17 Computer Modeling of Quantum Liquids and Crystals M. H. KALOS, P. A. WHITLOCK, and D. M. CEPERLEY
Downloaded by UNIV OF TEXAS AT DALLAS on July 11, 2016 | http://pubs.acs.org Publication Date: June 1, 1978 | doi: 10.1021/bk-1978-0086.ch017
Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012
There are a number of many-body systems which exhibit quantum effects on a macroscopic scale. These include liquid and crystal states of both He-3 and He4, the electron gas, and neutron matter which probably constitutes the interior of pulsors. In addition, "nuclear matter" - a hypothetical extensive system of nucleons has been studied for the insight one may gain into the nature of finite nuclei. The theoretical studies of these systems have by now a long history, but are by no means concluded. In the last few years, significant advances have been made. This has come in part from the maturity of and gradual unification of many-body theory, in part from the development and application of powerful new expansion procedures, especially varieties of hypernetted-chain equations (1) and finally to the growing power of computer simulation methods for quantum systems. This article is intended as a review of some recent development in computational methods for extensive quantum systems, and of the relation between results so obtained to the evolution of other theoretical work. C o m p u t a t i o n a l m o d e l l i n g o f quantum many-body s y s tems i s n o t e s p e c i a l l y n o v e l . The e a r l y h i s t o r y o f M o n t e C a r l o m e t h o d s i n c l u d e d many p r o p o s a l s f o r t h e solution of Schrodinger s equation with intended a p p l i c a t i o n t o t h e many-body p r o b l e m . Unfortunately, the c o m p u t a t i o n a l p o w e r a v a i l a b l e was n o t a d e q u a t e t o d o more t h a n s i m p l e e x e r c i s e s . T h e f i r s t work i n w h i c h a s i g n i f i c a n t c o n t r i b u t i o n t o t h e o r y was made was t h a t o f W. L . M c M i l l a n (2_) who n o t e d t h a t t h e g e n e r a l s a m p l i n g a l g o r i t h m o f M e t r o p o l i s e t a l . (3)developed to t r e a t e q u i l i b r i u m c h e m i c a l systems c o u l d be used e q u a l l y w e l l t o o b t a i n v a r i a t i o n a l estimates o f the energy o f a m a n y - b o d y s y s t e m when t h e t r i a l f u n c t i o n h a s a product form. S i n c e t h e n , a l a r g e number o f s i m i l a r 1
0-8412-0463-2/78/47-086-219$05.00/0 © 1978 American Chemical Society Lykos; Computer Modeling of Matter ACS Symposium Series; American Chemical Society: Washington, DC, 1978.
Downloaded by UNIV OF TEXAS AT DALLAS on July 11, 2016 | http://pubs.acs.org Publication Date: June 1, 1978 | doi: 10.1021/bk-1978-0086.ch017
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c a l c u l a t i o n s have been c a r r i e d o u t t r e a t i n g extensive s y s t e m s o f atoms f r o m h y d r o g e n t o n e o n . For a thorough r e v i e w o f t h e s e c a l c u l a t i o n s see r e f e r e n c e ( £ ) . I t i s l i k e l y t h a t c a l c u l a t i o n s o f t h i s k i n d w i l l be e v e n more u s e d i n t h e f u t u r e s i n c e t h e y a r e w e l l s u i t e d f o r modern m i n i c o m p u t e r s . We w o u l d l i k e t o e m p h a s i z e h e r e some a d d i t i o n a l m e t h o d o l o g i c a l developments and t h e i r r e s u l t s . The f i r s t i s the v a r i a t i o n a l treatment of f u l l y antisymmetrized t r i a l functions (5). The second i s the G r e e n ' s f u n c t i o n Monte C a r l o a l g o r i t h m ( £ , 6) w h i c h h a s , i n e f f e c t , made p o s s i b l e t h e n u m e r i c a l i n t e g r a t i o n of the S c h r o d i n g e r e q u a t i o n . F e r m i o n Monte C a r l o Consider
a
Hamiltonian
H = - ^ X Z v
2 i
+
1
Z I v ( r ±
