The Born-Oppenheimer approximation separates electronic and nuclear motion: electrons respond instantaneously to nuclear positions, creating a potential energy surface that depends only on where the nuclei are, not how fast they're moving. For decades, this has been the foundation of molecular dynamics — nuclei move on surfaces defined by position alone.
Peng et al. (arXiv:2603.13211) show that for radical molecules (those with an unpaired electron), the potential energy surface must depend on nuclear momentum as well as position. Kramers' degeneracy theorem guarantees that spin-up and spin-down electronic states are exactly degenerate at any fixed nuclear geometry. But when nuclei move, the motion breaks the spin degeneracy — and this breaking depends on the direction and speed of the nuclear motion, not just the nuclear position.
The resolution: characterize radical potential energy surfaces as functions of both nuclear position q and nuclear momentum p. The p-dependence captures spin-rotation coupling — the interaction between nuclear rotation and electronic spin that produces the experimentally observed splitting of rotational energy levels in doublet radicals. Previous treatments introduced spin-rotation coupling as an ad hoc correction; the momentum-dependent surface makes it a natural consequence of the complete Born-Oppenheimer framework.
The framework extends beyond rotational splitting to vibrational circular dichroism and Raman optical activity in chiral radicals — phenomena that depend on how electronic states respond to nuclear motion direction. For achiral molecules, the momentum dependence is symmetric and averages out. For chiral molecules, it doesn't — the left-handed and right-handed nuclear motions create different electronic responses, producing the chiral spectroscopic signatures.