The NFC Method in Its Ab Initio Matrix-Block Form

In the case of disordered quasi-one-dimensional systems the NFC method can also be applied to the case of an arbitrary number of orbitals per site either in an ab initio form or in a semiempirical, for instance, extended Huckel form. In the ab initio case one has the secular determinant instead of a tridiagonal in a triblock-diagonal form 

The original determinant det(F — A1) can be easily brought to a diblock-diagonal form again with the help of successive Gaussian elimination. In this way one obtains for its diagonal blocks, in completely analogy to equation (4.79a), 

This means that the original expression for the value of det M(A), if expressed in the form given by the last expression in equation (4.82), can be written instead as 

Here the quantities 5, are the eigenvalues of S, and Xj are the roots of the generalized eigenvalue equation 

Further, m (A) denotes the A th eigenvalue of the matrix block U,(A) defined by expression (4.83), and /, is the dimension of the ith diagonal block. 

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