Linear-scaling quantum Monte Carlo technique with non-orthogonal localized orbitals

Abstract

We have reformulated the quantum Monte Carlo (QMC) technique so that a large part of the calculation scales linearly with the number of atoms. The reformulation is related to a recent alternative proposal for achieving linear-scaling QMC, based on maximally localized Wannier orbitals (MLWO), but has the advantage of greater simplicity. The technique we propose draws on methods recently developed for linear-scaling density functional theory. We report tests of the new technique on the insulator MgO, and show that its linear-scaling performance is somewhat better than that achieved by the MLWO approach. Implications for the application of QMC to large complex systems are pointed out.The quantum Monte Carlo (QMC) technique [1] is becoming ever more important in the study of condensed matter, with recent applications including the reconstruction of semiconductor surfaces [2], the energetics of point defects in insulators [3], optical excitations in nanostructures [4], and the energetics of organic molecules [5]. Although its demands on computer power are much greater than those of widely used techniques such as density functional theory (DFT), its accuracy is also much greater for most systems. With QMC now being applied to large complex systems containing hundreds of atoms, a major issue is the scaling of the required computer effort with system size. In other electronic-structure techniques, including DFT, the locality of quantum coherence [6] suggests that it should generally be possible to achieve linear-scaling, or O(N ) operation, in which the computer effort is proportional to the number of atoms N . Very recently, a procedure has been suggested [7] for achieving at least partial linear scaling for QMC, based on the idea of "maximally localized Wannier functions" [8]. The purpose of this report is to propose and test a simpler alternative method, which appears to have important advantages.The O(N ) techniques that have been developed for tight binding (TB) [9], DFT [10, 11] and Hartree-Fock [12] calculations all depend ultimately on the fact that the density matrix ρ(r, r ′ ) associated with the single-electron orbitals decays to zero as |r − r ′ | → ∞, and the manner of this decay has been extensively studied ([13] and references therein). Briefly, the decay is algebraic for metals and exponential for insulators, with the decay rate increasing with band gap, so that there is more to be gained from O(N ) techniques for wide-gap insulators. Equivalently, the extended orbitals used in most conventional techniques can be linearly combined to form localized orbitals, which again decay exponentially in insulators. (For a review of early localized-orbital methods in quantum chemistry, see Ref. [14].) Orthogonal Wannier functions are one form of localized orbitals, but it has long been recognized that stronger localization can be achieved by going to 1

\mathcal{O}(N) methods in electronic structure calculations

“…A local, grid-based non-orthogonal basis also used in linear scaling methods are blip functions (or b-splines) [41], which can be shown to be a form of finite element. They have also recently been used for linear scaling Quantum Monte Carlo calculations [56]. The psinc functions mentioned above, which are periodic bandwidth-limited delta functions, are another local basis set [52]; interestingly, almost the same functions were derived in the context of optimal local basis sets [57].…”

Linear and nonlinear susceptibilities from diffusion quantum Monte Carlo: Application to periodic hydrogen chains

“…These results and approach can be expected to be used as reference for testing novel first-principles approaches for the evaluation of dielectrics properties. Although DMC calculations of susceptibilities are computational demanding, they can benefit, in terms of computational speed, from the use of order-N methods based on localized orthogonal [17,39] or nonorthogonal basis sets [40,41]. Therefore, it would be of great interest to determine the accuracy of the calculated dielectric properties when such approximations are used.…”

The quantum Monte Carlo method

“…The number of such orbitals contributing at a point in space is independent of N which leads to the required improvement in scaling. Two different groups using the CASINO code have shown that this approach is extremely effective, namely Williamson, Hood, Grossman, and Reboredo [15,67], and Alfè and Gillan [68]. An impartial evaluation of the two different methods [69] showed that the latter was superior, and this was the approach finally adopted for the production version of CASINO.…”