Extended Data Fig. 3: Illustration of the zero-field splitting and explanation of the asymmetric lineshape.

a, Schematic illustration of the anisotropic nature of magnetic dipole–dipole interaction (black field lines) between the two spins (red arrows) constituting the triplet state, as shown for the case of pentacene (grey molecular skeleton). b, For a spherical density distribution of the two electrons, the three triplet states T_X, T_Y and T_Z are degenerate. However, for an oblate density, the probability distribution of the electrons’ mutual distance differs in the different spatial directions. In this case, because of the anisotropy of the dipole–dipole interaction, the alignment of the spins with respect to the spatial directions matters and T_Z splits off in energy. For a probability distribution that differs in all three dimensions, the degeneracy of all three states (T_X, T_Y and T_Z) is lifted. In these considerations, the x, y and z directions refer to the high-symmetry directions of the molecule, as depicted²¹. c, The hyperfine interaction in protonated pentacene can be described as an effective magnetic field B_HFI created by the nuclei acting on the electron spins. Owing to the random orientation of the 14 proton nuclear spins (bottom), B_HFI will fluctuate around zero-field and its probability distribution for the z component is depicted (orange, bottom). The T_X and T_Y states show a hyperbolic energy dependence (top) as a function of a magnetic field in the z direction, B_HFI,Z. Weighting the different transition frequencies (blue double-headed arrows) with the probability distribution of B_HFI,Z gives rise to the asymmetric lineshape as schematically illustrated for the T_X–T_Z transition by the projection onto the energy axis (orange, top left). Of all directions, the z component is the most important one for the hyperfine-related lineshape (see Methods).