Variational theory of linearized methods

The Kohn-Rostoker variational principle [204] implies variational equations in the extended basis / , / . Each energy-independent ACO basis function j k is defined by one fixed function pk modified by a sum of f11 functions with coefficients co lk. Suppressing L-indices, the KKR/MST equations indexed by 4 K in cell r/x for the coefficients in fk are 

Matrices C, S, C and S here are to be considered as rectangular matrices. The internal sums over solid-harmonic L-indices should be carried to convergence. The L, L indices of the square matrix co are basis function indices and may have a smaller range. 

If the KKR functional A were treated as a functional of the coefficient matrix co, the derived variational equations would be a set of linear equations of the form S1 J]v[- ] = 0, where the bracketed term is the same as in Eqs. (7.10). The solution of these simplified equations for a given value of X,L and all values of pt, L is a column vector of the o -matrix. These simplified equations were tested by empty-lattice calculations on an fee space-lattice [280]. 

The Schlosser-Marcus variational principle [359] provides an alternative that does not use structure constants. On substituting the expansion of an energy-independent ACO into the SM variational functional, the variational equations indexed by in cell r/( are 

These difficulties with the linearized VCM can be resolved by using the extended functional, defined over the global matching surface a in Subsection 7.2.4, above, 

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