The Koopmans Theorem

There is some small print to the derivation the orbitals must not change during the ionization process. In other words, the orbitals for the cation produced must be the same as the orbitals for the parent molecule. Koopmans (1934) derived the result for an exact HF wavefunction in the numerical Hartree-Fock sense. It turns out that the result is also valid for wavefunctions calculated using the LCAO version of HF theory. 

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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... 

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. 

The term D of Eq. (3.14) is called the delocalization stabilization, which is usually positive. This term comes from the electron delocalization between the molecules A and B. The physical meaning of the denominator of each term in the right side of Eq. (3.14) can be discussed in relation to the Koopmans theorem 58)... 

In the simplest frozen orbital approach, both IE and EA values can be approximated as the negative of the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) energies, respectively, following the Koopmans theorem. A better way is to calculate the energies of the system and its cationic and anionic counterparts separately and then estimate fx and 17 from Equations 12.4 and 12.5, respectively. 

The X-ray photoelectron spectrum of the core ionization of an atom in a molecule consists of peaks and bands corresponding to transitions to various excited states. None of these transitions corresponds to the formation of the Koopmans theorem frozen-orbital ionic state, which is a completely hypothetical state. However, the center of gravity of the various peaks and bands lies at the energy corresponding... 

If the primary peak is the only peak with a binding energy less than the Koopmans theorem energy (this is usually the case), then an increase in the primary relaxation energy must be accompanied by an increase in the quantity 2 ( )- KTX/ for the shake-up and shake-off bands, where the intensities If are... 

The evaluation of elements such as the M n,fin s is a very difficult task, which is performed with different levels of accuracy. It is sufficient here to mention again the so called sudden approximation (to some extent similar to the Koopmans theorem assumption we have discussed for binding energies). The basic idea of this approximation is that the photoemission of one-electron is so sudden with respect to relaxation times of the passive electron probability distribution as to be considered instantaneous. It is worth noting that this approximation stresses the one-electron character of the photoemission event (as in Koopmans theorem assumption). 

In the Koopmans theorem Umit the photoemission of one-electron from an atom or a core in a solid is given by a single Une, positioned at the eigenvalue of the electron in the initial state. The intensity of this line depends on the cross-section for the event, which is determined by the one-electron atomic wavefunctions Wi ( j m)(-Eb) and Pfln(nM, m )(Ekin) (where the atomic quantum numbers are indicated as well as the eigenvalues En,i,m = Eb and E dn of the initial and final state) (the overlap integral of (13)... 

Direct calculation of the ionization potential by LCAO-Xa, HAM/3 and Green s function techniques or via the Koopmans theorem by ab initio techniques. 

I drew attention in Chapter 12 to the fact that the Xa orbitals did not satisfy the nice properties of standard HF-LCAO ones the Koopmans theorem is not valid, and so on. The same is true of all density functional KS-LCAO calculations. In practice, it usually turns out that the KS-LCAO orbitals are very similar to ordinary HF-LCAO ones, which must mirror the fact that exchange-correlation effects are only a minor part of the total electronic energy. So the orbitals are often analysed as if they were ordinary HF orbitals (Figure 13.4). 

Table 6.6 Some ionization energies (eV). The AE values (cation energy minus neutral energy) correspond to adiabatic, and the Koopmans theorem values to vertical IEs. The ab initio energies are MP2(fc)/6-31G (Table 5.17). Experimental values are adiabatic, from [115] (CH3OH and CH3COCH3) and [116] (CH3SH)...

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