QUANTUM MECHANICAL EFFECTS FROM RING POLYMER … · 6. Dynamics of the solvated electron (2008). 7. Diffusion of H isotopes in transition metals (2009). 8. Competing quantum effects

QUANTUM MECHANICAL EFFECTS FROM RING POLYMER MOLECULAR DYNAMICS

Department of Chemical Engineering, Massachusetts Institute of Technology

77 Massachusetts Avenue, Cambridge, MA 02139

01.26.2013

2005 MSc in Chemistry (Moscow State University) 2009 PhD in Theoretical Chemistry (Moscow State University) 2009-11 Newton International Fellow (Royal Society/University of Oxford) 2011- Princeton CEFRC Fellow (Princeton/MIT)

Dr. Yury V. Suleymanov

Outline

1. Ring polymer molecular dynamics 2. Ring polymer reaction rate theory • Chemical reactions: - A+BC (H+H2, Cl+HCl, F+H2,Mu+ H2) - H+CH4

3. RPMDrate code

1. Ring polymer molecular dynamics Based on the classical isomorphism:a

][ HetrZ β−=∫ ∫ −= ),(

)2(1

nnnnHnnn eddZ qpqp β

πwhere

a D. Chandler and P. G. Wolynes, JCP 74, 4078 (1981). ./1 and /with

)()(21

2),(

1

21

22

nnn

n

jjjjn

jnnn

n

qVqqmm

pH

βωββ

ω

==

+−+=∑

=−qp

Single quantum particle

Classical n-bead ring polymer

)ˆ(2ˆ

)ˆ,ˆ(2

qVm

pqpH +=

QM partition function: Classical partition function:

where

map each quantum particle onto an effective classical system of N beads coupled via harmonic springs and each under external potential V

∞→n

1. Ring polymer molecular dynamics Based on the classical isomorphism:a

][ HetrZ β−=∫ ∫ −= ),(

)2(1

nnnnHnnn eddZ qpqp β

πwhere

a D. Chandler and P. G. Wolynes, JCP 74, 4078 (1981). ./1 and /with

)()(21

2),(

1

21

22

nnn

n

jjjjn

jnnn

n

qVqqmm

pH

βωββ

ω

==

+−+=∑

=−qp

Single quantum particle

Classical n-bead ring polymer

)ˆ(2ˆ

)ˆ,ˆ(2

qVm

pqpH +=

QM partition function: Classical partition function:

where

map each quantum particle onto an effective classical system of N beads coupled via harmonic springs and each under external potential V

∞→n π8/)(TRG Λ=

mkThT π2/)( =Λ

(Proof) [ ] ( )[ ] ./ where, netretrZ n

nHH n ββββ === −−

,1211 qeqqeqdqdqZ Hn

Hn

nn ββ −−∫∫=

So

where

∫ −−−− −= )(/)(2/ 12

21 jnjjn qVqqipmpdpe ββ

π

[ ])(2/)(2/1

21

2221 jjjnn qVqqm

n

em +−− −

= ωβ

βπ

π

[ ] ann

qVqqmmpj

jjjnjnedp ./1 with ,21 )(2/)(2/ 2

122

βωπ

ωβ =≡ +−+− −∫

a M. Parrinello and A. Rahman, JCP 80, 860 (1984).

Path Integral Molecular Dynamics (PIMD) • The ring polymer trajectories

as a sampling tool to calculate exact values of static equilibrium properties such as

[ ].1 AetrZ

A Hβ−=

pn

qn

Note: - any classical method (MC, MD) can be used to sample multidimendional

configuration integrals: PIMD - Molecular Dynamics - PIMD: Time is only a parameter for the exploration of phase space

(considered as unphysical dynamics).

Examples: • thermodynamic energy • heat capacity • Helmholtz free energy

ββ

∂∂

−=)(ln ZE

2

2

2B

ln1β∂

∂=

∂∂

=Z

TkTE

Cv

ZTkF lnB−=

Ring Polymer Molecular Dynamics (RPMD)

The RPMD approximation is simply:

,)()()2(

1)(~0

),(00

00∫ ∫ −≈ tnnH

nAB BAeddZ

tc nn qqqp qpβ

π

where

.)(1)( and )(1)(11∑∑==

==n

jjn

n

jjn qB

nBqA

nA qq

(Classical molecular dynamics in an extended phase space)

pn

qn

• The same ring polymer trajectories to calculate approximate Kubo-transformed correlation functions of the form

[ ].)()0(1)(~0

)(∫ −−−=β

λλβλβ

tBeAetrdZ

tc HHAB

.)()(1)(0∫ −=β

λλβ

thiFtrdtc ),(lim)(

1)( side-flux tcTQ

Tkt

r∞→

=

).( TSqq −= hh

Examples: • Chemical reaction rates

Step operator: Flux operator:

• Diffusion coefficients ,)(1)(

0velocity-velocity∫

∞

= dttcd

TD .)()(1)(0∫ −=β

λλβ

tmpi

mptrdtc

• Neutron scattering ,),(

21),(

functionscattering

teintermedia

structuredynamicincoherent dttkFewkS iwt∫

∞

∞−

−=π

,1),(particle

1

)()(

0particle∑∫=

⋅+−⋅−=N

j

tiii jj eedN

tkF rkrk λβ

λβ

- energy transfer (from neutron to liquid)

- momentum transfer jie rk⋅− )(ti je rk⋅+and - correlated density operators

• Dipole absorption spectroscopy

),(Volume3

)()(0

2

wIc

wwwnε

πβα =

∫+∞

∞−−

−−−

= ),(2

)1()( tCdtew

ewI dipoledipoleiwt

w

πβ

β,)()(1)(

0∫ −=β

µλµλβ

tidtC

,)( )( HH eeiF λλβλ −−−≡− F,)( // iHtiHt eeth −+≡ h

)(wα)(wn

- Beer-Lambert absorption coefficient

- frequency-dependent refraction index

Golden rule of TDPT:

Properties of RPMD:

1. Exact in the limit as t → 0.a 2. Exact in the classical (high temperature) limit.b

3. Exact in the harmonic limit (for linear A and/or B).b

4. Exact when A = 1 (the unit operator).c

5. Consistent with the QM equilibrium distribution.

a B. J. Braams and D. E. Manolopoulos, JCP 125, 124105 (2006). b I. R. Craig and D. E. Manolopoulos, JCP 121, 3368 (2004). c S. Habershon and D. E. Manolopoulos, JCP 131, 244518 (2009).

(exact QM rate coefficient for a parabolic barrier)

(exact QM in the low T limit)

𝑍 = 𝑒−𝛽𝐸0

RPMD:

(exact QM rate coefficient for a parabolic barrier)

(exact QM in the low T limit)

𝑍 = 𝑒−𝛽𝐸0

RPMD: But it neglects QM interference effects in the real-time dynamics (no 𝑒±𝑖𝑖𝑖/ℏ)

Example applications:

1. Quantum diffusion in liquid para-hydrogen (2005). 2. Quantum diffusion in liquid water (2005). 3. Neutron scattering from liquid para-hydrogen (2006). 4. Proton transfer in a polar solvent (2008). 5. Diffusion of H isotopes in water and ice (2008). 6. Dynamics of the solvated electron (2008). 7. Diffusion of H isotopes in transition metals (2009). 8. Competing quantum effects in liquid water (2009). 9. Hydrogen surface diffusion (2012). 10. Gas phase chemical reaction rates (2009-2012).

Example applications:

1. Quantum diffusion in liquid para-hydrogen (2005). 2. Quantum diffusion in liquid water (2005). 3. Neutron scattering from liquid para-hydrogen (2006). 4. Proton transfer in a polar solvent (2008). 5. Diffusion of H isotopes in water and ice (2008). 6. Dynamics of the solvated electron (2008). 7. Diffusion of H isotopes in transition metals (2009). 8. Competing quantum effects in liquid water (2009). 9. Hydrogen surface diffusion (2012). 10. Gas phase chemical reaction rates (2009-2012).

2. Ring polymer reaction rate theory Consider an atom-diatom reaction in the gas phase

A+BC→AB+C The exact QM rate coefficient is

),(~lim)(

1)( stcTQ

Tk fstr

∞→=

where

[ ].)()0(1),(~0

)(∫ −−−=β

λλβλβ

theFetrdstc HHfs

with

[ ]

=≡=0>s ,10<s ,0

)( and , shhhHiF

(Flux) (Side) NB: 1. k(T) is independent of s 2. cfs(t→0+)~0

2. Ring polymer reaction rate theory The classical limit rate coefficient is

),(lim)(

1)( stcTQ

Tk clfst

r

cl

∞→=

where

[ ] [ ]

)()()(

)2(1);(

00

),(

0000

t

Hfff

cl

shdt

dss

eddstcfs

qqq

qpqp

××

=−

∫ ∫

δ

πβ

Flux (t = 0) Side (t > 0) NB: 1. kcl(T) is independent of s 2. cfs(t→0+)≠0 → classical TST

2. Ring polymer reaction rate theory The RPDM rate coefficient is

);(lim)(

1)( stcTQ

Tk fstr

RPMD

∞→=

where

[ ] [ ]

)()()(

)2(1);(

00

),(

0000

t

Hnfnfnffs

shdt

dss

eddstc nn

qqq

qp qp

××

= −∫ ∫

δ

πβ

Flux (t = 0) Side (t > 0)

nnn

n

jjjjn

jn nqVqqm

mp

H βωββω /1 ,/ with ,)()(21

2),(

1

21

22

==

+−+=∑

=−qp

and

∑∑==

==n

jj

n

jj p

npq

nq

11

1 ,1

NB: 1. kRPMD(T) is independent of s 2. cfs(t→0+)≠0 → “quantum” TST (centroid density)

Computational strategy

0)(0 =qs

0)(1 =qs

- dividing surface in the asymptotic reactant valley

- dividing surface in the region of the reaction barrier


0)(0 =qs

0)(1 =qs



)(

0

),(

0

1

)(

1

1

0011

)();0(

);0();0(

);0();(

)(

sk

r

fs

ssp

fs

fs

s

fs

fsRPMD

QTST

TQstc

stcstc

stcstc

Tk +

+

+

+

→×

→

→×

→

∞→=

κ


0)(0 =qs

0)(1 =qs



)(

0

),(

0

1

)(

1

1

0011

)();0(

);0();0(

);0();(

)(

sk

r

fs

ssp

fs

fs

s

fs

fsRPMD

QTST

TQstc

stcstc

stcstc

Tk +

+

+

+

→×

→

→×

→

∞→=

κ

transmission coefficient for s1

kQTST(s0) ratio of ring polymer averages on two different dividing surfaces


0)(0 =qs

0)(1 =qs



)(

0

),(

0

1

)(

1

1

0011

)();0(

);0();0(

);0();(

)(

sk

r

fs

ssp

fs

fs

s

fs

fsRPMD

QTST

TQstc

stcstc

stcstc

Tk +

+

+

+

→×

→

→×

→

∞→=

κ

Rq −= ∞Rs )(0

2/12

0 214)(

= ∞

R

QTST Rskπβµ

π

Consider the following dividing surface s0

CBA

CBAR

ACB

CcBB

mmmmmm

,mmmm

CABBCA

+++

=

−++

=

+→+

)(µ

rrrR


0)(0 =qs

0)(1 =qs



2/12

),(

0

1

)(

1

1

214

);0();0(

);0();(

)(

011

×

→

→×

→

∞→= ∞

+

+

+ R

ssp

fs

fs

s

fs

fsRPMD Rstcstc

stcstc

Tkπβµ

π

κ

Two separate stages:

1. Calculate κ(s1) (dynamical) 2. Calculate p(s1,s0) (static)


0)(0 =qs

0)(1 =qs



Advantages: 1. Practical way to compute k(T) when the probability that a spontaneous thermal fluctuation will bring the system to the TS is very small 2. No need to calculate Qr(T)

2/12

),(

0

1

)(

1

1

214

);0();0(

);0();(

)(

011

×

→

→×

→

∞→= ∞

+

+

+ R

ssp

fs

fs

s

fs

fsRPMD Rstcstc

stcstc

Tkπβµ

π

κ


1

1

1

)]([)()();0();(

1011

0

)(

1

1 lims

tst

s

fs

fs shdt

dsfstcstc

qqq −

∞→+

=→

∞→

κ

0)(0 =qs

0)(1 =qs



2/163

1

2)(

21)(

∂∂

= ∑−

=

N

i iis q

sf qqπβµ

where


0)(0 =qs

0)(1 =qs



)]0()1([

),(

0

1

01

);0();0( WW

ssp

fs

fs estcstc −−

+

+ =→

→ β

W(ξ) is the centroid potential of mean force along the reaction coordinate

==

=−

=0)( ,0

0)( ,1)()(

)(0

1

10

0

qq

qqq

ss

sssξ

Umbrella integration along the reaction coordinate:

• H + H2 • D + H2 • Mu + H2

• H + CH4 • D + CH4 • Mu + CH4 • O + CH4

• Cl + HCl • F + H2

RESULTS: H+H2

Classical and RPMD potentials of mean force for the H+H2 reaction at 1000 and 300 K.

Classical (dashed) and RPMD (solid) transmission coefficients for the H+H2 reaction at 1000 and 300 K.

R. Collepardo-Guevara, Yu. V. Suleimanov, and D. E. Manolopoulos, J. Chem. Phys. (2009) 130, 174713.

RESULTS: H+H2

Arrhenius plot of quantum, RPMD, and classical rate coefficients for the distinguishable atom H+H2 reaction between 200 and 1500 K (quantum results stopping at 1000 K).

Deep quantum tunnelling • RPMD gives k(T) within a factor of 2 of the exact QM result when the classical rate is out by 3 orders of magnitude

Symmetric barrier. Tc = 345 K.


RESULTS: Cl+HCl

Classical and RPMD potentials of mean force for the Cl+HCl reaction at 1000 and 300 K

Classical (dashed) and RPMD (solid) transmission coefficients for the Cl+HCl reaction at 1000 and 300 K.


RESULTS: Cl+HCl

Arrhenius plot of quantum, RPMD, and classical rate coefficients for the Cl+HCl reaction between 200 and 1500 K (quantum results stopping at 1000 K).

Symmetric barrier. Tc = 320 K.


RESULTS: F+H2

Classical and RPMD potentials of mean force for the F+H2 reaction at 1000 and 300 K

Classical (dashed) and RPMD (solid) transmission coefficients for the F+H2 reaction at 1000 and 300 K.


RESULTS: F+H2

Arrhenius plot of quantum, RPMD, and classical rate coefficients for the distinguishable atom H+H2 reaction between 200 and 1500 K (quantum results stopping at 1000 K).

Asymmetric barrier. Tc = 264 K.


RESULTS: F+H2


))(,())(,()( pQTSTRPMD qstqsTkTk κ⋅=


RESULTS: F+H2


centroid formalism



Qualitative change in behaviour for iωb

→ ..TF

The deep tunnelling regime

.)(etc

RESULTS: F+H2


centroid formalism



symm

asym

RESULTS: Mu+H2

Classical minimum energy path (blue solid line) and vibrationally adiabatic curve for the (000) state (red dashed line) for the Mu+H2 reaction vs. the natural reaction coordinate.

R. P. de Tudela*, F. J. Aoiz, Yu. V. Suleimanov*, and D. E. Manolopoulos, J. Phys. Chem. Lett. (2012) 3, 493–497. (* - equally contributed to this work)

Classical minimum energy path

vibrationally adiabatic curve for the (𝟎𝟎𝟎) state

RESULTS: Mu+H2

Classical minimum energy path (blue solid line) and vibrationally adiabatic curve for the (000) state (red dashed line) for the Mu+H2 reaction vs. the natural reaction coordinate.


tunneling is suppressed

Classical minimum energy path

vibrationally adiabatic curve for the (𝟎𝟎𝟎) state

RESULTS: Mu+H2

Classical (dashed red line) and RPMD (blue solid line) potentials of mean force for the

Mu+H2 reaction at 1000 and 200 K.

Classical (red dashed line) and RPMD (blue solid line) transmission coefficients for the Mu+H2 reaction at

1000 and 200 K.


RESULTS: Mu+H2

Arrhenius plot of QM, RPMD, classical and QCT rate coefficients for the Mu+H2 reaction between 200 and 1000 K.


RESULTS: Mu+H2


kRPMD(T) is:

• simple to compute • equal to kQTST(T) in the limit as t→0+ • bounded from above by kQTST(T) in the limit as t → ∞ • independent of s(q) in the limit as t → ∞ • reliable at high temperatures • captures the zero-point energy effect • within a factor of 2-3 of the exact QM results in the deep

quantum tunneling regime

→ complex chemical reactions!

RESULTS: H+CH4

s0(q)=0 s1(q)=0

;)(0 Rq −= ∞Rs { })(),(),(),(max)(1 rsrsrsrss δγβα=q

[ ] HHrHCr HHrHCrs

xx

xxx

)()( )()()(−−−−

−−−−=∗∗

r

Yu. V. Suleimanov, R. Collepardo-Guevara, and D. E. Manolopoulos, J. Chem. Phys. (2011) 134, 044131.

RESULTS: H+CH4

Classical (dashed) and RPMD (solid) potentials of mean force for the H+CH4 reaction at 1000 and 200 K.

Classical (dashed) and RPMD (solid) transmission coefficients for the H+CH4 reaction at 1000 and 200 K.


RESULTS: H+CH4

Arrhenius plot of various approximate quantum mechanical rate coefficients for the H+CH4 reaction between 200 and 2000 K.



RESULTS: H+CH4

Percentage deviation of the QI, centroid QTST, CVT/μOMT, and RPMD rate coefficients from the more accurate MCTDH results in the temperature range from 225 to 400 K.


A+BC % error: • QI: ~10% • QTST: ~50% • CVT: ~10-15%

kRPMD(T) is:

• simple to compute • equal to kQTST(T) in the limit as t→0+ • bounded from above by kQTST(T) in the limit as t → ∞ • independent of s(q) in the limit as t → ∞ • reliable at high temperatures • captures the zero-point energy effect • within a factor of 2-3 of the exact QM results in the deep

quantum tunneling regime

→ complex chemical reactions!

3. RPMDrate

Comp. Phys. Comm., In Press (October 2012)

http://ysuleyma.scripts.mit.edu

Incorporated algorithms Bennett-Chandler factorization of the rate coefficient. [1] Bennett, C. H. In Algorithms for Chemical Computations, ACS Symposium Series No. 46; Christofferson, R. E., Ed.; American Chemical Society: Washington DC, 1977; p 63. [2] Chandler, D. Statistical mechanics of isomerization dynamics in liquids and the transition state approximation, J. Chem. Phys. 1978, 68, 2959. Umbrella integration along the reaction coordinate. [3] Kästner, J.; Thiel, W. Bridging the gap between thermodynamic integration and umbrella sampling provides a novel analysis method: “Umbrella integration” J. Chem. Phys. 2005, 123, 144104. [4] Kästner, J.; Thiel, W. Analysis of the statistical error in umbrella sampling simulations by umbrella integration J. Chem. Phys. 2006, 124, 234106. [5] Kästner, J. Umbrella integration in two or more reaction coordinates, J. Chem. Phys. 2009, 131, 034109. RATTLE algorithm for constrained molecular dynamics simulations. [6] Andersen, H. C. Rattle: A “velocity” version of the shake algorithm for molecular dynamics calculations, J. Comput. Phys. 1983, 52, 24. Andersen thermostat. [7] Andersen, H. C. Molecular dynamics simulations at constant pressure and/or temperature, J. Chem. Phys. 1980, 72, 2384. Colored-Noise, generalized Langevin equation thermostats. [8] Ceriotti, M.; Bussi, G.; Parrinello, M. Langevin equation with colored noise for constant-temperature molecular dynamics simulations, Phys. Rev. Lett. 2009, 102, 020601. [9] Ceriotti, M.; Bussi, G.; Parrinello, M. Nuclear quantum effects in solids using a colored-noise thermostat, Phys. Rev. Lett. 2009, 103, 030603. [10] Ceriotti, M.; Bussi, G.; Parrinello, M. Colored-noise thermostats a la carte, J. Chem. Theory Comput. 2010, 6, 1170. [11] Ceriotti, M.; Manolopoulos, D. E.; Parrinello, M. Accelerating the convergence of path integral dynamics with a generalized Langevin equation, J. Chem. Phys. 2011, 134, 084104.

3. RPMDrate

Example input file for the H + H2 reaction.

Yu. V. Suleimanov, J. W. Allen, and W. H. Green, Comp. Phys. Comm. 184 (2013) 833


3. RPMDrate

Classical (solid) and centroid (dashed) potentials of mean force for the H + H2, H + CH4, OH + CH4, H + C2H6 reactions at 300 K.


3. RPMDrate

Classical (solid) and RPMD (dashed) time-dependent transmission coefficients for the H + H2, H + CH4, OH + CH4, H + C2H6 reactions at 300 K.


3. RPMDrate

Classical (solid) and RPMD (dashed) time-dependent transmission coefficients for the H + H2, H + CH4, OH + CH4, H + C2H6 reactions at 300 K.

Method k(T = 300 K), cm3molecule-1

s-1 RPMD 4.28(-17)

QI 1.15(-16)

CVT/SCT 1.44(-16)

Experiment 2.54(-17) (295 K)

• OH + HO2 • RPMDrate v2.0 • …

http://ysuleyma.scripts.mit.edu/ Yu. V. Suleimanov et al. CPC, 2013

RPMDrate

• Mu + H2 • H + H2 • D + H2 • Heμ + H2 • Cl + O3 • ….

Prof. F. Javier Aoiz Group Universidad Complutense

de Madrid

Prof. Hua Guo Group University of New Mexico

• Mu + CH4 • H + CH4 • O + CH4 • Cl + CH4 • OH + CO • ….

Dr. Albert F. Wagner Argonne National

Laboratory

Dr Yohann Scribano Universite de Montpellier

• Ion-molecule reactions (D+ + H2…)

http://ysuleyma.scripts.mit.edu/


• OH + HO2 • RPMDrate v2.0 • …

http://ysuleyma.scripts.mit.edu/ Yu. V. Suleimanov et al. CPC, 2013

RPMDrate

• Mu + H2 • H + H2 • D + H2 • Heμ + H2 • Cl + O3 • ….

Prof. F. Javier Aoiz Group Universidad Complutense

de Madrid

Prof. Hua Guo Group University of New Mexico

• Mu + CH4 • H + CH4 • O + CH4 • Cl + CH4 • OH + CO • ….

Dr. Albert F. Wagner Argonne National

Laboratory

Dr Yohann Scribano Universite de Montpellier

• Ion-molecule reactions (D+ + H2…)



• Joshua W. Allen • William H. Green

• Stephen Klippenstein • Albert Wagner • David Manolopoulos • Rosana Collepardo-

Guevara • Javier Aoiz • Ricardo Perez de Tudela

• Hua Guo

Documents

QUANTUM MECHANICAL EFFECTS FROM RING POLYMER … · 6. Dynamics of the solvated electron (2008). 7. Diffusion of H isotopes in transition metals (2009). 8. Competing quantum effects