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

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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.