Muffin-Tin Orbitals in the ASA

In the limit where k goes to zero the conventional phase-shift notation 

The normalisation of x (r) an J (r) 1S such that their amplitudes at the sphere boundary are 1//S/2 where the factor /S/2 is introduced for reasons of notation. The augmented spherical Neumann function N(r/S) appearing in (6.11) may be defined by the = 0.limit as (5.24) 

The energy-independent muffin-tin orbital Xj (r - R) centred at R is especially designed to have a tail orthogonal to the core states inside the sphere centred at R . Therefore the function L1( , r - R ) appears in the 

The one-centre expansion (6.15) is specialised to the case where R = 0, and is valid inside the atomic sphere centred at the origin. It may be used to derive the LMTO equations and with the normalisation implied by (6.11) it is consistent with the secular matrices (5.46,47) in the ASA. In linear methods in band theory [6.2] Andersen presented the one-centre expansion in the form (6.15) and derived the LMTO formalism from that assumption. His LMTO formalism is equivalent to that presented here apart from the normalising factor [/S/2 (- -1)] 1 appearing in the definition (6.11) of the energy-dependent muffin-tin orbital. 

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In Chap.6 the atomic-sphere approximation is introduced and discussed, canonical structure constants are presented, and it is shown that the LMTO-ASA and KKR-ASA equations are mathematically equivalent in the sense that the KKR-ASA matrix is a factor of the LMTO-ASA secular matrix. In addition, we treat muffin-tin orbitals in the ASA, project out the i character of the eigenvectors, derive expressions for the spherically averaged electron density, and develop a correction to the ASA. 

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