[Wien] Normalization of Lattice harmonics in Wien2k

[Maurits W. Haverkort]

Dear all

I've been looking into the effects of non-spherical potentials in DFT
and run into the problem that I do not understand the normalization used
for the lattice harmonics in Wien2k. In order to see if I understand
correctly what Wien2k does I calculated two times Bcc Fe. Once with the
C4 axes in the x, y, and z direction and once I rotated the input file
and placed the C3 axes in the z direction (In order to do so I created a
super cell and lowered the appearing symmetry)

In a cubic symmetry I would expect that if the C4 axes is // x,y, and z,
the potential looks like V(r) (Y_{4,-4} +Y_{4,4}+\sqrt(14/5) Y_{4,0})
In a cubic symmetry with the C3 axes // z and a C2 axes // y and the C4
axes in the (-1,\sqrt{3},\sqrt{2}) direction I would expect the
potential to be V(r) (Y_{4,-3} - Y_{4,3} - \sqrt{7/10} Y_{4,0}).

The functions Y_{l,m} are the normalized spherical harmonics as defined
on http://mathworld.wolfram.com/SphericalHarmonic.html or
http://en.wikipedia.org/wiki/Spherical_harmonic

In the file case.vtotal I find:
For the struct file with the C4 axes in the z direction I find a ratio
of \sqrt{14/5} (for large or small r) between the two components of the
potential.
For the struct file with the C3 axes in the z direction I find a ratio
of 6/10 between the two components of the potential.

My question is how are the lattice harmonics as used in Wien2k defined
with respect to the normalized spherical harmonics.

What I found out so far:

The spherical harmonics used are (as in the subroutine ylm.f) defined to
be normalized i.e. <Ylm | Ylm>=1 and with the (additional) CS phase (-1)^m.

(Upto the factor of (-1)^m this is the same as on
http://mathworld.wolfram.com/SphericalHarmonic.html or
http://en.wikipedia.org/wiki/Spherical_harmonic)

One can define normalized tesseral harmonics, which are real functions as:

Z_{m}^{(l)}=N(m) (Y_{l,-Abs(m)} + s(m) Y_{l,Abs(m)})
with N(m)=1,\sqrt(1/2) i ^(1-Sign(m))/2
and s(m)=0,(-1)^m Sign(m)
for m=0 , m<>0 respectively.

(These are the standard real functions one likes to work with)

For potential expansions one not always likes to work with normalized
functions and renormalized spherical harmonics (or tesseral harmonics)
can be defined with an additional pre-factor:
C_(l,m) = \sqrt((4 \pi)/(2l+1)) Y_(l,m)

Thanks in advance!
Maurits

(Calculations done with Wien2k V8.1 linux MKL 9.1 ifort)