Rotational spectra of molecules in small Helium clusters: Probing superfluidity in finite systems F....

Rotational spectra of molecules in Rotational spectra of molecules in small Helium clusters:small Helium clusters:

Probing superfluidity in finite Probing superfluidity in finite systemssystems

F. Paesani and K.B. WhaleyF. Paesani and K.B. Whaley

Department of Chemistry andDepartment of Chemistry and

Pitzer Center for Theoretical ChemistryPitzer Center for Theoretical Chemistry

University of California, Berkeley, CA, 94720University of California, Berkeley, CA, 94720

$$$ NSF $$$ NSF

60th Symposium on Molecular Spectroscopy60th Symposium on Molecular SpectroscopyOhio State University, June 20-24, 2005Ohio State University, June 20-24, 2005

Spectroscopy in large Spectroscopy in large 44He dropletsHe droplets

courtesy A.F. Vilesovcourtesy A.F. Vilesov

• 44He droplets: ultracold environment (T≈ 0.15-0.4 He droplets: ultracold environment (T≈ 0.15-0.4 K)K) for high resolution spectroscopyfor high resolution spectroscopy• Free rotation in Free rotation in 44He He

• Rotational diffusion in Rotational diffusion in 33He He

Grebenev, Toennies and Vilesov, Grebenev, Toennies and Vilesov,

Science Science 279279, 2083 (1998), 2083 (1998)

• Superfluidity Superfluidity

BerkeleyBerkeleyUniversity of CaliforniaUniversity of California

Spectroscopy in small Spectroscopy in small 44He clustersHe clusters

OCS(OCS(44He)He)NN NN22O(O(44He)He)NN

Tang, Xu, McKellar and JägerTang, Xu, McKellar and JägerScience Science 297297, 2030 (2002), 2030 (2002)

Xu, Jäger, Tang and McKellarXu, Jäger, Tang and McKellarPhys. Rev. Lett. Phys. Rev. Lett. 9191, 163401 (2003, 163401 (2003))

Similar results also for :Similar results also for :• COCO22((44He)He)NN

Tang, McKellar, Mezzacapo and Moroni, Phys. Rev. Lett. Tang, McKellar, Mezzacapo and Moroni, Phys. Rev. Lett. 9292, 145503 (2004, 145503 (2004))

• CO(CO(44He)He)NN

McKellar, J. Chem. Phys. McKellar, J. Chem. Phys. 121121, 6868 (2004), 6868 (2004)

Simulations of doped Simulations of doped 44He clustersHe clusters

• Molecules:Molecules: OCS, NOCS, N22O and COO and CO22

• 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 Blume, Lewerenz, Niyaz and Whaley, Phys. Rev. E 5555, 3664 (1997), 3664 (1997)

Rotational spectrumRotational spectrum

2. Path-Integral Monte Carlo (PIMC)2. Path-Integral Monte Carlo (PIMC) Ceperley, Rev. Mod. Phys. Ceperley, Rev. Mod. Phys. 6767, 279 (1995), 279 (1995)

Superfluid propertiesSuperfluid properties

Potential Energy SurfacesPotential Energy Surfaces

He-NHe-N22OOChang, Akin-Ojo, Bukowski and Szalewicz, Chang, Akin-Ojo, Bukowski and Szalewicz, J. Chem. Phys. J. Chem. Phys. 119119, 11654 (2003), 11654 (2003)

He-COHe-CO22Yan, Yang and Xie, Yan, Yang and Xie, J. Chem. Phys. J. Chem. Phys. 109109, 10284 (1998), 10284 (1998)

He-OCSHe-OCSPaesani and Whaley, Paesani and Whaley, J. Chem. Phys. J. Chem. Phys. 121121, 4180 (2004), 4180 (2004)

Monte Carlo methodsMonte Carlo methods

HArot =Bx

∂2 j x

∂2 j y

∂2 jz

Vtot(R) = VA−B(Ri )i=1

∑ + VB−B(Rij )i< j

• Total interaction potentialTotal interaction potential

H=−1

∇A2 −HA

rot −1

∇i2 +V tot(R)

• Rotational Hamiltonian of molecule ARotational Hamiltonian of molecule A

• Hamiltonian for ABHamiltonian for ABNN systems systems

Projection Operator Imaginary Time Projection Operator Imaginary Time Spectral Evolution methodSpectral Evolution method

k ω( ) = φ0 A φnn

d(E0 −En +ω)

Λ k ω( ){ } =k(τ) =φ0 Aexp−(H−E0 )τ[ ]A

φ0 φ0

≡ A+ (0)A(τ)

k τ( ) =ΨT Aexp−(H−Eref )τ[ ]A

ΨT exp−(H−Eref )τ[ ] ΨT

• Laplace transformLaplace transform

• In practice: In practice: ΨΨTT ≈ ≈ 0 0 and Eand Erefref ≈ E ≈ E00

ˆ ˆˆ ˆ

1. D1. Djjmkmk((,,,,)) = molecular Wigner functions = molecular Wigner functions

2. 2. ,,,, = Euler angles = Euler angles molecule-fixed frame molecule-fixed frame space-fixed frame space-fixed frame

1. m=k=01. m=k=0 2. 2. DDjj

0000((,,,,)) P Pjj(cos(cos) ) j≈Jj≈J eigenfunctions for linear rotorseigenfunctions for linear rotors eigenfunctions for symmetric top rotors eigenfunctions for symmetric top rotors (k=0) (k=0)

• Present calculationsPresent calculations

• Â Â D Djjmkmk((,,,,))

ε( J ) τ( ) =−1

J( ) τ( )/ c1 ⎡ ⎣ ⎢

⎤ ⎦ ⎥

P k ω( )k τ( ), I ⎡ ⎣ ⎢

⎤ ⎦ ⎥∝P k τ( )k ω( ), I

⎡ ⎣ ⎢

⎤ ⎦ ⎥P k ω( ) I[ ]

k(ω) =Λ−1 k τ( ){ }

• Exponential fitExponential fit

k( J ) τ( ) = cn exp−En −E0( )τ[ ]n

• Maximum Entropy (based on Bayes’s statistics)Maximum Entropy (based on Bayes’s statistics)

N=1N=1POITSE: 0.657±0.002 cmPOITSE: 0.657±0.002 cm-1-1

Chang et al.: 0.65297 cmChang et al.: 0.65297 cm-1-1

N=2N=2POITSE: 0.419±0.006 cmPOITSE: 0.419±0.006 cm-1-1

N>2: J=1-3N>2: J=1-3

Rotational Excited States:Rotational Excited States:POITSE calculations for NPOITSE calculations for N22O(O(44He)He)NN

POITSE for N ≤ 2POITSE for N ≤ 2

POITSE for N > 2POITSE for N > 2

Rotational Excited States:Rotational Excited States:POITSE calculations for NPOITSE calculations for N22O(O(44He)He)NN

Rotational Spectrum for N > 5Rotational Spectrum for N > 5

POITSE for N ≤ 2POITSE for N ≤ 2

POITSE for N > 2POITSE for N > 2

Experiment: Tang, Xu, McKellar and Jäger, Science Experiment: Tang, Xu, McKellar and Jäger, Science 297297, 2030 (2002), 2030 (2002)Theory: Paesani, Viel, Gianturco and Whaley, Phys. Rev. Lett. Theory: Paesani, Viel, Gianturco and Whaley, Phys. Rev. Lett. 9090, 073401 (2003), 073401 (2003)

E = BE = BeffeffJ(J+1)J(J+1)

Theory

Spectroscopic Constants for OCS(Spectroscopic Constants for OCS(44He)He)NN

Theory vs ExperimentTheory vs Experiment

Theory

POITSE (exact)

Rigid coupling (approx.)

Theoretical interpretationTheoretical interpretationTransition from van der Waals complexes to quantum solvationTransition from van der Waals complexes to quantum solvation

Spectroscopic Constants for OCS(Spectroscopic Constants for OCS(44He)He)NN

E = BE = BeffeffJ(J+1)J(J+1)

Experiment: Xu, Jäger, Tang and McKellar, Phys. Rev. Lett. Experiment: Xu, Jäger, Tang and McKellar, Phys. Rev. Lett. 9191, 163401 (2003), 163401 (2003)Theory: Paesani and Whaley, J. Chem. Phys. Theory: Paesani and Whaley, J. Chem. Phys. 121121, 5293 (2004, 5293 (2004)

E = BE = BeffeffJ(J+1) - DJ(J+1) - DeffeffJJ22(J+1)(J+1)22

Distortion ConstantRotational Constant

Theory

Spectroscopic Constants for NSpectroscopic Constants for N22O(O(44He)He)NN

Distortion ConstantRotational Constant

Spectroscopic Constants for COSpectroscopic Constants for CO22((44He)He)NN

E = BE = BeffeffJ(J+1) - DJ(J+1) - DeffeffJJ22(J+1)(J+1)22

Experiment: Tang, McKellar, Mezzacapo and Moroni, Phys. Rev. Lett. Experiment: Tang, McKellar, Mezzacapo and Moroni, Phys. Rev. Lett. 9292, 145503 (2004), 145503 (2004)Theory: Paesani, Kwon and Whaley, Phys. Rev. Lett. Theory: Paesani, Kwon and Whaley, Phys. Rev. Lett. 9494, 153401 (2005), 153401 (2005)

Theory

Rotational Dynamics in Rotational Dynamics in 44He clustersHe clusters

• 11stst solvation shell solvation shell OCS: BOCS: Beffeff ≈ B ≈ Bdropletdroplet

NN22O, COO, CO22: B: Beff eff > B> Bdropletdroplet

??????

BBeffeff/B/B00 • decrease for small N decrease for small N “ “rigid” couplingrigid” coupling

• turnaround at N ≈ 5-9turnaround at N ≈ 5-9 ??????

Does this behavior reflect onset of superfluidity ???Does this behavior reflect onset of superfluidity ???

Path-Integral Monte Carlo methodPath-Integral Monte Carlo method

ZdRd ′ R ′ R∫ O R ρ R, ′ R ;( )

• Thermal density matrixThermal density matrix

ρ R, ′ R ;( ) = Re−H ′ R

ρB R, ′ R ;( ) =1

N!ρ R,P ′ R ;( )

∑Bose symmetryBose symmetry

=1/ kBT

• Implementation: Implementation: sampling ofsampling of

ρ R,P ′ R ;( ) = ...∫ dR1...dRM−1∫ ρ R,R1; τ( )...ρ RM−1,P ′ R ; τ( )

R,R1, ...,RM−1,P ′ R{ }

τ =/ MBerkeleyBerkeleyUniversity of CaliforniaUniversity of California

• Thermal averageThermal average

ffijij= superfluid fraction= superfluid fraction

||( ⊥)s =

4m 2kbT

I ||( ⊥)cl

A||( ⊥)A||( ⊥)

Superfluidity in PIMCSuperfluidity in PIMC

I ijcl =m dRρ R( ) R2dij −xix j( )∫

IIclcl= classical moment of inertia= classical moment of inertia

A = projected area of a pathA = projected area of a path macroscopic exchangesmacroscopic exchanges (Feynman, 1953)(Feynman, 1953)

I ij =∂ Λi

∂ω j

ωj =0

I ij =I ijcl(1−f ij

• Response to a slow rotation of an external fieldResponse to a slow rotation of an external field

• Superfluid fractionSuperfluid fraction

Superfluidity in PIMCSuperfluidity in PIMC

ρ||( ⊥)s R( ) =

4m2NkbT

I ||( ⊥)cl

A||( ⊥) R( ) ⋅A||( ⊥)

• Response to a slow rotation of an external fieldResponse to a slow rotation of an external field

• Superfluid densitySuperfluid density

• Superfluid fractionSuperfluid fraction

I ij =∂ Λi

∂ω j

ωj =0

I ij =I ijcl(1−f ij

||( ⊥)s =

4m 2kbT

I ||( ⊥)cl

A||( ⊥)A||( ⊥)

• direct correspondence between fdirect correspondence between f and B and Beffeff

- N ≤ 5: f- N ≤ 5: f negligible negligible decrease of B decrease of Beffeff van der Waals complexes van der Waals complexes - N > 5: increase of f- N > 5: increase of f increase of B increase of Beffeff onset of superfluidity onset of superfluidity - N ≥ 12: saturation of f- N ≥ 12: saturation of f saturation of B saturation of Beffeff

• ff||||≈1 for N ≥ 5 ≈1 for N ≥ 5 negligible component of J on the CO negligible component of J on the CO22 axis axis no Q-branch no Q-branch

Onset of Superfluidity in COOnset of Superfluidity in CO22((44He)He)NN

Rotational ConstantRotational Constant Superfluid FractionSuperfluid Fraction

Theory

T= 0.15 KT= 0.15 K

ParallelParallel

PerpendicularPerpendicular

Superfluid FractionSuperfluid Fraction

Total DensityTotal Density

Perpendicular Superfluid DensityPerpendicular Superfluid Density

Parallel Superfluid DensityParallel Superfluid Density

COCO22((44He)He)55

• parallel:parallel: localized in the global minimumlocalized in the global minimum• perpendicular: perpendicular: zero superfluid densityzero superfluid density

T= 0.15 KT= 0.15 K

ParallelParallel

COCO22((44He)He)99

• parallel:parallel: diffused around the global minimumdiffused around the global minimum• perpendicular: perpendicular: extending along the COextending along the CO22 axis axis

T= 0.15 KT= 0.15 K

ParallelParallel

COCO22((44He)He)1313

T= 0.15 KT= 0.15 K

ParallelParallel

COCO22((44He)He)1717

T= 0.15 KT= 0.15 K

ParallelParallel

Linear responseLinear response

POITSE

I ⊥ =I ⊥cl(1−f⊥

Beff =1

2I ⊥

POITSE

Linear response

Linear responseLinear response

I ⊥ =I ⊥cl(1−f⊥

SummarySummary

• Calculations of rotational excitations in Calculations of rotational excitations in 44He clustersHe clusters very good agreement with experimentsvery good agreement with experiments

• Calculations of superfluid propertiesCalculations of superfluid properties onset of superfluidity in COonset of superfluidity in CO22((44He)He)NN

direct relation to Bdirect relation to Beffeff

insight to local contributionsinsight to local contributions to superfluid densityto superfluid density

Paesani, Kwon and WhaleyPaesani, Kwon and WhaleyPRL PRL 9494, 153401 (2005), 153401 (2005)

Rotational spectra of molecules in small Helium clusters: Probing superfluidity in finite systems F....

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