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Effective Potential Approach to the Simulation of Large Para-hydrogen

Clusters and Droplets

Jing Yang, Christopher Ing, and Pierre-Nicholas Roy

University of Waterloo

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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?

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Hydrogen – the Beauty of Nature

Melting Point (K) Boiling Point (K) Triple Point (K)

14.01 20.00 13.80

• Simplest, lightest, most abundant element• Common form is the H2 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+

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Ways to Understand the Quantum System

• Direct Approach

- Restrict to small systems

• Alternative Approach – Path Integral Theory

Z = +

+

+ . . .

R. P. Feynman, Statistical Mechanics (W. A. Benjamin, New York, 1972).

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Drawbacks of Path Integral Simulations

• Time consuming (simulating 55 H2 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 N=20, P=128

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Centroid Variables

• Most classical variable in quantum system

Where q0 is the centroid position of the beads, τ is the imaginary time, q(τ) is the position changes with τ.

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, New York, Mcgraw-Hill, 1965

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Centroid Theory Application: Bulk Hydrogen System

• 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)

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Centroid Density Construction and Centroid Potential Mean force

From centroid pair density to calculate the effective potential

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Fitted Effective Potentials H2-H2 Dimer

U. Buck, F. huisken, A. Kohlhase, D. Otten, and J. Schaefer, J. Chem. Phys 78, 4439 (1983)

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Results – g(r) from PIMC and pseudopotential Simulations

N=20 T=1K N=20 T=2K

N=20 T=3K N=20 T=5K

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Conclusions• The centroid approach is accurate down to 5K• Below 5K, bosonic exchange and many-body

correlations should be consideredFuture Directions• Include exchange (Bose-Einstein statistics)• Calculate Raman Shifts (compare to experiments)• Dynamics (diffusion)• Hydrogen storage applications

S. J. Kolmann, B. Chan and M. J.T. Jordan, Chem. Phys. Lett. 467 (2008) 126–130

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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