Koopmans' theorem states that in closed-shell Hartree–Fock theory (HF), the first ionization energy of a molecular system is equal to the negative of the orbital energy of the highest occupied molecular orbital (HOMO).
[3][4][5] Therefore, the validity of Koopmans' theorem is intimately tied to the accuracy of the underlying Hartree–Fock wavefunction.
[citation needed] 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 determinant composed 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.
[3][4] The LUMO energy shows little correlation with the electron affinity with typical approximations.
[9] 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.
Koopmans’ theorem applies to the removal of an electron from any occupied molecular orbital to form a positive ion.
For example, the electronic configuration of the H2O molecule is (1a1)2 (2a1)2 (1b2)2 (3a1)2 (1b1)2,[10] where the symbols a1, b2 and b1 are orbital labels based on molecular symmetry.
[10] As explained above, the deviations are due to the effects of orbital relaxation as well as differences in electron correlation energy between the molecular and the various ionized states.
However the lowest ionization energy corresponds to removal of an electron from the 3σg bonding orbital.
In this case the deviation is attributed primarily to the difference in correlation energy between the two orbitals.
[11] It is sometimes claimed[12] that Koopmans' theorem also allows the calculation of electron affinities as the energy of the lowest unoccupied molecular orbitals (LUMO) of the respective systems.
However, Koopmans' original paper makes no claim with regard to the significance of eigenvalues of the Fock operator other than that corresponding to the HOMO.
Nevertheless, it is straightforward to generalize the original statement of Koopmans' to calculate the electron affinity in this sense.
Calculations of electron affinities using this statement of Koopmans' theorem have been criticized[13] on the grounds that virtual (unoccupied) orbitals do not have well-founded physical interpretations, and that their orbital energies are very sensitive to the choice of basis set used in the calculation.
Comparisons with experiment and higher-quality calculations show that electron affinities predicted in this manner are generally quite poor.
Koopmans' theorem is also applicable to open-shell systems, however, orbital energies (eigenvalues of Roothaan equations) should be corrected, as was shown in the 1970s.
[16] Later, the validity of Koopmans’ theorem for ROHF was revisited and several procedures for obtaining meaningful orbital energies were reported.
More generally, this relation is true even when the KS system describes a zero-temperature ensemble with non-integer number of electrons
[22] It can be argued that the vertical electron affinity is equal exactly to the negative of the sum of the LUMO energy and the derivative discontinuity.
[22][23][24][25] Unlike the approximate status of Koopmans' theorem in Hartree Fock theory (because of the neglect of orbital relaxation), in the exact KS mapping the theorem is exact, including the effect of orbital relaxation.
[29] A tuning procedure is able to "impose" Koopmans' theorem on DFT approximations, thereby improving many of its related predictions in actual applications.
[29][30] In approximate DFTs one can estimate to high degree of accuracy the deviance from Koopmans' theorem using the concept of energy curvature.
Dyson orbitals contain all information about the initial and final states of the system needed to compute experimentally observable quantities, such as total and differential photoionization/phtodetachment cross sections.