Relation Between the LMTO and KKR Matrices

In Chap.2 we introduced the concept of canonical bands based upon the KKR-ASA equations and used it to interpret energy bands calculated by the LMTO method. We did this because the KKR-ASA and LMTO-ASA methods are mathematically equivalent, as proven below. Specifically, we show that in a range around so narrow that the small parameter may be neglected the LMTO-ASA and KKR-ASA equations will lead to the same eigenvalues. 

To see that the matrix multiplications in (6.17) actually give (5.46,47) one must use (4.17) in the three-centre term. It now follows that if 

the KKR-ASA matrix 4 is a factor of the LMTO-ASA matrix, and the LMTO and KKR methods are equivalent in the neighbourhood of E, as we wished to prove. 

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