A matrix representation of the translation operator with respect to a basis set of exponentially declining functions

Abstract

The matrix elements of the translation operator with respect to a complete orthonormal basis set of the Hilbert space L2(R3) are given in closed form as functions of the displacement vector. The basis functions are composed of an exponential, a Laguerre polynomial, and a regular solid spherical harmonic. With this formalism, a function which is defined with respect to a certain origin, can be ’’shifted’’, i.e., expressed in terms of given functions which are defined with respect to another origin. In this paper we also demonstrate the feasibility of this method by applying it to problems that are of special interest in the theory of the electronic structure of molecules and solids. We present new one-center expansions for some exponential-type functions (ETF’s), and a closed-form expression for a multicenter integral over ETF’s is given and numerically tested.

Combining Slater-type orbitals and effective core potentials

“…[113][114][115][116][117][118][119][120][121][122][123]] , yet we have not found these alternative formulations to be particularly advantageous in the present case compared to the standard Barnett-Coulson scheme, Eq. (8).…”

Simplified derivation of a one‐range addition theorem of the Yukawa potential

“…But in Hilbert spaces, convergence can be defined in the mean or in the weak sense, but usually not in a pointwise sense. Hence, a pointwise convergence of the one-range addition theorem follows in a mathematically rigorous sense neither from this derivation nor from the numerical experiments reported in [lo], nor from the computer-algebra based transformation of the one-range addition theorem of Filter and Steinborn [8] into a more simple form equivalent to Eq. (23) as given by Fern6ndez Rico et al [lo].…”

“…Based on a one-range addition theorem for exponential-type orbitals given by Filter and Steinborn [8], computer algebra, and recurrence relations of Laguerre polynomials were used by Fernindez Rico et al [9,10] to obtain simpler expansions of these orbitals, especially a surprisingly simple one-range addition theorem [lo] for the Yukawa potential that subsequently was used to obtain representations and programs that were used for the evaluation of molecular integrals with a Though some numerical tests indicating the convergence of this expansion were given in [lo], an independent derivation of the addition theorem-using neither computer algebra nor a reordering of infinite sums-is desirable to confirm the validity of the expansion. This will be given in the present article.…”

Improved quadrature methods for the Fourier transform of a two‐center product of exponential‐type basis functions

“…The B functions [16] are the ETOS with the most simple Fourier transform [17]. B functions can be expressed as finite linear combinations of STOS and vice versa [16,18]. Later, Grotendorst and Steinborn [19] rederived the integral representation of Trivedi and Steinborn [5], analyzed its numerical properties, and presented test values obtained by the application of an automatic quadrature routine of the QUAD-PACK [20] family.…”