1

NABIL F. FARUK, HUI LI , J ING YANG, ROBERT J. LE ROY & PIERRE-NICHOLAS ROY

Simulation Studies of the Vibrational Dynamics of para-

Hydrogen Clusters

2

Background: Superfluidity

Superfluids lack viscosity and have infinite thermal conductivity

Display quantized vortices and Rollin film flowConsequence of Bose-Einstein Condensation

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 bosonHelium 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)

4

Molecular Superfluid

Spectroscopic studies of para-H2 clusters doped with CO2 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

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-H2

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 InertiaBottom: Superfluid Fraction

6

Spectroscopic Observables: Vibrational Shifts

Indicate the quality of the potential energy surface

Vibrational bands of dopants are perturbed by cluster molecules

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

N (N 1) V 1D0

(R)P N (R)dR

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)

2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

-25

-20

-15

-10

-5

0

5 Ab Initio 1D PES

Difference

Ground State

1st Excited State

R [Å]

V [

cm

-1]

8

Pure clusters can be probed with Raman spectroscopy

What if the dopant is another para-H2?It would be indistinguishable from the cluster

para-H2

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)

9

Potential Energy Surface

V6D →V4D: average over r1, r2 using monomer bond length wavefunctions

V4D →V1D: average over θ2, φ, θ1 using Adiabatic Hindered Rotor approach

r1

r2

6D PES 1D 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: Δν for H2-H2 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 stateMethod 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

5 10 15 20 25 30 35 40 45-6

-5

-4

-3

-2

-1

0

Raman Spectral Shifts

Experiment

L-J (DMC)

Ab Initio (PIGS)

L-J (PIGS)

N

Δν

[cm

-1]

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 DensitiesPIGS: Path Integral Ground State Densities

2 3 4 5 6 7 8

-5

-4

-3

-2

-1

0

1

2

3

4

51D Difference PES

Ab Initio

Toennies L-J

R [Å]

ΔV

[cm

-1]

12

Conclusions

The empirical and ab initio difference potentials are qualitatively different

Still need to improve ΔV1D(R); the error in the shifts however appears to be systematic

Once an accurate ΔV1D(R) is obtained, we can extend the prediction of the Raman shifts to para-H2 clusters not yet observed