Calculating excitation energies by extrapolation along adiabatic connections

In this paper, an alternative method to range-separated linear-response time-dependent density-functional theory and perturbation theory is proposed to improve the estimation of the energies of a physical system from the energies of a partially interacting system. Starting from the analysis of the Taylor expansion of the energies of the partially interacting system around the physical system, we use an extrapolation scheme to improve the estimation of the energies of the physical system at an intermediate point of the range-separated or linear adiabatic connection where either the electron–electron interaction is scaled or only the long-range part of the Coulomb interaction is included. The extrapolation scheme is first applied to the range-separated energies of the helium and beryllium atoms and of the hydrogen molecule at its equilibrium and stretched geometries. It improves significantly the convergence rate of the energies toward their exact limit with respect to the range-separation parameter. The range-separated extrapolation scheme is compared with a similar approach for the linear adiabatic connection, highlighting the relative strengths and weaknesses of each approach.

Figure 1

Helium ground-state energy (top) ${\mathcal{E}}_0^{\mu }$, and first $S$ (middle) and $P$ (bottom) excitation energies $\Delta {\mathcal{E}}_k^{\mu }={\mathcal{E}}_k^{\mu }-{\mathcal{E}}_0^{\mu }$ calculated without extrapolation (full lines), with first-order extrapolation (dashed), and second-order extrapolation (dot-dashed) as a function of $\mu$. The dotted horizontal lines are the physical energies. The colored regions represent errors of $\pm 10$ and $\pm 1$ mhartree for the ground-state and excitation energies, respectively.

Figure 2

Ground-state energy ${\mathcal{E}}_0^{\mu }$ (top) and excitation energies $\Delta {\mathcal{E}}_k^{\mu }={\mathcal{E}}_k^{\mu }-{\mathcal{E}}_0^{\mu }$ (bottom) of beryllium as a function of $\mu$. The unextrapolated energies are shown as full lines, the first-order extrapolated energies are plotted in dashed lines, and the second-order ones in dot-dashed lines. The energies of the physical system are given as horizontal dotted lines. An error of $\pm 50$ mhartree is colored around the physical ground-state energy and an error of $\pm 2$ mhartree is colored around the physical excitation energies.

Figure 3

Unextrapolated (full lines), first-order extrapolated (dashed lines), and second-order extrapolated (dot-dashed lines) excitation energies $\Delta {\mathcal{E}}_k^{\mu }={\mathcal{E}}_k^{\mu }-{\mathcal{E}}_0^{\mu }$ of the ${\mathrm{H}}_2$ molecule at the equilibrium internuclear distance $R_{\text{eq}}$ (top) and three times the equilibrium distance (bottom) as a function of $\mu$. The excitation energies of the physical system $\Delta E_k=\Delta {\mathcal{E}}_k^{\mu \to \infty }$ are plotted as horizontal dotted lines. An error of $\pm 1$ mhartree is colored around the physical excitation energies at equilibrium and an error of $\pm 5$ mhartree at stretched geometry.

Figure 4

Helium ground- and excited-state energy (top), $S$ excitation energies (middle), and $P$ excitation energies (bottom) calculated without extrapolation (full lines), with first-order extrapolation (dashed), and second-order extrapolation (dot-dashed) along the linear connection. The dotted horizontal lines are the physical energies. The colored regions represent errors of $\pm 1$ mhartree for the excitation energies.

Figure 5

Error in the ${}^3\mathrm{S}$ excitation energy along the range-separated and linear adiabatic connections for the helium atom as functions of $\mu$ and $\lambda$. The unextrapolated energies are given in full lines and the extrapolated ones in dashed lines. An error smaller than 1 mhartree around the physical limit is given by the colored region.