An efficient algorithm to calculate three-electron integrals for Gaussian-type orbitals using numerical integration

A novel method to numerically calculate three-electron integrals of explicitly correlated approaches has been developed and implemented. Coulomb operators of inter-electronic interactions are re-expressed as an integral identity, which is discretised. The discretisation of the auxiliary dimension separates the Cartesian x, y and z dependencies, transforming the integrals of Gaussian-type orbitals to a linear sum of products of three-dimensional intermediate integrals. The intermediate s-type integrals can be calculated analytically, whereas integrals of the higher angular-momentum functions are computed using recursion formulae. The three-electron integrals are obtained by two-dimensional numerical integration of the discretised auxiliary dimensions of the integral transformation of the Coulomb operators. Common sets of quadrature points and weights for all integrals can be used after a coordinate transformation. Calculations indicate that it is possible to achieve an overall accuracy of 10(-15)E(h) using the numerical approach. The same approach can be employed for calculating more general three-electron integrals in so far the operator can be accurately expanded in Gaussian-type geminals.