Green Matrix Formulation of the Problem

The best way to take into account also propagation during the iterations leading to self-consistency of a perturbation localized originally to one or a few cells is a general Green matrix formulation of the problem.  

Here Bi(k) and Ci(k) are the eigenvalues (bands) and eigenvectors (LCAO expansion coefficients of the Bloch fimction) of the one-electron equation for the infinite periodic system (see Chapter 1) 

Dyson s equation defines the Green matrix for the system with a perturbation in the form 

The impurity potential V describes the deviation from the perfect infinite system and, in the case of nonunit overlap, is given by 

In the case of finite-range deviation potentials only a subblock of the infinite matrix V is nonzero. The linear combination of atomic orbital (LCAO) expressions for the Hartree-Fock case are given below. Equation (4.113) can be solved formally to give 

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