Towards a differential equation for the nonrelativistic ground-state electron density of the He-like sequence of atomic ions

The early study of Schwartz [Ann. Phys. (N.Y.) 6, 156 (1959)] led to an explicit expression for the ground-state electron density $\rho \left(r\right)$ of He-like atomic ions with nuclear charge $Ze$ in the limit of large $Z$. Much later, Gál, March, and Nagy [Chem. Phys. Lett. 305, 429 (1999)] derived a third-order linear homogeneous differential equation satisfied by the Schwartz limiting density. Our aim here has been to solve a still correlated model more closely related to the He atom itself. Motivated by semiclassical studies (as by Handke [Phys. Rev. A 50, R3561 (1994)]), we have solved a fully quantal model in which the Coulomb repulsion energy $e^2∕r_{12}$ of the two electrons at separation $r_{12}$ is expanded in a one-center form about the atomic nucleus and only the $s$-wave term is retained. From the resulting analytical ground-state density, a differential equation has been derived, which is contrasted with that satisfied by the large-$Z$ limiting form of Schwartz. Suggestions are finally made as to the way the density of the He atomic ion series may be characterized, the ionization potential $I\left(Z\right)$ being proposed as crucial input data.

Figure 1

Plot of the ground-state density $\rho \left(r\right)$ from Eq. (2.11) for $Z=2$. The hydrogenic density is shown for comparison, and the role of screening is seen to be crucial.

Figure 2

Nonrelativistic ground-state density for He-like “$s$-wave” model with Eq. (2.11) used for $Z=92$. The result is now seen to be graphically indistinguishable from the hydrogenic limiting density shown.

Figure 3

X-ray scattering factor $f\left(k\right)$ derived from the density shown in Fig. 1 for $Z=2$, using the Fourier transform relationship (A1). The results of Eqs. (A2) and (A3) for small and large $k$, respectively, are indicated by the dashed lines.

Figure 4

X-ray scattering factor $f\left(k\right)$ derived from the density shown in Fig. 2 for $Z=92$, using Eq. (A1). Results of Eqs. (A2, A3) for small and large $k$, respectively, are indicated by the dashed lines.