Second order Møller-Plesset theory provides a remarkably simple form for the electron correlation energy with many desirable properties, e.g. it is size-consistent, free of self-interaction error, and scales with the fifth power of system size. However, MP2 exhibits well-known shortcomings including an incomplete description of dispersion interactions and sizable failures for transition metal chemistry. Herein, we first explore multiple physically justified forms of single-parameter regularization and then demonstrate that with appropriate parameter choice, regularized MP2 with Hartree-Fock reference orbitals yields high and transferable accuracy across a wide variety of noncovalent interactions (S22, S66, XB40, A24, and L7 test sets) and (mostly closed-shell) transition metal thermochemistry (metal-carbonyl dissociations and a subset of MOR41). We find that, especially for systems with interacting pi systems relevant to dispersion interactions and dative bonding, regularization serves to damp overestimated pair-wise additive contributions to the first-order amplitudes that affect correlation energy and charge-density. The optimal parameter values for the noncovalent and transition metal sets are 1.1 and 0.4 for two regularizers, $\kappa$ and $\sigma^2$, respectively. These two regularizers slightly degrade the accuracy of conventional MP2 for some small-molecule test sets which are well-known to be sensitive to charge-density distribution (radical stabilization energies, barrier heights, dipole moments, and polarizabilities), most of which have relatively large gaps. Due to the relatively straightforward implementations of nuclear gradient and other properties, we recommend $\kappa$-MP2 with $\kappa$ = 1.1 as a more accurate alternative to conventional MP2 and other related variants. Our results suggest that appropriately regularized MP2 models represent promising forms for the nonlocal correlation part of double hybrid density functionals, at no additional cost over conventional MP2.
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