On the relation between the Hartree-Fock and Kohn-Sham approaches

We show that the Hartree-Fock (HF) results cannot be reproduced within the framework of Kohn-Sham (KS) theory because the single-particle densities of finite systems obtained within the HF calculations are not v-representable, i.e., do not correspond to any ground state of a N non-interacting electron systems in a local external potential. For this reason, the KS theory, which finds a minimum on a different subset of all densities, can overestimate the ground state energy, as compared to the HF result. The discrepancy between the two approaches provides no ground to assume that either the KS theory or the density functional theory suffers from internal contradictions. PACS number(s): 31.15.Ew, 31.10.+z, 71.10.-w Typeset using REVTEX E–mail: [email protected] 1 The Hartree-Fock method (HF), first proposed in the pioneering works of Hartree and Fock [1,2] is known to be successful in calculating properties of electron systems, in particular, the ground state properties of atoms. Based on a variational principle, the HF method estimates the ground state energy E of an electron system from above, i.e., EHF ≥ E, where EHF is the ground state energy calculated within the HF method. If the ground state wave function of N electrons is approximated by a single N -electron Slater determinant, the HF solution delivers the minimum value EHF on the set of all such determinants. Agreement, or otherwise, with the HF results is often used to estimate the success of other approximate computational schemes. The Density Functional Theory (DFT) exploits the one-to-one correspondence between the singleparticle electron density and an external potential acting upon the system and relies on the existence of a universal functional F [ρ(r)] which can be minimized in order to find the ground state energy [3]. The Kohn-Sham (KS) theory goes further in reducing the problem of calculating ground state properties of a many-electron system in a local external single-particle potential to solving Hartreelike one-electron equations [3,4]. Successful solution of these equations allows to predict, at least in principle, the atomic, molecular, cluster and solid bodies binding energies, phonon spectra, activation barriers etc., see e.g. [5]. It is natural, therefore, to ask whether the HF ground state energy can be successfully reproduced in the Kohn-Sham approach. We note first that a universal density functional FHF [ρ] can be defined with the help of the constrained-search technique [6]. Had the explicit form of FHF [ρ] been available, the HF and the DFT approach would have yielded the same results for the ground state energy and the electron density. Unfortunately, the correspondence theorem [3] establishes the existence of the functional only in principle, and provides no unique practical recipe for its construction. Rather, for practical calculations one has to resort to the KS approach. Exhaustive calculations [7–9] of the ground state energies of different atoms show that, if the KS approach is used, the resulting energy EKSHF usually exceeds the energy EHF obtained by the HF method, EKSHF ≥ EHF . (1) The purpose of this Letter is to analyze the significance and implications of the inequality (1) for the KS method. Recently, there have been suggestions that this disagreement may point to intrinsic flaws in both the DFT and the KS theories. It is obvious that EKSHF cannot be identified with the exact DFT Hartree-Fock energy EDHF which, as already mentioned, must coincide with EHF . One might suspect, therefore, that an exact local exchange potential does not exist for ground states of typical atoms (see [8,9] and references therein). We will, however, argue that the discrepancy (1) between the HF and KS is due to the different domains on which the respective functionals are defined. More specifically, we will show that while a KS density is v-representative, a HF density is not, i.e., it cannot be obtained as the ground state density of any N non-interacting electrons in a local potential. As a result, the KS method simply delivers a minimum on a different class of electron densities, and its disagreement with the HF approach does not indicate the existence of any internal contradictions either in KS or DFT approach. We begin our proof with considering the HF ground state energy which is given by the equation