Density-matrix theory for the ground state of spin-compensated harmonically confined two-electron model atoms with general interparticle repulsion

For model two-electron atoms with harmonic confinement, the correlated first-order density matrix can be expressed in terms of the relative motion wave function ${\Psi }_R\left(r\right)$. Here we demonstrate that the probability density $P\left(r\right)$ associated with this wave function is directly related to the x-ray scattering factor $f\left(G\right)$. This latter quantity, in turn, is determined by the ground-state electron density $n\left(r\right)$. The Euler-Lagrange equation of the resulting density-matrix theory is thereby shown to take the form of a third-order integro-differential equation for $n\left(r\right)$ in which the probability density $P\left(r\right)={\Psi }_R^2\left(r\right)$ also appears. For two specific choices of the interaction between the two fermions under consideration, the above integro-differential equation derived here is shown to lead back to known linear homogeneous differential equations for the electron density. Finally, it is emphasized that specific equations summarized here will apply directly to theoretical study of the nonrelativistic ground-state electron density $n\left(r,Z\right)$ in the He-like ions with atomic number $Z$.

Figure 1

(Color online) Shows magnitude $z\left(r\right)$ of the vector field $\mathbf{z}\left(\mathbf{r}\right)$ entering the DVT Eq. (21) for the Moshinsky atom for parameter $\beta =3/2$ measuring the interparticle interaction strength.

Figure 2

(Color online) Comparison of correlated and independent-particle energy densities $4\pi r^2{t}_g\left(r\right)$ (continuous curve) given analytically in Eq. (B9) and $4\pi r^2{t}_w\left(r\right)$, respectively, $t_w\left(r\right)$ being the von Weizsäcker density [10].

Figure 3

(Color online) Comparison of different radial kinetic-energy densities $4\pi r^{2}t\left(r\right)$ and $4\pi r^2{t}_g\left(r\right)$ for the inverse square-law model with $\alpha =1$. $t\left(r\right)$ (continuous curve) is taken from Eq. (18), while $4\pi r^2{t}_g\left(r\right)$ is given in Eq. (B9). The areas under the two curves, of course, are equal, corresponding to the same total kinetic energy. Atomic units are used; the length $a$ being chosen as $2^{-1/2}$.