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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
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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:
[email protected] 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.,