Hybrid Density Functionals Tuned towards Fulfillment of Fundamental DFT Conditions

Hybrid exchange-correlation functionals (XC), e.g. PBE0 and HSE, have significantly improved the theoretical description of molecules and solids. Their degree of exact-exchange admixture ($\alpha )$ is in principle a functional of the electron density, but the functional form is not known. In this talk, I will discuss \textit{fundamental conditions} of exact density-functional theory (DFT) that enable us to find the optimal choice of $\alpha $ for ground-state calculations. In particular, I will discuss the fact that the highest occupied Kohn-Sham level of an $N$-electron system ($\varepsilon _{\mathrm{HOMO}}(N))$ should be constant for fractional particle numbers between $N$ and \textit{N-1 }[1,2] and equals the ionization potential (IP) [3, 4], as given by the total-energy difference. In practice, we realize this in three different ways. XC($\alpha )$ will be optimized (opt-XC) until it $(i)$ fulfills the condition: $\varepsilon_{\mathrm{HOMO}}(N) = \varepsilon _{\mathrm{HOMO}}$(\textit{N-1/2}) or the Kohn-Sham HOMO agrees with the ionization potential computed in a more sophisticated approach $\varepsilon _{\mathrm{HOMO}}(N) =$ IP such as \textit{(ii)} the $G_{\mathrm{0}}W_{\mathrm{0}}$@opt-XC method [5,6] or \textit{(iii)} CCSD(T) or full CI [6]. Using such an opt-XC is essential for describing electron transfer between (organic) molecules, as exemplified by the TTF/TCNQ dimer [5]. It also yields vertical ionization energies of the G2 test set of quantum chemistry with a mean absolute percentage error of only $\approx $3{\%}. Furthermore, our approach removes the starting-point uncertainty of \textit{GW} calculations [5] and thus bears some resemblance to the consistent starting point scheme [7] and quasiparticle self-consistent \textit{GW} [8]. While our opt-XC approach yields large $\alpha $ values for small molecules in the gas phase [5], we find that $\alpha $ needs to be 0.25 or less for organic molecules adsorbed on metals [9]. \\[4pt] [1] J. P. Perdew et al., PRL 1982.\\[0pt] [2] P. Mori-Sanchez et al., JCP 2006.\\[0pt] [3] M. Levy et al., PRA 1984.\\[0pt] [4] T. Stein et al., PRL 2010.\\[0pt] [5] V. Atalla et al., PRB 2013.\\[0pt] [6] N. A. Richter, et al., PRL 2013.\\[0pt] [7] T. K\"{o}rzd\"{o}rfer, N. Marom, PRB 2012.\\[0pt] [8] M. van Schilfgaarde et al., PRL 2006.\\[0pt] [9] O. T. Hofmann et al., NJP 2013.