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Physical Review Letters

Semiclassical Ground-State Energies of Many-Electron Systems

Brian R. Landry, Adam Wasserman, Eric J. Heller

Published Aug 4, 2009• 4 citations
Publisher

Abstract

A new semiclassical method is proposed to obtain accurate ground-state energies for many-electron systems. The method borrows its semiclassical character from Thomas-Fermi (TF) theory, but improves upon it by including exchange-correlation effects, at least approximately. We illustrate our method (correlated TF method) on simple models of 1D-interacting electrons, showing that it yields dramatic improvements over TF theory, particularly in the strongly correlated regime.

DOI
10.1103/physrevlett.103.066401

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A new semiclassical method is proposed to obtain accurate ground-state energies for many-electron systems. The method borrows its semiclassical character from Thomas-Fermi (TF) theory, but improves upon it by including exchange-correlation effects, at least approximately. We illustrate our method (correlated TF method) on simple models of 1D-interacting electrons, showing that it yields dramatic improvements over TF theory, particularly in the strongly correlated regime.
With the eigenfunctional theory, we study a general interacting electron system, and give a rigorous expression of its ground state energy which is composed of two parts, one part is contributed by the non-interacting electrons, and another one is represented by the correlation functions that are controlled by the electron correlation. Moreover, according to the rigorous expression of the ground state energy, an effective method beyond the local density approximation of the density functional theory is proposed.
With the eigenfunctional theory, we study a general interacting electron system, and give a rigorous expression of its ground state energy, which is composed of two parts: one part is contributed by the non-interacting electrons, and the other one is represented by the correlation functions that are controlled by the electron correlation. Moreover, according to the rigorous expression of the ground state energy, an effective scheme beyond the local density approximation of the density functional theory is proposed. As a simple example for a spin-1/2 XXZ chain, under the linear approximation in solving the equation of the phase field, the ground state energy obtained by the present scheme is quite close to that of the Bethe ansatz.
We report preliminary results of convergence properties for nonperturbative resolvent approximations to Anderson-like models of magnetic ions in metals. Our study is initially focused on the spin- 1/2 Anderson model for magnetic impurities, but the methods studied can include multiplet and crystal-field effects which are needed for more accurate descriptions of real systems. We will compare the nonperturbative Lanczos method (tridiagonalization) and similar truncation schemes to exact ground-state energies for the impurity model and assess the efficacy of these nonperturbative approaches to understanding the Anderson lattice, heavy fermions, and other strongly interacting electronic systems.
A new variational method is developed to calculate the ground state energy of Fermi systems with strong short-range correlations. A trial wave function of Gutzwiller's type contains additional variational parameters corresponding to configurations of pairs of nearest-neighbor sites. To evaluate the ground state energy, generalized Kikuchi's pseudo-ensemble method is used. The Hubbard model at half band-filling is investigated. The ground state energy of the paramagnetic phase is calculated for a chain, square and simple cubic lattices. It is shown that the short-range order lowers drastically the ground state energy of the Hubbard model at intermediate interaction strength. The paramagnetic phase of the Kondo-Hubbard model (S=1/2S=1/2S=1/2 and S=5/2S=5/2S=5/2) at half band-filling is investigated. The ground state energy, correlation functions and effective mass are calculated for chain, square and simple cubic lattices. A quantum phase transition was found for simple cubic lattice.
Semiclassical theories such as the Thomas-Fermi and Wigner-Kirkwood methods give a good description of the smooth average part of the total energy of a Fermi gas in some external potential when the chemical potential is varied. However, in systems with a fixed number of particles NNN, these methods overbind the actual average of the quantum energy as NNN is varied. We describe a theory that accounts for this effect. Numerical illustrations are discussed for fermions trapped in a harmonic oscillator potential and in a hard-wall cavity, and for self-consistent calculations of atomic nuclei. In the latter case, the influence of deformations on the average behavior of the energy is also considered.
In the application of the extended Thomas-Fermi (ETF) method to a many-fermion system, only those terms in the ft expansion will be included that give finite contribution to the energy. The ETF model yields the smooth part of the level density and other quantities. It will be derived from the E. Wigner-J. G. Kirkwood. It discusses the ETF density variational method and its use for obtaining in a self-consistent way the average parts of the energy and other observables of a many-fermion system in its ground state. Finally, a numerical smoothing of the level density may also be done, by replacing each delta function spike by an appropriate smoothing function, like a Gaussian or a Lorentzian. Since the TF approximation is obtained by replacing the sum over the quantum states by the integral over classical phase space, it is not surprising that it amounts to taking the inverse Laplace transform of the classical canonical partition function.