the details derivation of the local current?

[qqgu]

Hi， in the transport example, you show the local current calculation formula as,

can you tell me the details of derivation for this formula? in the orthogonal basis set, the local current should be H\rho , where is the -S\epsilon term com form? and how to calculate the \epsilon?

Thanks a lot.

[awvwgk]

in the orthogonal basis set, the local current should be H\rho , where is the -S\epsilon term com form

Probably from exactly this. The basis set is not orthogonal, therefore the overlap of the basis functions has to be accounted for.

[qqgu]

in the orthogonal basis set, the local current should be H\rho , where is the -S\epsilon term com form

Probably from exactly this. The basis set is not orthogonal, therefore the overlap of the basis functions has to be accounted for.

Thanks for the immediate response.
Yes, the non-orthogonal basis set is the reason of the additional terms -s\epsilon.
But i just don't know the derivation of this. \epsilon is referred to as energy-weighted density matrix, what exactly does it look like? something like $\int E*\rho(E) dE$ ?

[aradi]

When calculated with eigenvalues and eigenvectors, the density matrix is $P_{\mu,\nu} = \sum_i f_i c_{i,\mu} c_{i,\nu}^*$ and the energy weighted density matrix is $P^\epsilon_{\mu,\nu} = \sum_i f_i \epsilon_i c_{i,\mu} c_{i,\nu}^*$, with $\epsilon_i$ being the eigenvalue of the eigenstate $i$.