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

The detailed nature of the correlated first-order density matrix for the model atoms in the title for arbitrary interparticle interaction $u\left(r_{12}\right)$ is studied. One representation with contracted information is first explored by constructing the momentum density $\rho \left(\mathbf{p}\right)$ in terms of the wave function of the relative motion, say ${\Psi }_R\left(r_{12}\right)$, which naturally depends on the choice of $u\left(r_{12}\right)$. For $u\left(r_{12}\right)=e^2∕r_{12}$, the so-called Hookean atom, and for the inverse square law $u\left(r_{12}\right)=\lambda ∕r_{12}^2$, plots are presented of the above density $\rho \left(\mathbf{p}\right)$ in momentum space. The correlated kinetic energy is recovered from averaging $p^2∕2m$, $m$ denoting the electron mass, with respect to $\rho \left(\mathbf{p}\right)$. The second method developed is in coordinate space and expands the density matrix $\gamma ({\mathbf{r}}_1,{\mathbf{r}}_2)$ in Legendre polynomials, using relative coordinate ${\mathbf{r}}_1-{\mathbf{r}}_2$, center-of-mass coordinate $({\mathbf{r}}_1+{\mathbf{r}}_2)∕2$ and the angle, $\theta$ say, between these two vectors. For the Moshinsky atom in which $u\left(r_{12}\right)=\frac{1}{2}k{r}_{12}^2$ 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.

Figure 1

Ground-state density $\rho \left(p\right)$ for the three models of harmonically confined two-electron atoms $(k=\frac{1}{4})$. (Dashed line) Moshinsky model with $\beta =2$. (Solid curve) Hookean atom. (Dashed-point line) Inverse square law repulsion with $\alpha =2$.

Figure 2

(Color online) Wigner function ${\gamma }_w(c,p)$ for the Moshinsky model with $k=\frac{1}{4}$ in a.u. and $\beta =2$.

Figure 3

(Color online) $s$-state $(l=0)$ component ${\gamma }_0^w(c,p)$ of Wigner function ${\gamma }_w(c,p)$ in Eq. (34), for inverse square law $u\left(r_{12}\right)=\lambda ∕r_{12}^2$, with $k=\frac{1}{4}$ in a.u., $\lambda$ related to $\alpha$ in relation (21) and $\alpha =2$.

Figure 4

(Color online) $d$-state $(l=2)$ component ${\gamma }_2^w(c,p)$ of Wigner function ${\gamma }_w(c,p)$ in Eq. (34), for Inverse square law $u\left(r_{12}\right)=\lambda ∕r_{12}^2$, with $k=\frac{1}{4}$ in a.u., $\lambda$ being related to $\alpha$ as in relation (21) and $\alpha =2$.

Figure 5

Scattering factors $f\left(G\right)$, which are Fourier transforms of ground-state density $n\left(\mathbf{r}\right)$ as in Eq. (A1). (Dashed line) Moshinsky model with $\beta =2$. (Solid curve) Hookean atom. (Dashed-point line) Inverse square law repulsion with $\alpha =2$. The harmonic confinement potential in the three curves is equal, chosen in such a way $a=1$ in a.u.

Figure 6

Illustrates departures from idempotency for the Moshinsky model. (a) Integrated difference between sides of Eqs. (B3, B6), versus interaction strength represented by parameter $\beta$ in Eq. (12). (b) Plot of Eqs. (B4, B5) for $2n\left(\mathbf{r}\right)$ (dashed line) and $\int {\left[\gamma (\mathbf{r},{\mathbf{r}}^{\prime })\right]}^2$ $d{\mathbf{r}}^{\prime }$ (solid curve), with $a=1$ in Eq. (4) and $\beta =2$ in relation (12).