Construction of Wave Function Functionals

We recently proposed [1] expanding the space of variations in calculations of the energy by considering the approximate wave function $\Psi$ to be a functional of functions $\chi$, $\Psi = \Psi[\chi]$, rather than a function. A constrained search is first performed over all functions $\chi$ such that $\Psi[\chi]$ satisfies a physical constraint or leads to a known value of an observable. A rigorous upper bound to the energy is then obtained via the variational principle. In this paper we apply this idea to the ground state of the He atom by constructing $\Psi[\chi]$ that reproduce the exact expectations of the Hermitian single- and two-particle operators $W = \sum_{i} r_{i}^{n}, n = -2, -1, 1, 2$; $W =\sum_{i}\delta ({\bf r}_{i})$; $W=|{\bf r}_{1}-{\bf r}_{2}|^{m}, m=-1,-2,1,2$. The functionals are of the form $\Psi[\chi] = \Phi [1 - f(\chi)]$, where $\Phi$ is a prefactor and $f(\chi)$ a correlation factor. The $\Psi[\chi]$ (\emph{i}) lead to the exact expectation value of $W$; (\emph {ii}) are automatically normalized; and (\emph{iii}) provide a rigorous upper bound to the energy. [1] X.-Y. Pan, \textit {et al}, PRA \textbf{72}, 032505 (2005).