2D Orbital-Overlap Integral: [2+2] Photocycloaddition

The [2+2] cycloaddition — two alkenes fusing into a cyclobutane ring — is the textbook example of a reaction that orbital symmetry forbids thermally but allows photochemically. This 2D simulator makes the Woodward–Hoffmann argument computable rather than merely illustrated: each alkene's terminal p-orbital lobe is modelled as a Gaussian amplitude, and the overlap integral between the two approaching π-systems is numerically integrated every frame with the trapezoidal rule. Because the top alkene's HOMO is always symmetric but the bottom alkene's LUMO is antisymmetric in the ground state, the two terminus overlaps cancel exactly — the live overlap readout Σ sits at zero and the ring never closes. Promoting the bottom alkene to its excited π→π* state flips the sign of one lobe, turning both termini constructive at once, and Σ jumps to a clearly non-zero value exactly when the ring closes. Below the overlap diagram, a grid of many independent alkene pairs evolves under real excited-state kinetics — photon absorption, radiative and non-radiative decay, and collisional quenching by a tunable oxygen concentration — with live readouts for excited-state fraction, cumulative product yield, and photons absorbed.