Optimized Potential Method: Orbital-dependent Representations of the Exchange-Correlation Energy Functional

Optimized Potential Method: Orbital-dependent Representations of the Exchange-Correlation Energy Functional

Introduction

The Hartree energy and potential contain self-interaction contributions, which are cancelled exactly by the corresponding contributions to the exchange energy and potential in the standard Hartree-Fock scheme. In density functional theory (DFT) this cancellation is provided, in principle, by the exchange part of the exchange-correlation (xc) energy functional. In practice, that is for the local density approximation (LDA) and for gradient corrected functionals, one finds that this cancellation is only partial. The most obvious manifestation of the incomplete cancellation is the exponential decay of the LDA/GGA exchange potential in the asymptotic region of finite systems, which is in obvious contradiction to the -1/r-behavior of the exact exchange potential [1].

An extension of the DFT selfconsistency scheme with a strictly multiplicative xc-potential, which treats exchange exactly, was originally introduced by Talman and Shadwick [1] and has later been adopted as a DFT-method by Sahni, Gruenebaum and Perdew [2] as well as Langreth and Mehl [3]. While this so-called Optimized-Potential-Method (OPM) initially had been formulated only for the exact exchange energy, it is not restricted to this limit, but can be directly extended to the DFT correlation energy [4,5]. In this approach the xc-energy is represented in terms of the Kohn-Sham single-particle orbitals, rather than as an explicit functional of the density. As a consequence the selfconsistency procedure requires the simultaneous solution of the standard single-particle equations of DFT (Kohn-Sham equations) and an integral equation determining the multiplicative xc-potential (for a review see [6]).

We have solved these OPM equations with the exact exchange for atoms as well as for jellium spheres and used the results to analyze explicit density functionals for the exchange energy, as the gradient expansion and the generalized gradient approximation (GGA) [7-9]. For these studies it has been particularly advantageous to compare the x-only potentials from approximate functionals with the exact OPM standard as the potential contains much more information than the corresponding energy. On this basis a new GGA was developed [10] which reproduces exchange potentials much more accurately than conventional GGAs.

As the OPM is numerically much more involved than the standard DFT scheme only a limited number of applications beyond atoms has been reported until now [11-15]. Orbital-dependent DFT methods have, however, attracted considerable interest [16-20] since an efficient and accurate approximate solution of the crucial OPM integral equation for the xc-potential was put forward by Krieger, Li and Iafrate (KLI) [21]. In particular, the combination of the KLI approximation with the plane-wave pseudopotential approach allows the application of the exact exchange to complex systems. Suitable norm-conserving pseudopotentials for the exact exchange have been developed recently [19].

Essentially all these applications use the OPM only for the exchange functional, while correlation is either neglected or included in the form of the LDA or GGA. Attempts to take correlation into account on a semi-empirical level via the Colle-Salvetti functional [22] were not successful [17,13]. As a second semi-empirical orbital-dependent functional the self-interaction corrected (SIC) LDA [23] has been suggested for use within the OPM [24]. However, the SIC-LDA suffers from a unitarity problem (it is not invariant against unitary transformation of degenerate KS orbitals). Of course, this deficiency is not resolved by the OPM, so that a straightforward application of the SIC-LDA is not possible [6].

In recent years the focus has thus been on systematically derived orbital-dependent correlation functionals. The framework for these efforts is provided by many-body theory based on the Kohn-Sham auxiliary Hamiltonian [4,5,25], in particular by Kohn-Sham perturbation theory [5]. It was found that the lowest order contribution to the perturbation expansion of the exact ground state energy with respect to the Kohn-Sham system reproduces dispersion forces [25,18], in contrast to the LDA or GGA. On the other hand, correlation energies are overestimated [6,13,26]. Moreover, the corresponding correlation potential diverges in the asymptotic region of finite systems with their exponentially decaying densities [26,27]. In response to these difficulties partial resummation of the perturbation series has been suggested [11,28,29]. The properties of the resulting functionals are presently under investigation.

An extension of the OPM to the relativistic domain has been developed in our group [30,25] (for a review of the relativistic OPM see [31]). This approach allows the DFT treatment of the exact relativistic exchange, including the (retarded) Breit interaction, as well as of relativistic orbital-dependent correlation functionals. It has the particular advantage that a gauge-invariant handling of the exact exchange is possible even at the no-pair level [25]. A corresponding KLI-approximation has also been derived [32,25]. The exchange-only results obtained with the relativistic OPM for heavy atoms show the same overall picture as observed in the nonrelativistic limit.

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