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arXiv:0801.3787 · Physical Review B

Reduced density matrix functional for many-electron systems

S. Sharma, J. K. Dewhurst, Nektarios N. Lathiotakis, Eberhard K. U. Gross

Published Nov 24, 2008• 177 citations
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Abstract

Reduced density matrix functional theory for the case of solids is presented and an exchange-correlation functional based on a fractional power of the density matrix is introduced. We show that compared to other functionals, this produces more accurate behavior for total energies as a function of particle number for finite systems. Moreover, it captures the correct band-gap behavior for conventional semiconductors, as well as strongly correlated Mott insulators, where a gap is obtained in the absence of any magnetic ordering.

DOI
10.1103/physrevb.78.201103

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Reduced density matrix functional theory for the case of solids is presented and an exchange-correlation functional based on a fractional power of the density matrix is introduced. We show that compared to other functionals, this produces more accurate behavior for total energies as a function of particle number for finite systems. Moreover, it captures the correct band-gap behavior for conventional semiconductors, as well as strongly correlated Mott insulators, where a gap is obtained in the absence of any magnetic ordering.
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Density-Functional Theory (DFT), although widely used and very successful in the calculation of several observables, fails to correctly describe strongly correlated materials. In the first part of this work we, therefore, introduce reduced-density-matrix-functional theory (RDMFT) which is one possible way to treat electron correlation beyond DFT. Within this theory the one-body reduced density matrix (1-RDM) is used as the basic variable. Our main interest is the calculation of the fundamental gap which proves very problematic within DFT. In order to calculate the fundamental gap we generalize RDMFT to fractional particle numbers M by describing the system as an ensemble of an N and an N+1 particle system (with N≤ M≤N+1). For each fixed particle number, M, the total energy is minimized with respect to the natural orbitals and their occupation numbers. This leads to the total energy as a function of M. The derivative of this function with respect to the particle number has a discontinuity at integer particle number which is identical to the gap. In addition, we investigate the necessary and sufficient conditions for the 1-RDM of a system with fractional particle number to be N-representable. Numerical results are presented for alkali atoms, small molecules, and periodic systems. Another problem within DFT is the description of non-relativistic many- electron systems in the presence of magnetic fields. It requires the paramagnetic current density and the spin magnetization to be used as basic variables besides the electron density. However, electron-gas-based functionals of current-spin-density-functional Theory (CSDFT) exhibit derivative discontinuities as a function of the magnetic field whenever a new Landau level is occupied, which makes them difficult to use in practice. Since the appearance of Landau levels is, intrinsically, an orbital effect it is appealing to use orbital-dependent functionals. We have developed a CSDFT version of the optimized effective potential (OEP) method which allows for the use of explicitly orbital-dependent functionals in the context of CSDFT. We present the derivation of corresponding equations and show results for a quantum dot in external magnetic fields in the second part of this thesis.
We propose a method that incorporates explicit derivative discontinuity of the total energy with respect to the number of electrons and treats both delocalization and static correlation effects in density functional calculations. Our approach is motivated by the exact behavior of the ground state total energy of electrons and involves minimization of the exchange-correlation energy with respect to the Fock space density matrix. The resulting density matrix minimization (DMM) model is simple to implement and can be solved uniquely and efficiently. In a case study of KCuF3_33​, a prototypical Mott-insulator with strong correlation, LDA+DMM correctly reproduced the Mott-Hubbard gap, magnetic ordering and Jahn-Teller distortion.
Density functional theory provides the most widespread framework for the realistic description of the electronic structure of solids, but the description of strongly correlated systems has remained so far elusive. We consider a particular limit of electrons and ions in which a one-band description becomes exact all the way from the weakly correlated metallic regime to the strongly correlated Mott-Hubbard regime. We provide a necessary condition a density functional should fulfill to describe Mott-Hubbard behavior in this one-band limit and show that it is not satisfied by standard and widely used local, semilocal, and hybrid functionals. We illustrate the condition in the case of few-atom systems and provide an analytic approximation to the exact exchange-correlation potential based on a variational wave function which shows explicitly the correct behavior, combining in a neat way lattice and continuum methods.