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.
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.
In this paper, we present the rigorous expression of the ground state energy and study the phase transition of the three-dimensional homogeneous electron gas by the eigenfunctional theory. The ground state energy is decided completely by the pair distribution function $ g(r)$, which is, by definition, strictly a positive function. But when the density decreases, the electronic correction effect becomes strong from weak, the previous theories basing on one-particle approximation, such as: RPA, Hubbard and STLS ,\cite{4,9} can't insure
g(r) positive always, which implies that they are becoming invalid and overestimate the ground state energy. The eigenfunctional theory has a significant improvement over them, under the linear approximation in solving the equation of the phase field, the
g(r) obtained by the eigenfunctional theory is always positive and can well satisfy the normalization integral (
n0∫d3r[g(r)−1]=−1) which is one of the important features of
g(r). After obtaining
g(r) and calculating the ground state energy of the electron gas both in paramagnetic and ferromagnetic by
g(r), we observe a continuous phase transition from paramagnetic to the ferromagnetic occurring at
rs=19.9±0.8. This can be indirectly supported by the observation of a ferromagnetic state in doped hexaboride
(Ca1−xLaxBb),\cite{42} and is close to the result of G.Ortiz et.al (
rs=20±5).\cite{12}
The correlation energy of an inhomogeneous electron gas is ruled by the density correlation function
L. It obeys a Bethe-Salpeter equation (BSE), whose kernel is the functional derivative of the particle self-energy. It is solved on equal footing for two-particle and single-particle Green's functions within Hedin's GW approximation. According to the Hartree and exchange-correlation contributions to the kernel, the correlation expression decays into random-phase-approximation (RPA) -like and screened-exchange-like parts. The actual treatment of the screened potential
W in the BSE for
L leads to different second-order screened exchange corrections (SOSEX). Including correlation in the kernel even yields approximations beyond SOSEX. The progress beyond RPA is illustrated for spinless and spin-polarized homogeneous electron gases, indicating that the screening treatment is substantial.
Beyond the usual many body perturbation theory, with the eigenfunctional theory, we directly calculate the correlation function produced by the Coulomb interaction of electrons in the equation of one-electron Green function, and give the general expression of the non-local effective interaction potential in a Hartree-type potential, which is absent in all previous many body perturbation theories. This effective interaction potential originates from the quantum many body effect of the system, and it cannot be obtained directly by the usual perturbation expression approach. Moreover, using theoretical models, we demonstrate that this effective interaction potential can be used to characterize the electron correlation strength of the system.
We analyze the ground-state energy and correlation energy of the Heisenberg model as a function of spin, both in the ferromagnetic and the antiferromagnetic case, and in one, two and three dimensions. First, we present a comparative analysis of known expressions for the ground-state energy
E0(S) of homogeneous Heisenberg models. In the one-dimensional antiferromagnetic case we propose an improved expression for
E0(S), which takes into account Bethe-ansatz data for
S=1/2. Next, we consider inhomogeneous Heisenberg models (e.g., exposed to spatially varying external fields). We prove a Hohenberg-Kohn-like theorem stating that in this case the ground-state energy is a functional of the spin distribution and that this distribution encapsulates the entire physics of the system, regardless of the external fields. Building on this theorem, we then propose a local-density-type approximation that allows us to utilize the results obtained for homogeneous systems in inhomogeneous situations too. We conjecture a scaling law for the dependence of the correlation functional on dimensionality, which is well satisfied by existing numerical data. Finally, we investigate the importance of the spin-correlation energy by comparing results obtained with the proposed correlation functional to ones from an uncorrelated mean-field calculation, taking as our example a linear spin-density wave state.
We introduce a scheme through which the ground-state electronic density n and total energy Ey[n] of inhomogeneous interacting systems can be determined in a self-consistent (SC) way. It is based on a SC perturbation expansion for the single-particle Green function G. Our employed expansion is argued to be valid under all conditions provided that the ground-state charge density of the interacting system is pure-state non-interacting v representable. The formalism discussed in this paper has been applied to a quasi-one-dimensional periodic system. It has even been possible to carry out SC calculations with the dynamically screened exchange diagram (the so-called G W diagram, where W is the dynamically screened electron4ectron interaction function) for the selfenergy operator C. A coherent mode of interpretation of a number of our results entails that the overwhelming success of the G W framework in relatively accurately reproducing the experimental energy gaps of a number of semiconductors and insulators may be a direct consequence of non-selfconsistency effects; whereas a non-self-consistent calculation in our tests increases the value of the local-density approximation (LDA) gap in the range of 40-65% of the ‘experimental’ gaps, self-consistency effects reduce this range to 1 4 1 7 % . The same line of argument leads us to conclude that, in our model, 2635% of the LDA error in the gap energies is due to the absence of a well known discontinuity in the LDA exchange-correlation potential, and the remaining part is due to the Kohn-Sham eigenvalues not being identifiable with one-particle excitation energies. A natural conclusion that may be drawn from these observations is that a correct description of excitation energies (e.g. of the gap energy) cannot be achieved without including, at least approximately, the vertex part of C, which is entirely absent in the GW scheme. 5 1 . INTRODUCTION The ground-state (GS) density-functional theory (DFT) due to Hohenberg and Kohn (1964) provides a rigorous foundation for determining properties of systems in their ground states (for reviews see Lundqvist and March (1983), Jones and Gunnarsson (1989) and Dreizler and Gross (1990)). The central position in this theory is taken by the total-energy functional EY[n] which, owing to its unique absolute minimum a t the GS electronic density n, can be used to obtain not only the GS energy but also the GS charge density. tPresent address: Max-Planck-Institut fur Festkorperforschung, HeisenbergstraRe 1, 70569 Stuttgart, Federal Republic of Germany. 0141-8637/97 $12 .00