1. 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

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 -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).

4. Rovibrational spectrum Dunham formula of a vibrating rotor J.L. Dunham, Phys. Rev. 41, 721 (1932) l=1, m=04401.21 l=2, m=0-121.34 l=0, m=160.853 l=1, m=1-3.062 l=3, m=00.813 Y (cm -1 )

5. Molecular spectrum Q means  J=0 S means  J=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 T TP = 13.8 K at P TP =0.72 MPa At T=0 it is an hcp solid density = 0.026 molecules per Å 3 Energy 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  =0  J=2 line S 0 (0)

10. Small para-Hydrogen clusters: dimer Rovibrational absorption Splitting of  =1  J=2 line S 1 (0)

11. Small para-Hydrogen clusters: Conclusions on dimer 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-like or 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 Q 1 (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 N  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 Magical clusters

16. The two-body problem and H 2 -H 2 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(H 2 ) ≈ M(He)/2, but V min (H 2 ) ≈ 4 V min (He): Larger zero-point energy but more attraction B=4.311 K =5.13 Å B=0.0018 K =57.33 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) p 5, s T and w T are fairly independent of cluster size

21. DMC: characteristics    N=10N=20N=30 0.001183.470.05559.280.171006.40.3 0.0005185.910.06566.560.171020.00.4 Extrap  =0 186.720.09568.990.281024.50.5 0.0001186.930.06569.160.121025.20.2 0.00002186.720.03569.480.071024.80.1 Richardson extrapolation assumes a correction O(  2 ) Acceptable value for  =0.0001, without bias. Sampling up to T=10 K -1 Calculations: N-walkers=1000, Nsteps=10 5 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 distributions DMC 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 : 

29. 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

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

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 4 He 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 ? ?