Guaranteed Convergence of the Kohn-Sham Equations

A sufficiently damped iteration of the Kohn-Sham (KS) equations with the exact functional is proven to always converge to the true ground-state density, regardless of the initial density or the strength of electron correlation, for finite Coulomb systems. We numerically implement the exact functional for one-dimensional continuum systems and demonstrate convergence of the damped KS algorithm. More strongly correlated systems converge more slowly.

Figure 1

(a) The input and output densities for a single step of the Kohn-Sham scheme, as well as the exact density, of a one-dimensional, strongly correlated four atom, four electron system. (b) The energy of the system which interpolates between the input and output densities $E_v[n_{\lambda }]$, measured from the ground-state energy $E_v^{\mathrm{gs}}$. Also shown is the linear-response approximation with slope given by Eq. (12).

Figure 2

KS procedure for a moderately correlated four-electron system (four hydrogen atoms with $R=3$), showing the first few iterations. Using a fixed $\lambda =0.30$, we converge to $\eta <{10}^{-6}$ using Eq. (4) within 13 iterations.

Figure 3

Differences in the density $\eta$ using Eq. (4) and the energy with $\Delta E=E_v[n^{\prime }]-E_v^{\mathrm{gs}}$, for an ${\mathrm{H}}_2$ molecule with (a) $R=1.6$ and (b) $R=3$. In (b), the $\Delta E$ curves are omitted for clarity, but are like those in (a).

Figure 4

Starting an exact KS calculation of stretched ${\mathrm{H}}_2$ with a spin-polarized density still converges (with $\lambda =0.5$) to the correct spin-singlet density. For the same initial density, the KS calculation with the local spin-density (LSD) approximation [47] converges to the broken spin-symmetry solution shown.