In a recent paper A. Akbari, N. H. March, and A. Rubio [Phys. Rev. A 76, 032510 (2007)] investigated the one-electron reduced density matrix and the momentum density of several two-electron model atoms, including Hooke's atom. The method used by the authors for deriving an integral form of the momentum density is in fact the most suited for deriving a closed-form expression of the exact reciprocal form factor, in a special case. This function is important inasmuch as it reflects the off-diagonal side of the exact reduced density matrix.
The detailed nature of the correlated first-order density matrix for the model atoms in the title for arbitrary interparticle interaction
u(r12) 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(r12). For
u(r12)=e2∕r12, 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
p2∕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
(r1+r2)∕2 and the angle, $\ensuremath{\theta}$ say, between these two vectors. For the Moshinsky atom in which
u(r12)=21kr122 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.
Holas, Howard and March [Phys. Lett. A {\bf 310}, 451 (2003)] have obtained analytic solutions for ground-state properties of a whole family of two-electron spin-compensated harmonically confined model atoms whose different members are characterized by a specific interparticle potential energy u(
r12). Here, we make a start on the dynamic generalization of the harmonic external potential, the motivation being the serious criticism levelled recently against the foundations of time-dependent density-functional theory (e.g. [J. Schirmer and A. Dreuw, Phys. Rev. A {\bf 75}, 022513 (2007)]). In this context, we derive a simplified expression for the time-dependent electron density for arbitrary interparticle interaction, which is fully determined by an one-dimensional non-interacting Hamiltonian. Moreover, a closed solution for the momentum space density in the Moshinsky model is obtained.
Moshinsky introduced an exactly soluble model of a two-electron atom consisting of two spin-
21 particles interacting via harmonic forces and moving in a harmonic-oscillator potential. Here, the exact ground-state density $\ensuremath{\rho}(r)$ is related to the (also analytically known) Hartree-Fock density ${\ensuremath{\rho}}_{\mathrm{HF}}(r).$ The generalization to the off-diagonal matrix $\ensuremath{\gamma}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})$ is then effected, this being related to the idempotent ${\ensuremath{\gamma}}_{\mathrm{HF}}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})/2.$ This exact information on this ``model atom'' prompts us to propose an approximate form of $\ensuremath{\gamma}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})$ for the He-like ions, the
H2 molecule and, in general, all two-electron systems. $\ensuremath{\gamma}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})$ is constructed solely from the exact $\ensuremath{\rho}(\mathbf{r})$ and its Hartree-Fock counterpart. Some detailed treatment of the two-electron Hookean atom with spring constant
k=1/4 (atomic units) is also presented.