Solubility of the optimized-potential-method integral equation for finite systems

We provide a detailed analysis of the solubility of the optimized-potential-method (OPM) integral equation for the case of the orbital- and eigenvalue-dependent correlation energy functional $E_{\mathrm{c}}^{\mathrm{MP}2}$ obtained by second-order perturbation theory on the basis of the Kohn-Sham Hamiltonian. For this functional it was shown [Phys. Rev. Lett., 86, 2241 (2001)] that for free atoms no solution of the OPM equation can be found which satisfies the boundary condition $v_{\mathrm{c}}^{\mathrm{MP}2}(r\to \infty )=0$. On the other hand, there exists a proof that $v_{\mathrm{c}}^{\mathrm{MP}2}\left(r\right)$ decays like $1∕r^4$ [J. Chem. Phys., 118, 9504 (2003)]. Here we resolve the obvious contradiction by demonstrating that (i) the OPM equation cannot be solved if continuum states are present, (ii) the OPM equation cannot be solved for a free atom if only a finite number of Rydberg states are included in $E_{\mathrm{c}}^{\mathrm{MP}2}$, and (iii) the OPM equation does allow a solution satisfying $v_{\mathrm{c}}^{\mathrm{MP}2}(r\to \infty )=0$ in the case of finite systems with a countable spectrum (exemplified by an atom in a spherical box), if the complete spectrum is taken into account in the OPM procedure.

Figure 1

$G_{1s}(r,r)$ for the He atom: exact result (50) versus sum-over-states form (22), in which only a finite number of unoccupied Rydberg shells is included.

Figure 2

$Q_{\mathrm{c}}$ for the He atom: dependence of $Q_{\mathrm{c}}$ on the number of unoccupied Rydberg states included in $E_{\mathrm{c}}^{\mathrm{DD}}$. For comparison also $Q_{\mathrm{x}}$ is plotted.

Figure 3

Absolute value of $Q_{\mathrm{c}}$ for the He atom: dependence of individual components on the number of unoccupied Rydberg states included in $E_{\mathrm{c}}^{\mathrm{DD}}$. For comparison also $Q_{\mathrm{x}}$ is plotted.

Figure 4

As in Fig. 2, but $G_{nl}$ constructed from a finite number of Rydberg states only.

Figure 5

As in Fig. 3, but $G_{nl}$ constructed from a finite number of Rydberg states only.

Figure 6

$Q_{\mathrm{c}}$ for the He atom in a box: dependence of $Q_{\mathrm{c}}$ on the number of unoccupied Rydberg states included in $E_{\mathrm{c}}^{\mathrm{MP}2}$. For comparison also $Q_{\mathrm{x}}$ is plotted. The highest shell included is $n_{\max }=300$ ($l_{\max }=6$, $i_{\max }=6400$). The box radius is chosen to be $R_0=20\text{bohrs}$.

Figure 7

$Q_{\mathrm{c}}$ for the He atom in a box: range of cancellation between $Q_{\mathrm{c}}^{\mathrm{a}}$ and $Q_{\mathrm{c}}^{\mathrm{b}}$ as a function of the number of unoccupied shells $n$ included in $E_{\mathrm{c}}^{\mathrm{MP}2}$ ($l_{\max }=6$, $i_{\max }=6400$).

Figure 8

Ratio (60) for the He atom in a box: results for the MP2 functional ($n_{\max }=300$, $l_{\max }=6$, $R_0=20\text{bohrs}$).