= Projected hopping matrix element and sign convention
{title2=$-t_{ij}=\langle\phi_i|\hat p^2/(2m^*)+V|\phi_j\rangle$}
If a tight-binding <Hamiltonian> is written with coefficient $-t_{ij}$ multiplying $c_i^\dagger c_j$, projecting the one-electron <Hamiltonian> gives the displayed relation. An alternative convention defines the coefficient itself as the <hopping integral> and has no leading minus. <Orthogonality> removes constant scalar potentials from off-diagonal matrix elements, but does not remove a spatially varying potential. Orbital rephasing changes both matrix elements together and cannot repair an algebraic mismatch between these two conventions in a fixed basis.
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