[Wien] Constructing lattice harmonics from "real" spherical harmonics

[Gus Hart]

Dear WIEN users,

I would like to construct a lattice harmonic expansion for different point 
groups. In other words, I would like to expand a function f(theta,phi) in 
spherical harmonics, throwing away those terms that are zero by symmetry.

In principle this is easily done since it has already been done in the WIEN 
code (for the 32 crystallographic point groups) and tables are given in the 
user's guide (see 7.5.3, Tables 7.38 and 7.38). In these tables the 
non-zero "el-em" combinations are given. The caveat is that these 
correspond to Kara & Kurki-Suonio's "real" spherical harmonics, Y_lm+ and 
Y_lm-, which are defined as (see WIEN 1999 digest):

for M=2n (even):
Y_lm+ = 1/sqrt2 (Y_lm + 
Y_l-m)
Y_lm- = -i/sqrt2 (Y_lm - 
Y_l-m)

for M=2n+1 (odd):
Y_lm+ = -1/sqrt2 (Y_lm - 
Y_l-m)
Y_lm- = i/sqrt2 (Y_lm + 
Y_l-m)


So following the tables in the user's guide, the el=4 lattice harmonic for 
full cubic symmetry (point group m3m) is constructed by using Y_40+ & 
Y_44+. But a simple addition of these two terms does not work--for the 
resultant "lattice harmonic" (K4) to have cubic symmetry, they must be 
added as K4 = a*Y_40 + b*Y_44, where the ratio a/b must be sqrt(7/5). (I 
figured this out by trial and error but it can be verified by referring to 
table 1 in Altmann and Bradley's 1965 article in Rev. Mod. Phy.). The el=6 
cubic harmonic is similar--K6=a*Y_60+b*Y_64 where the ratio a/b is  -sqrt(1/7).

My question is simply this, using the tables in the user's guide to 
construct lattice harmonics, how can one find how much of each term to 
include? That is, how does one find the values, generally, for the 
coefficients a and b above? I tried to read the Kara & Kurki-Suonio paper 
to figure this out but I couldn't figure it out from there. Presumably 
everything I want to know is also contained in the 1965 two-part Rev. Mod. 
Phy. article by Altmann and Bradley, but I don't know enough group theory 
to glean out what I need.


Thanks for your help,
-Gus H.

++++++++++++++++++++++++++++
Gus Hart   (Rm. 312)
Department of Physics and Astronomy
PO Box 6010
Flagstaff AZ 86011-6010
tel. (928)523-0426
fax (928)523-1371
gus.hart at nau.edu