The Constrained Search Formulation

The second Hohenberg-Kohn theorem has two drawbacks. Firstly, it assumes that there is no degeneracy in the ground state, and secondly the density must be v-representable: it must arise from a wavefunction with a Hamiltonian able to be written in the form of equation (2.2). The specific conditions that make a density v-representable are unknown, but many `reasonable' densities have been shown to be non-v-representable [72,73].

A weaker constraint, that the density is N-representable, can be used if the Levy constrained-search is used [74,72]. A density is N-representable if it can be obtained from some antisymmetric wavefunction. The theory begins by showing how to distinguish the ground state wavefunction $\Psi_{0}$ from a wavefunction $\Psi_{\rho_{0}}$ that simply integrates to the ground state density $\rho_{0}({\bf r})$. The variational principle gives
$$\begin{align}\langle \Psi_{\rho_{0}}\vert\hat{H}\vert\Psi_{\rho_{0}} \rangle \ge \langle \Psi_{0}\vert\hat{H}\vert\Psi_{0} \rangle = E_{0}. \end{align}$$
Remembering that the potential energy due to the external field $v({\bf r})$ is a function of the density leads to
$$\begin{align}\langle \Psi_{\rho_{0}}\vert\hat{T}+\hat{V}_{ee}\vert\Psi_{\rho_{0}... ...e &\ge \langle\Psi_{0}\vert\hat{T}+\hat{V}_{ee}\vert\Psi_{0}\rangle. \end{align}$$
Thus the $\Psi_{0}$ is the wavefunction that integrates to $\rho_{0}$ and minimizes the expectation value of $\hat{T} + \hat{V}_{ee}$. Defining our universal functional as
$$\begin{align}F[\rho] = \min_{\Psi\to\rho}\langle \Psi\vert\hat{T}+\hat{V}_{ee}\vert\Psi\rangle, \end{align}$$
where $F[\rho]$ searches all $\Psi$ that yield the input density $\rho$, allows the energy to be expressed as
$$\begin{align}E_{0} &= \min_{\rho} \left[F[\rho] + \int v({\bf r})\rho({\bf r})\,d{\bf r}\right] \\ &= \min_{\rho} E[\rho] \end{align}$$
where
$$\begin{align}E[\rho]= F[\rho] + \int v({\bf r})\rho({\bf r})\,d{\bf r} \end{align}$$
which is a search over all N-representable densities. Thus, the v-representable problem has been removed. The restriction requiring no degeneracy has also been lifted. In degenerate systems the wavefunction giving $\rho({\bf r})$ will be selected. All that remains is to find an accurate form for the functional $E[\rho]$.