*Ab initio* Hartree-Fock (HF) and post-HF electron correlation methods, such as Møller−Plesset perturbation theory (MP) and coupled cluster theory (CC), are standard tools in computational chemistry nowadays, and various program packages are available for accurate calculations of properties of atoms and molecules. For solids, HF calculations are possible with, e.g., the program package CRYSTAL. However, the problem of an accurate treatment of electron correlation is not fully settled.

Various approaches have been devised that attack the problem directly, e.g. the local and density-fitted MP2 approach implemented in CRYSCOR, and the plane-wave-based MP2 code developed in VASP. Albeit, all of these applications are dealing with insulating systems.

An alternative way is to treat electron correlation through considering finite clusters. Here, two variations are possible. First, one may gradually approach the infinite system starting from finite clusters of increasing size, which are tractable using standard quantum-chemical correlation methods. For the transfer of information from the finite clusters to the infinite solid, an extrapolation scheme can be employed. Such schemes may be based on the spatial extensions of the clusters. Unfortunately, the cluster size needed for a reliable extrapolation is rather large, and this may be prohibitive for the application of post-HF methods. Other schemes achieve a more rapid convergence by combining high-level (e.g., MP2) results for clusters with lower-level (e.g., DFT) results for clusters and solids.

A branch of approaches is inspired by the many-body expansion. One of the approaches of this type is the method of increments, originally proposed by H. Stoll and further developed by other groups. In this approach, a periodic HF calculation is followed by a many-body expansion of the correlation energy, where the individual units of the expansion are either atoms or other domains of localised orbitals. Calculations based upon the method of increments have been performed on a variety of solids with band gaps. We have shown that the incremental scheme, after some reformulation, can also be applied when considering metals [1,2].

This approach can be extremely useful when studying weakly bound systems (e.g. interlayer interaction in graphite, physisorption, etc.). The related examples are physisorption of water on graphene [3], Xe/Mg(0001) [4], He/graphene [5], He/Mg(0001)[6].