Dear Mal,

Neither DFT nor semi-empirical predict correctly the IPs of molecules. The following text describes the methods that are better in predicting IPs of molecules.

Koopmans' theorem does not hold for DFT. which is considered not  to be an MO method. In DFT, a HOMO energy refers to the eigenvalue of the highest occupied Kohn-Sham orbital. Apparently, some interpretation of the Kohn-Sham orbital energies is possible. Being that B3LYP is a hybrid of HF and DFT, it seems as though Koopmans' theorem does not apply to this specific case.

Koopmans' theorem states that in closed-shell Hartree–Fock theory, the first ionization energy of a molecular system is equal to the negative of the orbital energy of the highest occupied molecular orbital (HOMO). Koopmans' theorem is exact in the context of restricted Hartree–Fock theory if it is assumed that the orbitals of the ion are identical to those of the neutral molecule (the frozen orbital approximation). Ionization energies calculated this way are in qualitative agreement with experiment – the first ionization energy of small molecules is often calculated with an error of less than two electron volts. Therefore, the validity of Koopmans' theorem is intimately tied to the accuracy of the underlying Hartree–Fock wave function. The two main sources of error are: orbital relaxation, which refers to the changes in the Fock operator and Hartree–Fock orbitals when changing the number of electrons in the system, and electron correlation, referring to the validity of representing the entire many-body wavefunction using the Hartree–Fock wavefunction, i.e. a single Slater determinantcomposed of orbitals that are the eigenfunctions of the corresponding self-consistent Fock operator. Empirical comparisons with experimental values and higher-quality ab initio calculations suggest that in many cases, but not all, the energetic corrections due to relaxation effects nearly cancel the corrections due to electron correlation.

A similar theorem exists in density functional theory (DFT) for relating the exact first vertical ionization energy and electron affinity to the HOMO and LUMO energies, although both the derivation and the precise statement differ from that of Koopmans' theorem. Ionization energies calculated from DFT orbital energies are usually poorer than those of Koopmans' theorem, with errors much larger than two electron volts possible depending on the exchange-correlation approximation employed. The LUMO energy shows little correlation with the electron affinity with typical approximations. The error in the DFT counterpart of Koopmans' theorem is a result of the approximation employed for the exchange correlation energy functional so that, unlike in HF theory, there is the possibility of improved results with the development of better approximations.

Kohn–Sham (KS) density functional theory (KS-DFT) admits its own version of Koopmans' theorem (sometimes called the DFT-Koopmans' theorem) very similar in spirit to that of Hartree-Fock theory.

A large number of theoretical methods and techniques, semiempirical as well as ab initio, are available for calculating the energies of molecular orbitals. However, in many cases these theoretical methods fail to provide the quantitative IPs correctly, or even to reproduce the correct orbital ordering. Cederbaum and his coworkers have shown that the combination of ab initio techniques with Green's function method can improve significantly the ability of theoretical calculations to accurately predict IPs, for a variety of systems. In this method many-body perturbation theory is used to derive the equations that calculate IPs including corrections for electron correlation and for orbital relaxation effects. Hartree-Fock solutions obtained from ab initio calculations serve as the zeroth order approximation in the perturbation series. Unfortunately, the application of ab initio-Green's function method is still prohibitively expensive for the large size molecules. To overcome this difficulty, it was suggested recently to couple the Outer-Valence Green's Function (OVGF) approach with semiempirical methods such as MNDO, AM1  and PM3 (denoted in this presentation as OVGF(corresponding semiempirical method)). The OVGF(semiempirical) method was recently successfully applied to calculate the IPs of a variety of nitrogen-containing heterocycles, such as substituted pyridines (30 compounds), pyrimidines, pyridazines and azoles. More recently the method was applied  to calculate the IPs of a series of substituted triazines, tetrazines, cyclic and linear polysilanes as well as for calculating the IPs of radicals and the electron affinities of various neutral compounds. In general it was found that inclusion of the Green's function method in the calculation improves substantially the quantitative agreement between theoretical and experimental data. A computer program combining the OVGF formalism with the NDDO based semiempirical methods is available from QCPE, and it was also incorporated in the new MOPAC-93 and AMPAC 4.5 packages.

For further information, you can use the following links:

http://pubs.rsc.org/en/Content/ArticleLanding/1993/P2/P29930000321#!divAbstract

http://yfaat.ch.huji.ac.il/David/c60.html

http://www.sciencedirect.com/science/article/pii/0166128085850582

https://www.unige.ch/sciences/chifi/wesolowski/public_html/dft_epfl_2007/opt_failures/jpca_2007_111_1554.pdf

Hoping this will be helpful,

Rafik

Concerning experimental verification, Rafik, we must differentiate between vertical and adiabatic transitions while measuring IPs and affinities.

Mr./Miss/Mrs. Ali,

The Koopmans' theorem is "IPs = -E(SCF,HOMO)". And those are HF energies. In this context most probably you would like to obtain IPs, using DFTs or semi-empirical methods. If yes, then you should take into consideration the comment by Mr. Kislyakov, If you compare experimental adiabatic (frequently case) with theoretical vertical IPs data, the deviation can be large then 0.3 eV, first. Second, computing adiabatic IPs by DFTs, for example, you should consider energies at 0 K, because of, those are values obtained by total atomization energies of species at 0K (IP(T=0) = ET(T=0)(neutral)-ET(T=0)(cation)).

There is important however to take into consideration, as well as, that semi-empirical methods operate with parametrization based on "training set" of reference molecules. So they cannot be representative to large set objects. The energetics can deviate up to 30% (enthalpy), particularly, computing H-bonding systems.

There is important to mention, as well as, which DFTs you have taken into consideration? M06-2X, M06-HF and LC-wPBE, for example, provide accurate data about energies of H-interacting systems, including intra- and intermolecular bonding type. You can pay attention to ref. 1. There are reports using PBE0 with SVP (Ref. 2), as well.

On the other hand, the Koopmans' theorem does not take into consideration electron correlation and final state orbital relaxation. The deviations towards experimental IPs data can be > 1-3 eV.

Or, in order to achieve accurate, IPs particularly, you can use W1 (even not W2) or G4 methods computing adiabatic IPs by equation above, or, GW family methods, obtaining vertical IPs. OVGF/P3 methods, for example (for IPs). Or CBS methods, particularly CBS-APNO. Those have shown RMS errors towards IPs within 0.2-0.02 eV for organics. As computational cost, methods of choice within the frame of this set methods, those are OVGF/P3 and CBSs ones.

For W1/W2, OVGF/P3 and CBSs methods, please pay attention to refs 3-5, for example.

[Ref.1] M. Pinheiro Jr., M. Caldas, P. Rinke, V. Blum, M. Scheffler, Length dependence of ionization potentials of transacetylenes: Internally consistent DFT/GW approach, Phys. Rev. B 92, 195134 (2015)

[Ref.2] I. Petrushenko, DFT Study on Adiabatic and Vertical Ionization Potentials of Graphene Sheets, Adv. Mater. Sci. Engineer. 2015, Article ID 262513

[Ref.3] J. Martin, G. de Oliveira, Towards standard methods for benchmark quality ab initio thermochemistry - W1 and W2 theory, J. Chem. Phys. 111, 1843 (1999)

[Ref.4] J.Ortiz, Partial third-order quasiparticle theory: Comparisons for closed-shell ionization energies and an application to the Borazine photoelectron spectrum, J. Chem. Phys. 104, 7599 (1996).

[Ref.5] J.Montgomery Jr., J.Ochterski, G.Petersson, A complete basis set model chemistry. IV. An improved atomic pair natural orbital method, J. Chem. Phys. 101, 5900 (1994)