= Solution
An effective one-electron <Hückel method> operator may be written
$$
\boxed{\hat h_{\rm H}=-\frac{\hbar^2}{2m_e}\nabla^2+v_1^{\rm eff}(\mathbf r-\mathbf R_1)+v_2^{\rm eff}(\mathbf r-\mathbf R_2).}
$$
The first term is one-electron kinetic energy. The effective site potentials represent attraction to the nuclei and any averaged screening implicit in the chosen parametrization. They replace the explicit many-electron interaction; they do not contain a coordinate-dependent two-electron operator $r_{12}^{-1}$.
For normalized hydrogen-centred orbitals, symmetry gives
$$
\boxed{\alpha=\langle\phi_1|\hat h_{\rm H}|\phi_1\rangle
=\langle\phi_2|\hat h_{\rm H}|\phi_2\rangle,\qquad
\beta=\langle\phi_1|\hat h_{\rm H}|\phi_2\rangle.}
$$
The <Hückel Coulomb parameter> $\alpha$ is an on-site one-electron energy, not the electron–electron Coulomb repulsion integral. The <Hückel resonance integral> $\beta$ couples the sites; with the usual same-phase atomic-orbital convention it is negative and stabilizes the bonding combination. For zero overlap, the two energies are $\alpha\pm\beta$.
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