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Physical Review A

Strong-interaction limit of density-functional theory

Michael Seidl

Published Dec 1, 1999• 156 citations
Publisher

Abstract

Electrons can have a given smooth density distribution $\ensuremath{\rho}(r),$ even if their Coulomb interaction is scaled to infinity. This strong-interaction limit of density-functional theory provides essential information on the correlation energy of real electron systems. The simple concept of strictly correlated electrons (SCE) is analyzed here as a model for that limit. SCE is solved exactly for any one-dimensional (1D) N-electron density and, in particular, for any 3D spherical two-electron system, such as the helium atom. Both the SCE interaction energy and the SCE external potential, which are obtained here as density functionals, obey all the relations known for the corresponding quantities in the unknown true strong-interaction limit. At large but finite interaction, the electrons are still strongly correlated, performing zero-point oscillations about the SCE limit.

DOI
10.1103/physreva.60.4387

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Electrons can have a given smooth density distribution \ensuremathρ(r),\ensuremath{\rho}(r),\ensuremathρ(r), even if their Coulomb interaction is scaled to infinity. This strong-interaction limit of density-functional theory provides essential information on the correlation energy of real electron systems. The simple concept of strictly correlated electrons (SCE) is analyzed here as a model for that limit. SCE is solved exactly for any one-dimensional (1D) N-electron density and, in particular, for any 3D spherical two-electron system, such as the helium atom. Both the SCE interaction energy and the SCE external potential, which are obtained here as density functionals, obey all the relations known for the corresponding quantities in the unknown true strong-interaction limit. At large but finite interaction, the electrons are still strongly correlated, performing zero-point oscillations about the SCE limit.
We re-adapt a spectral renormalization method, introduced in nonlinear optics, to solve the Kohn-Sham (KS) equations of density functional theory, with a focus on functionals based on the strictly-correlated electrons (SCE) regime, which are particularly challenging to converge. Important aspects of the method are: (i) the eigenvalues and the density are computed simultaneously; (ii) it converges using randomized initial guesses; (iii) easy to implement. Using this method we could converge for the first time the Kohn-Sham equations with functionals that include the next leading term in the strong-interaction limit of density functional theory, the so called zero-point energy (ZPE) functional as well as with an interaction-strength-interpolation functional that includes both the exact SCE and ZPE terms. This work is the first building block for future studies on quantum systems confined in low dimensions with different statistics and long-range repulsions, such as localization properties of fermions and bosons with strong long-range repulsive interactions in the presence of a random external potential.
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.
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.
A density-functional theory (DFT) of lattice fermion models is presented, which uses the single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ as basic variable. A simple, explicit approximation to the interaction-energy functional $W[\ensuremath{\gamma}]$ of the Hubbard model is derived from exact dimer results, scaling properties of $W[\ensuremath{\gamma}]$ and known limits. Systematic tests on the one-dimensional chain show a remarkable agreement with the Bethe-ansatz exact solution for all interaction regimes and band fillings. New results are obtained for the ground-state energy and charge-excitation gap in two dimensions. A successful description of strong electron correlations within DFT is achieved.
We consider the high-density-limit correlation energy E(c) in D > or = 2 dimensions for the (1)S ground states of three two-electron systems: helium (in which the electrons move in a Coulombic field), spherium (in which they move on the surface of a sphere), and hookium (in which they move in a quadratic potential). We find that the E(c) values are strikingly similar, depending strongly on D but only weakly on the external potential. We conjecture that for large D, the limiting correlation energy E(c) approximately -delta(2)/8 in any confining external potential, where delta = 1/(D-1).
In this work we consider a numerically solvable model of a two-electron diatomic molecule to study a recently proposed approximation based on the density functional theory of so-called strictly correlated electrons (SCE). We map out the full two-particle wave function for a wide range of bond distances and interaction strengths and obtain analytic results for the two-particle states and eigenenergies in various limits of strong and weak interactions, and in the limit of large bond distance. We then study the so-called Hartree-exchange-correlation (Hxc) kernel of time-dependent density functional theory which is a key ingredient in calculating excitation energies. We study an approximation based on adiabatic SCE (ASCE) theory which was shown to display a particular feature of the exact Hxc kernel, namely, a spatial divergence as function of the bond distance. This makes the ASCE kernel a candidate for correcting a notorious failure of the commonly used adiabatic local density approximation (ALDA) in the calculation of excitation energies of dissociating molecules. Unlike the ALDA, we obtain nonzero excitation energies from the ASCE kernel in the dissociation regime but they do not correspond to those of the true spectrum unless the interaction strength is taken to be very large such that the SCE theory has the right regime of validity, in which case the excitation energies become exact and represent the so-called zero-point oscillations of the strictly correlated electrons. The commonly studied physical dissociation regime, namely, large molecular separation at intermediate interaction strength, therefore remains a challenge for density functional approximations based on SCE theory.