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.
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.
In both molecular physics and condensed matter theory, deeper understanding of the correlation energy density
ϵc(r) remains a high priority. By adopting L\"owdin's definition of correlation energy as the difference between the exact and the Hartree-Fock values, here we propose two alternative routes to define this. One of these involves both exact and Hartree-Fock (HF) wave functions, while the second requires a coupling constant integration. As an exact analytical example of the first route, we treat the two-electron model atom of Moshinsky, for which both confinement potential and interactions are harmonic. Though the correlation energy density
ϵc(r) is known analytically, we also investigate numerically its relation to the exact ground-state density in this example.
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.
We take as a starting point the ground-state electron density in two-electron model atoms in which Coulomb confinement in the He atom is first replaced by harmonic restoring forces. Switching off electron–electron interactions, one readily constructs a third-order differential equation for the ground-state electron density, as in the recent work of March and Ludeña (2004 Phys. Lett. A 330 16). We then switch on two different model interactions, first in the so-called Hookean atom going back to Kestner and Sinanoglu (1962 Phys. Rev. 128 2687), in which e 2 / r 12 is retained as in He, and secondly in the Moshinsky (1968 Am. J. Phys. 36 52) atom in which Kr 2 12 /2 is switched on. Some analyticity properties of the low-order linear homogeneous differential equations which result are next studied. He-like atomic ions are then treated in the limit of large atomic number Z . In this latter case, one identifies both the electron–nuclear cusp, or equivalently Kato's theorem, and the corresponding electron–electron cusp in the ground-state spatial wavefunction Ψ( r 1 , r 2 ). A final comment concerns quantum information and entanglement in relation to the recent work of Amovilli and March (2004 Phys. Rev. A 69 054302).
The exact ground-state solutions for the model of two particles in a confining harmonic oscillator potential interacting through a repulsive harmonic oscillator force are used from the standpoint of geometric distances [Phys. Rev. Lett. 106, 050401 (2011).] between wave functions and densities. The distances from the noninteracting reference state are calculated at a specified confinement by increasing the coupling of the interparticle interaction. Based on the analytic expressions for coupling-dependent geometric measures, a discussion of the Hohenberg-Kohn mapping is given.