1. Quantum Monte Carlo simulations of helium clusters doped with molecular and ionic impurities Stefano Paolini CNR-INFM-Democritos National Simulation Center and Physics Department “G. Galilei” University of Padova The Towler Institute, Vallico Sotto, July 27 th, 2007 Quantum Monte Carlo in the Apuan Alps III

2. Rotational dynamics of helium solvated molecules: from small clusters toward the nanodroplet regime Part 1:

3. Acknoledgements : Stefano Baroni (SISSA & INFM-DEMOCRITOS)Stefano Baroni (SISSA & INFM-DEMOCRITOS) Paolo Cazzato (INFM-DEMOCRITOS)Paolo Cazzato (INFM-DEMOCRITOS) Stefano Fantoni (SISSA)Stefano Fantoni (SISSA) Saverio Moroni (SISSA & INFM-DEMOCRITOS)Saverio Moroni (SISSA & INFM-DEMOCRITOS) Giacinto Scoles (SISSA)Giacinto Scoles (SISSA)

4. G. Scoles and K. K. Lehmann, Science, 287 2429 (2000) He nanodroplets H 2 O@ 4 He N N ~10 4 4 He atoms Interest for the solvent: properties of quantum fluidsInterest for the solvent: properties of quantum fluids in confined systems in confined systems Interest for the impurity:Interest for the impurity: good spectroscopic matrix HENDI SPECTROSCOPY

5. Helium Nanodroplets Isolation spectroscopy from G. Scoles and K. K. Lehmann Science 287, 5462 (2000)

6. 4 He nanodroplets are superfluid Relative Depletion [%] Pure 3 He droplets T=0.15K Broad peak Experiment: (Toennies et al. Science, 1998) Wave Number Change [ cm -1 ] Relative Depletion [%] Pure 4 He droplets T=0.38K free rotor spectrum with increased inertia Superfluidity: response to an imposed rotation How small can a superfluid droplet be?

7. How does superfluidity start to show up? N = 1 - 8 J. Tang, Y. Xu, A.R.W. McKellar, and W. Jäger, Science, 297 2030 (2002) OCS@ 4 He N N-selective experiments: OCS@ 4 He N

8. Understanding the rotational dynamics What is the relation between structure and dynamics? What determines the increase of inertia? Can we predict the increase of the inertia? How does B saturate to the nanodroplet value, B eff ? Can we extrapolate B eff from the small size behavior?

9. Theory - previous scenario Models Suited for large droplets dynamical properties are indirectly derived from structural information (calculated by simulations) QMC results spurred the view that: B attains its asimptotic value fast for heavy rotors (e.g. OCS): before the 1 st solvation shell is completed slowly for light rotors (e.g. HCN): well beyond the 1 st solvation shell The reduction of B upon solvation is due to the molecular mass large reduction for heavy rotors small reduction for light rotors

10. Experiments do not validate this picture CO 2 @ 4 He N N 2 O@ 4 He N J. Tang et al. PRL (2004)W. Jager et al. JCP (2006) For some heavy molecules the convergence is slow For N 2 O (lighter than OCS) B reduction is larger than for OCS

11. Ground-state path integral Monte Carlo

12. Reptation quantum Monte Carlo Path probability : Random walk: Weight of the path: (S.Baroni and S. Moroni, Phys. Rev. Lett. 82, 4745 (1999))

13. Reptation quantum Monte Carlo Sampling the paths Metropolis test For large systems ( N > 50), bisection-multilevel algorithm is more efficient

14. Hamiltonian Trial function

15. Calculating the spectrum spectrum of He solvated molecules analytic continuation in imaginary time for a linear molecule

16. Elucidating the relation between the structure and the dynamics RQMC simulations: CO@He N : double-lined spectra

17. O C He density accumulation CO@ 4 He N – disentangling the spectra + Simulations Experiments a-type b-type

18. CO@ 4 He N – Structure Simulations HIGH LOW

19. CO@ 4 He N – Asymmetric structure He O C

20. CO@ 4 He N – Matrix dynamics

21. Convergence of B to the nanodroplet limit RQMC simulations: OCS@He N : a prototype of HEAVY ROTORS

22. OCS@ 4 He N - Structure HIGH LOW

23. OCS@ 4 He N – Rotational dynamics B converges slowly to the nanodroplet limit

24. Convergence of B to the nanodroplet limit RQMC simulations: HCN@He N : a prototype of LIGHT ROTORS

25. HCN@ 4 He N - Structure Density

26. HCN@ 4 He N – Rotational dynamics B converges fast to the nanodroplet limit nanodroplet value

27. HCN@ 4 He N – Matrix dynamics

28. Reduction of the rotational constant B eff /B gas = 33%B eff /B gas = 36% fudged OCS@He N Rotational dynamics real OCS@He N Rotational dynamics Simulations with fictitious inertia fudged-OCS = He-OCS potential + HCN inertia

29. Reduction of the rotational constant fictitious inertia vs real inertia B eff /B gas = 90%B eff /B gas = 81% fudged-HCN@He N real HCN@He N fudged-HCN = He-HCN potential + OCS inertia

30. Reduction of B upon solvation For a given potential B eff /B gas can increase with increasing B gas B gas f-OCS CO HCN f-HCN CO 2 N2ON2O OCS DCN

31. Conclusions RQMC a general tool for computational spectroscopy : - structure and dynamics (ground- and excited states properties) - computer experiments (simulations with fictitious inertia). The approach to the nanodroplet regime is slow for heavy rotors (OCS, N 2 O, CO 2 ). The decrease of the rotational constant is mostly due to the anisotropy and the strength of the potential, more than to the molecular weight.

32. Solid-like vs liquid-like behavior in 4 He clusters doped with alkali and alkaline-earth ions Part 2:

33. Work done with : Flavio Toigo and Francesco AncilottoFlavio Toigo and Francesco Ancilotto (Physics Department “G. Galilei”, University of Padova (Physics Department “G. Galilei”, University of Padova and INFM-Democritos NSC, Trieste, Italy). and INFM-Democritos NSC, Trieste, Italy). Stefano Baroni and Saverio MoroniStefano Baroni and Saverio Moroni (International School for Advanced Studies and INFM-Democritos NSC, Trieste, Italy). I also want to thank

34. Mobility experiments Experimental apparatus Be + is slower than other alkaline-earth ions Does Be + form a “snowball”? Be + mobility differs from that of other alkaline-earth ions Liquid helium Foerste et al., Z. Phys. B (1997)

35. Existing QMC calculations VMC (Shadow Wave Functions) - static correlations criterion Solid-like order in the first shell is found for all these ions 4 He clusters doped with Na +, K +, Cs +, Be +, Mg + Rossi et al. PRB(2004) 1 2 3 Cs + @He 64 Mg + @He 64

36. Dynamical correlations criterion A slow decaying indicates solid-like behavior Multipole moments imaginary-time correlations: Baroni and Moroni ChemPhysChem (2005) Used for clusters of para-hydrogen made of just one shell

37. Interactions and radial density distributions The potential well depth decreases with increasing the ion atomic number The potential minimum radius and the density maximum radius increase with increasing the ion atomic number

38. In Li + @He 70 the 1 st shell is solid Persistence of a rigid structure in the 1 st shell multipole correlations1 st shell He density

39. The 1 st shell of Na + @He 70 has an icosahedral structure Slow decaying for L=6 multipole correlations 1 st shell He density

40. Comparing alkaline-earth ions doped clusters Persistence of a rigid structure in the 1 st shell of Be + @He 70 multipole correlations 1 st shell He densities Be + @He 70 Mg + @He 70 Ca + @He 70

41. Conclusions The multipole dynamical correlations criterion is extensible to the case of clusters with more than one shell. Multipole time-correlations provide clearly distinct signals for snowball and bubble-like defects. Li + @He 70 and Na + @He 70 have solid first shell which move in a liquid environment. Mg + @He 70 and Ca + @He 70 form bubbles. Be + @He 70 shows a signature of a solid-like behavior of the first shell and forms a snowball.

42. Fluctuations of the inter-particles distances Radial density profiles Li + @He 70 Be + @He 70 Mg + @He 70 Berry parameter Berry, JCP (2001)

43. Rotational diffusion in the 1 st shell

44. Li + @He 8 is a solid-like cluster multipole correlations Persistence of a rigid-like structure 1 st shell He density static multipoles

45. CO@ 4 He N – Structure Simulations HCN@He N and CO@He N – similar structures

46. OCS@ 4 He N – Rotational dynamics B converges slowly to the nanodroplet limit recent experiments, Jäger, PRL (2006) expt our RQMC

47. Ground-state path integral Monte Carlo approaches exact ground state as Optimized trial function Compute expectation values: Use discretized path integral to represent Metropolis (reptation or bisection-multilevel) algorithm to sample paths time step exact results for R0R0 RMRM