Recent progress on representations for Coulomb integrals of exponential-type orbitals

Abstract

We present some recent work on Coulomb integrals with certain exponential-type orbitals (ETOs), the B functions (E. Filter and E.O. Steinborn, Phys. Rev. A, 18 (1978) 1). The main advantage of B functions is that they possess a very simple Fourier transform (E.J. Weniger and E.O. Steinborn, J. Chem Phys., 78 (1983) 6121), which results in relatively compact general formulas for molecular integrals derivable using the Fourier transform method. Because, in addition, these functions can be taken as a basis of the space of ETOs, molecular integrals for the more common Slater-type orbitals and other ETOs can be written as finite linear combinations of integrals with B functions. Various representations for Coulomb integrals of B functions with equal exponential parameters and three integral representations of Coulomb integrals of B functions with different exponential parameters are given, which could be derived recently. The integral representations are one-dimensional and contain more simple molecular integrals as part of the integrand. Upon insertion of various analytic representations of the latter integrals by finite sums a large number of different integral representations for the Coulomb integrals emerge. The structure of one of the integral representations is completely analogous to a one-dimensional integral representation (H.P. Trivedi and E.O. Steinborn, Phys. Rev. A, 27 (1983) 670) for the two-center overlap integral of B functions with different exponential parameters. Further, a Jacobi polynomial representation for the Coulomb integrals of B functions with different exponential parameters is discussed. Besides finite sums of functions regular at the origin this representation contains irregular solid harmonics and distributional contributions which are displayed explicitly.