TY - JOUR

T1 - Linear response time-dependent density functional theory of the Hubbard dimer

AU - Carrascal, Diego J.

AU - Ferrer, Jaime

AU - Maitra, Neepa

AU - Burke, Kieron

N1 - Funding Information:
DC and JF wish to thank funding support from the Spanish Ministerio de Economía y Competitividad via grant FIS2012-34858. NTM thanks the US National Science Foundation CHE-1566197 for support. KB acknowledges DOE grant number DE-FG02-08ER46496. All the authors have benefitted either directly or indirectly from the accumulated impact of Prof. Gross’s works. Some also acknowledge about 45 accumulated years of friendship and learning at the feet of Prof. E.K.U. (Hardy) Gross, who taught us (almost) everything we know about time-dependent density functional theory. We hope that this small contribution, demonstrating the exactness of TDDFT in the simplest possible case, might contribute to elucidating how the theory works to skeptics in many-body theory or ab-initio computational chemistry. While this paper (and, indeed, much of Prof. Gross’s work) might be regarded as FEPU (formally exact, practically useless), the proof of the RG theorem [1] was clearly anything but.
Publisher Copyright:
© 2018, EDP Sciences, SIF, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2018/7/1

Y1 - 2018/7/1

N2 - 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.

AB - 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.

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U2 - 10.1140/epjb/e2018-90114-9

DO - 10.1140/epjb/e2018-90114-9

M3 - Article

AN - SCOPUS:85049316161

SN - 1434-6028

VL - 91

JO - European Physical Journal B

JF - European Physical Journal B

IS - 7

M1 - 142

ER -