A highly efficient algorithm for electron repulsion integrals over relativistic four-component Gaussian-type spinors

In the previous studies, a highly efficient computational scheme has been proposed for the Dirac–Hartree–Fock and the Dirac–Kohn–Sham solutions using the generally contracted kinetically balanced Gaussian-type spinors (GTSs). Nevertheless, the calculations based on the full Dirac Hamiltonian are limited to small systems if they contain heavy elements. The bottleneck is the calculation of the two-electron repulsions over the four-component GTSs. The present paper presents an improved algorithm for evaluation of the four-component relativistic integrals. The new algorithm fully exploits the transfer relation of Head-Gordon and Pople (HGP) and the accompanying coordinate expansion (ACE) formulas of Ishida. The HGP transfer relation can reduce the four-component integrals into several common two-center integrals (p0|q0), which can be computed rapidly using the ACE method. The algorithm is implemented into the four-component program system REL4D. Benchmark calculations demonstrate that a good performance is achieved, particularly for the calculation of the (SS|SS) integrals.