Fig. 5. Modeling a conical intersection of ethylene.

Inset: Ethylene structure. a Total energies (relative to the ground state of the planar geometry E₀) of the ground state and first-excited singlet state as a function of torsion angle τ, with MR-CI⁵⁰ and TDDFT⁴⁸ results also plotted for comparison. TDDFT overestimates the barrier (the ground state at τ = 90°) and produces an unphysical cusp, while the MR-CI results which predict an avoided crossing are well reproduced by PauliNet. b Same as above but as a function of pyramidalization angle ϕ (τ = 90°), with the degeneracy of the two states producing a conical intersection. The arrows denote the conical intersection, with PauliNet (ϕ ~ 100°) closely matching the MR-CI result (ϕ ~ 96°). Note: The geometric parameters (bond lengths and angles) vary slightly between the torsion and pyramidalization experiments (see ref. ⁵⁰). (The numerical data can be found in Supplementary Table V; source data are provided as a Source Data file).