The ABINIT Users Mailing List ( CLOSED )

[Xavier Gonze <gonze@pcpm.ucl.ac.be>]

Dear Matteo and Fabien,

One should not forget that the goal is to compute < QP_i | r | QP_j  
>, and
not <QP_i | [\Sigma , x] | QP_j > .

So, if one knows all the < KS_i | r | KS_j >, one can derive the
< QP_i | r | QP_j > by using the proper transformation matrix from  
the | KS >
to the | QP >.

Xavier

On 23 Mar 2007, at 18:01, Matteo Giantomassi wrote:

> On Fri, 2007-03-23 at 15:00 +0100, Fabien Bruneval wrote:
> > Actually Matteo, the implementation of the correct commutator for
> > self-consistent GW could be done in the following way:
> >
> > First, calculate the commutator on the KS basis set (as usual)
> > and then, rotate them using the unitary matrix that transforms KS
> > wavefunctions into GW wavefunctions.
>
> Hi Fabien,
>
> Maybe I dind't understand well, but if you use the unitary matrix to
> rotate the matrix elements,  you get the same operator but in a
> different representation, schematically:
>
> <KS_i | [ H_KS , x] | KS_j > ==>    <QP_i | [H_KS ,x ] | QP_j >,
>
> while one should evaluate
>
> <QP_i | [\Sigma , x] | QP_j >
>
> which requires the knowledge of \Sigma(r, r') or equivalently
> \Sigma_{G,G'} (q).
>
> It's not so clear to me that these two quantites are equal, but maybe
> I'm missing something.
>
> Bye
> Matteo
>
> > By now, we are doing the opposite: we rotate first the wavefunctions
> > KS->GW and then we calculate the commutator, which is approximately
> > right for these wavefunctions.
> >
> > But I did not feel the need to implement it for all the systems I
> > studied so far.
> >
> >
> > fabien
> >
> >
> >
> > Matteo Giantomassi wrote:
> > > On Fri, 2007-03-23 at 10:19 +0100, Fabien Bruneval wrote:
> > >
> > >
> > > > Hi Deyu!
> > > >
> > > > This is subtle a point, but abinit code is right (as always?!).
> > > > Let me explain what the code does:
> > > >
> > > > You want to calculate terms such as < v | r | c >, where v  
> > > > stands for
> > > > valence wavefunctions, c for conduction wfs and where r is the  
> > > > position
> > > > operator.
> > > > The r operator is not periodic, but the bra and ket are. So the  
> > > > easiest
> > > > way out is to use the trick:
> > > >
> > > > < v | r | c > = < v | [ H , r ] | c > / (E_v - E_c).
> > > >
> > > > Therefore, the energies in the denominator are the one  
> > > > corresponding to
> > > > the Hamiltonian that you have introduced. So far, this  
> > > > Hamiltonian can
> > > > be anything, it just needs to have the periodicity of the crystal.
> > > >
> > > > In practice, we have chosen the Hamiltonian that corresponds to the
> > > > wavefunctions in the bra and the ket and, as a consequence, the  
> > > > energies
> > > > should be consistent with this choice.
> > > >
> > > > To summarize, when you use KS wavefunctions, like in G0W0,  
> > > > scissor+G0W0,
> > > > self-consistent GW on the energies-only, the code is perfectly
> > > > consistent and you should not change the energies in the  
> > > > denominator!
> > > >
> > > > I have to confess that there is a small inconsistency, when one  
> > > > performs
> > > > self-consistent GW calculation on the wavefunctions, because we  
> > > > still
> > > > use the commutator of the KS Hamiltonian and not the one of the GW
> > > > Hamiltonian. This "approximation" is pretty harmless, since the  
> > > > effect
> > > > of the commutator is really small in bulk systems. For  
> > > > nanostructures,
> > > > this may require some fix...
> > >
> > >
> > > Dear all,
> > >
> > > I agree with Fabien: it is true that in case of an update of the
> > > wavefunctions in \Chi_0, the treatment is not perfectly  
> > > consistent since
> > > the code uses quasiparticle energies but without taking into  
> > > account the
> > > commutator of \Sigma with the position operator x (strictly  
> > > speacking
> > > one should consider also \Sigma^*, if I'm not wrong).
> > > Anyway using the commutator of the KS hamiltonian should be a good
> > > approximation if the quasiparticle amplitudes are almost equal to  
> > > the KS
> > > wavefunctions.
> > > Maybe in more problematic systems where the self-energy operator  
> > > is not
> > > diagonal in the KS basis set, one should use the commutator of  
> > > \Sigma
> > > with x, but the implementation is not so straightforward. What do  
> > > you
> > > think?
> > >
> > > Best regards
> > > Matteo
> > >
> > >
> > > > If you want a reference on the commutator trick, you can find it in
> > > > Baroni and Resta, Phys. Rev. B 33, 7017-7021 (1986).
> > > >
> > > > Bye.
> > > >
> > > > Fabien
> > > >
> > > >
> > > >
> > > >
> > > >
> > > > deyulu@yahoo.com wrote:
> > > >
> > > > > Hello, abinit users:
> > > > >       I think there is a bug in the source code cchi0q0.F90.
> > > > > When computing polarizability chi0 at long wave length limit (q- 
> > > > > >0), the
> > > > > code uses the quasi-particle energies instead of KS orbital  
> > > > > energies.
> > > > > The quasi-particle energies could be simply KS orbital energies  
> > > > > plus a scissor
> > > > > operator or the output from a previous GW calculation.
> > > > > In the calculation of the G=0 component using perturbation  
> > > > > theory, however, KS orbital energies are used. In order to fix  
> > > > > this bug, one should change:
> > > > >
> > > > > rhotwx(:)=-rhotwx(:)/ediff
> > > > >
> > > > > to
> > > > >
> > > > > rhotwx(:)=-rhotwx(:)/egwdiff
> > > > >
> > > > >
> > > > >       Best
> > > > >       Deyu
> > > > >
> > > > > ****************************************************************** 
> > > > > *********
> > > > > Deyu Lu (Ph.D)
> > > > >
> > > > > 190 Chemistry Building
> > > > > University of California, Davis
> > > > > One Shields Avenue
> > > > > Davis, CA 95616
> > > > > Office phone: (530) 754-9663
> > > > > Group Webpage: http://angstrom.ucdavis.edu/
> > > > >
> > > > > ****************************************************************** 
> > > > > *********