1. Quantum Monodromy Quantum monodromy concerns the patterns of quantum mechanical energy levels close to potential energy barriers. Attention will be restricted initially to two dimensional models in which there is a defined angular momentum, with particular reference to the quasi-linear level structures of H 2 O at the barrier to linearity and the vibration-rotation transition as the H atom passes around P in HCP. The aim will be to show how the organisation of the energy level patterns reflects robust consequences of aspects of the classical dynamics, regardless of the precise potential energy forms. The first lecture will relate to assignment of the extensive computed highly excited vibrational spectrum of H 2 O. The second to modelling spectra close to saddle points on the potential energy surface.

2. Quantum monodromy in H 2 O Model Hamiltonian and quantum eigenvalues Bent and linear state assignments Classical motions Quantum monodromy defined and illustrated Assignment of the Partridge-Schwenke computed spectrum Relevance to Bohr-Sommerfeld quantization Localised quantum corrections

3. Model Hamiltonian

4. ε x y

5. Matrix elements in degenerate SHO basis

8. Bohr-Sommerfeld quantization Corresponds to Johns’s ‘bent state’ label v bent Alternative linear state label Both well defined for all states

10. Classical-Quantum correspondences arising from angle- action transformation (P R,R,P φ,φ)→(I R,θ,I φ,φ) Relates energy differences in monodromy plot to radial frequency ω R and ratio (Angle change ΔΦ over radial cycle)/(radial time period Δt)

11. Classical trajectories ε< 0 ε > 0 y xx

14. (0v 2 0) bending progression of H 2 O 10000 20000 30000 E/cm -1 kaka -20 020 -10 10

15. (0v 2 0) bending progression of H 2 O

17. Mathematical origin of monodromy dislocation

18. Quantum correction to Bohr Sommerfeld

21. Summary Pattern of quasi-linear eigenvalues analysed by semiclassical arguments Eigenvalue lattice contains a characteristic dislocation, regardless of the precise potential Classical trajectories explain sharp change in ε vs k at fixed v as sign of ε changes Application to vib assignment for H 2 O Term quantum monodromy explained Error in semiclassical theory quantified

22. Acknowledgements R Cushman introduced the idea at a workshop for mathematicians, physicists and chemists J Tennyson extracted and organised the data on H 2 O T Weston helped with the semiclassical analysis UK EPSRC paid for TW’s PhD