Symmetry Exploitation in Local MP2 Method for Periodic Systems

3 Symmetry Exploitation in Local MP2 Method for Periodic Systems 

SALWFs are partitioned into subsets (/) belonging to a class b, each transforming into itself under the operators of a local symmetry point-subgroup F/ of the full point group F, while the operations of the associated left- or right-cosets transform subset / into another equivalent one in the same class. As an example in the diamond crystal 

For the scope of numerical applications, even in those cases where localization gives rise to definite symmetry properties (as in the diamond case), accurate symmetry equivalences are only reached under strong numerical conditions in the localization algorithm. This can be, in several cases, very computationally demanding and a more efficient shortcut to obtain accurate symmetry equivalence together with a good localization character is therefore desirable. 

Due to the previously mentioned difficulties of an a priori prediction of what kind of symmetry behavior will be compatible with good localization properties under efficient criteria, in [201] an a-posteriori strategy was adopted based on a chemical analysis of the problem, which is made easier by the fact that WFs are expressed as a linear combination of atomic orbitals (AOs) it represents a compromise between the two criteria previously mentioned. Starting from a set of locahzed WFs, the main steps of the presented scheme are  

A criterion based on a Mulliken analysis is adopted again to subdivide atomic LWFs associated to the same irreducible atom A into shell-subsets (without loss of generality, A can be assumed to be in reference 0 cell). The bond LWFs are the representatives of double and triple bonds, and are assigned to the (AO —BM)-subset. Thus, part of the LWFs, classihed as SALWFs, only require numerical rehnement, or the subset including all the atypical ones. All others are grouped in subsets S, either of a shell or of a bond type, each one comprising ns members. 

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