Hartree-Fock theory 592 INDEX

We recognize this as the normal Fock matrix in closed shell Hartree-Fock theory, but only for those matrix elements where the second index, i, corresponds to an occupied orbital. The first index is arbitrary. The condition for an optimum wave function is then Fpi = Fip. This is trivially fulfilled when also p corresponds to an occupied orbital, since the wave function is invariant towards rotations among these orbitals. On the other hand, when p is a virtual orbital we have Fip = 0 (all density matrix elements in (4 42) are then zero). 

The average local ionization energy /(r) is probably the most effective single reaction index for analyzing regioselectivity and relative reactivity of S Ar. The /(r) function was first proposed by Sjoberg et al. in 1990 [82] and is rigorously defined within the frameworks of Hartree-Fock theory and KS-DFT by (Eq. 4.9)... 

If the Kohn-Sham orbitals [52] of density functional theory (DFT) [53] are used instead of Hartree-Fock orbitals in the reference state [54], the RI can become essential for the realization of electron propagator calculations. Modern implementations of Kohn-Sham DFT [55] use the variational approximation of the Coulomb potential [45,46] (which is mathematically equivalent to the RI as presented above), and four-index integrals are not used at all. A very interesting example of this combination is the use of the GW approximation [56] for molecular systems [54],... 

It has also been shown that for a series of benzene rings in paracyclophanes" and in benzenoid hydrocarbons," a very good correlation exists between the geometry-based index HOMA and the Hartree—Fock energy (HF) at the 6-3IG level of theory. It is clear that in all these cases the selection process favors a projection from a multidimensional space to a lower... 

While the expansion of hpq follows readily from the development for the Dirac operator above, the electron-electron interaction integrals must be considered separately. We also want to develop a Kramers-restricted Dirac-Hartree-Fock (KR-DHF) theory, but first we develop expressions for the general, Kramers-unrestricted case. In the developments below we follow the practice of giving only the basis function index in the integrals. 

The number of electrons delocalization (shared) between atomic basins Q,- and Dj can be measured by the delocalization index (Dl), 8(Q,-,Q ). For a closed-shell molecule, the DI at the Hartree-Fock level of theory is defined [19] ... 

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