- Email: [email protected]
Computation of wavefunctions for the helium isoelectronic sequence
C-19 C O M P U T E R PHYSICS COMMUNICATIONS 1 (1969) 167-180. N O R T H - H O L L A N D PUBLISHING C O M P . , A M S T E R D A M
COMPUTATION FOR
THE
HELIUM
OF
WAVEFUNCTIONS
ISOELECTRONIC
SEQUENCE
H. O. KNOX Department of Applied Mathematics and Theoretical P h y s i c s , The Queen's University of Belfast, Belfast. Northern Ireland R e c e i v e d 20 N o v e m b e r 1969
PROGRAM SUMMARY Title of program (32 c h a r a c t e r s m a x i m u m ) : T W O - E L E C T R O N W A V E F U N C T I O N S Catalogue number: ACQK Computer f o r which the program is designed and others upon which is it operable Computer: ICL 1907. Installation: Q u e e n ' s U n i v e r s i t y of B e l f a s t Operating s y s t e m or monitor under which the program is executed: QUBE P r o g r a m m i n g languages used: F O R T R A N High speed store required: 20.000 w o r d s . No. o f bits in a word: 24 Is the program overlaid? No No. of magnetic tapes required: One m a y be u s e d to s t o r e the r e s u l t s What other peripherals are used ? C a r d R e a d e r ; Line P r i n t e r No. of cards in combined program and lest deck: 1754 Keywords d e s c r i p t i v e o f problem and method o f solution: A t o m i c W a v e f u n c t i o n s , H e l i u m I s o e l e c t r o n i c S e q u e n c e , S t u r m i a n F u n c t i o n s . C o n f i g u r a t i o n I n t e r a c t i o n , I n v e r s e I t e r a t i o n . E i g e n v a l u e s . Bound SLate. S c h r ~ d i n g e r E q u a t i o n . A u t o i o n i z i n g State. R e s o n a n c e E n e r g y . E n e r g y L e v e l
Nature of the physical problem T h i s p r o g r a m c o m p u t e s w a v e f u n c t i o n s and e n e r g i e s of the h e l i u m i s o e l e c t r o n i c s e q u e n c e to any d e s i r e d a c c u r a c y . T h e w a v e f u n e t i o n s h a v e the c o r r e c t a s y m p t o t i c f o r m and t h e r e f o r e they a r e p a r t i c u l a r l y s u i t a b l e f o r u s e in s c a t t e r ing p r o b l e m s . T h e y h a v e b e e n u s e d to c a l c u l a t e line s t r e n g t h s by Knox and R u d g e [1].
Method of solution T h e w a v e f u n c t i o n is of 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 type
= 5~X(SMs.0102)
~ ~ a(nln2t) y ( l l l 2 L I l ' l t~2)r-~o Fnlll(~rl_rl) Fn212(ot2r2). nln 2 ~ i z
w h e r e . ~ is the a n t i - s y m m e t r i z i n g o p e r a t o r , X is the t w o - e l e c t r o n s p i n f u n c t i o n , i r e p r e s e n t s t h e o r b i t a l a n g u l a r m o m e n t a q u a n t u m n u m b e r s (lll2L) and Fnl(Otr) is a S t u r m i a n function. T h e m a t r i x f o r m u l a t i o n of the v a r i a t i o n a l p r i n ciple is u s e d , t h e e n e r g y c l o s e s t to the e x p e r i m e n t a l e n e r g y and the c o r r e s p o n d i n g w a v e f u n c t i o n b e i n g e x t r a c t e d u s ing an i n v e r s e p o w e r i n g m e t h o d d e s c r i b e d by W i l k i n s o n [2].
R e s t r i c t i o n s on the complexity o f the problem T h e s i z e of the w a v e f u n c t i o n is l i m i t e d by t h e s t o r a g e a v a i l a b l e and by r o u n d i n g e r r o r s in the c a l c u l a t i o n of t h e m a t r i x e l e m e n t s and in m a t r i x i n v e r s i o n . T h e m a j o r s t o r a g e r e q u i r e m e n t i s f o r f o u r m a t r i c e s w h o s e d i m e n s i o n e q u a l s the n u m b e r of b a s i s f u n c t i o n s u s e d ; the m a t r i x e l e m e n t s b e c o m e i n a c c u r a t e if n l , n 2 > 10: and it is n o t r e c o m m e n d e d t h a t the p r e s e n t m a t r i x i n v e r s i o n r o u t i n e be u s e d on m a t r i c e s l a r g e r t h a n 60 x 60.
Typical running t i m e T h e t i m e r e q u i r e d to c o m p u t e one e n e r g y and w a v e f u n c t i o n i n c r e a s e s r o u g h l y a s N 2 w h e r e N i s t h e n u m b e r of b a s i s f u n c t i o n s u s e d and f o r N = 10 it is about 22 s e c o n d s on an I C L 1907. T h e t i m e n e e d e d f o r d i f f e r e n t w a v e f u n c t i o n s of the s a m e s i z e m a y v a r y c o n s i d e r a b l y due to the u s e of an i t e r a t i v e r o u t i n e .
References [1] H. O. Knox and M. R. H. Rudge, J. P h y s . B (Set. 2) 2 (1969) 521. [21 J. H. W i l k i n s o n . T h e a l g e b r a i c e i g e n v a l u e p r o b l e m ( C l a r e n d o n P r e s s ,
O x f o r d . 1965).
COMPUTATION FOR
THE
HELIUM
OF
WAVEFUNCTIONS
ISOELECTRONIC
SEQUENCE
H. O. KNOX Department of Applied Mathematics and Theoretical P h y s i c s , The Queen's University of Belfast, Belfast. Northern Ireland R e c e i v e d 20 N o v e m b e r 1969
PROGRAM SUMMARY Title of program (32 c h a r a c t e r s m a x i m u m ) : T W O - E L E C T R O N W A V E F U N C T I O N S Catalogue number: ACQK Computer f o r which the program is designed and others upon which is it operable Computer: ICL 1907. Installation: Q u e e n ' s U n i v e r s i t y of B e l f a s t Operating s y s t e m or monitor under which the program is executed: QUBE P r o g r a m m i n g languages used: F O R T R A N High speed store required: 20.000 w o r d s . No. o f bits in a word: 24 Is the program overlaid? No No. of magnetic tapes required: One m a y be u s e d to s t o r e the r e s u l t s What other peripherals are used ? C a r d R e a d e r ; Line P r i n t e r No. of cards in combined program and lest deck: 1754 Keywords d e s c r i p t i v e o f problem and method o f solution: A t o m i c W a v e f u n c t i o n s , H e l i u m I s o e l e c t r o n i c S e q u e n c e , S t u r m i a n F u n c t i o n s . C o n f i g u r a t i o n I n t e r a c t i o n , I n v e r s e I t e r a t i o n . E i g e n v a l u e s . Bound SLate. S c h r ~ d i n g e r E q u a t i o n . A u t o i o n i z i n g State. R e s o n a n c e E n e r g y . E n e r g y L e v e l
Nature of the physical problem T h i s p r o g r a m c o m p u t e s w a v e f u n c t i o n s and e n e r g i e s of the h e l i u m i s o e l e c t r o n i c s e q u e n c e to any d e s i r e d a c c u r a c y . T h e w a v e f u n e t i o n s h a v e the c o r r e c t a s y m p t o t i c f o r m and t h e r e f o r e they a r e p a r t i c u l a r l y s u i t a b l e f o r u s e in s c a t t e r ing p r o b l e m s . T h e y h a v e b e e n u s e d to c a l c u l a t e line s t r e n g t h s by Knox and R u d g e [1].
Method of solution T h e w a v e f u n c t i o n is of 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 type
= 5~X(SMs.0102)
~ ~ a(nln2t) y ( l l l 2 L I l ' l t~2)r-~o Fnlll(~rl_rl) Fn212(ot2r2). nln 2 ~ i z
w h e r e . ~ is the a n t i - s y m m e t r i z i n g o p e r a t o r , X is the t w o - e l e c t r o n s p i n f u n c t i o n , i r e p r e s e n t s t h e o r b i t a l a n g u l a r m o m e n t a q u a n t u m n u m b e r s (lll2L) and Fnl(Otr) is a S t u r m i a n function. T h e m a t r i x f o r m u l a t i o n of the v a r i a t i o n a l p r i n ciple is u s e d , t h e e n e r g y c l o s e s t to the e x p e r i m e n t a l e n e r g y and the c o r r e s p o n d i n g w a v e f u n c t i o n b e i n g e x t r a c t e d u s ing an i n v e r s e p o w e r i n g m e t h o d d e s c r i b e d by W i l k i n s o n [2].
R e s t r i c t i o n s on the complexity o f the problem T h e s i z e of the w a v e f u n c t i o n is l i m i t e d by t h e s t o r a g e a v a i l a b l e and by r o u n d i n g e r r o r s in the c a l c u l a t i o n of t h e m a t r i x e l e m e n t s and in m a t r i x i n v e r s i o n . T h e m a j o r s t o r a g e r e q u i r e m e n t i s f o r f o u r m a t r i c e s w h o s e d i m e n s i o n e q u a l s the n u m b e r of b a s i s f u n c t i o n s u s e d ; the m a t r i x e l e m e n t s b e c o m e i n a c c u r a t e if n l , n 2 > 10: and it is n o t r e c o m m e n d e d t h a t the p r e s e n t m a t r i x i n v e r s i o n r o u t i n e be u s e d on m a t r i c e s l a r g e r t h a n 60 x 60.
Typical running t i m e T h e t i m e r e q u i r e d to c o m p u t e one e n e r g y and w a v e f u n c t i o n i n c r e a s e s r o u g h l y a s N 2 w h e r e N i s t h e n u m b e r of b a s i s f u n c t i o n s u s e d and f o r N = 10 it is about 22 s e c o n d s on an I C L 1907. T h e t i m e n e e d e d f o r d i f f e r e n t w a v e f u n c t i o n s of the s a m e s i z e m a y v a r y c o n s i d e r a b l y due to the u s e of an i t e r a t i v e r o u t i n e .
References [1] H. O. Knox and M. R. H. Rudge, J. P h y s . B (Set. 2) 2 (1969) 521. [21 J. H. W i l k i n s o n . T h e a l g e b r a i c e i g e n v a l u e p r o b l e m ( C l a r e n d o n P r e s s ,
O x f o r d . 1965).