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arXiv:1512.05455 · The Journal of Chemical Physics

Sign Learning Kink-based (SiLK) Quantum Monte Carlo for molecular systems

Xiaoyao Ma, Randall W. Hall, Frank Löffler, Karol Kowalski, Kiran Bhaskaran-Nair, Mark Jarrell, Juana Moreno

Published Jan 5, 2016• 0 citations
arXivHTMLPDF

Abstract

The Sign Learning Kink (SiLK) based Quantum Monte Carlo (QMC) method is used to calculate the ab initio ground state energies for multiple geometries of the H2O, N2, and F2 molecules. The method is based on Feynman's path integral formulation of quantum mechanics and has two stages. The first stage is called the learning stage and reduces the well-known QMC minus sign problem by optimizing the linear combinations of Slater determinants which are used in the second stage, a conventional QMC simulation. The method is tested using different vector spaces and compared to the results of other quantum chemical methods and to exact diagonalization. Our findings demonstrate that the SiLK method is accurate and reduces or eliminates the minus sign problem.

DOI
10.1063/1.4939145

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The Sign Learning Kink (SiLK) based Quantum Monte Carlo (QMC) method is used to calculate the ab initio ground state energies for multiple geometries of the H2O, N2, and F2 molecules. The method is based on Feynman's path integral formulation of quantum mechanics and has two stages. The first stage is called the learning stage and reduces the well-known QMC minus sign problem by optimizing the linear combinations of Slater determinants which are used in the second stage, a conventional QMC simulation. The method is tested using different vector spaces and compared to the results of other quantum chemical methods and to exact diagonalization. Our findings demonstrate that the SiLK method is accurate and reduces or eliminates the minus sign problem.
A Sign Learning Kink (SiLK) based Quantum Monte Carlo (QMC) method is used to calculate the ground state energies for H2O, N2 and F2 molecules. This method has two stages. The first (learning stage) reduces the minus sign problem by optimizing the states which are used in the second (QMC stage). I test the method in Single, Double excitations (SD), Single, Double, and Triple excitations (SDT), and Full Configuration Interaction (FCI) vector spaces. I also perform exact diagonalization in those vector spaces as a benchmark. In each vector space and for each molecule, I perform SiLK QMC for different bond lengths demonstrating that the SiLK QMC is applicable to many systems.
A kink-based expression for the canonical partition function is developed using Feynman’s path integral formulation of quantum mechanics and a discrete basis set. The approach is exact for a complete set of states. The method is tested on the 3×3 Hubbard model and overcomes the sign problem seen in traditional path integral studies of fermion systems. Kinks correspond to transitions between different N-electron states, much in the same manner as occurs in configuration interaction calculations in standard ab initio methods. The different N-electron states are updated, based on which states occur frequently during a Monte Carlo simulation, giving better estimates of the true eigenstates of the Hamiltonian.
A Green's function quantum Monte Carlo (GF QMC) method is described which renders possible the determination of the exact ground state energy of any π system in the computational framework of effective π Hamiltonians. An instruction to evaluate the growth estimator in the simultaneous presence of positive and negative phases is suggested. Numerical results for five annulene molecules with 4–12 atomic centres as well as 10 further π molecules with different topology (i.e., monocycles with methylene groups, bi- and polycycles) are given. Negative phases occur in 13 of the 15 π networks considered. The statistical QMC results are compared with data derived by conventional (many-body) diagonalization techniques. The interaction-free Hückel Hamiltonian has been adopted as most simple setup to test the applicability of the Monte Carlo approach. Many-body results have been derived for the Hubbard and Pariser-Parr-Pople variants of the π-electron Hamiltonian. It is demonstrated that GF QMC results can be derived whose accuracy does not depend on the presence of negative signs in the numerical simulation.
The shell model provides a powerful framework for nuclear structure calculations. The nucleons beyond an inert magic core are confined in a valence shell and interact through an effective two-body potential generally determined from the G-matrix method. However, the applicability of the shell model remains limited by the exponential growth of the many-body space with the number of valence nucleons and with the size of the one-body space. The quantum Monte Carlo (QMC) methods may thus be considered as a potentially attractive alternative to the direct diagonalization of the Hamiltonian matrix. The many-body problem is indeed reduced to a set of stochastic one-body problems describing independent particles in a fluctuating external field. Up to date, the QMC approaches of the shell model lead to the ground-state and the thermodynamic properties, but a detailed spectroscopy of nuclei is not reachable. Furthermore, they suffer from a common pathology of fermionic QMC methods, namely the sign/phase problem, which strongly contaminates the convergence of the calculation. In this context, we will present a new QMC method, providing the "yrast spectroscopy" of nuclei, and in which the sign/phase problem is managed through the constrained path approximation that is widely used in the ab initio nuclear calculations and in condensed matter physics.
High Resolution Image Download MS PowerPoint Slide Density matrix quantum Monte Carlo (DMQMC) is a recently developed method for stochastically sampling the N -particle thermal density matrix to obtain exact-on-average energies for model and ab initio systems. We report a systematic numerical study of the sign problem in DMQMC based on simulations of atomic and molecular systems. In DMQMC, the density matrix is written in an outer product basis of Slater determinants. In principle, this means that DMQMC needs to sample a space that scales in the system size, N, as O[(exp(N)) 2 ]. In practice, removing the sign problem requires a total walker population that exceeds a system-dependent critical walker population ( N c ), imposing limitations on both storage and compute time. We establish that N c for DMQMC is the square of N c for FCIQMC. By contrast, the minimum N c in the interaction picture modification of DMQMC (IP-DMQMC) is only linearly related to the N c for FCIQMC. We find that this difference originates from the difference in propagation of IP-DMQMC versus canonical DMQMC: the former is asymmetric, whereas the latter is symmetric. When an asymmetric mode of propagation is used in DMQMC, there is a much greater stochastic error and is thus prohibitively expensive for DMQMC without the interaction picture adaptation. Finally, we find that the equivalence between IP-DMQMC and FCIQMC seems to extend to the initiator approximation, which is often required to study larger systems with large basis sets. This suggests that IP-DMQMC offers a way to ameliorate the cost of moving between a Slater determinant space and an outer product basis.