y averaged; (c) the wavefunction can have a node at the s i n g u l a r i t y ; and (d) the generalized coalescence conditions are applicable to both electron-nucleus and e l e c t r o n - e l e c t r o n coalescence. Considering the manifold of c o n f i g -urations i n which the two coalesced p a r t i c l e s 23 i and j are within an a r b i t r a r i l y small distance of each other, and a l l other p a r t i c l e s are well separated, Pack and Byers Brown obtained the general 23 s o l u t i o n of the form i = E E E ^ C t m ' » . « ( 2 . I . 9 ) f r o M-t-X FE=0 where t = ti\ , (9 = 0 , <J> = (J)^- , Jj tMt do not depend on f , # or <p and ^m.^itf) are the surface harmonics. The superscript k comes from a power serie s expansion of the r a d i a l part of the s o l u t i o n . For t h i s p a r t i c u l a r form of §> , i n the v i c i n i t y of the coalescence, the -c 23 behavior of 2L i s given by r X { £ C ^ ( G * * ) [ « + ! m + X ) ~ , r ] (2.1.10) n\=-A-l where t = jtj , /l/j = vn^W-j/cwii + nVj) ; and A i s the smallest value of / f o r which "fj?rH. £ 0. The generalised coalescence conditions have 14 been applied by Chong to the s p e c i a l case of electron-nucleus coalescence f o r open-shell s e l f -consistent f i e l d o r b i t a l s i n atoms and molecules. Integral coalescence conditions were f i r s t derived and then used to construct constraints f o r the 15 v a r i a t i o n of the o r b i t a l s . Chong also derived i n t e g r a l electron-nucleus coalescence conditions f o r c o n f i g u r a t i o n - i n t e r a c t i o n wavefunctions. Recently, a physical i n t e r p r e t a t i o n of the cusp conditions f o r 24 molecular wavefunctions has been presented by Bingel In the case of electron-nucleus coalescence, the vector W-AJ that appears in equation (2.1.4) i s shown to be p a r a l l e l to the e l e c t r i c f i e l d produced by the other electrons and n u c l e i at the nucleus i n question. As in the case of electron-electron coalescence, the d i r e c t i o n of Si^ j remains undetermined. In t h i s work, we s h a l l only consider the s p a t i a l eigenfunction which s a t i s f i e s equation (2.1.10). The system i s described by a spinless Hamiltonian, our concern i s the electron-nucleus coalescence and hence only the i n t e g r a l electron cusp condition w i l l be examined. 25 2.2 On the use of the i n t e g r a l e lectron cusp condition as a constraint For an exact eigenfunction f£ of the Hamil-tonian H , Chong was able to obtain the follow-15 ing i n t e g r a l coalescence condition ( A + i><£| f/ .|£>= r<£Uj i > (2.2.D where (2.2.2) •;xj;fr. r.) r"* (2.2.3) In the case of A =0, which corresponds to a nodeless wavefunction, equation (2.2.1) i s reduced to < £ | f l * > = *<*|.t|*> (2.2.4) which i s the i n t e g r a l e lectron cusp condition, 26 }f i s equal to the atomic number, and the oper-ators ^ and are N Jf =^ SC £ " (2.2.5) N ^ = $ ^ 8 C r - j C ; ) ( 2 . 2 . 6 ) JL-I I t should be noted that an equation analogous to equation (2.2.4) can also be obtained from the electron-nucleus cusp condition given by Roothaan 19 and Weiss f o r the exact wavefunction for the ground state of helium (2.2.7) where the p a r t i a l d i f f e r e n t i a t i o n s are performed while t r e a t i n g t\ and £. as independent v a r i a b l e s and ^ii i s held f i x e d . A s t r a i g h t forward i n t e -25 gration of equation (2.2.7) w i l l lead to an expression s i m i l a r to equation (2.2.4). The i n t e g r a l 27 electron cusp condition given by equation (2.2.4) i s also equivalent to the cusp condition f o r the 22 21 electron density of Steiner and Bingel Since the exact function £ s a t i s f i e s the coalescence conditions, values of <T calcu l a t e d from approximate wavefunctions can be used as a measure of t h e i r deviation from the exact s o l u t i o n . For approximate wavefunctions, the electron cusp value F was defined and the use of the i n t e g r a l 15 electron cusp condition as a co n s t r a i n t was proposed The electron cusp value f and A . , the electron density at the nucleus f o r an approximate wavefunc-t i o n , are defined as r = i*>/<t|+> (2 .2 .8) A = ( f l ^ l t y ^ l t ) (2.2.9) and are evaluated at Jf =0. Comparison of F with cf shows how well equation (2.2.4) i s s a t i s f i e d by an approximate wavefunction. 28 14 15 2 3 I t has been reasoned ' ' that good electron-nucleus cusp values f o r approximate wave-functions should c o r r e l a t e with good values f o r e l e c t r o n i c properties near the n u c l e i . I t has 15 been suggested by Chong that a better value f o r /\ , the electron density at the nucleus may be obtained i f the approximate wavefunction i s cons-trained to s a t i s f y equation (2.2.4). In f a c t , 25 a recent work by Chong and Schrader shows that there i s a strong s t a t i s t i c a l c o r r e l a t i o n between the accuracies of A and I . This provides a basis for the use of the i n t e g r a l electron cusp condition as a co n s t r a i n t . A simple constraint operator i s (2 .2 .10) where (2 .2 .11) and *P i s the transpose of the complex conjugate of ^ . I f an approximate wavefunction ^ s a t i s f i e s 29 C = O (2.2.12) where C =<W|f> (2.2.13) then equation (2.2.4) i s s a t i s f i e d by ^ , and r •= r . In t h i s work, we study the a p p l i c a t i o n of the constraint in equation (2.2.12) by t e s t i n g i t on eleven 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-functions f o r the ground state of helium. Since the s a t i s f a c t i o n of the cusp condition may improve other properties which are s e n s i t i v e to ^ near * -2. the nucleus, the expectation values of T and r' (besides /\ , the electron density at the nucleus) are also examined. The e f f e c t s of using the electron density at the nucleus as a constraint, on the cusp value and expectation values of JT 2 and are also investigated. These expectation values are i n f a c t a sum of one-electron expectation values. 30 2.3 Calcul a t i o n s and r e s u l t s Three d i f f e r e n t types of c o n f i g u r a t i o n - i n t e r a c t i o n approximate wavefunctions for the ground state of helium are used. The f i r s t type of these functions i s designated by <p . They are constructed from a basis set i n the one-electron space containing only one Is o r b i t a l . The second type i s designated by ^ , they are constructed from a basis set i n the one-electron space containing two Is o r b i t a l s . The t h i r d type i s designated by ^ , they are constructed from a basis set i n the one-electron space containing more than two Is o r b i t a l s , the numeral subscript of each function s i g n i f y i n g the number of configurations i n that function. A l l configurations belong to a state and are constructed from the Slater-type atomic o r b i t a l s designated by | A , jt, m) = Nn e" " Y*m<*3,<l>) (2.3.1) where |\{n i s the normalization constant, -Av\ is_ Some of the r e s u l t s have been submitted f o r p u b l i c a t i o n . See r e f . (26). 31 the o r b i t a l exponent, YHTAC&'Q ) are the surface harmonics, , £ and yrL. are the p r i n c i p a l , azimuthal and magnetic quantum numbers r e s p e c t i v e l y . • Free V a r i a t i o n The eleven c o n f i g u r a t i o n - i n t e r a c t i o n approx-imate wavefunctions used i n the present work are summarized i n Tables I and I I . The o r b i t a l exponents, with the exception of ^ 1 have been crudely op-timized through an i t e r a t i v e procedure where the 2 7 change i n the energy i s approximated by a parabola The c o e f f i c i e n t s f o r the terms i n each function are calcul a t e d by the usual v a r i a t i o n method and they are given in Table X X I I I (Appendix B ) . In Table I I I , the r e s u l t s of free v a r i a t i o n calculations, are presented. The e n t r i e s are arranged i n order of decreasing energy £ , i . e . the lower the energy, the better the wavefunction according to the energy c r i t e r i o n . The cusp value P and /\ , the electron density at the nucleus, are calcu l a t e d according to equations (2.2.8) and (2.2.9). Pri m i t i v e i n t e g r a l s of the ^ and operators are f i r s t c a l c u l a t e d and l a t e r c o l l e c t i o n s and transformations Table I. Co n f i g u r a t i o n - i n t e r a c t i o n approximate ., wavefunctions for the ground state of helium. Each configuration belongs to a S state. Function Configurations ^ 3 ls 2+ls2s+2s 2 2 2 2 Is +ls2s+2s +2p 2 2 2 Is +Is2s+ls3s+2s +2s3s+3s 4>7h 2 2 2 2 Is +Is2s+ls3s+2s +2s3s+3s +2p lsls'+2s 2+2p 2 2 2 lsls'+2s+2s3s+3s 2 2 2 lsls'+2s +2s3s+3s +2p 2 2 2 lsls'+2s +2s3s+2s4s+3s +3s4s+4s lsls'+2s 2+2s3s+2s4s+3s 2+3s4s+4s 2+2p 2 v 3 l s l s ' + l s " l s " ' + l s 1 V l s V ^4 l s l s ' + l s " l s ' " + l s : L V l s V + 2 p 2 ° Ref. (28) b Ref. (13) 33 Table I I . O r b i t a l exponents of Slater-type atomic o r b i t a l s used in the c o n f i g u r a t i o n - i n t e r a c t i o n approximate wavefunctions shown in Table I. Function l s + 2s 3s 4s 2p <t> 4 0 7 f3 ft t 7 ? 8 ? 3 2 . 2 . 1. 1. 2 . 1. 1. 1. 2 . 1. 1. 1. 1. 1. 1. 2 . 1. 1. 1. 4. 3481 2514 908 575 1603 2 38 7 7398 3547 17621 20152 9298 3100 8874 3338 1482 4178 3603 6463 8741 1535 1.1407 2 .5036 1. 1 1.! 4. 3596 7260 8741 1535 1.6619 1.62 72 1.908 1.843 4.0883 5.1 908 955 3.3658 2.4864 5.1 3.3809 2.5325 3.3601 2.6116 2 .8809 3.0138 2.4797 2 2 475 4729 2 .47547 2.4756 2.4696 The o r b i t a l exponents are l i s t e d i n the order of the Is o r b i t a l s appear in Table I 34 Table I I I . Results of free v a r i a t i o n s ( in atomic units ). The % errors of the cusp value T and /\ , the electron density at the nucleus, from t h e i r exact values are given i n parentheses. Function E P ^ 3 -2 .876129 -1 .97625C-1 .188) 3 .59342(-0 . 758) ? 3 -2 .877406 -1 .99274C-0 .363) 3 .6134K-0 .206) * 4 -2 .877864 -2 .07H0( + 3 .555) 3 .69098(+1 .937) * 6 -2 .878296 -2 .00253(+0 .127) 3 .63099(+0 .280) * 7 -2 .878371 -2 .01147C+0 .573) 3 .65662(+0 .988) -2 .895655 -1 .96899(-l .551) 3 •57623C-1 .231) * 3 -2 .896548 -2 .1216K + 6 .080) 3 .68579C+1 . 793) 9>4 -2 .896686 -1 .94580C-2 .710) 3 .52533C-2 .638) t 5 -2 .897142 -1 .95l68(-2 .416) 3 .61539C-0 .151) V8 -2 .897696 -2 .01225(+0 .612) 3 .63465C+0 .381) *7 -2 .897840 -1 .98980C-0 .510) 3 .60150(-0 .535) c t a -2 .903724 -2 ( 0 ) 3 .62085C 0 ) a. Ref. (29) are made. The p r i m i t i v e i n t e g r a l s of ^ and Sly are given i n Appendix C. A l l the q u a n t i t i e s are i n atomic units and placed under corresponding headings in Table I I I . The % errors of V and A from t h e i r exact values are given i n parentheses. The % error i s defined as % error = calc u l a t e d value - exact value exact value (2.3.2) The exact value f o r the energy £ and A , the 2 9 electron density at the nucleus obtained by Pekeris are included i n the table f o r comparison. Constrained V a r i a t i o n I t should be noted that as the operator given by equation (2.2.5) i s not hermitian, hence the operator iP defined by equation (2.2.11) i s also not hermitian. The constraint operator , on the other hand, i s hermitian by d e f i n i t i o n since equation (2.2.10) i s used to construct >^ Using the f r e e v a r i a t i o n wavefunctions as a b a s i s , the constrained secular equation i s solved by the perturbation^ and parametrization^ methods. 3 6 Two d i f f e r e n t computing programs are written f o r these two d i f f e r e n t methods. The fre e v a r i a t i o n r e s u l t s are read i n as inputs. The perturbation approach gives a value f o r the Lagrangian m u l t i p l i e r ^\ , which becomes an i n i t i a l guess f o r the para-meter i n the parametrization approach. In p r a c t i c e we f i n d that, with the exception of <p/j. , the value of N from perturbation i s so good that further parametrization becomes unnecessary. The rapid rate of convergence of the pertur-bation s e r i e s i s demonstrated by the following example of <Py \ = +1.031756 x 10" 3 - 1.06279 x 10~ 5 + 1.1119 x 10" -1.099 x 10~ 9 + 9.41 x 10~ 1 2 - 6.07 x 10~ 1 4 = +1.021239 x 10~ 3 (2.3.3) as f o r the change in energy A E due to the constraint A E = 3.749950 x 10~ 5 - 1.855863 x 10~ 5 - 1.26147 x 10" -9.4319 x 10~ 1 0 - 6.7909 x 10~ 1 2 - 4.699 x 1 0 ~ 1 4 -3.109 x 10~ 1 6 + ... = 1.881378 x 10" 5 (2.3.4) The value of /\ given by the parametrization approach i s 0.00102124 and AE i s 0.000020. The accuracy i n the expectation value of the con s t r a i n t operator — 6 when the constraint condition i s s a t i s f i e d i s — 10 for both the perturbation and parametrization method. The r e s u l t s of constrained v a r i a t i o n a l c a l c u l -ations on the eleven approximate wavefunctions are summarized in Table IV. The % error of Z \ , the electron density at the nucleus, i s given i n parenthe-ses. The s a c r i f i c e i n energy A E i s i n s i g n i f i c a n t as shown in column two of the table. The constrained wavefunctions in. terms of the free v a r i a t i o n functions 38 Table IV. Results of constrained v a r i a t i o n s ( i n atomic u n i t s ). The % error of /\ , the electron density at the nucleus, from the exact value i s given in parentheses. Function 1 0 3 \ 10 5 A E A y 3 + 1 .392867 6 .01 3.64996(+0 .804) 03 + 6 .209249 8 .20 3.66617(+1 .252) - 1 .837415 23 . 76 3.58552(-0 .976) <Pe 0 .2876273 0 .13 3.62446C+0 .100) f i 0.0846074 0 .18 3.64547(+0 .680) ft + 1 .493586 8 .28 3.54972(-l .964) f3 - 2 .270682 50 .18 3.58426'(-l .010) 04 + 55 .6501 562 .18 3.94843C+9 .047) + 0 .5008557 4 . 38 3.62460( + 0. .104) fs - 0 .07278035 0 .16 3.62387(+0 .083) + 1 .0212 39 1 .88 3.62639(+0 .153) 39 used as basis are given i n Table V. 2.4 Discussion The s t a t i s t i c a l c o r r e l a t i o n between the accur-acies of the electron cusp value T* and /S. , the electron density at the nucleus, shown by Chong and 25 Schrader can also be seen from Table I I I . Our present free v a r i a t i o n a l r e s u l t s confirm t h e i r f i n d i n g q u a l i t a t i v e l y . A p l o t of the % error of A against the % error of Y from t h e i r exact values i s shown i n F i g . 2. A two-term function (p^ = b 1 ( l s 2 ) + b 2 ( 2 p 2 ) not l i s t e d in Tables I and II i s also shown i n F i g . 2 at the lower l e f t corner. The % error of /\ and P of t h i s two-term func-ti o n i s -15.494 and -15.503 r e s p e c t i v e l y . The dashed l i n e i s the perfect c o r r e l a t i o n curve. Our r e s u l t s also indicate that.an approximate wavefunction with a better energy does not n e c e s s a r i l y give a better value of f ^ A • For example, the energy of the wavefunction <f>£ i s -2 .878296, the % errors of f and /\ of t h i s function are +0,12 7 40 and +0.280 r e s p e c t i v e l y . On the other hand, the energy f o r i s -2.897696 which i s considerably better than that of <p£ , but the % errors of P and /\ of t h i s function are +0.612 and +0.381 re s p e c t i v e l y and are both higher than that of 0£ . This i s i n f a c t a c h a r a c t e r i s t i c of the v a r i a t i o n a l l y ---determined -approximate-wavefunctions with the energy minimum c r i t e r i o n and again i n agreement with Chong 25 and Schrader. In general, we f i n d that the a p p l i c a t i o n of the cusp constraint has a tendency to overcorrect the deviation of /\ , the electron density at the nucleus from the exact value. When the approx-imate function i s not f l e x i b l e enough, the value of A obtained from the constrained wavefunction may even be worse than that obtained from the free" v a r i a t i o n a l function. The function <$j_ i s an example of t h i s case. The % error of A obtained from the free v a r i a t i o n a l c a l c u l a t i o n i s -2.638, af t e r the cusp constraint i s imposed, i t becomes +9.047, the s a c r i f i c e in energy A £ i s 0.005625 a.u. and i s not too serious. The case of <p7 i s Table V. C o e f f i c i e n t s of constrained wavefunctions i n terms of free v a r i a t i o n functions. nction a2 a3 a4 a5 a6 a7 ? 3 0 .999982 -0 .005927 0 .001204 ^4 0 .998673 -0 .050557 0 .009804 -0 .000147 *6 0 .999999 -0 .000439 -0 .000007 0 .000428 -0 .000019 -0 .000024 * 7 0 .999997 0 .001606 0 .000025 0 .000189 -0 .001552 0 .000081 0 .000080 0 .999952 -0 .006222 0 .007582 0 .999985 -0 .003616 -0 .000377 -0 .001274 0 .999997 0 .001261 -0 .001663 0 .001274 -0 .000305 t l 0 .999999 0 .000147 -o .000017 -0 .000143 0 .000007 0 .000004 -0 .000215 % 0 .999999 0 .000099 -0 .000010 0 .000004 -0 .000101 -0 .000002 -0 .000002 0 .999995 - 0 .000295 0 .003283 % 0 .999991 -0 .002353 -0 .000116 0 .003406 "8 < c •H u o u u W 0 • • • j 1 • j s 1 • • • • • -5 -10 -15 -15 Jo~ -5 % Error i n P 0 + 5 F i g . 2 a more t y p i c a l example of overcorrection. The % error of A\ changes from -0.535 to +0.153 a f t e r the cusp constraint condition i s s a t i s f i e d , t h e s a c r i f i c e i n energy i n t h i s i s 0.0000188 and i s i n s i g n i f i c a n t . Table VI shows the behavior of various quant-i t i e s as the parameter X i s varie d i n the case of . The s a c r i f i c e i n energy due to the imposition of constraint i s given i n column two under the designation A E , the expectation value C of the constraint operator t3 defined by equation (2.2.10) i s shown i n column three. The values of f and j\ are contained in columns four and f i v e r e s p e c t i v e l y . The % errors of P and A. from t h e i r exact values are given i n parentheses. I t can be seen from Table VI that with the exception of A E » a l l other q u a n t i t i e s vary l i n e a r l y with the parameter X . A plot of the v a r i a t i o n s in C , % errors of P and A vs X Is shown in F i g . 3. The c o r r e l a t i o n between the accuracies of P and A f o r p a r t i c u l a r values of X i s shown i n F i g . 4. 44 By comparing the r e s u l t s shown in Table I I I and Table IV, we can see that when the approximate wavefunction i s f l e x i b l e enough, the calculated value of A 5 the electron density at the nucleus, i s considerably improved as one imposes the cusp c o n s t r a i n t . Our best r e s u l t appears i n the case of ^fg which i s also the most f l e x i b l e function of the eleven. As mentioned in section 2.2 of t h i s chapter the s a t i s f a c t i o n of the cusp condition i s expected to improve other properties which depend strongly on the wavefunction near the nucleus. For t h i s reason, we also examine the expectation values of r Z and T ' f o r some of our more f l e x i b l e wave-functions. These expectation values are i n f a c t a sum of the one-electron expectation values. The expectation values of r~* i s c l o s e l y r e l a t e d to the diamagnetic contribution to the s h i e l d i n g constant G~ * . For s states of atoms, with the nucleus as o r i g i n , the high-frequency term vanishes and the only c o n t r i b u t i o n to <Td i s the Lamb term (2.4.1) .45 Table VI. Typical behavior of various quantities as a function of X (as i l l u s t r a t e d by the case of 7^ ). The % errors of the electron cusp value F and A ? the electron density at the nucleus from t h e i r exact values are given i n parentheses. i o 3 A 1 0 5 A E 1 0 2 C -r A 0.00 0.0 3.6 72 1 .98980(-0.510)- 3 .60150C-0 .535) 0.20 0.7 2 .959 1 .99180(-0.410) 3 .60633(-0 .401) 0.40 1.2 2 .243 1 .99379(-0.310) 3 .61118(-0 .267) 0.60 1.6 1.524 1 .99579(-0.210) 3 .61606(-0 .132 ) 0.80 1.8 0.802 1 .99779(-0.111) 3 .62095(+0 .003) 1.02124 1.9 0.000 2 .00000(0.000) 3 .62639(+0 .153) 48 where cL i s the f i n e structure constant 1/137). The r e s u l t s , l i s t e d i n Table VII, are somewhat disap-pointing. The change i n the expectation values i s i n s i g n i f i c a n t . This suggests that i n order to improve -2 ^ the expectation values of f and V , some other constraint must be used. I t i s i n t e r e s t i n g to see what the e f f e c t s on the cusp velue P and the expectation values of £ Z and r w i l l be i f A. , the electron density at the nucleus, i s used as a c o n s t r a i n t . The constraint operator i n t h i s case i s simply defined as = X " A exact (2.4.2) where A e x a c t i s 3.62085 and i s the exact value of 29 the electron density at the nucleus given by Pekeris This p a r t i c u l a r constraint i s tested on three four-term configuration-.interaction approximate wavefunctions 4>4 , S^f- and % . The main reason of choosing these functions i s that the % error of A of these functions i s worse than the free v a r i a t i o n r e s u l t s a f t e r the cusp c o n s t r a i n t i s imposed. The r e s u l t s of 4 9 constrained v a r i a t i o n a l c a l c u l a t i o n s are summarized in Table V I I I . The c o e f f i c i e n t s of the constrained functions are given i n Table IX. Comparing to the free v a r i a t i o n r e s u l t s shown in Table I I I , the r e s u l t s given i n Table VIII i n d i c a t e that, i n general, the cusp value P i s improved when /\ , the electron density at the nucleus, i s cons-trained to be exact. In the case of ^ , the cusp value P i s s l i g h t l y worse a f t e r the constraint i s imposed. The s a c r i f i c e i n energy Ajr due to con-s t r a i n t i s again n e g l i g i b l e . The e f f e c t of t h i s con--2. -I s t r a i n t on the expectation values of JT and i s small as in the case of cusp c o n s t r a i n t . The changes in the expectation values of r"2 and x: ' due to the constraint are given in Table X. I t should be mentioned that the use of the e l e c -tron density at the nucleus as a constraint i s not p r a c t i c a l l y u s e f u l . The main objections are, f i r s t l y , i n general, the exact value of the electron density at the nucleus i s not known; secondly, the electron cusp value does not seem to be r e l a t e d to any physical properties which are of greater i n t e r e s t to chemists. 50 Table VIL Expectation values of r and r f o r the more f l e x i b l e wavefunctions in the case of cusp c o n s t r a i n t . Function 1  Free Constrained Free Constrained 4 e 12.0404 12.3031 3.37208 3.37153 t v 12.1081 12 .0985 3.38438 3.38389 ^8 12.0744 12.0659 3.38402 3.38367 <f>7 12.0100 12.0451 3.37354 3.37547 Exact 3 12.0341 3.37663 aRef. (29) 51 Table VIII. Results of constrained v a r i a t i o n s ( i n atomic units) f o r the approximate wavefunctions <P^_ , ^4 and ^4 . The % error of the cusp value from the exact value i s given i n parentheses. Function 1 0 3 X i o 4 d £ 0 4 7*4 f 4 -5.931842 -0.1888646 +2.022564 2.85 0.05 0. 71 -1.95836(-2.082) -1 .96825(-l.587) -2.03966(+1.983) Table IX. C o e f f i c i e n t s of in terms of free v a r i a t i o n constrained functions. wavefunctions Function a^ a^ a^ 4 0.999931 -0.011687 0.000857 0.000299 y > 4 0.999999 0.001025 -0.000009 0.000317 vp 4 0.999996 0.001152 -0.000088 -0.002408 53 Table X. Expectation values of r ~ and r ~ i n the case of using A i the electron density at the nucleus as a constraint . Function < r - 2 > Free Constrained Free Constrained 11.9222 12.1479 11.9961 12.1456 12.1158 12.0091 3 3 3 .36985 3.40244 .37326 3.41468 .38289 3.33734 Exact 3 12.0348 3.37668 aRef. (29) 54 As f o r the methods of solving the constrained secular equation, we use both the perturbation and parametrization method. In the case of ^7 , we have demonstrated the rapid rate of convergence of the perturbation s e r i e s . Unfortunately, t h i s i s not always the case. In f a c t , the perturbation approach does not guarantee a f a s t convergent perturbation s e r i e s s o l u t i o n . In contrast to <f>y , i n the case of ^ , we have X = +6.642482xl0~ 2-1.3303870xl0~ 2+3.16833xl0~ 3 -8.05 71xl0" 4+2.1094xl0~ 4-5.634xl0~ 4 = +5.563816xl0" 2 (2.4.3) 4 £ = +1.063135xl0~ 2-4.45247xl0" 3-4.9797xl0~ 4 -5.0809xl0~ 5-4.746xl0~ 6-4.189x10" 7 -3.908xl0~ 8 = 5.625xl0~ 3 (2.4.4) The advantage of t h i s approach i s that no a r b i t r a r y choice of parameter values i s needed. In the parametrization procedure, one solves the constrained secular equation f o r selected values of X and the cor r e c t value of A. i s obtained when the constraint condition i s s a t i s f i e d . In p r a c t i c e , one can choose a r b i t r a r i l y several values of A and obtain the corresponding expectation value C f o r the constraint operator . The approximate cor-r e c t value of A can then be obtained from the p l o t of A vs C . The advantage of t h i s procedure i s that one never stops u n t i l the cor r e c t value of A i s obtained when the constraint condition i s s a t i s f i e d to the desired degree of accuracy, e.g. - 10~ . The main disadvantage of t h i s approach i s the a r b i t r a r y choice of values of A , even with the help of the perturbation approach to obtain an i n i t i a l guess. For example, the value of A\ given by the pertur-bation approach i s +5.563816 x 10 , whereas the cor-r e c t value of Av obtained from the parametrization method i s +5.56501 x 10~ 2 i n the case of . Also, the choice of both the i n t e r v a l s f o r A and of the fac t o r by which t h i s i n t e r v a l i s decreased from one cycle to the next may be too a r b i t r a r y for workers 56 without previous experience i n constrained v a r i a t i o n s . 9 The p e r t u r b a t i o n - i t e r a t i o n method proposed by Chong w i l l be studied and discussed i n the next chapter. In short, we have learned from t h i s study: (a) that the wavefunction must be f l e x i b l e enough i n order to take advantage of the cusp constraints; -and (b) that the cusp c o n s t r a i n t has l i t t l e e f f e c t on the expertation values of IT 2 and c~l ; (c) that the c a l c u l a t e d value of the electron density at the nucleus improves when one imposes the cusp constraint on an approximate wavefunction with enough l i n e a r c o e f f i c i e n t s ; (d) the use of the electron density at the nucleus as a constraint i s not p r a c t i c a l and (e) as fo r the methods of solving the constrained secular equation, the perturbation method does not n e c e s s a r i l y guarantee rapid convergence i n the perturbation s e r i e s while the choice of values f o r the parameter X i s somewhat a r b i t r a r y in the parametrization approach. 57 CHAPTER I I I OFF-DIAGONAL HYPERVIRIAL THEOREMS AS CONSTRAINTS 3•1 Introduction The Heisenberg equation of motion i s the quantum mechanical equivalent of the equation of motion i n terms of Poisson bracket i n c l a s s i c a l mechanics. Consider the commutator of some operator U/ with the Hamiltonian , and l e t <//* and <A be eig en-functions of the H amiltonian with eigenvalues £*vi. 30 and r e s p e c t i v e l y . Then ^zt O & M <A-> - < ^ / C K / / X J [t&> (3.1.D where i s the commutator of ^ and and i s defined by C M ^ - J = \AJ-L - ~L ^  . I t should be noted that i n order to obtain equation(3.1.1), a time-independent operator i s assumed and we only consider 58 stationary state wavefunctions. Equation (3.1.1) i s the f a m i l i a r Heisenberg equation of motion i n the energy representation. I f we consider the expectation value of ~UJ~ f o r a stationary state n, then equation (3.1.1) i s reduced to diagonal form ICW", &llti> = O (3.1.2) 31 Equation (3.1.2) gives the h y p e r v i r i a l theorems or more p r e c i s e l y the diagonal h y p e r v i r i a l theorems. The off-diagonal terms of equation (3.1.1) are WntlttTAl IVn> -~ ( ^ - ^ ) < 7 L | K r l A > (3.1.3) 32 which give the off-diagonal h y p e r v i r i a l theorems The appl i c a t i o n s of the diagonal h y p e r v i r i a l theorems have been extensively investigated i n stationary 2 34-39 40-42 states ' and s c a t t e r i n g problems . On the other hand, however, very l i t t l e has been said about the off-diagonal h y p e r v i r i a l theorems. The p r o b a b i l i t y 43 of a given t r a n s i t i o n obtained from the Heisenberg equation of motion has been recognised as a sp e c i a l 2 12 32 case of the off-diagonal h y p e r v i r i a l theorems ' ' 1 Both the diagonal and off-diagonal h y p e r v i r i a l theorems given by equations (3.1.2) and (3.1.3) are, in general, only s a t i s f i e d by exact solutions of the Schrodinger equation. Approximate wavefunctions, such as those obtained from the v a r i a t i o n p r i n c i p l e , do not s a t i s f y these r e l a t i o n s i n general. But i t has been 2 31 33 34 conjectured 1 ' * that i f we compel them to s a t i s f y some of these conditions, i t should be possible to obtain a better wavefunction i n c e r t a i n desired respect, and hopefully to improve values predicted f o r some physical properties. Two methods have been proposed to make an approximate wavefunction s a t i s f y any p a r t i c u l a r diagonal h y p e r v i r i a l theorem. 33 These are the phase v a r i a t i o n and constrained var-g i a t i o n . Equation (3.1.2) then becomes (%,\t»r, 1111 ^ > = ° (3.1.4) In the case of the phase v a r i a t i o n method, based on 36 a perturbation approach Robinson proved that i f the 60 f i r s t - o r d e r perturbation c o r r e c t i o n to the expectation value of an a r b i t r a r y operator vanished, then the approximate wavefunction used s a t i s f i e d a c e r t a i n d i a -gonal h y p e r v i r i a l theorem and v i c e versa. The phase v a r i a t i o n method has been contrasted and compared with 16 the constrained v a r i a t i o n method by Byers Brown For convenience (but not necessary), Byers Brown considered the s p e c i a l case of phase v a r i a t i o n where the free v a r i a t i o n c o e f f i c i e n t s of the wavefunction are frozen, and from a perturbation expansion a n a l y s i s , Byers Brown was able to show that the phase v a r i a t i o n and c o n s t r a i n t v a r i a t i o n methods lead to changes equal i n magnitude but opposite i n sign f o r t h e i r e f f e c t s i n the energy and other properties. The formalism f o r the off-diagonal constrained v a r i a t i o n method has been developed f o r open-shell 12 s e l f - c o n s i s t e n t f i e l d theory by Benston and Chong The imposed constraints are based on the off-diagonal h y p e r v i r i a l theorems f o r the s p e c i a l cases of t r a n s i t i o n p r o b a b i l i t i e s . More re c e n t l y , the off-diagonal hyper-v i r i a l theorems have been suggested as constraints i n order to c a l c u l a t e hopefully better o s c i l l a t o r strengths 61 by the same authors . For approximate wavefunctions ^fal and ^ty^ , then equation (3.1.3) becomes <^|/I^^JMvv> ( ^ - ^ X t N l ^ ) (3.1.5) where £ ^ = < ^ / ^ / ^ > / < v f K / ^ > (3.1.6) - <U/&(T°^>/<^i^> (3.1.7) I t should be noted that i f the commutator 11^,7x^2 vanishes, V^ v and Y**- are usually chosen such that K^t\I^ I ^ V ^ i s zero, and equation. (3.1.5) i s t r i v i a l l y s a t i s f i e d . In the present work, we s h a l l r e s t r i c t to the study of the off-diagonal h y p e r v i r i a l theorems. Our only concern i s the cases where ^ does not commute with and thus equation (3.1.5) i s not s a t i s f i e d . We s h a l l apply the constrained v a r i a t i o n method to compel 62 our approximate wavefunctions to s a t i s f y equation (3.1.5) and our attention i s focused on atomic t r a n s i t i o n pro-b a b i l i t i e s . 3.2 On the use of the off-diagonal h y p e r v i r i a l theorems as constraints The theory on the use of the off-diagonal h y p e r v i r i a l theorems as constraints has been formul-13 ated by Chong and Benston . The p r i n c i p l e i s the same as that i n the diagonal case^. As has been pointed out i n the diagonal case^, the cost i n energy i s only of order & ±f the error of order A i n some property obtained from the free v a r i a t i o n function i s constrained to vanish. The major d i f f e r e n c e between these two cases comes from the d i f f e r e n c e i n the con-s t r a i n t operator. Let us consider approximate wavefunctions ^ and Z- with energy <2- and E f o r the lower and upper state r e s p e c t i v e l y . (In what follows, c a p i t a l l e t t e r s w i l l be used f o r the upper state and lower-case l e t t e r s for the lower s t a t e ) . Furthermore, l e t us consider the \ a r i a t i o n of with held f i x e d . The constrained v a r i a t i o n p r i n c i p l e can be written as <6t I ( # - £ ) / t > = O (3.2.1). where H. ~ A/ + A l S (3.2.2) and with the constraint condition C = O (3.2.3) In the diagonal case^, C . i s the expectation value of some constraint operator ~& C = <*/-6 / + ( (3.2.4) 13 but i n the off-diagonal case, C, i s taken as C = < ^ ( 9 | i > + < £ / ^ / ^ > (3.2.5) where * p . i s some operator of i n t e r e s t . The v a r i a t i o n of + 1 eads to a pseudodiagonal c o n s t r a i n t operator 13 defined by -6 = + 1^X^19 (3.2.6) then the off-diagonal constraint equation (3.2.5) becomes a pseudodiagonal c o n s t r a i n t . As i n the diagonal case, the v a r i a t i o n of leads to secular equation and the free v a r i a t i o n s o l -utions are used as a convenient b a s i s . In matrix notation, equation (3.2.1) becomes a pseudo-eigenvalue problem L -fl t A(^« + + J 1 - o (3.2.7) and equation (3.2.5) becomes C - 2^2. ta (3.2.8) where PK - < A f P | £ > (3.2.9) 65 and .P- , 4 are column vectors. As f o r the methods of so l v i n g the constrained secular equation (3 . 2.7), value of can be obtained from a perturbation approach or a parametrization 13 method . The perturbation approach in t h i s case d i f f e r s from normal ones i n that the perturbed Hamiltonian i s written as ft = Z X ^ f L * 0 -(3.2.10) n as compared to the usual one K = c } t x r ( 3 . 2 . 1 D and the nib-order perturbation energy i s (3.2.12) where 1*0 are the perturbation wavefunctions and are solutions to Z, l ^ - ^ j U - f c ) * O (3.2.13) 66 CS) The c a l c u l a t i o n of Q- from equation (3.2.12) requires ^ ^ , but a c t u a l l y , only up to H'( ^ i s needed. Reduction formulas f o r C 0 ^ are given by Chong and 13 Benston i n t h e i r appendix. In the parametrization method, f o r a p a r t i c u l a r value of X , equation (3.2.7) i s solved by i t e r a t i o n with the approximate solutions of fre e v a r i a t i o n s as basis set and as i n i t i a l guess of , e.g. (1, 0, 0 . . . ) . When 2: has converged, C i s cal c u l a t e d from equation (3.2.8). The correct value of /\ i s the one at which C i s zero. 3.3 Atomic t r a n s i t i o n p r o b a b i l i t i e s I t i s well known that the alternat e expressions for an e l e c t r i c d i p o l e • t r a n s i t i o n between atomic states which are t h e o r e t i c a l l y equivalent, usually give d i f f e r e n t values when approximate wavefunctions are 45-50 used . These three equivalent dipole operators 45 are 67 /*Lt = ZC4€/(jr^^ (3.3.2) a ^ ^ ) Z ( 5 ^ V ) (3.3.3) where r e f e r s to electron yU- , ^ £ has the magnitude (£-£-) and V i s the Columb p o t e n t i a l energy operator. I t should be noted that Afc i s p o s i t i v e when I^LL operates on £ but i s negative when H.2. operates on ^ . The expectation values of H'Li , Jft^and ^ 3 are usually r e f e r r e d to as dipole length dipole v e l o c i t y Mz. and dipole a c c e l e r a t i o n M3 r e s p e c t i v e l y . For exact wavefunctions, 32 • i t can be shown that < Y K / " l . / flt> = <1^l«U}fn> (3.3.4) <VW*U/<A> - <y^/ »U I tfi> (3.3.5) which i n f a c t are s p e c i f i c off-diagonal h y p e r v i r i a l theorems. I f the operator 'P which appears i n eq-uation (3.2.5) i s defined as 68 % = Ki. -Wj + l l , +.j= 1,2,3 (3.3.6) then constrained v a r i a t i o n s with Hi and 'PJ.S which give ^ and £ s a t i s f y equation (3.3.4) and equation (3.3.5). A double co n s t r a i n t with the constraint op-erator constructed from ^ and fi$ w i l l lead to a unique answer f o r the t r a n s i t i o n d i p o l e , i . e . dipole length, dipole v e l o c i t y and dipole acceleration w i l l be the same. This has been demonstrated by Chong 13 and Benston in t h e i r recent work assuming a constant t r a n s i t i o n energy AE An energy-independent off-diagonal h y p e r v i r i a l theorem can e a s i l y be obtained from equations (3.3.4) and (3.3.5) (3.3.7) which implies that (3.3.8) 69 In t h i s case, a single c o n s t r a i n t w i l l make and s a t i s f y equation (3.3.7). The object of the work described i n t h i s chapter i s t h r eefold: f i r s t l y , to study the use of the energy-independent off-diagonal h y p e r v i r i a l theorem as a constraint; secondly, to study the e f f e c t of i t e r a t i n g on the t r a n s i t i o n energy ^£ when solving the con-strained secular equation; and f i n a l l y , to extend the off-diagonal constraint to the i s o e l e c t r o n i c species L i + and Be +  .. 3.4 Calculations and r e s u l t s 52 x Eckart-type functions are used f o r f o r the 2^ "P state of He, L i + and B e + + 5 3 . ¥ = 1//2 (ls2p + 2pls) (3.4.1) The o r b i t a l exponents of the Slater-type atomic o r b i t a l s and energies are given i n Table XI. As for the l^S * Part of the r e s u l t s have been submitted f o r p u b l i c a t i o n , see r e f . (51). 70 lower state approximate wavefunctions ^ , i n the case of energy-independent off-diagonal h y p e r v i r i a l theorem as constraint, we use the five-term configuration-13 i n t e r a c t i o n wavefunction of Chong and Benston f o r He % = b 1 ( l s l s 1 ) + b 2 ( 2 s 2 ) + b 3 ( 2 s 3 s ) + b 4 ( 3 s 2 ) + b 5 ( 2 p 2 ) (3.4.2) The same function has been used i n Chapter II and, the o r b i t a l exponents,energy and the configuration mixing c o e f f i c i e n t s are given i n Tables I and II and i n Table XXIII (Appendix B) r e s p e c t i v e l y . In the case of energy-dependent off-diagonal h y p e r v i r i a l theorems as c o n s t r a i n t s , two medium-size functions are used f o r He, L i + and B e + + % = b 1 ( l s 2 ) + b 2 ( l s 2 s ) + b 3 ( l s 3 s ) + b 4 ( 2 s 2 ) + b 5 ( 2 s 3 s ) + b 6 ( 3 s 2 ) (3.4.3) ^7 = b 1 ( l s 2 ) + b 2 ( l s 2 s ) + b 3 ( l s 3 s ) + b 4 ( 2 s 2 ) + b 5 ( 2 s 3 s ) + b 6 ( 3 s 2 ) (3.4.4) +b ?(2p 2) The o r b i t a l exponents together with the energies are given i n Table XII. These exponents have been com-27 p l e t e l y optimized by an i t e r a t i o n procedure * 1 Table XI. Eckart-type functions f o r the 2 P state + + + of He, L i and Be Species O r b i t a l exponent Energy (a.u.) He Is : : 2.003024 -2.12239009 2p : : 0.482 36 3 L i + Is : : 3.006438 -4.98998887 2p : : 0.970210 Be + + Is : : 4.008219 -9.10605448 2p : ; 1.465022 *Taken from r e f . (53) 72 Table XII. O r b i t a l exponents of Slater-type atomic o r b i t a l s and energy ( i n atomic units) f o r the approx-imation wavefunctions ^ 6 and ^ 7 f o r the l^S ground state of He, L i + and B e + + . Function Species Is 2s 3s 2p £* (a.u.) He 1 .5596 2 .1233 2 .1209 - 2.878525 L i + 2 .4707 3 .1146 3 .6011 - 7.252073 Be + + 3 .4434 4 .2187 4 .8601 -13.626393 He 1 .5394 1 .8189 2 .0170 2 .4608 - 2.897902 L i + 2 .5346 3 .5180 3 .6041 3 .9243 - 7.272961 Be + + 3 .4330 4 .0768 4 . 7897 5 .3869 -13.647883 73 Using these exponents, the l i n e a r c o e f f i c i e n t s are optimised by energy minimization. The l i n e a r c o e f f i -c i e n t s are given i n Table XXIV (Appendix B). I t can be seen from Tables XI and XII that the error i n the energy of both and i s an order of magnitude greater than that of the Eckart-type Since the usual c r i t e r i o n f o r the accuracy of an approximate wavefunction i s the error i n energy; the Eckart-type functions ^ f o r the 2"^ P upper state seem to be better approximations than the l^S lower state functions ^ . This i s the basis f o r keeping $ f i x e d i n a l l c a l c u l a t i o n s . The p r i m i t i v e i n t e g r a l s needed f o r the evaluation of dipole length dipole v e l o c i t y M2. and dipole acceleration ^ 3 are given i n Appendix C. The energy-independent off-diagonal h y p e r v i r i a l theorem 2 Coulson suggested the use of energy-independent off-diagonal h y p e r v i r i a l theorems as c o n s t r a i n t s . A p a r t i c u l a r a p p l i c a t i o n i s to formulate the constraint as C =• M i - M , M 5 (3.4.9) 74 The corresponding c o n s t r a i n t operator "O f o r t h i s C i s straight-forward to construct as can be seen from equation (3.3.7). In t h i s case >^ i s given by = [J<J«f)]l¥><*|[£(^ (3.4.10) which i s i n f a c t a diagonal c o n s t r a i n t operator. Such a c o n s t r a i n t i s tested on the five-term approx-imate wavefunction ^5 used by Chong and Benston^ 3 f o r the l^S ground state of helium. The constrained secular equation i s solved by the p e r t u r b a t i o n - i t e r a t i o n 9 method , the energies and matrix elements of the con-s t r a i n t operator matrix obtained from free v a r i a t i o n wavefunctions are part of the input data required f o r the computing program. Other input data are a few c o n t r o l parameters. The r e s u l t s of c a l c u l a t i o n s are shown in Table XIII. The % error of dipole length {^ \{ , dipole v e l o c i t y tAz. and dipole acceleration from the exact value are given i n parentheses. For comparison, the r e s u l t s obtained by Chong and 13 Benston are also included i n the table. I t can be seen that the s a c r i f i c e in energy due to the c o n s t r a i n t 7$ i s not s i g n i f i c a n t . I t e r a t i o n on the t r a n s i t i o n energy A E For each approximate wavefunction, Chong and Benston assumed that the t r a n s i t i o n energy A E i s constant, at the free v a r i a t i o n value. Since the constraints cause a small s a c r i f i c e i n the energy of the ground state wavefunctions, the constrained func-tions they obtained do not, s t r i c t l y speaking, sat-i s f y the off-diagonal h y p e r v i r i a l theorems. Therefore, the e f f e c t of i t e r a t i n g on A E i s studied i n the present work. The free v a r i a t i o n r e s u l t s are summarized i n Table XIV. The t r a n s i t i o n energy . AE i s obtained 1 1 from the Eckart-type 2 P state function and the 1 S state function. In the case of s i n g l e constraint, i . e . Mjl = , the constraint operator defined by equation (3.2.6) becomes (3.4.11) where the Vila's have been defined by equations (3.3.1) to (3.3.3). In the case of double constraint, 76 i . e . M i = M r = M j , then we have two constraint operators ^ , = C * r L , - ^ J I ^ X ^ I + I ^ X ^ K ^ i - ^ ) (3.4.12) C H x - ^ l i X ^ 4 W X i - K w U - ^ a ) (3.4.13) and the constrained secular equation (3.2.7) becomes (3.4.14) the constraint condition i s s a t i s f i e d when C i =• ° and G _ ^  O (3.4.15) where ' G ^ -2£V<a and G_ =-2 f t3 <2= (3.4.16) For the s i n g l e constraint c a l c u l a t i o n s , the perturbation-9 i t e r a t i o n method i s used. On the other hand, f o r Table X I I I . Results of using the energy-independent off-diagonal h y p e r v i r i a l theorem as a con s t r a i n t on the function ^ 5 f o r the l^S ground state of He. The % error of M» , Mi and IM3 from the exact value are given i n parentheses. Constraint B(a.u.) none Exact,none 0.46306(+10.05) 0.40667(-3.35) 0.46238C+9.89) 0.42155(+0.18) 0.40667(-3.35) 0.42697(+1.47) 0.42078b 0.40483(-3.79) 0.40667(-3.35) 0.39427C-6.30) -2 .897142 -2 .829589 -2 .895329 -2 .903724C 3. Taken from r e f . (13) b Ref. (54) C Ref. (29) 78 Table XIV. Results of fr e e v a r i a t i o n (in atomic un i t s ) f o r the l^S ground state functions and ^ 7 of He, L i + and B e + + . The A E i s the l^'S - 21V t r a n s i t i o n energy. Species Function M» Mi M 3 A t 0.756135 0.775512 0.779742b 2.262084 2.282972 2.286424b 4.520339 4.541828 4.544754b a Ref. (54) b Ref. (55) He L i + Be + + *6 % Exact % ^7 Exact +6 Exact 0.41222 0.40362 0.31958 0.31255 0.24959 0.23930 0.41346 0.41758 0.42078* 0.30928 0.31380 0. 316131 0.24191 0.25685 0.24643} 0.49437 0.40193 0.33711 0.30345 0.25570 0.25387 79 double c o n s t r a i n t s , we used the method proposed by Chong and Benston"^ to obtain c o r r e c t values for A» and A*, from two coupled l i n e a r equations. In both s i n g l e and*constraint c a l c u l a t i o n s , the c o r r e c t values of K-s are obtained when the constraint conditions reach the desired accuracy — 10~ . Computing programs are written f o r the sing l e and double c o n s t r a i n t c a l c u l a t i o n s i n such a way that a f t e r solving the corresponding constrained secular equation and the constraint condition i s s a t i s f i e d , the new t r a n s i t i o n energy obtained from the constrained wavefunction i s used to c a l c u l a t e new values of Mz. and M 3 . A new constrained secular equation i s formed and solved i n * s i m i l a r manner. The steps are repeated u n t i l e i t h e r the i t e r a t i o n on AE has con-verged or i t has exceeded the preassigned number of c y c l e s . The values of Mt , M i and M 3 obtained from the f i r s t c ycle of i t e r a t i o n correspond to the constant t r a n s i t i o n energy assumption made by Chong 13 and Benston . A complete l i s t and d e t a i l s of the program i n the case of double constraint are given in Appendix D. The r e s u l t s of our i n v e s t i g a t i o n are shown i n Table XV. Values given i n parentheses are 80 f o r t h e a s s u m p t i o n o f c o n s t a n t . The c o e f f i c i e n t s f o r t h e s e v e n - t e r m d o u b l e c o n s t r a i n e d w a v e f u n c t i o n s a r e g i v e n i n T a b l e X V I I I . E x t e n s i o n t o L i + and B e + + The same approach o f i i t e r a t i n g on the t r a n s i t i o n e nergy A E i s ex t e n d e d t o t h e c o r r e s p o n d i n g 1 S t o 2^P t r a n s i t i o n i n L i + and B e + + . The a p p r o x i m a t e w a v e f u n c t i o n s used f o r t h e 1 1S ground s t a t e a r e % and ^7 and a r e o f t h e same f o r m as t h o s e used i n t h e ca s e o f He. The o r b i t a l e x p o n e n t s o f t h e S l a t e r - t y p e a t o m i c o r b i t a l s o f t h e s e f u n c t i o n s a r e g i v e n i n T a b l e X I I . The c o n f i g u r a t i o n m i x i n g c o e f f i c i e n t s a r e g i v e n i n T a b l e XXIV, A p p e n d i x B. E c k a r t - t y p e f u n c t i o n s a r e used f o r t h e 2^ "P upper s t a t e and a r e shown i n T a b l e X I . The r e s u l t s o f c o n s t r a i n e d v a r i a t i o n a r e summarized i n T a b l e s XVI and X V I I . V a l u e s g i v e n i n p a r e n t h e s e s a r e f o r the a s s u m p t i o n o f c o n s t a n t t r a n s i t i o n e n e r g y The c o e f f i c i e n t s f o r t h e sev e n - t e r m d o u b l e c o n s t r a i n e d w a v e f u n c t i o n s a r e g i v e n i n T a b l e X V I I I . 3.5 D i s c u s s i o n The r e s u l t s f o r t h e use o f t h e e n e r g y - i n d e p e n d e n t o f f - d i a g o n a l h y p e r v i r i a l theorem as a c o n s t r a i n t a r e Table XV. Results of c o n s t r a i n t v a r i a t i o n s ( i n atomic units) on K\J6 and ^ fo r the l^S ground state of He. The r e s u l t s f o r the assumption of constant AE are given i n parentheses. Function Constraint E (a.u.) M, M i M 5 none -2.878525 0.41222 0.41346 0 .49437 Mi = Mi -2.878516 0.41378 0.41378 0 .49435 (-2.878516) (0.41377) (0.41377) (0 .49434) M V = H 3 _+ — — — (-2.856586) (0.48846) (0.42740) (0 .48846) Mz= M 3 + ~+ - - -—+ —+ - - -^ 7 none -2 .897902 0.40362 0.41758 0 .40193 1 / M,= M i -2 .897090 0.42006 0.42006 0 .40726 (-2.897138) (0.41956) (0.41956) (0 .40626) M,= 1M3 -2 .897897 0.40260 0.4172 3 0 .40260 (-2.897238) (0.40260) (0.41723). (0 .40260) Mi= N 3 -2.897267 0.40232 .0.41484 0 .41484 . (-2.897238) (0.40230) (0.41443) (0 .41443) M, = r>v= M 3 -2 .896630 0.41759 0.41759 0 .41759 (-2.896650) (0.41663) (0.41663) (0 .41663) Exact none -2.903724 a 0.42078 b + No s o l u t i o n e x i s t s . a Ref. (29) b Ref. (54) Table XVI. Results of constrained v a r i a t i o n s ( i n atomic units) on and ^ f o r the l^S ground state of L i + . The r e s u l t s f o r the assumption of constant A E are given i n parentheses. Function Constraint E (a.u.) M i M r M 3 none -7.252073 0. 31958 0. 30928 0 .33711 M,= Hz -7.248232 0. 30750 0. 30750 0 .33879 (-7.247810) (0. 30685) (0. 30685) (0 .33766) M, = M 3 -7.242141 0. 33839 0. 31394 0 .33839 (-7.244804 (0. 33572) (0. 31213) • <° .33572) Mx= M* _+ — — — (-7.002679) (0. 39559) (0. 31855) (0 .31855) M,= M^ = M 3 _+ — — — (-6.177154)7 (0. 30889) (0. 30889) (0 .30889) none -7.272961 0. 312 55 0. 31380 0 .30345 M, = -7.272926 0. 31388 0. 31388 0 .30388 (-7.272926) (0. 31387) (0. 31387) (0 .30387) M 5 -7.272016 0. 30709 0. 31214 0 .30709 (-7.271961) (0. 30693), (0. 31196) (0 .30693 M 3 -7.271289 0. 31236 0. 31172 0 .31172 (-7.271213) (0. 31236) (0. 31144) (0 .31144) M,= M 3 -7.271280 0. 31164 0. 31164 0 .31164 (-7.271194') (0. 31132) (0. 31132 ) (0 .31132) none -7.279913 a 0. 316^ ' Exact + No s o l u t i o n e x i s t s . 0 Ref. (56) b Ref. (55) Table XVII. Results of constrained v a r i a t i o n s ( i n atomic un i t s ) on ^6 and % fo r the l^S ground state of B e + + . The r e s u l t s f o r the assumption of constant," AE are given i n parentheses. Function Constraint IE. (a.u.) M » . (v^ 3 ^7 Exact none -13.626393 0 .24959 0.24191 0 .25570 M,= Mz -13.621127 0 .24080 0.24080 0 .25666 (-13.620721) (0 .24046) (0.24046) (0 .25607) M,= M 3 -13.623868 0 .25557 0.24294 0 .25557 (-13.624088 (0 .25531) (0.24276) (0 .25531) M i = M 3 _+ — — — (-13.403609) (0 .29548) k (0.24729) (0 .24729) M,= _+ — — — (-12.653118) (0 .24261) (0.24261) (0 .24261) none -13.647883 0 .23930 0.25685 0 .25387 M i = Mz -13.647087 0 .24584 0.24584 0 .23980 (-13.647091) (0 .24582) (0.24582) (0 .23975) M,= M 3 -13.647499 0 .24377 0.24935 0 .24377 (-13.647502) (0 .24376) (0.24935) (0 .24376) Mi= M 3 -13.647602 0 .23684 0.26091 0 .26091 (-13.647598) (0 .23683)' (0.26091) (0 .26091) M i = -13.645272 0 .24573 0.24573 0 .24573 (-13.645172) (0 .24567) (0.24567) (0 .24567) none -13.655566 a 0.246 b No s o l u t i o n exists, Ref. (56) Ref. (55) Table XVIII. C o e f f i c i e n t s f o r the seven-term double constrained wavefunctions i n terms of fre e v a r i a t i o n functions. Species a^ a^ a^ a^ a^ ag a^ He 0.999778 0.016533 -0.001711 0.012628 -0.002530 0.000958 -0.000168 L i + 0.999926 -0.004842 -0.000290 0.011177 -0.000347 . 0.000209 -0.000103 B e + + 0.999953 0.001042 -0.000237 0.009205 -0.000700 -0.000696 -0.002463 85 somewhat disappointing as can be seen from Table XI I I . We f i n d that the constrained wavefunction gives the values of 0.46238, 0.42697 and 0.39427 f o r dipole length M i , dipole v e l o c i t y M2. and dipole a c c e l -eration M 3 ( i n atomic un i t s ) r e s p e c t i v e l y , compared 54 to exact value of 0.42078 . m the f i r s t place, the constraint i n equation (3.4.9) f a i l s to remove the undesirable ambiguity i n the c a l c u l a t i o n of t r a n s i t i o n moments. Secondly, the best moment ( M i i n t h i s case) obtained from the constrained wavefunction deviates more from the exact, value than the free v a r i a t i o n a l r e s u l t of 0.42155 does. Thus, t h i s energy-independent . of f-diagonal c o n s t r a i n t i s not useful i n improving c a l c u l a t i o n of t r a n s i t i o n p r o b a b i l i t i e s . In contrast to the energy-independent o f f -diagonal constraint, the use of the energy-dependent off-diagonal h y p e r v i r i a l theorems as constraints can remove the ambiguity i n the calcul a t e d t r a n s i t i o n p r o b a b i l i t i e s and y i e l d a unique value as have been 13 shown by Chong and Benston and also the present r e s u l t s given i n Tables XV, XVI, and XVII. Since the s a c r i f i c e i n energy on constraining i s generally 86 small, i t i s s u r p r i s i n g to discover that f o r some of the constraints on the six-term functions, the i t e r a t i o n on the t r a n s i t i o n energy A£ f a i l s to converge. On the other hand, when the approximate wavefunction i s f l e x i b l e enough, e.g. ^ , the i t e r a t i o n converges i n a few cycle*. Comparing with the r e s u l t s based on the assumption of constant AE } we see that the e f f e c t of i t e r a t i n g on AE i s small. From our study on the i s o e l e c t r o n i c species L i and Be , i t i s encouraging to f i n d that such r e l a t i v e l y simple seven-term wavefunctions can lead to t r a n s i t i o n moments which are not only unambiguous, but also i n good agreement with those r e s u l t i n g from 55 extensive v a r i a t i o n a l c a l c u l a t i o n s As for the methods of solv i n g constrained sec-9 ul a r equations the p e r t u r b a t i o n - i t e r a t i o n method seems to have the advantages of both the perturbation^ and parametrization method^ and does not su f f e r from the disadvantages of e i t h e r . I t i s quite s u r p r i s i n g to f i n d that the method proposed by Chong and Ben-ston^ for double constraints works so well and very few i t e r a t i o n s w i l l give the correct values for X ( and Xi. . A nonlinear constrained v a r i a t i o n or the simple 2 Is and l s l s ' wavefunction for the same t r a n s i t i o n (l^S - 2^P) i n He has been studied by Chong and 13 Benston . The t r a n s i t i o n energy was held f i x e d and only the o r b i t a l exponents were varied to s a t i s f y the constraint condition. Their r e s u l t s of single constraints i n d i c a t e that the s a c r i f i c e i n energy i s very serious and the r e s u l t i n g values of M| } M2_ and N 3 are worse than the fre e v a r i a t i o n value 13 I t has also been suggested that by imposing a semiempirical off-diagonal constraint, the calcula t e d t r a n s i t i o n moment can be made to agree with the experimental value. 88 _ - CHAPTER IV PERTURBATION-INDUCED CONSTRAINTS 4.1 I n t r o d u c t i o n I n p r e v i o u s c h a p t e r s , we s t u d i e d t h e c o n s t r a i n e d v a r i a t i o n method by c o n s i d e r i n g some s p e c i f i c exam-p l e s , namely: t h e i n t e g r a l e l e c t r o n c usp c o n d i t i o n ; t h e e n e r g y - i n d e p e n d e n t and energy-dependent o f f -d i a g o n a l h y p e r v i r i a l theorems as c o n s t r a i n t s i n o r d e r t o c a l c u l a t e h o p e f u l l y b e t t e r e l e c t r o n i c p r o p e r t i e s a t t h e n u c l e i and t r a n s i t i o n p r o b a b i l i t i e s . I n most c a s e s , o ur r e s u l t s a r e e n c o u r a g i n g , t h e s a c r i f i c e i n e n e r gy due t o t h e i m p o s i t i o n o f c o n s t r a i n t s i s s m a l l and e x p e c t a t i o n v a l u e s o f some o t h e r p r o p e r -t i e s a r e impro v e d . D i f f e r e n t methods o f s o l v i n g t h e c o n s t r a i n e d s e c u l a r e q u a t i o n have a l s o been s t u d -i e d and d i s c u s s e d . Thus, t h e purpose o f c o n s t r a i n e d v a r i a t i o n s has been e s t a b l i s h e d . However, i n a l l t h e examples we s t u d i e d , a p a r t i c u l a r c o n s t r a i n t o p e r a t o r must be c o n s t r u c t e d 89 i n a p a r t i c u l a r manner according to the p a r t i c u l a r prop-erty subject to c o n s t r a i n t . One would n a t u r a l l y look into the p o s s i b i l i t y of constructing the constraint operator i n a more general way. A very recent method 17 proposed by Chong and Benston "perturbation-induced constraints" i n which d i f f e r e n t c o n straint operators are constructed f o r d i f f e r e n t properties i s in f a c t the f i r s t attempt i n t h i s respect. The perturbation-induced constraints scheme i s f o r f i r s t - o r d e r properties such as permanent dipol e s , quadrupole moments, charge density at the nucleus, etc. These properties are the s t r a i g h t expectation values of some operators which are usually the sum of one-electron operator. The f i r s t - o r d e r properties can also be regarded as a 5 7 f i r s t - o r d e r perturbation energy There are two d i f f e r e n t approaches i n the per-turbation-induced constraints scheme (PIC), namely, the i t e r a t i v e method of off-diagonal perturbation-induced constraints (IMODPIC) and the diagonal per-turbation-induced constraints (DPIC). In the former approach (IMODPIC), the f i r s t - o r d e r wavefunction i s f i x e d and therefore i s constant f o r a l l properties. This approach i s analogous to the uncoupled Hartree-Fock perturbation method. In the l a t t e r approach (DPIC), the f i r s t - o r d e r wavefunction changes along with the zeroth-order wavefunction during the cons-t r a i n t process and i s d i f f e r e n t f o r d i f f e r e n t proper-t i e s . This approach i s therefore analogous to the coupled Hartree-Fock perturbation method. In t h i s chapter, we s h a l l t e s t the v a l i d i t y of both approaches of the perturbation-induced c o n s t r a i n t s . The theory w i l l f i r s t be given i n the following section and then l a t e r we s h a l l apply the theory to study the expectation values of t 1 , -I « r 4 Z r , t and x . These expectation values are the sum of one-electron expectation values. In:, each case, a d i f f e r e n t c o n s t r a i n t operator i s constructed i n order to improve hopefully the cor-responding expectation value. 4.2 Theory The basic idea i s to construct a constraint operator such that the f i r s t - o r d e r c o r r e c t i o n to the expectation value of some operator W obtained from an approximate wavefunction vanishes a f t e r the imposition of the co n s t r a i n t . The f i r s t -order c o r r e c t i o n W C , ) i s due to the expansion of the li m i t e d subspace by introducing more basis functions (to a complete set, in the l i m i t ) . The addition of more functions to the basis set produces a per-turbation. Let us consider the H i l b e r t space spanned by an orthonormal set of functions ^ which are i n f i n i t e i n number. The orthonormal condition i s $ | 3 > = = , 2 . . . . ca (4.2.1) Let us now consider the subspace spanned by a f i n i t e number of these functions -^^} where 1 ^ k ^ K. Let the true Hamiltonian H of the system i n t h i s subspace be diagonal, i . e . (4.2.2) where [ <r\> i s an approximate wavefunction for the ground state and with approximate energy £, The p r o j e c t i o n operator f o r the subspace i s simply A Since the H i l b e r t space we considered i s spanned by an orthonormal set of functions { ^ j " which are i n f i n i t e i n number, we can write OO • ' ~ £ I I (4.2.4) ot = f the complementary proj e c t i o n operator P of i s then given by "P - I - t> (4.2.5) oo = £l4>n><<M n=K*t Obviously, these two operators P and -p s a t i s f y P^, = 0 = -pV (4.2.6) The zeroth-order Hamiltonian ~f[y i n t h i s subspace i s defined by -b- - -p H -p K - I! efc l<fVX4> | (4.2.7) and therefore -&S I4>k> = ^ » ^ > (4.2.8) then 6^ i s an eigenvalue of "/L with eigenfunction I . The c o r r e c t i o n V to the zeroth-order Hamiltonian ~hs in t h i s subspace i s zero. Since the c o r r e c t i o n i s defined as V = H " X (4.2.9) then according to equations (4.2.2) and (4.2.7) _ 0 (4.2.10) 94 which i s equivalent to -f>V-jo = ° (4.2.11) t h i s implies that i n t h i s subspace, the zeroth-order Hamiltonian i s as good as the true one and hence the f i r s t - o r d e r perturbation energy due to the c o r r e c t i o n V i s also zero. F i r s t - o r d e r properties Since f i r s t - o r d e r properties are s t r a i g h t ex-pectation values of some operator W and are usually ca l c u l a t e d from approximate wavefunctions, i t i s important to c o r r e c t such approximate expec-ta t i o n values. Consider the expression f o r the true Hamiltonian H = 7 i + V (4.2.12) I t i s obvious that the V i n the above equation i s responsible f o r the corrections i n the zeroth-order Hamiltonian or i n the corresponding approximate wavefunctions. As expectation values can be regarded 95 as perturbation energies associated with an operator Vs| , then the corrections become a double pertur-bation to . We can define a f i c t i t i o u s Hamil-tonian by = X + v + W (4.2.13) then f o r f i r s t - o r d e r properties, the corresponding expectation value <W*> can be expanded by a double 4. ^ . 5 7 perturbation se r i e s where w0> = ( t t , 0 , i w l t > t <*|wl + cl°> x '(4.2 .16) (4.2.14) co) . W = <4>, | W | 4>,> (4.2.15) 96 I 4*1 y i s the zeroth-order approximate wavefunction \«4'(po^ and l^ 0 ' 0^ i s the f i r s t - o r d e r function when V i s the perturbation to . By means of Dalgarno's interchange theorem^, we can rewrite equation (4.2.16) as W10 = <v|/<">lvi4\> + < c f > l y l * " ' ^ (4.2.17) where Ity^'^is the f i r s t - o r d e r wavefunction when W i s the perturbation to -R/ Perturbation by V Let lis now consider the perturbation by V to the zeroth-order Hamiltonian ~A, due to the expansion of the basis functions. The f i r s t - o r d e r 57 perturbation equation i s U- <ifOO,)ltC<0>>+ (V-Ol4>.>- O (4.2.18) where £ t e o >_ £.( # The i s the f i r s t - o r d e r perturbation energy and i s zero from equations (4.2.10) and (4.2.11). The above equation can there-fore be written as 97 '-.C-fU - €,>lt°°'> + Vl4>,> - O (4.2.19) In general, the f i r s t - o r d e r wavefunction 14*T(0>> can be s p l i t into two parts |VJA , 0 >> = f M C l r t > + - p ^ O o ^ (4.2.20) where ^ l ^ ^ 0 ^ indicates contribution from the sub-space and Pit ° 0 >> outside the subspace. Further-more, l e t T>Wwy> = £ l*R>bk (4.2.21) and apply the operator -p> to the l e f t of equation (4.2.19) p ( f t - e , ) ! t < ( 0 )> -f ^ V I 4 > , > = ° $ 4 . 2 . 2 2 ) Since the term -f>V vanishes and = H-p , the above equation i s reduced to K ^*>(%~ e , ) b k = 0 (4.2.23) 98 which i s true i f and only i f bfc = 0 for 6^ ^ 6, This implies that the cont r i b u t i o n from the subspace to the f i r s t - o r d e r wavefunction \y<-l0>y ± s zero as expected. S i m i l a r l y , i f we apply the operator P to the l e f t of equation (4.2.19) we obtain ? (A - 6,>1^ C , 0 >> « PV !<!>,> = 0 (4.2.24) which can be reduced to ?K t ( 0 )> « e.'pVl^,) (4.2.25) Let us write (4.2.26) and substitute equation (4.2.19) into equation (4.2.18), then b n i s given by b n > <«Vi Wl4\> (4.2.27) Since -p\K\jLt°'>) i s zero, then the only contribution to the f i r s t - o r d e r wavefunction I ^  ""'^ comes from outside the subspace and W C ( 0 >> i s given by PI ^  °o:>> , i.e. = < PV!4\> (4.2.28) I t should be noted that as P V = P H , the above equation can be rewritten as For the diagonal perturbation-induced constraints (DPIC) approach, \ > of the above equation i s given by the constrained approximate wavefunction whereas f o r the i t e r a t i v e method of off-diagonal 100 perturbation-induced constraints approach (IMODPIC), \ 4*1 y i s the free v a r i a t i o n approximate wavefunction and therefore i s f i x e d and a constant. An alternate expression f o r the perturbation of X by W can e a s i l y be obtained by the Dalgarno's interchange theorem and equation (4.2.29) then becomes lt t o n> = < ?w|4>,> a e< H I *n><4>n)W !•<!>,> (4.2.30) A very useful expression i s obtained on sub-s t i t u t i n g equation (4.2.29) i n equation (4.2.16) which gives W(" = <£-;' PH<UvvM4>,> + <4),lw|^PH^> (4.2.31) We s h a l l see l a t e r that the above equation v / i l l lead to two d i f f e r e n t approaches, DPIC and IMODPIC of the perturbation-induced constraints scheme. 1 Free v a r i a t i o n c o r r e c t i o n (FVC) The free v a r i a t i o n f i r s t - o r d e r c o r r e c t i o n W(l) to W'"1 can at once be obtained from equation (4.2.31) W (° * £i"'<4>KWPH + HPW) I4>> = 6«' £ <4>,»{wl4v,><<UH + H 14>rtX<Uw}l<t>(> n = K + l (4.2.32) I t i s obvious from the above equation that the f i r s t - o r d e r c o r r e c t i o n VN 0' comes from i t e r a c t i o n s between the approximate wavefunction with functions outside the subspace. I t should be noted that i f l ^ i ^ i s an exact wavefunction, W (^ i s automatically zero. However, since ) 4*1 )• i s a free v a r i a t i o n approximate wavefunction, \ A I 0 > i s non-zero. We s h a l l r e f e r the f i r s t - o r d e r c o r r e c t i o n W ( 1 ob-tained by equation (4.2.32) as the free v a r i a t i o n . c o r r e c t i o n (FVC). Diagonal perturbation-induced constraints (DPIC) As have been mentioned i n the introduction of t h i s chapter, i n the present approach, the f i r s t -102 order wavefunction |ty C (*^ given by equation (4.2.29) changes along with the zeroth-order wavefunction l4>i> during the c o n s t r a i n t process. In t h i s case, we can rewrite equation (4.2.31) as £,W<4>,ICW.PH H?W)l^> (4.2.33) In order to make \ N ° ^ vanish, we want to f i n d a normalized constrained approximate wavefunction l ¥ i > such that Wc' = e;' < t l ( W P H + H P W ) I 1 ) = O (4.2.34) and the energy (?, i s also a minimum, where d = <¥,| W | (4.2.35) I t should be noted that the constraint of the present approach i s a diagonal one as can be seen from eq-uations (4.2.33) and (4.2.34). 103 In order to show that the co n s t r a i n t oper-ator ~& has the proper form as suggested by equation (4.2.33), i . e . ~6 - WP H + H?W (4.2.36) n=k+J l e t us assume that we have found a' set of ortho-normal ized constrained wavefunctions •[ "i! k j such that (4.2.37) where Kc = H -V X-6 (4.2.38) The proj e c t i o n operator *f> of equation (4.2.3) then becomes. -fr = £ l¥fcX**l (4.2.39) 104 We can conveniently express the constrained functions i n terms of the free v a r i a t i o n functions \*m> = £ \$*L>CLlm (4.2.40) then i t follows that -f>c = j> and Pc = P Let us now define the constrained e f f e c t i v e Hamiltonian by "Ac = f H^f (4.2.41) which i s equivalent to -ic = £ (4.2.42) and -£clS*> = £ K I * K > (4.2.43) = ( e K U c ^ ) l { R ) 105 where £ k = < I H ) f f e > (4.2.44) CKK = < i h l ^ l i h > (4.2.45) Since we know that only i s needed f o r the c a l c u l a t i o n of Wt4) , i n the DPIC approach, i s given by P l O = ?Vfc I*> (4.2.46) and the f i r s t - o r d e r c o r r e c t i o n W"' becomes W c = e ^ < H C W P V c + V c F W ! *,> (4.2.47) I t i s straight-forward to show that p V c = P H and Vc P = H P . We can then rewrite the above equation as W c ° = er' O L K W P H + HPW|^ (> (4.2.48) 106 which i s completely i d e n t i c a l to equation (4.2.34) except f o r the change i n 6t to €\ , hence the co n s t r a i n t operator must be that given by equation (4.2.36) and the co n s t r a i n t condition i s c„ = <4 I -6 I *> - 0 (4.2.49) and. therefore £, = e, . I t can be seen from equation (4.2.46) that i n t h i s approach, the f i r s t -order wavefunction IH' c<°> depends on W , and hence i s d i f f e r e n t f o r d i f f e r e n t properties. I t e r a t i v e method of off-diagonal perturbation-induced  constraints (IMODPIC) In contrast to the DPIC approach the f i r s t -order wavefunction I ao)} in the present approach i s f i x e d and i s the same f o r a l l the properties. In t h i s case, we can rewrite equation (4.2.31) as W ( ° = <*;' pH4>, Iwl4>,> + <4>(! wleT'pH^> (4.2.50) 107 In order to constrain'; the f i r s t - o r d e r c o r r e c t i o n W ( l * to vanish, we want to f i n d a normalized con-strained approximate wavefunction \$y such that (4.2.51) c O and that the energy e. = <ilH\i)> i s also a minimum. Since 14l> i s normalized, equation (4.2.51) i s equivalent to W c =- <$l±><^PH4>,lw!4-> +<£|w|6?pH<fc><£l4> - 0 (4.2.52) In order to s a t i s f y the above co n s t r a i n t , the con-s t r a i n t operator i s defined as -4> - (liX^PH^IW + W|^ 'PH<t>,X*lJ (4.2.53) 1 We can also express the constrained wavefunction l i > i n terms of the free v a r i a t i o n functions. K * - E l<*V><U (4.2.54) the matrix element of the constraint operator matrix i s then given by C K I = £ la*<<fc|HI<f>w><<Mw|<fe> (4.2.55) 4 <4 , f e i w | < f v v > < ^ n iH\<fc>a*) I t should be noted that i n t h i s IMODPIC approach, the constraint operator "CS given by equation (4.2.53) i s pseudodiagonal as i n the case of off-diagonal h y p e r v i r i a l theorems as co n s t r a i n t s . The f i r s t -order wavefunction 1^°°^ i s independent of W A s i m i l a r scheme " d i f f e r e n t screening con-stants f o r d i f f e r e n t properties" suggested by Dalgarno 59 and Stewart has been further studied by Sanders and H i r s c h f e l d e r ^ . The f i r s t - o r d e r perturbation c o r r e c t i o n to some f i r s t - o r d e r property i s made to vanish by using d i f f e r e n t screening constants. In the present PIC scheme, i n both DPIC and IMODPIC approaches, t h i s f i r s t - o r d e r perturbation c o r r e c t i o n i s made to vanish by means of con s t r a i n t s . Since the constraints are induced by perturbation, t h i s scheme i s - c a l l e d perturbation-induced c o n s t r a i n t s , and the constraints are d i f f e r e n t f o r d i f f e r e n t properties. 4.3 Calcul a t i o n s and r e s u l t s The five-term 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-function of the helium atom used i n our previous c a l c u l a t i o n s i s again chosen f o r the present study. The wavefunctions obtained from the dia g o n a l i z a t i o n of the free v a r i a t i o n a l secular equation are desig-wavefunction f o r the ground state of helium. A l l these f i v e functions are of the form <j>k = b l k ( l s l s ' ) + b 2 k ( 2 s 2 ) + b 3 k ( 2 s 3 s ) + b 4 k ( 3 s 2 ) + b 5 k ( 2 p 2 (4.3.1 110 Each term appearing i n equation (4.3.1) i s a S function. The f i r s t - o r d e r properties i n the present study are the expectation values of jf2 , r~' , * and r*2- obtained from 4>f . The p r i m i t i v e i n t e g r a l s are given i n Appendix C. In order to correct these approximate expec-t a t i o n values, we expand the H i l b e r t space by adding three more functions to the o r i g i n a l f i v e functions basis set. The a d d i t i o n a l functions are (2s,4s), 2 1 (3s,4s) and (4s ). They are also S functions and designated by <p^ , and <PQ r e s p e c t i v e l y . The o r b i t a l exponent of the 4s Slater-type atomic o r b i t a l i s o p t i m i z e d by the minifltisation of an eight-term function ^ 8 = b ^ l s l s ' )+b 2(2s 2)+b 3(2s3s)+b 4(3s 2) + b 5 ( 2 p 2 ) +b 6(2s4s)+b 7(3s4s)+b g(4s 2) (4.3.2) The o r b i t a l exponents f o r the Is, Is', 2s, 3s and 2p Slater-type atomic o r b i t a l s are the same as those used i n the five-term functions. The o r b i t a l ex-ponent f o r the 4s atomic o r b i t a l thus obtained i s 4.5558 with an energy -2.897274 a.u. Since in the formulation of theory, we assume an orthonormal set of functions, hence the Schmidt orthogonalization method i s used to ensure our eight function $ 's form an orthonormal set. Before we apply the Schmidt orthogonalization procedure, we f i r s t c a l c u l a t e the matrices f o r W = t l , -I 4 2 JT , r and f using the configuration (term) functions as basis from the eight-term wavefunction. The overlap matrix £ and the Hamiltonian matrix bi are also c a l c u l a t e d i n the same bas i s . A c o e f f i c i e n t matrix B i s then constructed i n order to transform the W matrices, the § matrix and the ti matrix into the nonorthogonal basis when k = 1, 2 ... 8. These 8 x 8 matrices can be p a r t i t i o n e d into four submatrices as shown below: 1 2 1 2 3 4 5 6 7 8 5 x 5 5 6 7 8 5 x 3 3 x 5 3 x 3 112 For the || matrix, the matrix elements i n the 5 x 5. submatrix are the configuration mixing c o e f f i c i e n t s f o r to <PS , the 3 x 3 submatrix i s diagonal with u n i t diagonal elements and the matrix elements i n the 3 x 5 and 5 x 3 submatrices are zero. A f t e r the transformation by the B matrix, the 5 x 5 submatrix of the overlap matrix S' = B^S E> becomes a u n i t matrix and a l l other sub-matrices have non-zero matrix elements. The trans-, formed Hamiltonian ma t r i x VA = ' B> H 8 has the same form but with free v a r i a t i o n energies as diagonal elements i n the 5 x 5 submatrix, the Hit matrix element i s the approximate energy obtained from (f>( . The W = t"2 , r"' , r and r + 2 matrices are also transformed into the W = £ W B matrices, the Wit matrix elements are the expec-t a t i o n values of if1 , r~' , t and t42 obtained from ^ . The S matrix i s then used to obtain a X matrix f o r the Schmidt orthogonalization purpose. The X matrix thus obtained has a 5 x 5 u n i t sub-matrice, a 3 x 5 zero submatrix and with non-zero 113 m a t r i x e l e m e n t s f o r t h e 5 x 3 and 3 x 3 s u b m a t r i c e s . The" M' and W' m a t r i c e s a r e t h e n t r a n s f o r m e d f r o m a n o n - o r t h o g o n a l b a s i s t o an o r t h o g o n a l b a s i s by t h i s "X m a t r i x i n t o H" = 1^ H' X a n < 3 W" = m a t r i c e s , as a check we a l s o a p p l y t h e t r a n s -f o r m a t i o n t o t h e m a t r i x and o b t a i n = T*S T = J . w h i c h i s a u n i t m a t r i x . A f t e r t h e above o r t h o g o n a l i z a t i o n p r o c e s s , t h e H and m a t r i c e s a r e t h e n used t o c o n s t r u c t c o n s t r a i n t o p e r a t o r s f o r W = r* , t"' , r and -C41 . DPIC The c o n s t r a i n t o p e r a t o r ~& d e f i n e d by e q u a t i o n (4.2.36) f o r t h e d i a g o n a l p e r t u r b a t i o n -i n d u c e d c o n s t r a i n t s (DPIC) becomes = EW^X<^!H H l ^ X ^ l w J (4.3.3) The system o f e q u a t i o n s t o be s o l v e d a r e [ 4 + AQ - U 3 « ~ ° (4.3.4) a + c a =. o (4.3.5) 114 I t should be noted that a l l the matrix representations are i n the {^ftj basis with 1 ^ k ^ 5 , the column vector S i s the s o l u t i o n f o r the constrained ground state approximate wavefunction. The matrix elements of the £ matrix are 8 * Z < < t V \ H l < } V X < M w!9^ > (4.3.6) n=6 and the constraint condition, equation (4.3.5) i s equivalent to s s 5 - o ' (4.3.7) 115 Equations (4.4.4) and (4.4.5) are solved by the 8 parametrization method . The c o r r e c t value of the parameter ^ i s obtained when the constraint con-—8 d i t i o n s a t i s f i e s the desired accuracy ^ 10" . The r e s u l t s are summarized i n Table XIX. IMODPIC The constraint operator ~& defined by equation (4.2.43) i n t h i s approach i s pseudodiagonal, the system of equations to be solved are C"4 + ACf i + + ) - e i J fi -.0 (4.3.8) C = Z £ + a (4.3.9) where ^ i s the con s t r a i n t column vector defined as 8 ?*• = £ <<M H|4>M><<j>«| Wltx> l ^ i ^ 5 M = 6 (4.3.10) and the matrix elements of the con s t r a i n t operator matrix are 116 = 2 {ak<4',!H|cfvv><4vv|vv(<t)Jt> M = £ L + < 4 V ( W ! ^ X 4 > J H | < P , > C U } (4.3.11) Equations (4.4.8) and (4.4.9) are then solved by the 8 parametrization method. As i n the case of using energy-independent off-diagonal h y p e r v i r i a l theorems as c o n s t r a i n t s , an i n i t i a l guess f o r the constrained wavefunction vector fl = (1,0,0,0,0) i s used. The co r r e c t value of A i s obtained when self-consistency i s achieved f o r the column vector & and the co n s t r a i n t condition, equation (4.3.9) s a t i s f i e s the desired —8 accuracy — 10" . The r e s u l t s of constraint v a r i a t i o n are summarized in Table XX. 1.17 4.4 Discussion Although the number of our basis function i s only 8 0 0 the r e s u l t should be i n d i c a t i v e about the v a l i d i t y of the proposed scheme. The r e s u l t s of our present studies on the expectation values of t " 1 , r " ' , £ and -C "* by the d i a -gonal perturbation-induced constraints are very discouraging as can be seen from Table XIX. In the case of W = r 5 the c o n s t r a i n t of r e q u i r i n g the f i r s t - o r d e r c o r r e c t i o n w"^  to vanish leads to the value 12.0498 which i s overcorrected as compared, to the exact value 12.0348 and the free v a r i a t i o n value 12.0240. In a l l other cases, W = r*' , r and r 4 2 - t the constraint of vanishing \Nc,i changes the expectation values i n the wrong d i r e c t i o n so that they become worse than the free v a r i a t i o n r e s u l t s . As i n other constrained v a r i a t i o n studies, the s a c r i f i c e i n energy i s small and i s of the order 10 (t.U. As have been mentioned i n the previous chapter, * These expectation values are i n f a c t a sum of one-electron expecfetaion values. 13=8 Table XIX. Results of diagonal perturbation-induced constraints (in atomic units) . W i n constraint < ^ "2> < <r> < r + 2 > E (a.u.) -6 none 12 .0240 3. 36649 1.88432 2 .49154 -2.897142 -2 12 .0498 3. 36930 1.88407 2 .49190 -2.897115 -1 r 12 .0245 3. 36309 1.88592 2 .49552 -2.897086 r 12 .1005 3. 36598 1.88765 2 .50274 -2.896913 r + 2 12 .0959 3. 36844 1.88634 2 .49933 -2 .896999 Exact, none a 12 .0348 3. 37668 1.85858 2 .38468 -2 .903724 a Ref. (29) 119 T a b l e XX, R e s u l t s o f i t e r a t i v e method o f o f f - d i a g o n a l p e r t u r b a t i o n - i n d u c e d c o n s t r a i n t s ( i n a t o m i c u n i t s ) . W i n c o n s t r a i n t < r - 2 > < r " 1 ) < r > < r + 2 > I I ( a . u. ) -6 none 12 .0240 3. 36649 1.88432 2 .49154 -2 .897142 -2 r 14 .4970 4. 73940 1.43338 1 .36607 -2 .158429 -1 r 9 .9265 2 . 95979 1.92002 2 .49410 -2 . 15842 9 r 10 .9780 3. 71764 1.70819 1 .99882 -2 .158429 + 2 r 11 .6330 3. 54289 1.80318 2 .27124 -1 .805892 E x a c t , n o n e 3 12 .0348 3. 37668 1.85858 2 .38468 -2 .903724 a R e f . (29) 120 the expectation value of t i s c l o s e l y r e l a t e d to the Lamb term of the diamagnetic s h i e l d i n g constant. +2 The expectation value of £ i s r e l a t e d to the Langevin term A- of the diamagnetic s u s c e p t i b i l i t y "X For atoms, with the nucleus as o r i g i n , the h i g h - f r e -quency contribution to "X? vanishes, and "X-1* i s given by A-% U = -/6^<0 . (4.4.1) where i s the f i n e structure constant (» 1/137). I t i s i n t e r e s t i n g to compare the constrained v a r i a t i o n r e s u l t s with the r e s u l t s obtained by inclu d i n g the free v a r i a t i o n c o r r e c t i o n given by equation (4.2.32). The f i r s t - o r d e r corrections W ) obtained for W r"2- , r" ' , r and r* 2" and the expectation values of these operators are l i s t e d in Table X X I . As compared to the exact values, the i n c l u s i o n of the free v a r i a t i o n f i r s t - o r d e r c o r r e c t i o n improves the expectation values of £ 1 , r. and t + i but not £ 2 . I t i s also i n t e r e s t i n g to f i n d that the changes i n the calcu l a t e d expectation values of these operators are opposite 121 in sign but are of the same order of magnitude f o r the diagonal perturbation-induced constraints and the i n c l u s i o n of the free v a r i a t i o n f i r s t - o r d e r c o r r e c t i o n . A comparison of these two approaches i s presented i n Table XXI. The r e s u l t s obtained from the i t e r a t i v e method of off-diagonal perturbation-induced constraints are even more disappointing as can be seen from Table XX. The s a c r i f i c e i n energy due to the constraint i s very large and the expectation values of t'z , z ' , r and f 4 1 c a l c u l a t e d from the constrained wavefunctions are further away from the exact values than the free v a r i a t i o n r e s u l t s . In the case of W = c ' , the constraint of the f i r s t - o r d e r c o r r e c t i o n W1'* to vanish leads to the value 2.95979 as compared to the free v a r i a t i o n r e s u l t 3.36649 and the exact value 3.37668. In a l l other cases, W = JT~2 , r and c 4 z" , the expectation values of the corresponding constraint operators are grossly overcorrected. The poor r e s u l t s obtained from the perturbation induced constraints scheme are mainly due to the f a i l u r e of the f i r s t - o r d e r wavefunction given by equation (4.2.29) v/hich can be written as t ^ > = £ i ^ X ^ U r ' H !4>,> (4.4.2) 122 Table XXI. Comparison of the e f f e c t s on expectation 2 1 +2 values of r i r , r and r obtained from the diagonal perturbation-induced constraints method (DPIC) and the free v a r i a t i o n c o r r e c t i o n method (FVC) w + <'W>free <W>DPIC <W>FVC <W>DPIC- <W>FVC-<W> free <W)free . -2 r 12.0240 12.0498 11.9889 +0.0258 -0.0351 -1 r 3.36649 3.36309 3.372 70 -0.00340 +0.00621 r 1.88432 1.88 765 1.87691 +0.00333 -0.00741 + 2 r 2 .49154 2.49933 •2.46895 +0.00779 -0.02259 + "^^^free s t a i n e d from the free v a r i a t i o n approx-imate wavefunction ^, 123 Let us express the approximate wavefunction ( ^ i ^ as a l i n e a r combination of i^-A.) , the eigenfunctions of the true Hamiltonian H with eigenvalues - E M X i > (4.4.3) and hence = B'lb,,t;> (4.4.4) Since the Ex. 's can be p o s i t i v e or negative, the r a t i o ) i s larger than one for some A. so that any impurity i n I > i s more prominent i n 61 Hl^i) and can not be p u r i f i e d by the power method I t i s the impurity i n H 1^) that causes the f a i l u r e of the f i r s t - o r d e r wavefunction. 124 CHAPTER V CONCLUDING REMARKS The hope of getting more accurate expectation values of some operators at the cost of n e g l i g i b l e s a c r i f i c e i n energy v i a the constrained v a r i a t i o n method has been r e a l i z e d i n c a l c u l a t i o n s already i n ^ l i t e r a t u r e and confirmed by the work of t h i s t h e s i s . Both the formulation and methodology of t h i s method have been investigated quite thoroughly. The purpose of constrained v a r i a t i o n has been well established. Unfortunately, up to now, there i s s t i l l no way to obtain a general constraint operator which i s d i f f e r e n t f o r d i f f e r e n t properties. To our great disappointment, the f i r s t attempt made by Chong and Benston to overcome t h i s d i f f i c u l t y f a i l s to produce f r u i t f u l r e s u l t s as shown by the r e s u l t s of our c a l c u l a t i o n s i n Chapter IV. This implies that f o r a p a r t i c u l a r physical property, one s t i l l has to construct the corresponding constraint operator i n some p a r t i c u l a r ways. One cannot pre-d i c t with absolute confidence what other properties w i l l also be improved i f a p a r t i c u l a r c o n straint i s imposed. So f a r , the constrained v a r i a t i o n method has been applied only to ground state wavefunctions of some r e l a t i v e l y simple systems, studies on excited states and extension to more complicated systems are planned i n t h i s research group. 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Eckart, Phys. Rev. _36, 878 (1930). 53. D. P. Chong, Mol. Phys. 12., 599 (1967). 54. B. S c h i f f and C. L. Pekeris, Phys. Rev. 134, A638 (1964). 55. W. L. Weiss, M. W. Smith and R. M. Glennon, Atomic t r a n s i t i o n p r o b a b i l i t i e s , V o l . I. Washington, D. C : National Bureau of Stan-dards, 1966. 56. C. L. Pekeris, Phys. Rev. 112, 1649 (1958). 57. J . 0. H i r s c h f e l d e r , W. Byers Brown and S. T. Epstein, Adv. Quantum Chem. JL, 255 (1964). 58. A. Dalgarno and A. L. Stewart, Proc. Roy. Soc. (London), A247, 245 (1958). 59. A. Dalgarno and A. L. Stewart, Proc. Roy. Soc. (London), A257, 534 (1960). 60. W. A. Sanders and J . O. H i r s c h f e l d e r , J . Chem. Phys. 42 , 2904 (1965) . I I 61. C. E. Froberg, Introduction to Numerical A n a l y s i s , (Addison-Wesley Publishing Company, Inc. London, 1966), p. 103. 62. L. Pauling and E. B. Wilson, Introduction to Quantum Mechanics, (McGraw-Hill Book Company, Inc. New York, 1935), p. 184. 130 APPENDIX A CALCULATIONS AND RESULTS FOR THE ILLUSTRATIVE EXAMPLE IN CHAPTER I The ground s t a t e w a v e f u n c t i o n "J* f o r t h e h e l i u m atom i n C h a p t e r I i s a p p r o x i m a t e d by $ = <pU) ( A . l ) where 0a> S (^/TF)^ (A.2) The f u n c t i o n *P i s a n o r m a l i z e d s c r e e n e d h y d r o g e n i c I s a t o m i c o r b i t a l , and i s t h e o r b i t a l e xponent. The energy IE ( i n a t o m i c u n i t s ) o f t h i s f u n c t i o n * i s 6 2 E = ( 2 7 / 8 -•-*> (A.3) The optimum v a l u e o f , o b t a i n e d by m i n i m i z i n g 131 k w i t h r e s p e c t t o 4 . i s 1.68 75. The c o r r e s -ponding v a l u e of £ i s -2.847656 a.u. The e l e c t r o n d e n s i t y a t the n u c l e u s i s g i v e n by A = <¥ l [ S ( £ - r , ) + K r - n > J I * > =o (A.4) where t>Cr-r/i)±s the t h r e e - d i m e n s i o n a l D i r a c d e l t a f u n c t i o n , by which we mean C47TCcl) «$Cr- r^) # A f t e r s u b s t i t u t i n g e q u a t i o n s ( A . l ) and (A.2) i n t o e q u a t i o n s ( A . 4 ) , we o b t a i n A -^2"<3A (A . 5 ) s i n c e SCJL- r ^ ) | <£cd> = (A.6) The e x p e c t a t i o n v a l u e <Ct J> o f t 1 i s g i v e n by = A 4* (A.l) 132 The calcu l a t e d values of E. > A and (in atomic units) as a function of the o r b i t a l ex-ponent ^ are shown i n Table XXII. The % absolute error i n £ , % errors i n A and C r~\> are given i n parentheses. The % error i s defined as calcul a t e d value - exact value -, n n % error = r 1 x 100 exact value (A.8) 29 The exact values obtained by Pekeris are also included i n the table. Table XXII. V a r i a t i o n s of the energy /c , /\ , the e l e c t r o n density at the nucleus and expectation value <C T~2/> of t'2 ( i n atomic u n i t s ) as a func-t i o n of the o r b i t a l exponent f o r the function $ . The absolute % error of £ , % error of A and from t h e i r exact values are given i n parentheses. 4 E (a.u . ) A 1.5 -2 .812500 (3 .14) 2 .14859 (-40 .66) 9.0000 (-25.21) 1.6 -2 .840000 (2 .20) 2 .60759 (-27 . 98) 10.2400 (-14.91) 1.6875 -2 .847656 (1 .93) 3 .05922 (-15 .51) 11.3906 (-5.35) 1.7 -2 .847500 (1 .94) 3 .12771 (-13 .62) 11.5600 (- 3.94) 1.8 -2 .835000 (2 .37) 3 .712 76 (+ 2 .54) 12.9600 ( + 7.69) 1.9 -2 .802500 (3 .49) •4 .36657 ( + 20 .60) 14.4400 (+19.99) 2.0 -2 .750000 (5 .29) 5 .09296 ( + 40 .66): 16.0000 (+32.96) Exa c t a -2 .903724 3 .62085 12.0341 a Ref. (29) 134 . APPENDIX B LINEAR COEFFICIENTS FOR THE FREE VARIATION APPROXIMATE WAVEFUNCTIONS The l i n e a r c o e f f i c i e n t s f o r the fr e e v a r i a t i o 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 approximate wavefunctions c •shown i n Table3 Q U Uand Table XXIV are obtained by the minimum energy p r i n c i p l e of the v a r i a t i o n method, the o r b i t a l exponents of the Slater-type atomic o r b i t a l s i n each configuration are f i r s t optimized through an i t e r a t i v e procedure where the change in 27 the energy i s approximated by a parabola . The ^"S term functions for a l l configurations appearing i n each c o n f i g u r a t i o n - i n t e r a c t i o n approximate wave-function have been normalized but they are s t i l l not orthogonal to one another. Table XXIII. Free v a r i a t i o n 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 wavefunctions for the ground state of helium (see Chapter I I ) . Function b l b2 b 3 b 4 b 5 b6 b 7 b8 * 3 0 .340437 0 .652512 0 .071488 <*>4 . 0 .388105 0 .634391 0 .030831 -0 .061574 <t>S 0 .602749 0 .362679 0 .283507 -0 .400788 0 .284906 * 7 1-.070135 .0 .064320 0 .229711 -0 .591299 0 .379038 -0.180157 -0 .061434 + 3 1 .02 48 78 -0 .032658 -0 .061682 f 4 1 .236481 -0 .096086 -0 .009778 -0 .180402 t s 1 .016443 -0 .040706 0 .061825 -0 .047821 -0 .061067 f 7 1 .130096 -0 .149146 0 .523482 -0 .375990 -1 .374314 2 .009287 -0 .801105 % 1 .142868 -0 .179034 0 . 722710 -0 .532130 -1 .796255 2 .689211 -1 .091074 -0.061500 0 .614139 0 .383652 0 .0082 26 ft 0 .499398 0 .499077 0 .005387 -0 .061710 Table XXIV. Free v a r i a t i o n 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 wavefunctions f o r the ground state of He, L i + and B e + + (see Chapter I I I ) . Species Function b l b2 b 3 b4 b 5 b6 b 7 He % 1 .120196 -0 .024385 0 .264659 -0 .334118 0 .114687 -0 .182767 % 1 .141047 0 .056922 0 .178097 -0 .768920 0 .633702 -0 .277082 -0 .061761 L i + % 1 .167961 0 .061415 -0 .003762 -0 .586270 0 .622765 -0 .293057 % 1 .104137 -0 .004793 0 .103163 -0 .218213 0 .121274 - 9 .134318 -0 .040711 Be + + ' % 1 .140402 0 .048050 -0 .028647 -0 .454971 0 .485178 -0 .213517 * 7 1 .146802 0 .074602 -0 .058677 -0 .579585 0 .662828 -0 .270069 -0 .030308 137 APPENDIX C PRIMITIVE INTEGRALS A l l t h e p r i m i t i v e i n t e g r a l s a r e o n e - e l e c t r o n i n t e g r a l s e v a l u a t e d u s i n g S l a t e r - t y p e a t o m i c o r b i t a l s as b a s i s ; t h e y a r e d e f i n e d as where r\ , , fit a r e t h e p r i n c i p l e , a z i m u t h a l , m a g n e t i c quantum member r e s p e c t i v e l y , i s t h e o r b i t a l e x p o n e n t , ^ijlnt^/^) i s t h e s u r f a c e h a r m o n i c s and Nn 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 The <M'£WI ff;lwiw> and ^njgWf *lm> i n t e g r a l s The o n e - e l e c t r o n o p e r a t o r s and ^ a r e d e f i n e d as Ji= s a - j c o ^ x.=o ( c . i ) 138 %A. = $ < r - & ) r = o (C.2) where 6Cr.-f;) i s the Dirac d e l t a function, and by'S(r-Xc)* we mean (C.3) where o ^ : = r's;HS<J^<ds<J<f> ( C . 4 ) and equation ( C . 4 ) becomes 139 The f i r s t part of the i n t e g r a l i n s i d e the bracket i s I ° otkeirw<3e whereas the second part of the i n t e g r a l i n s i d e the bracket i s A rr^'-0 - ^ " i re*-*)-'*"1* . r ^ O • ( o (C.7) crj4\e«"w \sc. with the orthonormal properties of the surface har-monics r T r z i r * • Jo J o C<9'(P)T^CS>/<f>) SU <$J$d<p = c V V (C . 8 ) the only nonvanishing matrix elements are 140 and </ 'oV| fr\2oo> = (J^ TT) M(/Na. (C.10) For the <n'^ WI^ {*JL>*> i n t e g r a l ^ Jo J o « W ^ C0,<p)sinS>d<9d(p (C . 11) From the orthonormal properties of the surface harmonics, equation ( C . l l ) i s reduced to 141 4-TT Vn'A/a ^ n'^l j (C.12) O erfAervst/se. The only nonvanishlng matrix element i s I t should be noted that only the s Slater-type atomic o r b i t a l s contribute to both the <C*'JLWI fa/nX^and Kyi'JlWI^(rtJUwy i n t e g r a l s . The dipole length M, , dipole v e l o c i t y Mz. and  dipole a c c e l e r a t i o n Ms i n t e g r a l The i n t e g r a l f o r the dipole length Mi i s 142 = <H? i'o\$ I * i o o> (C.14) The i n t e g r a l f o r the dipole v e l o c i t y Mx. i s KjC/'i^f %j j *X.w> /d£ , where 4 £ i s the t r a n s i t i o n energy. < H V O ' I f y j l n ( O > / A J = * ~<n!t'o'l %$l noo>/*£ ( C 1 5 ) The i n t e g r a l f o r the dipole a c c e l e r a t i o n Afe i s < r t '/V/ f^/^m>/c/t^y i where V i s the Columb pot e n t i a l and i s the t r a n s i t i o n energy. 143 = <n'i'o'! «oos/(M:f (C.16) where 2? i s the atomic number. The one-electron expectation value < r p> i n t e g r a l = A / „ W« /"V <*>'•<• * * P i - r ^ n ' * *M 3 f - A/*' AV Cn'-*-** p) C-^n'-^^H3M'+"^' (C.17) where p i s the power of 144 APPENDIX D A COMPUTING PROGRAM FOR THE OFF-DIAGONAL HYPERVIRIAL THEOREMS DOUBLE CONSTRAINT WITH ITERATION IN THE TRANSITION ENERGY T h i s c o mputing program i s f o r t h e o f f - d i a g o n a l h y p e r v i r i a l theorems d o u b l e c o n s t r a i n t c a l c u l a t i o n w i t h i t e r a t i o n i n t h e t r a n s i t i o n e n e rgy as d e s c r i b e d i n C h a p t e r I I I . A s i m p l i f i e d f l o w c h a r t o f t h i s program i s shown i n F i g . 5. I n t h e a c t u a l c a l c u l a t i o n t h e f o l l o w i n g s t e p s a r e t o be p e r f o r m e d : (1) I n p u t : The f r e e v a r i a t i o n a l r e s u l t s , i n c l u d i n g v a l u e s f o r d i p o l e l e n g t h Mf , d i p o l e v e l o c i t y M± , d i p o l e a c c e l e r a t i o n Ms and e n e r g i e s t o g e t h e r w i t h i n i t i a l guess f o r t h e c o n s t r a i n e d wavefunc-t i o n , e x c i t e d s t a t e e n e r g y and t h e number o f terms i n t h e a p p r o x i m a t e w a v e f u n c t i o n . R e s u l t s f r o m a s e p a r a t e o f f - d i a g o n a l s i n g l e c o n s t r a i n t program can a l s o be p a r t o f t h e i n p u t . 145 Solve s i n g l e constrained secular equation (3.2.7) Two s i n g l e constrained secular equations are to be solved by the p e r t u r b a t i o n - i t e r a t i o n method. The two constraints are C-i = M; -Mi. and Ci_ - Ma - Ms . In each case, the constrained wavefunction i s obtained by i t e r a t i o n method u n t i l s e l f - consistency i n a of equation (3.2.7) i s achieved and the c o r r e c t value of the perturbation parameter i s obtained when the constraint condition reaches the desired —8 accuracy ^ 10 . Obtain perturbation parameters f o r double c o n s t r a i n t The free v a r i a t i o n a l r e s u l t s together with the r e s u l t s obtained from (2) are used to obtain the perturbation parameters f o r double c o n s t r a i n t by solving equations (1.4.8) and (1.4.9). Solve double constrained secular equation (3.4.14) The constraint i n t h i s case i s M» = Mi. = A 7 j and the values for the perturbation parameters obtained from (3) are used to solve the double constrained secular equation. The constrained 146 wavefunction, vector a. of equation (3.4.14), i s found by a s e l f - c o n s i s t e n t i t e r a t i o n method and the correct values f o r the perturbation parameters are obtained when the constraint conditions (3.4.16) s a t i s f y the desired accuracy ^ 1 0 ~ 8 . (5) Obtain better values f o r the perturbation para-meters f o r double constraint Better values f o r the perturbation parameters for double constraint are obtained by an i t e r -ation method. In each i t e r a t i o n , the double constrained secular equation (3.4.14) i s solved t h r i c e and each time with d i f f e r e n t values for the perturbation parameters. The r e s u l t s are then used to solve equations (1.4.8) and (1.4.9). (6) Solve double constrained secular equation (3.4.14) The values f o r the perturbation parameters ob-tained from (5) are used to solve the double constrained secular equation once more. Results are obtained when there i s a s e l f - c o n s i s t e n t s o l u t i o n f o r the constrained wavefunction, i . e . vector a. of equation (3.4.14), and the 147 c o n s t r a i n t c o n d i t i o n s s a t i s f y t h e d e s i r e d a c c u r -acy ^ 10~ 8. (7) Comparison o f t h e o l d and t h e new t r a n s i t i o n e n e r g i e s The o l d t r a n s i t i o n e n e r g y i s t h a t o b t a i n e d f rom t h e p r e v i o u s w a v e f u n c t i o n and the new t r a n s i t i o n e n e r g i e s i s t h a t o b t a i n e d f rom t h e c o n s t r a i n e d w a v e f u n c t i o n . I f t h e d i f f e r e n c e between them I s l a r g e r t h a n one p a r t per m i l l i o n , then-new v a l u e s f o r d i p o l e l e n g t h M i , d i p o l e v e l o c i t y Mj_ and d i p o l e a c c e l e r a t i o n M 3 a r e c a l c u l a t e d and s t e p s (2) t o (7) a r e r e p e a t e d . (8) Answer I f t h e d i f f e r e n c e between t h e o l d and t h e new t r a n s i t i o n e n e r g i e s i s l e s s t h a n one p a r t p e r m i l l i o n and i f t h e number o f c y c l e o f i t e r a t i o n i n t h e t r a n s i t i o n e n e rgy i s l e s s t h a n 10, t h e n t h e s o l u t i o n o f t h e d o u b l e c o n s t r a i n t c a l c u l a t i o n has c o n v e r g e d . ( 9 ) S t o p I f t h e number o f i t e r a t i o n i n t h e t r a n s i t i o n e nergy i s l a r g e r t h a n 10, t h e r e i s no a c c e p t a b l e 148 s o l u t i o n f o r t h e d o u b l e c o n s t r a i n t and t h e c a l c u l a t i o n w i l l s t o p . The c a s e o f ^ f o r t h e ground s t a t e o f h e l i u m i s c hosen as an example f o r t h e use o f t h i s program. The i n p u t d a t a a r e summarized i n T a b l e XXV and t h e r e s u l t s f r om each i t e r a t i o n i n t h e t r a n s i t i o n e n e rgy a r e p r e s e n t i n T a b l e XXVI. I t s h o u l d be n o t e d t h a t t h e ' v a l u e s o f d i p o l e l e n g t h Mf , d i p o l e v e l o c i t y , and d i p o l e a c c e l e r a t i o n Ms o b t a i n e d from t h e f i r s t c y c l e o f i t e r a t i o n i n t h e t r a n s i t i o n e n e rgy c o r r e s p o n d t o t h e c o n s t a n t t r a n s i t i o n energy assump-13 t i o n made by Chong and B e n s t o n 149 input c y c l e > 10 -Kstop ) < 10 solve s i n g l e constrained secular equation obtain perturbation parameters f o r double constraint cycle; > 3 < 3 solve double constrained secular equation solve double c o n s t r a i n e d s e c u l a r equation Iresults| <10 -6 answer $ F i g . 5 T a b l e XXV. I n p u t d a t a f o r t h e o f f — d i a g o n a l d o u b l e c o n s t r a i n t i n t h e c a s e o f f o r t h e ground s t a t e o f h e l i u m . D i p o l e l e n g t h ;. 0.40361806 0.68675523 -0.06867955 0.11616026 -0.38715882 0.14138083 -0.03194480 D i p o l e v e l o c i t y 0.41757721 0.13629037 0.00418988 -0.23156477 -0.03390028' 0.01006627 -0.00236449 D i p o l e a c c e l e r a t i o n 0.40192935 0.00627670 -0.02619199 1.11880960 -0.02152021 0.04438162 -0.02653110 F r e e v a r i a t i o n a l e n e r g i e s -2.8979024 -1.4529145 -0.0886734 2.2174150 4U300944 5.4012556 11.172075 I n i t i a l guess c o n s t r a i n e d w a v e f u n c t i o n 1.0 0.0 0.0 0.0 0.0 0.0 0.0 2~P s t a t e e n e r g y ( n e g a t i v e ) -2.1223902 P e r t u r b a t i o n p a r a m e t e r s E x p e c t a t i o n v a l u e s f o r c o n s t r a i n t o p e r a t o r s X I O . ) X 2 ( A x ) C I CZ 0.00000000 0.00000000 -0.01395914 0.01564786 -0.10935587 0.00000000 0.00000000 0.01329238 0.00000000 0.08486056 . -0.01213303' 0.00000000 Table XXVI. Results from the off-diagonal double co n s t r a i n t c a l c u l a t i o n i n the case of t y f o r the ground state of helium. I t e r a t i o n E +(a.u.) A , A*. C , M, = W2."tf5 1 -2.896650 -0.09694012 0.07353477 -8.435x10" 7.494x10" u 0.41663 2 -2.896630 -0.10296513 0.06894421 -1.294xl0" 9 2.580xl0~ 1 0 0.41757 3 -2.896630 -0.10305854 0.06887332 -1.404xl0~ 1 0 - l . O l O x l O " 1 0 0.41759 The energy of the constrained wavefunction. $T-BFTC C C C C C c c c THIS PROGRAM DOES THE OFF-DIAGONAL HYPERVIRIAL DOUBLE CONSRTAI NT WITH ITERATION IN THE TRANSITION ENERGY -T-HE—eONS-TR-A-I-NT-OPER-AT-OR-I S - : ~ C = 0.5*< P + P*) WHERE P=<M1-M2) +<M2-M3) DIMENSION EP( 25">"»CCC< 2 5 )»A"( 2 5 H B < 2 5 ) >XM< 25) >YM(2 5 ) rZM " ( "2 5") " " "' DIMENSION Hl<25»25)»H2(25»25)>H3<25»25)»HH<25»25) DIMENSION EVEC125»25)»EVEC1<25»25)»EVEC2<25»25)»EVEC3(25»25) REAL"~"Ml( 25T»M2 ( 2 5 ) »M3 ( 2 5T» M12 < 2 5 ) *M23 C25') '• : : : REAL MX-(25 ) »MY(25) » M Z ( 25) REAL LAM » JAM '" COMMON /IN / -• XI ("3")"» X 2 ( 3 ) "»C 1"( 3 )"» C 2 ( 3 ) : : COMMON /OUT/ XXi»XX2 COMMON /CXX/ H ( 2 5 ' 2 5 ) FF=OVI -•- •- — " - " : ~ 1 READ 100 PRINT 100 READ-101"» N P S r : — ~ PRINT 101*NPSI NN=NPSI _.p R I N-T--1-O 2— " — : J ~ READ 103' <MX( I ) » I . = 1»NN) PRINT i03»<MX(I)»I=1»NN) . p R j 0 I f • : READ 103»<MY(I),1=1,NN) , PRINT i 0 3 » tMY(I )»I = 1»NN) PRINT-10 5 : READ 103»(MZ(I),1=1,NN) PRINT 103»<MZ(I)»I=1»NN) • ! P RI NT—106 - — — : £ READ 103 * t EP ( I ) » I = 1»NN ) I\J PRINT i 0 3 » t EPI I)»I = 1NN) ' ' - • . DO-..8oo--I-=l-»-NN M i d ) =MX( I ) M2 t I ) = M Y ( I ) 8 00 M3<T )=M^< I ) DO 400 1=1»NN DO 400 J=1»NN "4" 00 H-(T» J ) =0;0 DO 4-0 1 I = 1 * NN 401 H ( I » I )=EP( I ) PRINT--r0 7 ~~ DO 402 1 = 1 • NN 402 PRINT 108»tH(I»J)•J=l»NN) R E A D 1 1 6 » < CCC (T) VT= I > NN ) PRINT 109 PRINT 103'<CCC(I)»I=1»NN) PRINT'3OO READ 103'EE PRINT 103'EE •DO" 403r=l-»-3" 403 READ llO.Xi(I)»X 2(I)»C1<I)»C2(I) PRINT 121 D0-4T3--r=T»'3-413 PRINT llO»Xi(I)»X2<I)»C1<I)>C2<I) E0=EP(1) • D E I = ' E - E = E O • — M = 0 500 M=M+1 KK = 0 :  I F ( M . G T . l O ) GO TO 505 PRINT 111 D 0 --4-0 4~I-=1-»'N N / -?ro-4-—M"r2'(T)"s"M'i"{"r)~M2"m~i~" ~~—: : I P R I N T i 0 3 » ( M 1 2 < I ) » I = 1 » N N ) Id P R I N T 112 i-ff QO~~4"05 r="rvN'N ~ ' ; j z j 405 M 2 3 ( I ) = M 2 < I ) - M 3 < I ) ' A . P R I N T i 0 3 » 'M23< I ) » I = 1 » N N ) M '• J F ( M . E Q - . 1 ) G 0 T 0 4 0 8 ' ~~ — : : j * CALL C V S C < E P » C C C»M1'2 » L A M » A » N N ) AA=0.0 ; -D O ' 406 I = T » N N : : ~ ; 406 AA=AA+M23< I ) * A < I ) • X i ( i ) = 0 . 0 . j • X ' 2 ( l )=0 . 0 ^ ; C I (1)=M12 <1) ! . C 2 < 1 >=M2 3( 1 > I X i ( 2 ) = L A M ; — • , X2<2)=0.0 i , - C I _ ) =0 . 0 _ _ _ _ _ _ _ ! "C2 (2 )=AA ; : : : : : : ; CALL C V S C ( E P » C C C » M 2 3 » J A M » B » N N ) I . 33 = 0.0 . . ' _ ' j D O 4 0 7 i = i * N N : : ' : j 407 BB = BB+M12 < I ) *B( I ) !• -X I (3)=0.0 ; ' X'2'(3") '= JAM - ~ : j CI (3)=BB ! .-C2(3)=0.0 ' | "4"0 8 KK = KfC+1 : 1 ~ : CALL LAMBDA ' , • c I F ( K K . G T . 3 ) G O T O 409 " C f l S J C 3 U o | - G 3 ! n-:i~i C O j C M ! ? ! C M I 2 i »- ! Z> * U u u . #. rH CO #• r H C_ UJ > UJ > o u > c_ C M t o ti n II X x J - „ . . . )< X J - H r H H ii ii _ J 4- — — f H r H < £ t H CM r H U > U X X u u. u. * r H + co C M _: C M > Ki P u u V ,#« rvj C M (NJ U UJ > UJ < M X L U > _J o CO _> ll (\l <M CM > DC CO II II II C O C M C O r H fM _ C M z C M X X C O > C O u u u co CO #» CO c_ ro U LU > UJ co X LL — LL. UJ * > r H _ J > o !+ co oo o <J-o r H _ > — II II _ J ~ C M ( M ( \ J < r H U _ > O X U CM CM <\J CM _> > || || | CM r H (M CO CO < O X O u _ > u o X X u u C O C M u_ 'CM r H u_ i" !U U J > UJ X X i_i > CO U r H 0 r H CO 1 i . l 155 z 1 co C M _: r H u_ II t—1 rv C M X — X r H •» #• CNJ — •—I r H r : — _: r> u i_ r H LU — > r H zr X U J x 2:X — <^ -#^ .«> r H u u U J LU > > U J LU * * Z 2T + C M + m co i n o 11 o p O r H O r H C M •—I O • r H r H r H r H r H O O j j O II jll O II II — — o II 5: >-II _ l r - U _ _ _ _ _ _ < l - ~ rH •if <f <^ »—t t—( < c _ _ _ _ _ _ 0 5 : _ i o o _ _ U a . a a L _ Q . Q - Q X > - r N l Q Q X > -o o o r H -Z-|0|"(*I"T= ZM'CTITM"3"(~DT*EV E ' C ' Q 1 1 '. ~ * P R I N T H 7 P R I N T H8»XM( 1 ) »YM( 1 ) »ZM( 1 ) 'E 'E E = T f H ' ( T » T ) D E 2 = E E - E E E F = D E 1 / D E 2 ' IF'('Mo E Q ' V D G O " T ' 0 ~9"00 ~ I F ( A B S ( F / F S A V E - 1 . 0 ) . L . T . 1 . 0 E - 0 6 ) GO T O 6 0 0 DO 4 1 2 1 = 1 » N N M r(-iT=MX-(-n : : M2(I)=MY(I}#F M3(I)=MZ(I)*F*F F S A V E = F : " ' ' G O T O 5 0 0 P R I N T H 9 G 0 ~ T ' Q — i .- • : XXXXXXXXXXXX F O R M A T ( 5 5 H ) F O R M A T ( 1 5 ) ~ ~ ' " F O R M A T ( 4 H M l = ) F O R M A T ( E 1 6 • 8 ) F O R M 7 Y T T 4 H ~M2^ ) : " F O R M A T ( 4 H M 3 = ) F O R M A T ( 2 7 H F R E E V A R I A T I O N A L E N E R G I E S * ) F O R M A T T 2 8 H ~ 7 HE " " 'UN P E R T U R B E D H A M I L T ON I A N ) " " F O R M A T ( 5 E 1 6 . 8 ) F O R M A T ( 2 1 H I N I T I A L B A S I S V E C T O R ) F ' O R M A T ' t 4 ~ E T 6 T 8 " ) ' F O R M A T ( 3 0 H T H E D I F F E R E N C E V E C T O R S M 1 - M 2 ) T1'2 F O R M A T ( 3 0 H ~ T H E " " D 1F F E R E N C E~ V E C T OR S M 2 - M 31 ~ 1 1 3 F O R M A T ( 5 2 H T H E G R O U N D S j A T E E N E R G Y O F T H E C O N S T R A I N E D F U N C T I O N ) 1 1 4 F O R M A T ( 2 8 H 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 ) 1 1 5 F O R M A T ( 5 X » 7 H L A M B D A 1 ' » 1 I X » 2 H C 1 > l O X » 7 H L A M B D A 2 > 1 4 X , 2 H C 2 ) ' 1 1 6 F O R M A T ( 1 0 F 5 . 1 ) 1 1 7 F O R M A T ( 5 5 H A N S W E R F O R M l M2 M 3 I T 8 F O R M A T ' ( T 4 X - » 3 F 1 2 ' . 8") " " 1 1 9 F O R M A T ( 1 H O » 1 5 X > 2 0 H * * * N O C O N V E R G E N C E * * * ' ) 1 2 0 F 0 R M A T U E 1 6 . 8 ) T 2 1 FORMAT~(THT»~6X~>3H " X i~»~l3X"»"3H""~X"2 » "13X V3H"~CT»"r3X"» 3 "H~C2 ) 3 0 0 F O R M A T ( 1 6 H I P S T A T E E N E R G Y ) 7 0 0 S T O P $ I B F T C E C V S C S U B R O U T I N E C V S C ( E P » C C C • M D > L A M • C C » N N ) R E A L M D < 2 5 ) ' L A M " — D I M E N S I O N E P < 2 5 > » C C < 2 5 ) > C C C < 2 5 ) > C < 2 5 » 2 5 ) » P ( 2 5 » 2 5 ) > V E C T < 2 5 » 2 5 ) C N V E R G = 1 . 0 E - 0 5 N C Y = 1 0 C O U T = l . 0 E - 0 8 X O U T = 1 . 0 E - 0 8 N S U P = 0 — N P U N C H = 0 DO 2 6 I = 1 » N N 2 6 C C ( I ) =CCC ' ( I ) K = 0 1 1 C O N T I N U E < = K + 1  S A V E = C C < 1 ) DO 2 3 I = 1 > N N D 0 ~ 2 3 - - J = l - r N N 2 3 P ( I » J ) = M D < I ) * C C ( J ) DO 2 4 I = 1 » N N D 0 - 2 4 — J = T » N N 2"$ c"nvTr=0T5"*(P"("r» uy+pru T I T T ™ ~ " CALL C V I ( E P . » C » N N » N C Y » C O U T » X O U T » N S U P » N P U N C H » V E C T » X X X ) DO 25 I = 1 » N N 2"5 cc( T ) ~ = v E r r n T r r ~ ~ ^ I F < A B S ( S A V E - C C < 1 ) ) . L T . C N V E R G ) GO To 12 I F ( K . L T . l O ) GO TO 11 •G-Q~-T(y-1-3- - -12 LAM=XXX RETURN T3 PRINT 20 2 : : ' ' 202 FORMAT(1H0»15X»20H * * * N O CONVERGENCE***) RETURN E'N D - : — ~ SIBFTC SOLVE SUBROUTINE SOLVE<HH»EVEC»FM12»FM23»C»U » V»NPS I,Ml2»M23) R E"A Lr~MT2"{"2'5")"TM'23"("2'5") "J" • COMMON / C X X / H(25»25) DIMENSION C(25)»CC(25)»Pl2<25'25)»P23(25'25)»HHt25»25)»EVEC(25»25) CNVERG=TV0E-05 : " " ' - " ~ " ; DO 12 I=1»NPSI 12 C C ( I ) = C ( I ) K = Q : . 100 CONTINUE K = K+1 SAVE = CC (1 ) • ' : DO 30 I=1»NPSI DO 30 J=1»NPSI P12"(-rvU)=M12 ("I")-*"CC'("U") " 30 P 2 3 < I » J ) = M 2 3 ( I ) * C C ( J ) DO 330 I=1»NPSI DO-3 3 0"~J=T»-NP-SI : " ~ " 330 HH(I»J ) = H<I»J)+ U*0.5*< P12<I»J)+P12<J»I) ) + V * 0 . 5 * ( P 2 3 < I * J ) + P 2 3 ( ' J » I ) ) . C A"L L FAS~J A'C"( HH »E V E O ' 0»0» 0>NP ST> 2 5 ) DO 120 J =1 *NPS I 120 C C < J ) = E V E C < J » 1 ) I F ( A35 ( SAVE -CC< 1 ) ) . ' LT.CNVERG) GO TO 300 I F < K . L T , l O ) GO TO 100 GO TO 201 yOO FM 12 = 0 . 0 — FM23=0.0 DO 50 I = 1 » N P S I F M 1 2 = M12 ( rr*CC-(-i-r+-FMT2—' 50 FM23 = M 2 3 U )*CC ( I J+FM23 RETURN 2 01 PR I N'T—202 202 F O R M A T ( 1 H 0 » 1 5 X » 2 0 H * * * N O CONVERGENCE * * * ) RETURN - N D , J I B F T C C V i DECK SUBROUTINE C V l ( E P »C»NN >NCY>COUT,XOUT >NSUP»NPUNCH»VECT» " DIMENSION EP( 25 r»CC '25»2 5)>VECT( 25 »25 )" "~ COMMON /FACTT/ F A C T ( 8 ) COMMON / C I / N ' N A L L »NW'ANT ,KWANT ( 25 ) »NWF»NSUPP "COMMON V P E 7 ~ EPERT ( 25>7 ) COMMON /INOUT/ C 0 ( 7 ) » D L ( 7 ) C C P ERTU RB-ATI ON "APPROACH "INVERT S~A~M ACL'AUR IN "SER'I E S T 0 R~X X C C THIS PROGRAM THEN INVERTS A TAYLOR SER I E S FOR X . c- 1  C NEW LAMBDA = XXX = XX+X AND ONE I T E R A T E S . C C I-N-I-T-I-A t-lZ E- ~ C CALL GETFAC ND-=—2 5 ' ~ " NALL = 2 NWANT = 1 ICW ATTT'(Tr5~1 NWF = 0 NSUPP = 0 G~"=1V0 : CALL D P E R T O ( E P ' G ' C ) DO 2 K = 1»7 F = K C 0" ( " K " ) — s—F * " E P E RT-(T» K") CONTINUE CALL SERTNG (TTCTTCO'fKXr)-XXX = XX CALL GETCC < E P » X X X,c » N N » V E C T , C C C » E E E » O D D ) PRINT 2 0 0 - " • ~ - - • DO 80 KCY = l » NCY XX = XXX CALL GETDL(XX ) : CC = CCC D L ( 1 ) = CCC C AUL~"'S'ER'I N G — r 7 T 0 T D " C T X T ' XXX = xx+x CALL G E T C C ( E P » X X X » C » N N » V E C T » C C C » E E E » D D D ) I F ^NSUP. 'EQ.O ' ) ' GO To 50 ~ " ' PRINT 2 1 0 » K C Y , C C ' X X , X , X X X , C C C CONTINUE A C = ABS (CCC ) " -AX = A B S ( X ) IF ( A C . L T . C O U T ) GO TO 82 I T — W C T T T v X - O - t r T - ) G O - T O - 8 ^ : 8 0 C O N T I N U E P R I N T 2 2 0 •GO-T -0—816 ' " 8 2 P R I N T 2 3 0 G O T O 8 6 S A " — P R I N T ~ " 2 4 0 — • : C 8 6 P R I N T 2 1 0 » K C Y » C C » X X » X » X X X » C C C C A L L — D E L T A T X X X » E P ER"T> D E L E » 0 ) : P R I N T 2 5 0 P R I N T 5 0 0 » < V E C T ( K » 1 ) » K = 1 » N N ) P R I N - T ~ 2 6 0 » E E E V E P " < T T » D D D » D E L E ~ " I F ( N P U N C H . N E . l ) GO TO 9 9 P U N C H 5 0 0 » ( V E C T ( K » 1 ) » K = 1 » N N ) ~9 9 — C 0 N ' T T N U E ' 2 0 0 F 0 R M A T ( / » 4 H K C Y » 6 X » 3 H C C » 1 5 X , 3 H X X » 1 5 X , 2 H X » 1 4 X » 4 H X X X » 1 2 X , 4 H C C C ) 2 1 0 F O R M A T ( I 3 » 5 E 1 6 . 8 ) -2 2 0 — F ORMA 'T ( / » 2 OH" " E ' X ' I T S - ' B Y K C Y . E Q ' V N C Y >7 ) " : " ~ " 2 3 0 F O R M A T ( / > 2 0 H E X I T S B Y A C . L T . C O U T » / ) 2 4 0 F O R M A T ( / » 2 0 H E X I T S B Y A X . L T . X O U T > / ) 2 5 0 — F 0 R M " A " T ~ r / v r 9 H " ~ C O N S T R 7 V I N E D ' " V E C T O R »"/")" : . 2 6 0 F O R M A T ( / » l O H D E L T A E = » E 1 6 . 8 » 2 H - » E 1 6 . 8 » 2 H = » E l 6 . 8 » 6 H > C F » E 1 6 « 8 1 5 0 0 F O R M A T ( ( 5 E 1 6 . 8 ) ) E N D S I B F T C D P E R T O D E C K • S - U B R O U T I N E " D P E R TO ( E P ' G > V M A T ) D I M E N S I O N E P ( 2 5 ) ' V M A T ( 2 5 » 2 5 ) ' W M A T ( 2 5 ' 2 5 ) » D E L < 2 5 > D I M E N S I O N A V E C < 2 5 ) » B V E C ' ( 2 5 ) ' C V E C ( 2 5 ) ' T V E C < 2 5 » 2 5 ) C 0 M M 0 N - - / C 1 ' / N ' N ' A L L - > N W A N T . K W A N T ( 2 5 ) » NWF » N S U p - — - — : C O M M O N / P E / E P E R T ( 2 5 » 7 ) I N T E G E R Q 6 C L A M B D A I N T H I S P R O G R A M I S C A L L E D G i o i . C • • • ! T T "C N I S T H E T O T A L N U M B E R O F F U N C T I O N S " U S E D ' _ " ' ! z i C i ' ] . , C I F N A L L = 1 » A L L S T A T E S W I L L B E C O M P U T E D , A N D N W A N T A N D K W A N T N E E D i! ~C N O T ' ' ' 6 " E " ~ S P ' E " C T F T E D ~ " " : " \ * C I F N A L L . N E . l , T H E N N W A N T S P E C I F I E S T H E T O T A L N U M B E R O F S T A T E S W A N T E D j C K W A N T S P E C I F I E S T H E S T A T E S W A N T E D » • E . G . N A L L = 2 N W A N T = 2 i ~C :<v.'AMT ( 1 ) = 3 K W A N T ( 2 ) = 7 M E A N T H A T S T A T E S 3 A N D 7 O N L Y "W I L I T " B E : | C C A L C U L A T E D , c " C I T " T H E X P E R T U R B E D " W A V E F U N C T 1 0 N I S W A N T ' E D >~ S E T " NWF~""=" 1"* A N D ] ' | C T H E N O R M A L I Z E D T O T A L T R U N C A T E D W A V E F U N C T I O N F O R T H E Q S T A T E ! C I S S T O R E D I N T V E C ( » Q ) \ j "c ' : ' ~ - I C I F N S U P = 0 , P R I N T I N G W I L L B E S U P P R E S S E D i c . - • •; • N D = 2 5 ' I , I F ( N A L L . N E . l ) G O T O 7 3 | DO 7 4 K = 1 » N K W A N T ( K ) = K 7 4 - X O N T I ' NUE " C 7 3 C O N T I N U E D O " ' 7 1 K'W~= 1 T N W A"N'T ; " '' ' Q = K W A N T ( K W ) E 1 = V M A T ( Q , Q ) DO 2 L = 1 » N 2 W M A T ( K » L J = V M A T ( < » L ) DEL~( IC) = E P ( Q ) - E P ( K T ~ 1 ; ~ ' — ^ 7 ; ' " ~ W M A T ( K * K ) = V M A T ( K » K ) - E 1 • cn F I R S T O R D E R W A V E F U N C T I O N A V E C ( Q ) = 0 . 0 E 2 = 0 . 0 " E " 3 - =0 . 0 S 1 1 = 0 . 0 DO 1 0 0 K = 1 » N - I F ( K . E Q . Q ) GO TO 1 0 0 A V E C ( K ) = V M A T ( K » Q ) / D E L ( K ) 0 C O N T I N U E DO 3 K = 1 » N E 2 = E 2 + V M A T ( Q , K ) * A V E C ( K ) S 11 = S H + A V E C ( K ) * A V E C ( K ) DO 4 L = 1 » N E 3 = E 3 + A V E C ( K ) * W M A T ( K » L ) * A V E C ( L ) -CONT-T 'NUE S E C O N D O R D E R W A V E F U N C T I O N B V E C ( Q ) = - 0 . 5 * S 1 1 E 4 = 0 . 0 S 2 0 = B V E C ( Q ) S 2 1 = 0 . 0 S-2 2 = 0 „ 0 DO 1 8 K = 1 » N I F ( K . E Q . Q ) GO TO 1 8 - T E M P = 0 . - 0 — — • DO 5 L = 1 » N T E M P = T E M P + W M A T ( K * L ) * A V E C ( L ) -B V E C ( - K ) = T c M P / D E L ( K - ) :  U CJ LU LU > > CD CD j * * u u LU LU z > > * K CD r H ;+ + II rH f\J _ CM CM CO cO vO ! l l II rH CNJ O CM CM O CO CO CJ u LU LU > > CD CD : * * L J Z i— I— < < rs 3 ; * * LU LU > > :< CD :+ + <f- m LU LU ;II it <J- iTl O LU LU CM CO . * O CM CM LU \f) * sjc O LU 111 CM • O z z LU CM + I <f i n LU LU II II O O <t m CJ CJ LU LU cj z o ti-er LU Q cm o Q 164 o I LU U ^  u — LU V > — CD CJ # LU 'r- > U < .'«• * V CM i - LU I a 1— !< r - vO U U CJ i CM >~ 3 LU I i f l Z O z I I * i • -O ! II rH O O rH • ! — II LU e II O O O O ^ • O _ J II • _ II N O O U f f l - affi rn 11 11 LU • _ _ CO v 0 r ~ - > O L _ L U O L U + I-a ~ 5; 11 LU — II — 0. cj _ : LU > CJ LU > V _ CJ Cj LU LU > > U CJ * * * LU U CJ v_ 2: LU LU 2! — • > > > « • U rH < CQ rH LU II + + II > rH CM I CD ro co + O H vO rH II II rH LU rH CM II O CO CO O vO Q cO cO Q LU ON 00 165 i ! <M 1 CM : t/) ! + i i-H U J I CO | > ! ^ r - l * ! * f M O 1 if) • J _ J # f\l i CO — to V U J * U J —» [I cn t—i r — U U J 0 CM 1 5 : O J <M U J * in ro :+ to EN rH * ro c\i 4- 'JO U J O u r * >—1 U J * 0 r-> U J U J C\J « < u 3 UJ CM to Z Z 1 1 (V 1—1 M r-U J r - I— U J U J r-11 z 11 11 a: f- O 0 U J ill u U J U J a. rH O O O C\J p H M m i n vo r- — U J U J U J U J U J U J U J O 11 in 11 11 11 11 11 j * ~ j — ~ • o O O O O O O O C L v O ~ r ~ ~ r 3 h- t— h- r - i — t~ y- -n h-U J U J U J I U U J U J U J — -< 0 - C L Q L . C k t X Q . C L U - Q : U J U J U J U J U J U J UJ^->CL DC UJ Q . UJ rH r-II i j - UJ UJ 3 3 z z I—( I—I r - I— o CM o o o a UJ a 00 1 UJ 3 * * o r- r -r- DC rH UJ a UJ ti D. UJ II u CM CM o CM O r -o 5: O UJ h-— u o •> • i s l O Z UJ UJ h-+ UJ u u u P oc a a . o o u j u - K U U D O. o o rH O CM O O O CM U • CM UJ _ J UJ a to 3 UJ 3 3 Z Q Z '-0 1 •—• II •-! Z Z r— UJ I— 11 — —< z _i z rsl I I O L U O z a u Q u i CM O CM CM r F'v'N •SUPTEQTOT~G'0'~'T "0"~2 0 3'— PRINT 53 PRINT 54* DELE P ERTRUBATI ON WAVE FUNCTIONS PRTN'T-5'5 • PRINT 56 DO 13 K=1»N NZ-='K PRINT 57 >NZ,AVECtK)»BVEC<K)»CVEC<K) CONTINUE -CONTINUE CONTINUE T"0RWATTl'HOT7TXT3Tr^ FORMAT(/2X»I2»1P1E18.7) F O R M A T ( 1H 0 » / / I X » 1 4H TOTAL DELTA E) FORMAT C / i X V I P IE 18. 7) F O R M A T ( i H 0 f / i X * 3 2 H PERTURBATION W AVE FUNCTIONS) FORMAT(1H0 V1X58H FIRST ORDER SECOND ORDER 1RD ORDER) ' FORMAT(/1X»I2»1P1E17.7»1P2E19.7) FORMAT (1H0.//1X»6H EPERT) RETURN" • ' "• END SUBROUTINE LAMBDA COMMON/ I N / X 1 ( 3) » X2 ('3 >' »'C H 3 )"'» C2 ( 3 ) COMMON /OUT/ XXi»XX2 DIMENSION B(3»3) THIS PROCEDURE ASSUMES THAT C1<X_»X2) = AIO + A11*X_ + A12*X2 C2-(-X-r» X~2-)—=--A-20—+—A-2T*Xl—+—A 22* 'X 2 ' DO 4 K = 1» 3 B - ( - K - r n — r r O • B (K»2) = X 1 (K ) i B<K» '3) = X2<K) I CO NT IN U E" : : ; : i CALL INVERT ( B » 3 » 3 ' D E T »COND) ' ; PRINT 102» COND, DET ; A iO - = 0 . 0 — : 1 A l l = 0 . 0 ! A12 = 0 . 0 . • A 2 o-=-o-. o : j A21 = 0 . 0 J \ A22 = 0 . 0 -DO - 6 — K - = — 3 AIO = AIO + B< 1 » K ) * C K K I A l l = A l l + B ( 2 » K ) * C K K ) -A l-2"=~Al-2~+~B (-3-,-K t*C H K*)*"' 1 j A20 = A20 + B ( 1 > K ) * C 2 ( K ) A21 = A21 + B ( 2 , K ) * C 2 ( K ) "A22=—A22—+~B (~3"»"IC)~*C2 ( K )— : :—: — " " CONTINUE PRINT 103» A I O , A l l , A I 2 ! P RI N'T--10 4"»~~A 2 0 •»- "A 2 1 , " A 2 2 r ~ | SOLVE 1 ! •! TO? F O R M A T " 7 H ~ r O W ^ ^ 1 0 3 F O R M A T ( / » 6 H A l O = » 1 P 1 E 1 6 . 7 » 6 H A 1 1 = » 1 P 1 E l 6 . 7 » 6 H A 1 2 = » 1 P 1 E 1 6 . 7 ) 1 0 4 F O R M A T ( / » 6 H A 2 0 = » I P 1 E 1 6 . 7 » 6 H A 2 1 = » I P 1 E 1 6 . 7 > 6 H A 2 2 = » 1 P 1 E 1 6 . 7 ) rO-5 F O R M A T - ( - / T r S i T O R E T U R N E N D $ T B " F T C G E T F A T — D E C K — 1  S U B R O U T I N E G E T F A C C O M M O N / F A C T T / F A C T ( 8 ) F A C T (-1-)—=--l-i- 0 : — DO 2 K = 1 » 7 F K = K F A C T i K'+"D "-="- F K " * F A C T ( K ' ) ~ — ~ ' : 2 C O N T I N U E R E T U R N _ N D . : _ S I B F T C S E R I N G D E C K S U B R O U T I N E S E R I N G ( N » N S U P * D E * D ) -Q- j M E ' N S T O N " ' DE ( " 7 ' ) ' • C ' C 7 " ) ~ ' C N = N O . O F E L E M E N T S I N D E ( 7 ) » E . G . » N = 7 F O R A S E X T I C C C D E ( K + 1 ) I S T H E " C O E F F . O F X * * K I N ' T H E O R I G I N A L P O L Y N O M I A L ' ~ C C C H E M R U B B E R 1 9 6 4 M A T H T A B L E S P . 3 3 1 F = 1 . 0 / D E ( 2 ) Y = - F * D E d > : : C A l = 1 . 0 A 2 1 = F * D E ( 3 ) A-3-i-a P * D E"( '4-) : A 4 1 = F * D E ( 5 ) A 5 1 = F * D E ( 6 ) IFF ' : : : _ . _ _ : : j ; ; . -A"6-rs-p»-DE-(-7-) ~ 1 c • A22=A21*A21 I 0 L V A23=A22*A21 • | T r A-2-4--A-23"*A-2-l ~' zt A25=A24*A21 i " L'. • A26=A25*A21 l j : A 3 2='A31*A3 1 : : \ m A33=A32*A31 1 • A42=A41*A41 ! v C ( !")-= l v O : : : C ( 2 ) = - A 2 1 i ( M 3)=-A31 + 2.0*A22 DC082768 ! -. (.A-r=._ A 4 T + 5._ o *A31*A21-5.0*A2 3 — ' " : i C(5)=-A51+6.0*A41*A21+3.0*A32-21C0*A31*A22+l4.0*A24 | , C.(6)=-A61 + 7.0*A51*A2.1 + 7.0*A41*A31-28.0*A41*A22-28.0*A32*A21 ! 1 +"8 4 . 0 * A 31 •* A2 3 -42VO * A 2 5 : ' — ~ ~ . 1 IF(NSUP.EQ.O)GO TO 8 9 j . ' WRITE ( 6 ' 9 9 ) j _g-9 ^ Q .j. N £ ~ . , SUM =0.0 ' j ( N1 = N-1 • . . • | ":--DO-90-K-=-lTNT T = C ( K ) * Z Z = Z * Y - SUM- -= -SUM"+-T I F ( N S U P . E Q . O ) G O T O 9 0 W R I T E ( 6 » 1 0 2 ) K ' T » S U M 9-0 -CO N T-I N U E ~ 1 " :  D = S U M 9 9 F O R M A T ( 9 0 X » 1 8 H T E R M W I S E S U M M A T I O N » / / » 8 5 X » 1 H K » 6 X » 4 H T E R M » 1 4 X » 3 H S U M 1 0 2 F O R M A T ( 8 5 X » U » 2 E 1 8 . 7 ) R E T U R N E N D * • r - 1 CT> g VO r IJ s . . . . . g --STBFT-C-GE-T-eG D E C K — : : : : j !• S U B R O U T I N E G E T C C ( E P • X X » C » NN » V E C T , C C C » E E E » DDD ) | 0 ' i D I M E N S I O N H ( 2 5 » 2 5 ) ' T T — D I M E N S I 0 N E P ( 25 ) » C ( 2 5'» 2 5") » V E C T ( 2 5 » 2 5 ) ~ \ Z \ ND = 2 5 | ! D"0~4" K ' " ' = " T » N N " ' fi DO 2 L = 1» N N H ( K » L ) = X X * C ( K » L ) ' 2 C O N T T N U E " H ( K > K ) = E P ( K ) + X X * C < K » K ) | 4 C O N T I N U E O A L T " F A S - J A C " ( H » V E d > O v O , O T N N V N D ) : \ 1 Z = 0 . 0 I ' D " 0 — 8 "—K~^~ 1 T " N N ~ : ~ ' | Y = 0 . 0 1 . DO 6 L = 1 * N N ! Y ~ = " Y - + C " ( K T L 7 ) * V E C T " r L " » i r — : 6 C O N T I N U E Z = Z + V E C T ( K , 1 ) * Y 8 C O N T I N U E : " ~ ~ C C C = z E E E = H ( l » l J D " D D ~ " = ~ E E ' E - E P ( T J • R E T U R N E N D " S I B F T C G E T D L D E C K S U B R O U T I N E G E T D L ( X X ) C O M M O N / I N O U T / C O ( 7 ) » D L < 7 ) C O M M O N " / F A C T T / - F A C T ('8 ) ™ O K v. v D O - 8 _ — = - _ - » - ? Z = 0 . 0 DO 6 K = L » 7 M = K _ L + 1  B = F A C T ( K ) / ( F A C T ( L ) * F A C T ( M ) ) N = K - L — : Z-~=--Z + B * ( X-X**N)-*CO<X) — — ; — -6 C O N T I N U E D L ( L ) = Z 8 — C O N T I N U E ; " R E T U R N E N D S T B F T C D E L T A D E C K S U B R O U T I N E D E L T A < G » E P E R T , D E L E » N S U P ) D I M E N S I O N E P E R T ( 2 5 » 7 ) » T E R M < 7 ) > S U M < 7 ) . z . _ 0 a Q - -DO 2 K = l i 7 T E R M ( K ) = ( G * * . K ) * E P E R T ( 1 . K ) Z - = Z ' + T E R M ' ( K ) " S U M ( K ) = Z 2 C O N T I N U E — D E L E " - = - " Z -I F ( N S U P . . E Q . 0 ) GO T O 1 0 P R I N T 1 0 0 D O - 8 - = - - 1 - , ' 7 P R I N T 1 1 0 » K » T E R M ( K ) » S U M < K ) 8 C O N T I N U E 1 - 0 — C O N T I N U E 1 0 0 • F 0 R M A T ( / » 3 1 H T E R M W I S E P E R T U R B A T I O N E N E R G I E S * / » 2 H 1 4 H S U M » / ) — 1 - 1 - 0 — F O R M A - T ( - I - 3 - v 2 E T 6 v 8 ) " ~ R E T U R N E N D $ T " B F T - C - F 7 V S 3 A - C ~ S U B R O U T I N E F A S J A C ( A » S » N S i » N S 2 » N S 3 , N > M ) C C F A S J A C P E R F 0 R M S ' A M A T RT X D ' l A G O N A L I S A T 1 0 N U S I NG J A C 0 B I S M E T H O D . C M A T R I C E S T O B E D I A G O N A L I S E D M U S T B E S Q U A R E » R E A L A N D S Y M M E T R I C C . W I T H M A X I M U M S I Z E O F ( 4 - 0 * 4 0 ) . C T'H E "A R G U M E N T S'" 0 F T H E S U B R O U T I N E A R E A S ' F O L L O W S C A I S T H E M A T R I X T O B E D I A G O N A L I S E D A N D I S R E T U R N E D T O T H E M A I N C P R O G R A M A F T E R T H I S P R O C E D U R E * W I T H T H E S A M E N A M E . C - S — I - S — T H E - - E T G E N V E C T O R ' M A T R I X . C N S l I S A N U M E R I C A L S U P P R S S O R W H I C H P R E V E N T S T H E P R I N T - O U T O F T H E C M A T R I X P R I O R TO D I A G O N A L I S A T I O N . F O R T H I S P U R P O S E I T M U S T B E S E T C : =' 0 " O T H E R W I S E " " A N Y ' I N T E G E R W H I C H F I T S T H E I 3 * F O R M A T " W I ' L L ' S U F F l C E . C NS2 I S A S I M I L A R Q U A N T I T Y W H I C H S U P P R E S S E S T H E P R I N T - O U T O F T H E C E I G E N V E C T O R S A N D E I G E N V A L U E S C N"S 3 I S A ' S I M I L A R Q U A N T I T Y WH I C H " S U P P R E S S E S T H E P R I N T - 6 ' U T O F ' " T H E C F I N A L D I A G O N A L I S E D M A T R I X . C N I S T H E O R D E R O F T H E M A T R I X TO B E D I A G O N A L I S E D C M " T S ~ T H E " 0 ' R D E R " C F " T H E M A T R I X " A S " D I M E N S I O N E D I N T H E " M A I N ' C P R O G R A M C # # # # # • « • # Q * *'#-#"* #>"#"#"*"*"#"'*—~ c c ~DTMEN'ST"0"N"~A"""( 'M»"M")""»"5 ( M »"M) ' " " DO 1 5 0 I = 1 * N DO 1 5 0 J = 1 * N ~A" ( • X » T ) ^ A T T V T ) ' " : : C G E N E R A T E I D E N T I T Y M A T R I X A S F I R S T A P P R O X I M A T I O N TO S I F ( I - J ) 1 0 0 , 1 0 1 * 1 0 0 ro-o-s-(-i-v3-)—S~OTO GO TO 150 101 S(I»J)=1.0 150 "CONTINUE I F < N S l . E Q . O ) G O T 0 4 0 l WRITE(6»7) CA'LT"'''MATOUTT'A'TNTM') ' 401 CONTINUE SET INDICATOR WHICH CHECKS OFF-DIAGONAL ELEMENTS I-NDTC—=—0 COMPUTE I N I T I A L NORM 151 VI = 0.0 "DO10 6' T = — 1 *~"N DO 106 J=1*N I F ( I - J ) 1 0 7 . 106» 107 ro 7~v r--= -vi + A( i v j ) **2 — : 106 CONTINUE 325 IF(NS2.EQ.O) GO TO 707 .—-WRTTE'('6"»"l'r"' 707 CONTINUE VI = SQ R T ( V I ) COMPUTE FINAL NORM VF = V l * 0 . 1 E - 0 7 COMPUTE THRESHOLD NORM A N " ' = N ~ • " 128 VI = VI/AN SET UP SYSTEMATIC SEARCH T 3 7 ~ r G r - 5 - 1 •= : " 124 IQ = IQ + 1 I P = 0 r _ T n " P — = — r p — * ~ T IF(A tIP»IQ) ) 108» 120, 109 CM CM CM " rH CM rH rH o * CM O rH CM LU U z <c p M I— ft 1— o ct: i< t—4 or t — <z z: u. p o LU z r H CO O I Q Z < < ; C L -— St-0 ~ •H O 1 r. t—( Q . * rH r a 1 < u I Q ix. u. z oo a> CM O O rH o UJ a < rH z <!o CM * * < U l P i ~ . i CM I * O ' * • ! 15 rH 5L — < H- : oc + o LO ; CM * . * . * o : < I II U l I— £> C L Si >: < o _i u < * ro o m rH « I H H O I r> n z: 11 < 3 — Z s: u. o < w I/) o rH rH O ii P o o to < CM CM _ l * < + * I - O X DC • r -P CMt/) to — ; 5 : o o < 00 • < „ * < h-z a a a ui a -0 5 : to o 11 11 11 < : O h - h -L U I X 51 r - r -O LO u z> _ J o u X I— o a z < X I— C L L U X u. p to z UJ rH S Z rH uj y _j » U l r H r H II -^H or - t a o M U_ vO ;l l / ) r-H •—I Z r H — < I O U. r - Q >-. X * — I oc I— < o rH Ct: rH < * _ J CO 3 rH U I - H LU ». to O rH UJ rH X ~ t — o i - i LU I r -— < t 0 X * r — r~ u 1—1 s(c jr. — O - H r-H o !' < r - + X I— I— < J X ; r— * LO r- * a -* — •—< < — CL p < <c .1 n X X r— I— -0 U LO X * I Z O O U M M u — — •—I — - w ' U . LO to u. U l I '+ o U H - h -X X U . I— h-o u - 0 * * X — — r H Q . C L M H r— r. 1—1 < M i - ( Q . 2 — — —< tO to < UJ | X II :ll II X ~~ \- CM * X * I— h-U X : * 1— j— to X * j — — to p if. *-* <S) I CM * O CL I I — u I* — ix < * CM b + i • CM CM * II * CM r -* X * r~ r - U X * I— — if) O LO ~ - X o o »-•- M U > — * b < ~ r-, + — — CM < * i+ * CMh-* X * r -r - LO X * o o 1-H UJ 1 r - l « C L O U C L r - t tfv « a a O !—* •—1 »—* | ^ < C L < II — II O II r-* ! 174 p •—I C L I—I < ro rH rH rH rH rH rH — I— C L O rH r l LJL O r-< r-. — J -M Qi <C < < < 1 I r-- o <d »-H e» »—1 »—1 1 - a. p •-• C L o O r - i r - i — DC < < tO - ; o «v »—« t—i I—I O. CL O Q_ k—I 1—4 I—I 1—4 — < < < < ! ! — II II II II «J I •—« p •H < C3 " *^  r-. C L O C L C L r - i r - i t—i tO < < < < < < vO C TR AN SFORM" MA'T'RI X DO 123 I = .1» N A CI»IP ) = A ( I P , I ) r 2 - 3 - -A^(lXrQ " )~H -7 \ - ( IQV'I") — 120 I F ( I P - I Q - f l ) 121» 122,122 122 I F ( I Q - N ) 124, 125, 125 1"25 'TF r i N D r C ) 1 2 6 , " 127", 126 126 INDIC = 0 GO TO 137 12 7 IF ( V l - V F ) 129, 129, 1 2 8 — 129 NQ1 = N-1 DO 130 1=1,N —DO-—••r30"-j-~='-"i"'~N"- ~ I F ( I - J ) 130, 131T130 131 MARKER = 0 D 0™3 01"K'~- 1TN.QT DO 301 L= 1,NQ1 IF < K - L ) 301, 302» 3 0 l 30'2~I F (A ( K V D ' - "A < K+1TL + T) )~"30 T » - 3 0 I T " 3 03 303 MARKER = 1 HOLE = A ( K , L ) "A"< K, L7) —=—ATK + T» L + D A(K+1 ,L+1) = HOLE DO 700 IP = 1»N S ( I p » K ) = S(IPt K+l) S( I P , K + l ) = HOLE 700" CONTINUE : 301 CONTINUE IF (MARKER) 304, 304, 131 "C OUTpUT--'CORRRE'SPONDrNG~-COEFFTCTE.NTS 3 0 H ~ - I P ( - N S-2TEQ vO -)G OTO A 0 2 W R I T E < 6 » 5 . ) I » . A < I » J ) 4 0 2 C O N T I N U E I-F-(-N-S-2-i-E-QTrO-)-GO-T-OAO-3 D O 2 0 8 I P = 1 » N 2 0 8 W R I T E ( 6 » 3 ) J » I P » S ( I P » J ) 4 0 3 - C 0 N T I N U E 1 3 0 C O N T I N U E I F ( N S 3 . E Q . 0 ) G 0 TO 7 0 8 W R I T E R 6 > 9 ) -C A L L M A T O U T ( A » N » M ) 7 0 8 C O N T I N U E - C " " " F O R M A T " S T A T E M E N T S . . 1 F O R M A T ( / 2 2 X » 1 1 H E I G E N V A L U E S » 2 2 X » 1 2 H E I G E N V E C T 0 R S 3 F O R M A T ( 4 0 X , I 5 > I 5 » 3 X » 1 P E 2 0 . 8 ) 5 F O R M A T ( / 1 0 X , I 5 » 2 X , 1 P E 2 0 . 8 ) 7 F O R M A T ( 1 H 1 » 1 0 X , 1 2 H I N P U T M A T R I X ) 9 F O R M A T ( / * 1 0 X , 1 9 H D I A G O N A L I S E D M A T R I X ) R E T U R N -E N D S E N T R Y 

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