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In this paper methods for calculations of multi-center integrals of squaredCoulomb potentials and Slater-type orbitals (STO) are derived. These integralsare necessary for accurate lower bounds to energy levels of molecular systems.All multi-center integrals are reduced to fundamental integrals using the Gauntcoefficients and translation of STO. When the potential is the usual Coulombpotential, using the Laplace expansion or the Neumann expansion of thepotential the integrals can be calculated. However, for the squared Coulombpotentials such expansions are not known. For the fundamental one-center andtwo-center integrals with squared Coulomb potentials, by methods free from suchexpansions exact analytic expressions and expressions by one-dimensionalintegrals of analytic functions are derived. The methods mainly rely on theintegration in ellipsoidal coordinates, the Fourier transform, Hobson's theoremand expansion of differential operators by simple ones suitable for thecalculation. Numerical results by these expressions are given and compared.

Methods for accurate calculations of multi-center integrals of squared Coulomb potentials for lower bounds to energy levels of molecular systems