Lie Group Calculation of the Green Function of Disordered Systems
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LIE GROUP CALCULATION OF THE GREEN FUNCTION OF DISORDERED SYSTEMS CHRISTIAN BROUDER* *Laboratoire de Physique du Solide, Universit6 de Nancy 1, B.P.239 F-54506 Nancy, France, and Laboratoire pour l'Utilisation du Rayonnement Electromagn6tique, F-91405 Orsay, France ABSTRACT Within the framework of the muffin-tin multiple-scattering theory, the scattering path operators are given by the inverse of a matrix consisting of atomic t-matrices and a structural matrix. The influence of the displacement of an atomic centre on the structural matrix can be described analytically using Lie group techniques. From this analytical expression and the standard perturbation expansion of the Lippmann-Schwinger equation, it is possible to write the Green function of a disordered system as a series of terms which are averages over configurations. These averages can be calculated analytically from the moments of the interatomic distances. Special terms of this series are then summed up to infinity using Dyson equation. This formalism is computationally very effective to calculate electronic properties of systems with thermal or structural disorder. In this paper, the theoretical basis of this approach is briefly described and the convergence properties of the expansions are investigated. Introduction The calculation of the electronic structure and Green function of disordered systems is known to be a very difficult problem [1]. In the field of Low Energy Electron Diffraction, Rous and coil. [2, 3] investigated, up to first order, the perturbation of the scattering matrix due to atomic displacements (see also ref.[4]). In this paper, we propose a Lie algebra approach that enables us to calculate, up to any order, the influence of atomic displacement on the system Green function. The aim of this formalism is the calculation of temperature-dependent electronic structure, and this paper describes the first steps towards this goal. We present the Lie group approach for the continuum case (positive energies), the extension to bound states being briefly discussed at the end.
Lie algebra of translations The free space spherical wave functions can be displaced with a translation operator J, so that, for R = r + r'
jt(R)'Yt(h) =
Z jt,(r')Yt,'C(r)Jt,m,,tm(r)
(1)
In their seminal paper on the matrix elements of the translation 6perator, Danos and Maximon [5] show that 441r E•(rit+\-t'Ct"'M'
(2)
JMA, jA(r)Y,'A()
where, as usually, j\(r) is a spherical Bessel function, YV(V) is a spherical harmonic and is a Gaunt coefficient given, in terms of Wigner 3-j symbols, by the expression:
cMA.=
47r 1m
(-()'
0 0)
is -(m,
(3)
In the same paper, Danos and Maximon suggest that Jem,vm,(r) can be written as the exponential
of a matrix, in other words: Jmt,m,(r) = (exp(A.r))tmt,m, Mat. Res. Soc. Symp. Proc. Vol. 253. 91992 Materials Research Society
(4)
412
Lie's theorem states that Jtrm,(r) is the exponential of its derivative at 0. Therefore, a simple calculation [6-8] gives us the non-zero elements of the three matrices A', AV and AZ. AtX, m
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