Exact Density Functionals for Ground‐State Energies II. Details and Remarks

Abstract

Statements utilized or mentioned in the Part I are proved or discussed in detail. Especially it is shown that for finite-dimensional state spaces the strictly positive densities being strictly less than two are E-V representable and that the density functional by Lieb is differentiable at the E-V densities.Aussagen, die im Teil I benutzt oder erwahnt wurden, werderi bewiesen bzw. im Detail diskutiert. Insbesondere wird gezeigt, daB fur endlichdimensionale Zustttndsraume die strikt positiven Dichten, die strikt kleiner als zwei sind, E-V-erzeugbar sind sowie daB das Dichtefunktional von Lieb genau an den E-V-Dichten differenzierbar ist.The exact density functionals and several types of representability of one-particle densities have been introduced in Part I (Section 2) of this paper; for references [l to 221 also confer Part I. 'V-RepresentabilityThe distinction between E-, PS-, andD-V densities makes sense, since not every density For finite-dimensional state spaces, however, it has been proved for bosons in [23] that every strictly positive density is PS-V representable and the corresponding conjecture for fermions has been posed in the same paper. This conjecture can indeed be shown [24] by refining the considerations in [23], but we want to deal here with an essentially shorter proof.Theorem 5.1. For finite-dimensional state spaces every strictly positive density n ( r ) < 1 (or < 2 for spin degeneracy 2) is a fermion E-V density.Proof. We consider a system of N fermions (for notational convenience without spin degeneracy), where every particle has the state space RQ; q > N . The functional F,(n) as defined by (2.10) is convex. Hence, F,(n) has a continuous tangent functional at every ?z in the interior of the set of all fermion densities 8, = {(n(l), ... , %(a)) : 0 5 5 n ( r ) 5 1, C n ( r ) = N } . In the proof of theorem 6.1 we show that F L ( n ) possesses a continuous tangent functional exactly at E -V densities. Therefore, all densities from the interior of S, are E -V representable.

Role of exact exchange in thermally-assisted-occupation density functional theory: A proposal of new hybrid schemes

“…To rectify the above situation, KS-DFT has been extended to ensemble DFT [59,60], wherein ρ(r) is assumed to be noninteracting ensemble v s -representable, as it is associated with an ensemble of pure determinantal states of the noninteracting KS system at zero temperature. Accordingly, the orbital occupation numbers in ensemble DFT are 0, 1, and fractional (between 0 and 1) for the orbitals above, below, and at the Fermi level, respectively.…”

Spin in density‐functional theory

“…In practice, it is always assumed that this so-called v s -representability condition is fulfilled, even though this is not guaranteed and several counter-examples are known. [84,[107][108][109][110][111] For a detailed discussion of these subtle issues, see, for example, Refs. [85].…”

The spin-unrestricted molecular Kohn–Sham solution and the analogue of Koopmans’s theorem for open-shell molecules

“…It has been argued [22][23][24][25] that such a potential indeed will always exist, or at least that a local potential can be constructed whose corresponding Kohn-Sham density approaches the target density arbitrarily closely. These arguments for the total density and spin-restricted Kohn-Sham potential carry over unmodified to the separate spin densities and potentials (r), where each spin potential is determined up to a constant.…”