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The Journal of Chemical Physics

Shifted-contour auxiliary field Monte Carlo for ab initio electronic structure: Straddling the sign problem

Roi Baer, Martin Head‐Gordon, Daniel Neuhauser

Published Oct 15, 1998• 56 citations
Publisher

Abstract

The auxiliary field Monte Carlo (AFMC) technique has advantages over other ab initio quantum Monte Carlo methods for fermions, as it does not seem to require approximations for alleviating the sign problem and is directly applicable to excited states. Yet, the method is severely limited by a numerical instability, a numerical sign problem, prohibiting application to realistic electronic structure systems. Recently, the shifted contour auxiliary field method (SC-AFMC) was proposed for overcoming this instability. Here we develop a theory for the AFMC stabilization, explaining the success of SC-AFMC. We show that the auxiliary fields can be shifted into the complex plane in a manner that considerably stabilizes the Monte Carlo integration using the exact one-electron density. Practical stabilization can be achieved when an approximate Hartree–Fock density is used, proposing that an overwhelming part of the sign problem is removed by taking proper account of the Fermion mean-field contribution. The theory is demonstrated by application to H2.

DOI
10.1063/1.477300

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The auxiliary field Monte Carlo (AFMC) technique has advantages over other ab initio quantum Monte Carlo methods for fermions, as it does not seem to require approximations for alleviating the sign problem and is directly applicable to excited states. Yet, the method is severely limited by a numerical instability, a numerical sign problem, prohibiting application to realistic electronic structure systems. Recently, the shifted contour auxiliary field method (SC-AFMC) was proposed for overcoming this instability. Here we develop a theory for the AFMC stabilization, explaining the success of SC-AFMC. We show that the auxiliary fields can be shifted into the complex plane in a manner that considerably stabilizes the Monte Carlo integration using the exact one-electron density. Practical stabilization can be achieved when an approximate Hartree–Fock density is used, proposing that an overwhelming part of the sign problem is removed by taking proper account of the Fermion mean-field contribution. The theory is demonstrated by application to H2.
The shifted-contour auxiliary-field Monte Carlo (SCAFMC) approach has been recently developed by Rom, Charutz and Neuhauser as an extension of the auxiliary-field Monte Carlo (AFMC) method. AFMC replaces the difficult fully interacting electrons problem by an ensemble of simpler problems where the electrons interact with a fluctuating electric field but not with each other. SCAFMC is based on shifting the auxiliary-field contour of integration to pass through the (imaginary) stationary point, leading to numerical stability at long propagation times. The new approach converges to the full CI energy in electronic structure calculations (both ground and low-lying excited states). Here we expand the application of SCAFMC from atomic to molecular problems. First, we calculate ground-state energies of a highly correlated transition-metal system (Cr2) with a moderate (12 orbitals) active space size, and demonstrate that SCAFMC is able to extract the energies accurately. In addition, we use SCAFMC to calculate a C–C bond-stretch energy in ethane with complete active spaces of up to 28 orbitals.
Shifted contour (SC) auxiliary field Monte Carlo (AFMC) for electronic structure is described in considerable algorithmic detail. Several applications demonstrate the high accuracy achievable in molecular electronic structure, including excited states, using the techniques of correlated sampling and variational AFMC. SC-AFMC is also applicable to large Hubbard lattices of strongly correlated electrons. Applications are shown, up to lattices of size 16 ×16 with 202 electrons. Finally, we present a method for exact coarse graining of the AFMC integrand.
We explore a novel and straightforward solution to the sign problem that has plagued the Auxiliary-field Monte Carlo (AFMC) method applied to many-body systems for more than a decade. We present a solution to the sign problem that has plagued the Auxiliary-field Monte Carlo (AFMC) method for more than a decade and report a breakthrough where excellent agreement between AFMC and exact CI calculations for fully realistic nuclear applications is achieved. This result offers the capability, unmatched by other methods, to achieve exact solutions for large-scale quantum many-body systems.
The path integral formulation of quantum mechanical problems including fermions is often affected by a severe numerical sign problem. We show how such a sign problem can be alleviated by a judiciously chosen constant imaginary offset to the path integral. Such integration contour deformations introduce no additional computational cost to the Hamiltonian Monte Carlo algorithm, while its effective sample size is greatly increased. This makes otherwise unviable simulations efficient for a wide range of parameters. Applying our method to the Hubbard model, we find that the sign problem is significantly reduced. Furthermore, we prove that it vanishes completely for large chemical potentials, a regime where the sign problem is expected to be particularly severe without imaginary offsets. In addition to a numerical analysis of such optimized contour shifts, we analytically compute the shifts corresponding to the leading and next-to-leading order corrections to the action. We find that such simple approximations, free of significant computational cost, suffice in many cases. We present a simulation of C60{\mathrm{C}}_{60}C60​ fullerenes (buckyballs) that are successful over a wide parameter range.
Abstract This paper describes a different Q11C method for solving the Schrödinger equation, one unlike variational, diffusion, and Green’s function methods. It can be applied to many-body systems to produce solutions for ground states. For systems of fermions, antisymmetrization can be imposed to overcome the problem of node location, but the instability of the method is a serious shortcoming, so that the method as described is limited to treating relatively simple systems. Nevertheless, this auxiliary field quantum 1,fonte Carlo (AFMC) method was demonstrated to be successful for test cases reported in this paper, and in time has been improved to allow treatment of realistic systemsa. The method makes use of the imaginary-time propagation of an initial wavefunction toward the ground-state function. It requires use of a trial wavefunction which is the product of single-particle orbitals for bosons or an antisymmetrized product for fermions. A Hubbard-Stratonovich transformation of the propagator changes the many-body problem to a problem of a collection of single particles interacting with a fluctuating (auxiliary) field which replaces the many-body interactions. The authors developed an integral expression for the energy which, after discretization, could be evaluated by a Metropolis Monte Carlo integration forward in time. The algorithm was tested for one-dimensional systems of particles: 6 to 20 bosons with varied interactions, 2 to 6 neutron-proton pairs, others with repulsive potentials. Importance sampling was found to be critical and computational requirements to be large. For spin systems on a lattice the AFMC method was considered feasible, but for general multidimensional systems the method appeared to be limited by its computation requirements.
The auxiliary‐field quantum Monte Carlo (AFQMC) method provides a computational framework for solving the time‐independent Schrödinger equation in atoms, molecules, solids, and a variety of model systems. AFQMC has recently witnessed remarkable growth, especially as a tool for electronic structure computations in real materials. The method has demonstrated excellent accuracy across a variety of correlated electron systems. Taking the form of stochastic evolution in a manifold of nonorthogonal Slater determinants, the method resembles an ensemble of density‐functional theory (DFT) calculations in the presence of fluctuating external potentials. Its computational cost scales as a low‐power of system size, similar to the corresponding independent‐electron calculations. Highly efficient and intrinsically parallel, AFQMC is able to take full advantage of contemporary high‐performance computing platforms and numerical libraries. In this review, we provide a self‐contained introduction to the exact and constrained variants of AFQMC, with emphasis on its applications to the electronic structure of molecular systems. Representative results are presented, and theoretical foundations and implementation details of the method are discussed. This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods Structure and Mechanism > Computational Materials Science Computer and Information Science > Computer Algorithms and Programming