Valence Atomic Orbital Centroids

The description presented so far has dealt primarily with the numbers of electrons associated with the individual atoms in a molecule. Now we examine the shape of these electron populations. The electron densities p and electron populations N are those of the valence region. 

Let us also introduce the corresponding expression obtained when the electron populations are kept at their proper values in the molecule under study, but the inverse-distance terms are replaced by their values (indicated by the superscript O ) of a model reference molecule  

The difference V describes a change in electrostatic potential at atom k. The corresponding change in potential energy, summed over aU atoms, is 

This is the quantity we shall use for our discussion. It is a convenient way of gaining insight into the function F [Eq. (10.26)] because 

Extensive numerical calculations indicate that/is (1) neghgible (say, 0.3 kcal/ mol) for saturated hydrocarbons but (2) signihcant for olefinic molecules (e.g., 40 kcal/mol for tetramethylethylene). The condition that/should vanish can be satisfied either because the various atomic contributions to/cancel or because the individual terms in the summation over k vanish. Since it seems unlikely that cancellation of terms associated to different atoms would take place systematically in a large number of molecules, we shall assume that, to a good approximation 

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Note that a distinction is made between electrostatic and polarization energies. Thus the electrostatic term, Ue e, here refers to an interaction between monomer charge distributions as if they were infinitely separated (i.e., t/°le). A perturbative method is used to obtain polarization as a separate entity. The electrostatic and polarization contributions are expressed in terms of multipole expansions of the classical coulomb and induction energies. Electrostatic interactions are computed using a distributed multipole expansion up to and including octupoles at atom centers and bond midpoints. The polarization term is calculated from analytic dipole polarizability tensors for each localized molecular orbital (LMO) in the valence shell centered at the LMO charge centroid. These terms are derived from quantum calculations on the... 

Localized Wannier functions (LWFs) have been calculated for three upper valence bands in SrTiOs and SrZrOs, represented mainly by O 2p, Sr 4p, and O 2s atomic states (in the case of SrZrOs the last two bands overlap considerably). A total of 15 crystalline orbitals have been used to generate, correspondingly, 15 LWFs per primitive unit cell in both crystals under consideration, three oxygen atoms occupy the same Wyckoff positions, and four LWFs can be attributed to each oxygen atom. It was found by calculations with CRYSTAL03 code [23] that the centroids of four functions are positioned near the center of one oxygen (at distances of about 0.3 A). 

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