Inexpensive Simulation of Nuclear Quantum E ects · 7 Michele Ceriotti Inexpensive Simulation of...

Inexpensive Simulation of

Nuclear Quantum Eects

Michele Ceriotti

EVS Meeting, 21/01/2014

Acknowledgements

David Manolopoulos

Tom Markland, Mariana Rossi

Joshua More

The Royal Society

The European Commission

Merton College

Giovanni Bussi

Jérôme Cuny, Ali Hassanali

Michele Parrinello

Swiss National Science Foundation

CSCS

2 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects

Nuclear quantum eects and the n (p)

NQEs are important for any vibrational mode for which ~ω/kBT > 1.[kB/~ ≈ 0.7cm−1/K]Very large eects on properties that depend on uctuationsDramatic when one measures the particle momentum distributionThe (quantum) kinetic energy is closely related to isotope-substitution ∆G


Ceriotti et al., PRB 82 (2010)Ceriotti & Markland, JCP 138 (2013)



Proton momentum distribution in Li2NH


Ceriotti et al., PRB 82 (2010)

Ceriotti & Markland, JCP 138 (2013)



∆A→B∆H→DG =

ˆ mD

mH

K (µ;A)− K (µ;B)

µdµ


Ceriotti et al., PRB 82 (2010)

Ceriotti & Markland, JCP 138 (2013)

Path integrals and evaluating n (p)

The path integral formalisms maps the (distinguishable particles) quantumpartition function onto a classical ring polymer Hamiltonian

HP =P∑i=1

[V (qi ) +

p2i2m

+1

2mω2P (qi − qi−1)2

]Many replicas of the physical system (computationally expensive!)Same particles are joined by harmonic springs, forming a loop

Path integrals provide straightforwardly congurational properties. PMDrequires open-path simulations that add complexity and cost.

One can however compute (relatively) easily⟨p2

⟩


Morrone et al., JCP 126 (2007)

Using colored noise to get NQEs

The nite-temperature density for a quantum harmonic oscillator is aGaussian, with a frequency and temperature dependent width

This is the same as a classical distribution at the eective temperatureT ? (ω) = (~ω/2kB) coth (~ω/2kBT )We can enforce an ω-dependent temperature by a generalized Langevindynamics based on colored (correlated) noiseWorks well also in anharmonic 1D examples


Ceriotti, Bussi, Parrinello, PRL (2009)

Using colored noise to get NQEs

The nite-temperature density for a quantum harmonic oscillator is aGaussian, with a frequency and temperature dependent width

This is the same as a classical distribution at the eective temperatureT ? (ω) = (~ω/2kB) coth (~ω/2kBT )We can enforce an ω-dependent temperature by a generalized Langevindynamics based on colored (correlated) noiseWorks well also in anharmonic 1D examples

Quartic double well

A proton in a QDW,distance betweenminima: 2Å,comparison betweenexact and quantumthermostat results


Ceriotti, Bussi, Parrinello, PRL (2009)

PIGLET - fast and accurate NQEs

You can use fancy stochastic dynamics to accelerate convergence of pathintegral simulations (5x speedup)

Also works for getting⟨p2⟩. Should be easy to get open path, but not yet

developed...


Ceriotti, Manolopoulos, Parrinello, JCP (2011); Ceriotti & Manolopoulos, PRL (2012)

Quantum hydrogen-bond uctuations

Describe the hydrogen bond with structural parameters that captureuctuationsQuantum eects trigger a signicant fraction of (transient) self-ionizationevents!The motion along the PT coordinate is strongly coupled with electronicuctuations (Wannier centers analysis, NMR).

cos Α

Ν @ÞD

PPmax

10-4 10-3 10-2 10-1 11

-2 -1 0

0.6

0.8

1

Classical


Kumar et al. JCP 2007

M. Ceriotti, J. Cuny, M. Parrinello, D. Manolopoulos, PNAS 2013



cos Α

Ν @ÞD

PPmax

10-4 10-3 10-2 10-1 11

-2 -1 0

0.6

0.8

1

-2 -1 0

Classical PIGLET






PPmax

10-4 10-3 10-2 10-1 11

O-X

O'-X'

-1 -0.5 0 0.5

Ν @ÞD

0.3

0.4

0.5

0.6

d@ÞD

0 0.5 1

PPmax




Isotope substitution & isotope fractionation

Equilibrium isotope eects are uniquely determined by the quantum natureof nuclei

Isotope fractionation between dierent phases1

Relevant to geochemistry (water cycle)2

Determining historical record of temperatures3


Hphase1 +Dphase2∆G Dphase1 +Hphase2

∆G ∝ˆ mD

mH

T1 (µ)

µ− T2 (µ)

µdµ

1Ceriotti & Markland JCP 2013, 2Worden et al., Nature 2007, 3Petit et al., Nature 1999

Isotope substitution & isotope fractionation

Equilibrium isotope eects are uniquely determined by the quantum natureof nuclei

Isotope fractionation between dierent phases1

Relevant to geochemistry (water cycle)2

Determining historical record of temperatures3


1Ceriotti & Markland JCP 2013, 2Worden et al., Nature 2007, 3Petit et al., Nature 1999

Surface-specic isotope eects

Surface-specic eects: dierential segregation of H/D at a water-vaporinterface comparison with VSFS experiments.

0 :100

25 : 75

50 : 50

75 : 25

100 : 0

H : D

2600 2700 2800 2900Ωq @ cm- 1 D

0 :100

25 : 75

50 : 50

75 : 25

100 : 0

H : D

3600 3700 3800Ωq @ cm- 1 D

I SF

G




sH - xH

sD - xD

0 0.2 0.4 0.6 0.8 1xD , H

- 6

- 4

- 2

0

2

4

6

HsD

,H

-x

D,

HL´

100


Deuterium is depleted on the dangling bonds at the water vapour interfaceof H2O/HOD/D2O mixture.




Collaboration with X. Chen, Boston University, J. Phys. Chem. C (2013)

Transient anisotropic Gaussian approx.

Conventional (easy) PIMD allows one to compute the kinetic energy tensor〈pαpβ〉. However only averages make sense!If the atomic environment changes slowly, one can compute a reasonableestimate of the principal components of the KE tensor as a moving average

One can use the average principal components to compute a multi-variateGaussian approximant to the PMD

〈pαpβ〉 =kBT

2δαβ +

1

4P

⟨∑i

(q(i)α − qα

) ∂V

∂q(i)β

+(q

(i)β − qβ

) ∂V

∂q(i)α

⟩


Ceriotti & Manolopoulos, PRL (2012)




Tαβ (t; ∆t) =1

∆t

ˆ t+∆t

t−∆t

Tαβ (t)

[1− |t − t ′|

∆t

]dt ′



NQEs in GPa water

Simulation of water at 750K and 10GPa. BLYP+D3, CP2K with i-PI

interface, PIGLET with 4 beads and NPT.

Modular design of i-PI makes it easy to interface it with any electronicstructure code

Robust implementation, NPT+PIMD via symmetric Trotter splitting andfully stochastic thermostatting


Ceriotti, More, Manolopoulos, Comp. Phys. Comm. 2014

Self dissociation at 10GPa

Quantum eects have a small (but noticeable) eect on density

Pressure promotes water dissociation. Presence of a large concentration ofcharged species, particularly with NQEs

10GPa,750K

classicalquantum

1.6 1.65 1.7Ρ @gcm3D

0

10

20h

HΡL

8 10 12Pint @GPaD

0

0.1

0.2

0.3

hHP i

ntL






0 1 2 3 4 5 6

æ

æ æ æ æ æ æ0

0.5

PHcla

ssic

alL

0 1 2 3 4 5 6

æ

æ

æ

æ æ æ æ

0 1 2 3 4 5 6n+-

0

0.2

0.4

PHqu

an

tum

L



Inhomogeneity and PMD

How much dierent are the kinetic energies of dierent species?Protonic H3 has much smaller Kz , but there is too little of it (1.6%) tomake a real dierenceEven if the mix was 1:1 H2:H3, it is impossible to tell the dierencebetween a mixture and a best-t mean proton scenario!

H3H2H1

0 5 10 15 20 25

p @Þ-1D

0

0.02

0.04

0.06

0.08

0.1

0.12

nHpL




Kinetic energy [meV]; TAG, ∆ = 25 fs. kBT/2 = 32.3meV

H1 H2 H3 H O1 O2 O3 O

Kx 36.8 37.2 39.7 37.2 32.5 32.2 32.8 32.5Ky 42.9 44.8 46.7 44.8 35.1 35.8 36.1 35.8Kz 99.1 96.2 80.5 96.0 39.6 40.6 40.6 40.6K 178.8 178.2 166.9 178.0 107.1 108.6 109.5 108.6




HnHH3+H2L2HpL-nHpLLnHpLHnfitHpL-nHpLLnHpL

HnH3HpL-nH2HpLLnHpL

0 5 10 15 20 25

p @Þ-1D

-5

0

5

10

15

%D

nHpL


A theoretician's drivel on DINS

Good progress in theoretical modeling of NQEs. Still some work needed tosimplify getting exact PMD from simulations

Transient anisotropic Gaussian approximation is not bad, and can beapplied easily to a conventional (or accelerated) PIMD simulation

In general, a multi-variate model of n (p) captures a lot of the physics...

Would be interesting to re-analyze multi-bump n (p) experimental resultsin terms of MVGMIt is very hard to capture inhomogeneity, as it is matched by an eectivesingle-kind model

Challenges for DINS experiments!

Directionally-resolved n (p) to avoid spherical averagingSite-selective probing of n (p)?


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