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The Journal of Chemical Physics

Fractional spin in reduced density-matrix functional theory

N. Helbig, Giannoula Theodorakopoulos, Nektarios N. Lathiotakis

Published Aug 5, 2011• 9 citations
Publisher

Abstract

We study the behavior of different functionals of the one-body reduced density matrix (1RDM) for systems with fractional z-component of the total spin. We define these systems as ensembles of integer spin states. It is shown that, similarly to density functional theory, the error in the dissociation of diatomic molecules is directly related to the deviation from constancy of the atomic total energies as functions of the fractional spin. However, several functionals of the 1RDM show a size inconsistency which leads to additional errors. We also investigate the difference between a direct evaluation of the energy of an ensemble of integer-spin systems and a direct minimization of the energy of a fractional-spin system.

DOI
10.1063/1.3615955

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We study the behavior of different functionals of the one-body reduced density matrix (1RDM) for systems with fractional z-component of the total spin. We define these systems as ensembles of integer spin states. It is shown that, similarly to density functional theory, the error in the dissociation of diatomic molecules is directly related to the deviation from constancy of the atomic total energies as functions of the fractional spin. However, several functionals of the 1RDM show a size inconsistency which leads to additional errors. We also investigate the difference between a direct evaluation of the energy of an ensemble of integer-spin systems and a direct minimization of the energy of a fractional-spin system.
Electronic states with fractional spins arise in systems with large static correlation (strongly correlated systems). Such fractional-spin states are shown to be ensembles of degenerate ground states with normal spins. It is proven here that the energy of the exact functional for fractional-spin states is a constant, equal to the energy of the comprising degenerate pure-spin states. Dramatic deviations from this exact constancy condition exist with all approximate functionals, leading to large static correlation errors for strongly correlated systems, such as chemical bond dissociation and band structure of Mott insulators. This is demonstrated with numerical calculations for several molecular systems. Approximating the constancy behavior for fractional spins should be a major aim in functional constructions and should open the frontier for density functional theory to describe strongly correlated systems. The key results are also shown to apply in reduced density-matrix functional theory.
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.
For a linear combination of electron densities of degenerate ground states, it is shown that the value of any energy functional is the ground state energy, if the energy functional is exact for ground state densities, size consistent, and translational invariant. The corresponding functional of kinetic and interaction energy is the linear combination of the functionals of the degenerate densities. Without invoking ensembles, it is shown that the energy functional of fractional number electrons is a series of straight lines interpolating its values at integers. These results underscore the importance of grand canonical ensemble formulation in density functional theory.
Abstract In reduced density matrix functional theory, approximate total energy expressions in terms of the one‐body reduced density matrix (1RDM) are minimized under the appropriate N‐representability subsidiary conditions. The optimal 1RDM is in general nonidempotent. Measures of nonidempotency are the total charge of weakly occupied states and correlation entropy given in terms of the eigenvalues of the 1RDM. In this work, we calculate these quantities for several systems using different 1RDM functionals and compare with accurate results from a multideterminant method. Our goal is to assess 1RDM functionals with respect to the quality of the obtained 1RDM. © 2012 Wiley Periodicals, Inc.
Standard approximations for the exchange-correlation functional have been found to give big errors for the linearity condition of fractional charges, leading to delocalization error, and the constancy condition of fractional spins, leading to static correlation error. These two conditions are now unified and extended to states with both fractional charge and fractional spin to give a much more stringent condition: the exact energy functional is a plane, linear along the fractional charge coordinate and constant along the fractional spin coordinate with a line of discontinuity at the integer. Violation of this condition underlies the failure of all known approximate functionals to describe the gaps in strongly correlated systems. It is shown that explicitly discontinuous functionals of the density or orbitals that go beyond these currently used smooth approximations is the key for the application of density functional theory to strongly correlated systems.
The exact conditions for density functionals and density matrix functionals in terms of fractional charges and fractional spins are known, and their violation in commonly used functionals has been shown to be the root of many major failures in practical applications. However, approximate functionals are designed for physical systems with integer charges and spins, not in terms of the fractional variables. Here we develop a general framework for extending approximate density functionals and many-electron theory to fractional-charge and fractional-spin systems. Our development allows for the fractional extension of any approximate theory that is a functional of G(0), the one-electron Green's function of the non-interacting reference system. The extension to fractional charge and fractional spin systems is based on the ensemble average of the basic variable, G(0). We demonstrate the fractional extension for the following theories: (1) any explicit functional of the one-electron density, such as the local density approximation and generalized gradient approximations; (2) any explicit functional of the one-electron density matrix of the non-interacting reference system, such as the exact exchange functional (or Hartree-Fock theory) and hybrid functionals; (3) many-body perturbation theory; and (4) random-phase approximations. A general rule for such an extension has also been derived through scaling the orbitals and should be useful for functionals where the link to the Green's function is not obvious. The development thus enables the examination of approximate theories against known exact conditions on the fractional variables and the analysis of their failures in chemical and physical applications in terms of violations of exact conditions of the energy functionals. The present work should facilitate the calculation of chemical potentials and fundamental bandgaps with approximate functionals and many-electron theories through the energy derivatives with respect to the fractional charge. It should play an important role in developing accurate approximate density functionals and many-body theory.