1. NABIL F. FARUK, HUI LI, JING YANG, ROBERT J. LE ROY & PIERRE-NICHOLAS ROY Simulation Studies of the Vibrational Dynamics of para- Hydrogen Clusters 1

2. Background: Superfluidity Superfluids lack viscosity and have infinite thermal conductivity Display quantized vortices and Rollin film flow Consequence of Bose-Einstein Condensation 2 Image retrieved from http://upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Liquid_helium_Rollin_film.jpg/678px- Liquid_helium_Rollin_film.jpg

3. Superfluid Clusters and Droplets Helium-4 is a boson Helium clusters have been studied for superfluity using OCS, N2O, CO2 dopant molecules by observing their spectra What about molecular superfluids? para-H2 is also a boson! J. Tang, Y. Xu, A.R.W. McKellar and W. Jager, Science 297, 2030 (2002) 3

4. Molecular Superfluid Spectroscopic studies of para-H 2 clusters doped with CO 2 and CO have shown that they indeed have superfluid behaviour that varies with the cluster size There are two spectroscopic properties used to probe clusters, the vibrational band origin shift of the dopants and their rotational constants 4 H. Li, R.J. Le Roy, P.-N. Roy and A.R.W. McKellar, Phys. Rev. Lett. 105, 133401 (2010)

5. Spectroscopic Observables: Rotational Constants Expect that a dopant’s moment of inertia increases when hydrogen molecules attach to it from the cluster Turn arounds in this behaviour are observed at certain cluster sizes corresponding to large fractions of frictionless superfluid para-H 2 5 H. Li, R.J. Le Roy, P.-N. Roy and A.R.W. McKellar, Phys. Rev. Lett. 105, 133401 (2010) Top: Rotational Constant, Inversely  to Moment of Inertia Bottom: Superfluid Fraction

6. Spectroscopic Observables: Vibrational Shifts Indicate the quality of the potential energy surface Vibrational bands of dopants are perturbed by cluster molecules 6 H. Li, N. Blinov, P.-N. Roy and R.J. Le Roy, J. Chem. Phys 130, 144305 (2009) H. Li, R.J. Le Roy, P.-N. Roy and A.R.W. McKellar, Phys. Rev. Lett. 105, 133401 (2010)

7. Predicting the Shifts Adiabatic-hindered-rotor approximation to produce a reduced- dimensional difference potential energy surface (PES) Monte Carlo simulations to obtain the radial probability distribution of hydrogen molecules in clusters Vibrational frequency shift for cluster of size N G. Tejeda, J.M. Fernandez, S. Montero, D. Blume and J.P. Toennies, Phys. Rev. Lett. 92, 223401 (2004) H. Li, P.-N. Roy, and R.J. Le Roy, J. Chem. Phys. 133, 104305 (2010) 7

8. Pure clusters can be probed with Raman spectroscopy What if the dopant is another para-H 2 ? It would be indistinguishable from the cluster para- H 2 Toennies and co-workers provided experimental results for the Raman shifts Also fitted their data to a Lennard-Jones model to give an empirical difference potential curve: G. Tejeda, J.M. Fernandez, S. Montero, D. Blume and J.P. Toennies, Phys. Rev. Lett. 92, 223401 (2004) 8

9. Potential Energy Surface 9 V 6D → V 4D : average over r 1, r 2 using monomer bond length wavefunctions V 4D → V 1D : average over θ 2, φ, θ 1 using Adiabatic Hindered Rotor approach r1r1 r2r2 6D PES1D PES R.J. Hinde, J. Chem. Phys. 128, 154308 (2008) H. Li, P.-N. Roy, and R.J. Le Roy, J. Chem. Phys. 133, 104305 (2010)

10. Results: Δ ν for H 2 -H 2 Dimer The shift is from having both molecules in the ground state to one in the first vibrationally excited state and the other in the ground state 10 Method Δν (cm-1) Observed-0.400 Toennies: L-J ΔV(R)-0.417 Hinde: ab initio ΔV(R)-0.405 Our Work: ab initio ΔV(R)-0.347 G. Tejeda, J.M. Fernandez, S. Montero, D. Blume and J.P. Toennies, Phys. Rev. Lett. 92, 223401 (2004) R.J. Hinde, J. Chem. Phys. 128, 154308 (2008)

11. Results: Shifts for Varying Cluster Sizes 11 G. Tejeda, J.M. Fernandez, S. Montero, D. Blume and J.P. Toennies, Phys. Rev. Lett. 92, 223401 (2004) J. E. Cuervo and P.-N. Roy, J. Chem. Phys. 125, 124314 (2006) J. E. Cuervo and P.-N. Roy, J. Chem. Phys. 128, 224509 (2008) DMC: Diffusion Monte Carlo Densities PIGS: Path Integral Ground State Densities

12. Conclusions The empirical and ab initio difference potentials are qualitatively different Still need to improve ΔV 1D (R); the error in the shifts however appears to be systematic Once an accurate ΔV 1D (R) is obtained, we can extend the prediction of the Raman shifts to para-H2 clusters not yet observed 12