Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2021 / Paper 2A / Section B / 24 / a

For clamped nuclei and SI units, the many-electron electronic Hamiltonian isThe terms are electron kinetic energy, electron–nucleus attraction, and electron–electron repulsion. The nuclear kinetic operator is omitted in the Born-Oppenheimer approximation. The nuclear repulsionis a constant at fixed separation and is added to electronic energies to form the total potential curve. Some conventions include it in ${\hat{H}}_e$ as a constant; the electronic-energy convention used below excludes it so that the two one-electron energies can be compared without double-counting nuclear repulsion.

An effective one-electron Hückel method operator may be writtenThe 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 givesThe 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 $\alpha \pm \beta$.