tight binding Hamiltonian

$\displaystyle H$	$\textstyle =$	$\displaystyle T + V_{\rm external}+V_{\rm ion}+ V_{\rm Coulomb} + V_{\rm exchange-correlation}$	(1)
$\displaystyle H \vert\psi_\nu\rangle$	$\textstyle =$	$\displaystyle E_\nu \vert\psi_\nu\rangle$	(2)
$\displaystyle \vert\psi_\nu\rangle$	$\textstyle =$	$\displaystyle \sum_i c_{\nu i}\vert \phi_i \rangle$	(3)
$\displaystyle H_{ij}$	$\textstyle =$	$\displaystyle \langle \phi_i\vert H \vert \phi_j\rangle$	(4)
$\displaystyle S_{ij}$	$\textstyle =$	$\displaystyle \langle \phi_i \vert \phi_j \rangle$	(5)

Not necessary that $\phi_i$ is orthogonal. $V_{\rm external}$ comes from, e.g., bias voltage.