[Wien] density expansion for cubic case and non-cubic

[Peter Blaha]

This is explained in eg. the KURKI-Suonio paper.
For the cubic case the "lattice harmonics" are more complicated
linear combinations (with some distinct sqrt-factors) of the complex
spherical harmonics and even M=0 and M=4 terms are "added" to ONE
lattice harmonics.

明文美 schrieb:
> Dear Prof. Blaha;
>      I am sorry that I did not express myself clearly the first time.
>      thanks for your kind reply last time, but I am so sorry to ask you 
> somewhat entry-level
> queations again.
>      now I guess the charge density in MT sphere is expanded in the 
> basis of
> real spherical harmonic.but I still have a question as the following: 
>      I find some different treatments of the multipole moments between 
> cubic and
> non-cubic cases in lapw0.F when I read the source code file in SRC_lapw0.
> Seemingly for the non-cubic case the coefficients of density is 
> straightly transformed
> f! rom real spherical haromonic representation to complex sphereical 
> harmonic representation,
> however, for cubic case, it go through some complex transformations 
> distinctly from non-cubic 
> transformation as the code below indicats. I feel that for cubic case, 
> the density is not
> expanded according to real spherical harmonics,but I don't know exactly 
> how it is expanded for
> cubic cases,also, it is not expanded in the another sphere set of 
> real spherical harmonics K(l,j),
> because based on this following code, there exists term like 
> a1**2,a1*a2, a1*a3 and so on, not
> the case of directly transfroming K(l,j) to Y(L,M).K(l,j) is linear 
> combination of y(l,m) 
>  
>    ## for non-cubic case
>    QQ(LLMM,JATOM)=QQ(LLMM,JATOM)*fc(llmm,jatom)
>           
>   ## for cubic case 
>            c1=c_kub(ABS(lm(1,i,jatom)),lm(2,i,jatom))
>            c2=c_kub(ABS(lm(1,i,jatom)),lm(2,i,jatom)+4)
>            c3=c_kub(ABS(lm(1,i,jatom)),lm(2,i,jatom)+8)
>            qq(i,jatom)=a*c1*c1/sq1 + b*c1*c2/sq1 + &
>       &n! bsp;         c*c1*c3/sq1
>            qq(i+1,jatom)=a*c1*c2/sqrt2 + b*c2*c2/sqrt2  &
>                 + c*c2*c3/sqrt2
>            qq(i+2,jatom)=a*c1*c3/sqrt2 + b*c2*c3/sqrt2  &
>                 + c*c3*c3/sqrt2
>  
>    I have read the recommended papar in Userguide in lapw2 program about 
> the lattice harmonic,
> but the still cannot figure out this question,so, could you give me some 
> your idea about this issue?
> thanks very much !
>   
> Regards!
> Wenmei Ming
> &nb! sp;     
>  
>     &nbs p;  
>  
> 
>  > Date: Mon, 17 Dec 2007 21:21:10 +0100
>  > From: Peter Blaha <pblaha at theochem.tuwien.ac.at>
>  > Subject: Re: [Wien] charge density in CLM files
>  > To: A Mailing list for WIEN2k users <wien at zeus.theochem.tuwien.ac.at>
>  > Message-ID: <4766DA36.9040605 at theochem.tuwien.ac.at>
>  > Content-Type: text/plain; charset=GB2312
>  >
>  > 1) I hope you agree: the electron density (psi^* psi) must be a "real"
>  > function.
>  > The usual definition of spherical harmonics gives them as a complex
>  > function. However, one can always form some linear combination and
>  > obtain "real"-spherical harmonics. This is what is used here. More
>  > specifically, we use "lattice" harmonics, i.e. a symmetry adapted
>  > selection of some spherical harmonics.
>  >
>  > 2) This "syntax" is connected with the selection of "lattice harmonics".
>  > Of course a "negative"-l does not mean that ! l is negative, but 
> indicates
>  > a particular form of linear combination. For more details see the UG or
>  > the original papers (cited in the UG).
>  >
>  > > 1) why the density in MT sphere given in case.clmXX files are all
>  > > real values
>  > > rather than complex values for all radial points. Based on the
>  > > userguide, these values
>  > > represent the coefficients of spherical harmonic
>  > > expansion,according to this,
>  > > these coefficients should be complex values not real ones. so ,I
>  > > want to know
>  > > what the so-called "density" for MT sphere in clm file really
>  > > mean !!
>  > >
>  > > 2) occasionaly,some of the lattice harmonic indexes for L listed in
>  > > case.in2 are negative,
>  > > and the possible LM listed in the table in the LAPW2 chapter in
>  > > userguide
>  > > also indicate that L can negative,furthermore, I r! ead the
>  > > related source codes
>  > > and find that some programs are firmly connected with this
>  > > issue. thus, I am puzzled about this,because L should not be 
> negative (>=0)
>  > >
>  > > I am eager to figure out what is going on about the two issues,
>  > > can you give me some ideas
>  > > or let me know the details of both of them.
>  > >
>  > > thanks very much !
>  > >
>  > > Ming Wenmei
>  > >
> 
> 
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-- 

                                      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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