Effective Potential Approach to the Simulation of Large Para-hydrogen Clusters and Droplets Jing Yang, Christopher Ing, and Pierre-Nicholas Roy University of Waterloo 1

Motivation Develop computational methods to study many-body quantum systems Apply those methods to weakly-bound quantum clusters at low temperatures Specifically focus on hydrogen Why hydrogen? 2

Hydrogen – the Beauty of Nature Melting Point (K)Boiling Point (K)Triple Point (K) Simplest, lightest, most abundant element Common form is the H 2 molecule Important role in quantum theory development Potential alternate fuel; storage is a current challenge Bulk properties (those will be different for clusters) e- p+ 3

Ways to Understand the Quantum System Direct Approach -Restrict to small systems Alternative Approach – Path Integral Theory 4 Z = R. P. Feynman, Statistical Mechanics (W. A. Benjamin, New York, 1972).

Drawbacks of Path Integral Simulations Time consuming (simulating 55 H 2 molecules at low T could take several weeks or even months) – This is because a large number of beads is needed to obtain converged results – Long simulation runs are required to converge results within acceptable statistical error bars 5 N=20, P=128

Centroid Variables Most classical variable in quantum system Where q 0 is the centroid position of the beads, τ is the imaginary time, q(τ) is the position changes with τ. 6 R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, New York, Mcgraw-Hill, 1965

Centroid Theory Application: Bulk Hydrogen System 7 Pseudopotential of bulk liquid parahydrogen at 14 K and 25 K Works for even low temperature in clusters? M. Pavese and G. A. Voth, Chem. Phys. Lett. 249, 231 (1996)

Centroid Density Construction and Centroid Potential Mean force 8 From centroid pair density to calculate the effective potential

Fitted Effective Potentials H 2 -H 2 Dimer 9 U. Buck, F. huisken, A. Kohlhase, D. Otten, and J. Schaefer, J. Chem. Phys 78, 4439 (1983)

Results – g(r) from PIMC and pseudopotential Simulations 10 N=20 T=1K N=20 T=2K N=20 T=3KN=20 T=5K

Conclusions The centroid approach is accurate down to 5K Below 5K, bosonic exchange and many-body correlations should be considered Future Directions Include exchange (Bose-Einstein statistics) Calculate Raman Shifts (compare to experiments) Dynamics (diffusion) Hydrogen storage applications 11 S. J. Kolmann, B. Chan and M. J.T. Jordan, Chem. Phys. Lett. 467 (2008) 126–130

Acknowledgements Prof. P.-N. Roy and Prof. R. J. LeRoy (University of Waterloo) Prof. Hui Li (University of Jilin) Roy group Sharcnet Research supported by the Natural Sciences and Engineering Research Council of Canada 12