Lower bound on the coefficient of an exchangelike energy for a single bound electron in a screened Coulomb potential

The spurious self-interaction energy, which is one-half the integrated electrostatic interaction energy of a normalized electronic charge distribution with the potential field generated by itself, is calculated analytically for a $1s$-like electron charge distribution governed by an attractive screened Hulthen-type model potential. The exact result obtained is used, via an energetic cancellation constraint and a convenient Hölder’s integral inequality, to deduce an analytical lower bound for the screening-dependent positive coefficient of a negative exchange-like energy term applied routinely in the local-density approximation to density functional theory with Coulomb potentials. The bare Coulomb case is investigated by Sobolev’s integral inequality as well.

Figure 1

The solid and dashed curves in the upper panel refer to $E_{\mathrm{si}}(x)/Z$ and $H(x)/Z$, respectively. These energies from Eqs. (8) and (12) are given in atomic units. The lower panel is devoted to the ${\beta }_h(x)=E_{\mathrm{si}}(x)/H(x)$ ratio, which represents the screening dependence of the coefficient of an exchange-like energy term.