Quantum Monte Carlo simulations of fermions. A mathematical analysis of the fixed-node approximation

Mathematical Models and Methods in Applied Sciences

Volumn 16, Issue 9, 2006, Pages 1403-1440

  Cancès, Eric ^a   Jourdain, Benjamin ^a   Lelièvre, Tony ^a  

^a UNIVERSITÉ PARIS EST   (France)

Author keywords

High dimensional partial differential equations; Monte Carlo methods; Particle methods; Quantum chemistry; Quantum Monte Carlo; Schr dinger equation

Indexed keywords

EID: 33747689830     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202506001583     Document Type: Article

References (33)

1. D. Alfè and M. Gillan, J. Phys. Cond. Mat. 16 (2004) 305.
2. S. Albeverio, Y. Kondratiev and M. Röckner, Strong Feller properties for distorted Brownian motion and applications to finite particle systems with singular interaction, in Finite, and Infinite. Dimensional Analysis in Honor of Leonard Gross, Contemp. Math., Vol. 317 (Amer. Math. Soc., 2003), pp. 15-35.
3. A. Aspuru-Guzik and W. A. Lester, Jr., Quantum Monte Carlo methods for the solution of the Schrödinger equation for molecular systems, in Handbook of Numerical Analysis, Special volume, Computational Chemistry, Vol. X, eds. Ph. G. Ciarlet and G. Le Bris (North-Holland, 2003), pp. 485-536.
4. A. Assaraf and M. Caffarel, Computing forces with Quantum Monte Carlo, J. Chem. Phys. 113 (2000) 4028-4034.
5. A. Assaraf and M. Caffarel, Zero-variance zero-bias principle for observables in quantum Monte Carlo: Application to forces, J. Chem. Phys. 119 (2003) 10536-10552.
6. R. Assaraf, M. Caffarel and A. Khelif, Diffusion Monte Carlo with a fixed number of walkers, Phys. Rev. E 61 (2000) 4566-4575.
7. M. Caffarel and P. Claverie, Development of a pure diffusion quantum Monte Carlo method using a full generalized Feynman-Kac formula. I. Formalism, J. Chem. Phys. 88 (1988) 1088.
8. E. Cancès, M. Defranceschi, W. Kutzelnigg, C. Le Bris and Y. Maday, Computational quantum chemistry: A primer, in Handbook of Numerical Analysis, Special volume, Computational Chemistry, Vol. X, eds. Ph. G. Ciarlet and C. Le Bris (North-Holland, 2003), pp. 3-270.
9. T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations (Oxford Univ. Press, 1998).
10. D. M. Ceperley, Fermion nodes, J. Stat. Phys. 63 (1991) 1237-1267.
11. C. Filippi and S. Fahy, Optimal orbitals from energy fluctuations in correlated wave functions, J. Chem. Phys. 112 (2000) 3523-3531.
12. C. Filippi and C. J. Umrigar, Phys. Rev. B 61 (2000) R16291.
13. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 3rd edn. (Springer, 1998).
14. P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136 (1964) B864-B871.
15. W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev. 140 (1965) A1133-A1138.
16. O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow (Gordon and Breach, 1987).
17. D. Lamberton and B. Lapeyre, Introduction au Calcul Stochastique Appliqué à la Finance (Ellipse, 1997).
18. Ed. W. A. Lester, Jr., Recent Advances in Quantum Monte Carlo Methods (World Scientific, 1997).
19. Eds. W. A. Lester, Jr., S. M. Rothstein and S. Tanaka, Recent Advances in Quantum Monte Carlo Methods, Part II (World Scientific, 2002).
20. I. N. Levine, Quantum Chemistry, 4th edn. (Prentice Hall, 1991).
21. P.-L. Lions, Remarks on mathematical modelling in quantum chemistry, in Computational Methods in Applied Sciences (Wiley, 1996), pp. 22-23.
22. E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Commun. Math. Phys. 53 (1977) 185-194.
23. S. Manten and A. Luchow, J. Chem. Phys. 119 (2003) 1307.
24. D. Marx and J. Hutter, Ab initio molecular dynamics: Theory and implementation, in Modern Methods and Algorithms of Quantum Chemistry, ed. J. Grotendorst (NIC FZ Zülich, 2000), pp. 301-449.
25. R. McWeeny, Methods of Molecular Quantum Mechanics, 2nd edn. (Academic Press, 1992).
26. Eds. M. P. Nightingale and C. J. Umrigar, Quantum Monte Carlo Methods in Physics and Chemistry, NATO Science Series C: Mathematical and Physical Sciences, Vol. 525 (Kluwer, 1999).
27. M. Reed and B. Simon, Methods of Modern Mathematical Physics -IV: Analysis of Operators (Academic Press, 1978).
28. D. Revuz and M. Yor, Continuous Martingales and Brownian Motion (Springer-Verlag, 1991).
29. L. C. G. Rogers, Smooth transition densities for one-dimensional diffusions, Bull. London Math. Soc. 17 (1985) 157-161.
30. D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes (Springer-Verlag, 1997).
31. C. J. Umrigar, K. G. Wilson and J. W. Wilkins, Optimal trial wave functions for quantum Monte Carlo calculations, Phys. Rev. Lett. 60 (1988) 1719-1722.
32. A. J. Williamson, R. Q. Hood and J. C. Grossman, Linear scaling quantum Monte Carlo calculations, Phys. Rev. Lett. 87 (2001) 246406.
33. G. M. Zhislin, Discussion of the spectrum of Schrödinger operators for systems of many particles, Trudy Moskovskogo Mat. Obscestva 9 (1960) 81-120, in Russian.