Optimum and efficient sampling for variational quantum Monte Carlo

Journal of Chemical Physics

Volumn 133, Issue 17, 2010, Pages

  Trail, J R ^a   Maezono, Ryo ^a  

^a JAPAN ADVANCED INSTITUTE OF SCIENCE AND TECHNOLOGY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

AB INITIO METHOD; CENTRAL LIMIT THEOREM; CONVENTIONAL METHODS; EFFICIENT SAMPLING; ESTIMATION AND OPTIMIZATION; EXPECTATION VALUES; HEAVY-TAILED; INFINITE VARIANCES; LOWER LIMITS; MANY-BODY SYSTEMS; MONTE CARLO; OPTIMUM PARAMETERS; PROBABILITY DENSITIES; QUANTUM MECHANICS; QUANTUM MONTE CARLO; SAMPLING STRATEGIES; STATISTICAL PROPERTIES; THEORETICAL BASIS; VARIATIONAL QUANTUM MONTE CARLO;

COMPUTATIONAL EFFICIENCY; DENSITY FUNCTIONAL THEORY; MONTE CARLO METHODS; OPTIMIZATION; PROBABILITY DENSITY FUNCTION; RANDOM ERRORS; SAMPLING; WAVE FUNCTIONS;

QUANTUM EFFICIENCY;

EID: 78650667164     PISSN: 00219606     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.3488651     Document Type: Article

References (36)

1. W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Rajagopal, Rev. Mod. Phys. RMPHAT 0034-6861 73, 33 (2001). 10.1103/RevModPhys.73.33
2. J. F. Traub and A. G. Wershulz, Complexity and Information (Cambridge University Press, Cambridge, 1998).
3. D. W. Stroock, Probability Theory: An Analytic View (Cambridge University Press, Cambridge, 1993).
4. J. H. Curtiss, Ann. Math. Stat. AASTAD 0003-4851 12, 409 (1941). 10.1214/aoms/1177731679
5. G. Samorodnitsky and M. S. Taqqu, " Stable Non-Gaussian Random Processes (Chapman & Hall, Florida, 1994); see also http://academic2. american.edu/jpnolan/stable/stable.html.
6. M. Dewing and D. M. Ceperley, Recent Advances in Quantum Monte Carlo Methods II (World Scientific, Singapore, 2002).
7. R. L. Coldwell, Int. J. Quantum Chem. IJQCB2 0020-7608 S11, 215 (1977).
8. S. A. Alexander, R. L. Coldwell, H. J. Monkhorst, and J. D. Morgan III, J. Chem. Phys. JCPSA6 0021-9606 95, 6622 (1991). 10.1063/1.461532
9. C. Attaccalite and S. Sorella, Phys. Rev. Lett. PRLTAO 0031-9007 100, 114501 (2008). 10.1103/PhysRevLett.100.114501 (Pubitemid 351415953)
10. R. Assaraf, M. Caffarel, and A. Khelif, J. Phys. A: Math. Theor. 40, 1181 (2007). 10.1088/1751-8113/40/6/001
11. J. R. Trail, Phys. Rev. E PLEEE8 1063-651X 77, 016703 (2008). 10.1103/PhysRevE.77.016703 (Pubitemid 351388532)
12. J. R. Trail, Phys. Rev. E PLEEE8 1063-651X 77, 016704 (2008). 10.1103/PhysRevE.77.016704 (Pubitemid 351388531)
13. D. Ceperley, M. Dewing, and C. Pierleoni, Bridging Time Scales: Molecular Simulations for the Next Decade (Springer-Verlag, Berlin Heidelberg, 2002).
14. 2 c22].
15. U. von Luxburg and V. H. Franz, Confidence Sets for Ratios: A Purely Geometric Approach To Fieller's Theorem (133), Max Planck Institute for Biological Cybernetics, Tübingen, Germany, 2004; E. Fieller, Biometrika BIOKAX 0006-3444 24, 428 (1932).
16. 2, and may replace m with (m-1).
17. R. J. Needs, M. D. Towler, N. D. Drummond, and P. López Ríos, J. Phys.: Condens. Matter JCOMEL 0953-8984 22, 023201 (2010). 10.1088/0953-8984/22/2/023201
18. A. Badinski, J. R. Trail, and R. J. Needs, J. Chem. Phys. JCPSA6 0021-9606 129, 224101 (2008). 10.1063/1.3013817
19. A. Badinski, P. D. Haynes, J. R. Trail, and R. J. Needs, J. Phys.: Condens. Matter JCOMEL 0953-8984 22, 074202 (2010). 10.1088/0953-8984/22/7/ 074202
20. N. D. Drummond, M. D. Towler, and R. J. Needs, Phys. Rev. B PLRBAQ 0556-2805 70, 235119 (2004). 10.1103/PhysRevB.70.235119 (Pubitemid 40270736)
21. P. López Ríos, A. Ma, N. D. Drummond, M. D. Towler, and R. J. Needs, Phys. Rev. E PLEEE8 1063-651X 74, 066701 (2006). 10.1103/PhysRevE.74. 066701 (Pubitemid 44958110)
22. C. Froese Fischer, G. Tachiev, G. Gaigalas, and M. R. Godefroid, Comput. Phys. Commun. CPHCBZ 0010-4655 176, 559 (2007). 10.1016/j.cpc.2007.01.006 (Pubitemid 46442716)
23. M. D. Brown, J. R. Trail, P. L. Rios, and R. J. Needs, J. Chem. Phys. JCPSA6 0021-9606 126, 224110 (2007). 10.1063/1.2743972 (Pubitemid 46961580)
24. E2 /r defined by Eq., provided the bivariate CLT is valid. We do not use unbiased estimates to define the minimized quantity since they do not change the discussion in this section, but do complicate the equations for general sampling.
25. M. P. Nightingale and V. Melik-Alaverdian, Phys. Rev. Lett. PRLTAO 0031-9007 87, 043401 (2001). 10.1103/PhysRevLett.87.043401
26. J. Toulouse and C. J. Umrigar, J. Chem. Phys. JCPSA6 0021-9606 126, 084102 (2007). 10.1063/1.2437215 (Pubitemid 46355599)
27. This is not entirely obvious since Eq. includes a total energy estimate, hence the averaged variables are neither independent or identically distributed. However, the bivariate CLT remains valid due to the contribution of correlation between samples at different times decaying relative to the contribution of samples at the equal times, as sample size increases. A derivation is analogous to showing that variance estimates become Normal.
28. M. Lax, W. Cai, and M. Xu, Random Processes in Physics and Finance (Oxford University Press, New York, 2006). 10.1093/acprof:oso/9780198567769.001. 0001
29. The computational budget for each system was 24 h of computational time for monitored optimization (12 h for optimization and 12 h for sample generation) followed by 24 h for the final accurate estimate, with calculations performed on a modest desktop with 2 quad-core processors.
30. L. Laaksonen, P. Pyykko, and D. Sundholm, Comput. Phys. Rep. CPHREF 0167-7977 4, 313 (1986). 10.1016/0167-7977(86)90021-3
31. J. Toulouse and C. J. Umrigar, J. Chem. Phys. JCPSA6 0021-9606 128, 174101 (2008). 10.1063/1.2908237 (Pubitemid 351644876)
32. F. Ruette, M. Sanchez, R. Anez, A. Bermudez, and A. Sierraalta, J. Mol. Struct.: THEOCHEM THEODJ 0166-1280 729, 19 (2005). 10.1016/j.theochem.2005.04. 024 (Pubitemid 41258142)
33. M. Puchalski and K. Pachucki, Phys. Rev. A PLRAAN 1050-2947 73, 022503 (2006). 10.1103/PhysRevA.73.022503 (Pubitemid 43270285)
34. S. J. Chakravorty, S. R. Gwaltney, E. R. Davidson, F. A. Parpia, and C. F. Fischer, Phys. Rev. A PLRAAN 1050-2947 47, 3649 (1993). 10.1103/PhysRevA.47. 3649
35. D. P. O'Neil and P. M. W. Gill, Mol. Phys. MOPHAM 0026-8976 103, 763 (2005). 10.1080/00268970512331339323 (Pubitemid 41673120)
36. L. A. Curtiss, P. C. Redfern, and K. Raghavachari, J. Chem. Phys. JCPSA6 0021-9606 126, 084108 (2007). 10.1063/1.2436888 (Pubitemid 46355605)