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arXiv:1506.02072 · Physical review. E

Using local operator fluctuations to identify wave function improvements

Kiel T. Williams, Lucas K. Wagner

Published Jul 25, 2016• 2 citations
arXivHTMLPDF

Abstract

A method is developed that allows analysis of quantum Monte Carlo simulations to identify errors in trial wave functions. The purpose of this method is to allow for the systematic improvement of variational wave functions by identifying degrees of freedom that are not well described by an initial trial state. We provide proof of concept implementations of this method by identifying the need for a Jastrow correlation factor and implementing a selected multideterminant wave function algorithm for small dimers that systematically decreases the variational energy. Selection of the two-particle excitations is done using the quantum Monte Carlo method within the presence of a Jastrow correlation factor and without the need to explicitly construct the determinants. We also show how this technique can be used to design compact wave functions for transition metal systems. This method may provide a route to analyze and systematically improve descriptions of complex quantum systems in a scalable way.

DOI
10.1103/physreve.94.013303

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A method is developed that allows analysis of quantum Monte Carlo simulations to identify errors in trial wave functions. The purpose of this method is to allow for the systematic improvement of variational wave functions by identifying degrees of freedom that are not well described by an initial trial state. We provide proof of concept implementations of this method by identifying the need for a Jastrow correlation factor and implementing a selected multideterminant wave function algorithm for small dimers that systematically decreases the variational energy. Selection of the two-particle excitations is done using the quantum Monte Carlo method within the presence of a Jastrow correlation factor and without the need to explicitly construct the determinants. We also show how this technique can be used to design compact wave functions for transition metal systems. This method may provide a route to analyze and systematically improve descriptions of complex quantum systems in a scalable way.
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A scheme of optimization of a trial wave function in the variational Monte Carlo calculation is developed. It provides a more precise description of the ground state for strongly correlated electronic systems with highly improved trial wave function. A general type of variational wave function constructed from a linear combination of Slater determinants multiplied by Jastrow correlation factors is proposed. They can explicitly contain low-energy fluctuations and long-range correlations. Using this optimization method, we examine the properties of these wave functions by applying them to a one-dimensional Hubbard model for which the exact solution is available for comparison. It is found that these wave functions prove to be substantial improvements compared to the conventional Jastrow-type wave functions.
Multideterminant wavefunctions, while having a long history in quantum chemistry, are increasingly being used in highly accurate quantum Monte Carlo calculations. Since the accuracy of QMC is ultimately limited by the quality of the trial wavefunction, multi-Slater determinants wavefunctions offer an attractive alternative to Slater-Jastrow and more sophisticated wavefunction ansatz for several reasons. They can be efficiently calculated, straightforwardly optimized, and systematically improved by increasing the number of included determinants. In spite of their potential, however, the convergence properties of multi-Slater determinant wavefunctions with respect to orbital set choice and excited determinant selection are poorly understood, which hinders the application of these wavefunctions to large systems and solids. In this paper, by performing QMC calculations on the equilibrium and stretched carbon dimer, we find that convergence of the recovered correlation energy with respect to number of determinants can depend quite strongly on basis set and determinant selection methods, especially where there is strong correlation. We demonstrate that properly chosen orbital sets and determinant selection techniques from quantum chemistry methods can dramatically reduce the required number of determinants (and thus the computational cost) to reach a given accuracy, which we argue shows clear need for an automatic QMC-only method for selecting determinants and generating optimal orbital sets.
Quantum Monte Carlo (QMC) methods have received considerable attention over past decades due to their great promise for providing a direct solution to the many-body Schrodinger equation in electronic systems. Thanks to their low scaling with the number of particles, QMC methods present a compelling competitive alternative for the accurate study of large molecular systems and solid state calculations. In spite of such promise, the method has not permeated the quantum chemistry community broadly, mainly because of the fixed-node error, which can be large and whose control is difficult. In this Perspective, we present a systematic application of large scale multideterminant expansions in QMC and report on its impressive performance with first row dimers and the 55 molecules of the G1 test set. We demonstrate the potential of this strategy for systematically reducing the fixed-node error in the wave function and for achieving chemical accuracy in energy predictions. When compared to traditional quantum chemistry methods like MP2, CCSD(T), and various DFT approximations, the QMC results show a marked improvement over all of them. In fact, only the explicitly correlated CCSD(T) method with a large basis set produces more accurate results. Further developments in trial wave functions and algorithmic improvements appear promising for rendering QMC as the benchmark standard in large electronic systems.
Quantum Monte Carlo (QMC) methods have received considerable attention over the last decades due to their great promise for providing a direct solution to the many-body Schrodinger equation in electronic systems.Thanks to their low scaling with number of particles, QMC methods present a compelling competitive alternative for the accurate study of large molecular systems and solid state calculations.In spite of such promise, the method has not permeated the quantum chemistry community broadly, mainly because of the fixed-node error, which can be large and whose control is difficult.In this Perspective, we present a systematic application of large scale multi-determinant expansions in QMC, and report on its impressive performance with first row dimers and the 55 molecules of the G1 test set.We demonstrate the potential of this strategy for systematically reducing the fixed-node error in the wave function and for achieving chemical accuracy in energy predictions.When compared to traditional quantum chemistry methods like MP2, CCSD(T), and various DFT approximations, the QMC results show a marked improvement over all of them.In fact, only the explicitly-correlated CCSD(T) method with a large basis set produces more accurate results.Further developments in trial wave functions and algorithmic improvements appear promising for rendering QMC as the benchmark standard in large electronic systems.
We present and motivate an efficient way to include orbital-dependent many-body correlations in trial wave functions of real-space quantum Monte Carlo methods for use in electronic structure calculations. We apply our new orbital-dependent backflow wave function to calculate ground state energies of the first row atoms using variational and diffusion Monte Carlo methods. The systematic overall gain of correlation energy with respect to single determinant Jastrow-Slater wave functions is competitive with the best single determinant trial wave functions currently available. The computational cost per Monte Carlo step is comparable to that of simple backflow calculations.