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arXiv:cond-mat/0204276 · Journal of Physics A Mathematical and General

Variational perturbation approach to the Coulomb electron gas

Sang Koo You, Noboru Fukushima

Published Sep 2, 2003• 5 citations
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Abstract

The efficiency of the variational perturbation theory (You et al 2000 Phys. Rev. C 62 045503) formulated recently for many-particle systems is examined by calculating the ground state correlation energy of the 3D electron gas with the Coulomb interaction. The perturbation beyond a variational result can be carried out systematically by the modified Wick's theorem which defines a contraction rule about the renormalized perturbation. Utilizing the theorem, variational ring diagrams of the electron gas are summed. As a result, the correlation energy is found to be much closer to the result of the Green function Monte Carlo calculation than that of the conventional ring approximation.

DOI
10.1088/0305-4470/36/37/304

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The efficiency of the variational perturbation theory (You et al 2000 Phys. Rev. C 62 045503) formulated recently for many-particle systems is examined by calculating the ground state correlation energy of the 3D electron gas with the Coulomb interaction. The perturbation beyond a variational result can be carried out systematically by the modified Wick's theorem which defines a contraction rule about the renormalized perturbation. Utilizing the theorem, variational ring diagrams of the electron gas are summed. As a result, the correlation energy is found to be much closer to the result of the Green function Monte Carlo calculation than that of the conventional ring approximation.
A variational perturbation theory based on the functional integral approach is formulated for many-particle systems. Using the variational action obtained through Jensen-Peierls' inequality, a perturbative expansion scheme for the thermodynamic potential is established. A modified Wick's theorem is obtained for the variational perturbation expansions. This theorem allows one to carry out systematic calculations of higher order terms without worrying about the double counting problem. A model numerical calculation was carried out on a nucleon gas system interacting through the Yukawa-type potential to test the efficiency of the present method.
With a superhigh-efficiency numerical algorithm, we are able to self-consistently calculate Green's function in the renormalized-ring-diagram approximation for a two-dimensional electron system with long-range Coulomb interactions. The obtained ground-state energy is found to be in excellent agreement with that of the Monte Carlo simulation. The numerical results of the self-energy, the effective mass, the distribution function, and the renormalization factor of Green's function for the coupling constants in the range $0\ensuremath{\leqslant}{r}_{s}\ensuremath{\leqslant}30$ are also presented.
We report variational and diffusion quantum Monte Carlo ground-state energies of the three-dimensional electron gas using a model periodic Coulomb interaction and backflow corrections for N = 54, 102, 178, and 226 electrons. We remove finite-size effects by extrapolation and we find lower energies than previously reported. Using the Hellman-Feynman operator sampling method introduced in Gaudoin and Pitarke (2007 Phys. Rev. Lett. 99 126406), we compute accurately, within the fixed-node approximation, the separate kinetic and interaction contributions to the total ground-state energy. The difference between the interaction energies obtained from the original Slater-determinant nodes and the backflow-displaced nodes is found to be considerably larger than the difference between the corresponding kinetic energies.
In order to properly understand the utility of many-body perturbation theory as applied to finite systems, we use the configuration interaction approach to compute the electronic self-energy. The validity of the commonly used GWGWGW approximation from many-body perturbation theory is tested by comparing the self-energy explicitly computed according to the diagrammatic expansions to that from the inversion of the Dyson equation. It is constructed as a functional of the interacting Green function GGG and screened Coulomb interaction WWW. WWW is explicitly computed through the diagonalization of the many-body Hamiltonian and, thus, takes polarization effects into account beyond the random phase approximation. The Na9+\mathrm{Na}_{9}{}^{+}Na9​+ cluster and the dissociation of the C2{\mathrm{C}}_{2}C2​ molecule are discussed as examples. We find that the GWGWGW approximation yields accurate results for weakly correlated systems with a predominantly single-determinant ground state such as Na9+\mathrm{Na}_{9}{}^{+}Na9​+ clusters. The C2{\mathrm{C}}_{2}C2​ molecule is a pathological system with multideterminantal ground state where the GWGWGW approach ceases to be valid.
We study the accuracy of analytical wave-function-based many-body methods derived by energy minimization of a Jastrow-Feenberg ansatz for electrons (the ``Fermi hypernetted chain/Euler Lagrange'' approach). Approximations to avoid the complexity of the fermion problem are chosen to parallel successful boson theories and to be computationally efficient. For the three-dimensional homogeneous electron gas, we calculate the correlation energy, the pair distribution function, and the static structure function in comparison with simulation results. We also present a variant of theory which is interpreted as an approximate, self-consistent sum of ladder and ring diagrams of perturbation theory. The theory performs particularly well in the highly dilute density regime.
We recently proposed the correlation matrix renormalization (CMR) theory to treat the electronic correlation effects [Phys. Rev. B 2014, 89, 045131 and Sci. Rep. 2015, 5, 13478] in ground state total energy calculations of molecular systems using the Gutzwiller variational wave function (GWF). By adopting a number of approximations, the computational effort of the CMR can be reduced to a level similar to Hartree-Fock calculations. This paper reports our recent progress in minimizing the error originating from some of these approximations. We introduce a novel sum-rule correction to obtain a more accurate description of the intersite electron correlation effects in total energy calculations. Benchmark calculations are performed on a set of molecules to show the reasonable accuracy of the method.