Impurity subspace

The range in coordinate space of the defect potential f/(r) determines the impurity subspace actually needed to describe the disturbance in Eq. (5.4). 

Equation (5.7) can be used to evaluate the matrix elements of G within the impurity subspace, and then, if needed, any other matrix element of G. 

When the defect potential C/(r) has a short-range nature, the order of the de-tenninantal equation (5.9) is manageable since the operator 1 — G (E)U needs to be represented only in the impurity subspace. 

The Green s-function formalism for impurities in its fully self-consistent formulation or in some simplified version has been used to treat short-range defect potentials. In this case the operator equations can be represented by a small basis set, restricted essentially to the impurity subspace. In addition to the matrix elements of U, one must calculate the matrix elements of G°( ). The latter are independent of the impurity disturbance and need only be calculated in the impurity subspace. Since the operator refers to the perfect crystal, it can be diagonalized with the standard methods of band the-... 

MOs) and MO energies of the impurity states are calculated by the DV-Xa molecular orbital method (18) in the second step, the many-electron Hamiltonian, in which the interactions between two electrons are exactly described, is diagonalized within a subspace spanned by Slater determinants made up of the obtained one-electron MOs. 

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