Numerically-specified orbitals

The coordinate—spin representation of the Hartree—Fock equation (5.19) is 

This is an extension of the Schrodinger equation for a bound state to include the nonlocal potential 

The notations for integration and the corresponding -function refer to the set X of coordinate—spin variables. The first term of (5.27) is the direct potential due to the screening of the nucleus for the electron in orbital rj) by the other electrons. The second term is the exchange potential. Note that the term for C) = fi) cancels between the direct and exchange potentials. The potential v x, x) is the two-electron Coulomb potential r — r In the relativistic case there are additional magnetic terms —a a r — r that are small and make little difference. The relativistic treatment of electron—electron interaction can only be approximate. It is discussed by Brown (1952). 

For the nonrelativistic case with neglect of spin—orbit coupling we separate the space and spin parts of the coordinate—spin representation of the orbital 

The spin functions vanish from the formalism due to their orthonormality. The exchange term however has a factor (v v ) restricting its effect to pairs of electrons with the same spin projection. Equn. (5.26) becomes an integrodifferential equation in coordinate space, which is reduced to an equation in the radial variable by the methods of sections 3.3 and 4.3. The coordinate-space Hartree—Fock equation for a closed-shell structure is 

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Figure 5.3. SHMO orbitals for cyclopropenyl, cyclobutadiene, cyclopentadienyl, and benzene. The energies are in units of f relative to a. Two alternative but equivalent representations are shown for the degenerate n orbitals of cyclobutadiene. Sizes of the 2p orbitals are shown proportional to the magnitudes of the coefficients whose numerical values are given. Coefficients not specified may be obtained by symmetry.

Let us return now to the original aim of localizing molecular orbitals. The basic expectation is that interaction between distant localized orbitals turns out to be negligible. Before specifying more precisely what interaction in the above sense means, it is necessary to define the concept of distance between localized MOs. We will discuss here some alternatives that have all been tested in our numerical calculations. 

In these identities the symbols i are the occupation numbers of the orbitals specified in the electronic configuration for the particular atom. In general, for many-electron atoms other than the simplest ones, these definitions of the screening constants lead to much better agreements with Hartree-Fock-Slater numerical results such as the output of hs.exe, for the atoms of the typical elements in the Periodic Table. 

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