Particle Number and Probability Density Functional Theory, and A-representability

In Hohenberg-Kohn (HK) density functional theory (DFT), the functional $F_{HK}[\rho]$ of the density $\rho({\bf{r}})$ representing the expectation of the electron-interaction and kinetic energy operators is universal. Knowledge of $F_{HK} [\rho]$ by itself is insufficient to obtain the energy: the electron number $N$ is primary. By emphasizing this primacy of $N$, we rewrite the energy $E$ as a nonuniversal functional of $N$ and probability density $p({\bf{r}}): E = E[N, p]$, with $p ({\bf{r}})$ satisfying the constraints of normalization to unity and positivity. A particle number $N$ and probability density $p ({\bf{r}})$ functional theory is constructed, and examples of exact functionals provided. The concept of $A$-representability is introduced as the set of functions $\psi_{p}$ that lead to quantum mechanical $p({\bf{r}})$ as the expectation of the probability density operator. We show via the Harriman and Gilbert constructions that the $A$- and $N$-representable probability density $p({\bf{r}})$ sets are equivalent, with the latter defined as $p({\bf{r}}) = \rho({\bf{r}})/N$.