Local MP2 Equations for Periodic Systems

The local MP2 electron-correlation method for nonconducting crystals [109] is an extension to crystalline solids of the local correlation MP2 method for molecules (see Sect. 5.1.5), starting from a local representation of the occupied and virtual HF subspaces. The localized HF crystalline orbitals of the occupied states are provided in the LCAO approximation by the CRYSTAL program [23] and based on a Boys localization criterion. The localization technique was considered in Sect. 3.3.3. The label im of the occupied localized Wannier functions (LWF) Wim = Wj(r — Rm) includes the type of LWF and translation vector Rm, indicating the primitive unit cell, in which the LWF is centered (m = 0 for the reference cell). The index i runs from 1 to A i, the number of filled electron bands used for the localization procedure the correlation calculation is restricted usually to valence bands LWFs. The latter are expressed as a linear combination of the Gaussian-type atomic orbitals (AOs) Xfiif Rn) = Xfin numbered by index = 1. M M is the number of AOs in the reference cell) and the cell n translation vector 

AOs constitute the nonorthogonal basis set used for the solution of the HF crystaUine equations. To exploit the translation invariance property of LWFs the matrix = 

As in the molecular case (Sect. 5.1.5) for the virtual space the projected AOs (PAOs) are used. To generate PAOs the projector P = / (/ onto the occupied 

HF space is expressed in terms of the density matrix in an AO representation, P, which is calculated very accurately by the CRYSTAL program via BriUouin-zone integration over all canonical occupied crystaUine orbitals /  

The projector onto the virtual space is hence obtained Q = 1 — P. By acting with Q on any one of the local functions Xiin = Xp of ft original AO set, a Q-projected AO is obtained, or simply PAO  

---