Hückel Molecular Orbitals & the Frost Circle

This simulator solves the Hückel molecular-orbital problem exactly for cyclic π-conjugated carbon rings from cyclopropenyl (N=3) up to cyclooctatetraenyl (N=8), using the closed-form eigenvalues E_m = α + 2β·cos(2πm/N). A 3D ring of p_z orbitals renders the real node pattern of whichever molecular orbital you select — lobe colour shows phase, lobe size shows the LCAO coefficient magnitude — while a side energy-level diagram fills orbitals by the Aufbau principle for the π-electron count you set with the charge control. Live readouts track total π energy, the HOMO–LUMO gap, and whether the current ring/charge combination satisfies Hückel's 4n+2 rule for aromaticity, letting you directly compare textbook cases like benzene's closed-shell stabilisation against cyclobutadiene's antiaromatic open shell or the aromatic cyclopentadienyl anion and tropylium cation.