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Journal of Computer Chemistry Japan

Application of Importance Sampling to Basis Quantum Monte Carlo Method

Toru YAGI, Umpei NAGASHIMA

Published Jan 1, 2012• 0 citations
Publisher

Abstract

Basis Quantum Monte Carlo(BQMC)法でImportance Samplingを行うための定式化を行い，1次元の調和振動子内における2個のフェルミ粒子の系に対する基底状態を求めた．通常のBQMC計算で得られる粒子の平均分布から波動関数やQuantum Force，及び局所エネルギーを求め，これらを用いてImportance Sampling付きのBQMC計算を行った．Importance Samplingを行うことにより，分散が小さくより高い精度でエネルギーを求めることができた．

DOI
10.2477/jccj.2012-0005

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Basis Quantum Monte Carlo(BQMC)法でImportance Samplingを行うための定式化を行い，1次元の調和振動子内における2個のフェルミ粒子の系に対する基底状態を求めた．通常のBQMC計算で得られる粒子の平均分布から波動関数やQuantum Force，及び局所エネルギーを求め，これらを用いてImportance Sampling付きのBQMC計算を行った．Importance Samplingを行うことにより，分散が小さくより高い精度でエネルギーを求めることができた．
Basis Quantum Monte Carlo(BQMC)法を用いて1次元と3次元の調和振動子内における2個のフェルミ粒子の系の基底状態の計算を行った。従来より広く利用されている拡散量子Monte Carlo法では、用いる節面の情報を事前に与えるためのガイド関数を利用するが、BQMC法ではこのガイド関数を用いることなしに、反対称性の問題を持つ系の計算を行うことができた。また、BQMC法において、いくつか異なるエネルギー計算手法を用いて平均エネルギーを求め、BQMC法による基底状態のエネルギー計算精度を比較した。
Schrödinger 方程式の虚時間発展に対する行列積解法についての検討を行なった。理論的な基礎とした基底量子モンテカルロ（Basis Quanutum Monte Carlo, BQMC）法では，粒子の拡散を表す遷移確率と生成消滅に対応する分岐項の行列が具体的な形で与えられている。このBQMC 法に対して，モンテカルロ計算を行わず，行列の積を繰り返すという方法を用い，水素原子の基底状態エネルギーと波動関数を求めた。計算に用いるクーロン項のパラメータを最適化することで，高い精度でエネルギーを求めることができた。また，本手法で求めた波動関数を解析解と比較し，良い結果が得られることを確認した。特に，原子核上で尖点を持つ構造が得られた。
The standard method of quantum Monte Carlo for the solution of the Schrödinger equation in configuration space can be described quite generally as devising a random walk that generates-at least asymptotically-populations of random walkers whose probability density is proportional to the wave function of the system being studied. While, in principle, the energy eigenvalue of the Hamiltonian can be calculated with high accuracy, estimators of operators that do not commute the Hamiltonian cannot. Bilinear quantum Monte Carlo (BQMC) is an alternative in which the square of the wave function is sampled in a somewhat indirect way. More specifically, one uses a pair of walkers at positions x and y and introduces stochastic dynamics to sample phi(i)(x)t(x,y)phi(j)(y), where phi(i)(x) and phi(j)(y) are eigenfunctions of (possibly different) Hamiltonians, and t(x,y) is a kernel that correlates positions x and y. Using different Hamiltonians permits the accurate computation of small energy differences. We review the conceptual basis of BQMC, discuss qualitatively and analytically the problem of the fluctuations in the branching, and present partial solutions to that problem. Finally we exhibit numerical results for some model systems including harmonic oscillators and the hydrogen and helium atoms. Further research will be necessary to make this a practical and generally applicable scheme.
We examine Gaussian-basis Monte Carlo method (GBMC) introduced by Corney and Drummond. This method is based on an expansion of the density-matrix operator rho by means of the coherent Gaussian-type operator basis Lambda and does not suffer from the minus sign problem. The original method, however, often fails in reproducing the true ground state and causes systematic errors of calculated physical quantities because the samples are often trapped in some metastable or symmetry broken states. To overcome this difficulty, we combine the quantum-number projection scheme proposed by Assaad, Werner, Corboz, Gull and Troyer in conjunction with the importance sampling of the original GBMC method. This improvement allows us to carry out the importance sampling in the quantum-number-projected phase-space. Some comparisons with the previous quantum-number projection scheme indicate that, in our method, the convergence with the ground state is accelerated, which makes it possible to extend the applicability and widen the range of tractable parameters in the GBMC method. The present scheme offers an efficient practical way of computation for strongly correlated electron systems beyond the range of system sizes, interaction strengths and lattice structures tractable by other computational methods such as the quantum Monte Carlo method.