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Physical review. B, Condensed matter

Solution of an inverse problem: Maximum-overlap Jastrow function of the Lennard-Jones Bose fluid

G. L. Masserini, L. Reatto

Published Nov 1, 1984• 15 citations
Publisher

Abstract

We determine for the Lennard-Jones Bose fluid the Jastrow function that has the maximum overlap with the exact ground state as determined by the Green's-function Monte Carlo method. We solve the resulting inverse problem with an iterative predictor-corrector method using integral equations and Monte Carlo methods, respectively. The maximum-overlap Jastrow function gives also a good value for the energy.

DOI
10.1103/physrevb.30.5367

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We determine for the Lennard-Jones Bose fluid the Jastrow function that has the maximum overlap with the exact ground state as determined by the Green's-function Monte Carlo method. We solve the resulting inverse problem with an iterative predictor-corrector method using integral equations and Monte Carlo methods, respectively. The maximum-overlap Jastrow function gives also a good value for the energy.
The problem of how to extract information on the structure of the exact ground-state wave function of a Bose fluid from the results of Green's-function Monte Carlo computations is considered. Use of the criterion of maximum overlap of the exact ground state with an extended Jastrow function leads to certain equalities for correlation functions. By using methods borrowed from theories of classical fluids various schemes are proposed that permit the determination of the "best" Jastrow function. The case of Fermi statistics is also considered.
A more powerful version of a numerical method published before is given which permits the integration of the Schr\"odinger equation for a large ensemble of particles. A new method for obtaining quantum expectation values is outlined. Numerical results are tabulated for the energy of the ground state of a system of 32 bosons interacting by Lennard-Jones potentials and with periodic boundary conditions. This is a commonly used model of liquid He4{\mathrm{He}}^{4}He4. The present results are substantially lower and closer to experiment than variational estimates based on a Jastrow-type wave function.
Imaginary-time correlation functions calculated by quantum Monte Carlo (QMC) are analyzed using the maximum entropy method (MaxEnt) to determine the ground-state energy and spectral overlap function. In contrast to earlier applications of MaxEnt, the data is obtained from importanced-sampled zero-temperature quantum Monte Carlo simulations. The analysis includes two steps. First, that spectral overlap function and ground state energy which maximizes the entropy and agrees with the QMC correlation functions is obtained. Then the errors in the energy are evaluated by averaging over all the possible images (average MaxEnt method), the multidimensional integrals being computed using the Metropolis algorithm. The central feature of this approach is that all the information present in the correlation functions is used in the only way consistent with fundamental probabilistic hypotheses. This allows us to fully exploit the information contained in the correlation functions at small imaginary times, thus avoiding large statistical fluctuations associated with large imaginary times. In addition, the computed errors include both the statistical errors and systematic extrapolation errors. The method is illustrated with a harmonic oscillator and the four-electron LiH molecule.