Koopman’s approximation

It is commonly known as Koopman s theorem. It is also called Koopman s approximation. 

These results are connected to the shell model by Koopmans s approximation, which asserts in a form suitable for our discussion that the ionization energy of an electron is the negative of the energy of the Hartree orbital of the electron ... 

The energy-level diagram (Fig. 5.20) is drawn by showing the negative of each ionization energy value as the energy of an orbital, in accordance with Equation 5.12 and Koopmans s approximation. 

Other properties can be computed from the wave function but as is well known the precision is not as good as in the case of the total energy. If the first ionization energy is calculated using Koopman s approximation, that is to say if it is taken as equal to — 5 a value of about 14 ev is obtained. Experiment gives 11 ev. Furthermore the wave function W corresponds to a dipole moment of 1 -82 D, the experimental value being 1 -46. 

For the molecules of Figures 3.13 and 3.14 the exited energies AE of anthocyanidins for two series of oscillating strengths (I II) are in Table 3.24 collected along with highest occupied and lowest unoccupied molecular orbitals HOMO, LUMO and with Koopman s approximation... 

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... 

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. 

Koopmans s theorem is not valid for Xa calculations, but Slater s transition state concept applies. This approximation allows the interpretation of electronic transitions in terms of 1-electron orbitals and yet includes electronic relaxation effects (208). The virtual (unoccupied) orbitals of Xa theory have a physical significance and can be used to discuss excitations of the electronic system, because the same potential due to the iV - 1 other electrons affects both occupied and unoccupied... 

Unfortunately, the results are not as good as might have been expected. Whether the discrepancies originate from some inadequacy of the SCF calculation or from the invalidity of Koopman s theorem is, however, an open question. It is probably, to a first approximation only, that the removal of an electron from a molecular system can be considered as a "vertical process which leaves the distribution and energies of the... 

If a photon has sufficient energy, on bombarding atoms or ions it can cause electron emission. To a first approximation (Koopman s theorem)... 

Finally, dispersion energy is obtained from an approximate London-type expression, with local static polarizabilities and average excitation energy of the monomer, evaluated using Koopman s theorem. An empirical correction factor is then applied to take into account the error in the evaluation of the average excitation energy and the approximate character of the London expression. 

The linear acenes, benzene to pentacene, are used as examples of the CURES-EC procedure. The results obtained utilizing MINDO/3 and AMI are compared. In addition to calculating the Ea by subtracting the energies of the optimized form, the LUMO of the neutral is compared with the experimental Ea. The electron affinity of hexacene has been estimated from the electronegativity and experimental ionization potential. As a further example of the use of CURES-EC, both the ionization potential and electron affinity of heptacene are estimated. The Ea of octacene and nova-cene are calculated for comparison to values obtained by using Koopman s theorem and a semi-empirical method based on a variable-parameter modification of the Pariser Parr Pople (PPP) approximation to the Hartree Fock equation [10]. 

HOMO Highest Occupied Molecular Orbital. A molecular orbital calculation yields a set of eigenvalues or energy levels in which all the available electrons are accommodated. The highest filled energy level is called the HOMO. The next higher energy level which is unoccupied because no more electrons are available is the LUMO or Lowest Unoccupied Molecular Orbital. On the basis of Koopman s theorem the HOMO and LUMO of a molecule can be approximated as its ionization and electron affinity, respectively. 

It is interesting to note that the Koopmans s theorem prediction for BH (i.e., the 1C, unrelaxed Cl result) is also very accurate for this basis set. This indicates that the effects of orbital relaxation and the changes in correlation approximately cancel for this example. This cancellation may be at least partially responsible for the excellent accuracy of the EOM ionization potential for BH for all systematic approximation schemes. 

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