Charge density atom-centered expansion

The nature of the charge density parameters to be added to those of the structure refinement follows from the charge density formalisms discussed in chapter 3. For the atom-centered multipole formalism as defined in Eq. (3.35), they are the valence shell populations, PLval, and the populations PUmp of the multipolar density functions on each of the atoms, and the k expansion-contraction parameters for... 

Here P and Plm are monopole and higher multipole populations / , are normalized Slater-type radial functions ylm are real spherical harmonic angular functions k and k" are the valence shell expansion /contraction parameters. Hartree-Fock electron densities are used for the spherically averaged core and valence shells. This atom centered multipole model may also be refined against the observed data using the XD program suite [18], where the additional variables are the population and expansion/contraction parameters. If only the monopole is considered, this reduces to a spherical atom model with charge transfer and expansion/contraction of the valence shell. This is commonly referred to as a kappa refinement [19]. 

With this asymptotic density model (ADM) an advantage over the standard multipole expansion can be achieved, so that the MESP is well approximated not only outside the molecule, but also near the atoms. This scheme has been implemented in SINDOl for first- and second-row elements. The evolving expansion has been truncated after the cumulative dipole moments terms. The three-center integrals have been approximated on the NDDO level. In order to achieve good agreement with ab initio based MESPs, the atomic hybrid moments had to be scaled. In the application to. solvation energies the atomic electronic charges had to be scaled too. 

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