Natural Density Pairs and the Basis-Set Locality of Operators

The work in this thesis can be divided into two distinct studies, the first of which involves the construction of the natural density pair (NDP) expansion of a general phase-space function. We focus on the Wigner function and its NDP expansion in a finite Gaussian basis, giving results for the first ten atoms calculated with several basis sets commonly used in electronic structure calculations. It is found that in all cases the NDP expansion converges too slowly to be of use in the visualization of the Wigner function. The rest of this thesis is concerned with the locality of operators in different basis sets. A measure of operator locality is introduced and used to study the kinetic energy, Fock, exchange, and one-electron charge density matrices defined by several common Gaussian basis sets. It is found that these matrices are very nearly equal to the matrices of local operators. In addition, a method of analyzing the locality of the exchange operator via the exchange supermatrix is presented and found to also indicate that the exchange operator is nearly local in these bases. Similar studies of the locality of the kinetic energy operator in numerous basis sets of orthogonal polynomials and orthogonal trigonometric functions were undertaken in an effort to understand what basis-set properties lead to the near locality of this operator. We find that the near locality of the kinetic energy operator in commonly -used Gaussian bases is largely (if not entirely) a result of the form of the basis functions. In general it is found that the locality of the kinetic energy operator decreases with increasing basis size; however many basis sets involving exponentials seem to behave differently for reasons we do not yet understand. For each basis studied we find that our locality measure converges to a constant value, the canonical limit, as the basis set approaches completeness. All of the one-dimensional bases studied are found to have the same canonical limit, while no such common value is found for pseudo one-dimensional and three-dimensional bases.