Use of the differential virial theorem to estimate the spatial variation of the exchange-correlation force $-\partial V_{XC}\left(r\right)∕\partial r$ in the ground states of the spherical atoms He and Be

We use the differential virial theorem (DVT) directly to display the approximate spatial dependence of the exchange-correlation (XC) force in He and Be, applying an exact integral constraint on the XC force, recently established by March and Nagy. In He, an analytic ground-state density $n\left(r\right)$, combined with the DVT plus the von Weizsäcker single-particle kinetic energy, suffices to determine an approximate XC force. For Be, the XC force is calculated for the semiempirical fine-tuned Hartree-Fock density, as proposed by Cordero et al. [Phys. Rev. A 75, 052502 (2007)]. However, for the single-particle kinetic energy, following Dawson and March, a phase $\theta \left(r\right)$ must be obtained by solving numerically a nonlinear pendulumlike equation.

Figure 1

(Color online) The exchange-correlation force $f_{XC}$ for the He atom using the density obtained from the Chandrasekhar wave function. Blue, dotted line: Green’s variational parameters and $Z=2$. Red, dashed line: Green’s variational parameters and $Z=1.963$ effective nuclear charge. Black, solid line: Howard’s consistent parameters, $Z=2$.

Figure 2

(Color online) The phase factor $\Theta \left(r\right)$ for the Be atom as the result of the numerical solution of the corresponding differential equation using the fine-tuned HF density.

Figure 3

(Color online) The exchange-correlation force $f_{XC}$ for the Be atom from the fine-tuned HF density. Blue, solid line: the Kohn-Sham force from the force equation. Red, dashed line: the $\mathrm{X}\alpha$ force with $\alpha =0.7$ for comparison.