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

Abstract

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.

Keywords:

• three-electron integrals
• numerical integration

Acknowledgements

We acknowledge Olli Lehtonen and Susi Lehtola for their helpful comments. This research has been supported by the Academy of Finland through its Computational Science Research Programme (LASTU) and within project 137460. CSC – the Finnish IT Center for Science is thanked for computer time. We also acknowledge the Magnus Ehrnrooth Foundation for financial support.

Notes

^a See Equation (26). ^b See Equation (17) ^c See Equation (25). ^d See Equation (10)

R _a =R _d =(1.5a₀, 0, 0), R _b =R _c =R _e =R _f =(0, 0, 0), ζ _a = ζ _b = ζ _c = ζ _d = ζ _e = ζ _f = 10.