Eur. Phys. J. B (2018) 91: 142
https://doi.org/10.1140/epjb/e2018-90114-9

Regular Article

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

¹ Department of Physics, Universidad de Oviedo, 33007 Oviedo, Spain
² Nanomaterials and Nanotechnology Research Center, CSIC/Universidad de Oviedo, Oviedo, Spain
³ Department of Physics, Hunter College, City University of New York, New York, NY 1006, USA
⁴ Department of Chemistry and of Physics, University of California, Irvine, CA 92697, USA

^a e-mail: ferrer@uniovi.es

Received: 1 March 2018
Received in final form: 3 May 2018
Published online: 2 July 2018

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.