Abstract
Gross–Oliveira–Kohn density-functional theory (GOK-DFT) is an extension of DFT to excited states where the basic variable is the ensemble density, i.e. the weighted sum of ground- and excited-state densities. The ensemble energy (i.e. the weighted sum of ground- and excited-state energies) can be obtained variationally as a functional of the ensemble density. Like in DFT, the key ingredient to model in GOK-DFT is the exchange-correlation functional. Developing density-functional approximations (DFAs) for ensembles is a complicated task as both density and weight dependencies should in principle be reproduced. In a recent paper [K. Deur et al., Phys. Rev. B 95, 035120 (2017)], the authors applied exact GOK-DFT to the simple but nontrivial Hubbard dimer in order to investigate (numerically) the importance of weight dependence in the calculation of excitation energies. In this work, we derive analytical DFAs for various density and correlation regimes by means of a Legendre–Fenchel transform formalism. Both functional and density driven errors are evaluated for each DFA. Interestingly, when the ensemble exact-exchange-only functional is used, these errors can be large, in particular if the dimer is symmetric, but they cancel each other so that the excitation energies obtained by linear interpolation are always accurate, even in the strongly correlated regime.
Similar content being viewed by others
References
A.K. Theophilou, J. Phys. C 12, 5419 (1979)
E.K.U. Gross, L.N. Oliveira, W. Kohn, Phys. Rev. A 37, 2805 (1988)
E.K.U. Gross, L.N. Oliveira, W. Kohn, Phys. Rev. A 37, 2809 (1988)
E.K.U. Gross, L.N. Oliveira, W. Kohn, Phys. Rev. A 37, 2821 (1988)
O. Franck, E. Fromager, Mol. Phys. 112, 1684 (2014)
A. Pribram-Jones, Z.H. Yang, J.R. Trail, K. Burke, R.J. Needs, C.A. Ullrich, J. Chem. Phys. 140, 18A541 (2014)
Z.H. Yang, J.R. Trail, A. Pribram-Jones, K. Burke, R.J. Needs, C.A. Ullrich, Phys. Rev. A 90, 042501 (2014)
K. Pernal, N.I. Gidopoulos, E. Pastorczak, Adv. Quantum Chem. 73, 199 (2016)
Z.H. Yang, A. Pribram-Jones, K. Burke, C.A. Ullrich, Phys. Rev. Lett. 119, 033003 (2017)
A. Nikiforov, J.A. Gamez, W. Thiel, M. Huix-Rotllant, M. Filatov, J. Chem. Phys. 141, 124122 (2014)
M. Filatov, WIREs Comput. Mol. Sci. 5, 146 (2015)
M. Filatov, M. Huix-Rotllant, I. Burghardt, J. Chem. Phys. 142, 184104 (2015)
M. Filatov, F. Liu, K.S. Kim, T.J. Martínez, J. Chem. Phys. 145, 244104 (2016)
M. Filatov, T.J. Martínez, K.S. Kim, J. Chem. Phys. 147, 064104 (2017)
T. Gould, S. Pittalis, Phys. Rev. Lett. 119, 243001 (2017)
T. Gould, L. Kronik, S. Pittalis, arXiv:1801.06314 (2018)
J.P. Perdew, M. Levy, Phys. Rev. B 31, 6264 (1985)
M. Levy, A. Nagy, Phys. Rev. Lett. 83, 4361 (1999)
M. Levy, A. Nagy, Phys. Rev. A 59, 1687 (1999)
A. Nagy, M. Levy, Phys. Rev. A 63, 052502 (2001)
P.W. Ayers, M. Levy, Phys. Rev. A 80, 012508 (2009)
P.W. Ayers, M. Levy, A. Nagy, Phys. Rev. A 85, 042518 (2012)
E. Pastorczak, N.I. Gidopoulos, K. Pernal, Phys. Rev. A 87, 062501 (2013)
B. Senjean, S. Knecht, H.J.Aa. Jensen, E. Fromager, Phys. Rev. A 92, 012518 (2015)
B. Senjean, E.D. Hedegård, M.M. Alam, S. Knecht, E. Fromager, Mol. Phys. 114, 968 (2016)
N. Gidopoulos, P. Papaconstantinou, E. Gross, Phys. Rev. Lett. 88, 033003 (2002)
E. Pastorczak, K. Pernal, J. Chem. Phys. 140, 18A514 (2014)
M.M. Alam, S. Knecht, E. Fromager, Phys. Rev. A 94, 012511 (2016)
M.M. Alam, K. Deur, S. Knecht, E. Fromager, J. Chem. Phys. 147, 204105 (2017)
K. Deur, L. Mazouin, E. Fromager, Phys. Rev. B 95, 035120 (2017)
D.J. Carrascal, J. Ferrer, J.C. Smith, K. Burke, J. Phys.: Condens. Matter 27, 393001 (2015)
J.C. Smith, A. Pribram-Jones, K. Burke, Phys. Rev. B 93, 245131 (2016)
E. Fromager, Mol. Phys. 113, 419 (2015)
B. Senjean, M. Tsuchiizu, V. Robert, E. Fromager, Mol. Phys. 115, 48 (2017)
B. Senjean, N. Nakatani, M. Tsuchiizu, E. Fromager, arXiv:1710.03125 (2017)
E.H. Lieb, Int. J. Quantum Chem. 24, 243 (1983)
M. Levy, Proc. Natl. Acad. Sci. 76, 6062 (1979)
M. Levy, Phys. Rev. A 52, R4313 (1995)
J. Hubbard, Proc. R. Soc. Lond. USA A 276, 238 (1963)
D. Carrascal, J. Ferrer, J. Smith, K. Burke, J. Phys.: Condens. Matter 29, 019501 (2016)
P.F. Loos, P.M. Gill, Phys. Rev. Lett. 103, 123008 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Contribution to the Topical Issue “Special issue in honor of Hardy Gross”, edited by C.A. Ullrich, F.M.S. Nogueira, A. Rubio, and M.A.L. Marques.
Rights and permissions
About this article
Cite this article
Deur, K., Mazouin, L., Senjean, B. et al. Exploring weight-dependent density-functional approximations for ensembles in the Hubbard dimer. Eur. Phys. J. B 91, 162 (2018). https://doi.org/10.1140/epjb/e2018-90124-7
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjb/e2018-90124-7