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We present a general formula for the tight-binding representation of momentummatrix elements needed for calculating the conductivity based on theKubo-Greenwood formula using atomic orbitals, which are in general notorthogonal to other orbitals at different sites. In particular, the positionmatrix element is demonstrated to be important for delivering the exactmomentum matrix element. This general formula, applicable to both orthonormaland non-orthonormal bases, solely needs the information of the position matrixelements and the ingredients that have already contained in the tight-bindingrepresentation. We then study the anomalous Hall conductivity in the standardexample, ferromagnetic bcc Fe, by a first-principles tight-binding Hamiltonian.By assuming the commutation relation $\hat{\vec{p}} = (i m_e/\hbar)[\hat{H},\hat{\vec{r}}]$, the obtained frequency-dependent Hall conductivity isfound to be in good agreement with existing theoretical and experimentalresults. Better agreement with experiments can be reached by introducing areasonable bandwidth renormalization, evidencing the strong correlation among3$d$ orbitals in bcc Fe. Since a tight-binding Hamiltonian can bestraightforwardly obtained after finishing a first-principles calculation usingatomic basis functions that are generated before the self-consistentcalculation, the derived formula is particularly useful for thosefirst-principles calculations.

Tight-binding calculations of optical matrix elements for conductivity using non-orthogonal atomic orbitals: Anomalous Hall conductivity in bcc Fe