Hückel method

A model of conjugated $\pi$ molecular orbitals with an orthonormal atomic basis, one-centre energies ${\alpha }_i$, and resonance matrix elements ${\beta }_{ij}$ between selected neighbouring sites. Solving $H\mathbf{c}=E\mathbf{c}$ gives orbital energies and coefficients. Its simplicity makes symmetry and electron filling explicit, but quantitative charge predictions can depend strongly on the chosen parameters and omitted orbital relaxation.

With one orthonormal site orbital per cube vertex, common site energy $\alpha$, and coupling $\beta$ along every edge, the one-electron Hamiltonian is $\alpha I+\beta A$, where $A$ is the cube adjacency matrix. Its eigenvalues are $\alpha +3\beta ,\alpha +\beta ,\alpha -\beta ,\alpha -3\beta$ with multiplicities $1,3,3,1$.

One proof labels each vertex by three signs. The adjacency operator flips one sign at a time, so a product of $k$ coordinate signs is an eigenfunction of $A$ with eigenvalue $3-2k$. There are $\left(\genfrac{}{}{0ex}{}{3}{k}\right)$ such products. For $\beta <0$, eight electrons fill the lowest four spatial molecular orbitals, yielding total one-electron energy $8\alpha +12\beta$.

For a four-site cycle with site energy $\alpha$, three resonance integrals $\beta <0$ and one weakened integral $n\beta$, $0<n<1$, put $D=\sqrt{(n-1{)}^2+4}$. In the independent-electron Hückel method, filling the two or four lowest spin orbitals gives total electronic energies $E_2=2\alpha +\beta (n+1+D)$ and $E_4=4\alpha +2\beta D$. At the square limit $n=1$, these become $2\alpha +4\beta$ and $4\alpha +4\beta$. The weakened-edge distortion raises the two-electron sum but lowers the four-electron sum. Near $n=1$ the latter change is $\beta (1-n{)}^2/2+O((1-n{)}^4)$, so its first-order term cancels. This electronic comparison omits structural repulsion and other restoring forces, and does not determine the open-shell spin at the square limit.

For a bipartite pi system with a common diagonal energy $\alpha$, the orbital spectrum is paired about $\alpha$. If $H=\alpha I+K$ and $P$ changes the sign on one sublattice, then $PKP=-K$; an eigenvector of energy $\alpha +ϵ$ is mapped to one of energy $\alpha -ϵ$. Their site-probability magnitudes are equal.

For real orthonormal site functions, $p_{ij}={\sum }_k{f}_k{c}_{ik}{c}_{jk}$, where $f_k$ is orbital occupation. It measures the off-diagonal pi density matrix; a complete C–C bond also contains a sigma contribution.

The diagonal one-centre energy ${\alpha }_i$ assigned to a pi-basis orbital. This one-electron parameter is distinct from a two-electron Coulomb repulsion integral in Hartree-Fock theory.

An off-diagonal Hamiltonian matrix element between selected neighbouring pi-basis functions. The usual bonding convention has $\beta <0$; non-neighbour matrix elements are neglected.

With equal on-site energy $\alpha$ and equal pair interaction $\beta <0$, the three-site Hamiltonian has one eigenvalue $\alpha +2\beta$ and a degenerate pair $\alpha -\beta$. This model explains symmetry and filling but omits ingredients needed for a quantitative dissociation energy.

The change in energy on allowing a specified localized orbital reference to delocalize across a conjugated system. In a Hückel method calculation, the reference must use the same one-centre parameters and bond types as the delocalized molecule. With the signed convention $E_{\pi }-E_{\mathrm{localized}}$, stabilization is negative; its positive magnitude is often also called the delocalization energy, so the sign convention must be stated.

In an orthonormal Hückel method basis, an occupied molecular orbital with occupancy $f_k$ contributes $f_k|c_{ik}{|}^2$ electrons to site $i$. Adding one electron to a fixed orbital changes the site population by $|c_i{|}^2$. A coefficient sign indicates orbital phase, not charge sign. A net atomic charge also needs a reference population and treatment of the electrons outside the model.