Ionization potentials theorem

This is known as the ionization potential theorem. Equation 7.7 between max and ionization potential I can also be obtained alternatively by looking at the asymptotic behavior of the density of a many-electron system. For atoms and molecules, the asymptotic decay of the density is given as [6-11]... 

Lindgren I, Salomonson S, Moller F (2005) Construction of accurate Kohn-Sham potentials for the lowest states of the helium atom Accurate test of the ionization-potential theorem, Int J Quant Chem, 102 1010-1017... 

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. 

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. 

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... 

Despite the strong MO mixings indicated by the Ae splittings, one may question to what extent the MO energy variations are reflected in measurable physical properties. As described in Section 3.2.4, the interactions of filled NBOs lead to symmetric second-order energy shifts with no net effect on total energy, wavefunc-fion, and other properties. However, the assumptions of Koopmans theorem imply that the vertical ionization potential (IP) is related to HOMO orbital energy by... 

The average value of the orbital energy can be calculated from the assumed wave function. If the wave function were exact, the orbital energy would indeed be constant, as required. For the approximate T°, the average orbital energy is equal to the ionization potential /, according to Koopmans theorem. In Equation 11.13, the electron affinity A has dropped out. 

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. 

Accepting the approximate validity of Koopmans theorem (129), UPS can be said to give information about the energies of the occupied orbitals in the ground state. Two UPS studies of push-pull ethylenes have been reported (130,131). In ref. 130, the ionization potentials (IPs) of 119 are discussed, where A denotes... 

Natural orbital functional vertical ionization potentials obtained from the extended Koopmans theorem. 

The photoelectron spectrum is frequently discussed in terms of Koopmans theorem, which states that the ionization potentials (IPs) are approximately related to the energies of the canonical orbital found in molecular orbital calculations.106. The relationship is approximate because two factors are neglected the change in the correlation energy, and the reorganization energy, which is a consequence of the movement of electrons in response to the formation of a cation. The two quantities are approximately equal and opposite. 

As discussed at the end of Chapter 3, one group orbital of a methyl or methylene group will always have the correct nodal characteristics to interact with an adjacent n orbital or with an adjacent spn orbital in fashion. The degree of interaction may be inferred from the energies of the orbitals, which may in turn be obtained by measurements of ionization potentials and application of Koopmans theorem. Thus, the methyl groups adjacent to the n bond in (Z)-2-butene (ionization potential IP = 9.12 eY [63]) raise the energy of the n orbital by 1.39 eV relative to that of ethylene (IP = 10.51 eV [87]). A similar effect is observed in cyclohexene [64]. 

Thus the ionization potential corresponding to removal of the electron from occupied is just the negative of that MO s energy. This observation is known as Koopmans theorem. One can similarly show that the energy of the lowest unoccupied MO is an estimate of the electron affinity of the molecule. In fact, ionization potentials estimated by Koopmans theorem are fairly accurate, but the electron affinities calculated this way are much less so. 

In the Fock operator, the core Hamiltonian h( 1) does not depend on the orbitals, but the Coulomb and exchange operators (1) and ( 1) depend on ( 1). If (1,2,3,..., Ne) is constructed from the lowest energy Ne orbitals, one has the lowest possible total electronic energy. By Koopmans theorem, the negative of the orbital energy is equal to one of the ionization potentials of the molecule or atom. 

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