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A first principles theoretical study of the hydration structure and dynamics of an excess proton in water clusters of varying size and temperature

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A first principles theoretical study of the hydration structure and dynamics of an excess proton in water clusters of varying size and temperature
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  A first principles theoretical study of the hydration structure and dynamicsof an excess proton in water clusters of varying size and temperature Arindam Bankura 1 , Amalendu Chandra ⇑ Department of Chemistry, Indian Institute of Technology, Kanpur 208016, India a r t i c l e i n f o  Article history: Received 12 May 2011In final form 6 July 2011Available online 18 July 2011 Keywords: Excess protonWater clusters  Ab  initio molecular dynamicsSurface solvationVibrational spectrumProton transfer a b s t r a c t We present a theoretical study of the structure and dynamics of protonated water clusters by means of quantum chemical calculations and  ab initio  molecular dynamics simulations. We have considered theclusters H + (H 2 O) n  for four different sizes corresponding to  n  =5, 9, 17 and 21. We have first looked atthesolvationstatesoftheexcessprotoninseverallowerenergystructuresoftheseclusterswithaspecialinterest infindingitssurfaceversusinteriorstatesanditshydrogenbondingenvironment. Subsequently,we have investigated the finite temperature behavior of these clusters through  ab initio  simulations. Wehave looked at vibrational spectral features with special emphasis given to the spectral features of freeOH (deuterated) modes and their dependence on donor–acceptor hydrogen bonding states of the watermolecules. Wehavealsoinvestigatedthemechanismandkineticsofprotontransfereventsintheseclus-ters by using a population correlation function approach.   2011 Elsevier B.V. All rights reserved. 1. Introduction Studies of neutral and charged water clusters have been a topicof great current interest because of their fundamental importanceinmanyareas of sciencesuchassolvationchemistry, biochemistryandatmosphericscience[1–4].Protonatedwaterclusters,inpartic-ular, are highly relevant in biological sciences, especially in caseswhere water molecules in a finite-sized environment are expectedto behave differently from bulk water. The condensation of waterdropletsonionsisofcentralimportanceinatmosphericchemistry.Formation of the noctilucent clouds can take place in the atmo-sphere in presence of protonated water clusters through ion-induced nucleation mechanism [5]. For an acid–base reactionoccurring in an aqueous medium of the type HX ð aq Þ  ¼ H þð aq Þ þ X ð aq Þ ,accurate characterization of the structural and dynamical aspectsof the solvated proton is a rather difficult task. Studies of proton-atedwaterclustersinthiscontextcanprovideveryusefulinforma-tion of the structural and dynamical behavior of these excessprotons. Indeed, there have been a rather large number of studieson protonated water clusters, both experimentally and theoreti-cally, over past 2–3 decades. Mass-spectrometric studies [6–8]provided evidence for the existence of protonated water clustersH + (H 2 O) n , someof whicharemorestablethanothers. Forexample, n  =21,28,30,37,50 . . .  are knownas ‘magic numbers’ [6–11]. Thesemagic number clusters have long been believed to be polyhedralcagestructuresenclosingeitherawatermoleculeoranH 3 O + ionin-side the cage. Information on hydrationstructure of an excess pro-ton in H + (H 2 O) n  has been experimentally obtained by highresolution infrared (IR) spectroscopy [12–19]. The structural infor-mationcontentisquitelimitedinthehydrogenbondedOHstretch-ingregionbecauseof thebreadthandcongestionof theabsorptionbandsinthespectrawhereasthenon-hydrogen-bondedorfreeOHstretching band is relatively sharp and well separated fromthe broad hydrogen-bonded OH absorption at 3000  3600cm  1 .The frequency of a free OH stretching band appeared at3600  3750cm  1 and the presence of these free OH modescontains useful information about the structure of these clusters.Furthermore, the free OHfrequencies are different for water mole-cules in different hydrogen bonding environments. For example, athree-coordinated H 2 O acting as a double-proton-acceptor andsingle-proton-donor (AAD), a two-coordinated H 2 O acting as a sin-gle-proton-acceptor/single-proton-donor (AD) or double-proton-acceptor (AA), and a one-coordinated H 2 O acting as a single-pro-ton-acceptor (A) or a single-proton-donor (D) have different fre-quencies for their free OH modes. Using this concept, Yeh et al.[12] and Jiang et al. [13] conducted pioneering IR spectroscopic investigations on small to mediumsized H + (H 2 O) n  ( n 6 8) clusters.They reported vibrational spectra of these clusters and interpretedtheirresultswiththehelpof  ab initio calculations.Headrickandco-workers [15] showed how the vibrational spectrum of protonatedwaterclustersevolvesinthesizerangefrom n  =2to11duetolocalenvironment of the excess proton. From the distinct spectralchanges in the free-OH stretching region, it is found that the 0301-0104/$ - see front matter   2011 Elsevier B.V. All rights reserved.doi:10.1016/j.chemphys.2011.07.008 ⇑ Corresponding author. Fax: +91 512 2597436. E-mail address:  amalen@iitk.ac.in (A. Chandra). 1 Present address: Department of Chemistry, Institute for Computational MolecularScience, Temple University, PA 19122, USA. Chemical Physics 387 (2011) 92–102 Contents lists available at ScienceDirect Chemical Physics journal homepage: www.elsevier.com/locate/chemphys  structureoftheclusterionsretainsachainliketopologyupto n  =6,thesingle-ringisomerfirstappearsat n  =7andthecage-liketopol-ogy first appears at n=11. Shin et al. [16] and Miyazaki et al. [17] recordedtheIRspectraof protonatedwaterclustersH + (H 2 O) n  with n  up to 27. These authors independently showed that the spectralfeaturesofthedanglingOHstretchofwatermoleculeschangewithincreasingclustersizeanditwasconcludedthatthreedistinctmor-phological forms exist in protonated water clusters. Protonatedclusters with less than ten water molecules form two dimensionalchainandringtypestructures.Forclusterscontaining10–21watermolecules, the hydrogen-bond network forms a two-dimensionalnet type structure to three-dimensional cage like structure withincreasing number of water. The magic number cluster H + (H 2 O) 21 has been the subject of substantial interest to experimentalists[6–11,16,17]. Johnson and co-workers [16] reported that the infra- redspectrumofH + (H 2 O) 21 inthefreeOHstretchingregionhasonlya single line at  3695cm  1 , thus showing that all the water mole-cules of H + (H 2 O) 21  with a dangling OH are in similar binding state(AAD). However, in the study of Miyazaki and co-workers [17] onH + (H 2 O) 21  in a warmer molecular beam, the presence of twoabsorption bands in the same frequency range were found whichmeant the existence of two different binding types of water mole-cules in the cluster with a dangling OH. More recently, Mikamiand coworkers [19] reported an IR spectroscopic study of size-se-lected protonated water clusters H + (H 2 O) n  over a wider range of  n  =15–100.Size-selective protonatedwater clusters have also been investi-gated theoretically [3,13–16,19–47] in recent years. In order toelucidate the hydration structure of the excess proton and itstransfer mechanism, a large number of different models have beenemployed ranging from empirical potentials [3,20–34] to  ab initio [13–16,19,27,34–47] studies. Hodges and wales [21] obtained the global-minimum structures of protonated water clusters for n  =2–21.Theircalculationsshowedapropensityinstructuraltran-sition from tree-like geometries at  n  =2–4 to four-membered andfive-membered rings at  n P 5, to cage-like configurations at n P 9, and finally to a distorted pentagonal dodecahedral cageenclosing a neutral water molecule at  n  =21. While a great dealof efforts have been given on investigations of the global minimastructures of different sized protonated clusters either by  ab initio methods or by empirical models, less attention has been paid toinvestigate the finite temperature behavior of these clusters. Itmay be noted that the protonated water clusters in laboratoryexperiments or in atmosphere are not at zero temperature. Schin-dleretal.[48]estimatedthetemperatureoftheirclustertobenear140K, while Jiang et al. [13] reported their cluster temperature tobe about 170K. The temperature of polar stratospheric cloud par-ticles has been estimated to be  190K [49]. In order to study thefinite temperature behavior of these clusters, several molecularsimulations [3,10,23–25,27,28,31–39,44,45] have been carriedoutonH + (H 2 O) n  overdifferenttemperatureranges.Specialempha-sis has been placed on solid-like to liquid-like phase transitions byseveral authors using both Monte Carlo [3,10,23–25,27] andmolecular dynamics [28,31–39,43–45] methods. To understandthe thermal properties of protonated water clusters, simulationsin the grand canonical ensemble have also been carried out at250K [28]. Singer et al. [25] and Christie et al. [26] examined the temperaturedependenceof thetopologyof   n  =6, 8and16clustersand they found that the major structural transitions involve reor-ganization of structures with treelike topology to ring-containingstructures. Klein and coworkers [8] focused on examining finitetemperature effects on the structure of H + (H 2 O) n  ( n  =5–22) usingthe parallel tempering method with OSS2 model [30]. Their calcu-lated trends of global minima structures were found to be qualita-tivelysimilartotheresultsbyHodgesandWales[21].However,allthe clusters were found to undergo significant structural changesdue to finite temperature effects. The structural and vibrationalproperties of the protonated clusters at finite temperature havealso been studied by Kim and co-worker [46] for  n 6 10 by usingquantum chemical calculations and  ab initio  molecular dynamicssimulations. Their calculated vibrational spectra were found to beconsistentwithexperimentalobservations.Recently,thestructure,dynamics, and vibrational spectra of a hydrated proton in waterclusters of   n =21, 31, 41, 51 have been studied at two differenttemperatures by using the  ab initio  atom-centered density matrixpropagation simulations [34]. Based on the results of these finitetemperature simulations, it was concluded that the protonatedspecies in these clusters resides on the surface rather than beingcompletelysolvatedintheinterior.Adetailedanalysisofthevibra-tional properties of protonated clusters were also carried out andcompared with the experimental findings. It was concluded thatthedynamicalandfinite-temperatureeffectsplayasignificantroleindeterminingthevibrationalspectralpropertiesofthesesystems.Another important aspect of these protonated water clusters istherateof protontransferinsuchfinite-sizedsystemsandtowhatextent the mechanismof proton transfer in these finite-sized clus-ters differs from that in bulk water. Majority of the existing theo-retical studies on protonated water clusters focused on either theglobal minimumstructures of these clusters or on how the overallstructures change due to finite temperature effects. A few theoret-ical calculations have looked at the dynamics and mechanism of proton transfer in these clusters [31,34,45]. Much more work isneededinthisareatohaveabetterunderstandingofthehydrationand translocation of excess protons in nano-sized hydrogenbonded clusters. One of our aims in this work is to provide a uni-fied study of the structural and vibrational properties and alsothe rate and mechanism of proton transfer in these protonatedclusters at finite temperature. In this context, it would also beworthwhile to investigate how the cluster properties depend onthe cluster size and temperature.  Ab initio  molecular dynamicsmethods appear to be particularly useful in the elucidation of theproton transport mechanism which can depend strongly on boththelocal solvationenvironmentof theionandthe hydrogenbond-ing fluctuations in the surrounding water molecules [50–52]. Inthis work, we have investigated some of the lower energy struc-tures of selected protonated water clusters by means of quantumchemical calculations and then the dynamical properties of theseclusters are investigated by  ab initio  molecular dynamics simula-tions. We have also calculated the probability distributions of hydrogen bonding states of water molecules and the hydroniumion and also vibrational frequencies fromfinite temperature simu-lations. In addition, the rates of proton transfer processes havebeen calculated by using the population correlation function ap-proach [53,54]. The rate of proton transfer is found to vary signif-icantly with cluster size for the clusters considered in this study.Therestofthepaperisorganizedasfollows. Thecomputationaldetailsofquantumchemicaland abinitio moleculardynamicscalcu-lationsaredescribedinSection2.Ourresultsoftheoptimizedgeom-etries, hydration structures and vibrational power spectra arepresentedinSection3.Thekineticsandmechanismofprotonmigra-tion in the present clusters are discussed in Section 4. Finally, ourmainresultsandconclusionsarebrieflysummarizedinSection5. 2. Computational methods We first carried out all-electron quantumchemical calculationsof H + (H 2 O) n  ( n  =5, 9, 17 and 21) clusters to find the location of theexcessprotoninsomeofthelowenergyisomersoftheclusters.Allquantum chemical calculations were done at the post-Hartree–Fock  ab initio  level using the Möller–Plesset second order (MP2)perturbation theory that includes electron correlation effects and  A. Bankura, A. Chandra/Chemical Physics 387 (2011) 92–102  93  alsoatdensityfunctionaltheory(DFT)levelusingthegradientcor-rected exchange–correlation functionals of BLYP[55,56] andB3LYP[56,57]. The Gaussian atomic basis set 6–31+G ⁄ is used forthe molecular orbital expansionwhich contains extra diffuse func-tions. The calculations were carried out using the GAUSSIAN 03program [58]. In calculating the stabilization energies of the pro-tonatedclusters,weaccountedforthebasissetsuperpositionerror(BSSE) by using the so-called counterpoise method [59].The  ab initio  molecular dynamics simulations were performedby means of the Car–Parrinello method [60,61] and the  CPMD  code[62]. A simple cubic box of length L containing an excess protonimmersed in a cluster of n water molecules was taken in isolationfor carrying out the simulations. Four different cluster sizes of  n  =5, 9, 17 and 21 with box lengths of   L  =15, 16, 18 and 18.52Å,respectively, were used. In each case, the cluster was kept in thecentral region of the box and the length of the box was taken tobe sufficiently large so that there was enough empty space aroundthe cluster in all directions. The electronic structure was repre-sented within the Kohn–Sham (KS) formulation [63] of densityfunctional theory within a plane wave basis. The core electronswere treated via the Troullier–Martins normconserving pseudopo-tentials [64] and the plane wave expansion of the KS orbitals wastruncated at a kinetic energy of 70Ry. A time step of 5.0a.u. alongwith an fictitious electron mass parameter  l  =800a.u. were em-ployed. In order to reduce the importance of nuclear quantum ef-fects associated with the protons, all protons in the system weregiven the mass of deuterium. Also, the use of deuterium mass inplace of hydrogen ensures that electronic adiabaticity and energyconservation are maintained throughout the simulations for thechosen values of the fictitious electronic mass parameter and timestep. We note that this mass assignment does not affect the struc-turalproperties,hencewewillcontinuetousethenomenclatureof ‘H’ rather than ‘D’. However, dynamical properties such as intra-molecular vibrational features change due to deuterium massassignment and, hence, an explicit mention of deuterium is madewhen such dynamical results are discussed. The  ab initio  MD sim-ulations have been performed using the BLYP [55,56] functional.Wenotethatthisfunctionalhasbeenusedinmanyearlier ab initio molecular dynamics studies of hydrogen bonded systems such aswater, [65–69], methanol [70] and also ammonia [71–73]. We equilibrated the systems for about 8ps in NVT ensemble usingNose–Hoover chainmethodat 300K and, thereafter, we continuedthe runs in NVE ensemble for another 20–25ps for calculations of thevariousstructuralanddynamicalquantities.For n  =21,wealsoperformed an additional simulation at a lower temperature of 150K. 3. Structural properties and vibrational frequencies In this section, we present the structural and vibrational spec-tral properties of different protonated water clusters consideredhere. It is known from previous studies that the lowest energystructures of these protonated water clusters transform from onedimensional chain to three dimensional cage through two dimen-sional ring type structures with increasing cluster size. Here wefirstdescribesomeofthelowerenergystructuresthatwehaveob-tained from quantum chemical calculations at BLYP, B3LYP andMP2 levels. Subsequently, the results of our  ab initio  moleculardynamics simulations will be discussed.Inordertoidentifytheoxygentowhichtheexcess protonis at-tached for a given configuration, we first assign to each oxygen itstwo nearest hydrogen atoms. Then the hydrogen that is left out isidentifiedastheexcessprotonanditisthenassignedtoitsnearestoxygen and this particular oxygen is identified as the hydroniumoxygen O ⁄ [45,52,53]. The hydration structure of the hydroniumion in our molecular dynamics simulations is investigated by cal-culating the number of water molecules in its first hydration shelli.e., thecoordinationnumberN O ⁄  andalsoby countingthenumberof hydrogen bonds that are donated through its three hydrogensi.e.hydrogenbondnumberN HB .Thehydrogenbondsarecalculatedbyusingacut-offof2.5Åforthehydrogen–oxygendistanceoftwoneighbouring molecules and an oxygen–oxygen cut-off distance of 3.5Å is used for the first solvation shell to calculate the coordina-tion number. We note that in our finite temperature simulations,the excess proton is not attached to a given oxygen for the entiretrajectory due to the presence of proton transfer events. Hence,the index of the hydronium oxygen O ⁄ changes along the simula-tion trajectory.We have also calculated the distributions of hydrogen bondingstates of all water molecules present in the clusters. The state of awater molecule in the clusters can be very different from that inthe bulk phase. A water molecule in the bulk phase is mostly four-foldcoordinatedwhereitacceptstwohydrogenbondsanddonatestwo hydrogen bonds i.e. double-acceptor and double-donor(AADD) type whereas water molecules in clusters can be one tofour-coordinated. Depending on the number of acceptor–donorhydrogen bonds, the state of a water molecule in a cluster can beone-coordinated A or D, two-coordinated AA, DD or AD andthree-coordinated AAD or ADD type. The vibrational properties of the clusters are found out by the calculating the Fourier transformof the velocity-velocity time correlation functions of all the atomswhich are obtained from  ab initio  simulations. The velocity timecorrelation functions of the hydrogen atoms captures signaturesof their hydrogen bonding states. The Fourier transform of thevelocity time correlation function produces the power spectrumor the vibrational density of states S  ð x Þ ¼ Z   1 0 C  ð t  Þ cos x tdt  :  ð 1 Þ We note that the hydrogen bonded OD stretching frequencies aremuchlessthanthatofthefreewaterorfreeODvibrationalfrequen-cies. Normally, the non hydrogen bonded or free OD frequenciesappear above 2450cm  1 whereas the hydrogen bonded ODfrequencies typically appear below 2400cm  1 .  3.1. H  + (H   2 O) 5 Wefirstdiscusstheresultsof quantumchemicalcalculationsof geometry optimizations. Out of the many geometries that we haveexplored for H + (H 2 O) 5 , we have shown the four most stable struc-turesinFig. 1. These include oneopen, noncyclic, chaintypestruc-ture HW5I, two four membered ring type structures of symmetricHW5II and asymmetric HW5III and one five membered ring typestructure HW5IV. In the chain type structure, three water mole-cules are A type and one water molecule is AD type. In the fourmembered ring structures, one A type, one AA type and two ADtype water molecules are found to be present. One AA type andtwo AD type water molecules are present in the five memberedring. The hydronium ion forms threefold coordinated Eigen cationH 9 O þ 4  in the chain type and four membered ring type structureswhereas it exists in a two-coordinated state in the five memberedring.BothB3LYPandMP2calculationsfavourthesymmetricsinglefour membered ring (HW5II) structure over the chain type (HW5I)and five membered ring type (HW5IV) structure (see Table 1). It isfoundthatHW5IIismorestablethanHW5I,HW5IIIandHW5IVby0.65, 2.35 and 4.32kcal/mol, respectively, at the B3LYP level. Thering type HW5III structure is found to transform to chain typeHW5I structure in our MP2 calculations. Also, for MP2, the fourmember ring type HW5II is more stable than HW5I and HW5IVstructuresby0.98and5.23kcal/mol,respectively.OurBLYPresultsalso show that the ring type (HW5II) structure is more stable than 94  A. Bankura, A. Chandra/Chemical Physics 387 (2011) 92–102  the other isomers. We note in this context that Jiang et al. [13] re-ported the chain type structure to be more stable than the ringtype structure whereas Hodges et al. [21] have reported the ringtype structure to be the global minimum structure for this cluster.At finite temperature, configurations evolve from one structureto another giving rise to a distribution of hydrogen bonding statesofwatermolecules.Suchadistributionasobtainedfromour ab ini-tio  simulations is shown in Fig. 2(a). We see that three water mol-ecules are in A state, one water molecule remains in AD state andthe hydroniumion is in DDD state. Thus, the chain type structuresare found to be preferred at finite temperature. Fig. 3(a) shows thepower spectra of vibrational frequencies of the H + (H 2 O) 5  cluster.The sharpest features appearing near 2450cm  1 and 2535cm  1 are attributed to symmetric and asymmetric stretching vibrationsof single coordinated A type water molecules. For the two-coordi-nated AD type water molecules, free OD stretching frequency ap-pears at   2515 cm  1 and the hydrogen bonded OD frequencyappears at less than 2400cm  1 . The OH stretching modes thatareinvolvedinhydrogenbondsarered-shiftedandare alsobroad-ened considerably with respect to the free OH stretching bands.The experimental spectra for the symmetric and asymmetric OHstretching bands appear at 3650 and 3740cm  1 for one-coordi-nated A type H 2 O molecules and the free OH stretching band ap-pears at 3715cm  1 for two-coordinated AD type H 2 O molecules[17]. A comparison of the experimental vibrational spectra of H + (H 2 O) 5  with our calculated power spectra clearly favours domi-nanceof HW5I type structures as was alsosupported bythe distri-butions of hydrogen bonding states (Fig. 2(a)). In Fig. 4, we have shownthehydroniumoxygenindexO ⁄ ,thevaluesofthecoordina-tion number ð N O  Þ and hydrogen bond number (N HB ) of the hydro-nium ion along the simulation trajectory. The average values of these hydration numbers are: N O   ¼ 2 : 92 and N HB  =2.92. We ob-served both from quantum chemical as well as from  ab initio molecular dynamics calculations that the hydronium ion prefersto donate three hydrogen bonds through its three hydrogens butit does not accept any hydrogen bond through its oxygen. Boththree-coordinated H 9 O þ 4  (Eigen cation) and a shared proton H 5 O þ 2 Fig. 1.  Low energy structures of protonated water clusters H + (H 2 O) n  for  n  =5 and 9. The hydronium oxygen is the one that is covalently bound to three hydrogens.  Table 1 The stabilization energies of various optimized structures of H + (H 2 O) n  clusters asobtained from all-electron quantum chemical calculations. All calculations are donewith the 6–31 + G ⁄ basis set. The energy values are expressed in units of kcal/mol. Isomer BLYP B3LYP MP2HW5I   90.32   92.09   87.53HW5II   90.74   92.74   88.51HW5III   88.56   90.39 –HW5IV   86.89   88.42   83.28HW9I   140.55   144.88   137.31HW9II   140.08   144.62   136.97HW9III   140.25   144.49   135.84HW9IV   137.85   141.71   135.40HW9V   138.95   142.58   134.19HW9VI   134.84   138.90   132.82HW17I   228.45   238.40   226.30HW17II   229.70   238.27   223.63HW17III   226.73   235.20   221.03HW21I   274.11   293.11   276.69HW21II   262.72   282.81   267.43HW21II   266.58   286.27   270.34  A. Bankura, A. Chandra/Chemical Physics 387 (2011) 92–102  95  (Zundel cation) are observed in the simulation with relativeweights of 95% and 5%, respectively. For Eigen and Zundel cations,the average distances between the hydronium oxygen and thenearest water oxygen  ð R  O  O Þ  in the first solvation shell are foundto be 2.54 and 2.44Å, respectively.  3.2. H  + (H   2 O) 9 In Fig. 1, we have shown some of the low energy structures of H + (H 2 O) 9  cluster that are found in the present study throughgeometry optimizations. The corresponding stabilization energiesare included in Table 1. The lowest energy isomer (HW9I) is foundtohaveathree-dimensionaldistortedcubicstructurewithtwofiveand four four-membered rings which is in agreement with previ-ous studies [8,21]. This isomer has six dangling hydrogens out of which one comes from the AD type water and other five belongto the five AAD type waters. Another stable structure (HW9II) isfound to have a three-dimensional structure with seven danglinghydrogens from seven waters out of which three belong to two-coordinated AD type and other four belong to three-coordinatedAADtypewaters.TheHW9IIisomerisonly0.3–0.4kcal/molhigherinenergythantheHW9I at BLYPandB3LYPlevels. TheMP2 calcu-lationspredict a strongerstabilityof theHW9I isomer thanHW9II.All other isomers have eight dangling hydrogens. HW9III andHW9IV have three dimensional structures whereas isomers of HW9V and HW9VI are found to have two-dimensional net or treetype structures. Our quantum chemical calculations predict thatthe HW9III isomer is higher in energy than HW9I by only 0.25–0.5kcal/mol, while the isomer HW9IV is found to be 2–3kcal/mol higher in energy. The isomer HW9III contains two sixmembered andtwo five membered rings whereall the eight watermolecules have one free hydrogen, five of them act as a two-coor-dinated AD type and the other three belong to three-coordinatedAAD type waters. We note that Karthikeyan et al. [46] reportedtheHW9IIIisomer tobethelowestenergystructureatB3LYPlevelwhile their MP2 calculations showed that the isomer HW9I is themost stable one. The two-dimensional HW9V and HW9VI isomersare found to be 2–6kcal/mol higher in energy than that of theHW9I isomer. The isomer HW9V also has eight dangling hydro-gens, five of them are due to AD type waters and rest three belongtoAADtypewatermolecules.TheisomerHW9VIhasthreeAAtypeand two AD type water molecules, hence all the dangling hydro-gens belong to two-coordinated water molecules. 03690369036903690369NilAAADDDDDDADAADADDAADD n = 5    A  v  e  r  a  g  e  n  u  m   b  e  r  p  e  r  c  o  n   f   i  g  u  r  a   t   i  o  n n = 17 n = 21 n = 21 (150 K) (a)(b)(c)(d) n = 9 (e) Fig. 2.  Distributions of the hydrogen bonding states of water molecules and thehydronium ion in protonated water clusters at finite temperature. The results arefor: (a) H + (H 2 O) 5 , (b) H + (H 2 O) 9 , (c) H + (H 2 O) 17 , (d) H + (H 2 O) 21 , all at 300K and (e)H + (H 2 O) 21  at 150K. 20002200240026002800Wave number (cm -1 )    A  r   b   i   t  r  a  r  y  u  n   i   t  s  n = 5 n = 17 n = 21 n = 9 (a)(b)(c)(d)(e)  n = 21 (150 K) Fig. 3.  Vibrational power spectra of H + (H 2 O) n  in the stretching region of OH (withdeuterium mass for hydrogen atoms). (a) H + (H 2 O) 5 , (b) H + (H 2 O) 9 , (c) H + (H 2 O) 17 , (d)H + (H 2 O) 21 , all at 300K and (e) H + (H 2 O) 21  at 150K. 024680246051015202530 0246Time (ps)    I  n   d  e  x  o   f   O   *   N    O   *    N    H   B (a)(b)(c) Fig. 4.  Changes of the (a) index of the hydronium oxygen O ⁄ , (b) coordinationnumber (N O ⁄ ) and the (c) hydrogen bond number (N HB ) of the hydronium ion alongthe simulation trajectory for H + (H 2 O) 5  at 300K.96  A. Bankura, A. Chandra/Chemical Physics 387 (2011) 92–102
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