See discussions, stats, and author profiles for this publication at: http://www.researchgate.net/publication/270516079Internal Coordinate Density of State fromMolecular Dynamics SimulationARTICLEinJOURNAL OF COMPUTATIONAL CHEMISTRY JANUARY 2015Impact Factor: 3.6 DOI: 10.1002/jcc.23822DOWNLOADS17VIEWS182 AUTHORS, INCLUDING:Pin-Kuang LaiUniversity of Minnesota Twin Cities5 PUBLICATIONS 15 CITATIONS SEE PROFILEAvailable from: Pin-Kuang LaiRetrieved on: 31 July 2015InternalCoordinateDensityofStatefromMolecularDynamicsSimulationPin-KuangLaiandShiang-Tai Lin*The vibrational density of states (DoS), calculated fromtheFourier transform of the velocity autocorrelation function,provides profound information regarding the structure anddynamic behavior of a system. However, it is often difficulttoidentifytheexact vibrational modeassociatedwithaspe-cific frequency if the DoS is determinedbasedonvelocitiesinCartesiancoordinates. Here, theDoSis determinedbasedon velocities in internal coordinates, calculated fromCarte-sian atomic velocities using a generalized Wilsons B-matrix.TheDoSininternal coordinates allows for thecorrect detec-tionof freedihedral rotations that may bemistakenas hin-dered rotation in Cartesian DoS. Furthermore, thepronouncedenhancement of lowfrequency modes inCarte-sian DoS for macromolecules should be attributed to thecoupling of dihedral and angle motions. The internal DoS,thus deconvolutes the internal motions andprovides fruitfulinsights tothedynamicbehaviors of asystem. VC2015WileyPeriodicals, Inc.DOI:10.1002/jcc.23822IntroductionThenormal modes, or vibrational densityof states(DoS), pro-vides profound insights to the structure details,[1,2]dynamicbehaviors,[3,4]andthermodynamic properties[58]of asystem.Someexperimental methodsincludeinfraredandRamanscat-teringareoftenusedtoobtainvibrationDoS.[2,9]Thenormalmodesareawayof representingthedynamicsof atomsinasystemusingacollectionof independent oscillatorymotions.At lowenough temperatures, the systemis trapped in thequadratic(harmonic)potential surfacenearsomeenergymini-mum, andthenormal modescanbecalculatedfromthecur-vaturesofthepotential surface, thatis, theHessianmatrix(thesecond derivative of the potential energy with respect toatomic positions at equilibriumgeometry).[1012]The squareroot of the eigenvalues of the Hessian matrix are the fre-quency of thenormal modes, andthecorrespondingeigen-vectorsprovidethedirectionof atomicmovementsassociatedwitheachmode.Asthetemperatureincreases, thepotential surfacemaynolonger be harmonic andthe Hessianmatrix calculations canbecomeinadequate. Under suchcircumstances, effectivenor-mal modes can still be determined fromthe covariance ofatomicpositionfluctuationsunder thequasiharmonicapproxi-mations.[13,14]Someeffortsweremadetoincludeanharmoniceffectsinsuchanalysis.[15,16]Another representation of dynamic of a systemusing aneffectiveharmonic vibrationis thepower spectral density ofthe mass-weightedvelocity, or the DoS.[17]In this case, thedynamic behavior of each atomic motion is represented bysuppositionof sinusoidal oscillations usingtheFourier series.Therefore, thevelocityspectral densityindicates thedistribu-tionof thevibrationinthefrequencydomain. Notethat theDoSagrees withthenormal modes (eigenvalues) determinedfromtheHessianmatrix for harmonic systems. However, thecollective motion of atoms (eigenvectors) associated with aspecificfrequency(eigenvectors of theHessianmatrix) is lostintheDoScalculations.Onepossiblewaytoreconstructthelinkbetweenthedirec-tionof atomicmotionandtheDoSistodeterminetheveloc-ity spectrumusingthe internal coordinates, whichcouldbedifferent fromthatusingtheCartesiancoordinates. For exam-ple, whenarigidrotor rotates at aconstant angular velocityx, thevelocityspectral densityisadeltafunctionatfrequencyx/2p basedonlinear velocity (v 5xd/2sin(xt) withd beingthelengthof therigidrod), whereas it is adeltafunctionatzero frequency based on angular velocity. Note that in thelimit wherethemodebecomesharmonic(e.g., rigidrotorsinthesolidstateat lowtemperatures), theDoScalculatedfromdifferentcoordinatesystemsareidentical.Classical molecular dynamicssimulationsthat integratestheNewtons equations of motionbasedoninternal coordinateshavebeenproposed.[1820]Unfortunately, suchsimulationsareoften less efficient compared to those based on Cartesiancoordinatesbecausereducedmassassociatedwitheachinter-nal modecanbetimedependent(especiallyforflexiblemole-cules). The recalculation of the reduced mass significantlyincreases the computational cost. Therefore, most popularmolecular simulation packages[2124]performintegration ofequationsof motionintheCartesiancoordinates. Therefore, itis desirabletohavearobust approachtoobtaintheDoSininternal coordinatesfrommolecular dynamicssimulation(MD)performedinCartesiancoordinates.P.-K. Lai, S.-T. LinDepartmentofChemical Engineering, National TaiwanUniversity, Taipei10617, TaiwanE-mail: stlin@ntu.edu.twContract grant sponsor: National Science Council of Taiwan; Contractgrantnumber: NSC1012628-E-002014-MY3VC2015WileyPeriodicals, Inc.Journal ofComputational Chemistry2015, DOI: 10.1002/jcc.23822 1FULL PAPER WWW.C-CHEM.ORGDespite of its importance, the transformation of atomicvelocityinCartesiancoordinatestovelocitiesininternal coor-dinates is nontrivial. Unlike the normal modes, the internalcoordinates are not orthogonal to eachother[25]andover-complete (i.e., internal coordinates that canbe definedaregreater thanthedegreeof freedom). Themost well-knownapproachis theWilsons method,[26,27]wherea3N33NB-matrix is constructedtoconvert Cartesiandisplacements ofall Natoms inasystemtointernal displacements. Recently,van Houteghem et al.[28]proposed a velocity projectionmethodtoobtainbond-stretchingvelocities by consideringcontributions of Cartesian atomic velocities parallel to thebond direction. However, the extension of the method toother typesofinternal modeshasnotbeendemonstrated. Inthiswork, weappliedtheWilsonsmethodtoconvertvelocityin different coordinate systems through the B-matrix. Theadvantageof this methodis that all internal modes canbeobtainedsimultaneouslyandthecompleteDoSfrominternalvelocity canbe constructed. We validated, thus determinedinternal velocitiesusingnumerical differentiationof displace-ment ininternal coordinates, andcomparedthedifferencesofDoSdeterminedfromvelocityindifferentcoordinates. Ourresultsshowthat theinternal DoShasmanyadvantagesandprovides morephysical insights comparedtoCartesianDoSalone.MethodsandTheoryThevibrationalDoSThevibrational DoS, or velocityspectrumof acomponent, isdefinedasthemass-weightedsumof velocityspectral densityfromallatomsinthesystem,[17]St52kTXNkj51X3b51 mjsbj t (1)where mjis the mass of atomj. Nkis the total number ofatoms of molecule k. The velocity spectral density sbj t ofatomj inthebthcoordinate(b5x, y, andzintheCartesiancoordinate)isdeterminedfromthesquareoftheFouriertrans-formofthevelocitiesassbj t5 lims!112s

s2svbj te2i2pttdt

2(2)TheDoScanbealsocalculatedfromtheFouriertransformofthevelocityautocorrelationfunction.[17]Theintegrationof Stgives thetotaldegreesoffreedomofthesystem.10St53N (3)CartesianvibrationalvelocityForpolyatomicspecies, thevelocityof anatomj containedina molecule k at a time instant t can be decomposed intotranslational, rotational, andvibrationalvelocities.[6]vj;tott5vj;trnt1vj;rott1vj;vibt (4)The translational velocity (vj,trn) is set tobe the center ofmassvelocityofthemoleculevj;trn5Xmivi;totXmi(5)wherethesummationsrunover all atomscontainedinmole-culek. Theangularvelocity(x)isdeterminedfromtheangularmomentum(L)andtheinverseofprinciplemomentsofinertiatensor(I )L5Xmjrj3vj;tot5I x (6)whererjisthepositionvectorofatomj tothecenterofmassofthemolecule. Therotational velocityisthenobtainedfromvj;rot5x3rj(7)andthevibrational velocityisobtainedfromvj;vib5vj;tot2vj;trn2vj;rot(8)InternalvibrationalvelocityfromWilsonsmethodThevibrational velocityobtainedfromeq. (8) aretheatomicvelocitywiththecenter of masstranslationandrotationcon-tributions removed. Here, wewouldliketoexpress theintra-molecular vibrations (vj,vib) in terms of contributions frombondstretching, anglebending, dihedral torsion, andsoforth.TheconversionfromCartesianvelocitytointernal velocitycanbeachievedbyWilsons B-matrix.[26,27]Assuminginfinitesimalamplitudesof vibration, theatomicdisplacement inCartesiancoordinates(Dx) canbeconvertedtothoseintheininternalcoordinates(Dq)[26]Dq5BDx (9)whereBisa3Nk33Nkmatrix(Nkisthenumberof atomsinmoleculek)whoseelementsarethechangeininternal coordi-natescorrespondingtoaninfinitesimal perturbationinCarte-siancoordinates, thatis,Bab5@qa@xb(10)Notethat bdenotesanyoneofthe3NkdegreesoffreedominCartesiancoordinates (x, y, or z directions), andadenotesanyoneof bonds, angles, anddihedralsdegreesof freedomsininternal coordinates(q). Theanalytical formof elementsinB, thatis, Bab, isfairlycomplicated, especiallyfor dihedral tor-sion, whichrequires tediousalgebra[29]andmaybecomefor-midablefor largemolecules. Inthiswork, Babaredeterminedfromnumerical differentiationintermsof thechangeininter-nal coordinates with respect to perturbation in CartesianFULL PAPER WWW.C-CHEM.ORG2 Journal ofComputationalChemistry2015, DOI: 10.1002/jcc.23822 WWW.CHEMISTRYVIEWS.COMcoordinates (see Appendix for detail). Once the B matrix isavailable, the velocity inthe internal coordinates ( _ q) canbecalculatedas[30]_ q5B _ x (11)wherethevelocity _ xinCartesiancoordinatesisequivalent tovvibobtainedfromvelocity decompositionof theMDtrajec-tory. Equation(11) followsfromthetimederivativeof eq. (9),assumingBisconstantoveraninfinitesimal timeinterval. Thevalidityofthisassumptionisexaminedusingnumerical deriva-tiveof displacements ininternal coordinates describedinthenextsection.InternalvibrationalvelocityfromnumericalmethodInthe Wilsons method, the internal velocities are determinedfromeq. (11). Itisalsopossibletodeterminetheinternal veloc-itiesdirectlywithoutresortingtoCartesian velocities. Thiscanbeachievedby calculatingthe time derivatives of bonds, angles,anddihedral torsions fromthe trajectory of a MDsimulation.The definitionofbondBj(inlength),angleAj(inradian), and tor-sionTj(inradian)areillustratedinFigure1andarecalculatedasBj5jjbjjj5jjrj112rjjj (12)Aj5cos21

bj21 bjjjbj21jjjjbjjj

(13)Tj5cos21

bj223bj21 bj213bjjjbj223bj21jjjjbj213bjjj!(14)Thevelocityofeachmodeisobtainedfromnumericaldiffer-entiationasfollows_Bjt10:5s 5Bjt1s2Bjts(15)_Ajt10:5s5Ajt1s2Ajts(16)_Tjt10:5s5Tjt1s2Tjts(17)whereatimeinterval s 5 4fsisusedinourstudy.DoSfromvelocityininternalcoordinatesInWilsons method, the3Nk26internal degrees of freedom(or 3Nk25for linear molecule) consists Nk21bonds, Nk22angles, andNk23dihedrals. However, itisoftenthecasethatmanymoreinternalmodescanbedefined(overcompleteness).Forexample, ethane(Nk58)containssevenbonds(sixCHandoneCC), 12angles(sixHCHandsixHCC), andninedihedrals(nineHCCH), althoughthereshouldonlybesixuniqueanglesandfiveuniquedihedrals. Notethattheenergychangeassoci-atedwiththedisplacementofeachofthese28internalmodes(forcefieldparameters) needstobespecifiedinaMDsimula-tion. Intheoriginal Wilsonsmethod, oneneedstoselect theuniqueangles anddihedrals (andthecenter of mass transla-tionandrotation) toformasquarematrixB[eq. (10)]. Thisisnecessary if one were to evaluate the G-matrix where theinverse of B is required. However, all (more than 3Nk26modes) internal modes canbeincludedintheB-matrix, andthevelocityassociatedwitheachmodecanbeobtained. Wehaveincludedall theinternal modes inthecalculationof B-matrix in this work and rescaled their contributions to theoverall internalDoS, thatis,SIntvibt52kXNBa51satTaNk21NB1XNAa51satTaNk22NA1XNDa51satTaNk23ND" #(18)whereTais thetemperatureof aninternal modea, andNB,NA, andNDarethenumber of bonds, angles, anddihedrals,respectively. ThescalingofDoSisduetoovercompletenessofinternal modes. This scalingensures that theDoSintegrationwouldbeequal tointernal degreeof freedom[eq. (3)]. TheDoSofeachinternal modeisdefinedassat5 lims!112s

s2sIap_ qate2i2pttdt

2(19)whereIaisthereducedmassassociatedwithinternal modea.For a dynamic system, Iais time and structure dependent.Note that integrating eq. (19) over frequency and applyingParsevalstheoremgives10satdt512hIa _ q2ai512kTa(20)In the next section, we discuss the evaluation of thereducedmass.MassofinternalmodeThemass associatedwitheachinternal modeisneededforthe evaluationof kinetic energy andtemperature associatedwiththemode. Massandtemperaturearealsoimportant forthecalculationofthermodynamicpropertiesfromtheDoS.[58]Thereducedmassforbondandanglecanbedeterminedana-lyticallyfordiatomicandsometriatomicmolecules.[31,32]How-ever, thereisnoanalytical expressionfor thereducedmassofdihedral torsion, andvariousapproximations[33]wereproposedfor this purpose. In this work, we propose a new, simplemethod for Iafromthe mass of atoms associated with theinternal mode a.Figure1. Schematicrepresentationofbond(B), angle(A), andtorsion(T).FULL PAPER WWW.C-CHEM.ORGJournal ofComputational Chemistry2015, DOI: 10.1002/jcc.23822 3Bond-reducedmass. Thereducedmassforbondthatbelongstoatomsj andj 11(Fig.2a)canbedeterminedexactlyasIbond5mjmj11mj1mj11(21)Angle-reducedmass. Theharmonicmotionof anglebendingisconsideredasthestretchingoftworeducedbonds(thetwosquares inFig. 2b). Thereducedmass for anglethat belongstoatomj 21, j, andj 11isIangle5lj21lj11lj211lj11(22)wherelj215mj21mjmj211mjjjrj21;jjj2(23)lj115mj11mjmj111mjjjrj11;jjj2(24)Torsion-reducedmass. FromNewmanprojection, thedihedraltorsionis similar toangle-bendingmotion, providedthat thebondvector isprojectedperpendicular torotational axis(seeFig. 2c). Therefore, thereducedmassfor torsionthat belongstoatomj 21, j, j 11, j 12 can be evaluatedas that of anangleItorsion5lj21lj12lj211lj12(25)wherelj215mj21mjmj211mjjjrj21;jjjsinAj2(26)Aj5cos21

rj21;j rj;j11jjrj21;jjjjjrj;j11jj!(27)lj125mj12mj11mj121mj11jjrj12;j11jjsin Aj112(28)Aj115cos21

rj12;j11 rj12;j11jjrj12;j11jjjjrj12;j11jj!(29)Theaccuracyof this methodis testedfromcorrespondingtemperatureofeachmodebyequipartitiontheorem. Thetem-peratureassociatedwithaninternal modeacanbecalculatedasTa5k(30)where hIa _ q2aiistwicethekineticenergyassociatedwithinter-nal mode a, and the angle brackets denotes ensembleaverage. For a thermally equilibrated system(i.e., equiparti-tion[34,35]is satisfied), the temperature determinedfromanydegreesof freedomof thesystemisthesameasthesystemtemperatureT. Therefore, theratioof TaandTisthusanindi-cationofappropriatenessoftheproposedinternal mass.ca5TTa(31)As thereducedmass proposedineqs. (21), (22), and(25)assumes no coupling between different internal modes, thedeviationof cashouldbeclosetounity for small moleculesandmay deviate fromunity for large molecules where cou-plingofreducedmassbecomeimportant.However, it is notedthat althoughtemperature is relatedwithmass, their valuescancel outfor total internal DoSineq.(18). Inother words, theperformanceof DoSis irrelevant ofmass, butasstatedearlier, accuratemassandtemperatureareimportantforsubsequentanalysisusingDoS.Notethat theWilsons methodalsoprovides anequivalentmass calculationinamatrixform. Thismassmatrixhasbeenusedinaninternal coordinatemolecular dynamicsaswell.[18]This is obtainedfromthefact that thekinetic energy(EK) isindependentofcoordinatesystem, thatis,2EK5_ xTM _ x5_ qTB-1TMB-1_ q5_ qTG-1_ q (32)whereMisthemassmatrix(Mjl5mjdjlwithmjbeingthemassofatomassociatedwithdegreeoffreedomj), GistheWilsonsG-matrix,[27]whose inverse provides the effective mass ofinternal modesG215B21TMB21(33)AsthematrixG21isnotdiagonal, themassassociatedwitheachinternal modecannotbeobtainedfromthismatrix.ComputationalDetailsThe vibrational DoS of oxygen, water, hydrogen peroxide,methanol, hexane, andubiquitinareevaluatedbasedonveloc-ity in Cartesian and internal coordinate systems. Table 1Figure2. Schematicrepresentationforcalculationof thereducedmassforbond(a), angle(b), anddihedral torsion(c).FULL PAPER WWW.C-CHEM.ORG4 Journal ofComputationalChemistry2015, DOI: 10.1002/jcc.23822 WWW.CHEMISTRYVIEWS.COMsummarizesall thesystemsandconditionsstudiedinthiswork.Oxygen, water, andhydrogenperoxidemolecules areusedtorepresent standarddiatomic, triatomic, andtetratomicmodels.Thedetailsof their forcefieldparametersareprovidedinSup-portingInformation. Theinteractionsfor methanol andhexanearedescribedbytheOPLS-AAforcefield,[36,37]andubiquitinisdescribed by AMBER03 force field.[38]Open software GRO-MACS[21]isusedformoleculardynamicssimulations.Forsimula-tions in pure liquid phase, 1000 molecules are randomly insertedinto a 3D periodic box of desired density (0.58 g/cm[3]) using thePackmol program.[39]Each system was first stabilized with energyminimizationandthensimulatedunder constant temperature(300 or 800 K) and constant volume for 1 ns with a timestep of 1fsforequilibration. Anadditional 200-pssimulationwassubse-quently performed and the trajectory recorded every 4 fs for theDoS analysis. Note that the simulation procedure for ubiquitin (1ubiquitin submerged in 5738 TIP3P water[40]) follows Ref. [41] TheV-rescalealgorithminGROMACSisappliedwithtimeconstant0.1 ps to control the temperature. The nonbond and electrostaticcutoff are both set to 10 A. Particle-Mesh Ewald (PME) is used tocalculatelongrangeinteractions. TheFourier spacingis1.2A,andthePMEorder is4. Oncethesimulationiscompleted, thetotal atomic velocity in each recorded timestep is further decom-posed into translational, rotational, and vibrational velocities [eqs.(5)(8)]. The vibrational velocities are then converted into internalvelocities by Wilsons B-matrix [eq. (11)]. The elements of B-matrixare calculated from the derivative of internal coordinate displace-ment withrespect toperturbationof Cartesiancoordinatebycentral difference method. In other words, the change of internalcoordinates (qa) corresponding to the small displacement ofdegreeof freedomxbinCartesiancoordinate(bothxb1handxb2 h with h51026A) are used to determine Bab[eq. (10)]. ThereportedDoShereareaveragedfrom10DoScalculationsusingthefinal 200-pstrajectory(i.e., 20pseach). Theminimumandmaximum frequencies are 0 and 4168.44 cm21, respectively, witha resolution of 1.112 cm21.ResultsandDiscussionDoSofoxygenOxygenisusedasthesimplesttypeofsystemswithoneinternalmotion, bondstretching. Figure3comparestheDoSof oxygenevaluatedusingCartesianandinternal velocities. It canbeseenthat evenfor thesimplest caseDoSdeterminedfromdifferentcoordinate systems differ. In Cartesian coordinates, the DoS issplitintotwopeaksaroundtheactualbondstretchingfrequency(698cm21). Notethat thetheoretical frequencyobtainedusingk=lp=2pis694cm21. Thisisthewell-knownrovibrational cou-plingindiatomicspecies.[42]However,onlyonepeakatthesamebondstretchingfrequencyisobservedwhentheinternalvelocityisusedforthecalculationofDoS. Inthiscase, thebondstretch-ingvelocitydeterminedfromtheWilsonsB-matrixisidentical tothevelocityprojectionofatomicvibrationalongthebonddirec-tion. Inother words, thevelocity component perpendicular tobonddirection(resultinginrotation) isremoved. Asaresult, weobserveacleanpeakof bondstretchingintheinternal DoSofdiatomicspecies. Theinternal DoShas theadvantageof distin-guishingpeaksfrom rotation and internal vibrations.DoSofwaterThenextsimplest possiblecaseisatriatomicspecies, suchaswater, whichcontains twobondstretchings andone angle-bendingmodes. Figure4illustratesthattheDoSfromthetwocoordinatesystemsaresimilarwiththebendingmodelocatedat 1440 cm21andsymmetric stretchingat 3675 cm21, andasymmetricstretchingat3700cm21. However, itispossibletodecomposetheinternal DoStocontributions fromeachindi-vidual degreeof freedom: onebondbendingandtwoequiva-lent bond stretchings. In such a case, each internal bondTable1. Systemsstudiedinthiswork.System EnsembleNum. ofmolecules ConditionOxygen NVT 1000 300K, 5.531025m3/molFlexiblewater NVT 1000 300K, 2.753 1025m3/molHydrogenperoxideNVT 1000 300K, 5.531025m3/molMethanol NVT 1000 300K/800K, 5.531025m3/molHexane NVT 1000 300K/800K, 1.531024m3/molUbiquitin(inwater)NPT 1(5738) 310K, 1atmFigure3. DoS(bottom) andits integration(top) of oxygenat 300Kand5.531025m3/mol. (Blue: bond; green: Cartesian).FULL PAPER WWW.C-CHEM.ORGJournal ofComputational Chemistry2015, DOI: 10.1002/jcc.23822 5stretchingcontainsbothsymmetricandasymmetricmodes. Itisnotedthat asmall bendingpeakexist inthebond-stretchingfrequency (around 3650 cm21). This implies that the normalmodeof bondstretchinginvolvedsymmetric bondstretchingand a small portion of internal angle motion. Similarly, oneobserves the internal bond stretching (blue curve) having averysmall peakat1450cm21, implyingthatthebondstretch-ings (normal mode) are accompanied with angle bending.These results are consistent with the fact that the internalmodes arenotorthogonal, andthenormalmodes arecombina-tions of theinternal modes. Thecomparisonof CartesianDoSandinternal DoSthusallowsforidentificationofall theinternalmodes associatedwithacertainnormalmode.DoSofhydrogenperoxideHydrogenperoxideisusedasanexampleofthesimplestmole-culethat has internal rotation. Thetorsional modeis relativelysoft comparedtoangleandbondmotions andis responsibleforconformationchangesoflargespecies. TherearesixinternalmodesinH2O2, onedihedral torsion(HOOC), twoanglebend-ings(HOO), andthreebondstretchings(twoHO, oneOO). WiththeaidoftheintegrationofDoS(topfigureofFig. 5), itcanbeseen that the system exhibit one degree of freedom at700cm21, another oneat 950cm21, twomoreat 1320cm21,andremainingtwo at 3790 cm21. Althoughthe overall DoSfromthetwocoordinatesystems aresimilar, theinternal DoSallows for the identification of normal modes with internalmolecular degree of freedom. For example, the lowest fre-quencymodeat700cm21comesfromthetorsionaldegreesoffreedom(orangecurve).Thesecondpeakat950cm21inCarte-sianDoSinvolvesOObondstretching, inadditional toasmallportionof anglebending. Thethirdpeakat 1320cm21corre-sponds mostly twoHOOangle bendingandsome OObondstretching (see inset plot), and the highest frequency peakat 3790 cm21comes from OH stretching. Note that thesummation of degree of freedom integration for internalDoSuptohighestfrequencyshouldbeequal tothatfromCar-tesianDoS.DoSofmethanolThesimulationof methanol moleculeis usedtoexaminethefreerotorbehaviorof dihedral torsion(HACAOAH). Methanolcontainssixatoms, fivebonds(threeCH, oneCO, andoneOH),Figure 4. DoS (bottom) and its integration (top) of flexible waterat 300 K and 2.75 31025m3/mol. (Blue: bond; red: angle; green:Cartesian).Figure5. DoS(bottom) andits integration(top) of hydrogenperoxideat300Kand5.531025m3/mol. (Blue: bond; red: angle; orange: dihe-dral; green: Cartesian; purple: OObond; pink: OHbond).FULL PAPER WWW.C-CHEM.ORG6 Journal ofComputationalChemistry2015, DOI: 10.1002/jcc.23822 WWW.CHEMISTRYVIEWS.COMFigure6. DoS(bottom)anditsintegration(top)ofmethanolata)300Kandb)800Kand5.5 3 1025m3/mol, respectively. (Blue: bond; red: angle; orange:dihedral; green: Cartesian). [Colorfigurecanbeviewedintheonlineissue, whichisavailableatwileyonlinelibrary.com.]Figure7. DoS(bottom) anditsintegration(top) ofhexaneata) 300Kandb) 800Kand1.531024m3/mol, respectively. (Blue: bond; red: angle; orange:dihedral; green: Cartesian). [Colorfigurecanbeviewedintheonlineissue, whichisavailableatwileyonlinelibrary.com.]FULL PAPER WWW.C-CHEM.ORGJournal ofComputational Chemistry2015, DOI: 10.1002/jcc.23822 7sevenangles(threeHCH, threeHCO, andoneCOH), andthreedihedrals (three HCOH). Figures 6a and6billustrate the DoSmethanolat300and800K,respectively. Itcanbeseenthatthenormal modes(greencurve) below800cm21constitutesolelyfromthedihedral torsion(orangecurve). At lowtemperature(300K), thedihedral torsionis under hinderedrotation(peaksat 100and650cm21) withasmall fractionof diffusional rota-tion(indicatedbyafiniteintensityat zerofrequency). As thetemperatureincreasesto800K, thelowfrequencypeakdisap-pears and the internal rotation diffusion is enhanced signifi-cantly. At high temperature (800 K), the dihedral torsion(HACAOAH)mayovercomethetorsionalbarrierandbecomeafreeinternal rotor. Althoughtheredshiftofthehindereddihe-dral rotationis capturedby the CartesianDoS, the enhance-mentofrotationaldiffusioncannotbe observed. Thisshowstheadvantage of the internal DoS for analyzingthe transitionofinternalrotationsof conformationallyflexiblemolecules.DoSofhexaneHexane is chosen as a representative for species containingmultipledihedral torsions. Figure7presentsDoSofliquidhex-aneat300and800K, respectively. ThetorsionDoSshowslessdiffusivemotioncomparedtothatofmethanol duetoamoresignificant sterichindranceof hexane. Another significant dif-ferencebetweenmethanol andhexanecasesisobservedfromCartesianDoSat lowfrequency. For methanol, theCartesianDoSis smaller thanthat frominternal DoSat lowfrequency,whileforhexane, theoppositebehaviorisshown(greencurveFigure 8. DoS (bottom) and its integration (top) of united-atomhexanewithfixedbondat 300K. (Red: angle; orange: dihedral; green: Cartesian;purple: angle1dihedral). [Color figurecanbeviewedintheonlineissue,whichisavailableatwileyonlinelibrary.com.]Figure9. Thetorsional DoSofthefirsttofifthdihedral angles(ae, respec-tively)forhexaneat300(blue)and800K(red). [Colorfigurecanbeviewedintheonlineissue, whichisavailableatwileyonlinelibrary.com.]FULL PAPER WWW.C-CHEM.ORG8 Journal ofComputationalChemistry2015, DOI: 10.1002/jcc.23822 WWW.CHEMISTRYVIEWS.COMabovetheorangecurve). Thesignificantlyenhancedlowfre-quencynormal modesisexaminedusingunited-atomhexane(DREIDINGforce field[43]) with all bonds fixed. In this case,thereareseveninternal degreesof freedom: four CCCanglesandthreedihedraltorsions.Figure8showstheDoSfortheunitedhexanefromtheCar-tesianandinternal velocities. Fromtheinternal DoS, thethreedihedral torsionconcentratedaround140cm21, andthefourangle bending appear at 300 (one degree of freedom), 340(onedegreeof freedom), and410cm21(twodegreesof free-dom). Itcanbeseenthattheintegrationof theCartesianDoS(greencurve) isalmost alwayshigher thanthat of theinternalDoS(purplecurve) before400cm21. Thedifferencesbetweenthetwoarecompensatedbytheanglebendingat 410cm21,wheretheCartesianDoS shownone degreeof freedomandtheinternal DoSshowstwodegreesoffreedom. Therefore, theenhancedlowfrequencynormal modes (e.g.,