On a probabilistic interpretation of shape derivatives of Dirichlet groundstates with application to Fermion nodes
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 44 (2010) no. 5, pp. 977-995.

This paper considers Schrödinger operators, and presents a probabilistic interpretation of the variation (or shape derivative) of the Dirichlet groundstate energy when the associated domain is perturbed. This interpretation relies on the distribution on the boundary of a stopped random process with Feynman-Kac weights. Practical computations require in addition the explicit approximation of the normal derivative of the groundstate on the boundary. We then propose to use this formulation in the case of the so-called fixed node approximation of Fermion groundstates, defined by the bottom eigenelements of the Schrödinger operator of a Fermionic system with Dirichlet conditions on the nodes (the set of zeros) of an initially guessed skew-symmetric function. We show that shape derivatives of the fixed node energy vanishes if and only if either (i) the distribution on the nodes of the stopped random process is symmetric; or (ii) the nodes are exactly the zeros of a skew-symmetric eigenfunction of the operator. We propose an approximation of the shape derivative of the fixed node energy that can be computed with a Monte-Carlo algorithm, which can be referred to as Nodal Monte-Carlo (NMC). The latter approximation of the shape derivative also vanishes if and only if either (i) or (ii) holds.

DOI : https://doi.org/10.1051/m2an/2010049
Classification : 60H30,  65C35,  65C05,  35P99
Mots clés : Schrödinger operator, groundstate, shape derivatives, Feynman-Kac formula, quantum Monte-Carlo methods, Fermion nodes, fixed node approximation
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title = {On a probabilistic interpretation of shape derivatives of {Dirichlet} groundstates with application to {Fermion} nodes},
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Rousset, Mathias. On a probabilistic interpretation of shape derivatives of Dirichlet groundstates with application to Fermion nodes. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 44 (2010) no. 5, pp. 977-995. doi : 10.1051/m2an/2010049. http://www.numdam.org/articles/10.1051/m2an/2010049/

[1] R. Assaraf and M. Caffarel, A pedagogical introduction to Quantum Monte Carlo, in Mathematical Models and Methods for Ab Initio Quantum Chemistry, M. Defranceschi and C. Le Bris Eds., Lecture Notes in Chemistry 74, Springer (2000). | Zbl 0992.81002

[2] R. 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.

[3] R. Assaraf, M. Caffarel and A. Khelif, Diffusion Monte-Carlo with a fixed number of walkers. Phys. Rev. E 61 (2000) 4566-4575.

[4] A. Badinski and R.J. Needs, Total forces in the diffusion Monte Carlo method with nonlocal pseudopotentials. Phys. Rev. B 78 (2008) 035134.

[5] A. Badinski, P.D. Haynes and R.J. Needs, Nodal Pulay terms for accurate diffusion quantum Monte Carlo forces. Phys. Rev. B 77 (2008) 085111.

[6] E. Cancès, B. Jourdain and T. Lelièvre, Quantum Monte-Carlo simulations of Fermions. A mathematical analysis of the fixed-node approximation. Math. Mod. Meth. Appl. Sci. 16 (2006) 1403-1440. | Zbl 1098.81095

[7] E. Cancès, C. Le Bris and Y. Maday, Méthodes mathématiques en chimie quantique : Une introduction. Springer-Verlag (2006). | Zbl 1167.81001

[8] M. Casalegno, M. Mella and A.M. Rappe, Computing accurate forces in quantum Monte Carlo using Pulay's corrections and energy minimization. J. Chem. Phys. 118 (2003) 7193-7201.

[9] D.M. Ceperley, Fermion nodes. J. Stat. Phys. 63 (1991) 1237-1267.

[10] D.M. Ceperley and B.J. Alder, Ground state of the electron gas by a stochastic method. Phys. Rev. Lett. 45 (1980) 566-569.

[11] D. Ceperley, G.V. Chester and M.H. Kalos, Monte-Carlo simulation of a many-fermion study. Phys. Rev. B 16 (1977) 3081-3099.

[12] C. Costantini, E. Gobet and N. El Karoui, Boundary sensitivities for diffusion processes in time dependent domains. Appl. Math. Optim. 54 (2006) 159-187. | Zbl 1109.49043

[13] P. Del Moral, Feynman-Kac Formulae, Genealogical and Interacting Particle Systems with Applications. Springer Series Probability and its Applications, Springer (2004). | Zbl 1130.60003

[14] P. Del Moral and L. Miclo, Branching and Interacting Particle Systems approximations of Feynman-Kac formulae with applications to nonlinear filtering. Lecture Notes Math. 1729 (2000) 1-145. | Numdam | Zbl 0963.60040

[15] P. Del Moral and L. Miclo, Particle approximations of Lyapounov exponents connected to Schrödinger operators and Feynman-Kac semigroups. ESAIM: PS 7 (2003) 171-208. | Numdam | Zbl 1040.81009

[16] A. Doucet, N. De Freitas and N.J. Gordon, Sequential Monte-Carlo Methods in Practice. Series Statistics for Engineering and Information Science, Springer (2001). | Zbl 0967.00022

[17] A. Doucet, P. Del Moral and A. Jasra, Sequential Monte Carlo samplers. J. Roy. Stat. Soc. B 68 (2006) 411-436. | Zbl 1105.62034

[18] C. Filippi and C.J. Umrigar, Correlated sampling in quantum Monte Carlo: A route to forces. Phys. Rev. B 61 (2000) R16291-R16294.

[19] J. Garcia Melian and J.S. De Lis, On the perurbation of eigenvalues for the p-laplacian. C. R. Acad. Sci. Paris, Sér. 1 332 (2001) 893-898. | Zbl 0989.35103

[20] D. Gildbarg and N.S. Trudinger, Elliptic Partial Differential Equation of Second Order. Springer-Verlag (1983). | Zbl 0562.35001

[21] B.L. Hammond, W.A. Lester and P.J. Reynolds, Monte Carlo Methods in ab initio quantum chemistry. World Scientific (1994).

[22] H. Hongxin and S. Liu, An improved algorithm of fixed-node quantum Monte Carlo method with self-optimization process. J. Mol. Struct. Theochem 726 (2005) 93-97.

[23] I. Karatzas and S.E. Shreve, Brownian motion and stochastic calculus, Graduate Texts in Mathematics 113. Second edition, Springer-Verlag, New York (1991). | Zbl 0734.60060

[24] T. Kato, Perturbation theory for linear operators, Grundlehren der Mathematischen Wissenschaften 132. Second edition Springer-Verlag, Berlin (1976). | Zbl 0342.47009

[25] M. Reed and B. Simon, Methods of modern mathematical physics. IV. Analysis of operators. Academic Press (Harcourt Brace Jovanovich Publishers), New York (1978). | Zbl 0242.46001

[26] M. Rousset, On the control of an interacting particle estimation of Schrödinger ground states. SIAM J. Math. Anal. 38 (2006) 824-844. | Zbl 1174.60045

[27] J. Toulouse and C.J. Umrigar, Optimization of quantum Monte Carlo wave functions by energy minimization. J. Chem. Phys. 126 (2007) 084102.

[28] J. Toulouse, R. Assaraf and C.J. Umrigar, Zero-variance zero-bias quantum Monte Carlo estimators of the spherically and system-averaged pair density. J. Chem. Phys. 126 (2007) 244112.

[29] C.J. Umrigar and C. Filippi, Energy and variance optimization of many-body wave functions. Phys. Rev. Lett. 94 (2005) 150201.

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