Projected hopping matrix element and sign convention ($-t_{ij}=⟨{\varphi }_i|{\hat{p}}^2/(2m^{\ast })+V|{\varphi }_j⟩$)

If a tight-binding Hamiltonian is written with coefficient $-t_{ij}$ multiplying $c_i^{†}{c}_j$, 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.