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Expand the field operator in orthonormal localized orbitals, $\hat{\psi }(\mathbf{r})={\sum }_i{\varphi }_i(\mathbf{r})c_i$. For a one-electron continuum Hamiltonian $h_0={\hat{p}}^2/(2m^{\ast })+V(\mathbf{r})$, projection givesThe coefficient in the displayed lattice Hamiltonian is $-t_{12}$. Thus the projected hopping matrix element and sign convention requireOrthogonality states $\int {\varphi }_1^{\ast }{\varphi }_2=0$; it does not imply $\int {\varphi }_1^{\ast }V{\varphi }_2=0$ for a spatially varying potential. If the off-diagonal potential is assumed zero, for example when it is a constant, the hopping coefficient is $h_{12}=K_{12}$. This is the kinetic integral requested by the paper, but it equals $-t_{12}$ under the paper's own leading-minus convention. A convention writing the hopping term with a plus sign would instead call $h_{12}$ itself $t_{12}$.

The printed positive equality is not a consequence of the stated orthogonality assumption and the displayed sign convention. Orthogonality does not diagonalize a varying potential gives an explicit counterexample to the underlying inference. The general projected relation above states the supported result; setting $V_{12}=0$ still leaves $-t_{12}=K_{12}$ in the question's convention. Rephasing an orbital changes both $h_{12}$ and the coefficient of the corresponding operator together; it does not remove this algebraic sign mismatch in a fixed convention. The spectrum in the remaining two-site calculation depends on $|t{|}^2$, so those parts are unaffected.