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Journal of Computer Chemistry Japan

Basis Quantum Monte Carlo Calculation of 2-Fermion System in a 1 or 3-Dimensional Harmonic Potential.

Toru YAGI, Umpei Nagashima

Published Jan 1, 2009• 1 citation
Publisher

Abstract

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。

DOI
10.2477/jccj.h2112

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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。
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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.
The Quantum Transfer Matrix and Its Application to Quantum Spin Chains (K Kubo) Transfer Matrices in Quantum Many-Body Systems (T Koma) Monte Carlo Calculations of Elementary Excitation (M Takahashi) The Decoupled Cell Method of Quantum Monte Carlo Calculation (S Homma) Decoupled Cell Monte Carlo Study of the Critical Properties of the Spin-1/2 Ferromagnetic Heisenberg Model in Three Dimensions (R J Creswick & C J Sisson) Variational Monte Carlo Studies of Correlated Electrons (H Shiba) Quantum Monte Carlo Simulation of Multiband Fermion Systems and Its Application to Superconductivity (K Kuroki & H Aoki) Quantum Monte Carlo in the Infinite Dimensional Limit (M Jarrell et al) Aspects of the Sign Problem (J H Samson) Ground-State Projection Using Auxiliary Fields (S Fahy) Fermion Simulations of Correlated Systems (M Imada) Dirty Bosons in 2D: Phases and Phase Transitions (N Trivedi & M Makivi'c). (Part Contents).
Two distinct types of Quantum Monte Carlo (QMC) calculations are applied to electronic structure problems such as calculating potential energy curves and producing benchmark values for reaction barriers. First, Variational and Diffusion Monte Carlo (VMC and DMC) methods using a trial wavefunction subject to the fixed node approximation were tested using the CASINO code.[1] Next, Full Configuration Interaction Quantum Monte Carlo (FCIQMC), along with its initiator extension (i-FCIQMC) were tested using the NECI code.[2] FCIQMC seeks the FCI energy for a specific basis set. At a reduced cost, the efficient i-FCIQMC method can be applied to systems in which the standard FCIQMC approach proves to be too costly. Since all of these methods are statistical approaches, uncertainties (error-bars) are introduced for each calculated energy. This study tests the performance of the methods relative to traditional quantum chemistry for some benchmark systems. References: [1] R. J. Needs et al., J. Phys.: Condensed Matter 22, 023201 (2010). [2] G. H. Booth et al., J. Chem. Phys. 131, 054106 (2009).
The chapter describes the principles of how Quantum Monte Carlo (QMC) uses random numbers to compute properties of a non-relativistic quantum mechanical system. In a many-body system, this involves computation of multidimensional integrals, which can be very effectively calculated using Monte Carlo methods. The spin of particles will be added to this framework, along with identical particles as symmetric or antisymmetric wave functions. An introduction to orbitals follows, and a discussion of the characteristics of boson and fermion ground states.