[Wien] 2 ways to calculate bindning energy

[B. Yanchitsky]

Lyudmila Dobysheva wrote:
 > Tuesday 11 March 2008 21:02 B. Yanchitsky has written:
 >> Ev_super = E[N-1] - (N-1)*E[1].      (1)
 >> Ev_atom = Ei - E[1].                          (2)
 >
 > I cannot quite catch the problem: do you expect these values equal?
 > If you multiply by n-1
 > (N-1)Ev_atom = (N-1)Ei - (N-1)E[1].                 (2')
 > in the first formula you have energy of N-1 atoms in a defect supercell as a
 > first term in subtraction. In the second formula you have energy of N-1 isolated
 > (far from each other) atoms as a first term. What puzzles you? Or I do not understand something?

This message might appear twice, I'm resending it with a shorter attachment. (Bogdan)

Yes, I think the values should be equal, at least approximately.
When something is wrong with number of atoms difference counts for tens of Ry,
depending on Z. Probably what wrong may be here, factor like 1/2, but I do not see for it.
I wrote binding/formation energy in the way it's positive, probably better is negative.
Consider the lattice model, where atoms are at lattice sites and interactions are only
between nearest neighbors, number of neighbors is z, interaction potential is v.
Then total energy of N atoms is just E[N] = N*Ei + N*z/2*v,
E[1] = E[N]/N = Ei + z/2*v,
(z/2 - is number of bonds), formation energy per atom is Ef = z/2*v
When 1 atom is removed z bonds are also removed, total energy is
E[N-1] = (N-1)*Ei + N*z/2*v - z*v = (N-1)*Ei + (N/2-1)*z*v =
(N-1)*Ei + (N-2)*z/2*v = (N-1)*Ei + (N-1)*z/2*v - z/2*v
Thus
z/2*v = Ef = E[N-1] - ( (N-1)*Ei+(N-1)*z/2*v ) = E[N-1] - (N-1)E[1]

Second way, isolated atom, is through thermodynamic, formation energy
is an energy needed to combine atoms, i.e.
integrated p*dV = dE/dV*dV = E[1] - E[i]

Sorry, I did not provide struct and scf files, they are in attachment


-- 
Bogdan Yanchitsky
Institute of Magnetism
Vernadsky Blvd., 36-b
03142  Kiev
UKRAINE

Tel. (+380-44) 444 34 20
Fax. (+380-44) 444 10 20

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