Muffin-Tin Orbitals and One-Centre Expansion

The starting point is now the energy-independent muffin-tin orbitals (6.11, 12) in the atomic-sphere approximation, i.e. 

We note that the augmented spherical Neumann Ntjl(r/St) and Bessel J (r) functions are in this case defined from a tail of radial dependence (r/S ) rather than (r/S) as in the energy-dependent orbital (8.1). The addition theorem (6.13,8.7) which represents the expansion in the sphere at q of the tail of the muffin-tin orbital centred at q must therefore be changed to include the correct radial dependences, i.e. (r/St). From (6.13,8.7) we find 

In this addition theorem the spherical harmonics iV are included in N.  

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