Koopman’s theorem

In studying molecular orbital theory, it is difficult to avoid the question of how real orbitals are. Are they mere mathematical abstractions The question of reality in quantum mechanics has a long and contentious history that we shall not pretend to settle here but Koopmans s theorem and photoelectron spectra must certainly be taken into account by anyone who does. 

Another technique for obtaining an ionization potential is to use the negative of the HOMO energy from a Hartree-Fock calculation. This is called Koopman s theorem it estimates vertical transitions. This does not apply to methods other than HF but gives a good prediction of the ionization potential for many classes of compounds. 

The consistent total energy makes it possible to compute singlet-triplet gaps using RHF for the singlet and the half-electron calculation for the triplet. Koopman s theorem is not followed for half-electron calculations. Also, no spin densities can be obtained. The Mulliken population analysis is usually fairly reasonable. 

Koopman s theorem a means for obtaining the ionization potential from a Hartree-Fock calculation... 

Paradoxical violations of Koopman s theorem with special reference to the 3d transition elements and the lanthanides. R. Ferreira, Struct. Bonding (Berlin), 1976, 31, 1-22 (73). 

In the uncorrelated limit, where the many-electron Fock operator replaces the full electronic Hamiltonian, familiar objects of HF theory are recovered as special cases. N) becomes a HF, determinantal wavefunction for N electrons and N 1) states become the frozen-orbital wavefunctions that are invoked in Koopmans s theorem. Poles equal canonical orbital energies and DOs are identical to canonical orbitals. 

The poles correspond to Koopmans s theorem.) The inverse-propagator matrix and its zeroth-order counterpart therefore are related through... 

Koopman s theorem is found to be valid only in the case of the vanadium (dA) complex 9. The amount of orbital reorganization is increasing considerably in the series 9-14-15-16. It is also found that strong metal-ligand interaction combined with low symmetry leads to extensive delocalization of the outer valence MO s especially in the cylobutadiene complex 16, only two orbitals with >80% metal character are found for which the convenient term essential 3d metal orbital would be justified. 

One of the main aims of such computations is the prediction and rationalization of the optoelectronic spectra in various steric and electronic environments by either semiempirical or ab initio methods or a combination of these, considering equilibrium structures, rotation barriers, vibrational frequencies, and polarizabilities. The accuracy of the results from these calculations can be evaluated by comparison of the predicted ionization potentials (which are related to the orbital energies by Koopman s theorem) with experimental values. 

Dyson orbitals from one isomer to the other. A higher peak at 10.86 eV, with intensity closer to that of the first one, is assigned to the 7r4 hole the Dyson orbital is approximately conserved after the proton transfer. Note that Koopmans s theorem (KT) results predict the wrong order of states for the enol form. A third a hole for both isomers is in reasonable agreement with the experimental peak at 11.32 eV. Finally, the feature at 13.3 eV is assigned to the 775 hole. 

Koopman s theorem, 25 12 Korringga-Kohn-Rostoker, method, 34 246... 

An observable energy for a selected set of electrons follows in a straightforward manner from the sum of their orbital energies less the electron-electron interactions, which are counted twice in this sum. This procedure, reflecting the spirit of Koopman s theorem, is nothing new in Hartree-Fock theory. 

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