Adiabatic connection for near degenerate excited states

In density functional theory, the adiabatic coupling may be treated perturbatively. This Görling-Levy perturbation theory can be applied to both nondegenerate and degenerate states. Formulas are given to second order. Some excitations are shown to be poorly described to first order (i.e., in exact exchange), no matter how weakly correlated the system is. Results are demonstrated on a simple model system, two fermions in a one-dimensional parabolic well with a contact interaction.

Figure 1

The lowest singlet levels of our model with $\lambda =1$, both exactly and in the KS system.

Figure 2

The adiabatic connection of the second and third excited states energies when $\lambda =0.4$. The crosses indicate the exact energies, the dotted straight lines are first order nondegenerate GLPT, the solid curves are first order near-degenerate GLPT and the dotted curves are the results of standard perturbation theory (first order), and the dashed curves are second order GLPT.

Figure 3

The adiabatic connection of the transition frequencies when $\lambda =0.4$ for second and third excited states. Notation same as Fig. 2.

Figure 4

The adiabatic connection for the energy when $\lambda =0.4$ for the first excited state. The cross indicates the exact energy, the solid line is the first order GLPT approximation, the dotted line is the standard perturbation theory results, and the dashed line is the second order GLPT approximation.