Xa theory

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. 

In Xa theories, the relationship between the electronie energy the orbital energy e,-, and the oceupation number v/ of orbital i is... 

We now direct our attention to the calculation of the a i parameters. The first and second derivatives, dE /dNk)° and d Ef /dNl)°, are most conveniently obtained from SCF-Xa theory [174], whieh offers the advantage of permitting calculations for any desired integer or fractional electron population. It is, indeed, important to account for the fact that these derivatives depend on N. The difficulty is that calculations of this sort cannot be performed directly for atoms that are actually part of a molecule. So one resorts to model free-atom calculations to mimic the behavior of atoms that are in a molecule but do not experience interactions with the other atoms in the host molecule (Table 10.3). 

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... 

Fournier, R. Andzelm, J. Salahub, D. R., to be published in Xa theory, eq. (5) can be further simplified by using the linear relationship between the exchange-correlation energy density, and the exchange-correlation potential, In general LSDF theory, this simple relationship does not apply. 

The intense x-ray beams produced by synchroton sources have enabled development of nondestructive x-ray absorption spectroscopy (XAS) techniques that can provide information on oxidation states and local chemical environments of elements in a variety of environmental materials (Fendorf and Sparks, 1996 Fendorf, 1999 Senesi and Loffredo, 2005). XAS theory is discussed in detail in several books (e.g., Teo, 1986 Koningsberger and Prius, 1988 Lytle, 1988). 

The quest for various approximations for the exchange-correlation energy density/(p) has spanned decades in quantum chemistry and was recently reviewed [92]. Here, we will present the red line of its implementation, as it will be further used for the current applications. The benchmark density functional stands as the Slater exchange approximation, derived within the so-called Xa theory [179] ... 

The broken-symmetry approach was developed within the framework of unrestricted Hartree-Fock theory or spin-polarized density functional theory, i.e., Xa theory. " It is based on the assumption that the energy difference between the singlet and triplet states resulting from the interaction between two 5 = 1/2 spin centers is given by ... 

L. Noodleman, D. Post, and E. J. Baerends, Chem. Phys., 64,159 (1982). Symmetry Breaking and Ionization from Symmetry Equivalent Inner Shells and Lone Pairs in Xa Theory. 

Finding that D30 systematically underestimates F30, Slater also proposed that the last term in (25) be increased by roughly 10%. The resulting semiempirical model is called Xa theory and was used widely for many years. 

In the Xa theory, the exchange potential depends on an adjustable a parameter which is evaluated throughout the Periodic Table [19]. In an extension of this method, the a para-... 

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