That the external potential ${v}_{\mathit{ext}}(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})$ of a system of electrons is determined uniquely by the ground-state density is one of the central statements of the first Hohenberg-Kohn theorem. It is known that the validity of this statement extends to densities $n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})$ with noninteger particle number [i.e., $n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})$ integrates to a number that is not an integer] if the functional derivative of ${T}_{s}[n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})]+U[n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})]+{E}_{xc}[n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})]$ exists or (without relying on the existence of functional derivatives) if the ground-state energy is a strictly convex function of the particle number. In the present article, a proof that relies neither on the existence of the above functional derivative nor on the strict convexity of the ground-state energy is presented. The fact that the density determines the external potential leads to a noncrossing theorem for ground-state densities. The noncrossing theorem produces knowledge as to what the integer-particle-number ground-state densities of a system cannot be. The noncrossing theorem produces inequalities that the functional derivatives of the exchange-correlation energy functional ${E}_{xc}[n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})]$ and the noninteracting kinetic energy functional ${T}_{s}[n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})]$ must fulfill.
That the external potential {v}_{\mathit{ext}}(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}}) of a system of electrons is determined uniquely by the ground-state density is one of the central statements of the first Hohenberg-Kohn theorem. It is known that the validity of this statement extends to densities n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}}) with noninteger particle number [i.e., n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}}) integrates to a number that is not an integer] if the functional derivative of {T}_{s}[n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})]+U[n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})]+{E}_{xc}[n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})] exists or (without relying on the existence of functional derivatives) if the ground-state energy is a strictly convex function of the particle number. In the present article, a proof that relies neither on the existence of the above functional derivative nor on the strict convexity of the ground-state energy is presented. The fact that the density determines the external potential leads to a noncrossing theorem for ground-state densities. The noncrossing theorem produces knowledge as to what the integer-particle-number ground-state densities of a system cannot be. The noncrossing theorem produces inequalities that the functional derivatives of the exchange-correlation energy functional {E}_{xc}[n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})] and the noninteracting kinetic energy functional {T}_{s}[n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})] must fulfill.
Based on the Schrodinger equation, exact expressions for the non-relativistic particle energy in the local external field and the external field potential are derived as inhomogeneous density functionals. On this basis, it is shown that, when considering more than two noninteracting electrons, the energy of such a system cannot be an inhomogeneous density functional. The result is extended for the system of interacting electrons. This means that the Hohenberg-Kohn lemma which assert that in the ground state to each inhomogeneous density corresponds only one potential of the external field cannot be a justification of the existence of the universal density functional in the general case. At the same time, statements of the density functional theory remain valid when considering any number of noninteracting ground-state bosons due to the Bose condensation effect.
The Hohenberg-Kohn theorem is extended to the case that the external potential is nonlocal. It is shown that, in this more general case, a nondegenerate ground-state wave function is a universal functional of the one-particle density kernel $\ensuremath{\mu}(x,{x}^{\ensuremath{'}})$, but probably not of the particle density $n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})=\ensuremath{\Sigma}{s}^{}\ensuremath{\mu}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}s,\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}s)$. The variational equations for the local and nonlocal cases are compared. The former must be replaced by a variational equation for an equivalent system of noninteracting particles, following a prescription of Kohn and Sham, in order to obtain a Schr\"odinger-like form, and contains only local potentials. The latter may be obtained directly in Schr\"odinger-like form, but the exchange-correlation potential is nonlocal. If the nonlocal pseudo-Hamiltonian exists [i.e., if the functional derivative $\frac{\ensuremath{\delta}E}{\ensuremath{\delta}\ensuremath{\mu}(x,{x}^{\ensuremath{'}})}$ exists for a nondegenerate ground-state density kernel], then the eigenfunctions of the pseudo-Hamiltonian are natural spin orbitals, and all partially occupied orbitals ($0<〈{\ensuremath{\varphi}}_{i}|\ensuremath{\mu}|{\ensuremath{\varphi}}_{i}〉<1$) belong to the same degenerate eigenvalue of the pseudo-Hamiltonian. Finally, it is shown, as a corollary of Coleman's theorem for N-representable density kernels, that any finite non-negative differentiable function is an N-representable particle density.
The analysis in this paper shows that the Hohenberg-Kohn theorem is the constellation of two statements: (i) the mathematically rigorous Hohenberg-Kohn lemma, which demonstrates that the same ground-state density cannot correspond to two different potentials of an external field, and (ii) the hypothesis of the existence of the universal density functional. Based on the obtained explicit expression for the nonrel-ativistic particle energy in a local external field, we prove that the energy of the system of more than two non-interacting electrons cannot be a functional of the inhomogeneous density. This result is generalized to the system of interacting electrons. It means that the Hohenberg-Kohn lemma cannot provide justification of the universal density functional for fermions. At the same time, statements of the density functional theory remain valid when considering any number of noninteracting ground-state bosons due to the Bose condensation effect. In the framework of the density matrix functional theory, the hypothesis of the existence of the universal density matrix functional corresponds to the cases of noninteracting particles and to interaction in the Hartree-Fock approximation.
It is shown that the Hohenberg-Kohn lemma and theorem are direct consequences of the statement that the ground state energy (or free energy) of a system of interacting particles in an external field is a unique functional of the potential of this field. This means that, if the Hohenberg-Kohn theorem is valid, the nonuniform density in the equilibrium system and the external field potential are biunique functionals. In this case, the nonuniform density is intimately related to the inverse response function. On this basis, a regular procedure can be constructed for determining the density functional for the free energy or ground state energy.
For a many-electron system, whether the particle density $\ensuremath{\rho}(\mathbf{r})$ and the total current density j(r) are sufficent to determine the one-body potential V(r) and vector potential A(r) is still an open question. For the one-electron case, a Hohenberg-Kohn theorem exists formulated with the total current density. Here we show that the generalized Hohenberg-Kohn energy functional ${\mathcal{E}}_{{V}_{0},{\mathbf{A}}_{0}}(\ensuremath{\rho},\mathbf{j})=\ensuremath{\langle}\ensuremath{\psi}(\ensuremath{\rho},\mathbf{j}),H({V}_{0},{\mathbf{A}}_{0})\ensuremath{\psi}(\ensuremath{\rho},\mathbf{j})\ensuremath{\rangle}$ can be minimal for densities that are not the ground-state densities of the fixed potentials V0 and A0. Furthermore, for an arbitrary number of electrons and under the assumption that a Hohenberg-Kohn theorem exists formulated with $\ensuremath{\rho}$ and j, we discuss the possibility of a variational principle in total current-density-functional theory such as that of Hohenberg-Kohn.
Based on the Schrodinger equation, exact expressions for the non-relativistic particle energy in the local external field and the external field potential are derived as inhomogeneous density functionals. On this basis, it is shown that, when considering more than two noninteracting electrons, the energy of such a system cannot be an inhomogeneous density functional. This means that the Hohenberg-Kohn lemma which assert that in the ground state to each inhomogeneous density corresponds only one potential of the external field cannot be a justification of the existence of the universal density functional in the general case. At the same time, statements of the density functional theory remain valid when considering any number of noninteracting ground-state bosons due to the Bose condensation effect.