Bond orders atomic overlap matrices

An original and complete solution of that problem has been offered by Komorowski and Lipinski [30,31]. Their analysis is based on the resolution of the Hartrec-Fock eigcnproblem for a molecule on the basis of atomic orbitals. If Fu denote elements of the energy matrix, Su arc overlap integrals and P stand for the elements of the bond order matrix, the following results for atomic electronegativity and hardness are obtained. 

Apart from choosing the force field for the classical subsystem and the level for quantum computations, the only latitude left to the user is the choice of the frontier bond characteristics. These may be extracted from a computation on a model molecule containing the bond of interest. In such a case, only one orbital needs to be localized so that one can use a so-called external localization criterion which is based upon a property of the bond of interest. The Weinstein-Pauncz criterion, which maximizes the overlap population between the two atoms defining the bond, appears to be quite convenient for this purpose. This localized orbital usually contains minor contributions of atoms other than the X and Y atoms. In order to obtain a strictly localized orbital, these contributions are discarded and the orbital renormalized. If the reference frame is different from the frame used for the molecule on which the LSCF computation is being performed, a simple matrix transformation generates the SLO expression in the LSCF basis set and the contribution of the orbital to the Pl density matrix is straightforward. 

Here is the overlap matrix and P xv the bond order matrix. Mulliken suggested that each overlap term is equally partitioned among the atoms to which the orbitals and V belong. This results in a unique definition of electronic contributions Na ... 

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