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.
Within the framework of zero-temperature Fock-space density-functional theory (DFT), we prove that the G\^ateaux functional derivative of the universal density functional, ${\ensuremath{\delta}{F}^{\ensuremath{\lambda}}[\ensuremath{\rho}]∕\ensuremath{\delta}\ensuremath{\rho}(\mathbit{r})\ensuremath{\mid}}_{\ensuremath{\rho}={\ensuremath{\rho}}_{0}}$, at ground-state densities with arbitrary normalizations $(⟨{\ensuremath{\rho}}_{0}(\mathbit{r})⟩=n∊{\mathcal{R}}_{+})$ and an electron-electron interaction strength $\ensuremath{\lambda}$, is uniquely defined, but is discontinuous when the number of electrons
n becomes an integer, thus providing a mathematically rigorous confirmation for the ``derivative discontinuity'' initially discovered by Perdew et al. [Phys. Rev. Lett. 49, 1691 (1982)]. However, the functional derivative of the exchange-correlation functional is continuous with respect to the number of electrons in Fock space; i.e., there is no ``derivative discontinuity'' for the exchange-correlation functional at an integer electron number. For a ground-state density ${\ensuremath{\rho}}_{0,n}^{v,\ensuremath{\lambda}}(\mathbit{r})$ of an external potential $v(\mathbit{r})$, we show that ${\ensuremath{\delta}{F}^{\ensuremath{\lambda}}[\ensuremath{\rho}]∕\ensuremath{\delta}\ensuremath{\rho}(\mathbit{r})\ensuremath{\mid}}_{\ensuremath{\rho}={\ensuremath{\rho}}_{0,n}^{v,\ensuremath{\lambda}}}={\ensuremath{\mu}}_{\mathrm{SM}}^{n}\ensuremath{-}v(\mathbit{r})$, where the constant ${\ensuremath{\mu}}_{\mathrm{SM}}^{n}$ is given by the following chain of dependences: ${\ensuremath{\rho}}_{0,n}^{v,\ensuremath{\lambda}}(\mathbit{r})\ensuremath{\mapsto}[v]\ensuremath{\mapsto}{E}_{0}^{v,\ensuremath{\lambda}}(n)\ensuremath{\mapsto}{\ensuremath{\mu}}_{\mathrm{SM}}^{n}={\ensuremath{\partial}{E}_{0}^{v,\ensuremath{\lambda}}(k)∕\ensuremath{\partial}k\ensuremath{\mid}}_{k=n}$. Here
[v] is the class of the external potential $v(\mathbit{r})$ up to a real constant, and ${\ensuremath{\mu}}_{\mathrm{SM}}^{n}$ is the chemical potential defined according to statistical mechanics. At an integer electron number
N, we find that there is no freedom of adding an arbitrary constant to the value of the chemical potential ${\ensuremath{\mu}}_{\mathrm{SM}}^{N}$, whose exact value is generally not the popular preference of the negative of Mulliken's electronegativity, $\ensuremath{-}\frac{1}{2}(I+A)$, where
I and
A are the first ionization potential and the first electron affinity, respectively. In addition, for any external potential converging to the same constant at infinity in all directions, we resolve that ${\ensuremath{\mu}}_{\mathrm{SM}}^{N}=\ensuremath{-}I$. Finally, the equality ${\ensuremath{\mu}}_{\mathrm{DFT}}={\ensuremath{\mu}}_{\mathrm{SM}}^{n}$ is rigorously derived via an alternative route, where ${\ensuremath{\mu}}_{\mathrm{DFT}}$ is the Lagrangian multiplier used to constrain the normalization of the density in the traditional DFT approach.
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.
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.
It is shown that the process of conventional functional differentiation does not apply to functionals whose domain (and possibly range) is subject to the condition of integral normalization, as is the case with respect to a domain defined by wave functions or densities, in which there exists no neighborhood about a given element in the domain defined by arbitrary variations that also lie in the domain. This is remedied through the generalization of the domain of a functional to include distributions in the form of , where is the Dirac delta function and is a real number. This allows the determination of the rate of change of a functional with respect to changes of the independent variable determined at each point of the domain, with no reference needed to the values of the functional at different functions in its domain. One feature of the formalism is the determination of rates of change of general expectation values (that may not necessarily be functionals of the density) with respect to the wave functions or the densities determined by the wave functions forming the expectation value. It is also shown that ignoring the conditions of conventional functional differentiation can lead to false proofs, illustrated through a flaw in the proof that all densities defined on a lattice are -representable. In a companion paper, the mathematical integrity of a number of long-standing concepts in density functional theory are studied in terms of the formalism developed here.
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.