[Wien] Normalization of Lattice harmonics in Wien2k

[Peter Blaha]

For "non-cubic" symmetry ("negative" atomic numbering in case.struct) we
use the tesserial harmonics as you have given below in your first 
definition (without the sqrt 4pi /2l+1 factor.

For "cubic harmonics", there is in principle only ONE independent C4 
coefficient, which is multiplied with a complicated linear combination
of Y40, Y44 and Y4-4.
You can find the definition eg. in lapw0 (or lapw5) in some subroutines
(common /norm_cub/)
The individual potential components c40 and c44 may not obay these 
rules, since the xcpot part is splitted "arbitrarely".


Maurits W. Haverkort schrieb:
> 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)
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