Density functional theory together with the Kohn-Sham (KS) scheme represent an efficient framework to recover the ground state density and energy of a many-body quantum system from an auxiliary system (one-body with a local potential that is a density functional). However, to fully achieve establishing the KS scheme, a general proof of the validity of the so-called
v-representability conjecture is still required. In this article, we propose such a proof. We reduce the demonstration to proving (1) that the KS potential is differentiable for all non-interacting
v-representable densities and (2) that its derivative is bounded. Then, we demonstrate points (1) and (2) applying static linear response to the non-interacting system.
By utilizing the knowledge that a Hamiltonian is a unique functional of its ground-state density, the following fundamental connections between densities and Hamiltonians are revealed: Given that ${\ensuremath{\rho}}_{\ensuremath{\alpha}}, {\ensuremath{\rho}}_{\ensuremath{\beta}},\dots{},{\ensuremath{\rho}}_{\ensuremath{\omega}}$ are ground-level densities for interacting or noninteracting Hamiltonians
H1,H2,…,HM (
M arbitrarily large) with local potentials
v1,
v2,
…,
vM, but given that we do not know which $\ensuremath{\rho}$ belongs with which
H, the correct mapping is possible and is obtained by minimizing $\ensuremath{\int}d\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}} [{v}_{1}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}){\ensuremath{\rho}}_{\ensuremath{\alpha}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})+{v}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}){\ensuremath{\rho}}_{\ensuremath{\beta}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})+\ensuremath{\cdots}{v}_{M}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}){\ensuremath{\rho}}_{\ensuremath{\omega}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})]$ with respect to optimum permutations of the $\ensuremath{\rho}$'s among the
v's. A tight rigorous bound connects a density to its interacting ground-state energy via the one-body potential of the interacting system and the Kohn-Sham effective one-body potential of the auxiliary noninteracting system. A modified Kohn-Sham effective potential is defined such that its sum of lowest orbital energies equals the true interacting ground-state energy. Moreover, of all those effective potentials which differ by additive constants and which yield the true interacting ground-state density, this modified effective potential is the most invariant with respect to changes in the one-body potential of the true Hamiltonian. With the exception of the occurrence of certain linear dependencies,
a density will not generally be associated with any ground-state wave function (is not wave function
v representable) if that density can be generated by a special linear combination of three or more densities that arise from a common set of degenerate ground-state wave functions. Applicability of the "constrained search" approach to density-functional theory is emphasized for non-
v-representable as well as for
v-representable densities. In fact, a particular constrained ensemble search is revealed which provides a general sufficient condition for non-
v representability by a wave function. The possible appearance of noninteger occupation numbers is discussed in connection with the existence of non-
v representability for some Kohn-Sham noninteracting systems.
Aiming at a unified treatment of correlation and inhomogeneity effects in superconductors, Oliveira, Gross, and Kohn proposed in 1988 a density functional theory for the superconducting state. This theory relies on the existence of a Kohn- Sham scheme, i.e., an auxiliary noninteracting system with the same electron and anomalous densities of the original superconducting system [Phys. Rev. Lett. 60, 2430 (1988)]. However, the question of noninteracting
v-representability has never been properly addressed and the existence of the Kohn-Sham system has always been assumed without proof. Here, we show that indeed such a noninteracting system does not exist at zero temperature. In spite of this result, we also show that the theory is still able to yield good results, although in the limit of weakly correlated systems only.
This note is intended for expanding the details on the derivation and properties of density functional theory, in hope to make them more systematic, better motivated, and step-by-step for readers new to the domain. The note starts with basic concepts in quantum mechanics, then takes the step towards many-body systems using the tools of second quantization and Fock space, with some highlights on properties of the Coulomb system. Given these general technical preparations, the Hartree-Fock method is naturally unrolled, with expressions for various cases and quantities. Density functional theory is then presented, with a motivating reasoning of the Kohn-Sham formulation, and a connection and comparison of the expressions with the Hartree-Fock method. Lieb's celebrated 1983 paper on functional analysis on density functionals is also summarized.
The State--Specific Kohn--Sham Density Functional Theory [arXiv:physics/0506037] is used to derive the Kohn-Sham exchange-correlation potential $\vxc$ and exchange-correlation energy $\Eco$ as explicit functionals of
vs and
ϕ1, where
vs is the local, one-body potential from the Kohn--Sham equations, and
ϕ1 is the spinless one-particle density matrix from the Kohn--Sham noninteracting state, say $|ϕ_1\ran$. In other words, $|ϕ_1\ran$ is the ground state eigenfunction of the noninteracting Schrödinger equation with the one-body potential
vs. For simplicity, we only consider noninteracting states that are closed-shell states and interacting states that are nondegenerate, singlet ground-states.
All downfolding procedures have to start from an approximate treatment of the full Hilbert space. For strongly correlated solids, density functional theory (DFT) within the Kohn–Sham scheme provides a suitable starting point. In this chapter, we describe density functional theory and discuss the different merits and problems of Kohn–Sham DFT. Since it is not the main topic of the book, we will focus on the conceptual ideas and formulation and on how DFT is used in practice as a starting point for many-body treatments. DFT is such a well developed theory that there exist a wealth of literature on the topic. There are a number of lucid reviews such as that by Jones and Gunnarsson [1] and more recent reviews by Jones [2] and Becke [3]; the last two contain a fascinating historical account of DFT. A collection of articles with contributions from pioneers in the field was given by Dreizler and da Providencia [4] and many others, which are too numerous to be listed here. A Nobel Lecture [5] is another source of inspiration.