VALENCE MULTIPOLE EXPANSION: AN EXTENSION OF THE VECTORIAL BOND-VALENCE MODEL

The bond-valence model (BVM) has proven an excellent tool for evaluating and modeling proposed crystal structures, because instead of focusing on individual bond lengths (as in most molecular mechanics force fields,) it dictates acceptable combinations of bond lengths incident to a given atom. Since the BVM only deals quantitatively with bond lengths, however, it cannot address the spatial distribution of ligands. It has been proposed that this gap can be partially bridged by treating bonds as vectors in the direction from cation to anion, and with magnitude equal to the bond valence. When the vectors incident to many cations are summed, they tend toward zero. We have recently shown that vectorial bond-valence sum magnitudes are also fairly predictable for atoms whose coordination shells are non-centrosymmetrically distorted by electronic structure effects, including lone-pair and second-order Jahn-Teller effects. The predictors are simply the bond-valence magnitudes of the strongest incident bonds.

The vectorial bond-valence sum can be thought of as a valence dipole moment, describing the lopsidedness of the spatial distribution of bond valence about a central atom. A dipole moment is the second term in a multipole expansion, which, in truncated form, is commonly used to approximate functions that depend on angles. The norms of the multipole expansion terms through the quadrupole moment are predictable based on the strongest bond-valence magnitudes, and can be used as single-parameter descriptors to constrain coordination shell geometries, even in cases where electronic structure effects (e.g., first-order Jahn-Teller effects) cause centrosymmetric distortions.