Ionization processes and Koopmans theorem

The photoionization process with which we shall be concerned in both UPS and XPS is that in Equation (8.4) in which only the singly charged is produced. The selection mle for such a process is trivial - all ionizations are allowed. 

When M is an atom the total change in angular momentum for the process M + /zv M+ + e must obey the electric dipole selection mle Af = 1 (see Equation 7.21), but the photoelectron can take away any amount of momentum. If, for example, the electron removed is from a d orbital ( = 2) of M it carries away one or three quanta of angular momentum depending on whether Af = — 1 or +1, respectively. The wave function of a free electron can be described, in general, as a mixture of x, p, d,f. wave functions but, in this case, the ejected electron has just p and/ character. 

In molecules, also, there is no restriction on the removal of an electron. The main difference from atoms is fhaf, since fhe symmefry is lower, fhe MOs fhemselves are mixfures of s,p, d,f. AOs and fhe ejected electron is described by a more complex mixture of s,p, d,f. character. 

The negative sign is due to fhe convention fhaf orbifal energies are negative. 

Af fhe level of simple valence fheory Koopmans fheorem seems to be so self-evidenf as to be scarcely worth sfafing. Flowever, wifh more accurate fheory, fhis is no longer so and greaf inferesf attaches to why Equation (8.5) is only approximately fme. 

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