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

[Peter Blaha]

> 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?

Just have a look e.g. into   SRC_lapw5/sum.f

This lists clearly the coefficients for the "cubic harmonics" and you can
also see how they are combined.

Regards


                                      P.Blaha
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Peter BLAHA, Inst.f. Materials Chemistry, TU Vienna, A-1060 Vienna
Phone: +43-1-58801-15671             FAX: +43-1-58801-15698
Email: blaha at theochem.tuwien.ac.at    WWW: http://info.tuwien.ac.at/theochem/
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