By Andreas Kissavos.
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Extra resources for Development and application of Muffin-Tin Orbital based Green's function techniques to systems with magnetic and chemical disorder
21) T (V ) = (1 − V G0 )−1 V = (V −1 − G0 )−1 . 22) with the formal solution: One can also write this in terms of the Green’s functions: G = G0 + G0 V G −1 = (G−1 0 −V) = G0 + G0 T (V )G0 . 20) where each V i is one of the cell potentials. 24) j i i i Vj + .... V i G0 Vi+ V i] = T = T[ By grouping all terms containing only one cell potential, we can rewrite the equation above in terms of the cell t-matrices: ti + V i] = T[ i i tj + . . 26) i i,j and it can be shown by iteration that the T ij s satisfy: T ij = ti δij + ti G0 T kj .
It is given by (suppressing the integrations): T ij = tˆi δij + tˆi G0 tˆj (1 − δij ) + tˆi G0 tˆk G0 tˆj + . . 8) k=i which can be found by iteration. Although we have assumed point scatterers so far, all results are valid for any geometrical form of the scatterers as long as they do not overlap . So far, we have kept everything time dependent, since it is very nice to think in terms of moving waves. In condensed matter theory, many phenomena are time independent, and this can of course also be treated with scattering theory.
44) where ΩIR is the interstitial at R and Ωov R is the overlapping part of the potential cell at R. This means that we need the full potential for parts of the unit cell, but since the calculation of this is very time consuming, we here use the SCA. 45) 3 where WR = 4π(wR − s3R )/3. Both the spherical part of the muffin-tin potential and the muffin-tin constant are thus given by the spherical average of the full potential. Now it is just to construct this average potential. 46) R where ZR is the protonic charge at R.