The Atomic Xa Model

By analogy with solid-state studies, Slater had the idea of writing the atomic Hartree-Fock eigenvalue equation 

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 conceive the occupation number v. In standard HF theory, we deal with doubly occupied, singly occupied and virtual orbitals for which v = 2, 1 and 0 respectively. In solid-state theory, it is conventional to think about the occupation number as a continuous variable that can take any value between 0 and 2. 

In Xa theories, the relationship between the electronic energy ee, the orbital energy e , and the occupation number v of orbital i is 

If we make the assumption that the total energy is a quadratic function of occupation number v, then a quick calculation shows that the ionization energy is given by the orbital energy calculated when that orbital is half occupied. 

A separate HF-Xa calculation is therefore needed in order to calculate each ionization energy. What we do is to place half an electron in the orbital from which the electron is supposedly ionized and re-do the HF-Xa calculation. The hypothetical state with a fractional electron is sometimes called an Xa transition state, a phrase borrowed from chemical kinetics. We treat the transition state by UHF or ROHF methods according to personal preference. 

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