Efficient evaluation of three-centre two-electron integrals over London orbitals

Abstract

The nonperturbative calculation of molecular properties in magnetic fields requires the evaluation of integrals over complex-valued Gaussian-type London atomic orbitals (LAOs). With these orbitals, the calculation of four-centre electron-repulsion integrals (ERIs) is particularly demanding, because their permutational symmetry is lowered, and because complex algebra is required. We have implemented the resolution-of-the-identity (RI) approximation for LAOs in the TURBOMOLE program package. With respect to LAOs, employing the RI approximation is particularly beneficial, because the auxiliary basis set may always be chosen to be real-valued. As a consequence, the two-centre integrals in the RI approximation remain real-valued, and the three-centre integrals possess the same permutational symmetry as their real-valued counterparts. Compared to a direct calculation of four-centre ERIs over LAOs, using the RI approximation thus not only reduces the scaling of the integral evaluation, but also increases the efficiency by an additional factor of at least two. By using other well-established methods such as Cauchy–Schwarz screening, the difference-density approach, and Pulay's direct inversion in the iterative subspace (DIIS), the efficiency of nonperturbative calculations in magnetic fields can be increased even further.

GRAPHICAL ABSTRACT

Keywords:

• London atomic orbitals
• Boys function
• resolution-of-the-identity approximation

Acknowledgments

The authors would also like to thank Yannick J. Franzke (Karlsruhe) for help with the OpenMP parallelisation and Florian Weigend (Karlsruhe) for help with the RI-K approximation. We also would like to thank Tom J. P. Irons and Andrew M. Teale (Nottingham) for providing a set of integral values for verifying our code.

Disclosure statement

No potential conflict of interest was reported by the author(s).

ORCID

Wim Klopper  http://orcid.org/0000-0002-5219-9328

Additional information

Funding

A.P. gratefully acknowledges financial support by Fonds der Chemischen Industrie and Studienstiftung des deutschen Volkes.