Semi-localized orbital

Mueller, C. R., and Eyring, H., J. Chan. Phys. 19, 1495, Semi-localized orbitals. I. The hydrogen molecule. Combination of Inui (1941) and Coulson-Fischer (1949). 

The above considerations suggest a third type of expansion, proposed by Coulson and Fischer [5] and later by Mueller and Eyring [6], in which the MOs are replaced by semi-localized orbitals of the form... 

If a vector space representation of electronic states is chosen, that is, a basis-set expansion, two types of basis sets are needed. One for the many-electron states and one for the one-particle states. For the latter, two choices became popular, the molecular orbital (MO) [9] and valence bond (VB) [10] expansions. Both influenced the understanding and interpretation of the chemical bond. A bonding analysis can then be performed in terms of their basic quantities. Although both representations of the wave function can be transformed (at least partially) into each other [11,12], most commonly an MO analysis is employed in electronic structure calculations for practical reasons. Besides, a VB description is often limited to small atomic basis sets as (semi-)localized orbitals are required to generate the VB structures [13]. If, however, diffuse functions with large angular momenta are included in the atomic orbital basis, a VB analysis suffers from their delocalization tails. As a consequence, the application of VB methods can often be limited to organic molecules. 

Higuchi, J., J. Chem. Phys. 27, 825, (ii) Semi-localized bond orbital treatment of the allyl radical. Extension of VB. 

The nA— obc and nc— oab delocalizations lead to semi-localized (NLMO) orbitals cuab c and cuA Bc, which can be written as... 

Figure 4.45 A metal-ligand m,—orbital splitting diagram depicting interaction of the metal-atom d NAO and ligand nL NBO to form semi-localized NLMOs of the coordination complex, with splitting energy Aed. = < d/NLMO — fd> (NAO).

A two-pronged approach has been discussed for dealing with electron correlation in large systems (i) An extension of zeroth-order full-valence type MCSCF calculations to larger systems by radical a priori truncations of SDTQ-CI expansions based on split-localized orbitals in the valence space and (ii) the recovery of the remaining dynamic correlation by means of a theoretically-based simple semi-empirical formula. 

In the last three cases above the authors have made the radial form of the potential dependent on the angular part of the wavefunction on which it operates, recognizing that the potential experienced by an electron in, for example, the 3p orbital of chlorine is different from that in the 3s. Such potentials are termed semi-local. This dependence is particularly important when there are valence orbitals in an atom which have angular momenta which are not present in the core, e.g. the 3d orbital of the first row of transition metals. 

As expected from simple MO considerations, the radical cations of five-membered heterocycles, e.g. the blue species formed from furan, pyrrole, thiophene, and their alkyl derivatives, are n ions. The semi-occupied orbital is the n orbital with the heteroatom in the nodal plane (ai), see Scheme 2, structure 1. In radical cations of a,ca-bis-(l-pyrrolyl) alkanes the charge remains localized on a single ring, rather than being delocalized over both units [5, 10, 11]. 

In this study, we have chosen the supermolecule approach and have used the semi-empirical quantum mechanical method called PCILO (Perturbative Configuration Interaction using Localized Orbitals) (16) to calculate intermolecular interactions. This method has recently been used successfully to calculate the intermolecular energies and geometries of hydrogen-bonded dimers of hydrocarbons and water (17,18). H-bonded complexes are particularly well characterized by this method (19). 

For these norm-conserving pseudopotentials, a different potential needs to be applied on each orbital depending on its angular momentum. These pseudopotentials then have a semi-local form ... 

---