quantum monte carlo simulations of helium clusters doped with molecular and ionic impurities stefano...
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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
•