Linear response time-dependent density functional theory of the Hubbard dimer

TL;DR

This paper investigates the exact frequency-dependent kernel in time-dependent density functional theory for the Hubbard dimer, highlighting limitations of the adiabatic approximation and proposing an improved density-functional approximation for accurate excitation energies.

Contribution

It provides the exact form of the kernel for the Hubbard dimer, demonstrates the use of Casida's method with this kernel, and introduces an interpolation approach for weak and strong correlation regimes.

Findings

Exact kernel form derived for the Hubbard dimer

Casida's method yields exact oscillator strengths with the kernel

Interpolation approach improves transition frequency predictions

Abstract

The asymmetric Hubbard dimer is used to study the density-dependence of the exact frequency-dependent kernel of linear-response time-dependent density functional theory. The exact form of the kernel is given, and the limitations of the adiabatic approximation utilizing the exact ground-state functional are shown. The oscillator strength sum rule is proven for lattice Hamiltonians, and relative oscillator strengths are defined appropriately. The method of Casida for extracting oscillator strengths from a frequency-dependent kernel is demonstrated to yield the exact result with this kernel. An unambiguous way of labelling the nature of excitations is given. The fluctuation-dissipation theorem is proven for the ground-state exchange-correlation energy. The distinction between weak and strong correlation is shown to depend on the ratio of interaction to asymmetry. A simple interpolation between carefully defined weak-correlation and strong-correlation regimes yields a density-functional approximation for the kernel that gives accurate transition frequencies for both the single and double excitations, including charge-transfer excitations. Many exact results, limits, and expansions about those limits are given in the appendices.