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

[whoming]

Dear USERS and Gus.H,
         I have much difficulty in understanding the article written by Kara & Kurki-Suonio. 
Thank god that I had read this mail! I think my problem will be a piece of cake to you.
        I am going to find out how the WIEN get the projected-DOS( e. g. dz2, dxy, dyz, etc).
Would you kindly tell me is there any relationship between the el-em combinations and 
d-splitting or p-splitting in projecting DOS?
        Thank you very much!
Yours,
Whoming
----- Original Message ----- 
From: "Gus Hart" <gus.hart at NAU.EDU>
To: <wien at zeus.theochem.tuwien.ac.at>
Sent: Tuesday, July 01, 2003 9:04 AM
Subject: [Wien] Constructing lattice harmonics from "real" spherical harmonics


> 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
> 
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