High errors in frontier orbital energy calculation using DFT

Question

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{equation} \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} \end{equation}

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:

3. 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.

Answer 1

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.