Nonlocal Density Functionals for the Exchange-Correlation Energy: Gradient Expansion and Generalized Gradient Approximation

Nonlocal Density Functionals for the Exchange-Correlation Energy:
Gradient Expansion and Generalized Gradient Approximation

Brief introduction

The most simple approximation for the exchange-correlation (xc) energy functional, the crucial quantity of density functional theory, is the local density approximation (LDA). For given local density n(r), the LDA replaces the true xc-energy density at each point r in space by the xc-energy density of a homogeneous electron gas of the same (global) density n 0=n(r). Although the LDA is very successful for a large variety of problems, it has become clear that the description of many interesting systems in atomic physics (see e.g.[1]), quantum chemistry [2,3] and condensed matter physics [4-6] requires nonlocal corrections to the LDA.

The gradient expansion (GE), in which in addition to n(r) also its gradients are used for the density functional representation of the xc-energy, is the most systematic nonlocal extension of the LDA. The lowest order contribution to the GE, depending on [grad(n)]2, is rigorously determined by the long wavelength limit of the linear response function of the homogeneous electron gas [7-14]. However, the lowest order GE was found to be inadequate for the description of atoms [7-10] and solids [15], in spite of the fact that the characteristic gradients, which control the convergence of the GE, are reasonably small in these systems (apart from the asymptotic regime of atoms). The benefits of the error cancellation both between exchange and correlation as well as between different regions of space, which is responsible for the accuracy of the LDA, are lost. This imbalance reflects the fact that the lowest order GE does not satisfy the sum rule for the xc-hole [16], in contrast to the LDA. In addition, the asymptotic region of finite systems with their exponentially decaying density can not be treated appropriately by the GE: In this region the gradient correction leads to a diverging xc-potential, consistent with the exponential divergence of the characteristic gradients.

The first principles evaluation of higher order gradient terms requires knowledge of higher order contributions to the long wavelength expansion of the linear response functions and of nonlinear response [13,17-19]. As a consequence the available information on higher order gradient corrections is very limited. Moreover, applications including such terms are technically rather involved, if the higher gradients have to be handled numerically. As one has little hope to resolve all the deficiencies of the lowest order gradient correction in this way, the order by order extension of the GE has not been pursued any further in recent years.

Gradient corrected functionals have received renewed attention since the introduction of generalized gradient approximations (GGAs) [20-26]. GGAs can be interpreted as semi-empirical partial resummations of the complete GE, including only terms depending on grad(n(r)) , but no higher density gradients. It has been demonstrated that GGAs give excellent atomic ground state energies [2,3,27] and significantly improve dissociation energies and bond lengths of small molecules [2,3,27-29]. Similarly, the cohesive properties of many solids are reproduced more accurately by GGAs [27,30-32]. Most notably, however, GGAs are able [15] to correct one of the most well known deficiencies of the LDA, i.e. its failure to give the correct ferromagnetic bcc ground state for metallic iron. In addition, the GGA xc-potential remains finite in the asymptotic regime of finite systems (by construction of the functional).

Energy differences like ionization potentials and electron affinities, on the other hand, are not always improved by GGAs [27,33]. GGAs neither provide a description of atomic negative ions nor are they able to deal with dispersion forces. In the case of solids GGAs do not give results which are consistently superior to those of the LDA [6,17,34-38]. For instance, GGAs have problems with predicting the transition metal oxides FeO and CoO to be antiferromagnetic insulators [6,37]. Furthermore, the differences between the results obtained with the various proposed GGAs are often almost as large as those between individual GGAs and the LDA, so that the optimum form of a GGA is not obvious.

In order to identify the properties of GGAs which are responsible for their ambiguous performance we have analyzed GGAs for the exchange-only (x-only) energy, using atoms as testing ground. In contrast to comparisons of GGA results with experimental data, which suffer from fortuitious error cancellation between exchange and correlation and additional approximations (e.g. required for the numerical treatment of complex systems), the restriction to atoms in the x-only limit allows a comparison with exact reference results, obtained via the so-called Optimized Potential Method (for an overview see [39]). Our studies exhibited inherent limitations of GGAs, which are most obvious in the exchange potential. In particular, we have found that due to their semilocal form GGAs are not able to (accurately) reproduce

(see illustrative plots of exchange potentials). Corresponding studies of atomic correlation potentials [43-45] revealed that the GGA correlation potential is quite different from the exact result [46], in spite of the excellent GGA correlation energies.

In response to these problems a new form of GGA has been constructed [42], placing more emphasis on the quality of the local xc-potential rather than on the global xc-energy. The resulting functional reproduces atomic exchange potentials much more accurately than all other GGAs. In addition, it is the first GGA which predicts a small gap for FeO and CoO [37], emphasizing the importance of the exchange potential for these Mott insulators. However, the improvement of the exchange potential is accompanied by a loss of accuracy for the exchange energy: The subtle cancellation of local errors on which the excellent energies of standard GGAs are based is lost as soon as the GGA kernel is constructed to give the optimum local accuracy. This points at the limitations of the functional form of the GGA.

It seems questionable whether GGA-type functionals including higher gradients of the density [47] can overcome these difficulties. For that reason fully nonlocal xc-functionals have attracted more and more interest in recent years. Two strategies may be distinguished: On the one hand, orbital-dependent (implicit) density functionals have been have been suggested as next generation of DFT (for an overview see [39]). As a purely density-dependent alternative a linear response type functional has been put forward [48], which is currently under investigation.

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