A Linear-Scaling Integral-Direct Explicitly Correlated Second-Order Møller–Plesset Approach

We present an integral-direct, iteration-free, linear-scaling, local explicitly correlated second-order Møller–Plesset (MP2-F12) approach, extending our previous local MP2 method [J. Chem. Theory Comput. 2016, 12, 4897]. The correlation contributions for individual electron pairs are computed within domains defined by the corresponding localized orbitals, while the correlation energies of spatially distant electron pairs are determined via multipole expansions. All the various types of integrals are computed and transformed directly, thereby precluding the need for integral storage and yielding asymptotically constant memory as well as negligible disk I/O demand. Another competitive advantage is the implementation of the 2B MP2-F12 ansatz, the most complete one so far with local approximations, enabling excellent basis set convergence even with double-ζ basis sets. Our validation studies indicate that the approach recovers at least 99.9% of the canonical MP2-F12 correlation energy and yields reaction energies with a mean error of less than 1 kJ/mol. With respect to the complete basis set limit of MP2, the error of our local approach is just slightly larger than that of canonical MP2-F12. With the new local MP2-F12 approach, we were able to compute the correlation energy for a small protein containing 644 atoms, which is the largest system ever considered in an explicitly correlated calculation.