Hi Peter,
In the integrals below, \rho is just the electronic charge density
(without nuclei).
Thus c \int{\rho] does NOT vanish and gives c * NE (number of
electrons).
However, if rho comes from electronic states, each eigenvalue is
shifted by the constant c
and thus the sum of eigenvalues cancels the c * NE term
However, when I add a background charge to neutralize the unit
cell, this does not come
from any eigenvalue, so if I handle this in the usual way, \rho
will now integrate to
NE + Q, and I get an extra c * Q term, which is not compensated by
an eigenvalue.
Actually, in the integrals below, \rho is the *total* (electronic +
nuclear) charge, which must be net neutral to have a well-defined
total energy (otherwise energy diverges).
With regard to the present question on charged-cell calculations, the
point is just that the calculation must be performed on a neutralized
cell in order to have well-defined total energy. So the Kohn-Sham
calculation is performed on a neutral cell, whether or not the
physical system is charged, and the corrections for non-neutrality, if
any (e.g., Makov-Payne, Eq. (15)), are added after.
So as long as the neutralizing charge enters all potential and energy
expressions along with the physical charge, so that all expressions
operate on a net-neutral total, the Kohn-Sham total energy must be
invariant to arbitrary constants in V (because the total Coulomb
energy is).
John
John Pask schrieb:
Dear Peter,
Yes, the background charge must be taken into account as part of
the net-neutral total charge in order to have well-defined total
energy. Then as long as the compensation charge is then in exactly
the same way as the remaining physical charge (i.e., enters all
the same integrals), then the arbitrary constant in potential
should not matter since:
\int{ \rho (V + c)} = \int{ \rho V} + c \int{ \rho} = \int {\rho
V},
independent of arbitrary constant c.
John
On Feb 24, 2010, at 11:54 PM, Peter Blaha wrote:
Is the question regarding the computation of total energy per
unit cell in an infinite crystal with non-neutral unit cells? If
so, then the total energy diverges -- and so is not well-
defined. (So neutralizing backgrounds must be added in such
cases to obtain meaningful results, etc.)
Yes, this is the question and yes, of course we add a positive or
negative background.
We are quite confident that the resulting potential is ok, but the
question is if there
is a correction to the total energy due to the background charge.
I believe: yes (something like Q * V-col_average / 2), but my
problem is that V-coul
is in an infinite crystal only known up to an arbitrary constant
and thus this correction
is arbitrary.
--
-
Peter Blaha
Inst. Materials Chemistry, TU Vienna
Getreidemarkt 9, A-1060 Vienna, Austria
Tel: +43-1-5880115671
Fax: +43-1-5880115698
email: pblaha at theochem.tuwien.ac.at
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--
-
Peter Blaha
Inst. Materials Chemistry, TU Vienna
Getreidemarkt 9, A-1060 Vienna, Austria
Tel: +43-1-5880115671
Fax: +43-1-5880115698
email: pblaha at theochem.tuwien.ac.at
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