SMALL CLUSTERS OF

para-HYDROGEN

Jesús Navarroand

Rafael GuardiolaIFIC and Universidad de

Valencia

14 th International Conference on Recent Progressin Many Body Theories

Barcelona, July 16-20 2007

The Hydrogen molecule Bound system of two hydrogen atoms Two species: >Para-Hydrogen : nuclear spins

coupled to S=0, so space symmetric >Ortho-Hydrogen: Nuclear spins

coupled to S=1, so space antisymmetric

As an elementary constituent, both cases correspond to a BOSON

Properties of the molecule Mass : 2.0198 amu Equilibrium distance: R=1.4 bohr Electronic binding energy (without lowest

vibrational correction) D=38293.04 cm-1

Dissociation energy (including zero-point motion) Theory: D=36118.06 cm-1 L.

Wolniewicz, J. Chem. Phys. 103, 1792 (1995) Experiment: 36118.062(10) cm-1

Y.P. Zhang, C.H. Cheng, J.T. Kim, J. Stanojevic, and E.E. Eyler, Phys. Rev. Lett. 92, 203003 (2004).

Rovibrational spectrumDunham formula of a vibrating rotorJ.L. Dunham, Phys. Rev. 41 , 721 (1932)

l=1, m=0 4401.21

l=2, m=0 -121.34

l=0, m=1 60.853

l=1, m=1 -3.062

l=3, m=0 0.813

Y (cm-1)

Molecular spectrum

Q means DJ=0

S means DJ=2

Transitions depleted because of Bose symmetry and Spin. Dipolar transitions do not exist and higher electromagnetic orders are requested

Properties of the extended system

The energy difference between oH and pH is 170.50 K: at room temperature equilibrium hydrogen is 75% ortho and 25% para

Enrichment of para-H is slow, requires magnetic anisotropies to change the ortho spin (magnetically active catalysts)

The critical point for para-H is Tc = 33 K and Pc= 1.3 MPa

Properties of the extended system (contd.) The triple point where hydrogen

begins to solidify under saturated vapor pressure is TTP = 13.8 K at PTP=0.72 MPa

At T=0 it is an hcp solid density = 0.026 molecules per Å3 Energy per particle 93.5 K M.J.Norman, R.O.Watts and U. Buck, J. Chem. Phys. 81, 3500 (1984).

Small para-Hydrogen clusters:dimer A. Watanabe and H.L. Welsh

Phys.Rev.Lett. 13, 810 (1964) A.R.W. McKellar

J.Chem.Phys. 92, 3261 (1990) A.R.W. McKellar

J. Chem. Phys. 95, 3081 (1991) Technique: Infrared Absorption by

gas at 20 K

Small para-Hydrogen clusters:dimer Pure

rotational absorption splitting of Dn=0 DJ=2 line S0(0)

Small para-Hydrogen clusters:dimer

Rovibrational absorption Splitting of Dn=1 DJ=2 line S1(0)

Small para-Hydrogen clusters:Conclusions on dimer

Proof of the existence of bound state

Determination of excitation spectrum, both bound and resonant levels

Para-Hydrogen clusters: Motivation

They have been detected We have some expertise in dealing

with clusters Interesting questions raised: are quantal or classical? Some clusters are magical Solid-like or liquid-like?

Raman shifts and identification of small clusters G. Tejeda, J.M. Fernández, S.Montero, D. Blume

and J. P. Toennies Phys. Rev. Lett. 92, 223401 (2004)

Measurement of Q1(0) Raman shift of para-Hydrogen molecules in small clusters

Alternative method to mass diffraction Intermolecular effects on intramolecular

interaction J. van Kranendonk and G. Karl

Rev.Mod.Phys. 40, 531 (1968) studies the effect in hcp solid.

Pictures of the experimental set

Experimental results

N Dn cm-1

2 -0.400

3 -0.822

4 -1.521

5 -1.594

6 -1.904

7 -2.316

8 -2.350

Ortho-Hydrogen impurities

Mag

ical

clu

sters

The two-body problem and H2-H2 interacion

U. Buck, F. Huisken, A. Kohlhase, D. Otten, and J. Schaeffer J. Chem. Phys. 78, 4439 (1983) I.F. Silvera, V.V. Goldman, J. Chem. Phys. 69, 4209 (1978)

M(H2) ≈ M(He)/2, but Vmin(H2) ≈ 4 Vmin(He): Larger zero-point energy but more attraction

B=4.311 K

<r>=5.13 Å

B=0.0018 K

<r>=57.33 Å

Theoretical analysis: previous work P. Sindzingre, D.M.Ceperley and M.L.Klein,

Phys.Rev.Lett. 67, 14 (1992) (13, 18 and 33) Daphna Scharf, Michael L. Klein and Glenn J.

Martyna, J. Chem.Phys. 97, 3590 (1992) (13, 19, 33, and 34)

Michele A. McMahon, Robert N. Barnett, and K. Birgitta Whaley, J. Chem.Phys. 99, 8816 (1993) (N=7)

Michele A. McMahon, K. Birgitta Whaley, Chem. Phys. 182, 119 (1994) (6, 7, 13, 33)

E. Cheng, Michele A. McMahon, and K. Birgitta Whaley, J. Chem. Phys. 104, 2669 (1996) (N=7)

Theoretical analysis: recent work Rafael Guardiola and Jesús Navarro

Phys. Rev. A 74, 025201 (2006) DMC - BUCK Javier Eduardo Cuervo and Pierre-Nicholas Roy

J.Chem.Phys. 125, 124314 (2006) PIGS – BUCK & SILVERA

Fabio Mezzacapo and Massimo Boninsegni Phys.Rev.Lett. 97, 045301 (2006) PIMC - SILVERA

Fabio Mezzacapo and Massimo Boninsegni Phys.Rev. A 75, 033201 (2007) PIMC - SILVERA

S. A. Khairallah, M. B. Sevryuk, D.M.Ceperley and J. P. Toennies Phys.Rev.Lett. 98, 183401 (2007) PIMC - SILVERA

Present situation

DMC and PIMC in agreement for N<25

but large discrepancies for N between 30 and 40

Our action: revise DMC

Importance sampling trial function for DMC

Two- and three-body Jastrow correlationsK.E.Schmidt, M.A.Lee and m.H.Kalos, Phys.Rev.Lett. 47, 807(1981)

p5, sT and wT are fairly independent of cluster size

DMC: characteristics

t (K-

1)N=10 N=20 N=30

0.001 183.47 0.05 559.28 0.17 1006.4 0.3

0.0005 185.91 0.06 566.56 0.17 1020.0 0.4

Extrap t =0

186.72 0.09 568.99 0.28 1024.5 0.5

0.0001 186.93 0.06 569.16 0.12 1025.2 0.2

0.00002

186.72 0.03 569.48 0.07 1024.8 0.1Richardson extrapolation assumes a correction O(t2)

Acceptable value for t=0.0001, without bias. Sampling up to T=10 K-1

Calculations: N-walkers=1000, Nsteps=105

Error control: Statistical analysis of 10 independent runs to avoid statistical correlations

VMC versus DMC

Three-body correlations provide more than 50% of the missing variational energy

Dissociation energy and magical clusters: DMC

Magical N=13 observed

Magical? N=36 for BUCK potential

Silvera: less statistics, results scattered

BUCK and SILVERA qualitatively similar

Comparison DMC-PIMC

Clear disagreement between DMC and PIMC calculations

The origin of the disagreement

Is DMC too strongly constrained by the importance sampling wave function?

Have PIMC calculations too optimistic error estimates?

NOT: different potentials NOT: poor DMC statistics

One-body distributionsDMC and VMC(Jastrow-2)

Conclusion: well defined geometrical shells, even in VMC.

Trial function is liquid-like but reveals signs of shells

Shells actually constructed by DMC algorithm

Comparison with He clusters

Para-Hydrogen Helium

Shell occupancy

Centroids : c

Widths : s

About the structure of shells Radii of shells grow slowly but

steadily:Elastic shells

Widths of gaussians (error bars) fairly constant

After N=50 the particle at the center dissapears, and reappears near N=70

Inner shells with non constant number of particles

Pair distribution functionsparahydrogen helium

Parahydrogen has a crystal-like structure, absent in Helium

N=2 N=30

A long way to a classical system

Definite analysis of clusters would require …

To find a very good variational wave function

Variations on the variational wave function: shells

A model with shells: add one-body terms

Or with a quenching parameter

Variations on the variational wave function: solid-like

Nosanov-like wave function

Both approaches give rise to a minimal gain in energy.

Open question! Lack of imagination?

FINAL COMMENTS Hydrogen clusters are fascinating, with

a richness of properties not found in the more familiar 4He clusters.

Open problems: >PIMC calculations should be revisited >Other variational forms for DMC

should be experimented. >One should fill the gap between T=0

and non null temperatures by studying the excitation spectrum of clusters.

Excitation spectrum: preliminary

Levels for L=2 to 6 MagicClusters?

Thanks for your patience

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