DFT Calculations, Atomic Ionization Potentials — Which Exchange-Correlation Functional to Use, to Preserve Koopmans Theorem?

I have a program which can perform density-functional calculations for atoms, given a density functional.

Of course the simplest form of exchange potential to use is one relevant for a uniform electron gas (i.e. the original Kohn-Sham exchange, proportional to $n_e^{1/3}$). A good correlation functional is also available. However when I performed calculations for some atoms the energy of the highest occupied orbital differs greatly from the ionization potential (off by a few eVs!).

That the energy of the highest occupied orbital equals the ionisation potential of the neutral atom is known as Koopmans' Theorem. I get a feeling from the literature that DFT can be quite accurate. So what's wrong?

Interestingly when I use the Slater exchange functional, coupled with a correction suggested by Skillman (i.e. replacing the potential by $1/r$ when the overall potential drops below that value), the results improve significantly. (See the book "Atomic Structure Calculations" by Herman Skillman). Well, maybe I should follow this recipe, but it seems the procedure is quite ad-hoc.

My question is, are there any functionals which will give reasonable values for the atomic ionisation potentials? I do not want to implement methods like the optimised effective potential method since the latter is not readily generalised to a finite-temperature scenario. Thanks.

• The only thing which is wrong is that you somehow misinterpret the fact that DFT, as you said, can be quite accurate. Yes, it can, but it is not guaranteed to. In general the accuracy of DFT depends on many factors, but for prediction of -IP by HOMO eigenvalues it is well known that DFT performs rather poorly: absolute errors of few eVs are in fact quite usual. – Wildcat Dec 18 '14 at 17:06
• See, e.g. this paper and references there in. – Wildcat Dec 18 '14 at 17:08
• Thanks for the paper. Very informative indeed. It seems that KMLYP is best from the point of view of conformance to Koopmans' theorem. You are right; absolute errors of a few eVs are in fact quite usual. – Jamie Dec 19 '14 at 1:01
• journals.aps.org/prb/abstract/10.1103/PhysRevB.90.075135 – user23061 May 25 '16 at 8:14

1 Answer

I think the most widely-used approach at the moment comes from Roi Baer and Leeor Kronik and others, e.g.

Basically the idea is to tune the range-separation parameter $\gamma$ between short-range and long-range electrostatic effects in a range-separated hybrid functional. You find a match such that Koopmans' theorem holds.

The result is an "optimally-tuned" range-separated hybrid DFT functional (OT-RSH).

It's not perfect, for example, it violates size consistency: J. Chem. Phys. (2013) 138, 204115

• Have yet to read the paper. But from what you said, "You find a match such that Koopmans' theorem holds", it seems that the method is even more computationally intensive than the original, naive DFT. (For each potential obtain the self-consistent solutions (for both the atom and the ion?); for each pair of solutions, compute the ionization potential, and the energy of the highest occupied orbital (of the atom); if the two do not agree, tune the parameter; otherwise stop.) – Jamie Dec 19 '14 at 1:04
• This may be true in general, but it seems that for comparisons across related molecules, one can find a reasonable $\gamma$. The authors probably cringe at that suggestion, but errors ~ 0.1 eV for an intermediary parameter in the second paper, if I remember correctly. – Geoff Hutchison Dec 19 '14 at 2:42
• My other comment is that Koopmans' theorem may not always hold in absolute terms, but there's usually a linear correlation between eigenvalues and ionization potentials with low residual errors. – Geoff Hutchison Dec 19 '14 at 2:48