A method is presented for relating a ground-state density distribution to the effective single-particle potential which has this density for its ground state for a noninteracting system of particles. The method is useful for systems consisting of a small number of spinless fermions and in the case of two particles a second-order, nonlinear differential equation relates the density and the effective potential. The question of
v representability is addressed through examples of densities for one- and three-dimensional systems. It is found that densities of the form ${e}^{\ensuremath{-}ar}$ for two-particle systems are not ground-state
v representable. The added flexibility of three dimensions over one allows an effective potential which does not necessarily have the chosen density as its ground state, but as its lowest nondegenerate state.
We consider the v-representability of the particle density for a noninteracting system of spinless fermions by introducing the idea of proper order of a set of energy levels. It is shown that if
E1(\ensuremath{\lambda}),
E2(\ensuremath{\lambda}), and
E3(\ensuremath{\lambda}) are three energy levels associated with some local potential ${\mathit{V}}_{\ensuremath{\lambda}}$(r) that is a continuous function of \ensuremath{\lambda}=(${\ensuremath{\lambda}}_{1}$,${\ensuremath{\lambda}}_{2}$,${\ensuremath{\lambda}}_{3}$) over all possible points \ensuremath{\lambda}, where ${\ensuremath{\lambda}}_{\mathit{i}}$ is the occupation number of the ith state and M=${\ensuremath{\lambda}}_{1}$+${\ensuremath{\lambda}}_{2}$+${\ensuremath{\lambda}}_{3}$ is the total number of particles distributed over the three levels, then there must be at least one \ensuremath{\lambda} for which the three levels are in so-called proper order, in which the levels below the highest occupied level are filled. This result provides a basis for the proof of ensemble v-representability of some N-particle density for which the ground-state degeneracy of the system is no more than three. As examples, three- and two-dimensional central systems are examined, and an N-particle central density is shown to be ensemble v-representable for small N (N\ensuremath{\le}14 and N\ensuremath{\le}9 for three- and two-dimensional cases, respectively). The implications for density-functional theory are discussed.
A particle-density distribution is said to be v-representable if it is the ground-state density for some potential v(r). By developing the idea of proper order, proposed earlier [J. Chen and M. J. Stott, Phys. Rev. A 44, 2816 (1991)], to any number of energy levels which are involved in possible ground-state configurations, we show that any reasonable density with an arbitrary number of spinless fermions is either pure-state or ensemble v-representable. This result supports the general applicability of the Kohn-Sham method [Phys. Rev. 140, A1133 (1965)] in density-functional theory.
The treatment of degenerate states within Kohn-Sham density functional theory is a problem of long-standing and current interest. We propose a solution to this mapping from the interacting degenerate system to that of the noninteracting fermion model whereby the equivalent density and energy are obtained via the unifying physical framework of quantal density functional theory. We describe the quantal theory of both ground and excited degenerate states, and for the cases of both pure state and ensemble v-representable densities. The quantal description further provides a rigorous physical interpretation of the corresponding Kohn-Sham energy functionals of the density, ensemble density, bidensity and ensemble bidensity, and of their respective functional derivatives. We conclude with examples of the mappings within the quantal theory.
It is shown that if
n(r) is the discrete density on a lattice (enclosed in a finite box) associated with a nondegenerate ground state in an external potential
v(r) (i.e., is "
v-representable"), then the density $n(r)+\ensuremath{\mu}m(r)$, with
m(r) arbitrary (apart from trivial constraints) and $\ensuremath{\mu}$ small enough, is also associated with a nondegenerate ground state in an external potential ${v}^{\ensuremath{'}}(r)$ near
v(r); i.e., $n(r)+\ensuremath{\mu}m(r)$ is also
v-representable. Implications for the Hohenberg-Kohn variational principle and the Kohn-Sham equations are discussed.
In density-functional theory (DFT) of the ground state, a density distribution
no(r) is called v-representable (VR) if it is the ground-state density in some external potential. [It is known that not all ``reasonable''
n0(r) are VR.] In DFT of time-dependent linear response of a nondegenerate ground state, a similar question arises: Is a response density,
n1(r,\ensuremath{\omega})${e}^{\mathrm{\ensuremath{-}}i\ensuremath{\omega}t}$, VR, i.e., is it the response to some perturbing potential
v1(r,\ensuremath{\omega})${e}^{\mathrm{\ensuremath{-}}i\ensuremath{\omega}t}$ [E.K.U. Gross and W. Kohn, Phys. Rev. Lett. 55, 2850 (1985)]? In the present paper we show that (1) if the frequency \ensuremath{\omega}${\ensuremath{\omega}}_{\mathrm{min}}$ (the lowest excitation frequency), the answer is affirmative; and (2) if \ensuremath{\omega}>${\ensuremath{\omega}}_{\mathrm{min}}$, the answer is not necessarily affirmative, as demonstrated by counterexamples. [We exhibit ``reasonable'' functions
n1(r,\ensuremath{\omega})${e}^{\mathrm{\ensuremath{-}}i\ensuremath{\omega}t}$ which, at isolated frequencies, are not VR.] Implications for time-dependent DFT of linear response are discussed.