There are several approximations to the exchange-correlation functional in density-functional theory, which accurately predict total energy-related properties of many-electron systems, such as binding energies, bond lengths, and crystal structures. Other approximations are designed to describe potential-related processes, such as charge transfer and photoemission. However, the development of a functional which can serve the two purposes simultaneously is a long-standing challenge. Trying to address it, we employ in the current work the ensemble generalization procedure proposed by Kraisler and Kronik [Phys. Rev. Lett. 110, 126403 (2013)]. Focusing on the prediction of the ionization potential via the highest occupied Kohn-Sham eigenvalue, we examine a variety of exchange-correlation approximations: the local spin-density approximation, semi-local generalized gradient approximations, and global and local hybrid functionals. Results for a test set of 26 diatomic molecules and single atoms are presented. We find that the aforementioned ensemble generalization systematically improves the prediction of the ionization potential, for various systems and exchange-correlation functionals, without compromising the accuracy of total energy-related properties. We specifically examine hybrid functionals. These depend on a parameter controlling the ratio of semi-local to non-local functional components. The ionization potential obtained with ensemble-generalized functionals is found to depend only weakly on the parameter value, contrary to common experience with non-generalized hybrids, thus eliminating one aspect of the so-called “parameter dilemma” of hybrid functionals.

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135.

Here and below, it is assumed that the ground states of the system of interest and of its ion are not degenerate, or that the degeneracy can be lifted by applying an infinitesimal external field.

136.

The fact that only contributions from the N0 − 1- and the N0-states are included relies on the conjecture that the series E(N0) for N0 ∈ ℕ is a convex, monotonously decreasing series. In other words, all ionization energies I(N0) : = E(N0 − 1) − E(N0) are positive, and higher ionizations are always larger than the lower ones: I(N0 − 1) > I(N0). This conjecture, although strongly supported by experimental data, remains without proof, to the best of our knowledge.4,41,88

137.

Janak’s theorem49 states that the ith KS eigenenergy, εi, equals ∂E/∂fi—the derivative of the total energy of the interacting system, E, with respect to the occupation of the ith level, fi. It can be shown that with the exact xc functional the ho eigenenergy, εho, has to equal εho = ∂E/∂α = ∂E/∂N = E(N0) − E(N0 − 1) = − I, i.e., it equals the negative of the IP.

138.

For the LSDA, relaxation runs have been performed for all molecules. It was found that the experimental bond length lies within the numerical error range of the relaxed bond length in all cases. We checked that the ho and e − ho energy values for the relaxed geometries agree with the ones at experimental geometries within 0.002 hartree, except for H2, NH, and F2, where the difference reaches 0.005 hartree. For the ISOcc functional, similar relaxation checks were performed, as described in Ref. 85.

139.

In this context, we note that the values reported by some of us in Ref. 87 for the relaxed H2 molecule, namely, the e-ho energy, εe-ho, and as a result—the fundamental gap of the ion, Eg, are slightly different upon closer observation. In fact, at the relaxed bond length of L = 1.45 Bohr, these values are εe-ho = 0.618 hartree =1.236 Ry and Eg = 0.671 hartree =1.341 Ry and not 1.223 Ry and 1.320 Ry, respectively. The difference originates from retrieving the value for εe-ho directly, not relying on the chemical potential μ calculated in the DARSEC program, with a temperature of 1K.

140.

We recall that the latter does not produce a strictly piecewise linear energy curve E(N), but there typically remains some concavity, which is attributed to the implicit dependence of E(N) on α via the KS orbitals. This concavity affects the value of εe-ho. However, even in case E(N) would be exactly piecewise linear, εe-ho would reproduce −IΔSCF rather than the experimental IP.

141.

Note that the combination of the EXX functional with the standard form for the Hartree functional results in an intrinsically ensemble-generalized functional if the ground state is described by an ensemble comprised of two pure many-electron states. This is the case throughout this work as we describe the ionization process by extracting an electron from a specific spin-channel. If the number of many-electron states is larger than two (as is the case, e.g., if both spin channels are fractionally occupied), then the EXX is not intrinsically ensemble-generalized, but an appropriate ensemble generalization, proposed in Ref. 91, is available.

142.

When calculating the NH molecule with the LSDA or PBEh(a) using values of 0 ≤ a ≲ 0.55, the global εho and εe-ho do not belong to the same spin channel, a behavior that has not been observed in any other system in our test set.

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