Green matrix method chain

However, to do this for every state is practically impossible in the case of really long chains (N> 1000, where N is the number of sites). Therefore, a technique is needed that gives a rough idea as to whether or not the states are localized in a certain eneigy region without explicitly calculating them. Such a method based on the calculation of Green matrix elements of the chain will be examined in the next section. 

These are iterative procedures that allow the calculation and the Green s function matrix elements without explicit diagonalization of the Hamiltonian [18]. In the present case the Hamiltonian is factorized into double chains, and the renormalization method can be conveniently and efficiently applied, since its implementation simply requires the handling and the inversion of small matrices of rank two. For more details and elaboration of the renormalization procedures, see, for example, Ref. [23]. 

In order to find the local density of states at the surface, one requires a formalism not in terms of delocalized orbitals as we have used so far, but in terms of local quantities. Such a formalism is provided by the Green s function method, which Is also a very convenient tool for the study of chemisorption. Here we will introduce its use by applying it to the open and the closed chain problems discussed before. Within the LCAO basis set description and mean field electron approximation (see section 2.7.3.4), the Green s function is a matrix element that satisfies the following set of equations. 

We have shown earlier that Gn, the Green s function for the end atom of the open chain, can also be found by less laborious methods. Once Gu E) is known, the resolvent method enables calculation of the other matrix elements of G. 

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