1. Spectroscopic signatures of bond- breaking internal rotation in HCP. Mark S Child and Matt P Jacobson Oxford University UK UK EPSRC

2. Outline Classical origin and validity of the polyad approximation for a Hamiltonian with the angular periodicity of HCP. Vibrational/rotational level stucture in lowest (pure bending) polyad components.

3. Spherical pendulum Hamiltonian Quantum states, semiclassical theory and validity of the polyad approximation Part 1

4. The model H C P θ Expressed in trigonometric form

5. Hamiltonian components Spherical pendulum (Bend) CP stretch Scaled parameter values Fermi resonance coupling Energy unit ~ 147 cm -1

6. Spherical pendulum states

7. Periodic orbits Bifurcation diagram showing onset and frequencies of periodic orbits Classical Fourier components closely related to ‘spherical pendulum’ matrix elements Pendulum frequency variation Semiclassical considerations Points are Quantum level separations

8. Polyad attributes Fraction of eigenstate not attributable to a single 2:1 polyad Reduced bending energy Vertical columns indicate ‘good polyads’ vs

9. Improved bending potential and rotation-vibration coupling parameters Part 2

10. Extended RKR bending potential with bending frequency plot

11. Bending energy vs vibrational angular momentum HCP ‘monodromy’ plot

12. Coriolis splittings at n b =10 and n b =40

13. Vibration rotation constants

14. Conclusions Trigonometric (spherical potential) Hamiltonian form imposes necessary periodicity. Eventually invalidates any harmonic oscillator based representation. 2:1 Fermi polyad model valid almost up to saddle point, provided matrix elements take account of angular periodicity. RKR based bending potential predicts large energy variation of vib-rotn parameters – in line with experimental observations.

15. RKR procedure

16. Quantum monodromy plot Note shape change as unit cell is transported around O

17. Accuracy of polyad approximation Notes Barrier max at E=100 units 1 unit ~ 140 cm -1