A quantum generalization of intrinsic reaction coordinate using path integralcentroid coordinatesMotoyuki Shiga and Hiroshi Fujisaki Citation: J. Chem. Phys. 136, 184103 (2012); doi: 10.1063/1.4709723 View online: http://dx.doi.org/10.1063/1.4709723 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v136/i18 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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THE JOURNAL OF CHEMICAL PHYSICS 136, 184103 (2012)

A quantum generalization of intrinsic reaction coordinate using pathintegral centroid coordinates

Motoyuki Shiga1 and Hiroshi Fujisaki2,3

1Center for Computational Science and E-systems, Japan Atomic Energy Agency, 5-1-5 Kashiwanoha,Kashiwa, Chiba 277-8587, Japan2Department of Physics, Nippon Medical School, 2-297-2, Kosugi-cho, Nakahara, Kawasaki 211-0063, Japan3Molecular Scale Team, Integrated Simulation of Living Matter Group, Computational Science ResearchProgram, RIKEN, 2-1 Hirosawa, Wako 351-0198, Japan

(Received 15 March 2012; accepted 17 April 2012; published online 9 May 2012)

We propose a generalization of the intrinsic reaction coordinate (IRC) for quantum many-body sys-tems described in terms of the mass-weighted ring polymer centroids in the imaginary-time pathintegral theory. This novel kind of reaction coordinate, which may be called the “centroid IRC,”corresponds to the minimum free energy path connecting reactant and product states with a leastamount of reversible work applied to the center of masses of the quantum nuclei, i.e., the centroids.We provide a numerical procedure to obtain the centroid IRC based on first principles by combiningab initio path integral simulation with the string method. This approach is applied to NH3 moleculeand N2H−

5 ion as well as their deuterated isotopomers to study the importance of nuclear quan-tum effects in the intramolecular and intermolecular proton transfer reactions. We find that, in theintramolecular proton transfer (inversion) of NH3, the free energy barrier for the centroid vari-ables decreases with an amount of about 20% compared to the classical one at the room temper-ature. In the intermolecular proton transfer of N2H−

5 , the centroid IRC is largely deviated from the“classical” IRC, and the free energy barrier is reduced by the quantum effects even more drastically.© 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4709723]

I. INTRODUCTION

Ab initio molecular dynamics (AIMD) simulations havebecome a standard practice allowing theoretical insights intovast applications in chemistry.1 In principle, this approach al-lows one to follow the dynamics of a given many-body molec-ular systems on the ground state Born-Oppenheimer (BO) po-tential energy surface, which is obtained from first principlesby on-the-fly ab initio electronic structure calculations. How-ever, a challenging issue remains in the AIMD simulations inorder to explore complex chemical reactions, since the statis-tical ensemble of all the possible reaction pathways would bea formidable task to compute. This is true particularly whenthe reaction is a rare event. Furthermore, an additional diffi-culty arises for the reactions taking place in multidimensionalspace where the reaction coordinates cannot be predefined inan obvious manner. Therefore, as an alternative to this “di-rect” AIMD approach, it would be useful to develop a theoret-ical model which enables the rough estimation of complex re-action pathways, while keeping the chemical accuracy and themethodological generality of ab initio simulation technique.

Fukui’s intrinsic reaction coordinate2 (IRC) is a conve-nient concept, which is well accepted in the field of theoreticalchemistry, giving a characteristic reaction pathway on a com-plex and multidimensional potential energy surface. The con-ventional “classical” IRC corresponds to the minimum energypath in the space of mass-weighted nuclear coordinates, con-necting a reactant minimum and a product minimum by wayof a saddle point that is accessible with the least amount of BOpotential energy. According to the transition state theory,3 the

saddle point thus obtained can then be associated to the tran-sition state, from which the reaction rate k can be derived as

k = ν exp

(− �A

kBT

), (1)

where ν is the attempting frequency along the IRC, �A isthe free energy difference between the reactant and the saddlepoint, kB is the Boltzmann constant, and T is the temperature.Conventionally, �A is estimated as the sum of the BO poten-tial barrier �V and the quantum harmonic correction, whichis based on the normal mode analysis at the reactant and thesaddle point. However, assuming the classical IRC should notbe appropriate when nuclear quantum effects are important,such as the case of proton transfer reactions. The reaction pathcould be altered by zero-point vibration and tunneling motionof the proton which, in turn, could be coupled to other flexibledegrees of freedom, such as the hydrogen bond stretching. Infact, it has been argued that the classical IRC may fail to es-timate the reaction rates correctly.4 Therefore, it is desirableto extend the IRC concept to quantum systems, which will bethe focus of this paper.

The centroid coordinate in Feynman’s imaginary-time path integral (PI) formulation of quantum statisticalmechanics5, 6 is a variable that naturally describes the meanposition of a quantum particle. The PI theory establishes anisomorphism between the quantum statistics of a particle andthe classical statistics of a ring polymer coupled by harmonicsprings,7 and the centroid is the center-of-mass of the ringpolymer. To the authors’ knowledge, Gillan has been the first

0021-9606/2012/136(18)/184103/11/$30.00 © 2012 American Institute of Physics136, 184103-1

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184103-2 M. Shiga and H. Fujisaki J. Chem. Phys. 136, 184103 (2012)

one who explicitly introduced the effective free energy withrespect to the reaction coordinate described in terms of thecentroid position of the nucleus.8, 9 This idea has been suc-cessful in explaining clearly the mechanism on how the quan-tum diffusion of hydrogen in metals becomes important asthe temperature becomes low. (However, it is known to workwhere the temperature is not too low so that the deep tun-neling is unimportant; see also the development of instantonmethod which should be more suitable in the low temperatureregime.10–16) Based on this idea, Voth et al. have formulatedquantum transition state theory (QTST) on a rigid theoreti-cal foundation.17, 18 After these breakthroughs, many variantsof QTST have been proposed.19–26 In the combination with abinitio, semi-empirical, or QM/MM calculations of the BO sur-face, the QTST has been employed in various applications in-cluding the proton transfer reactions in small molecules27 andenzymatic reactions in proteins.4, 19, 28, 29 While most of theseprevious studies have discussed the reaction rates, they havenot shed much light on the reaction pathway itself. In princi-ple, the QTST requires only the information of the partitionfunctions of the reactant and transition state regions. How-ever, in order to go further to analyse the reaction mechanism,it is required to characterize the quantum reaction pathway.

In this paper, we propose a way to extend the IRC con-cept based on the idea by Gillan rooted in QTST as men-tioned above. We consider an effective free energy surfacewhich is identical to the reversible work necessary to displacethe centroid coordinates. Now, we define a pathway in whichthe reaction occurs with the least amount of the effective freeenergy via a steepest descent path with respect to the mass-weighted centroid coordinates. We demonstrate that this path-way, called the “centroid IRC,” can be computed from firstprinciples using the string method (SM)30, 31 combined withab initio path integral simulation32–36 where the BO potentialenergy and forces are computed on the fly. What distinguishesfrom the definitions of reaction path in previous studies8–11 isthat the centroid IRC is defined in the space of mass-weightedcentroid coordinates, instead of regular Cartesian coordinates.In this way, the centroid IRC passes in the direction of thetransition vector, i.e., the negative frequency eigenvector, atthe saddle point of the centroid free energy surface. In thecase of proton transfer reactions, it is reasonable that the pathcan be affected by the masses of heavy atoms whose motionsare coupled to the reactive coordinate.

The string method, which may be considered as a variantof the nudged elastic band method,37 is a convenient approachto optimize a minimum energy path (or IRC) when the tran-sition state is not known a priori, and it has been employedsuccessfully in various applications.38–42 The IRC calculationrequires much less computational effort than more sophisti-cated path ensemble methods43–45 in which all the swarm ofdynamical trajectories should be collected. This is a crucialfactor for any path search method that is to be combined withab initio electronic structure calculations. Note that, in princi-ple, the calculation of IRC does not require any specificationsto characterize the reaction coordinate, such as bond lengthsor angles, which are sometimes not trivial. This feature is incontrast to the case of constrained molecular dynamics46, 47

and metadynamics.48

This paper is organized as follows. In Sec. II, we startfrom the basic ideas of the centroid IRC. We explain in de-tail on how to obtain numerically the centroid IRC by com-bining the ab initio PI simulation and the string method. InSec. III, using the method described above, we study thequantum nature of the intramolecular proton transfer (inver-sion) reaction of ammonia molecule and the intermolecularproton transfer reaction of deprotonated ammonia dimer. Wecompare the pathways of classical and centroid IRCs for thesereactions as well as the free energy profiles along them. Wegive an analysis of the free energy components which areresponsible in these reactions. These results are interpretedby means of a simple inverted harmonic model which is de-scribed in the Appendix. In Sec. IV, we summarize our workand discuss the future prospects.

II. THEORY

A. Intrinsic reaction coordinate

The IRC is the steepest descent path in the mass-weighted coordinate space of N atoms, {Q1, . . . , QN}. In thisspace, the Cartesian coordinate of each atom I, RI = (XI, YI,ZI), is weighted by its nuclear mass MI as

QI =√

MI RI . (2)

The IRC is a downhill trajectory from a transition state to anequilibrium state following the differential equation,

dQI

dτ= FI

||F|| , (3)

where FI ≡ − ∂V∂QI

is the force associated with the BO poten-

tial V whose norm is ||F|| ≡√∑N

I=1 |FI |2 and τ is the metric

of the path having the unit of (mass)1/2 × (length). At the tran-sition state, the IRC passes in the direction of the transitionvector, i.e., the eigenvector associated with the only negativeeigenvalue of the Hessian matrix at the saddle point.

Fukui has pointed out that IRC corresponds to the nuclearmotion from a given point along the path with an infinitesimalvelocity. This view can be reinterpreted as the zero tempera-ture limit of the underdamped Langevin equation describingBrownian dynamics,49

MI

dRI

dt= − 1

γ

∂V

∂RI

+√

2kBT MI

γf (t), (4)

where γ is the friction coefficient. For T → 0, the effect ofrandom force f(t) vanishes, and therefore

dQI

dt= 1

γFI , (5)

which is identical to Eq. (3) except for the metric conversiondt/γ = dτ /||F||. Note that this identity holds only in the mass-weighted coordinate space.

B. Centroid free energy and centroid IRC

Now, our purpose is to construct a model in parallel tothe IRC concept for systems that obey quantum statistical

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184103-3 M. Shiga and H. Fujisaki J. Chem. Phys. 136, 184103 (2012)

mechanics, which may be called the “centroid IRC.” In theimaginary-time path integral theory,5, 6, 50, 51 the partition func-tion of N quantum nuclei can be expressed in terms of themass-weighted coordinates Q as

Z = limP→∞

[N∏

I=1

(P

2πβ¯2

) 3P2∫

dQ(1)I · · ·

∫dQ(P )

I

]

× exp (−βVeff [Q]) , (6)

where β = 1/kBT is the inverse temperature and P is the Trot-ter number. The effective potential is

Veff [Q] ≡P∑

s=1

[N∑

I=1

P

2¯2β2

(Q(s+1)

I − Q(s)I

)2

+ 1

PV(Q(s)

1 , . . . , Q(s)N

)](7)

with the cyclic boundary condition, Q(P+1)I = Q(1)

I . This hasthe same form as that of the classical N ring polymers withP beads interacting by the spring constant P

¯2β2 , where thefluctuation of the polymers reflects the Heisenberg uncer-tainty principle. It is worthwhile to note that the mass depen-dence in Eqs. (6) and (7) is now completely absorbed into themass-weighted coordinates Q(s)

I = √MI R(s)

I . Based on thisclassical isomorphism, Gillan8, 9 has introduced a free energy,in the context of QTST which is associated with the reactionrates of quantum systems. This free energy corresponds ex-actly to the reversible work necessary to move the center-of-mass positions, i.e., the centroids, of the quantum nuclei, eachof which is described as a ring polymer. Here, we define sucha free energy surface with respect to the mass-weighted cen-troid coordinates

QcentI = 1

P

P∑S=1

Q(s)I (8)

from a reference point to a given point in configuration space,while all the other degrees of freedom of the ring polymerfluctuation are integrated out. We thus define the centroid freeenergy as

A (q1, . . . , qN ) ≡ −β−1 log ρ (q1, . . . , qN ) , (9)

where the centroid distribution function

ρ (q1, . . . , qN ) ≡⟨

N∏I=1

δ(Qcent

I − qI

)⟩pi

≡ Z−1 limP→∞

[N∏

I=1

(P

2πβ¯2

) 3P2∫

dQ(1)I · · ·

∫dQ(P )

I

]

× exp (−βVeff [Q])

[N∏

I=1

δ(Qcent

I − qI

)](10)

describes the probability density of finding the centroid coor-dinates, respectively, at the positions (q1, . . . , qN). Now we areable to identify the stationary points on the surface A, with thezero-gradient condition ∂A

∂qI= 0 for all I. It will be called the

equilibrium state if the eigenvalues of Hessian matrix ∂2A∂qI ∂qJ

are all positive and the transition state if only one of them is

negative. The transition vector at the saddle point can also bedefined as the direction of the eigenvector of the Hessian ma-trix with a negative eigenvalue. Therefore, the centroid IRCcan be naturally defined as the steepest descent pathway start-ing from the transition state, moving toward the direction ofsuch a transition vector, and ending at the equilibrium state onthe surface A.

It is useful to show that the centroid IRC reduces to theconventional IRC in the classical limit ¯→ 0 or in the high-temperature limit β → 0. In these limits, the strong springconstant P

¯2β2 makes the ring polymer shrink toward its center

of mass, Q(s)I = Qcent

I = qI for each nucleus I, otherwise theeffective potential in Eq. (7) becomes too high. In this case,we recover the classical probability density

ρcl (q1, . . . , qN ) =⟨

N∏I=1

δ (QI − qI )

⟩cl

= Z−1cl

(1

2πβ¯2

) 3N2

exp (−βV (q1, . . . , qN )) (11)

and the classical free energy

Acl (q1, . . . , qN ) ≡ −β−1 log ρcl (q1, . . . , qN )

= V (q1, . . . , qN ) + const (12)

which is equal to the potential energy V apart from aredundant constant. Therefore, in the classical and high-temperature limit, the steepest descent path on A is the sameas that on V , i.e., the conventional “classical” IRC. Note thatthis result can also be directly obtained by setting to P = 1 inEqs. (6)–(10), which is useful for actual calculations.

A peculiar aspect of the centroid IRCs is the mass andtemperature dependence. The centroid IRCs for different iso-topomers are completely different, and they pass through dif-ferent equilibrium and transition states. The centroid IRCs aredifferent at different temperatures because the centroid freeenergy A depends on temperature. In contrast, the classicalIRCs for different isotopomers are different in pathways butthey pass through the same equilibrium and transition states.The classical IRCs obviously do not depend on the tempera-ture because it is just a topological property of the potentialenergy surface.

C. String method for the centroid IRC

The force felt by the centroid variables is defined by thenegative gradient of Eq. (9),

FI ≡ − ∂A

∂qI

=−⟨(

∂Veff∂qI

)∏NJ=1 δ

(Qcent

J − qJ

)⟩pi⟨∏N

J=1 δ(Qcent

J − qJ

)⟩pi

= −⟨∂Veff

∂qI

⟩pi at Qcent=q

= −⟨

1

P

P∑s=1

∂V

∂Q(s)I

⟩pi at Qcent=q

,

(13)

where the subscript pi represents the path integral averageusing the effective potential, Eq. (7). This can be called the

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184103-4 M. Shiga and H. Fujisaki J. Chem. Phys. 136, 184103 (2012)

FIG. 1. A flow chart of the calculation of centroid IRC.

“centroid mean force” because it corresponds to the averageforce acting on the ring polymer whose centroid is fixed. Onemight think that the centroid IRC can be obtained by the nu-merical integration of Eq. (13) starting from a transition state.However, it is difficult to locate the transition state because thecentroid mean force is evaluated from the statistical average,leading to a convergence problem due to a small statisticalerror.

The string method developed by E et al.30, 31 is an effi-cient approach to compute minimum energy path in multi-dimensional space without a priori knowledge of the equilib-rium and transition states. This method enables one to look foran optimal path described by evenly spaced discrete images inan automatic manner using the information of the force actingon the all the images. The algorithm tailored for the presentapplication is as follows, see also Fig. 1. (i) Create an initialguess of the series of images (1 ≤ α ≤ L) of the system com-posed of 3N centroid coordinates. A good guess would be theimages along the classical IRC. (ii) For each image α, the cen-troid mean force, FI(τα), is computed by a sufficiently long PIsimulation, whereas all the centroid coordinates, qI(τα), arekept fixed. In the case of path integral hybrid Monte Carlo(PIHMC),52–55 the statistical accuracy is relevant to the num-ber of HMC steps, Nhmc, which is the product of the numberof MD steps per trial move Nmd/mc and the number of trialmoves Nmc. In the present work, Nmd/mc is set to 1. The effi-ciency can be controlled by the MD step size dtmd such thatthe acceptance ratio is about 50%–70%. (iii) For each image,the centroid coordinates are shifted in the steepest descentdirection

qI (τα) ←− qI (τα) + FI (τα)

||F(τα)||dτsm (14)

where dτ sm represents the step parameter of the string methodto control the shift of the centroids for all the images. (iv) Allthe images except the ones at both ends (2 ≤ α ≤ L − 1) arerearranged to be evenly spaced with respect to the arc lengthin mass-weighted coordinates along the path,

�α =∫ τα+1

τα

dτ

√√√√ N∑I=1

|qI (τ ) − qI (τα)|2 = const, (15)

where the integral can be evaluated from the spline interpola-tion of the coordinates of images qI(τα) (1 ≤ α ≤ L). Accord-ing to this procedure, the images eventually converge to thepositions along the centroid IRC by iterating Steps (ii)–(iv),for Nsm times, and the obtained path should be stable if dτ sm

is taken to be small enough. The both end images convergeto the equilibrium states of the reactant and the product sincethey simply follow the steepest descent trajectory as long astheir initial guesses are inside the relevant basins. It can beshown that the solution, i.e., the evenly spaced images alongthe IRC, is stable in the above procedure, because the positionof images will be unchanged only for this solution. As longas dτ sm is small enough, the evenly spaced images along theIRC will be shifted in Step (iii) in parallel to the IRC, andthus, the course of the path connecting these images will beunchanged, and therefore, they will be shifted right back tothe original position in Step (iv).

The string method has been implemented into the in-house version of the PIMD++ code.56 This code has theinterface with the electronic structure program packageTURBOMOLE57 allowing “on-the-fly” ab initio PIHMC cal-culations on the BO potential energy surface. To evaluate thecentroid mean forces, ab initio PIHMC calculations with fixedcentroids have been carried out in parallel with respect to allthe images.

In this paper, our method is applied to two simple chemi-cal reactions, the inversion reaction of NH3 molecule and theproton transfer reaction of N2H−

5 ion. For 41 images, ab initioPIHMC simulations based on the second-order Trotter expan-sion have been carried out using the similar method as de-scribed in Ref. 55 with the normal mode representation of thering polymer. The only difference is that the fictitious massesof the centroids were set extremely heavy to quench the mo-tion of the centroids.

The ab initio electronic structure calculations were basedon Hartree-Fock (HF) theory and resolution-of-the-identityapproximation of second-order Møller-Plesset perturbationtheory (RIMP2 (Ref. 58)) with the SV(P) basis set59 forNH3 and N2H−

5 , respectively. These settings were chosenaccording to the data of static ab initio calculations. Thepotential barrier of NH3 using HF, density functional the-ory (DFT) with B3LYP functional (Ref. 60), RIMP2, andRICC2 (Ref. 61) theories with the SV(P) basis set arefound to be �V = 5.6, 5.1, 5.4, and 5.4 kcal/mol, respec-tively, and they all agree well with the RICC2 result withthe larger aug-cc-pVTZ basis set,62 �V = 4.9 kcal/mol.For N2H−

5 , the results of HF, DFT(B3LYP), RIMP2, andRICC2 theories with SV(P) basis set are �V = 4.9, 0.9,2.1, and 2.2 kcal/mol, respectively, while the RICC2 resultwith the larger aug-cc-pVTZ basis set is �V = 2.9 kcal/mol.In this case, as the HF calculation overestimates whilethe DFT(B3LYP) underestimates �V , we decided to useRIMP2.

The detailed information for setting up the simulationsis provided in Table I. For each case, the total number ofab initio calculations needed is PLNhmcNsm. For one sim-ulation of N2H−

5 , this amounts to 16 × 41 × 500 × 50= 16 400 000 iterations, and therefore, it took about twomonths using a quadcore personal computer (PC) machine.

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184103-5 M. Shiga and H. Fujisaki J. Chem. Phys. 136, 184103 (2012)

TABLE I. Simulation setups.

Parameter Symbol NH3 N2H−5

Number of images L 41 41Trotter number P 1, 16, 32, 64 1, 16MD step size (fs) dtmd 2.0 2.5MD steps per trial move Nmd/mc 1 1HMC cycles Nhmc 500 500SM step size (a.u.) τ sm 0.1 0.1SM updates Nsm 40 50Temperature (K) T 300, 600, 900, 1200 300

Electronic structure theory V HF/SV(P) RIMP2/SV(P)

Note that the calculations for beads or images can be easilyparallelized and the computation can be much quicker whenwe use a PC-cluster system.

III. RESULTS AND DISCUSSIONS

A. Ammonia molecule

Before carrying out the production runs, we have checkedthe statistical convergence of the centroid mean force since itis relevant to the numerical accuracy of the centroid IRC aswell as the centroid free energy. In other words, the number ofPIHMC steps, Nhmc, should be set adequately so as to estimatethe centroid mean force with a good balance between accu-racy and efficiency. For this purpose, we have examined therunning average of the approximate free energy differencesamong the neighboring images,

dAα ∼ −N∑

I=1

FI (τα) · (qI (τα+1) − qI (τα)) . (16)

The statistical convergence of {dAα} measures that of the cen-troid mean forces {FI(τα)} for a given set of the centroid con-figurations {qI(τα)}. Figure 2 shows the running average ofdAα for each of the 41 images of NH3 inversion reaction afterthe convergence to the centroid IRC. It can be seen that all ofthem are numerically stable within about 500 PIHMC steps,although the complete convergence is reached after few thou-sand steps. As a compromise, we have chosen Nhmc = 500 inthe present study.

The centroid IRC of the inversion reaction of NH3 is ob-tained after all the centroid coordinates of all the images havebeen optimized by the string method. In Fig. 3, we show theequilibrium and transition state structures obtained from thefirst (or the last) image and the central (21st) image, respec-tively. In this case, it is found that the centroid IRC differsonly slightly from the classical IRC. The NH distance of thecentroid IRC is slightly longer in both the equilibrium andtransition state structures, compared to those of the classicalIRC.

In Figs. 4–6, we show the centroid free energy profile cal-culated by means of thermodynamic integration (TI) method,

Aα = A1 −∫ τα

τ1

N∑I=1

FI (τ ) · dqI (τ )

dτdτ, (17)

-0.04

-0.02

0

0.02

0.04

0 2000 4000 6000 8000 10000 12000 14000

- dA

[har

tree

/boh

r]

Number of steps

FIG. 2. The running averages of the free energy difference dAα from thecalculation of NH3 at 300 K with P = 16. Each line represents each of the 41images.

where the centroid mean force FI(τ ) and dqI (τ )dτ

are obtainedfrom the spline interpolation of those of the images 1 ≤ α

≤ L along the centroid IRC. This profile has the minimum atthe first/last image corresponding to the equilibrium, and themaximum at the 21st image corresponding to the transitionstate.

We have also calculated the classical IRC by the stringmethod with 41 images. It has been confirmed that the resulthas perfectly matched with the conventional algorithm, i.e.,the steepest descent trajectory from the transition state ob-tained by the numerical integration of Eq. (3). In the classicalIRC, the potential energy difference between the equilibriumand the transition state is 5.62 kcal/mol. This value has beenreproduced by means of the TI method, �V = V ts − V eq

≡ V ({qI (τ21)}) − V ({qI (τ1)}) along the IRC pathway ob-tained by the string method with 41 images. Therefore, as theTI works quite accurately with 41 images for the potential en-ergy of classical IRC, we could expect that this is also thecase for the free energy of the centroid IRC, i.e., �A = Ats

− Aeq ≡ A({qI(τ 21)}) − A({qI(τ 1)}).In Fig. 4, we show the centroid free energy profile of NH3

inversion reaction at 300 K with different Trotter numbers (thenumber of beads). The free energy profiles are found nearlythe same with the Trotter number P = 16 or larger. Therefore,we employ P = 16 for the rest of this study. We believe that

FIG. 3. Molecular structures of (a) the equilibrium state and (b) the transitionstate. The NH bond length and the HHH–N out-of-plane distances of theclassical IRC of NH3 (black numbers) and the centroid IRC of NH3 (rednumbers) and ND3 (blue numbers) calculated at 300 K with P = 16 are alsoshown.

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184103-6 M. Shiga and H. Fujisaki J. Chem. Phys. 136, 184103 (2012)

0 10 20 30 400

1

2

3

4

5

6

image no.

A[k

cal/

mo

l]64 beads32 beads16 beadsclassical

FIG. 4. Trotter number dependence on the free energy profile of NH3 in-version reaction. Potential energy profile of classical IRC (black line) withP = 1, and the centroid free energy profile of optimized centroid IRC at300 K with P = 16 (red line), P = 32 (blue line), and P = 64 (green circles)are shown as functions of image number.

P = 16 is sufficient to study the deuterated systems andsystems at higher temperatures where quantum effects aresmaller.

In Fig. 5, we compare the centroid free energy profilesalong the centroid IRC at different temperatures. We can seethat the free energy barrier is lower than that of the classi-cal IRC due to the quantum effects. However, the differencediminishes as the temperature is higher with less quantum ef-fects. The quantum effect lowers the barrier by 0.8 kcal/molat 300 K, while the quantum effect is very small at 1200 K. InFig. 6, we show the free energy profiles for the isotopomers,NDH2, ND2H, and ND3. As many proton are substituted bydeuterons, the free energy barrier gets higher. This is reason-able because the deuterated systems tend to behave closerto the corresponding classical systems, since deuteron has alarger mass and thus less quantum effects than proton.

In Table II, we summarize the free energy barriers of NH3

at different temperatures and the deuterated isotopomers at300 K. These results can be compared with the free energiesbased on the simple harmonic analysis (HA) (Ref. 63)

�Aha = V ts +3N−7∑i=1

kT log

[2 sinh

(¯ωts

i

2kT

)]

−V eq −3N−6∑i=1

kT log

[2 sinh

(¯ω

eqi

2kT

)], (18)

0 10 20 30 400

2

4

6

image no.

A[k

cal/

mo

l]

300 K600 K900 K1200 Kclassical

FIG. 5. Temperature dependence on the free energy profile of NH3 inversionreaction. Potential energy profile of classical IRC (black line) and the centroidfree energy profile of optimized centroid IRC at 300 K (red line), 600 K(green line), 900 K (blue line), and 1200 (purple) with P = 16 are shown asfunctions of image number.

0 10 20 30 400

2

4

6

A[k

cal/

mo

l]

image no.

NH3NH2DNHD2ND3classical NH3

FIG. 6. Isotope effect on free energy profile of NH3 inversion reaction. Po-tential energy profile of classical IRC (black line) and the centroid free en-ergy profile of optimized centroid IRC of NH3 (red line), NH2D (green line),NHD2 (blue line), and ND3 (purple line) at 300 K with P = 16 are shown asfunctions of image number.

where ωeqi and ωts

i are the normal mode frequencies of theequilibrium and transition states, respectively, of the classicalIRC. It could be seen that HA is qualitatively correct; it givesthe same tendency as the free energy barrier calculated alongthe centroid IRC. In the quantitative sense, however, the freeenergy barrier is underestimated in HA, which implies thatHA overestimates the quantum effects in the present case.

An advantage of the centroid IRC calculation is that itcan estimate the parameters in QTST, Eq. (1). In Table II, welist the free energy barrier, �A, as well as the real frequency atthe equilibrium state, νeq, and the imaginary frequency at thetransition state, ν ts, associated with the reactive mode. Thevalues of νeq and ν ts have been obtained by fitting the cen-troid free energy profile along the centroid IRC direction toa quadratic function around the 1st and 21st images, respec-tively. Since the quantum effects make the slope of the profile

TABLE II. Results for inversion reaction of ammonia.

Quantity �A νeq νts

(unit) (kcal/mol) (cm−1) (cm−1)

Centroid IRC:NH3, 300 K 4.8 950a 850ib

NH3, 600 K 5.4 950a 900ib

NH3, 900 K 5.5 1000a 900ib

NH3, 1200 K 5.6 1000a 900ib

NH2D, 300 K 5.0 800a 750ib

NHD2, 300 K 5.1 750a 700ib

ND3, 300 K 5.2 650a 650ib

Classical IRC:NH3 5.6 1151c 915id

Harmonic analysis:NH3, 0 K 4.4 1151c 915id

NH3, 300 K 4.4 1151c 915id

NH3, 600 K 4.5 1151c 915id

NH3, 900 K 4.7 1151c 915id

NH3, 1200 K 5.0 1151c 915id

NH2D, 300 K 4.5 1061c 824id

NHD2, 300 K 4.6 956c 749id

ND3, 300 K 4.8 866c 681id

aFit to harmonic function at the equilibrium state of centroid IRC.bFit to harmonic function at the transition state of centroid IRC.cNormal mode analysis at the equilibrium state of classical IRC.dNormal mode analysis at the transition state of classical IRC.

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184103-7 M. Shiga and H. Fujisaki J. Chem. Phys. 136, 184103 (2012)

FIG. 7. Molecular structures of (a) the equilibrium state and (b) the transitionstate. The NH*, N· · ·H*, and N· · ·N distances of the classical IRC of N2H−

5(black numbers) and the centroid IRC of N2H−

5 (red numbers) and N2D−5

(blue numbers) calculated at 300 K with P = 16 are also shown.

more gentle, the absolute values |νeq| and |ν ts| are smaller asthe temperature is lower. This temperature effect is not in-cluded in the conventional HA where the |νeq| and |ν ts| valuesare constant. The |νeq| and |ν ts| values are smaller by deuteronsubstitution, since the normal mode frequencies depend on thenuclear mass. This tendency of isotope effect is similar to thecase of HA, but the frequencies are found to be systematicallylower than the results of HA.

B. Deprotonated ammonia dimer ion

Figure 7 shows the equilibrium and transition state struc-tures of the intermolecular proton transfer reaction of theN2H−

5 and D2N−5 ions at 300 K. In the case of the N2H−

5 ion,the equilibrium structure of the centroid free energy exhibitsnotable difference from the corresponding classical one. TheNN separation is much shorter and the NH* bond (where H*refers to the bonded proton) is much longer in the quantumcase compared to the classical case. This is a typical natureof the low barrier hydrogen bonds that the quantum effectsshift the bonded proton towards the center of hydrogen bondand shorten the distance between the heavy atoms attached toit.64–67 This tendency is also observed for the N2D−

5 ion, butthe quantum effects are smaller compared to the N2H−

5 ion. InFig. 8, the classical and the centroid IRC are plotted as a func-tion of the N–N separation rNN and the difference of N–H*distances, rNH∗ − rNH∗ . In this plot, the arc length of the cen-troid IRC is shorter than that of the classical IRC, because theequilibrium structure is already closer to the transition statestructure. For the same reason, the arc length of the centroidIRC of the N2H−

5 ion is shorter than that of the N2D−5 ion.

-0.5 0 0.5

2.7

2.8

2.9

rNH* - rH*N [Å]

r NN

[Å]

N2H5- classical IR C

N2D5- centroid IRC

N2H5- centroid IRC

FIG. 8. The trajectory of the classical IRC of N2H−5 (black line), the centroid

IRC of N2H−5 at 300 K (red line), and the centroid IRC of N2D−

5 at 300 K(blue line), projected onto two-dimensional space with respect to the N· · ·Ndistance, rNN, and the difference of the NH* distances, rNH∗ − rH∗N.

0 10 20 30 40

0

1

2

image no.

A[k

cal/

mo

l]

N2H5-

N2D5-

N2H5- classical

N2H5- non- opt

FIG. 9. Profiles of the N2H−5 proton transfer reaction are shown as functions

of image number. The potential energy profile of classical IRC (black line)and the centroid free energy profile of optimized centroid IRCs of N2H−

5 (redline) and N2D−

5 (blue line) is calculated at 300 K with P = 16. The centroidfree energy profile on the classical unoptimized IRC of N2H−

5 (green line) iscalculated at 300 K with P = 16.

In Fig. 9, we show the centroid free energy profiles forthe N2H−

5 and N2D−5 ions. We can see that the free energy

barrier of the centroid IRC is much lower than that of the clas-sical IRC, by 1.7 kcal/mol and 1.2 kcal/mol for the N2H−

5 andN2D−

5 ions, respectively. This reduction of free energy barrierdue to the quantum effects is more drastic than the case ofNH3 and their isotopomers in Fig. 6. Here, it might be inter-esting to study the large deviation of the centroid IRC fromthe classical one. So, we computed the non-optimized (“non-opt”) centroid free energy profile where the centroid imagesare fixed to the classical IRC without optimizing the reactionpath. As plotted in Fig. 9, such a non-optimized profile re-sults in a wavy curve, and it is quite different from that forthe centroid IRC. This suggests that the centroid IRC shouldbe optimized to obtain the free energy properly. Here, thestarting points (e.g., the equilibrium structures) for the ther-modynamic integration of the optimized and non-optimizedpathways are different. As we have not computed the freeenergy difference between these starting points, we cannotsimply compare the barrier heights of the optimized and non-optimized pathways.

The values of the free energy barrier along with the fre-quencies ν ts and νeq for the N2H−

5 and N2D−5 ions are listed in

Table III. We can see that the values of |νeq| and |ν ts| calcu-lated by the centroid IRC are smaller than those by the clas-sical IRC, reflecting the quantum effects on the free energycurvature. In particular, |ν ts| from the centroid IRC is foundto be much smaller than that from the classical IRC due to thedrastic damping of the free energy profile.

In order to study the centroid free energy Aα in more de-tail, we decompose it into three different contributions fromthe kinetic energy, the potential energy, and the entropy. Thekinetic energy part can be calculated using the form of thevirial estimator68, 69

Kα=3NkBT

2+ 1

2P

⟨P∑

s=1

N∑I=1

(Q(s)

I −QcentI

)· ∂V

∂Q(s)I

⟩pi at Qcent=q(τα)

(19)

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184103-8 M. Shiga and H. Fujisaki J. Chem. Phys. 136, 184103 (2012)

TABLE III. Results for proton transfer reaction of deprotonated ammonia dimer.

Quantity �A �K �V −T�S νeq νts

(unit) (kcal/mol) (kcal/mol) (kcal/mol) (kcal/mol) (cm−1) (cm−1)Centroid IRC:N2H−

5 , 300 K 0.4 − 0.7 0.7 0.4 300a 350ib

N2D−5 , 300 K 0.9 − 0.7 1.2 0.5 250a 500ib

N2H−5 , 300 K (non-opt) 0.3 − 0.9 0.8 0.4 200ia 150ib

Classical IRC:N2H−

5 2.1 0.0 2.1 0.0 350a 1000ib

244c, 2342d 1006ie

Harmonic analysis:N2H−

5 , 0 K − 0.4 244c, 2342d 1006ie

N2H−5 , 300 K 0.0 244c, 2342d 1006ie

N2D−5 , 300 K 0.8 226c, 1718d 723ie

aFit to harmonic function at the equilibrium state of the centroid IRC.bFit to harmonic function at the transition state of the centroid IRC.cNormal mode analysis at the equilibrium state of classical IRC (N-N stretching mode).dNormal mode analysis at the equilibrium state of classical IRC (N-H* or N-D* stretching mode).eNormal mode analysis at the transition state of classical IRC.

and the potential energy part can be calculated from

Vα =⟨

1

P

P∑s=1

V(

Q(s)I

)⟩pi at Qcent=q(τα)

. (20)

The sum corresponds to the internal energy, Uα = Kα + Vα .These formulas are the same as those used in the conventionalpath integral simulations except the condition that the centroidcoordinate, Qcent, is constrained to that of each image, q(τα).We can derive Eq. (19) from the definition of Aα in Eq. (9)using a thermodynamic relation Kα ≡ −∑N

i=1 mi∂Aα

∂miand the

integration by parts. Equation (20) is derived from anotherthermodynamic relation Uα ≡ ∂

∂β(βAα). Defining the entropy

as Sα ≡ − ∂Aα

∂T, the remaining part can be estimated from the

relation

−T Sα = Aα − Kα − Vα. (21)

In Figs. 10(a) and 10(b), we show the calculated resultsfor each component of the centroid free energy, Kα , Vα , and−TSα , for the N2H−

5 and N2D−5 ions, respectively. Although

the statistical noise of each component is relatively large, atleast the qualitative change from the equilibrium to the tran-sition state is clear from this figure. The kinetic energy Kα

0 10 20 30 40

0

1

image no.

ener

gy[k

cal/

mo

l]

VK-TSA

(a)

0 10 20 30 40

0

1

image no.

ener

gy[k

cal/

mo

l]

VK-TSA

(b)

FIG. 10. The profiles of free energy components of (a) N2H−5 proton trans-

fer and (b) N2D−5 deuteron transfer reaction as functions of image number α.

Total free energy �Aα (black), kinetic energy contribution �Kα (green), po-tential energy contribution �Vα (red), and the entropy contribution −T�Sα

(blue).

decreases considerably as approaching the transition state ofthe centroid IRC. This is in contrast to the case of the clas-sical IRC in which the kinetic energy has a constant valueof 3NkBT/2. Meanwhile, the potential energy Vα increases asapproaching the transition state, but the amount of increaseis not as much as that in the case of the classical IRC. Theentropy contribution −TSα increases slightly as approachingthe transition state. The relative changes from the equilibriumto the transition state, �Kα , �Vα , and −T�Sα , are listed inTable III. It is interesting that �Kα and −T�Sα do not differmuch between N2H−

5 and N2D−5 , and therefore, the �Vα is

mainly responsible for the isotope effect on the centroid freeenergy barrier in this case.

In order to explain the results obtained above, we showthe analytical expression for each of the centroid free energycontributions in one-dimensional harmonic model in the Ap-pendix. The quantum harmonic correction to the free energy,Kqhc, Vqhc, and −TSqhc, at the temperature T = 300 K is de-picted in Fig. 11 as a function of the frequency ω of the har-monic potential. We can see that the values of Kqhc and Vqhc

are positive for the real values of ω corresponding to the equi-librium state, while they are negative for the imaginary values

-1000 0 1000

-0.5

0

0.5

ω [cm-1]

ener

gy[k

cal/

mo

l]

iω [cm-1]

A(iω) A(ω)U(iω) U(ω)-TS(iω) -TS(ω)K (iω), V (iω) K (ω), V (ω)

frequen cy

FIG. 11. Quantum harmonic corrections of one-dimensional harmonic os-cillator to the free energy, Aqhc (blue), the potential and kinetic energy con-tributions, Vqhc and Kqhc (light blue), and the entropy contribution, −TSqhc(green), as a function of frequency, ω. The solid and dotted lines correspondto the real and imaginary frequency regions, respectively.

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184103-9 M. Shiga and H. Fujisaki J. Chem. Phys. 136, 184103 (2012)

of ω corresponding to the transition state. Thus, the quantumeffects have a negative contribution to �K and �V . In con-trast, the value of −TSqhc is negative for real ω, while it ispositive for imaginary ω, and thus the quantum effects havea positive contribution to −T�S. Actually, this is what wehave found in the results of the N2H−

5 and N2D−5 ions. Note,

of course, that the one-dimensional harmonic model is toosimplified, and it is useful only for qualitative discussions. Asevere limitation of this model is that the free energy curvediverges at the imaginary frequency ω = 2π i/β¯, which cor-responds to about 1300i cm−1 when the temperature is 300 K.

IV. CONCLUDING REMARKS

In this paper, we have introduced the centroid IRC whichis able to characterize the chemical reactions of quantummany-body systems. The centroid IRC is defined as the min-imum free energy path in the space of mass-weighted cen-troid coordinates in the imaginary-time path integral theory. Anumerical algorithm has been proposed in order to calculatethe centroid IRC based on first principles by combining theab initio path integral simulation and the string method.

Using this method, we have studied the quantum natureof the intramolecular and intermolecular proton transfer re-actions of the NH3 molecule and the N2H−

5 ion, respectively,as well as their deuterated isotopomers. Comparing the cen-troid IRC to the classical IRC, we found that the structuralaspects of the intramolecular proton transfer of NH3 are sim-ilar to those of the classical system. However, due to the nu-clear quantum effects, the barrier height and the curvature ofthe free energy profile of NH3 are decreased considerably forthe centroid IRC compared to the classical one. The nuclearquantum effect is diminished by the deuteron substitution orby the increase of temperature, as expected.

We have found that the quantum effects have muchgreater influence on the inter-molecular proton transfer ofN2H−

5 . There is a huge drop of the free energy barrier calcu-lated from the centroid IRC compared to the classical counter-part. According to the analysis of free energy decomposition,we have seen that due to the quantum effects not only the po-tential energy but also the kinetic energy is decreased in thetransition state region. We have also seen that the course ofthe centroid IRC of N2H−

5 is much shorter in distance than thecentroid IRC of N2D−

5 as well as the classical IRC of N2H−5 .

This can be ascribed to the fact that the quantum effects makethe N· · ·N distance shorter and NH* distance longer at theequilibrium, approaching toward the transition state. This ob-servation should be relevant to the Ubbelohde effect, the lat-tice expansion of hydrogen bonded crystals upon deuteronsubstitution,70–73 as have been mentioned in previous theo-retical studies.64, 65, 67

The classical and centroid IRC can be generally used forreactions between the potential and free energy minima, re-spectively, where the reaction coordinate is defined automati-cally in full dimensional space. Using the mass-weighted co-ordinates, it is possible to extract the parameters for QTSTsuch as the free energy barrier height, �A, and the curvaturesat the equilibrium and transition states, νeq and ν ts. In these re-spects, it is quite different from the PI simulations combined

with the constraints or biased potentials in which the reac-tion coordinate should be predefined in a chemically intuitivemanner.27, 74

On the other hand, there should be a room for modifi-cation to the present IRC approach for the applications in thecondensed phases, e.g., chemical reactions in solutions, wherethe reaction coordinates must be defined in a reduced dimen-sion. In such cases, we could divide the whole system into theactive region composed of N solute atoms and the inactive re-gion composed of N′ solvent atoms, the latter is the degreesof freedom being integrated out. Then, the average influenceof inactive region can be absorbed into the effective potential

Veff[Q]=−β−1 log

[N+N ′∏

I=N+1

(P

2πβ¯2

) 3P2∫

dQ(1)I · · ·

∫dQ(P )

I

]

× exp

{−β

P∑s=1

[N+N ′∑I=1

P

2¯2β2

(Q(s+1)

I − Q(s)I

)2

+ 1

PV(Q(s)

1 , . . . , Q(s)N+N ′

)]}. (22)

The IRC on this surface will be the extension of the free en-ergy gradient method75, 76 to quantum systems.

In this work we have only treated the gas phase sys-tems due to the limitation in our present computer environ-ment. However, it should be noted that our method will beapplicable, in principle, to larger molecular systems with theuse of highly parallel computers. Instead of a full ab initioapproach, the QM/MM method utilising the multiple timescale method77 may help reducing the computational cost,for instance, in the applications to proton transfer reactionsin proteins and enzymes, which is currently an active areaof research.

ACKNOWLEDGMENTS

The authors thank the Alexander von Humboldt Foun-dation for providing the opportunity to start our collab-oration at the AvH meeting in Berlin, 2007. This workwas also supported by Grants-in-Aid for Scientific Research(KAKENHI) 22018024, 22750023, 23350010 (M.S.), and22540421 (H.F.), and by funds for Research and Developmentof the Next-Generation Integrated Simulation of Living Mat-ter, a part of the Development and Use of the Next-GenerationSupercomputer Project of the Ministry of Education, Culture,Sports, Science, and Technology (MEXT).

APPENDIX: ONE DIMENSIONAL HARMONIC MODEL

Let us consider a quantum particle moving on a one-dimensional harmonic potential V (x) = m

2 ω2x2 whose centerof mass is constrained to the origin, x = 0. The quantum par-tition function for this system can be described in terms of thepath integral representation as

Z=∮

D′x(u) exp

[− 1

¯

∫ β¯

0

{m

2(x(u))2+m

2ω2(x(u))2

}du

],

(A1)

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184103-10 M. Shiga and H. Fujisaki J. Chem. Phys. 136, 184103 (2012)

where∮D′x(u) stands for the closed path integral in the pres-

ence of a constraint

1

β¯

∫ β¯

0x(u)du = 0. (A2)

Using the identity for delta function,

δ

(1

β¯

∫ β¯

0x(u)du

)= 1

2π

∫ ∞

−∞dk exp

(ik

β¯

∫ β¯

0x(u)du

),

(A3)

we obtain an expression in terms of the closed path inte-gral

∮Dx(u) where the constraint is now included in the

integrand,

Z =∫ ∞

−∞

dk

2π

∮Dx(u) exp

[− 1

¯

∫ β¯

0

{m

2(x(u))2

+m

2ω2(x(u))2 − i

kx(u)

β

}du

]. (A4)

This is similar to the diagonal matrix element of theFeynman kernel of a harmonic oscillator to which a constantforce, f = ik/β, is applied,

K(z, z) =∫ z

z

Dx(u) exp

[− 1

¯

∫ β¯

0

{m

2(x(u))2

+m

2ω2(x(u))2 − i

kx(u)

β

}du

]

=√

mω

2π¯ sinh β¯ωexp

[−mω(cosh β¯ω − 1)

¯ sinh β¯ω

×(

z + k

imω2β

)2

− k2

2mω2β

]. (A5)

Using this formula, we obtain the quantum partition function

Z =∫ ∞

−∞

dk

2π

∫ ∞

−∞dzK(z, z) =

√mβ

8π

ω

sinh(β¯ω/2). (A6)

Meanwhile, in the classical limit, the partition functionZ becomes

Zcl = 1

2π¯

∫ ∞

−∞dp exp

(−β

p2

2m

)=√

m

2πβ¯2. (A7)

Therefore, the free energy difference associated with thequantum and classical partition functions is

Aqhc=−β−1 log Z+β−1 log Zcl=−β−1 log

{β¯ω/2

sinh(β¯ω/2)

}.

(A8)

Therefore, we define Aqhc as the quantum harmonic correctionof the free energy for one-dimensional oscillator.

From the derivatives of Aqhc, we obtain the quantum har-monic corrections of the kinetic energy,

Kqhc = −m∂Aqhc

∂m= 1

4

(¯ω

tanh β¯ω− kT

), (A9)

the potential energy,

Vqhc = ω2 ∂Aqhc

∂ω2= 1

4

(¯ω

tanh β¯ω− kT

), (A10)

the internal energy,

Uqhc = ∂(βAqhc

)∂β

= 1

2

(¯ω

tanh β¯ω− kT

), (A11)

and the entropy term

−T Sqhc = T∂Aqhc

∂T= −kT log

(β¯ω

2

sinh β¯ω2

)

− 1

2

(¯ω

tanh β¯ω− kT

). (A12)

From these expressions, we can reconfirm the thermodynamicrelations Uqhc = Kqhc + Vqhc and Aqhc = Uqhc − TSqhc. InEq. (A11), we find that the quantum harmonic correctionof the internal energy, Uqhc, is the difference of the aver-age energy of classical and quantum harmonic oscillators, kT

2

and ¯ω2 coth β¯ω, respectively. In the zero temperature limit(T → 0), the quantum correction should correspond to the en-ergy of zero-point vibration of the oscillator, and indeed, wefind that Kqhc = Vqhc = ¯ω/4. The quantum correction to theentropy term should vanish, since they should be zero both inthe quantum and classical cases, and indeed, −TSqhc → 0 asT → 0. Meanwhile in the limits of high temperature (T → ∞)or the flat potential surface (ω → 0), the quantum correctionshould vanish, since they are classical limits. Indeed, we findthat Kqhc = Vqhc = −T Sqhc = 0 in these limits.

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