At the lowest level of approximation, the ionization energy is provided by Koopmans' theorem.

Koopmans' theorem is also applicable to open-shell systems.

The corresponding eigenvalues are interpreted as ionization potentials via Koopmans' theorem.

Koopmans' theorem applies to the removal of an electron from any occupied molecular orbital to form a positive ion.

Ionization energies are linked approximately to orbital energies by Koopmans' theorem.

The required atomic orbital energies can come from calculations or directly from experiment via Koopmans' theorem.

The Kohn-Sham orbital energies ε, in general, have little physical meaning (see Koopmans' theorem).

Koopmans' early works on the Hartree-Fock theory are associated with the Koopmans' theorem, which is very well known in quantum chemistry.

The predicted MO energies as stipulated by Koopmans' theorem correlate with photoelectron spectroscopy.

Therefore, the validity of Koopmans' theorem is intimately tied to the accuracy of the underlying Hartree-Fock wavefunction.

A tuning procedure is able to "impose" Koopmans' theorem on DFT approximations thereby improving many of its related predictions in actual applications.

In approximate DFTs one can estimate to high degree of accuracy the deviance from Koopmans' theorem using the concept of energy curvature.

Hybrid orbitals cannot therefore be used to interpret photoelectron spectra, which measure the energies of ionized states, identified with delocalized orbital energies using Koopmans' theorem.

Because the orbital energies are non-unique in the more general restricted open-shell case, this equation only holds true for specific choices of orbital energies (see Koopmans' theorem).

It is sometimes claimed that Koopmans' theorem also allows the calculation of electron affinities as the energy of the lowest unoccupied molecular orbitals (LUMO) of the respective systems.

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.

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).

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).

It was previously believed that this was only in the case for removing the unpaired electron, but the validity of Koopmans' theorem for ROHF in general has been proven provided that the correct orbital energies are used.

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.

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.

Calculations of electron affinities using this statement of Koopmans' theorem have been criticized 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.

While Koopmans' theorem was originally stated for calculating ionization energies from restricted (closed-shell) Hartree-Fock wavefunctions, the term has since taken on a more generalized meaning as a way of using orbital energies to calculate energy changes due to changes in the number of electrons in a system.

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.