PHYSICAL REVIEW E 89, 033304 (2014) Calculation of space localized properties in correlated quantum Monte Carlo methods with reweighting: The nonlocality of statistical uncertainties Roland Assaraf and Dominik Domin Laboratoire de Chimie Th´ eorique, CNRS-UMR 7616, Universit´ e Pierre et Marie Curie Paris VI, Case 137, 4 place Jussieu, 75252 Paris Cedex 05, France (Received 3 December 2013; published 10 March 2014) We study the efficiency of quantum Monte Carlo (QMC) methods in computing space localized ground state properties (properties which do not depend on distant degrees of freedom) as a function of the system size N . We prove that for the commonly used correlated sampling with reweighting method, the statistical fluctuations σ 2 (N ) do not obey the locality property. σ 2 (N ) grow at least linearly with N and with a slope that is related to the fluctuations of the reweighting factors. We provide numerical illustrations of these tendencies in the form of QMC calculations on linear chains of hydrogen atoms. DOI: 10.1103/PhysRevE.89.033304 PACS number(s): 02.70.Ss, 31.15.−p I. INTRODUCTION Many important quantities of chemical or physical interest are localized in space; that is, they do not depend on spatially distant degrees of freedom. For example, the forces exerted on any particular nucleus in a molecule are barely influenced by the presence of neutral molecules that are very far away from the molecule of interest. One would expect that the distant neutral molecules would represent irrelevant degrees of freedom in computation of the force experienced by the nucleus and could be eliminated in the computation of such property. Quantum chemistry exploits locality in many different ways, for example, the Lewis description of covalent bonding [1] and the closely related valence bond theory [2,3]. Deterministic computational methods often exploit locality to tackle large systems. The most obvious strategy would be to study a fragment (for example, in a protein) and rely on the transferability of the results to a larger system. This would also be the idea behind coarse graining and hybrid methods. [4,5] However, calculations on the full system are more robust since they do not depend on this transferability hypothesis. Exploiting the locality property to lower the cost of a calculation on the entire system is the strategy involved, for example, in local electronic structure methods [6–9]. The latter scale down the computational cost as a function of the system size N (number of particles for a rather homogeneous system at a given scale, or a collection of identical systems). In such methods, molecular orbitals can be localized on single or spatially adjacent atoms according to a variety of criteria [10–12]. For large molecular systems, locality enables local correlation methods to reduce the computational scaling up to a linear dependence with N [13,14]. Quantum Monte Carlo (QMC) methods [15–20], a powerful set of stochastic techniques for solving the Schr¨ odinger equation, are increasingly used for electronic structure cal- culations on molecular systems. This is primarily because of the moderate computational scaling [O(N 3−4 )] and the perceived high accuracy to compute total energies. In the past decade there has been a similar drive to formulate and program linear-scaling QMC algorithms [21–27]. The focus of such works was to generate Monte Carlo sample configurations and evaluate energies with a reduced computational cost. However, another important factor in the numerical efficiency of a Monte Carlo method is the size of the statistical fluctuations in a calculated property. The purpose of this paper is to understand the behavior of QMC statistical fluctuations of spatially localized properties as a function of N . We show that conventional methods (correlated sampling methods) do not have the locality property regarding the statistical fluctuations. Many properties, such as the force on a nucleus, dipole moment, or substitution energy, can be written as a difference of two ground state energies E λ − E 0 . E 0 and E λ are, respectively, the ground state energies of the Hamiltonians H 0 and H λ . H λ is a small perturbation of H 0 and λ is a small perturbation parameter, H λ = H + λO. (1) When considering a localized property, O depends mainly on the positions of particles lying in a small region of the space. We then have to compute a small difference in energies E λ − E 0 or the energy derivative, 〈O〉= dE λ dλ     λ=0 ≃ E λ − E 0 λ . (2) Computing these differences (2) from independent energy calculations is particularly inefficient in QMC. Since energy is size extensive, the statistical uncertainty on the energy usually behaves as √ N . Consequently, the statistical uncertainty on such a calculation of (2) is σ i (λ,N,M) ∝ 1 λ  N M . (3) This formula is valid asymptotically, for small λ, large system size N , and large sample size M. It is obvious from (3) that the smaller λ is the less efficient independent energy calculations are. The so-called correlated sampling with reweighting methods [28–31], which are popular strategies to compute small differences of energies or properties, seem to be much more suitable methods. As we show later these methods encompass improved estimators which are built using the Hellmann-Feynman theorem [28,32–37]. We prove in this paper that the statistical uncertainty in correlated sampling with reweighting methods behaves as σ c (λ,N,M) ∝  N M . (4) 1539-3755/2014/89(3)/033304(10) 033304-1 ©2014 American Physical Society