[Wien] Bad formation energies for the charged vacancies

[Peter Blaha]

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 above.

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.

PS: In response to another posting, since Q is +1 and -1 (and we assume that V-could does not change
too much), the corrections would have opposite sign for the positively and negatively charged cell !!

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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> 
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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
-----------------------------------------