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["Matteo Giantomassi" <Matteo.Giantomassi@uclouvain.be>]

> Hi,
>
> I am generating a matrix Hamiltonian over different Bloch states. When
> integrating over different bands I roughly obtain the expected result,
> i.e.
> identity matrix to the accuracy of about 1e-5.
>
> However when solving <nk|nk'> my off diagonals are not near 0 (i.e. 1e-1)
> when
> the values of k and k' are close. Theoretically states across different k
> should be orthogonal. Is this definitely also the case in abinit? Has
> anybody
> ever had similar problems?
>
> Thank you.
> David
>

 Dear David,

 are you taking into account the phase factor e^(-i(k-k').r) in the
 numerical evaluation of the off diagonal matrix elements?
 For the diagonal matrix elements, there's a cancellation of the phases
 and the integration can be done by just integrating the product
 of the lattice periodic part u_(nk) . u(n'k) inside the unit cell.

 The same trick cannot be used when k and k' are different.
 A numerical test will require an explicit and expensive integration
 over the entire volume of the crystal described with Born von Karman
 boundary conditions.
 Fortunately, group theory and linear algebra tell you that
 the overlap between states of different crystalline momentum
 is exactly zero.

 BTW:
 You might consult Appendix D of Ashcroft and Mermin where
 several useful properties of the integrals involving plane waves.
 are discussed.

 Best Regards
 Matteo Giantomassi