An Efficient Scaled Opposite-Spin MP2 Method for Periodic Systems

We develop SOS-RILT-MP2, an efficient Gaussian-based periodic scaled opposite-spin second-order Møller-Plesset perturbation theory (SOS-MP2) algorithm that utilizes the resolution-of-the-identity approximation (RI) combined with the Laplace transform technique (LT). In our previous work [Goldzak, T. <i>J. Chem. Phys.</i> 2022, 157, 174112], we showed that SOS-MP2 yields better predictions of the lattice constant, bulk modulus, and cohesive energy of 12 simple semiconductors and insulators compared to conventional MP2 and some of the leading density functionals. In this work, we present an efficient SOS-MP2 algorithm that has a scaling of <i>O</i>(<i>N</i>⁴) with the number of atoms <i>N</i> in the unit cell, and a reduced scaling with the number of <i>k</i>-points in the Brillouin zone of the dominant scaling step in terms of unit cell complexity. We implemented and tested our algorithm on both molecular and solid-state systems, confirming the predicted scaling behavior by systematically increasing the number of atoms, the size of the basis set, and the density of <i>k</i>-point sampling. Using the benzene molecular crystal as a case study, we demonstrated that SOS-RILT-MP2 achieves significantly improved efficiency compared to conventional MP2. This efficient algorithm can be used in the future to study complex materials with large unit cells as well as defect structures.