Restricted Hartree-Fock LCAO Method for Crystals

5 Restricted Hartree-Fock LCAO Method for Crystals 

In this section, we examine the main modifications in the Hartree-Fock Roothaan equations (4.33), it being necessary to take into account the translation symmetry of periodic systems. The first most important difference appears in the LCAO representation of the crystalline crbitals (CO) compared to the molecular orbitals (MO). 

For the periodic system the AO symmetrization over the translation snbgroup of the space gronp is made giving the Bloch sums of AOs  

the index fi labels all AOs in the reference primitive unit cell (p = 1,2. M) and Rn is the translation vector of the direct lattice (for the reference primitive cell Rn = 0). The summation in (4.53) is snpposed to be made over the infinite direct lattice (in the model of the infinite crystal) or over the inner primitive translations ii of the cyclic cluster (in the cyclic model of a crystal). In the latter case, the sum of the two inner translation vectors fi -I- R = Ri may appear not to be the inner translation of the cychc cluster. However, the subtraction of the translation vector A of the cyclic cluster as a whole (in the cyclic model the vector A is 

The Bloch sum of AO (4.53) satisfies the Bloch theorem. To check this let us apply the translation operator t (for the fixed translation Rm) to the sum (4.53) 

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