Valence shell population parameter

A simple modification of the IAM model, referred to as the K-formalism, makes it possible to allow for charge transfer between atoms. By separating the scattering of the valence electrons from that of the inner shells, it becomes possible to adjust the population and radial dependence of the valence shell. In practice, two charge-density variables, P , the valence shell population parameter, and k, a parameter which allows expansion and contraction of the valence shell, are added to the conventional parameters of structure analysis (Coppens et al. 1979). For consistency, Pv and k must be introduced simultaneously, as a change in the number of electrons affects the electron-electron repulsions, and therefore the radial dependence of the electron distribution (Coulson 1961). 

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... 

In subsequent work by the same authors, non-neutrality of the standard atoms was allowed by addition of a transferable PD valence-shell monopole population, with neutrality being maintained by a slight adjustment of the hydrogen charges, and k parameters refined after the transfer (Pichon-Pesme et al. 1995b). 

The parameters Pim , Pcore, and k can be refined within a least square procedure, together with positional and thermal parameters of a normal refinement to obtain a crystal structure. In the Hansen and Coppens model, the valence shell is allowed to contract or expand and to assume an aspherical form [last term in (11)], as it is conceivable when the atomic density is deformed by the chemical bonding. This is possible by refining the k and k radial scaling parameters and population coefficients Pim of the multipolar expansion. Spherical harmonics functions yim are used to describe the deformation part. Several software packages [68-71] are available for multipolar refinement of the electron density and some of them [68, 70, 72] also compute properties from the refined multipolar coefficients. 

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]. 

Mossbauer spectroscopy is well suited to such a study, since two major parameters of a spectrum can be related directly to the populations and changes in population of the valence shell orbitals. The isomer shift relates to the total electron density on the atom, and the quadrupole splitting reflects any asymmetry in the distribution of electron density. Thus, each parameter is capable of giving information about bonding from their combination, it is often possible to make quite detailed analyses. 

Where n and are the populations of s and p electrons, the coefficient A quantifies the direct effect of the valence shell s and pi electrons and their shielding of the core electrons, and the coefficients B, C and D quantify the mutual shielding of the s electrons, the shielding of p electrons by s electrons, and the shielding of core electrons by p electrons respectively. The signs of B, C and D would therefore be expected to be opposite to that of A. It is usually impractical to employ so many parameters and a simpler form... 

Yet the properties (bond distances and bond energies) of hypervalent compounds are consistent with Lewis structures with five or six bonding electron pairs in the valence shell of the central atoms. What is the source of this discrepancy The wavefunctions obtained by such ah initio calculations do not constitute exact solutions to the Schodinger equation and molecular orbitals orbital energies and population parameters are nonobservable quantities that cannot be checked by experiment. 

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