# High errors in frontier orbital energy calculation using DFT

In a recent paper by Faber et . al., Mean Absolute Error and DFT (B3LYP) error (relative to the experiments) for 9 properties of QM9 dataset molecules has been reported. $$$$\begin{array}{l|ccccccccc} {} & {\mu} & {\alpha} & {\varepsilon_{\text {HOMO }}}& {\varepsilon_{\text {LUMO }}} & {\Delta \varepsilon} & {\left\langle\mathrm{R}^{2}\right\rangle} & {\text { ZPVE }} & { U_{0}} & {C_{v}} \\ \hline & {\text { Debye }} & {\text { Bohr }^{3}} & {\text { eV }} & {\text { eV }} &{\text { eV }} & {\text { Bohr }^{2}} & {\text { eV }} & {\text { eV }} &{\text {cal/molK}}\\ \hline {\mathrm{MAD}} & {1.17} & {6.29}& {0.439} & {1.05} & {1.07} & {203} & {0.717} & {8.19} & {3.21} \\ {\mathrm{DFT}} & {0.10} & {0.4}& {2.0} & {2.6} & {1.2} &{-} & {0.0097} & {0.10} & {0.34} \\ \hline\end{array}$$$$

1. I was wondering why the DFT results for frontier orbital energies have such high errors in comparison with atomization energy $$U_0$$. For instance, the error reported for $$\varepsilon_{\text {HOMO }}$$ is $$2.0$$ while the breadth of values is $$0.439$$. Now compare this with the error $$0.1$$ for $$U_0$$ with breadth of values for the dataset reported as $$8.19$$, which is much more diverse.

2. A study on different DFT methods has been conducted by Zhang and Musgrave, which discusses inaccurate computation of frontier orbital energies. Some works have proposed the use ML methods to predict these values with very low errors. Since DFT is Ab initio, would suggesting to use ML methods even make sense?

Edits:

1. In the first paper (by Faber et al.) we read

Frontier molecular orbital energies (HOMO, LUMO and HOMO-LUMO gap) can not be measured directly. However, for the exact (yet unknown) exchange-correlation potential, HOMO and LUMO eigenvalues correspond to the negative of the vertical ionization potential (IP) and electron affinity (EA), respectively.

Based on this, I was wondering if DFT is the main approach to compute frontier orbital energies or this paper is just trying to find it using DFT in order to include the errors for DFT method.

• Since all the methods are tested and optimized for atomization energies, it is not surprising that those are the best-reproduced numbers whatever methods you use while other properties (eg electron density, orbital energies...) can be much worse. – Greg Nov 8 at 4:14
• I'd like to add the small comment that DFT is not really considered an ab initio method, so you should not really be surprised that it is less accurate in cases than ML. – Ezze Nov 8 at 14:20
• It is known (and some people have given analysis why) that range-separated hybrid density functionals (such as CAM-B3LYP and $\omega$B97 and related) give better results for orbital energies as measured by excited states performance. – TAR86 Nov 9 at 15:57
• @Blade Range-separated hybrids have the same formal scaling behavior as hybrid DFs such as B3LYP. Implementation details and different behavior in the estimation of negligible terms in practice mean additional costs on the order of 20% (in my estimation). – TAR86 Nov 11 at 5:43
• @Greg - I certainly wouldn't argue with you about using ML to fit garbage data. I'm just clarifying that the ML methods are tailored for fitting each property. I think you were commenting that the DFT methods are tested and optimized mostly for atomization energies (thermochemical properties) which is true, but not clear from your initial comment. – Geoff Hutchison Nov 15 at 13:22

IMHO, this particular comparison is a strange one.

You reference "Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations" by Zhang and Musgrave, and there are other similar comparisons between density functional orbital methods and ionization potentials and electron affinities.

In general, these Kohn-Sham DFT benchmark papers show that most functionals don't compute orbital energies that directly compare with ionization potentials and electron affinities. That is, in Hartree Fock, Koopmans' theorem suggests that the orbital energy of the HOMO should correspond to the negative of the ionization potential (-IP). In principal, for the exact density functional, a similar property should hold.

What does hold true (e.g., the Zhang and Musgrave paper) is that you can easily calibrate orbital eigenvalues from Kohn-Sham DFT methods to counteract systematic issues. Usually a linear regression is sufficient. (Computing the actual vertical ionization potential using two DFT calculations is generally even more accurate.)

Coming back to your question, I wouldn't be surprised if the absolute errors in orbital eigenvalues are mixed from DFT methods. But that's not how most research uses them - they either perform a calibration (e.g., linear regression) or are comparing multiple different molecules. In the latter case, the relative errors are much smaller (usually ~0.2 eV, IIRC - I'll look for some sources).

I believe the overall argument in the Faber et. al paper is that modern ML methods are very good. In some properties that's probably correct.

• In reference to @Greg 's comment above, the other issue is that density functional methods are often optimized and tested for ground-state thermochemical properties like atomization energies, so other properties (like orbital eigenvalues) may not be as reliable. – Geoff Hutchison Nov 15 at 13:23