SMALL%20CLUSTERS%20OF%20para-HYDROGEN

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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

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The Hydrogen molecule

• Bound system of two hydrogen atoms
• Two species
• gtPara-Hydrogen nuclear spins coupled to
  S0, so space symmetric
• gtOrtho-Hydrogen Nuclear spins coupled to S1,
  so space antisymmetric
• As an elementary constituent, both cases
  correspond to a BOSON

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Properties of the molecule

• Mass 2.0198 amu
• Equilibrium distance R1.4 bohr
• Electronic binding energy (without lowest
  vibrational correction) D38293.04 cm-1
• Dissociation energy (including zero-point motion)
• Theory D36118.06 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).

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Rovibrational spectrum
Dunham formula of a vibrating rotor J.L. Dunham,
Phys. Rev. 41 , 721 (1932)
l1, m0 4401.21
l2, m0 -121.34
l0, m1 60.853
l1, m1 -3.062
l3, m0 0.813
Y (cm-1)
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Molecular spectrum
Q means DJ0 S means DJ2 Transitions depleted
because of Bose symmetry and Spin. Dipolar
transitions do not exist and higher
electromagnetic orders are requested
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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

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Properties of the extended system (contd.)

• The triple point where hydrogen begins to
  solidify under saturated vapor pressure is
  TTP 13.8 K at
  PTP0.72 MPa
• At T0 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).

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Small para-Hydrogen clustersdimer

• 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

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Small para-Hydrogen clustersdimer

• Pure rotational absorption splitting of Dn0 DJ2
  line S0(0)

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Small para-Hydrogen clustersdimer

• Rovibrational absorption Splitting of Dn1 DJ2
  line S1(0)

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Small para-Hydrogen clustersConclusions on dimer

• Proof of the existence of bound state
• Determination of excitation spectrum, both bound
  and resonant levels

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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?

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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.

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Pictures of the experimental set
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Experimental results
Ortho-Hydrogen impurities
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
Magical clusters
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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

B0.0018 K ltrgt57.33 Å
B4.311 K ltrgt5.13 Å
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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)
  (N7)
• 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) (N7)

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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

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Present situation

• DMC and PIMC in agreement for Nlt25
• but large discrepancies for N between 30 and 40
• Our action revise DMC

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Importance sampling trial function for DMC
Two- and three-body Jastrow correlations K.E.Schmi
dt, M.A.Lee and m.H.Kalos, Phys.Rev.Lett. 47,
807(1981)
p5, sT and wT are fairly independent of cluster
size
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DMC characteristics
t (K-1) N10 N20 N30
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 0.1
Richardson extrapolation assumes a correction
O(t2) Acceptable value for t0.0001, without
bias. Sampling up to T10 K-1 Calculations
N-walkers1000, Nsteps105 Error control
Statistical analysis of 10 independent runs to
avoid statistical correlations
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VMC versus DMC
Three-body correlations provide more than 50 of
the missing variational energy
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Dissociation energy and magical clusters DMC
Magical N13 observed Magical? N36 for BUCK
potential Silvera less statistics, results
scattered BUCK and SILVERA qualitatively similar
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Comparison DMC-PIMC
Clear disagreement between DMC and PIMC
calculations
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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

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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
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Comparison with He clusters
Para-Hydrogen Helium
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Shell occupancy
Centroids c Widths s
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About the structure of shells

• Radii of shells grow slowly but steadily
• Elastic shells
• Widths of gaussians (error bars) fairly constant
• After N50 the particle at the center dissapears,
  and reappears near N70
• Inner shells with non constant number of
  particles

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Pair distribution functionsparahydrogen
helium
N2 N30
Parahydrogen has a crystal-like structure, absent
in Helium
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A long way to a classical system
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Definite analysis of clusters would require

• To find a very good variational wave function

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Variations on the variational wave function
shells

• A model with shells add one-body terms

Or with a quenching parameter
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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?
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FINAL COMMENTS

• Hydrogen clusters are fascinating, with a
  richness of properties not found in the more
  familiar 4He clusters.
• Open problems
• gtPIMC calculations should be revisited
• gtOther variational forms for DMC should be
  experimented.
• gtOne should fill the gap between T0 and non
  null temperatures by studying the excitation
  spectrum of clusters.

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Excitation spectrum preliminary

• Levels for L2 to 6

Magic Clusters?
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Thanks for your patience
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