Koopmans

So, within the limitations of the single-detenninant, frozen-orbital model, the ionization potentials (IPs) and electron affinities (EAs) are given as the negative of the occupied and virtual spin-orbital energies, respectively. This statement is referred to as Koopmans theorem [47] it is used extensively in quantum chemical calculations as a means for estimating IPs and EAs and often yields results drat are qualitatively correct (i.e., 0.5 eV). 

In the spirit of Koopmans theorem, the local ionization potential, IPi, at a point in space near a molecule is defined [46] as in Eq. (54), where HOMO is the highest occupied MO, p( is the electron density due to MO i at the point being considered, and ej is the eigenvalue of MO i. 

This quantity is found to be related to the local polarization energy and is complementary to the MEP at the same point in space, making it a potentially very useful descriptor. Reported studies on local ionization potentials have been based on HF ab-initio calculations. However, they could equally well use semi-empirical methods, especially because these are parameterized to give accurate Koopmans theorem ionization potentials. 

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. 

Koopmans proposed that the orbital shucture of a cation M" " ought to be nearly the same as that of the molecule that engenders it, so that the amount of energy necessary to remove electrons from a stable molecule by hitting it with high-energy photons... 

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... 

Mass spectrometry can be used to determine ionization potentials by the method of Lossing (283). The values obtained can be compared with those found by photoelectron spectroscopy and those calculated by CNDO/S (134) or ab initio (131) methods using the Koopman theorem approximation. The first and second, ionization potentials concern a ir... 

Koopmans, T. On the Relation of Wave Functions and Eigenvalues to Individual Electrons of Atoms Physica (Utrecht) 1 104, 1933. 

Af fhe level of simple valence fheory Koopmans fheorem seems to be so self-evidenf as to be scarcely worth sfafing. Flowever, wifh more accurate fheory, fhis is no longer so and greaf inferesf attaches to why Equation (8.5) is only approximately fme. 

In most cases of closed-shell molecules Koopmans theorem is a reasonable approximation but N2 (see Section 8.1.3.2b) is a notable exception. For open-shell molecules, such as O2 and NO, the theorem does not apply. 

Because of the general validity of Koopmans theorem for closed-shell molecules ionization energies and, as we shall see, the associated vibrational sttucture represent a vivid illustration of the validity of quite simple-minded MO theory of valence electrons. 

MO calculations of the SCF type for Nj place the state below the X g state. This discrepancy is an example of the breakdown of Koopmans theorem due to deficiencies in fhe calculations. 

D. N. Blewitt, J. F. Yohn, R. P. Koopman, and T. C. Brown, International Conference on Hapor Cloud Modeling American Institute of Chemical... 

Kojic acid — see also Pyran-4-one, 5-hydroxy-2-hydroxymethyl-, 3, 611 acylation, 3, 697 application, 3, 880 occurrence, 3, 692 reactions, 3, 714, 715 with amines, 3, 700 with phenylhydrazine, 3, 700 synthesis, 3, 810 Kokusagine occurrence, 4, 989 Kokusaginine occurrence, 4, 989 synthesis, 4, 990 Koopmans theorem, 2, 135 Kostanecki-Robinson reaction chromone and coumarin formation in, 3, 819-821 mechanism, 3, 820 flavones, 3, 819... 

Goldwire, Jr., H. C., H. C. Rodean, R. T. Cederwall, E. J. Kansa, R. P. Koopman, J. W. McClure, T. G. McRae, L. K. Morris, L. Kamppiner, R. D. Kiefer, P. A. Urtiew and C. D. Lind. 1983. Coyote series data rejwrt LLNL/NWC 1981 LNG spill tests, dispersion, vapor bum, and rapid-phase-transition. Lawrence Livermore National Laboratory Report UCID-I9953. Vol. 2. 

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