Statistical-like signature of molecular basis sets

This paper defines the construction of a sequence of functions that might be used to geometrically characterize any GTO set employed in any molecular calculation. Such functions characterizing a basis set act as molecular statistical-like elements: the arithmetic mean or centroid of the basis set, and also variance, skewness, and kurtosis, corresponding to the first terms of a possibly larger sequence of descriptor moments. Once the moment functions are built, then they can be integrated over their electron variables. In this manner, a set of statistical-like scalars can be obtained, which will be uniquely associated with a chosen basis set. Therefore, such condensed moment scalars provide a unique numerical signature, which can be attached to any basis set employed in a molecular environment. Once obtained, basis set condensed statistical-like signatures could be used for comparison purposes. The overlap matrix of the basis set, which acts as a Gram matrix and a metric matrix, appears as the fundamental reference from where the statistical-like information can be obtained for a given basis set. Several numerical examples are provided.