Although resonant x-ray scattering of molecules fulfills strict electronic symmetry selection rules, as now firmly proven by spectra of diatomic molecules, the accumulated body of data for polyatomic molecules indicates that an apparent breaking of these rules represents the common situation rather than the exception. The CO2 molecule provides a good example of symmetry breaking, with the oxygen x-ray emission spectra showing strong intensity for transitions that are forbidden by the parity selection rule. We present time-independent and time-dependent theories for frequency-dependent resonant x-ray scattering beyond the Born–Oppenheimer approximation in order to explore under what circumstances one can anticipate symmetry breaking in the spectra of polyatomic molecules. The theory starts out from the Kramers–Heisenberg dispersion relation and is generalized for vibrational degrees of freedom and for nonadiabatic coupling of the electronic (vibronic) states, including the frequency dependency of the scattering cross section. Different limiting cases and few-level models are considered. The symmetry breaking is proven to be the result of pseudo-Jahn–Teller-like vibronic coupling between near-degenerate core-excited states. Thus vibronic interaction over the antisymmetric vibrational mode between the “bright” 1σg−12πu1 and “dark” 1σu−12πu1 intermediate states of CO2 allows transitions otherwise forbidden. The measurements and theory demonstrate that the symmetry-selective character of the resonant x-ray emission is strongly frequency dependent. The strong intensity of “dipole-forbidden” transitions in the π* oxygen K spectrum at resonance is reduced monotonically with the detuning of the excitation energy from resonance, and the spectra become “symmetry purified.” Simulations with full vibronic coupling predict this feature of the x-ray scattering experiment and a few-level model explains the energy dependence of the symmetry selection and the symmetry purification at large detuning energies in the limit of narrowband photon excitation.

Cesar, A., Gel’mukhanov, F., Luo, Y., Ågren, H., Skytt, P., Glans, P., Guo, J., Gunnelin, K., & Nordgren, J. (1997). Resonant x-ray scattering beyond the Born–Oppenheimer approximation: Symmetry breaking in the oxygen resonant x-ray emission spectrum of carbon dioxide. The Journal of Chemical Physics, 106(9), 3439–3456.