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 27th, 2007Quantum Monte Carlo in the Apuan Alps III

Rotational dynamics of

helium solvated molecules: from small clusters

toward the nanodroplet regime

Part 1:

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)

G. Scoles and K. K. Lehmann, Science, 287 2429 (2000)

He nanodropletsHe nanodroplets

H2O@4HeN

N ~104 4He atoms

•Interest for the solvent:Interest for the solvent: properties of quantum fluids properties of quantum fluids in confined systemsin confined systems

•Interest for the impurity:Interest for the impurity: good spectroscopic matrix HENDI SPECTROSCOPY

Helium Nanodroplets Isolation Helium Nanodroplets Isolation spectroscopyspectroscopy

from G. Scoles and K. K. Lehmann Science 287, 5462 (2000)

4He nanodroplets are superfluidR

ela

tiv

e D

ep

leti

on

[%

]

• Pure 3He droplets

•T=0.15K •Broad peak

Experiment: (Toennies et al. Science, 1998)

Wave Number Change [ cm -1]

Re

lati

ve

De

ple

tio

n

[%]

• Pure 4He droplets

• T=0.38K

• free rotor spectrum with increased inertia

•Superfluidity: response to an imposed rotation

How small can a superfluid droplet be?

How does superfluidity How does superfluidity start to show up? start to show up?

• N = 1 - 8

J. Tang , Y. Xu, A.R.W. McKellar, and W. Jäger, Science, 297 2030 (2002)

• N-selective experiments: OCS@OCS@44HeHeNN

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, Beff?

Can we extrapolate Beff from the small size behavior?

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 1st solvation shell is completed slowly for light rotors (e.g. HCN): well beyond the 1st solvation shell

The reduction of B upon solvation is due to the molecular mass large reduction for heavy rotors

small reduction for light rotors

Experiments do not validate this picture

CO2@4HeN N2O@4HeN

J. Tang et al. PRL (2004) W. Jager et al. JCP (2006)

For some heavy molecules the convergence is slow

For N2O (lighter than OCS) B reduction is larger than for OCS

Ground-state path integral Monte Carlo

•

•

•

•

Reptation quantum Monte Carlo

•Path probability :

• Random walk:

• Weight of the path:

(S.Baroni and S. Moroni, Phys. Rev. Lett. 82, 4745 (1999))

Reptation quantum Monte Carlo

• Sampling the paths

• Metropolis test

• For large systems ( N > 50), bisection-multilevel algorithm is more efficient

Hamiltonian

€

H =P 2

2M+

J 2

2I+

pi2

2m+ VHe−I (ri,θ i)

i=1

N

∑i=1

N

∑ + VHe− He (rij )i< j

∑

€

ΨT = exp − u1

i=1

N

∑ (ri,θ i) − u2(rij )i< j

∑ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

Trial function

€

u1(r,θ) = PL (cosθ) fL (r)L

∑

Calculating the spectrum

• spectrum of He solvated molecules

• analytic continuation in imaginary time

• for a linear molecule

Elucidating the relation between

the structure and the dynamics

RQMC simulations:

• CO@HeN: double-lined spectra

O

CHe density accumulation

CO@4HeN – disentangling the spectra

+ SimulationsExperiments

a-type

b-type

CO@4HeN – Structure

Simulations

HIGH

LOW

CO@4HeN – Asymmetric structure

He

He

O

C

CO@4HeN – Matrix dynamics

Convergence of B to the nanodroplet limit

RQMC simulations:

• OCS@HeN: a prototype of HEAVY ROTORS

OCS@4HeN - Structure

HIGH

LOW

OCS@4HeN – Rotational dynamics

B converges slowly to the nanodroplet limit

Convergence of B to the nanodroplet limit

RQMC simulations:

• HCN@HeN: a prototype of

LIGHT ROTORS

HCN@4HeN - Structure

Density

HCN@4HeN – Rotational dynamics

B converges fast to the nanodroplet limit

nanodroplet value

HCN@4HeN – Matrix dynamics

Reduction of the rotational constant

Beff/Bgas = 33% Beff/Bgas = 36%

fudged OCS@HeN

Rotational dynamics

real OCS@HeN

Rotational dynamics

Simulations with fictitious inertia• fudged-OCS = He-OCS potential + HCN inertia

Reduction of the rotational constant

fictitious inertia vs real inertia

Beff/Bgas = 90% Beff/Bgas = 81%

fudged-HCN@HeNreal HCN@HeN

• fudged-HCN = He-HCN potential + OCS inertia

Reduction of B upon solvation

For a given potential Beff/Bgas can increase with increasing Bgas

Bgas

f-OCS

COHCN

f-HCN

CO2

N2O

OCS

DCN

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, N2O, CO2).

• The decrease of the rotational constant is mostly due to the anisotropy and the strength of the potential, more than to the molecular weight.

Solid-like Solid-like vsvs liquid-like liquid-like behavior in behavior in 44He clusters He clusters

doped with doped with alkali and alkaline-earth alkali and alkaline-earth

ionsions

Part 2:Part 2:

Work done with:• Flavio Toigo and Francesco Ancilotto Flavio 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

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

€

μ =⟨vdrift⟩

E

Liquid helium

Foerste et al., Z. Phys. B (1997)

Existing QMC calculations

• VMC (Shadow Wave Functions) - static correlations criterion

• Solid-like order in the first shell is found for all these ions

• 4He clusters doped with Na+, K+, Cs+, Be+, Mg+

Rossi et al. PRB(2004)

1

23

Cs+@He64

Mg+@He64

Dynamical correlations criterion

• A slow decaying indicates solid-like behavior

• Multipole moments imaginary-time correlations:

Baroni and Moroni ChemPhysChem (2005)

€

cL (τ ) =

⟨QLM *(0)Q L

M (τ )⟩M

∑

⟨QLM *(0)Q L

M (0)⟩M

∑

€

QLM =

4π

2L +1d

r r ρ(

r r )∫ rLYL

M (ϑ ,ϕ )*

• Used for clusters of para-hydrogen made of just one shell

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

In Li+@He70 the 1st shell is solid

Persistence of a rigid structure in the 1st shell

multipole correlations 1st shell He density

The 1st shell of Na+@He70

has an icosahedral structure

Slow decaying for L=6

multipole correlations

1st shell He density

Comparing alkaline-earth ions doped clusters

Persistence of a rigid structure in the 1st shell of Be+@He70

multipole correlations

1st shell He densitiesBe+@He70 Mg+@He70 Ca+@He70

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+@He70 and Na+@He70 have solid first shell which move in a liquid environment.

• Mg+@He70 and Ca+@He70 form bubbles.

• Be+@He70 shows a signature of a solid-like behavior of the first shell and forms a snowball.

Fluctuations of the inter-particles distances

•Radial density profiles

€

ΔB (rcut ) =1

Ncut (Ncut −1)

rij2

RW− rij RW

2

rij RWi, j

N

∑paths

€

ΔB (rcut ) =1

Ncut (Ncut −1)

rij2

RW− rij RW

2

rij RWi, j

N

∑paths

Li+@He70 Be+@He70 Mg+@He70

•Berry parameterBerry, JCP (2001)

Rotational diffusion in the 1st shell

€

D(τ ) =1

N

r r i(0) −

r r i(τ )[ ]

2

i=1

N

∑

€

D(τ ) =1

N

r r i(0) −

r r i(τ )[ ]

2

i=1

N

∑

Li+@He8 is a solid-like clustermultipole correlations

€

D(τ ) =1

N

r r i(0) −

r r i(τ )[ ]

2

i=1

N

∑

€

D(τ ) =1

N

r r i(0) −

r r i(τ )[ ]

2

i=1

N

∑

Persistence of a rigid-like structure1st shell He density

static multipoles

CO@4HeN – Structure

Simulations

HCN@HeN and CO@HeN – similar structures

OCS@4HeN – Rotational dynamics

B converges slowly to the nanodroplet limit

recent experiments, Jäger, PRL (2006)

expt

our RQMC

Ground-state path integral Monte Carlo

• approaches exact ground state as

•

• Optimized trial function

• Compute expectation values:

Use discretized path integral to represent

€

e−β ˆ H

Metropolis (reptation or bisection-multilevel) algorithm to sample paths

time step

exact results for

€

ε =β / M

€

ε → 0R0

RM

•