**Introduction**

There are many ways to approximate the correlation energy of periodic systems like crystalline solids. One way is to use wave function based approaches like many body perturbation theory. This approach provides the opportunity to systematically improve the approximation of the correlation energy. However, it also comes along with a very high computational effort. All wave function based methods suffer from a steep scaling of the computation time with respect to the system size. Therefore, finding new algorithms that reduce the computational cost is of great importance. In this work we present a novel and low-complexity approach to calculate the correlation energy of large periodic systems in second-order Møller-Plesset pertubration theory (MP2). The scaling of the computation time is reduced from *N*^{5} to *N*^{4} and the parallelization efficiency of the algorithm is close to the ideal case. It is therefore possible to treat larger systems containing up to several hundrets of atoms on supercomputers.

**Abstract**

We present a low-complexity algorithm to calculate the correlation energy of periodic systems in second-order Møller-Plesset (MP2) perturbation theory. In contrast to previous approximation-free MP2 codes, our implementation possesses a quartic scaling, *O*(*N*^{4}), with respect to the system size *N* and offers an almost ideal parallelization efficiency. The general issue that the correlation energy converges slowly with the number of basis functions is eased by an internal basis set extrapolation. The key concept to reduce the scaling is to eliminate all summations over virtual orbitals which can be elegantly achieved in the Laplace transformed MP2 formulation using plane wave basis sets and fast Fourier transforms. Analogously, this approach could allow us to calculate second order screened exchange as well as particle-hole ladder diagrams with a similar low complexity. Hence, the presented method can be considered as a step towards systematically improved correlation energies.

**Original publication:** Schäfer, T., Ramberger, B., Kresse, G., 2017. Quartic scaling MP2 for solids: A highly parallelized algorithm in the plane wave basis. The Journal of Chemical Physics 146, 104101. DOI:10.1063/1.4976937