Koopmans theorem

The physical meaning of orbital energies is clarified by Koopmans theorem (Koopmans 1934). For orbital pi, the orbital energy of the Hartree-Fock equation is represented by 

Tsuneda, Density Functional Theory in Quantum Chemistry. DOI 10.1007/978-4-431-54825-6 7. Springer Japan 2014 

Moreover, the energy after removing one electron from orbital (pi is derived as 

The ionization potential, which is the energy difference from Eo to E, is, therefore, proven to be 

It is easily proven that the Koopmans theorem is established for unoccupied orbitals (Szabo and Ostlund 1996). The energy after adding one electron to an unoccupied orbital (pa is derived from Eq. (7.2) as 

Despite the fact that the total electronic energy is not given by the sum of SCF one-electron energies, it is still possible to relate the e, s to physical measurements. If certain assumptions are made, it is possible to equate orbital energies with molecular ionization energies or electron affinities. This identification is related to a theorem due to Koopmans. 

For the neutral molecule, which we assume is a closed-shell system. 

To compare this with E of (11-23) we should remove the remaining index restriction. We do this by allowing / to equal in the sum and simultaneously subtracting the new terms thus produced  

This illustrates that, within the context of this simplified model, the negative of the orbital energies for occupied HE MOs are to be interpreted as ionization energies. 

Another way to see the relation between 7° and — k, is to recognize that the physical interactions lost upon removal of an electron from cpk, are precisely those that constitute 6 t,[SeeEq. (11-15).] 

The canonical MOs are convenient for the physical interpretation of the Lagrange multipliers. Consider the energy of an A-electron system and the corresponding system with one electron removed from orbital number k, and assume that the MOs are identical for the two systems (eq. (3.32)). 

The last two sums are identical and the energy difference becomes 

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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. 

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... 

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. 

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... 

The orbitals and orbital energies produced by an atomic HF-Xa calculation differ in several ways from those produced by standard HF calculations. First of all, the Koopmans theorem is not valid and so the orbital energies do not give a direct estimate of the ionization energy. A key difference between standard HF and HF-Xa theories is the way we eoneeive the occupation number u. In standard HF theory, we deal with doubly oecupied, singly occupied and virtual orbitals for which v = 2, 1 and 0 respectively. In solid-state theory, it is eonventional to think about the oecupation number as a continuous variable that can take any value between 0 and 2. 

In this particular example, the Xa orbital energies resemble those produced from a conventional HF-LCAO calculation. It often happens that the Xa ionization energies come in a different order than HF-LCAO Koopmans-theorem ones, due to electron relaxation. 

The Huckel methods perform the parameterization on the Fock matrix elements (eqs. (3.50) and (3.51)), and not at the integral level as do NDDO/INDO/CNDO. This means that Huckel methods are non-iterative, they only require a single diagonalization of the Fock (Huckel) matrix. The Extended Huckel Theory (EHT) or Method (EHM), developed primarily by Hoffmann again only considers the valence electrons. It makes use of Koopmans theorem (eq. (3.46)) and assigns the diagonal elements in the F... 

The matrix elements between the HF and a doubly excited state are given by two-electron integrals over MOs (eq. (4.7)). The difference in total energy between two Slater determinants becomes a difference in MO energies (essentially Koopmans theorem), and the explicit formula for the second-order Mpller-Plesset correction is... 

Felsche J (1973) The Crystal Chemistry of the Rare-Earth Silicates. 13 99-197 Ferreira R (1976) Paradoxial Violations of Koopmans Theorem, with Special Reference to the 3d Transition Elements and the Lanthanides. 31 1-21 Fichtinger-Schepman AMJ, see Reedijk J (1987) 67 53-89... 

Eq. (90) can be called an extension of Koopmans theorem to radicals. Recently, Richards (107) discussed the approximate nature of Koopmans theorem in treatments of closed-shell systems, and most probably his arguments apply here also. 

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