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SMALL CLUSTERS OF para- HYDROGEN

SMALL CLUSTERS OF para- HYDROGEN. Jesús Navarro and Rafael Guardiola IFIC and Universidad de Valencia 14 th International Conference on Recent Progress in Many Body Theories Barcelona, July 16-20 2007. The Hydrogen molecule. Bound system of two hydrogen atoms Two species:

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SMALL CLUSTERS OF para- HYDROGEN

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  1. SMALL CLUSTERS OF para-HYDROGEN Jesús Navarro and Rafael Guardiola IFIC and Universidad de Valencia 14 th International Conference on Recent Progressin Many Body Theories Barcelona, July 16-20 2007

  2. 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

  3. 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-1L. 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).

  4. Rovibrational spectrum Dunham formula of a vibrating rotor J.L. Dunham, Phys. Rev. 41 , 721 (1932) Y (cm-1)

  5. 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

  6. 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

  7. 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 Å3Energy per particle 93.5 K M.J.Norman, R.O.Watts and U. Buck, J. Chem. Phys. 81, 3500 (1984).

  8. 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

  9. Small para-Hydrogen clusters:dimer • Pure rotational absorption splitting of Dn=0 DJ=2 line S0(0)

  10. Small para-Hydrogen clusters:dimer • Rovibrational absorption Splitting of Dn=1 DJ=2 line S1(0)

  11. Small para-Hydrogen clusters:Conclusions ondimer • Proof of the existence of bound state • Determination of excitation spectrum, both bound and resonant levels

  12. 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-likeor liquid-like?

  13. 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.

  14. Pictures of the experimental set

  15. Experimental results Ortho-Hydrogen impurities Magical clusters

  16. 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=0.0018 K <r>=57.33 Å B=4.311 K <r>=5.13 Å

  17. 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)

  18. 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

  19. Present situation DMC and PIMC in agreement for N<25 but large discrepancies for N between 30 and 40 Our action: revise DMC

  20. Importance sampling trial function for DMC Two- and three-body Jastrow correlations K.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

  21. DMC: characteristics Richardson 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

  22. VMC versus DMC Three-body correlations provide more than 50% of the missing variational energy

  23. 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

  24. Comparison DMC-PIMC Clear disagreement between DMC and PIMC calculations

  25. 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

  26. 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

  27. Comparison with He clusters Para-Hydrogen Helium

  28. Shell occupancy Centroids : c Widths : s

  29. About the structure of shells • Radii of shells grow slowly but steadily: Elastic shells • Widths of gaussians (errorbars) fairly constant • After N=50 the particle at the center dissapears, and reappears near N=70 • Inner shells with non constant number of particles

  30. Pair distribution functionsparahydrogen helium N=2 N=30 Parahydrogen has a crystal-like structure, absent in Helium

  31. A long way to a classical system

  32. Definite analysis of clusters would require … To find a very good variational wave function

  33. Variations on the variational wave function: shells A model with shells: add one-body terms Or with a quenching parameter

  34. 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?

  35. 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.

  36. Excitation spectrum: preliminary • Levels for L=2 to 6 Magic Clusters?

  37. Thanks for your patience ? ?

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