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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
3 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
Ceriotti et al., PRB 82 (2010)Ceriotti & Markland, JCP 138 (2013)
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
3 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
Ceriotti et al., PRB 82 (2010)Ceriotti & Markland, JCP 138 (2013)
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
3 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
Ceriotti et al., PRB 82 (2010)Ceriotti & Markland, JCP 138 (2013)
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
Proton momentum distribution in Li2NH
3 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
Ceriotti et al., PRB 82 (2010)
Ceriotti & Markland, JCP 138 (2013)
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
∆A→B∆H→DG =
ˆ mD
mH
K (µ;A)− K (µ;B)
µdµ
3 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
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
⟩
4 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
Morrone et al., JCP 126 (2007)
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
⟩
4 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
Morrone et al., JCP 126 (2007)
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
⟩
4 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
Morrone et al., JCP 126 (2007)
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
⟩
4 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
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
5 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
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
5 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
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
5 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
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...
6 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
Ceriotti, Manolopoulos, Parrinello, JCP (2011); Ceriotti & Manolopoulos, PRL (2012)
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...
6 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
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
7 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
Kumar et al. JCP 2007
M. Ceriotti, J. Cuny, M. Parrinello, D. Manolopoulos, PNAS 2013
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
-2 -1 0
Classical PIGLET
7 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
Kumar et al. JCP 2007
M. Ceriotti, J. Cuny, M. Parrinello, D. Manolopoulos, PNAS 2013
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).
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
7 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
Kumar et al. JCP 2007
M. Ceriotti, J. Cuny, M. Parrinello, D. Manolopoulos, PNAS 2013
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).
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
7 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
Kumar et al. JCP 2007
M. Ceriotti, J. Cuny, M. Parrinello, D. Manolopoulos, PNAS 2013
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
8 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
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
8 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
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
8 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
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
9 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
Surface-specic isotope eects
Surface-specic eects: dierential segregation of H/D at a water-vaporinterface comparison with VSFS experiments.
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
9 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
Deuterium is depleted on the dangling bonds at the water vapour interfaceof H2O/HOD/D2O mixture.
Surface-specic isotope eects
Surface-specic eects: dierential segregation of H/D at a water-vaporinterface comparison with VSFS experiments.
9 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
Collaboration with X. Chen, Boston University, J. Phys. Chem. C (2013)
Surface-specic isotope eects
Surface-specic eects: dierential segregation of H/D at a water-vaporinterface comparison with VSFS experiments.
9 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
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
⟩
10 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
Ceriotti & Manolopoulos, PRL (2012)
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
10 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
Ceriotti & Manolopoulos, PRL (2012)
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
Tαβ (t; ∆t) =1
∆t
ˆ t+∆t
t−∆t
Tαβ (t)
[1− |t − t ′|
∆t
]dt ′
10 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
Ceriotti & Manolopoulos, PRL (2012)
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
10 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
Ceriotti & Manolopoulos, PRL (2012)
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
10 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
Ceriotti & Manolopoulos, PRL (2012)
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
11 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
Ceriotti, More, Manolopoulos, Comp. Phys. Comm. 2014
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
11 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
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
12 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
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
12 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
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
12 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
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
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
12 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
Ceriotti, More, Manolopoulos, Comp. Phys. Comm. 2014
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
13 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
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!
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
13 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
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!
HnHH3+H2L2HpL-nHpLLnHpLHnfitHpL-nHpLLnHpL
HnH3HpL-nH2HpLLnHpL
0 5 10 15 20 25
p @Þ-1D
-5
0
5
10
15
%D
nHpL
13 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects
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)?
14 Michele Ceriotti Inexpensive Simulation of Nuclear Quantum Eects