Introducing the scattering path operator

Here ERL are the zeroes of the potential function DaRL z). In this way, the poles of D is exactly cancelled by the poles of the last term in Eq.(4.30). 

It is sometimes possible to treat these states as core states, but often, especially when alloying with another element, hybridization broadens the valence band and the semi-core states and creates a continuous band, and in this case the semi-core states have to be treated as valence states. 

In a generic EMTO calculation, the slope matrix is calculated at an expansion center and later Taylor expanded to the energy points on the integration contour  

One way to improve on the convergence of the expansion is to use another expansion center at a large negative (nw)2, but this creates some problems when trying to join the two expansions. A better way is to use information from another energy point in a two-center Taylor expansion  

These conditions lead to a system of linear equations for an+i,an+2,. Solving these equations, we obtain a (n + m+ l)th order expansion for Sa. In Paper I, this expansion is examined and compared with the old one-center Taylor expansion. The results show a marked improvement in the expansion of the fee slope matrix, see Fig.(4.2), and we also find some cases where it is necessary to use the two-center expansion in order to produce correct results. The error in Fig.(4.2) is defined as 

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