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The Journal of Chemical Physics

Classical path approximation for one-electron Green’s function in a magnetic field and the density functional model

Shiwei Li, Jerome K. Percus

Published Sep 15, 1990• 5 citations
Publisher

Abstract

This paper is an extension of our previous work on density functional models of electrons in magnetic fields. A full ‘‘classical path’’ (CP) one-body Green’s function is used in the calculation of the kinetic functional and the one particle density. This treatment leads to a more precise expression of the density functional theory in the strong B case, in which magnetic fields are much stronger than the external scalar potential. In the semiclassical limit t→0, the CP Green’s function is checked by an exact solvable system, interactionless electrons in a central harmonic potential and a constant magnetic field. It turns out that the Green’s function is correct through the order O(t2). Furthermore, in the high B limit (but still t→0), the leading terms in B of the CP Green’s function are the same as those of the exact Green’s function in the entire t-expansion series.

DOI
10.1063/1.458759

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This paper is an extension of our previous work on density functional models of electrons in magnetic fields. A full ‘‘classical path’’ (CP) one-body Green’s function is used in the calculation of the kinetic functional and the one particle density. This treatment leads to a more precise expression of the density functional theory in the strong B case, in which magnetic fields are much stronger than the external scalar potential. In the semiclassical limit t→0, the CP Green’s function is checked by an exact solvable system, interactionless electrons in a central harmonic potential and a constant magnetic field. It turns out that the Green’s function is correct through the order O(t2). Furthermore, in the high B limit (but still t→0), the leading terms in B of the CP Green’s function are the same as those of the exact Green’s function in the entire t-expansion series.
A density-functional model of atoms in strong magnetic fields is proposed along lines parallel to Hohenberg-Kohn-Sham density-functional theory. We derive a Thomas-Fermi-Weiz\"acker-like (TFW) functional by using the classical-path approximation to the relevant one-particle Green's function. We also address the correlation effect between the effective potential and the magnetic field. The TFW-like model provides a simple way of calculating ground-state energies and charge densities of atoms in magnetic fields, and can be systematically improved.
Several widely used methods for the calculation of band structures and photo emission spectra, such as the GW approximation, rely on many-body perturbation theory. They can be obtained by iterating a set of functional differential equations (DEs) relating the one-particle Green's function (GF) to its functional derivative with respect to an external perturbing potential. In this work, we apply a linear response expansion in order to obtain insights into various approximations for GF calculations. The expansion leads to an effective screening while keeping the effects of the interaction to all orders. In order to study various aspects of the resulting equations, we discretize them and retain only one point in space, spin and time for all variables. Within this one-point model we obtain an explicit solution for the GF, which allows us to explore the structure of the general family of solutions and to determine the specific solution that corresponds to the physical one. Moreover, we analyze the performances of established approaches like GW over the whole range of interaction strength, and we explore alternative approximations. Finally, we link certain approximations for the exact solution to the corresponding manipulations of the DE which produces them. This link is crucial in view of a generalization of our findings to the real (multidimensional functional) case where only the DE is known.
A new approach for the calculation of ground states of many-electron systems is proposed via an integral formulation of the Hohenberg-Kohn-Sham density-functional theory. Only equations for the total electron density are involved; orbitals are not employed. Exchange and correlation effects are incorporated. In place of the set of single-electron equations, the total electron density is explicitly expressed in terms of the Kohn-Sham effective local potential through multidimensional integrations. The development is based on the first-order density matrix as obtained from the one-body Green's function in polygonal and Fourier path-integral representations. This might open up the possibility of ab initio calculations for molecules with very many electrons. It also provides explicit solutions to two long-standing problems: electron kinetic energy and momentum density as functionals of the total electron density. The formulation can also be used in calculations for general fermions.
The asymptotic expression for the one-particle Green's function in the strong coupling limit of the double degenerate Hubbard model containing multiplet terms is obtained for arbitrary band filling within the irreducible Green's function approach. An explicit analytic expression is obtained for the mean-field Greens's function, conserving the sum rules for the first few spectral moments. Stability of homogeneous phases is analised in terms of a single flip spin and orbital excitations. It is shown that degeneracy has an additional enhancing effect on correlation driven interband transfer of spectral weight. Discovery of superconductivity in fullerides and of materials with colossal magnetoresistance during recent years have increased the interest for electron systems with physical properties being essentially dependent on orbital degeneracy and strong correlation effects. In many cases the orbitally degenerate Hubbard model (DHM) incorporates the key features of such materials [1], although there are even fewer exact results established for DHM than for the nondegenerate Hubbard model (NHM). In view of complexity of correlated systems, corrections to the atomic limit of one-particle excitation spectrum and spectral weights of correlated bands found by Harris and Lange for NHM [2] by means of perturbation theory based on canonical transformations provide an important exact result which serves as a test for different approximations. A similar derivation for DHM has not been carried out yet. In the present paper the Green's function for the strong coupling limit of the DHM is obtained by means of irreducible Green's function approach [3] - [6] which is closely related to canonical perturbation theory because of its spectral moment conserving construction. As was explicitly proved in [7] for the NHM, the first couples of spectral moments of correlated bands obtained within mean-field decoupling are identical to those of canonical perturbation theory to first order in / tU ( t is the n.n. hopping amplitude and U the Coulomb repulsion).
The exact reduced density-matrix functional is derived from the Luttinger-Ward functional of the single-particle Green's function. Thereby, a formal link is provided between diagrammatic many-body approaches using Green's functions on the one hand and theories based on many-body wave functions on the other. This link can be used to explicitly construct approximations for the density-matrix functional that are equivalent to standard diagrammatic resummation techniques and to nonperturbative dynamical mean field theory in particular. Contrary to functionals of the Green's function, the exact density-matrix functional is convex and thus provides a true minimum principle which facilitates the calculation of the grand potential and derived equilibrium properties. The benefits of the proposed Green's-function-based density-matrix functional theory for geometrical structure optimization of strongly correlated materials are discussed.
A systematic method is presented for deriving the Thomas-Fermi equation for an atom and the quantum corrections from the many-body description. The novel feature of the method is that it does not require any a priori assumptions about the assignment of electrons to fully occupied single-particle states or about the distribution of electrons in phase space, but shows instead that the distribution which is usually assumed, or derived from the assumption of fully occupied single particle states, is a direct consequence of specifying that the many particle system is in its ground state. The procedure used in the derivation is the expansion of the mixed position-momentum representation of the Green's function in a series of powers of $\ensuremath{\hbar}$. The lowest order term is found to correspond with the Thomas-Fermi density. The form of the higher order terms, which are to be considered as corrections to zeroth order term, depends on the approximations made in the many-body equations for obtaining the Green's function. This paper deals only with the Hartree-Fock approximation, but the methods presented here allow generalization to other approximations which can include correlation effects.