Momentum density and spatial form of correlated density matrix in model two-electron atoms with harmonic confinement

Abstract

The detailed nature of the correlated first-order density matrix for the model atoms in the title for arbitrary interparticle interaction $u({r}_{12})$ is studied. One representation with contracted information is first explored by constructing the momentum density $\ensuremath{\rho}(\mathbf{p})$ in terms of the wave function of the relative motion, say ${\ensuremath{\Psi}}_{R}({r}_{12})$, which naturally depends on the choice of $u({r}_{12})$. For $u({r}_{12})={e}^{2}∕{r}_{12}$, the so-called Hookean atom, and for the inverse square law $u({r}_{12})=\ensuremath{\lambda}∕{r}_{12}^{2}$, plots are presented of the above density $\ensuremath{\rho}(\mathbf{p})$ in momentum space. The correlated kinetic energy is recovered from averaging ${p}^{2}∕2m$, $m$ denoting the electron mass, with respect to $\ensuremath{\rho}(\mathbf{p})$. The second method developed is in coordinate space and expands the density matrix $\ensuremath{\gamma}({\mathbf{r}}_{1},{\mathbf{r}}_{2})$ in Legendre polynomials, using relative coordinate ${\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{2}$, center-of-mass coordinate $({\mathbf{r}}_{1}+{\mathbf{r}}_{2})∕2$ and the angle, $\ensuremath{\theta}$ say, between these two vectors. For the Moshinsky atom in which $u({r}_{12})=\frac{1}{2}k{r}_{12}^{2}$ only the $s$ term $(l=0)$ contributes to the Legendre polynomial expansion. The specific example we present of the inverse square law model is shown to be characterized by the low-order terms $(s+d)$ of the Legendre expansion. The Wigner function is finally calculated analytically for both Moshinsky and inverse square law models.