Evaluation of vorticity expansion approximation of current-density functional theory by means of Levy’s asymptotic bound

The density functional theory DFT provides a powerful and reliable approach to the electronic structures of manyelectron systems.1,2 The DFT has been extended so that various physical quantities can be chosen as basic variables.3–5 For instance, the spin-density functional theory,6,7 the current-density functional theory CDFT ,8–11 and LDA+U method12–15 are regarded as examples of such extensions. In the CDFT, we choose the paramagnetic current density as a basic variable in addition to the electron density. Therefore, the CDFT is useful for describing the ground-state properties of systems such as open shell atoms, inhomogeneous electronic systems in an external magnetic field, and f-electron materials where an orbital current is induced from both the strong spin-orbit interaction and the intra-atomic Coulomb interaction. For the practical use of the CDFT, one has to develop the approximate form of the exchange-correlation energy functional. Many attempts to develop the approximate form have been presented so far.8,9,16–29 We have already suggested two strategies for treating the approximate form.22 One is to start with the coupling-constant expression of the exchangecorrelation energy functional of the CDFT.22 By using this expression, the local-density approximation LDA , averagedensity approximation, and weighted-density approximation have been proposed similarly to those in the conventional DFT.22 Another strategy is to utilize as constraints exact relations that are satisfied with the exchange and correlation energy functionals.22 This strategy is analogous to the one used for developing the generalized gradient approximation30–32 in the conventional DFT. To date, many exact relations have been derived by means of the virial theorem, uniform and nonuniform scaling properties.18,19,22,25,28 Along the latter strategy, we have recently proposed the vorticity expansion approximation VEA for the exchange and correlation energy functionals.29 Many exact relations that are derived from uniform and nonuniform scalings of electron coordinates are satisfied with the VEA formulas.29 Due to the well-behaved forms, the VEA formulas can well reproduce the exchange and correlation energies of the homogeneous electron liquid under a uniform magnetic field.29 However, Levy’s asymptotic bound,33,34 which is regarded as an important constraint and has been very effectively used in developing the Perdew-Burke-Ernzerhof PBE functional of the conventional DFT,32 has not been taken into account in developing the VEA of the CDFT.29 Levy’s asymptotic bound is such that the correlation energy functional for the uniformly scaled density approaches a constant in the limit of infinite scaling factor.33,34 This bound is not satisfied with the LDA of the conventional DFT because the short-range correlation term35,36 causes the logarithmic divergence in the limit of infinite scaling factor. The additional term that expresses the effect of the density gradient was devised in the PBE functional so as to avoid such divergence.32 Thus, this bound has played an important role in the development of the PBE functional. However, this bound has not yet been proven for the correlation energy functional of the CDFT. Also in the CDFT, this exact property would be true and very useful for developing, modifying, or evaluating the approximate forms of the exchange and correlation energy functionals. In this paper, we derive Levy’s asymptotic bound in the framework of the CDFT and evaluate both the VEA and LDA formulas. The organization of this paper is as follows. In Sec. II, we shall give a proof of Levy’s asymptotic bound for the correlation energy functional of the CDFT. In Secs. III and IV, it is shown that Levy’s asymptotic bound is satisfied with the VEA correlation energy functional while it is not with the CDFT-LDA one. Finally, a conclusion is given in Sec. V.