Analytical calculations of molecular integrals for multielectron R-matrix methods
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Schematic diagram of a molecule enclosed by an -matrix sphere of radius . Exchange between the scattered and atomic electrons is only important within the sphere. The bound target basis functions centered on nuclei and (only two nuclei are shown for clarity) have negligible amplitude outside the -matrix sphere. The basis functions for the scattered electron are positioned at the center of the sphere and have significant amplitude on the -matrix sphere. The various integrals of the scattered electron in the outer region must be subtracted off when evaluating Hamiltonian matrix elements over the volume inside the -matrix sphere.
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[(a)–(e)] Electron repulsion tail integrals as a function of the -matrix radius for -, -, -, -, and -type continuum orbitals. All of the numerical parameters for each of the plots were taken from Eqs. (47) and (48). The open circles correspond to a full six-dimensional numerical integration of the tail integrals, and solid lines are the result of the analytical summations over the product of Eqs. (34) and (46). The analytical expressions are in excellent agreement with the multidimensional numerical integrations.
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