J. Mulak, NONINTEGRAL EXPANSION METHOD OF F(R)Y-L(M)(THETA,PHI) FUNCTIONS ABOUTA DISPLACED CENTER, Journal of alloys and compounds, 219, 1995, pp. 316-321
The transformation properties of f(R)Y-M(M)(Theta,Phi) functions under
translation of the reference system along the z axis, apart from thos
e in relation to rotation of the system, have to be known to formulate
the effective one-electron crystal held hamiltonian. The classical so
lution of the problem for a general radial function f(R) in the form o
f expansion into the spherical harmonic series has been given by Sharm
a in 1976. Another method avoiding integration but making use of the s
imple translational behaviour of the multipole functions R(-(L+1))Y-L(
M)(Theta,Phi), and reduction of the Kronecker products of spherical ha
rmonics, is presented. The method allows one to find the expansion of
the f(R)Y-L(M)(Theta,Phi) function by means of the 3-j symbols only. T
he method is applicable to a wide class of radial functions which can
be expanded into a power series. Moreover, in the case of crystal fiel
d potential, it gives a clear survey of the contributions of each loca
l potential moment to the crystal field parameters.