Multiple scattering can in principle be expressed in terms of the so-called scattering path operator τ (z), which in turn is defined by two kinds of quantities, namely single site t matrices t (z) and so-called structure constants G0 (z):

τ (z) = ³ t (z)−1 −G0 (z)

´−1 The quantities τ (z), t (z) and G0 (z) are matrices with rows and columns labelled by sites defined by location vectors Rn,

τ (z) = {τnm (z)} , t (z) = {tn (z) δnm} , tn (z) = {tnΛΛ0(z)}

G0 (z) = {Gnm0 (z) (1− δnm)} , Gnm0 (z) = © Gnm0,ΛΛ0 (z)

ª each matrix element being itself a matrix labelled by angular momentum quantum numbers, namely

Λ = ½ L = (cm) , non-relativistic Q = (κμ) , relativistic

The location vectors Rn can refer to atomic positions in an infinite (bulk), semi-infinite (bulk with surface) or finite solid (cluster) system, or specify positions in the vacuum. From the scattering path operator the single particle Green’s function can directly be obtained and therefore all related physical properties. In the presence of three-or two-dimensional translational symmetry use can be made of lattice Fourier transformations. In Fig. 10.1 all the options available in using a multiple scattering scheme

are compiled.