[Wannier] center of wannier

Nicola Marzari marzari at MIT.EDU
Mon Aug 24 23:20:42 CEST 2009

Mohammad R. Rasoulian wrote:
> Dear Nicola,
>       my trial wave is gn(X)=sigma L (Cn,L*Ul(r)YL(r)) 
> that r=X-t is the position relative to the center of the atom  L=(l,lz)
> and Cn,L control the angular distribution of gn(X) and type of the 
> orbital for example s or sp3 or ... .
> Ul(r) is radial part of wave function.this is for atom center trial 
> Wannier function .
> for where the center of trial Wannier function do not coincide with 
> the center of atom gn(X) should be constructed  as a linear 
> combination of two localized orbital .
> but for example for diamond that has fcc structure and two atom in basis .
> the Ul(r) for atoms in (0,0,0) and in (0.5,0.5,0) and in (0,0.5,0.5) 
> are similar so for similar orbitals the gn(X)s are not different .
> therefor the linear combination of gn(X) for each of that atom are one 
> result . in fact it is not different which of them we should apply for 
> center ? and it shows that we can't define the center of trial wannier 
> function exactly? and for definite center we can't apply the role of 
> the center.
> Am i right?
> How we can calculate gn(X) for bond center type exactly ? or how we 
> can highlight the position of center? or what is the way that each 
> atom give us the different center exactly?

Dear Mohammed,

suppose you put a "spherical" gaussian in 0,0,0 (U is gaussian, and C is 
constant), and one in
0.25, 0.25, 0.25 (the second atom in the basis). We'll call these, for 
futrher reference,
g_a and g_b. If you choose as a linear of these two their sum 
(normalized) you will
have a center in 0.125, 0.125, 0.125.

Admittedly, I'm not sure I understood your questions very well, in 
particular from
"threfor the linear combination of gn(X)..." onwards.

Still, the most important point for you to consider/think is this: you 
need the projection
of Bloch orbitals onto trial functions that might have centers not on 
atoms. You can
only calculate projections of Bloch orbitals onto trial functions that 
are atom centered.

Linear combinations of trial functions are *not* atom centered, and 
scalar products
are linear operators, so the linear combination of the scalar products 
that you
are able to calculate wiill give you the scalar product with a 
bond-centered trial
function. In the language above, you know how to calculate <psi|g_a>, and
<psi|g_b>, so the semisum of these two, divided by sqrt(2), is something you
know how to calculated, and it gives you <psi| (g_a+g_b)/sqrt(2)>; note that
g_a+g_b is a bond-centered trial function.

Make sure you go over this carefully !

Best luck,


Prof Nicola Marzari   Department of Materials Science and Engineering
13-5066   MIT   77 Massachusetts Avenue   Cambridge MA 02139-4307 USA
tel 617.4522758 fax 2586534 marzari at mit.edu http://quasiamore.mit.edu 

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