[Pw_forum] K Points and Band structures

[Lorenzo Paulatto]

In data 11 settembre 2009 alle ore 13:45:43, Shaptrishi Sharma  
<sh.shapt at gmail.com> ha scritto:

> Can anybody please help me in understanding what are the k points ?? I  
> have read books a lot but its difficult to understand.

They are eigenvalues of the translation operator, T(R) where  
R=n1*a1+n2*a2+n3*a3; a1,a2,a3 are the cell basis vectors and n1,n2,n3  
integer numbers. The eigenvectors are, of course, the Bloch wavefunctions.

Because the hamiltonian is periodic it commutes with the translation  
operator, you can diagonalize both the hamiltonian and the translation  
operator at the same time. As a consequence each Bloch wavefunction has a  
well-defined energy (hamiltonian's eigenvalue) and k-point (translation  
eigenvalue). Using both eigenvalues you can classify the states without  
ambiguity, except where the bands cross.

> And also  how do we choose k points while performing a band structure
> calculation in quantum espresso when we are having 330 atoms.

It does not depend on the number of atoms.

You have to test the convergence at fixed smearing. E.g. you choose a  
smearing that's small enough for you, than you increase the number of  
k-points until total energy converges. You may then try again for a  
smaller/larger smearing and see if the results are consistent.

I would advise starting with a smaller system, it would take ages to do it  
on 330 atoms. Keep in mind that the number of k-points needed for an  
accurate sampling is directly proportional to the size of the Brillouin  
zone, hence inversely proportional to the size of the cell.

E.g. if you estimate that 6x6x6 k-points converge a certain calculation  
than 3x3x3 k-points will be converge a 2x2x2 supercell to exactly the same  
level.

Best regards


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Lorenzo Paulatto
SISSA  &  DEMOCRITOS (Trieste)
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