Rotational spectra of molecules in small Helium clusters: Probing superfluidity in finite systems F. Paesani and K.B. Whaley Department of Chemistry and Pitzer Center for Theoretical Chemistry University of California, Berkeley, CA, $$$ NSF 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005

Spectroscopy in large 4 He droplets courtesy A.F. Vilesov 4 He droplets: ultracold environment (T≈ K) 4 He droplets: ultracold environment (T≈ K) for high resolution spectroscopy for high resolution spectroscopy Free rotation in 4 He  Free rotation in 4 He  Rotational diffusion in 3 He  Rotational diffusion in 3 He  Grebenev, Toennies and Vilesov, Science 279, 2083 (1998) Superfluidity  Superfluidity  Berkeley University of California 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005

Spectroscopy in small 4 He clusters OCS( 4 He) N N 2 O( 4 He) N Tang, Xu, McKellar and Jäger Science 297, 2030 (2002) Xu, Jäger, Tang and McKellar Phys. Rev. Lett. 91, (2003 ) Similar results also for : CO 2 ( 4 He) N CO 2 ( 4 He) N Tang, McKellar, Mezzacapo and Moroni, Phys. Rev. Lett. 92, (2004 ) Tang, McKellar, Mezzacapo and Moroni, Phys. Rev. Lett. 92, (2004 ) CO( 4 He) N CO( 4 He) N McKellar, J. Chem. Phys. 121, 6868 (2004) McKellar, J. Chem. Phys. 121, 6868 (2004) Berkeley University of California 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005

Simulations of doped 4 He clusters Molecules: Molecules: OCS, N 2 O and CO 2 OCS, N 2 O and CO 2 Methods: Methods: 1. Projection Operator Imaginary 1. Projection Operator Imaginary Time Spectral Evolution (POITSE) Time Spectral Evolution (POITSE) Blume, Lewerenz, Niyaz and Whaley, Phys. Rev. E 55, 3664 (1997) Blume, Lewerenz, Niyaz and Whaley, Phys. Rev. E 55, 3664 (1997)  Rotational spectrum  Rotational spectrum 2. Path-Integral Monte Carlo (PIMC) 2. Path-Integral Monte Carlo (PIMC) Ceperley, Rev. Mod. Phys. 67, 279 (1995)  Superfluid properties  Superfluid properties Berkeley University of California 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005

Potential Energy Surfaces He-N 2 O Chang, Akin-Ojo, Bukowski and Szalewicz, J. Chem. Phys. 119, (2003) He-CO 2 Yan, Yang and Xie, J. Chem. Phys. 109, (1998) He-OCS Paesani and Whaley, J. Chem. Phys. 121, 4180 (2004) Berkeley University of California 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005

Monte Carlo methods Total interaction potential Total interaction potential Rotational Hamiltonian of molecule A Rotational Hamiltonian of molecule A Berkeley University of California 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005 Hamiltonian for AB N systems Hamiltonian for AB N systems

Projection Operator Imaginary Time Spectral Evolution method Berkeley University of California 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005 Laplace transform Laplace transform In practice:  T ≈  0 and E ref ≈ E 0 In practice:  T ≈  0 and E ref ≈ E 0 ˆ ˆˆ ˆˆ ˆˆ ˜ ˜

1. D j mk ( , ,  ) = molecular Wigner functions 1. D j mk ( , ,  ) = molecular Wigner functions 2. , ,  = Euler angles 2. , ,  = Euler angles molecule-fixed frame  space-fixed frame molecule-fixed frame  space-fixed frame 1. m=k=0 1. m=k=0 2. D j 00 ( , ,  )  P j (cos  )  j≈J 2. D j 00 ( , ,  )  P j (cos  )  j≈J eigenfunctions for linear rotors eigenfunctions for linear rotors eigenfunctions for symmetric top rotors (k=0) eigenfunctions for symmetric top rotors (k=0) Present calculations Present calculations Projection Operator Imaginary Time Spectral Evolution method Berkeley University of California 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005 Â  D j mk ( , ,  ) Â  D j mk ( , ,  )

Exponential fit Exponential fit Projection Operator Imaginary Time Spectral Evolution method Berkeley University of California 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005 Maximum Entropy (based on Bayes’s statistics) Maximum Entropy (based on Bayes’s statistics) ˜ ˜ ˜ ˜ ˜

N=1 POITSE: 0.657±0.002 cm -1 Chang et al.: cm -1 N=2 POITSE: 0.419±0.006 cm -1 N>2: J=1-3 Rotational Excited States: POITSE calculations for N 2 O( 4 He) N POITSE for N ≤ 2 POITSE for N > 2 Berkeley University of California 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005

Rotational Excited States: POITSE calculations for N 2 O( 4 He) N Rotational Spectrum for N > 5 Berkeley University of California 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005 POITSE for N ≤ 2 POITSE for N > 2

Experiment: Tang, Xu, McKellar and Jäger, Science 297, 2030 (2002) Theory: Paesani, Viel, Gianturco and Whaley, Phys. Rev. Lett. 90, (2003) E = B eff J(J+1)  Exp. Theory Spectroscopic Constants for OCS( 4 He) N Theory vs Experiment Berkeley University of California 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005

 Exp. Theory POITSE (exact)  Rigid coupling (approx.) Theoretical interpretation Transition from van der Waals complexes to quantum solvation Berkeley University of California 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005 Spectroscopic Constants for OCS( 4 He) N Theory vs Experiment E = B eff J(J+1)

Experiment: Xu, Jäger, Tang and McKellar, Phys. Rev. Lett. 91, (2003) Theory: Paesani and Whaley, J. Chem. Phys. 121, 5293 (2004 Theory: Paesani and Whaley, J. Chem. Phys. 121, 5293 (2004) E = B eff J(J+1) - D eff J 2 (J+1) 2 Distortion ConstantRotational Constant  Exp.  Theory Spectroscopic Constants for N 2 O( 4 He) N Theory vs Experiment Berkeley University of California 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005

Distortion ConstantRotational Constant Spectroscopic Constants for CO 2 ( 4 He) N Theory vs Experiment E = B eff J(J+1) - D eff J 2 (J+1) 2 Experiment: Tang, McKellar, Mezzacapo and Moroni, Phys. Rev. Lett. 92, (2004) Theory: Paesani, Kwon and Whaley, Phys. Rev. Lett. 94, (2005)  Exp. Theory Berkeley University of California 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005

Rotational Dynamics in 4 He clusters 1 st solvation shell 1 st solvation shell OCS: B eff ≈ B droplet OCS: B eff ≈ B droplet N 2 O, CO 2 : B eff > B droplet N 2 O, CO 2 : B eff > B droplet  ???  ??? B eff /B 0 decrease for small N decrease for small N  “rigid” coupling  “rigid” coupling turnaround at N ≈ 5-9 turnaround at N ≈ 5-9  ???  ??? Berkeley University of California 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005 Does this behavior reflect onset of superfluidity ???

Path-Integral Monte Carlo method Thermal density matrix Thermal density matrix Bose symmetry Implementation: sampling of Implementation: sampling of Berkeley University of California 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005 Thermal average Thermal average ˆ ˆ

f ij = superfluid fraction Superfluidity in PIMC I cl = classical moment of inertia A = projected area of a path  macroscopic exchanges  macroscopic exchanges (Feynman, 1953) (Feynman, 1953) s Berkeley University of California 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005 Response to a slow rotation of an external field Response to a slow rotation of an external field Superfluid fraction Superfluid fraction

Superfluidity in PIMC Berkeley University of California 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005 Response to a slow rotation of an external field Response to a slow rotation of an external field Superfluid density Superfluid density Superfluid fraction Superfluid fraction

direct correspondence between f  and B eff direct correspondence between f  and B eff - N ≤ 5: f  negligible  decrease of B eff  van der Waals complexes - N ≤ 5: f  negligible  decrease of B eff  van der Waals complexes - N > 5: increase of f   increase of B eff  onset of superfluidity - N > 5: increase of f   increase of B eff  onset of superfluidity - N ≥ 12: saturation of f   saturation of B eff - N ≥ 12: saturation of f   saturation of B eff f || ≈1 for N ≥ 5  negligible component of J on the CO 2 axis  no Q-branch f || ≈1 for N ≥ 5  negligible component of J on the CO 2 axis  no Q-branch Onset of Superfluidity in CO 2 ( 4 He) N Rotational Constant Superfluid Fraction  Exp. Theory T= 0.15 K Berkeley University of California 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005 Parallel Parallel  Perpendicular

Onset of Superfluidity in CO 2 ( 4 He) N Superfluid Fraction Total Density Perpendicular Superfluid Density Parallel Superfluid Density CO 2 ( 4 He) 5 parallel: parallel: localized in the global minimum localized in the global minimum perpendicular: perpendicular: zero superfluid density zero superfluid density T= 0.15 K Berkeley University of California 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005 Parallel Parallel  Perpendicular

CO 2 ( 4 He) 9 parallel: parallel: diffused around the global minimum diffused around the global minimum perpendicular: perpendicular: extending along the CO 2 axis extending along the CO 2 axis Onset of Superfluidity in CO 2 ( 4 He) N Parallel Superfluid Density Perpendicular Superfluid Density Total Density Superfluid Fraction T= 0.15 K Berkeley University of California 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005 Parallel Parallel  Perpendicular

CO 2 ( 4 He) 13 parallel: parallel: diffused around the global minimum diffused around the global minimum perpendicular: perpendicular: extending along the CO 2 axis extending along the CO 2 axis Berkeley University of California 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005 Parallel Superfluid Density Perpendicular Superfluid Density Total Density Onset of Superfluidity in CO 2 ( 4 He) N Superfluid Fraction T= 0.15 K Parallel Parallel  Perpendicular

CO 2 ( 4 He) 17 parallel: parallel: diffused around the global minimum diffused around the global minimum perpendicular: perpendicular: extending along the CO 2 axis extending along the CO 2 axis Berkeley University of California 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005 Onset of Superfluidity in CO 2 ( 4 He) N Superfluid Fraction T= 0.15 K Parallel Superfluid Density Perpendicular Superfluid Density Total Density Parallel Parallel  Perpendicular

Linear response Experiment: Tang, McKellar, Mezzacapo and Moroni, Phys. Rev. Lett. 92, (2004) Theory: Paesani, Kwon and Whaley, Phys. Rev. Lett. 94, (2005)  Exp. POITSE Berkeley University of California 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005 Onset of Superfluidity in CO 2 ( 4 He) N

 Exp. POITSE Linear response Berkeley University of California 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005 Onset of Superfluidity in CO 2 ( 4 He) N Experiment: Tang, McKellar, Mezzacapo and Moroni, Phys. Rev. Lett. 92, (2004) Theory: Paesani, Kwon and Whaley, Phys. Rev. Lett. 94, (2005) Linear response

Summary Calculations of rotational excitations in 4 He clusters Calculations of rotational excitations in 4 He clusters  very good agreement with experiments  very good agreement with experiments Calculations of superfluid properties Calculations of superfluid properties  onset of superfluidity in CO 2 ( 4 He) N  onset of superfluidity in CO 2 ( 4 He) N  direct relation to B eff  direct relation to B eff  insight to local contributions  insight to local contributions to superfluid density to superfluid density Paesani, Kwon and Whaley PRL 94, (2005) Berkeley University of California 60th Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2005