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 >>> ----------------------------------------- >>> _______________________________________________ >>> Wien mailing list >>> Wien at zeus.theochem.tuwien.ac.at >>> http://**zeus.theochem.tuwien.ac.at/mailman/listinfo/wien >>> >> _______________________________________________ >> Wien mailing list >> Wien at zeus.theochem.tuwien.ac.at >> http://*zeus.theochem.tuwien.ac.at/mailman/listinfo/wien > > -- > ----------------------------------------- > 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 > ----------------------------------------- > _______________________________________________ > Wien mailing list > Wien at zeus.theochem.tuwien.ac.at > http://*zeus.theochem.tuwien.ac.at/mailman/listinfo/wien >