Molecular dynamics integration and molecular vibrational theory. III. The infraredspectrum of waterMatej Praprotnik and Dušanka Janežič Citation: The Journal of Chemical Physics 122, 174103 (2005); doi: 10.1063/1.1884609 View online: http://dx.doi.org/10.1063/1.1884609 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/122/17?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Molecular dynamics simulation of liquid methanol. I. Molecular modeling including C–H vibration and fermiresonance J. Chem. Phys. 134, 024509 (2011); 10.1063/1.3514139 Quantum dynamical effects in liquid water: A semiclassical study on the diffusion and the infrared absorptionspectrum J. Chem. Phys. 131, 164509 (2009); 10.1063/1.3254372 Water structure, dynamics, and vibrational spectroscopy in sodium bromide solutions J. Chem. Phys. 131, 144511 (2009); 10.1063/1.3242083 Comparison of path integral molecular dynamics methods for the infrared absorption spectrum of liquid water J. Chem. Phys. 129, 074501 (2008); 10.1063/1.2968555 Vibrational dynamics of DNA. III. Molecular dynamics simulations of DNA in water and theoretical calculations ofthe two-dimensional vibrational spectra J. Chem. Phys. 125, 114510 (2006); 10.1063/1.2213259

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Molecular dynamics integration and molecular vibrational theory.III. The infrared spectrum of water

Matej Praprotnik and Dušanka Janežiča!

National Institute of Chemistry, Hajdrihova 19, 1000 Ljubljana, Slovenia

sReceived 17 November 2004; accepted 11 February 2005; published online 29 April 2005d

The new symplectic molecular dynamicssMDd integrators presented in the first paper of this serieswere applied to perform MD simulations of water. The physical properties of a system of flexibleTIP3P water molecules computed by the new integrators, such as diffusion coefficients, orientationcorrelation times, and infraredsIRd spectra, are in good agreement with results obtained by thestandard method. The comparison between the new integrators’ and the standard method’sintegration time step sizes indicates that the resulting algorithm allows a 3.0 fs long integration timestep as opposed to the standard leap-frog Verlet method, a sixfold simulation speed-up. Theaccuracy of the method was confirmed, in particular, by computing the IR spectrum of water inwhich no blueshifting of the stretching normal mode frequencies is observed as occurs with thestandard method. ©2005 American Institute of Physics. fDOI: 10.1063/1.1884609g

I. INTRODUCTION

Molecular dynamicssMDd simulations are among themain theoretical methods of investigating complex molecularsystems, e.g., water.1 In recent years, MD simulations haveprovided a substantial amount of data about water structuraland dynamical properties.2–9 In the MD simulation, the clas-sical equations of motion for an assembly of interacting par-ticles are solved, usually numerically. However, computingtime is a serious handicap in this regard. Several MD inte-gration approaches were derived10–16 to overcome this prob-lem. The most used is the symplectic second-order leap-frogVerlet sLFVd method.17 Using this method, a water system offlexible TIP3P water molecules18,19 can be accurately inte-grated by 0.5 fs integration time step.

Water plays a unique role among liquids due to its ubiq-uity, importance for life, and also anomalous liquid proper-ties. Therefore it represents one of the most interesting andchallenging systems to study for the MD simulation. TheMD simulations of liquid water have been hindered by theshort time step allowed by integration methods1,20 owing tostrong intramolecular and intermolecular interactions in thesystem, which generate atom motion on a very short timescalesfemtosecondd. New semianalytical symplectic MD in-tegrators introduced in the first paper of this series,21 how-ever, allow the use of longer time steps owing to the analyti-cal treatment of high-frequency molecular vibrations.

Our newly developed MD integrators are different toother approaches, which also allow longer time steps,12–16 inthat we split the whole Hamiltonian of the system into twoparts corresponding to the fast and slow molecular motions,respectively, and that the high-frequency molecular motions,i.e., vibrations, are resolved analytically by normal coordi-

nates. This approach allows using a long MD integrationtime step and also accurate calculations of the physical prop-erties of the system.

In this paper we present a series of MD simulations of awater system of flexible TIP3P water molecules18,19 usingnew integrators21 with a long integration time step to suc-cessfully reproduce liquid water’s structual and dynamicalproperties at 300 K. Results show that the new integratorsallow an integration time step up to six times longer than thestandard leap-frog Verlet method and are especially efficientfor calculating the vibrational IR spectra because of theiranalytical treatment of high-frequency molecular vibrations.

II. METHODS

A. Overview of integration methods

A series of MD simulations of planar water moleculesusing the new MD integrators described in the first paper ofthis series21 were performed. The newly developed integratorSISM and its three derivatesfmultiple time stepping SISMsSISM-MTSd, equilibrium SISM sSISM-EQd, equilibriumSISM-MTS sSISM-MTS-EQdg sRef. 21d are explicit second-order symplectic methods, which enables them to resolve theequations of motion, in part analytically, with a long integra-tion time step. These methods, derived within the Lie grouptheory,22 are especially suitable to compute the IR spectra ofsystems of flexible molecules with one equilibrium configu-ration and no internal rotation, e.g., water molecules. As ref-erence methods the LFV and LFV-EQsRef. 21d were chosenbecause they are a time reversible symplectic second-ordermethods with only one variable parameter—the integrationtime step size.

B. Diffusion coefficient

The diffusion coefficient, which characterizes transla-tional motion of water molecules, is defined as1

adAuthor to whom correspondence should be addressed. Electronic mail:dusa@cmm.ki.si

THE JOURNAL OF CHEMICAL PHYSICS122, 174103s2005d

0021-9606/2005/122~17!/174103/10/$22.50 © 2005 American Institute of Physics122, 174103-1

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D =1

3E

0

`

Cstddt, s1d

whereCstd is the oxygen velocity autocorrelation function,

Cstd = kvistd ·vis0dl =1

Mmok=1

M

oi=1

m

vistkd ·vistk + td s2d

andvistd the velocity of the oxygen atom in theith moleculeat time t. Angular brackets denote the average over allmoxygens and allM time origins.23 The diffusion coefficients1d is, in the long time limit, equal to the correspondingdiffusion coefficients calculated either from the hydrogen ve-locity autocorrelation function or water molecule’s center-of-mass velocity autocorrelation function.2

The diffusion coefficient can be also calculated from thedisplacements of oxygen atoms using the Einstein relation23

D =1

3limt→`

kur istd − r is0du2l2t

, s3d

wherer istd is the position of the oxygen atom in theith watermolecule at timet and averaging is performed over all oxy-gen atoms and all choices of time origin. Also, in the longtime limit, the diffusion coefficient determined in this way isequal to the corresponding diffusion coefficients calculatedeither from the trajectory of the hydrogen atoms or the tra-jectory of the water molecules’ centers of mass.2

C. Orientational correlation times

Rotational motion of the molecules is described bysingle molecule orientational autocorrelation functions1,24 ofthe corresponding intramolecular vectors

Clastd = kPlsnastd ·nas0ddl, s4d

wherePl is the lth degree Legendre polynomial andna is aunit vector pointing along thea axis in the internal coordi-nate system of an individual molecule.2,25 In our study wehave used three differenta axes: thea=HH axis pointsalong the vector connecting two hydrogen atoms in a watermolecule; thea=m axis points along the dipole moment of awater molecule; and thea=' axis points along the directionperpendicular to the molecular plane.

Single molecule correlation times can be determined bywriting the functionss4d in the form

Clastd = exps− t/tl

ad, s5d

where tla is the corresponding correlation time1,25 In the

present work, the correlation timestla were determined from

ln Clastd in the time interval from 1.0 ps to 2.0 ps by the lin-

ear regression method.26

D. IR spectrum

The infrared sIRd absorption spectrum is obtained byFourier transformation of the dipole moment autocorrelationfunction as23,27,28

Isvd ~ E0

`

kM std ·M s0dlcossvtddt, s6d

where Isvd is the spectral density,M std is the total dipolemoment of the system at timet, andv is the vibration fre-quency.M std is equal to the sum of all the individual dipolemoments of the molecules in the simulation box23,27

M std = oi=1

m

mistd, s7d

wheremistd is the dipole moment of theith molecule at timet and m is the number of molecules in the simulation box.The autocorrelation function of the dipole moment is givenby29,30

kM std ·M s0dl =Koj=1

n

ejr jstd ·oj=1

n

ejr js0dL , s8d

wheren is the number of all the atoms in the system,ej is thefixed electric charge of thej th atom, andr jstd is the positionvector of thej th atom at timet. The angular brackets repre-sent an average taken over all time origins. The IR spectrumcan be also obtained using the electrical flux-flux autocorre-lation function

Koj=1

n

ejv jstd ·oj=1

n

ejv js0dL=Ko

j=1

n

ejdr jstd

dt·o

j=1

n

ejdr js0d

dt L=KdM std

dt·

M s0ddt

L , s9d

which corresponds the autocorrelation function of the timederivative of the dipole moment.29,30 The spectral density isthen calculated as29–31

Isvd ~ E0

` KdM stddt

·dM s0d

dtLcossvtddt

=E0

`Koj=1

n

ejv jstd ·oj=1

n

ejv js0dLcossvtddt. s10d

Since the calculated IR spectrum using Eq.s10d differs fromthe IR spectrum obtained by Eq.s6d only by its intensity,29

we have used Eq.s10d in our work for calculating the IRspectrum.

III. COMPUTATIONAL DETAILS

As a representative from the class of the planar mol-ecules, for which the SISM is most efficient, we have pickedthe watersH2Od molecule. The equilibrium configuration isdepicted in Fig. 1, in which atoms 1 and 3 denote hydrogenatoms and atom 2 denotes the oxygen atom in the H2O mol-ecule.

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A. Water model

We have chosen the flexible TIP3P model18 as the modelfor the water molecule. The original TIP3P model is changedin such a way that it allows for the flexibility of bonds andangles and also each of the hydrogens was given a small vander Waals radius to prevent the “electrostatic catastrophe” atsmall distances between water molecules.19,32

The model Hamiltonian is18,19

H = oi

pi2

2mi+

1

2 obonds

kbsb − b0d2 +1

2 oangles

kusu − u0d2

+ oi. j

eiej

4pe0r ij+ o

i. j

4«i jFSsi j

r i jD12

− Ssi j

r i jD6G , s11d

where i and j run over all atoms,mi is the mass of theithatom, pi =mivi is the linear momentum of theith atom,b0

and u0 are reference values for bond lengths and angles,respectively,kb andku are corresponding force constants,ei

denotes the charge on theith atom,e0 is the dielectric con-stant in vacuum,r ij is the distance between theith and j thatoms, and«i j andsi j are the corresponding constants of theLennard-Jones potential. The van der Waals and Coulombinteractions between the atoms of the same molecule are notcalculated explicitly because they are already taken into ac-count by the bond stretching and angle bending terms in theHamiltonians11d. The parameters of the model are given inTable I.

B. Vibrational potential energy and internalcoordinate system

The vibrational potential energy is the sum of the vibra-tional potential energies of all of the molecules in the system

Vvib = ok=1

m

Vvibk=

1

2 obonds

kbsb − b0d2 +1

2 oangles

kusu − u0d2,

s12d

whereVvibkis the vibrational potential energy of thekth mol-

ecule in the system andm is the number of molecules in thesystem.

The potential functionVharm, which is developed fromEq. s12d in the same way as in the previous paper,33 is

Vharm= Vstretch+ Vbend, s13d

whereVstretch and Vbend are quadratic potential functions interms of relative Cartesian displacement coordinates, whichrepresent the harmonic approximations for the bond stretch-ing and angle bending potentials, respectively.21 The expres-sions for Vstretch and Vbend are obtained by projecting thedisplacements of atoms along and perpendicular to the bondsbetween the atoms in each molecule.34

Vstretch is then

Vstretch=12kbfssDx1 − Dx2,Dy1 − Dy2,Dz1 − Dz2d

· s− sinb,cosb,0dTd2 + ssDx3 − Dx2,Dy3

− Dy2,Dz3 − Dz2d · ssinb,cosb,0dTd2g

= 12kbff− sDx1 − Dx2dsinb + sDy1 − Dy2dcosbg2

+ fsDx3 − Dx2dsinb + sDy3 − Dy2dcosbg2g s14d

andVbend is

Vbend=1

2kuF 1

b0fsDx1 − Dx2,Dy1 − Dy2,Dz1 − Dz2d

· s− cosb,− sinb,0dTg +1

b0fsDx3 − Dx2,Dy3

− Dy2,Dz3 − Dz2d · scosb,− sinb,0dgG2

=1

2kuF 1

b0f− sDx1 − Dx2dcosb − sDy1 − Dy2dsinb

+ sDx3 − Dx2dcosb − sDy3 − Dy2dsinbgG2

=1

2

ku

b02fsDx3 − Dx1dcosb

+ s2Dy2 − Dy1 − Dy3dsinbg2. s15d

The vibrational frequencies corresponding to normalmodes of vibration of the flexible TIP3P water model18,19areobtained in the same way as in the previous paper33 and arereported in Table II. We checked the accuracy of vibrationalfrequencies numerically obtained in this way with the corre-sponding analytically computed vibrational frequencies.34

TABLE I. Parameters of the flexible TIP3P model of the H2O moleculesRefs. 18 and 19d. The quantitye0 is the elementary charge.

Parameter Value

bOH0=b0 0.9572 Å

u0 104.52°kbOH

=kb 900.0 kcal/mol/Å2

ku 110.0 kcal/mol/ rad2

eH1=e1 0.417e0

eO=e2 −0.834e0

eH2=e3 0.417e0

sOO 3.1507 ÅsHH 0.40 ÅeOO −0.152 073 kcal/moleHH −0.045 98 kcal/mol

FIG. 1. Description of the positions of atoms in the equilibrium configura-tion of a water molecule. The orthogonal unit vectorsf1 and f2 are the twounit vectors, which define the internal coordinate system of a molecule.

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The moving internal coordinate system is for each mol-ecule in the system defined with the same procedure as in theprevious paper.33 The elements of the matrixF−1/2, which isrequired to determine the unit vectors of the internal coordi-nate system of a molecule in the SISM,21 are explicitly writ-ten as

F−1/2s1,1d =d + Fs2,2d

ds, s16d

F−1/2s2,2d =d + Fs1,1d

ds, s17d

F−1/2s1,2d = −Fs1,2d

ds, s18d

F−1/2s2,1d = F−1/2s1,2d, s19d

where d=ÎFs1,1dFs2,2d−Fs1,2d2 ands=Î2d+Fs1,1d+Fs2,2d.35

C. Simulation protocol

All MD calculations are carried out for the system of256 flexible TIP3P water molecules at the system densityr=1.0 g/cm3 and T=300 K. The corresponding simulationbox size isa=19.7 Å. Periodic boundary conditions are im-posed to overcome the problem of surface effects; the mini-mum image convention is used.1 The Coulomb interactionsare truncated using the force-shifted potential36 at a cut-offdistancerof f=8.5 Å.37 The Lennard-Jones interactions areshifted by adding the termCij r ij

6 +Dij to the potential, whereCij and Dij are chosen such that the potential and force arezero atr ij =rof f.

19 The initial positions and velocities of theatoms are chosen at random. The system is then equilibratedfor 20 ps, where the velocities are scaled every 500 integra-tion steps to correspond to a temperature of 300.0 K. After-wards an additional 30 ps of equilibration is carried out atthe constant total energy of the system so that the velocitycorresponds to the Maxwell distribution at 300 K. Simula-tions are then performed for 100 ps using different integra-tors and/or integration time steps.

IV. RESULTS AND DISCUSSION

As in the previous paper33 the error in total energyDE/Edefined as

DE

E=

1

Mok=1

M UE0 − Ek

E0U , s20d

whereE0 is the initial energy,Ek is the total energy of thesystem at the integration stepk, andM is the total number ofintegration steps, was monitored for all methods.

In Fig. 2sad, in which the error in total energy for thesystem of water molecules for the LFV, SISM, and SISM-MTS is displayed, it can be observed that the error in totalenergy for a 1.25 fs integration time step for the SISM cor-responds to the same error as for the 0.5 fs integration timestep for the LFV. The estimated value for the maximal ac-ceptable integration time step for the LFV is 0.5 fs since theperiod of the antisymmetric bond stretching is equal to 9.9 fsssee Table IId. This means that for the same level of accuracy,the SISM allows the use of up to a two and a half timeslonger integration time step than the LFV. The jump in theerror in total energy, which can be observed in Fig. 2sad for a2.5 fs integration time step, can possibly be explained bynonlinear instabilities occurring when the integration stepsize is a fourth of the period of the fastest stretching vibra-tions in water molecules.38

For the system of water molecules, a high amount ofanharmonic forces occurs due to the strong anharmonic po-tential describing the interactions in the system. However, asin the case of the hydrogen peroxide simulation,33 the SISM-

TABLE II. Experimental vibrational frequencies of the H2O moleculesRef.47d and normal mode frequencies of the H2O molecule determined by nor-mal mode analysis using parameters from Table I.

Normal mode1/l scm−1d

sexperimentda1/l scm−1d

stheorydb

Antisymmetric O–H stretch 3756 3381Symmetric O–H stretch 3652 3334

Angle bending 1595 1743

aExperimental vibrational frequencies.bNormal mode frequencies.

FIG. 2. sad Error in the total energy of the system of 256 water moleculeswith r=1.0 g/cm3 at T=300 K using the LFV, SISM, and SISM-MTS forM =1000. sbd Error in total energy of the system of 256 water moleculeswith r=1.0 g/cm3 at T=300 K using the LFV-EQ, SISM-EQ, and SISM-MTS-EQ for M =1000.

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MTS with dt=0.5 fs for the integration of motions generatedby high-frequency anharmonic interactions is less accuratethan the SISMfFig. 2sadg.

The error in total energy for the LFV-EQ, SISM-EQ, andSISM-MTS-EQsRef. 21d is depicted in Fig. 2sbd. The totalenergy is defined in this case as21

H = Tspd + Vvibsqd + Vnbfdsqdg, s21d

wheredsqdPR3n are the equilibrium positions of atoms inall molecules of the system, given by the standard theory ofmolecular vibrations.21,39–41 Since the vibrational potentialVvib used in the LFV-EQ is the same as in the LFV, theestimated value of the maximal acceptable time step for theLFV-EQ is also 0.5 fs.

From the results presented in Fig. 2sbd we can see thatthe error in total energy for the LFV-EQ with a 0.5 fs inte-gration time step corresponds to the error for the SISM-EQwith a 2.0 fs integration time step and 3.0 fs for the SISM-MTS-EQ. The SISM-EQ therefore allows the use of a fourtimes longer integration time step than the LFV-EQ for thesame level of accuracy, whereas the SISM-MTS-EQ allowsthe use of even up to six times longer time steps than theLFV-EQ. The energy drift for integration time step’s sizeslonger than 3.0 fs is due to nonlinear instabilities, which oc-cur when the integration step size is a third of the period ofthe fastest motion in the system as explained in Ref. 38.

The analytical solution of the equations of motion doesnot exist and the difference between trajectories calculatedwith different sizes of integration time steps and/or differentintegrators grows exponentially with time.1 Therefore the ac-curacy of trajectories calculated by the new methods can notbe checked by direct comparison of the calculated trajecto-ries to the corresponding trajectories computed by the stan-dard methods. Statistical correctness of the trajectories com-puted by the new methods can only be checked bycomputing different structural and dynamical properties ofthe molecular system and comparing them with correspond-ing properties calculated by the standard methods.

A. Structural properties

1. Radial distribution functions

The structure of the system of water molecules waschecked by calculating site-site radial distribution functionsgOOsrd for oxygens,gOHsrd for oxygens and hydrogens indifferent water molecules, andgHHsrd for hydrogens in dif-

ferent water molecules. ThegOOsrd, gOHsrd, andgHHsrd werecomputed by the LFV, SISM, SISM-MTS, and LFV-EQ,SISM-EQ, and SISM-MTS-EQ from 100 ps long trajectoriesof the system of 256 flexible TIP3P water molecules withr=1.0 g/cm3 at T=300 K using different sizes of the inte-gration time step. Radial distributions functions calculatedby the LFV using a 0.5 fs integration time step has beentaken as a reference for the corresponding functions calcu-lated by the SISM and SISM-MTS, whereas the radial dis-tribution functions calculated by the LFV-EQ using a 0.5 fsintegration time step have been taken as a reference for thecorresponding functions obtained by the SISM-EQ andSISM-MTS-EQ. The positions and heights of the first maxi-mum ofgOOsrd, gOHsrd, andgHHsrd, which are in good agree-ment with the published calculated results in theliterature,25,37 are given in Table III. The results reported inTable III show a good agreement between radial distributionfunctions calculated by the SISM and SISM-MTS using a2.0 fs integration time step and the corresponding referencefunctions. The level of agreement with the experimental re-sults depends mostly on the applied model of the water mol-ecule and the “cut-off” function and not on the applied nu-merical integrator and/or the length of the integration timestep.42

Since the site-site radial distribution functions are mostlyforce field dependent, they can be used to check the effect ofthe calculation of the electrostatic and van der Waals poten-tials in the equilibrium positions of the atoms as in the LFV-EQ, SISM-EQ, and SISM-MTS-EQ. The positions andheights of the first maximum ofgOOsrd, gOHsrd, andgHHsrdare given in Table IV for the LFV-EQ using a 0.5 fs integra-tion time step and the SISM-EQ and SISM-MTS-EQ using a3.0 fs integration time step. We can again report a goodagreement of the radial distribution functions calculated bythe SISM-EQ and SISM-MTS-EQ using a 3.0 fs integrationtime step with corresponding reference functions.

B. Dynamical properties

The motion of water molecules in bulk can be character-ized by dynamical properties of an individual molecule suchas the diffusion coefficient and the orientational time con-stants, which can be also determined by MD simulation.24

TABLE III. Comparison ofgOOsrd, gOHsrd, andgHHsrd calculated by the LFV, SISM, and SISM-MTS. We givethe positions and heights of the first maximum of the calculated functions and the corresponding experimentalvaluessRef. 42d.

LFV s0.5 fsd SISM s2.0 fsd SISM-MTS s2.0 fsd Expt.

rOO1sÅd 2.8 2.8 2.8 2.88

gOOsrOO1d 2.75 2.77 2.75 3.09

rOH1sÅd 1.8 1.8 1.8 1.85

gOHsrOH1d 1.26 1.26 1.24 1.39

rHH1sÅd 2.5 2.5 2.5 2.45

gHHsrHH1d 1.21 1.21 1.21 1.26

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1. Diffusion coefficient

The calculated values of the diffusion coefficient by Eq.s1d using the LFV, SISM, and SISM-MTS are reported inTable V, and suggest that the description of translational mo-tion of molecules is independent of the choice of the numeri-cal method and/or the size of the integration time step. Thediffusion coefficients calculated by different integratorsand/or different sizes of the integration time step are in goodagreement.

A similar conclusion holds for the obtained diffusion co-efficients using the LFV-EQ, SISM-EQ, and SISM-MTS-EQ,which are reported in Table VI. The diffusion coefficientvalues are increased in comparison to the results reported inTable V according to results from Ref. 15. However, thecalculated diffusion coefficients given in Table V are alsoapproximately two times larger than the experimental value.This is in accordance with the values obtained from a similarMD simulation in Ref. 37. The disagreement between calcu-lated and experimental diffusion coefficients is due to thewater model and not due to the applied numerical integrator.Good agreement between calculated diffusion coefficientsobtained by different integrators is expected since all of themtreat the molecules’ translational degrees of freedom in thesame way.

The calculated values of the diffusion coefficient ob-tained by virtue of Eq.s3d using different integrators anddifferent sizes of the integration time step are reported inTables V and VI. The obtained values are in good agreementwith the values calculated by using Eq.s1d. From the ob-

tained results we can conclude that the translational motionof water molecules is correctly described by the SISM and itsderivatives.

2. Orientational correlation times

The correlation timestla determined by Eqs.s4d ands5d

using the LFV, SISM, and SISM-MTS using different sizesof the integration time step are reported in Table VII. Theresults obtained by different integrators are in good agree-ment for all sizes of integration time steps. This proves thatthe rotational motion of water molecules is also correctlydescribed by the SISM and SISM-MTS using a 2.0 fs inte-gration time step. The values of the calculated orientationalcorrelation times given in Table VII are lower than the ex-perimental values, which is in accordance with the resultsfrom similar calculations reported in the literature.2,24,25De-spite that the experimental data suggests the rotation of asingle water molecule in a bulk system is isotropic,25,43 theobtained correlation timestl

' calculated by all numericalmethods are shorter thantl

HH andtlm, which was also found

by other authors.25

Similar conclusions are also valid for the results ob-tained using the LFV-EQ, SISM-EQ, and SISM-MTS-EQ,which are reported in Table VIII. From the results in TableVIII we can conclude that the rotation of a water molecule iscorrectly described by the SISM-EQ and SISM-MTS-EQeven with a 3.0 fs long integration time step.

Comparing the results in Tables VII and VIII we canobserve that the values oftl

a are lower when the equilibrium

TABLE IV. Comparison ofgOOsrd, gOHsrd, andgHHsrd calculated by the LFV-EQ, SISM-EQ, and SISM-MTS-EQ. We give the positions and heights of the first maximum of the calculated functions and the correspondingexperimental valuessRef. 42d.

LFV-EQ s0.5 fsd SISM-EQ s3.0 fsd SISM-MTS-EQs3.0 fsd Expt.

rOO1sÅd 2.8 2.8 2.8 2.88

gOOsrOO1d 2.54 2.55 2.55 3.09

rOH1sÅd 1.9 1.9 1.9 1.85

gOHsrOH1d 1.07 1.07 1.07 1.39

rHH1sÅd 2.5 2.5 2.5 2.45

gHHsrHH1d 1.17 1.17 1.17 1.26

TABLE V. Diffusion coefficientD for the system of 256 flexible TIP3P water molecules calculated using Eq.s3d sMSDd and Eq.s1d sVACd and various integration time steps. Experimental value ofD for liquid water atT=300 K is 2.4310−9 m2/s sRef. 37d.

Dt sfsd

a D

10−9 sm2/sd b D

10−9 sm2/sd c D

10−9 sm2/sd

MSD VAC MSD VAC MSD VAC

0.5 4.5 4.6 4.5 4.5 4.7 4.71.0 4.4 4.4 4.6 4.5 4.6 4.61.5 4.8 4.9 4.5 4.6 4.6 4.82.0 4.9 5.0 4.6 4.8 4.5 4.6

aCalculated by LFV.bCalculated by SISM.cCalculated by SISM-MTS.

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positions of atoms are used to calculate the electrostatic andvan der Waals potential energies. Comparing the diffusioncoefficient given in Tables V and VI and relaxation timest1

m

for the dipole moment reported in Tables VII and VIII we seethat shortert1

m, when calculating the electrostatic and van derWaals potential energies with atoms’ equilibrium positions,are associated with larger diffusion coefficients. This is be-cause the overall motion in liquid water is governed by thecoupled translational and rotational motions of water mol-ecules within first hydration shell.24 If we calculate the prod-uct s6Dt1

md1/2 from corresponding data in Tables V–VIII, weobtain the value of 2.5Å for the average displacement inde-pendently from the manner of calculating the electrostaticand van der Waals potential energies. These values corre-spond to the average distance between two watermolecules,24 which are equal for both examples as can beseen from the results in Tables III and IV.

C. IR spectrum

The vibrational and rotational motions of molecules arethose which involve energies that produce the spectra in theinfrared region. Therefore, our newly developed MD integra-tion methods are particulary suitable for computing the IRspectra because rotational, translational, and vibrational mo-tions are resolved analytically, independent of the MD inte-gration time step.

The IR spectra calculated by Eq.s10d of bulk water atT=300 K using the LFV, SISM, and SISM-MTS with inte-gration time steps of lengths from 0.5 fs to 2.0 fs are shownin Figs. 3 and 4.

Figure 3sad demonstrates that the IR spectra calculatedby different numerical integrators using a 0.5 fs integrationtime step are in good agreement. These IR spectra weretaken as a reference for comparison with calculated IR spec-tra using longer integration time steps. The magnitudes of the

TABLE VI. Diffusion coefficientD for the system of 256 flexible TIP3P water molecules calculated using Eq.s3d sMSDd and Eq.s1d sVACd and various integration time steps. Experimental value ofD for liquid water atT=300 K is 2.4310−9m2/s sRef. 37d.

Dt sfsd

a D

10−9 sm2/sd b D

10−9 sm2/sd c D

10−9 sm2/sd

MSD VAC MSD VAC MSD VAC

0.5 6.3 6.3 6.4 6.5 6.4 6.41.0 6.6 6.6 6.4 6.3 6.4 6.41.5 6.7 6.7 6.6 6.7 6.4 6.72.0 6.7 7.1 6.5 6.5 6.5 6.72.5 6.6 6.4 6.4 6.8 6.8 6.93.0 ¯ ¯ 6.3 6.6 6.8 7.1

aCalculated by LFV-EQ.bCalculated by SISM-EQ.cCalculated by SISM-MTS-EQ.

TABLE VII. Orientational correlation timestla for the system of 256 flexible TIP3P water molecules atT

=300 K obtained by using various integration time steps. The corresponding experimental values oftla at T

=300 K aret2HH=2.1 pssRef. 48d, t1

m=7.5 pssRef. 24d, t2'=2.4 pssRef. 43d.

Dt a

atla spsd btl

a spsd ctla spsd

l =1 l =2 l =1 l =2 l =1 l =2

HH 1.7 0.9 1.6 0.8 1.6 0.80.5 m 2.3 0.9 2.2 0.9 2.2 0.8

' 1.3 0.7 1.3 0.7 1.3 0.7

HH 1.6 0.9 1.7 0.9 1.6 0.81.0 m 2.2 0.9 2.2 0.9 2.2 0.8

' 1.3 0.7 1.3 0.7 1.3 0.6

HH 1.6 0.8 1.6 0.8 1.6 0.81.5 m 2.1 0.8 2.1 0.8 2.2 0.9

' 1.2 0.6 1.3 0.6 1.3 0.6

HH 1.5 0.8 1.7 0.9 1.6 0.82.0 m 2.1 0.8 2.2 0.9 2.2 0.9

' 1.2 0.6 1.3 0.7 1.3 0.6

aObtained by LFV.bObtained by SISM.cObtained by SISM-MTS.

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spectral peaks were scaled so that the intensities of the higherpeak positioned above the frequency of 3300 cm−1 and thestretching peak at 3408 cm−1 in the experimental spectrumare equal. The double peak at 3300 cm−1 corresponds to thenormal modes of vibration of a water molecule describingthe antisymmetric and symmetric bond stretchesssee TableII d. The single peak at 1775 cm−1 corresponds to the anglebending normal mode of a water moleculessee Table IId. TheIR band between 300 cm−1 and 900 cm−1 arises from to thelibrational motion of water molecules.2

When using a 1.0 fs integration time step, the high-frequency double peak at 3300 cm−1 in the IR spectrum cal-culated by the LFV already shifts to the higher frequencies asshown in Fig. 3sbd. The observed blueshift suggests thatwhen using a 1.0 fs integration time step, the LFV can nolonger accurately describe the high-frequency vibrationalmotions of atoms in a water molecule, which has been al-ready predicted by estimating the integration time step sizefrom the error in the total energy shown in Fig. 2sad. Thisphenomenom is even more evident in Fig. 4 for the cases of

TABLE VIII. Orientational correlation timestla for the system of 256 flexible TIP3P water molecules atT

=300 K obtained by using various integration time steps. The corresponding experimental values oftla at T

=300 K aret2HH=2.1 pssRef. 48d, t1

m=7.5 pssRef. 24d, t2'=2.4 pssRef. 43d.

Dt a

atla spsd btl

a spsd ctla spsd

l =1 l =2 l =1 l =2 l =1 l =2

HH 1.2 0.6 1.2 0.6 1.2 0.60.5 m 1.5 0.6 1.6 0.6 1.5 0.6

' 0.9 0.5 0.9 0.5 0.9 0.5

HH 1.2 0.6 1.1 0.6 1.2 0.61.0 m 1.5 0.6 1.4 0.6 1.5 0.6

' 0.9 0.5 0.9 0.5 0.9 0.5

HH 1.2 0.6 1.2 0.6 1.2 0.61.5 m 1.5 0.6 1.5 0.6 1.5 0.6

' 0.9 0.5 0.9 0.5 0.9 0.5

HH 1.1 0.6 1.2 0.6 1.2 0.62.0 m 1.4 0.6 1.5 0.6 1.5 0.6

' 0.8 0.4 0.9 0.5 0.9 0.5

HH 1.1 0.6 1.2 0.6 1.2 0.62.5 m 1.4 0.6 1.5 0.6 1.5 0.6

' 0.9 0.5 0.9 0.5 0.9 0.5

HH ¯ ¯ 1.2 0.6 1.1 0.63.0 m ¯ ¯ 1.5 0.6 1.5 0.6

' ¯ ¯ 0.9 0.5 0.9 0.5

aObtained by LFV-EQ.bObtained by SISM-EQ.cObtained by SISM-MTS-EQ.

FIG. 3. sColord IR spectrumsarbitrary unitsd of bulk water atT=300 K calculated by LFV, SISM, and SISM-MTS using an integration time step ofsad 0.5 fsand sbd 1.0 fs.

174103-8 M. Praprotnik and D. Janežič J. Chem. Phys. 122, 174103 ~2005!

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1.5 fs and 2.0 fs integration time steps where the peak at1775 cm−1 also starts shifting toward higher frequencies.Peaks in corresponding IR spectra, which are calculated bythe SISM and SISM-MTS, however, remain at the same po-sitions as corresponding peaks in the reference IR spectracalculated using the integration time step of 0.5 fs. Thisproves that owing to the analytical description of high-frequency molecular vibrations, the later are accurately de-scribed by the SISM and SISM-MTS also using a 2.0 fsintegration time step.

Comparing the results in Fig. 5sad, in which the calcu-lated IR spectra of bulk water are shown calculated by theLFV-EQ, SISM-EQ, and SISM-MTS-EQ using a 2.5 fs inte-gration time step, with those in Fig. 3sad we can see that thepeaks corresponding to the bond stretching and angle bend-ing have remained in the same positions in IR spectra com-puted by the SISM-EQ and SISM-MTS-EQ. This is becausethe electrostatic and van der Waals interactions are not cal-culated between the atoms belonging to the same water mol-

ecule. The intensity of the band corresponding to the libra-tional motion of water molecules is, however, slightlydecreased in comparison with the IR spectrum in Fig. 3sad.Again, a blueshift of the the double peak at 3300 cm−1 canbe observed in IR spectrum computed by the LFV-EQ.Therefore we can conclude that the LFV-EQ is also not ableto accurately describe the high-frequency molecular vibra-tions using integration time steps longer than 0.5 fs. On thecontrary, the IR spectra computed by the SISM-EQ andSISM-MTS-EQ show that these methods accurately describehigh-frequency molecular vibrations even when using up to a3.0 fs integration time step as can be observed from Fig.5sbd.

The calculated IR spectra as well as diffusion coeffi-cients and rotational correlation times prove that the SISMand SISM-MTS correctly describe the translational, rota-tional, and vibrational degrees of freedom of water mol-ecules up to a 2.0 fs integration time step. The error in totalenergy, however, shows that the size of the maximal allowed

FIG. 5. sColord IR spectrumsarbitrary unitsd of bulk water atT=300 K calculated by LFV-EQ, SISM-EQ, and SISM-MTS-EQ using an integration time stepof sad 2.5 fs andsbd 3.0 fs.

FIG. 4. sColord IR spectrumsarbitrary unitsd of bulk water atT=300 K calculated by LFV, SISM, and SISM-MTS using an integration time step ofsad 1.5 fsand sbd 2.0 fs.

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integration time step is 1.25 fs for the SISM and SISM-MTSin comparison with 0.5 fs for the LFV. From the computeddata one can also conclude that the SISM-EQ and SISM-MTS-EQ also correctly integrate the translational, rotational,and vibrational motion of water molecules even up to a3.0 fs integration time step. However, the error in total en-ergy shows that the size of the maximal acceptable integra-tion time step is 2.0 fs for the SISM-EQ and 3.0 fs in theexample of the SISM-MTS-EQ, which is a sixfold speed-upin comparison with LFV-EQ.

V. CONCLUSIONS

We have presented a number of MD simulation of liquidwater using newly introduced symplectic MD integratorswhich combine MD integration with the standard theory ofmolecular vibrations.39–41 The computed structural and dy-namical properties of the studied systems are in good agree-ment with the corresponding properties computed by theLFV using an accordingly short integration time step. Due tothe analytical treatment of high-frequency molecular vibra-tions, the new methods are especially efficient for computingIR spectra since no blueshifting of the bond stretching nor-mal mode peaks occurs, as is the case when using the LFVintegrator. We also point out the ability of using a 3.0 fsintegration time step with the SISM-MTS-EQ integrationmethod at the same accuracy as employing the standard LFVmethod with a 0.5 fs integration time step. This is a sixfoldsimulation speed-up compared to the standard method for thesimulation of a bulk water system.

We believe that new integration methods could become amethod of choice in spectroscopic calculations31,44–46wherethe standard theory of molecular vibrations is applied to ana-lyze the dynamics of the studied system from already com-puted MD trajectories. With our methods the informationabout the energy distribution of normal modes and the en-ergy transfer between them is obtained without additionalcomputations as opposed to the standard MD integrationmethods because the normal mode dynamics is computedalong with the computation of the trajectory of a molecularsystem.

ACKNOWLEDGMENTS

The authors thank Dr. M. Hodošček and Dr. F. Merzelfor helpful discussions and to U. Borštnik for careful readingof the manuscript. This work was supported by the Ministryof Education, Science and Sports of Slovenia under GrantNos. P1-0002 and J1-6331.

1M. P. Allen and D. J. Tildesley,Computer Simulation of LiquidssClaren-don, Oxford, 1987d.

2R. W. Impey, P. A. Madden, and I. R. McDonald, Mol. Phys.46, 513s1982d.

3J. Marti, J. Chem. Phys.110, 6876s1999d.4A. Luzar, J. Chem. Phys.113, 10663s2000d.5M. Boero, K. Terakura, T. Ikeshoji, C. Liew, and M. Parrinello, Phys. Rev.Lett. 85, 3245s2000d.

6B. L. de Groot and H. Grubmuller, Science294, 2353s2001d.

7F. Merzel and J. C. Smith, Proc. Natl. Acad. Sci. U.S.A.99, 5378s2002d.8M. Matsumoto, S. Saito, and I. Ohmine, NaturesLondond 416, 409s2002d.

9Ph. Wernet, D. Nordlund, U. Bergmann, M. Caualleri, M. Odelius, H.Ogasawara, L. A. Naslund, T. K. Hirsch, L. Ojamäe, P. Glantzel, L. G. M.Pettersson, and A. Nilsson, Science304, 995 s2004d.

10J. P. Ryckaert, G. Ciccotti, and H. J. C. Berendsen, J. Comput. Phys.23,327 s1977d.

11H. C. Anderson, J. Comput. Phys.52, 24 s1983d.12M. E. Tuckerman, G. J. Martyna, and B. J. Berne, J. Chem. Phys.93,

1287 s1990d.13M. E. Tuckerman, B. J. Berne, and G. J. Martyna, J. Chem. Phys.97,

1990 s1992d.14B. Garcia-Archilla, J. M. Sanz-Serna, and R. D. Skeel, SIAM J. Sci. Com-

put. sUSAd 20, 930 s1998d.15J. A. Izaguirre, S. Reich, and R. D. Skeel, J. Chem. Phys.110, 9853

s1999d.16R. D. Skeel and J. A. Izaguirre, inComputational Molecular Dynamics:

Challenges, Methods, Ideas, Lecture Notes in Computational Science andEngineering, Vol. 4, edited by P. Deuflhard, J. Hermans, B. Leimkuhler, A.E. Mark, S. Reich, and R. D. SkeelsSpringer, Berlin, 1999d.

17L. Verlet, Phys. Rev.159, 98 s1967d.18W. L. Jorgensen, J. Chandrasekhar, J. D. Madura, R. W. Impey, and M. L.

Klein, J. Chem. Phys.79, 926 s1983d.19P. J. Steinbach and B. R. Brooks, J. Comput. Chem.15, 667 s1994d.20A. R. Leach,Molecular Modelling, 2nd ed.sPearson, Harlow, 2001d.21D. Janežič, M. Praprotnik, and M. Merzel, J. Chem. Phys.122, 174101

s2005d, this issue.22J. M. Sanz-Serna and M. P. Calvo,Numerical Hamiltonian Problems

sChapman & Hall, London, 1994d.23A. R. Leach,Molecular Modelling, 2nd ed.sPearson, Harlow, 2001d.24A. Wallqvist and B. J. Berne, J. Phys. Chem.97, 13841s1993d.25D. van der Spoel, P. J. van Maaren, and H. J. C. Berendsen, J. Chem. Phys.

108, 10220s1998d.26W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling,Nu-

merical Recipes: The Art of Scientific ComputingsCambridge UniversityPress, Cambridge, 1987d.

27B. Guillot, J. Chem. Phys.95, 1543s1991d.28B. Borštnik, D. Pumpernik, D. Janežič, and A. Ažman, inMolecular In-

teractions, edited by H. Ratajczak and W. J. Orville-ThomassWiley,1980d.

29B. Boulard, J. Kieffer, C. C. Phifer, and C. A. Angell, J. Non-Cryst. Solids140, 350 s1992d.

30P. Bornhauser and D. Bougeard, J. Phys. Chem. B105, 36 s2001d.31M. Praprotnik, D. Janežič, and J. Mavri, J. Phys. Chem. A108, 11056

s2004d.32D. Janežič and B. R. Brooks, J. Comput. Chem.16, 1543s1995d.33M. Praprotnik and D. Janežič, J. Chem. Phys.122, 174102s2005d, pre-

ceding paper.34L. D. Landau and E. M. Lifshitz,Course of Theoretical Physics: Mechan-

ics, 3rd ed.sPergamon, Oxford, 1976d, Vol. 1.35G. A. Natanson, Chem. Phys. Lett.121, 343 s1985d.36C. L. Brooks III, B. M. Pettitt, and M. Karplus, J. Chem. Phys.83, 5897

s1985d.37M. Prevost, D. van Belle, G. Lippens, and S. Wodak, Mol. Phys.71, 587

s1990d.38Q. Ma, J. A. Izaguirre, and R. D. Skeel, SIAM J. Sci. Comput.sUSAd 24,

1951 s2003d.39C. Eckart, Phys. Rev.47, 552 s1935d.40E. B. Wilson, J. C. Decius, and P. C. Cross,Molecular Vibrations

sMcGraw-Hill, New York, 1955d.41J. D. Louck and H. W. Galbraith, Rev. Mod. Phys.48, 69 s1976d.42A. K. Soper and M. G. Phillips, Chem. Phys.107, 47 s1986d.43B. Halle and H. Wennerström, J. Chem. Phys.75, 1928s1981d.44R. Rey, J. Chem. Phys.104, 2356s1996d.45R. Rey, Chem. Phys.229, 217 s1998d.46R. Rey, J. Chem. Phys.108, 142 s1998d.47Handbook of Chemistry and Physics, edited by D. LidesCRC, Boca Ra-

ton, FL, 1995d.48J. Jonas, T. DeFries, and D. J. Wilbur, J. Chem. Phys.65, 582 s1976d.

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