Electron Correlation in the Condensed Phase from a Resolution of Identity Approach Based on the Gaussian and Plane Waves Scheme

The second-order Meller-Plesset perturbation energy (MP2) and the Random Phase Approximation (RPA) correlation energy are increasingly popular post-Kohn Sham correlation methods. Here, a novel algorithm based on a hybrid Gaussian and Plane Waves (GPW) approach with the resolution-of-identity (RI) approximation is developed for MP2, scaled opposite-spin MP2 (SOS-MP2), and direct-RPA (dRPA) correlation energies of finite and extended system. The key feature of the method is that the three center electron repulsion integrals (mu nu broken vertical bar P) necessary for the RI approximation are computed by direct integration between the products of Gaussian basis functions mu nu and the electrostatic potential arising from the RI fitting densities P. The electrostatic potential is obtained in a plane waves basis set after solving the Poisson equation in Fourier space. This scheme is highly efficient for condensed phase systems and offers a particularly easy way for parallel implementation. The RI approximation allows to speed up the MP2 energy calculations by a factor 10 to 15 compared to the canonical implementation but still requires O(N-5) operations. On the other_ hand, the combination of RI with a Laplace approach in SOS similar to MP2 and an imaginary frequency integration in dRPA reduces the computational effort to O(N-4) in both cases. In addition to that, our implementations have low memory requirements and display excellent parallel scalability up to tens of thousands of processes. Furthermore, exploiting graphics processing units (GPU), a further speedup by a factor similar to 2 is observed compared to the standard only CPU implementations. In this way, RI-MP2, RI-SOS-M132, and RI-dRPA calculations for condensed phase systems containing hundreds of atoms and thousands of basis functions can be performed within minutes employing a few hundred hybrid nodes. In order to validate the presented methods, various molecular crystals have been employed as benchmark systems to assess the performance, while solid LiH has been used to study the convergence with respect to the basis set and system size in the case of RI-MP2 and RI-dRPA.