Relativistic two-component transformation inclusive contraction scheme on atomic mean field and its scalable approximation

We propose a two-component transformation-inclusive contraction (TIC) on the atomic mean field (TICamf) method as an efficient and accurate approach for relativistic many-electron systems. TICamf integrates into the contraction of basis functions a two-component transformation that incorporates electron repulsion potentials. This approach yields contracted atomic orbital integrals with significantly improved accuracy compared with that of the conventional TIC method, while maintaining the effectively zero computational cost of the two-component transformation. To further enhance computational efficiency by accelerating the evaluation of small-component-type two-electron integrals, which is a bottleneck in TICamf calculations, we introduced the relativistic RI-V (rRI-V) approximation for two-electron integrals. In addition, rRI-V approximation improves the overall scaling of the integral evaluations, including contributions from large components to O(N3). For the auxiliary basis sets required in the rRI-V scheme, we propose three generation schemes: DFc, DFp, and DFb. Numerical validation of these methods demonstrates that the TICamf approach achieves an accuracy comparable to that of the four-component methods in terms of both total energy and equilibrium internuclear distances. Furthermore, despite approximating a scaling of O(N3), the TICamf(rRI-V) method yields total energies comparable to those of the four-component methods across all three types of auxiliary basis sets. Among these, TICamf(rRI-V)-DFb demonstrated particularly high accuracy also at equilibrium internuclear distances.