[Pw_forum] epsilon.x and scissor shift renormalization of momentum operator

[Eduardo Ariel Menendez Proupin]

Hi all,
Recently a  reviewer has warned me about a problem of the dielectric
function calculation that I performed using the epsilon code, where I
applied the scissor shift (option shift in the input) to correct the
infamous gap problem.  Following the referee's criticism I got to an article
by Rodolfo del Sole and Rafaello Girlanda, PRB 48, 11780 (1993), when a
scissor shift  is applied to correct the LDA underestimation of the
transition energies, the momentum matrix elements <c|e.p|v> must be
multiplied by (Ec-Ev)/(Ec0-Ev0), (see unnumbered equation after Eq. (15) in
the cited paper) , where Ec0-Ev0 is the LDA transition energy and
Ec-Ev=Ec0-Ev0 +\Delta  ( \Delta is the scissor shift). Equivalently, one can
replace the \omega^2 in the denominator by (\omega-\Delta/\hbar)^2  (Eq. 17
and 19 of the above article) . As shown by del Sole and Girlanda (and
repeated many times in the literature)  this is a simple approximation for
the fact that "exact" transition energies Ec-Ev arise from non-local
self-energy (e.g. from GW) and for non-local one electron hamiltonians the
interaction with an electromagnetivc wabe is not longer A.p.

It is very easy to set this correction in the code epsilon.x, for example ,
in subroutine eps_calc one can change

               etrans = ( et(iband2,ik) -et(iband1,ik) ) * RYTOEV + shift
.......
                   epsi(:,iw) = epsi(:,iw) + dipole(:,iband1,iband2) *
intersmear * w* &
                                             RYTOEV**3 * (focc(iband1,ik))/
&
                                  (( (etrans**2 -w**2 )**2 + intersmear**2 *
w**2 )* etrans )

                   epsr(:,iw) = epsr(:,iw) + dipole(:,iband1,iband2) *
RYTOEV**3 * &
                                             (focc(iband1,ik)) * &
                                             (etrans**2 - w**2 ) / &
                                  (( (etrans**2 -w**2 )**2 + intersmear**2 *
w**2 )* etrans )

by the following

               etrans = ( et(iband2,ik) -et(iband1,ik) ) * RYTOEV + shift
               shift_renorm = ( etrans/(etrans-shift) )**2

                   epsi(:,iw) = epsi(:,iw) + dipole(:,iband1,iband2)
*shift_renorm* intersmear * w* &
                                             RYTOEV**3 * (focc(iband1,ik))/
&
                                  (( (etrans**2 -w**2 )**2 + intersmear**2 *
w**2 )* etrans )

                   epsr(:,iw) = epsr(:,iw) + dipole(:,iband1,iband2)
*shift_renorm* RYTOEV**3 * &
                                             (focc(iband1,ik)) * &
                                             (etrans**2 - w**2 ) / &
                                  (( (etrans**2 -w**2 )**2 + intersmear**2 *
w**2 )* etrans )

and similarly in a few other parts of the code.

This is equivalent to replace epsilon_2(omega) by epsilon2(omega-
shift/hbar) , at least for small broadening parameter so that the "delta
function" is sharp  ,  and it corrects automatically the real part of the
dielectric function.

This is valid for LDA, and in principle not valid for GGA, although I
suspect that in practice gives the same results. In fact

I would appreciate a feedback to this comment.

Best regards

-- 


Eduardo Menendez Proupin
Departamento de Química Fisica Aplicada
Facultad de Ciencias
Universidad Autónoma de Madrid
28049 Madrid, Spain
Phone: +34 91 497 6706

On leave from: Departamento de Fisica, Facultad de Ciencias, Universidad de
Chile URL: http://fisica.ciencias.uchile.cl/~emenendez

"*Padece, espera y trabaja para gentes que nunca conocerá y que a su vez
padecerán, esperarán y trabajarán para otros, que tampoco serán felices,
pues el hombre ansía siempre una felicidad situada más allá de la porción
que le es otorgada. Pero la grandeza del hombre está precisamente en querer
mejorar lo que es. En imponerse Tareas. En el Reino de los Cielos no hay
grandeza que conquistar, puesto que allá todo es jerarquía establecida,
incógnita despejada, existir sin término, imposibilidad de sacrificio,
reposo y deleite. Por ello, agobiado de penas y de Tareas, hermoso dentro de
su miseria, capaz de amar en medio de las plagas, el hombre puede hallar su
grandeza, su máxima medida en el Reino de este Mundo*".
Alejo Carpentier, El reino de este mundo, (1949).

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