A density-functional theory for ensembles of unequally vreighted states is formulated on the basis of the generalized Rayleigh-Ritz principle of the preceding paper. From this formalism, two alternative approaches to the computation of excitation energies are derived, one equivalent to the equiensemble method proposed by Theophilou [J. Phys. C 12, S419 (1979)], the other grounded on an expression relating the excitation energies to the Kohn-Sham single-particle eigenvalues. I. INraODUCTIONDensity-functional theory was originally developed' as a ground-state formalism. For excited states, a straightforward extension on the basis of the RayleighRitz principle is possible only for the lowest-energy state in each symmetry class. '" A more general approach, applicable to arbitrary excited states, has been proposed by Theephilou, s who extended the theory to equiensem bles of the lowest M eigenstates, equally weighted. Using a variational principle for the ensemble energy, he demonstrated that the ensemble density uniquely determines the external potential and that the correct density of a given system can be obtained by solving selfconsistently a set of Kohn-Sham (KS) -like equations. The exchange-correlation functional E~[p] arising in this formalism has recently been investigated, a quasilocal-density approximation for E~h aving been derived by identifying the equiensemble with a thermal ensemble. In this paper, we shall develop a density-functional theory for ensembles of fractionally occupied states. In these ensembles, the M states are weighted unequally. The extended Rayleigh-Ritz principle presented in the preceding paper, hereafter referred to as I, enal)les us to generalize Theophilou's ensemble, diff'erent weights w"w2, . . . , w being assigned to the lowest M eigenstates. A density-functional formalism for such an ensemble, parametrized by M distinct weights, can be constructed. For practical applications, however, it is more convenient to de5ne the weights as functions of a single, real parameter w. Thus, for example, in the case of a nondegenerate spectrum, we assign to the highest-energy state in the ensemble the weight w (i.e. , choose wst --w) and assign to each other state the weight (1 -w)/(M -1) [i.e., choose w, =w2 --. --wst, =(1 -w)/(M -1)]. The condition w& &w2& . -&w~, required by the variational principle in paper I, implies that 0 & w & 1 lM. For w =1lM, this deSnition of the weights ensures that Theophilou's formalism for an equiensemble of M states be recovered, all weights being equal to 1/M. Similarly, for w=O, the formalism for an equiensemble of M -1 states is obtained, all weights being equal to 1/(M -1). In these two limits, therefore, Ref. 6 provides an approximation for the exchange-correlation energy functional. Our analysis leads to an exact expression relating the excitation energies to the KS energies. In a subsequent paper, hereafter referred to as III, we shall show that, with a simple approximation, the expression for E", in Ref. 6 turns the formal relation into a practical, a...
By shifting the reference system for the local-density approximation (LDA) from the electron gas to other model systems one obtains a new class of density functionals, which by design account for the correlations present in the chosen reference system. This strategy is illustrated by constructing an explicit LDA for the one-dimensional Hubbard model. While the traditional ab initio LDA is based on a Fermi liquid (the electron gas), this one is based on a Luttinger liquid. First applications to inhomogeneous Hubbard models, including one containing a localized impurity, are reported. 71.15.Mb, 71.10.Pm, 71.10.Fd, 71.27.+a Density-functional theory (DFT) [1] is the basis of almost all of todays electronic-structure theory, and much of materials science and quantum chemistry. Many-body effects enter DFT via the exchange-correlation (xc) functional, which is commonly approximated by the localdensity approximation (LDA) [1]. The essence of the LDA is to locally approximate the xc energy of the inhomogeneous system under study by that of the homogeneous electron gas. This electron gas plays the role of a reference system, whose correlations are transfered by the LDA into the DFT description of the inhomogeneous system. The most popular improvement upon the LDA are generalized gradient approximations [2], whose basic philosophy is to abandon the requirement of homogeneity of the reference system. This system, however, is normally still the interacting electron gas [2].In the present paper we propose to explore a different paradigm for the construction of novel density functionals: instead of sticking to the electron gas as a reference system, and abandoning homogeneity, it may sometimes be advantageous to do the reverse: stick to homogeneity (and thus to the LDA) but abandon the electron gas as a reference system. The new reference system is chosen such that it accounts for the correlations present in the inhomogeneous system under study.The only requirement for the reference system is that in the absence of any spatially varying external potential its xc energy must be known exactly or to a high degree of numerical precision. Besides the electron gas (or Jellium model) there are many other physically interesting model systems that satisfy this criterium. Most notably among these is a large class of low-dimensional models which can be solved exactly by Bethe Ansatz (BA) techniques or bosonisation (in one dimension, e.g., the repulsive and the attractive Hubbard model, the hard-core Fermi and Bose gases, the Heisenberg, the supersymmetric t-J, and the Tomonaga-Luttinger model [3,4]). The solutions to these models in the homogeneous case can be used instead of the electron gas to construct LDA functionals that can then be applied to study these models also in inhomogeneous situations. The main advantage offered by a DFT treatment of such models is the gain in simplicity that arises from mapping the inhomogeneous interacting many-body system onto a noninteracting auxiliary system, which is diagonalized much more easily...
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