Koopmans’ approximation

The inner-shell ionization energies are eolleeted in Table 2 and eompared with the Koopmans estimates (which are seen to be seriously in error) and the best available experimental values. Whilst the Koopmans approximation is clearly incapable of giving good ionization energies and must therefore he used with caution in predicting the chemical shifts in going from one molecule to another, the ionization energies based on (2) are rather satisfactory. 

For a closed-shell molecule, Koopmans approximation identifies an IE with the negative of the SCF orbital energy (equation 5) if this were the case, a PE spectrum would provide an exact mapping of the occupied orbitals of a molecule (Figure 3). 

Calculated values Hartree-Fock-Roothaan (using Koopmans Approximation) Basis Set STO-3G 14.7 15.6... 

The MS-A"a method has been very successful for the calculation of ionization energies for both gas-phase molecules and molecular cluster models of solids. DeAlti et al. (1982) have employed both the MS-Aa method and the LCAO-Aa scheme of Sambe and Felton (1975) in the transition-state procedure for calculating ionization potentials for a number of molecules where the Koopmans approximation within Hartree-Fock theory gives the wrong ordering. They found that the LCAO-Aa ionization potentials were more accurate than the Hartree-Fock Koopmans values, while the MS-Aa results typically showed errors of one or two electron volts. Representative results are given for ozone (O3) in Table 3.8. 

The primary experimental data obtained from PE spectra are ionization energies and it is the well-known Koopmans approximation which relates ionization energies to orbital energies. It states that the ith ionization energy is given by the negative of the ith SCF... 

In cases where Koopmans approximation fails badly, the direct way of calculating ionization energies, the method of Green s functions, has been applied successfully, although we are not aware of its use in case of organometallic compounds. Applications of a simplified Green s function approach to transition metal complexes, however, have been reported The scattered-wave method , on the contrary, has wide applications in describing metal-metal bonds and the nature of multicentre interactions within metal-lacycles. ... 

CALCULATED VERTICAL IONIZATION ENERGIES OF COFj, BASED ON KOOPMANS APPROXIMATION... 

The measured E value is directly proportional to the difference Eb(IE) = Ef — E,. The final state in PES consists of an ion and the outgoing photoelectron. The electronic structure of material is often described by approximate, one-electron wavefunctions (MO theory). MO approximation neglects electron correlation in both the initial and final states, but fortunately this often leads to a cancellation of errors when Ef, is calculated. A related problem arises when one tries to use the same wavefunctions to describe 4q and I f. This frozen orbital approximation is embedded in the Koopmans approximation (or the Koopmans theorem as it is most inappropriately called), equation 4,... 

Ionization potentials were computed using Koopmans approximation, which states that for closed-shell systems the ionization potential is the negative of the corresponding molecular orbital eigenvalue. These results are summarized in Table 3, where comparison is made between MNDO, AMI, and PM3. The average errors for 256 compounds are 0.78 eV (MNDO), 0.61 eV (AMI), and 0.57 eV (PM3). 2... 

Ionization potentials of molecules containing H and elements of the first and second row are reasonably well estimated using the INDO/S model and Koopmans approximation. Generally the first three or four IPs can be predicted this way within 0.5 eV, and the INDO/S model seems to be among the more accurate models for predicting IPs. A far more accurate procedure to obtain... 

Note the Koopmans approximation and the delta-SCF methods both break down after the first two ionization potentials. 

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