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arXiv:0908.0669 · Physical Review Letters

Density-Functional Theory for Strongly Interacting Electrons

Paola Gori‐Giorgi, Michael Seidl, Giovanni Vignale

Published Oct 12, 2009• 106 citations
arXivHTMLPDF

Abstract

We present an alternative to the Kohn-Sham formulation of density-functional theory for the ground-state properties of strongly interacting electronic systems. The idea is to start from the limit of zero kinetic energy and systematically expand the universal energy functional of the density in powers of a "coupling constant" that controls the magnitude of the kinetic energy. The problem of minimizing the energy is reduced to the solution of a strictly correlated electron problem in the presence of an effective potential, which in our theory plays the same role as the Kohn-Sham potential plays in the traditional formulation. We discuss several schemes for approximating the energy functional, and report preliminary results for low-density quantum dots.

DOI
10.1103/physrevlett.103.166402

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We present an alternative to the Kohn-Sham formulation of density-functional theory for the ground-state properties of strongly interacting electronic systems. The idea is to start from the limit of zero kinetic energy and systematically expand the universal energy functional of the density in powers of a "coupling constant" that controls the magnitude of the kinetic energy. The problem of minimizing the energy is reduced to the solution of a strictly correlated electron problem in the presence of an effective potential, which in our theory plays the same role as the Kohn-Sham potential plays in the traditional formulation. We discuss several schemes for approximating the energy functional, and report preliminary results for low-density quantum dots.
We use the exact strong-interaction limit of the Hohenberg-Kohn energy density functional to approximate the exchange-correlation energy of the restricted Kohn-Sham scheme. Our approximation corresponds to a highly nonlocal density functional whose functional derivative can be easily constructed, thus transforming exactly, in a physically transparent way, an important part of the electron-electron interaction into an effective local one-body potential. We test our approach on quasi-one-dimensional systems, showing that it captures essential features of strong correlation that restricted Kohn-Sham calculations using the currently available approximations cannot describe.
In the widely-used Kohn-Sham density functional theory for the manyelectron ground state, only the density functional for the exchange-correlation energy is approximated. While the exact density functional minimizes at the exact ground-state energy for any system, widely used approximations can at best describe normally-correlated ground states. A “proper” approximation for a given problem is defined as one that well describes relevant normally-correlated states. Strong correlation, present in many quantum materials, can arise due to abnormally high degeneracy of the Kohn-Sham occupied and unoccupied one-electron energies near the Fermi level. In such situations, the approximate functional may lower its energy by spontaneously breaking symmetries that the ground-state density shares with the many-electron Hamiltonian, reducing the degeneracies that lead to strong correlation. Here the Levy constrained search is invoked to argue that the total energy of a “proper” functional with maximal spontaneous symmetry breaking is bounded from below by the exact total energy, and that the bound may be close to the extent that the abnormal degeneracies are removed. This is a consequence of a variational principle, and it may not hold in non-variational but more accurate approaches to normal correlation such as coupled cluster theory. Since the correlation energy of any one-electron state is zero, a widely “proper” approximate functional should be nearly or exactly self-interaction free, and progress toward a proper self-interaction correction is reviewed here. The time-centered physical interpretation of symmetry breaking is also discussed.
Abstract Density functional theory (DFT) is usually formulated in terms of the electron density as a function of position n(r). Here we discuss an alternative formulation of DFT in terms of the orbital occupation numbers {n α } associated with a local-orbital orthonormal basis set {ϕ α }. First, we discuss how the building blocks of DFT, namely the Hohenberg–Kohn theorems, the Levy–Lieb approach and the Kohn–Sham method, can be adapted for a description in terms of {n α }. In particular, the total energy is now a function of {n α }, E[{n α }], and a Kohn–Sham-like Hamiltonian is derived introducing the effects of the electron–electron interactions via effective potentials, { V α eff = ∂ E e e [ { n β } ] / ∂ n α } . In a second step we consider the Hartree and exchange energies and discuss how to describe them, in the spirit of a DFT approach, in terms of the orbital occupation numbers. In this contribution special attention is paid to the description of the (intra-atomic) correlation energy and corresponding correlation potentials {V corr,α }. For this purpose, a model system is analyzed in detail, whereby an atomic Hamiltonian interacts with the environment via a simplified model; the use of this model allows us to obtain the correlation energy and potentials (in terms of {n α }) for different cases corresponding to low, intermediate and high electron correlations.
We formulate the inverse Kohn–Sham equations based on the density equation theory. The present formulation relies on neither action functionals nor force balance equation, but is constructed by comparing the first-order density equation for the electron interacting real system and that for the electron non-interacting Kohn–Sham system. The derived inverse Kohn–Sham integral equation requires information of an exact ground-state electron density to calculate an exact Kohn–Sham potential for a ground state and it requires an exact time-dependent electron density and a time-derivative of the exact time-dependent first-order density matrix to calculate an exact time-dependent Kohn–Sham potential. A possible algorithm to construct the Kohn–Sham potential for a ground state is suggested as well as a solution algorithm for the time-dependent inverse Kohn–Sham problem without resorting to any approximations. These two solution algorithms, in combination, suggest a way to perform an exact time-dependent density functional calculation as a predictive method.
We propose to expand the territory of density functional theory to strongly correlated electrons by reformulating the Kohn-Sham scheme in the representation of fractionalized particles. We call it the "KS^{*} scheme." Using inhomogeneous t-J chains as a test bed, we show that the KS^{*} scheme with simple local density approximation is able to achieve accurate ground-state energy and density distribution comparable to the density matrix renormalization group method, while the computational complexity is much lower.
A rigorous derivation of the density functional in the Hohenberg-Kohn theory is presented. With no assumption regarding the magnitude of the electric coupling constant e2e^2e2 (or correlation), this work provides a firm basis for first-principles calculations. Using the auxiliary field method, in which e2e^2e2 need not be small, we show that the bosonic loop expansion of the exchange-correlation functional can be reorganized so as to be expressed entirely in terms of the Kohn-Sham single-particle orbitals and energies. The excitations of the many-particle system can be obtained within the same formalism. We also explicitly demonstrate at zero-temperature the single-particle limit, the weak-coupling limit of the energy functional, and its application to homogeneous electron gas.