Ab initio investigation of the lower energy candidate structures for (H2O)5 + waterclusterZhen-Long Lv, Kai Xu, Yan Cheng, Xiang-Rong Chen, and Ling-Cang Cai Citation: The Journal of Chemical Physics 141, 054309 (2014); doi: 10.1063/1.4891721 View online: http://dx.doi.org/10.1063/1.4891721 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Optimal geometries and harmonic vibrational frequencies of the global minima of water clusters (H2O) n , n =2–6, and several hexamer local minima at the CCSD(T) level of theory J. Chem. Phys. 139, 114302 (2013); 10.1063/1.4820448 Infrared predissociation spectroscopy of ammonia cluster cations ( N H 3 ) n + ( n = 2 – 4 ) produced by vacuum-ultraviolet photoionization J. Chem. Phys. 125, 164320 (2006); 10.1063/1.2360279 High-level ab initio calculations for the four low-lying families of minima of ( H 2 O ) 20 . II. Spectroscopicsignatures of the dodecahedron, fused cubes, face-sharing pentagonal prisms, and edge-sharing pentagonalprisms hydrogen bonding networks J. Chem. Phys. 122, 134304 (2005); 10.1063/1.1864892 Structure and vibrational spectra of H + (H 2 O) 8 : Is the excess proton in a symmetrical hydrogen bond? J. Chem. Phys. 113, 5321 (2000); 10.1063/1.1288918 Ab initio studies of anionic clusters of water pentamer J. Chem. Phys. 113, 2697 (2000); 10.1063/1.1301497

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THE JOURNAL OF CHEMICAL PHYSICS 141, 054309 (2014)

Ab initio investigation of the lower energy candidate structures for (H2O)5+

water clusterZhen-Long Lv,1,2 Kai Xu,1 Yan Cheng,1,a) Xiang-Rong Chen,1,a) and Ling-Cang Cai31Institute of Atomic and Molecular Physics, College of Physical Science and Technology, Sichuan University,Chengdu 610064, China2School of Physics and Engineering, Henan University of Science and Technology, Luoyang 471023, China3National Key Laboratory for Shock Wave and Detonation Physics Research, Institute of Fluid Physics,Chinese Academy of Engineering Physics, Mianyang 621900, China

(Received 9 June 2014; accepted 18 July 2014; published online 7 August 2014)

The particle swarm optimization method in conjunction with density functional calculations is usedto search the lower energy structures for the cationic water clusters (H2O)5

+. Geometry optimization,vibrational analysis, and infrared spectrum calculation are performed for the most interesting clustersat the MP2/aug-cc-pVDZ level. The relationships between their structural arrangements and their en-ergies are discussed. According to their relative Gibbs free energies, their energy order is determinedand four lowest energy isomers are found to have a relative population surpassing 1% below 350K. Studies reveal that, among these four isomers, one new cluster found here also contributes a lotto the experimental infrared spectrum. Based on topological analysis and reduced density gradientanalysis, some meaningful points are found by studying the structural characteristics and the bondingstrengths of these cationic water clusters: in the first solvation shell, the central H3O+ motifs mayhave a stronger interaction with the OH radical than with the water molecules. The interaction in thesecond solvation shell may also be stronger than that in the first solvation shell, which is opposite toour intuition. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4891721]

I. INTRODUCTION

Water, a common substance in the world, is indispensablefor living things and our daily life. Most biological processesand chemical reactions are proceeded with the participationof water.1, 2 Thus, understanding the properties and functionsof liquid water is important for exploring the mechanismsunderlying these processes. Because studying the propertiesof water clusters can provide much useful information forliquid water, many researchers devoted themselves to thisfield. As a result, a large number of isomers of water clusterswere found3, 4 and various physical properties of water clus-ters were investigated.5–8 Recently, the ionized water clustershave also become a hot object of experimental and theoreti-cal studies owing to their unique characteristics in compari-son with their neutral counterparts.9–13 Researches reveal thatthe ionization of water plays an important role in some pho-tocatalytic reactions.14–16 Nuclear radiation can ionize neutralwater clusters into water cluster cations,17, 18 which can takeplace in the operation of nuclear power plants. Nevertheless,the information of ionized water clusters or called cationicwater clusters is still rather limited.

In the past, the most intensively studied cationic waterclusters is (H2O)+ 19–22 and (H2O)2

+.23–28 As for the largercationic water clusters, the progress of the relevant studiesis much slower. In 1986, Haberland and Langosch29 mea-sured the mass spectra of (H2O)n

+ (n = 3–8) and concludedthat these clusters cannot be obtained by ionization of wa-

a)Authors to whom correspondence should be addressed. Electronicaddresses: ycheng@scu.edu.cn and x.r.chen@qq.com.

ter clusters in vacuum, but can be grown in the cold envi-ronment of a supersonic beam. In theoretical respect, somestudies reveal that quite a few exchange-correlation function-als within the density functional theory have drawbacks be-cause they give an incorrect results that the hemi-bondedstructures are more stable than the proton transfer ones,30–32

which makes the study of cationic water clusters difficult.Novakovskaya and Stepanov33 obtained some structures ofthe cationic water clusters (H2O)n

+ (n = 2–4) by directlyoptimizing their ionized neutral counterparts at the UHF/4-31++G∗∗ level. Born-Oppenheimer local-spin-density func-tional molecular dynamics method was also used to studythe energetics and geometrical structures of cationic waterclusters (H2O)n

+ (n = 2–5), which found both “hydrazine-based” and “disproportionated ion-based” isomers for theseclusters.34 Angel and Stace35 studied the collision inducedfragmentation patterns for (H2O)n

+ (n = 2–6) by supersonicmass spectrum experiment. Their results show that a compet-itive loss of OH and water molecule exists during the disso-ciation of their parent clusters. Structure transformation dy-namics of the ionized neutral clusters (H2O)n (n = 5–6) werestudied at HF/6-31++G∗∗ level, which reveals some possiblemechanisms in the reorganization of the H-bonded fragmentsin both clusters.36 Infrared spectroscopies of water clusterradical cations (H2O)n

+ (n = 3–11) were not well known till2011 and some possible structures of these clusters were si-multaneously suggested to give a theoretical description forthese experimental results.37 Guided by it, Do and Besley38

predicated a series of candidate structures for the cationicwater clusters (H2O)n

+ (n = 3–9) by using the basin hop-ping search algorithm in combination with quantum chemical

0021-9606/2014/141(5)/054309/8/$30.00 © 2014 AIP Publishing LLC141, 054309-1

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calculations. Lately, the dynamics and structural changes ofthe ionized neutral water clusters (H2O)n (n = 2–6) were fur-ther studied by the density functional method, second-orderMøller-Plesset method, and the coupled cluster method39

which confirmed the existence of the hydronium H3O+ cationand the OH radical in these ionized cationic water clusters.

It seems that the study for the small-numbered cationicwater clusters is enough, but our studies indicate that it is farfrom satisfactory. Many questions are still open for answers.For examples, are all the lower energy clusters found? Whichkind of bonding scheme is more favorable in these clusters?Are there any underlying regulations existing in forming theseclusters? To account for these questions, in this work, we usecationic water cluster (H2O)5

+ as a example to conduct therelevant studies. We first give a broad search for the possi-ble structures for the cationic water cluster (H2O)5

+, and thenperform structural and energetical analyses on the lower en-ergy clusters, in particular the four lowest energy ones, to getdeep insights for further studying this kind of clusters.

II. COMPUTATIONAL DETAILS

The possible structures with minimum energies of(H2O)5

+ were searched by the crystal structure analysis byparticle swarm optimization code (CALYPSO).40 CALYPSOperforms structure evolution utilizing the particle swarm op-timization (PSO) algorithm. The PSO method can be viewedas a distributed behavior algorithm that performs multidimen-sional search, which is intelligent because in the process theevolution of each structure is guided by the best local orglobal structure in the swarm, and each structure can learnexperience from its own evolving history to adjust its evolv-ing speed and direction. This unbiased and intelligent methodcan help individuals in the swarm locate themselves to thelocal or global minima of the potential surface quickly.40 Tomake it more efficiently for structurally searching, CALYPSOemploys point group symmetries to enable structural diver-sity and to avoid generating liquid-like clusters for large sys-tems. It also introduced the bond characterization matrix toget ride of similar structures. In addition, Metropolis criterionis adopted to further ensure the structural evolution towardsthe low-energy regions of the potential energy surface.41 Inthe searching process, the first generation was randomly gen-erated by CALYPSO, the structures of which were optimizedat the B3LYP/6-31+G∗ level by using the GAUSSIAN 09package,42 and then a certain number of clusters having lowerenergies evolved into the next generation with some new ran-domly generated structures. This procedure proceeded untilgiven criteria were met. After that, the most desirable struc-tures were further optimized without constraining their sym-metries at the second-order Møller-Plesset (MP2) level withthe aug-cc-pVDZ basis set. This computational level is foundto be rather accurate for studying different types of cation-water clusters.43–45 Harmonic vibrational frequencies of theseclusters were calculated to confirm their stability and to gettheir zero-point vibrational energies (ZPVE). In determiningtheir total energies, a scaling factor of 0.9615 (Ref. 46) wasadopted to approximately account for the anharmonicity ofthe vibrations.

Formation energy is known as a reasonable quality tomeasure the interaction strength within a supermolecule.Commonly, it can be calculated through subtracting thetotal energy of a supermolecule by the energies of theconstituent molecules. Here we use the scheme proposedby Novakovskaya and Stepanov33 to calculate the forma-tion energy of these (H2O)5

+ clusters by the formula:�E = E[(H2O)5

+]+BSSE-E(H3O+)-E(OH)-3E(H2O), whereE[(H2O)5

+], E(H3O+), E(OH), and E(H2O) are their respec-tive energies calculated at their ground sates. The basis setsuperposition error (BSSE) was obtained by the method pro-posed by Boys and Bernardi.47 All the calculations were car-ried out at the MP2/aug-cc-pVDZ level if not specified. Topo-logical and reduced density gradient (RDG) analyses werecarried out by using the Multiwfn program48 and all the clus-ter structures were drawn by the VMD code.49

III. RESULTS AND DISCUSSION

A. Structure analysis and their energy order

The obtained 12 lower energy structures of (H2O)5+ clus-

ter are depicted in Fig. 1. Through comparison, we classifythese clusters into four types: the first type is the chain-likestructures (denoted W1 and W2, similarly hereafter) with thefragment H3O+ being coordinated by two water molecules;the second is the branch-like structures (W3, W4, and W5)with the H3O+ being threefold coordinated; the third isformed by a four-numbered ring and a side tail (W6 to W10),in which the H3O+ motifs are either twofold coordinated (W6and W7) by water molecules or threefold coordinated (W8 toW10); the forth is formed by five-numbered ring (W11 andW12), differing from each other by the location of the OHradicals.

The calculated relative energies at 0 K at the MP2/aug-cc-pVDZ level are included in Table I. Fig. 2 is plotted toshow their energy differences illustratively. Lee and Kim39

reported the energy order W10, W5, W8, W12, and W1 at theMP2/CBS level without the ZPVE correction and W5, W10,W8, W1, and W12 with this correction. From Table I, we findthat the energy order presented here agrees basically withtheirs while the difference is that more structures are reportedhere. Our study indicates that the clusters W10, W5, W8, W9,and W4 are the five most stable ones without the ZPVE cor-rection, and their relative energies are −0.48, 0.00, 0.67, 1.06,1.30 kcal/mol, respectively. When the scaled ZPVE correc-tion is considered, the turn changes to W5, W10, W3, W4, andW8, and their relative energies are respectively 0.00, 0.04,1.19, 1.30, and 1.64 kcal/mol. These energy arrangementsalso agree with those reported by Do and Besley38 at theCCSD(T)/aug-cc-pVDZ level, where they gave a order of W5,W8, W4 for the ZPVE uncorrected case, and W5, W4, W8 forthe ZPVE corrected case with the frequencies obtained at theMP2/aVDZ level. When the BSSE is included, the energy or-der turns to W5, W10, W4, W3, and W8, and now the energydifferences are 0.00, 0.19, 1.28, 1.31, 2.16 kcal/mol, respec-tively. It can be seen that the ZPVE correction is important fordeciding the energy order of these clusters, as found in study-ing the neutral water cluster50 and protonated water clusters.51

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FIG. 1. Cationic water clusters (H2O)5+ predicated by the particle swarm optimization method.

The calculation also reveals that the effect of BSSE is not soserious for (H2O)5

+ cluster. But it is generally thought thatenergies of water clusters with BSSE are more reliable than

TABLE I. Calculated different types of energies (kcal/mol) with respect tothese of the cluster W5 and other quantities of the (H2O)5

+ clusters obtainedat the MP2/aug-cc-pVDZ level. �Ee: relative electronic energy; E0: ZPVEcorrected relative energy; �E: BSSE corrected total relative energy; EF: for-mation energy; EHB: average hydrogen bonding energy; N: the number ofHBs in the corresponding cluster; Iad: ZPVE corrected adiabatic ionizationpotential (eV).

�Ee �E0 �E EF N EHB Iad

W1 3.80 3.04 3.29 −70.69 4 −17.67 9.48W2 3.82 3.26 3.52 −70.46 4 −17.62 9.49W3 1.45 1.19 1.31 −72.68 4 −18.17 9.40W4 1.30 1.30 1.28 −72.70 4 −18.18 9.41W5 0.00 0.00 0.00 −73.98 4 −18.50 9.35W6 3.69 4.28 4.83 −69.16 5 −13.83 9.54W7 3.42 4.26 5.06 −68.93 5 −13.78 9.54W8 0.67 1.64 2.16 −71.82 5 −14.36 9.42W9 1.06 2.16 2.64 −71.34 5 −14.27 9.45W10 −0.48 0.04 0.19 −73.79 5 −14.76 9.35W11 3.90 4.59 5.36 −68.62 5 −13.72 9.55W12 3.48 2.89 3.56 −70.42 5 −14.08 9.48

without it,52 so the BSSE correction was considered in thepresent study.

Interesting phenomena can be found by careful inspec-tion and comparison: The first one is that these clusters (W3,W4, W5, W8, W9, and W10) with the H3O+ motifs being three-fold coordinated or say having a filled first solvation shell tendto have a lower energy. And in the two lowest energy clusters(W5 and W10), the H3O+ motifs are both directly bounded bythree water molecules in the first solvation shell with the OH

FIG. 2. Energy (kcal/mol) of the twelve (H2O)5+ clusters with respect to the

cluster W5. �Ee: relative electronic energy; �E0: ZPVE corrected relativeenergy; �E: BSSE corrected total relative energy.

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radical located in the second shell, which indicates that thiskind of arrangement is favorable for lowering the energy ofthese clusters. The second one is that these with unfilled firstsolvation shell always have a high energy, especially the oneshaving a ring structure with the OH radical located outside thefirst solvation shell, for example, W6, W7, and W11. The thirdone is that no OH radical is directly bonded to the H3O+ mo-tifs in the clusters with a unfilled first solvation shell (W1, W2,W6, W7, W11, and W12), which gives us a impression that theinteraction between H3O+ and the water molecule is strongerthan that between H3O+ and the OH radical.

The formation energy of these cationic clusters was cal-culated by the method previously presented. It is apparent thatthese formation energies have a similar changing trend as thetotal energies or the relative energies do (compare �E and EFin Table I). It can be seen that W5 has the maximum formationenergy in magnitude with a value of −73.98 kcal/mol, andW11 has a minimum one with the value of −68.62 kcal/mol.The average formation energy can be regarded as a indictor ofthe strength of the hydrogen bonds, which is defined as EHB= �E/N, where N is the number of hydrogen bonds (HBs)in the cluster. We also listed these values for these predicatedclusters in Table I. Obviously, in the water clusters studied,smaller numbers of HBs often correspond to a larger bondingenergy. The results also certify that the number of HBs is notthe decisive factor for the stability of these clusters.

Adiabatic ionization potential (AIP) is the energy differ-ence between the relaxed positively charged compound and itsneutral counterpart, which reflects the stability of the chargedcompound. For these (H2O)5

+ clusters, the AIPs can be ob-tained by the expression: Iad = E[(H2O)5

+]-E[(H2O)5], whereE[(H2O)5

+] and E[(H2O)5] denote the energies of the cationicwater cluster (H2O)5

+ and the neutral cluster (H2O)5 at theiroptimized states. It seems that the more stable a cationic wa-ter cluster is, the smaller an AIP it has. The calculated AIPsof these (H2O)5

+ clusters lie between 9.35–9.55 eV, whichgives a good comparison with the value 9.35 eV34 obtainedby using the Born-Oppenheimer molecular dynamics methodand the value 9.43 eV39 obtained at the CCSD(T)/aVDZlevel.

B. Temperature effect on the energy order

Above analyses are performed based on the energies ob-tained at 0 K, but in practice, experiments are often conductedat a finite temperature. So in this part we discuss the tempera-ture effect on the energies of these clusters.

Fig. 3 shows the relative change of the Gibbs free energywith the temperature with respect to the lowest energy clusterW5. It can be seen that the relative Gibbs free energies of W1,W2, W3, and W4 almost keep unchanged or slightly decreaseswhen the temperature rises, while others increase notably withthe rising temperature. Retrospecting on their structures, wefind that the former have a free open skeleton, which may in-duce their similar behaviors in temperature. For the latter, theyhave a four-numbered or five-numbered ring, so steric effector strain should exist in these clusters, leading to their notableenergy changes with the rising temperature. One can also find

FIG. 3. Gibbs free energies (BSSE corrected) of the twelve (H2O)5+ clusters

below 350 K with respect to the cluster W5.

that those clusters with ring structure have one more HB thanthe open skeleton ones. This reality indicates that cationic wa-ter clusters with a large number of HBs are easier to lost theirstabilities when temperature rises, which is a result of the en-tropy effect as suggested by Kim et al.53 One can also findthat W10 is still a member of the four lowest energy clustersbelow 350 K although its relative Gibbs free energy increasesquickly as the temperature goes up.

Previous studies indicate that temperature have a signif-icant influence on the relative populations of the neutral wa-ter clusters, regardless of them either being composed of dif-ferent number of water molecules54, 55 or being composed ofthe same number,53 so the temperature effect on the relativepopulations of the cationic water clusters (H2O)5

+ is antic-ipated. Here we calculated their relative populations at dif-ferent temperatures by using Boltzmann population formula:pi = e−�G

i/RT /

∑j e

−�Gj/RT , where pi is the relative popu-

lation of the ith cluster, �Gi is the Gibbs free energy of theith cluster with respect to the most stable one, R is the idealgas constant, and T is the thermal temperature. The relativepopulations at 180 K and 300 K are listed in Table II.

It can be seen that either at 0 K or 300 K, the most promi-nent (H2O)5

+ clusters are consistently these W5, W10, W4,and W3, all of which have a relative populations larger than1%. The energy order (BSSE corrected one) of them is W5,

TABLE II. Gibbs free energies (kcal/mol, BSSE corrected) of the twelve(H2O)5

+ clusters at 180 K and 300 K with respect to the cluster W5 and theirrelative Boltzmann populations.

�G180 K Pop. (%) �G300 K Pop.(%)

W5 0.000 89.39 0.000 76.99W10 0.983 5.72 1.692 4.50W4 1.221 2.93 1.158 11.01W3 1.374 1.92 1.455 6.69W8 2.850 0.03 3.653 0.17W1 3.122 0.01 3.110 0.42W2 3.463 0.00 3.588 0.19W9 3.703 0.00 4.820 0.02W12 4.507 0.00 5.505 0.01W6 5.634 0.00 6.575 0.00W7 6.009 0.00 7.170 0.00W11 6.572 0.00 7.977 0.00

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W10, W4, and W3 at 180 K, as that at 0 K, but it changes toW5, W4, W3, and W10 at 300 K. The energy order at 180 Kagrees with that reported by Mizuse et al.37 at the MPW1K/6-311++G(3df,2p) level, but W10 is missing and BSSE correc-tion is not considered there.

C. Infrared spectrum of the four lowestenergy clusters

As analyzed above, the clusters W5, W10, W4, and W3 arethe most stable ones up to 300 K and their relative populationsare all larger than 1% at the studied temperatures. This meansthat these clusters can coexist in this temperature range. Asis known that IR spectrum is often used as a powerful toolto verify the existence or to discriminate the structure of cer-tain water clusters.56 For (H2O)5

+ cluster, its IR spectrum hasrecently been measured by Mizuse et al.37 To confirm the va-lidity of our predication, we calculated the IR spectra of thefour lowest energy clusters. In reproducing the experimentalIR spectrum, we find that the B3LYP/6-311++G∗∗ level cangive a more desirable results than MP2/aug-cc-pVDZ at thehigh energy end, so in this part the former was used with ascaling factor 0.9679.57 The results are drawn in Fig. 4.

Mizuse et al.37 found two new bands appearing in theIR spectrum of the (H2O)5

+ clusters compared to that ofH+(H2O)5: one is at 3458 cm−1, and another is that locatedat 3553 cm−1. They suggested three lower energy structuresas candidates, W5 (denoted structure I there), W3 (structureII), and W4 (structure III), to explain the experimental IRspectrum. The frequency at 3458 cm−1 is assigned to the

FIG. 4. Simulated IR spectra for the four lowest energy clusters comparedwith the experimental result. The top panel is reprinted with permission fromMizuse et al., Chem. Sci. 2, 868 (2011). Copyright 2011 Royal Society ofChemistry.37

stretch of the terminal OH radical toward the water moleculeit connected in the cluster W5, while the frequency at 3553cm−1 is ascribed to the intermolecule vibration of the ter-minal OH radical of W5. The peak of W5 located at 2576cm−1 is attributed to the OH stretch of H3O+ toward the two-coordinated water, which is missing in the experiment be-cause this movement makes the proton be shared by two watermolecules, so the corresponding frequency shift to the lowerenergy part.37 Other characteristics of the experimental spec-trum were also assigned there. These account for the exper-imental IR spectrum well, but we find only with the clustersstructures I, II, and III (W5, W3, and W4), the broad intensepeak at about 2800 cm−1 cannot be very well reproduced. Thereason is that there the calculated relative populations for theclusters W5, W4, and W3 are 0.90, 0.06, and 0.04, respec-tively, at the experimental temperature of 180 K. So from theview point of the population, the contributions of the clustersW4 and W3 are quite limited, thence it is hard to give so broada peak at that frequency range.

However, our study shows that W10 has three intensivepeaks with the frequencies of 2741, 2812, and 2937 cm−1 inthat frequency region (the corresponding frequencies at theMP2/aug-cc-pVDZ level are 2749, 2813, and 2912 cm−1, re-spectively). Considering that the Gibbs free energy of W10is lower than these of W4 and W3 at 180 K, it should havean important contribution to the broad intense peak centeredat 2800 cm−1. In addition, the calculation shows that W10also has a vibrational frequency at about 3563 cm−1 (at theMP2/aug-cc-pVDZ level is 3570 cm−1), which should alsocontribute to the experimental peak found at 3553 cm−1. Thenext peak of W10 is located at 3628 cm−1, but no notablecounterpart exists in the IR spectrum (in fact a magnifiedexperimental spectrum gives a weak peak nearby) owing tothe small population of W10. Further analysis reveals that thethree intensive peaks at about 2800 cm−1 of W10 originatefrom the asymmetric and symmetric stretching vibrations ofOH fragments of the H3O+ motif. The frequencies located atabout 3570 cm−1 are simulated by the complicated relativemovements of the two water molecules and the OH radicalsituated in the right part of W10.

D. Topological analysis on the four lowestenergy clusters

Reduced density gradient (RGD), defined asRDG(r) = 1/2(3π2)1/3|∇ρ(r)|/ρ(r)4/3, is a dimensionlessquantity used to describe the difference between the actualelectron density and a homogeneous electron distribution.58

Johnson et al.59 found that the sign of the second Hessianeigenvalue of electron density can be used to discriminatethe type of interaction, and the strength of the interaction canbe deduced from the density on the non-covalent interactionsurface. This method provides a good way to distinguishvan der Waals interaction, hydrogen bond, and steric effect.The obtained RDGs at the bond critical points (BCPs) ofthe four lowest energy clusters are depicted in Fig. 5, wherethe blue, green, and red represent the strong attraction, vander Waals interaction, and strong repulsion, respectively.Figures 5(a)–5(d) show us that, the H3O+ motifs in these

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FIG. 5. (a)–(d) Reduced density gradient (RGD) figures of the four lowestenergy clusters, where 1, 2, 3, 4, and 5 denote the (3, −1) BCPs in the corre-sponding cluster. The blue, green, and red represent the strong attraction, vander Waals interaction and strong repulsion, respectively.

water clusters are apt to have a strong interaction with thewater molecules and OH radicals in the first solvation shell.Further inspection shows that if the OH radical acts only as aone-proton acceptor, it will have a stronger interaction withthe connected fragment than that acts as a double protonsacceptor [Fig. 5(b)]. The RGD also reveals a weak repulsion(steric) effect existing at the (3, 1) BCP of W10.

Although RGD can be employed to visually illustratethe interaction in a complex, it is hard to give a quantitivecomparison, especially when the interactions are of the samemagnitude. Studies find that weak interactions, such as hy-drogen bonds, can be classified and quantitively described bythe electron density ρ(r) and its second derivative ∇2ρ(r) atthe (3, −1) BCPs where the gradients of ρ(r) equal to zero.For example, covalent interaction typically has a large ρ(r)and a negative ∇2ρ(r) while non-covalent interactions suchas hydrogen bonding have a low ρ(r) and a positive ∇2ρ(r).60

To give a deeper investigation on the interaction within theseclusters, we calculated the electron densities and its secondderivatives at their (3, −1) BCPs for the four lowest energy(H2O)5

+ clusters. The results, together with the bond lengthsof these HBs are listed in Table III. Koch and Popelier61 pro-posed a set of criteria to judge the C–H. . . O hydrogen bond-ing interaction within a complex. The two most importantitems are that ρ(r) at the (3, −1) BCPs should lie in 0.002–0.035 a.u. and ∇2ρ(r) at these BCPs should lie in the rangeof 0.014–0.139 a.u. But from Table III, we find that most val-ues of these two quantities for these four clusters are greaterthan the corresponding criterion, which displays the failure ofthe above criteria to judge the hydrogen bonding interactionsin waters clusters. Parthasarathi et al.62 classified the inter-action within a complex into five types: for van der Waalsand weak interaction, the value of ρ(r) lies in 0–0.022 a.u.;for moderate HBs, it lies in 0.022–0.052 a.u.; for strong HBs,it lies in 0.052–0.091 a.u.; for very strong HB, it lies in0.091–0.120 a.u.; and beyond 0.120 a.u., it is for covalent

TABLE III. Electron densities ρ(r) (a.u.) and its second derivatives∇2ρ(r)(a.u.) at the (3, −1) bond critical points (BCPs) of the four lowestenergy clusters, together with their bond lengths dHB (Å), where Avg. repre-sents the average value.

BCP ρ ∇2ρ dHB

W5 1 0.056 0.176 2.5842 0.056 0.177 2.5833 0.072 0.165 2.5184 0.029 0.111 2.807

Avg. 0.053 0.157 2.623

W10 1 0.057 0.178 2.5742 0.061 0.175 2.5643 0.016 0.056 2.9624 0.016 0.057 2.9615 0.061 0.175 2.561

Avg. 0.042 0.128 2.724

W4 1 0.058 0.177 2.5712 0.059 0.177 2.5703 0.060 0.166 2.5804 0.044 0.161 2.669

Avg. 0.055 0.170 2.598

W3 1 0.038 0.141 2.7152 0.057 0.177 2.5793 0.090 0.114 2.4664 0.042 0.158 2.673

Avg. 0.057 0.147 2.608

(H2O)5 Avg. 0.038 0.140 2.731

bond. According to this sorting method, most interactions inthese four water clusters studied can be viewed as strong HBs.

The obtained values show us that the strongest HBs inW5, W4, and W3 are all located at the third BCP while thesein W10 are located at the second and the fifth BCPs. FromFig. 5 we can find that all these strongest HBs play a role inbridging the second solvation shell, which ensure the corre-sponding branches to be tightly bonded by the central H3O+

motifs. The weakest interactions in W5 and W4 are both lo-cated at the fourth BCP; for W10 it is located at the third andthe fourth ones; for W3 is at the first one. The case of W4tells us that the interaction between the H3O+ and OH rad-ical may be stronger than that between H3O+ and the watermolecules in the first shell. The case of W3 demonstrates usthat the weakest interaction in cationic water clusters (H2O)5

+

does not necessarily lie in the second shell, on the contrary, itcan also exist in the first shell. In the (H2O)5

+ clusters witha filled first solvation shell, the strength of the interactionswithin the complex depends on both the exact location (in thefirst or second shell) of the interaction and the concrete en-vironment of the constituent elements (a water molecule or aOH radical).

We also presented the corresponding average values ofthe most stable neutral (H2O)5 cluster in Table III. Com-parison reveals that all the average electron densities [at the(3, −1) BCPs, similarly hereafter] of the four lower energyclusters are greater than that of the neutral (H2O)5 clus-ter, while all the average lengths of the HBs are shorterthan that of the neutral (H2O)5 cluster, which implies thatremoving an electron from the neutral (H2O)5 cluster can

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054309-7 Lv et al. J. Chem. Phys. 141, 054309 (2014)

FIG. 6. Relationships between the electron densities and bond lengths (a),and between the electron densities and its second derivatives (b) for the fourlowest energy (H2O)5

+ clusters.

strengthen the hydrogen bonding interactions on the whole.Studies63, 64 suggest a exponential relationship existing be-tween the electron densities and the corresponding lengthsof HBs. For the four lowest energy (H2O)5

+ clusters, the de-pendence of their electron densities on the bond lengths isshown in Fig. 6(a), and the changing tendency of the sec-ond derivatives of the electron densities with ρ(r) is shownin Fig. 6(b). Fitting these data gives us a expression ρ(dHB)= 1879.731exp(−4.074dHB) + 0.006 and a correlation coef-ficient of 0.994 for the former, and gives a expression ∇2ρ(r)= −64.192ρ2(r) + 7.616ρ(r) − 0.051 together with a correla-tion coefficient of 0.993 for the latter. The quadratic behaviorof Fig. 6(b) suggests that with the enhancement of the hydro-gen bonding interaction, some covalent characteristics maybegin to appear.

IV. CONCLUSIONS

The particle swarm optimization method was used tosearch the lower energy structures of the cationic water clus-ters (H2O)5

+ in conjunction with density functional method.Geometry optimization, vibrational analysis, and infraredspectrum calculation are performed for the found cationic wa-ter clusters at the MP2/aug-cc-pVDZ level. The main conclu-sions of this work are as follows:

(1) Energy analyses show that zero-point vibration energy(ZPVE) plays an important role in determining the en-ergy order of these (H2O)5

+ clusters. And filled first sol-vation shell is more favorable for lowering the energy ofthese clusters.

(2) The calculated Gibbs free energies suggest that four low-est energy isomers, denoted W5, W10, W4, and W3, arethe most prominent clusters in the temperature range of0–350 K. And IR spectrum simulation confirms that themeasured IR spectrum can be reproduced better with thecluster W10.

(3) Reduced density gradient and topological analyses ofthe four lowest energy clusters indicate that removing anelectron may in whole strengthen the interaction withinthe (H2O)5

+ clusters. The interaction between the H3O+

motif and the water molecules is not necessarily strongerthan that between H3O+ and OH radical, which dependson where (in the first or second shell) they are locatedand what role they play in these clusters.

ACKNOWLEDGMENTS

The authors would like to thank the support bythe National Natural Science Foundation of China (GrantNos. 11174214 and 11204192) and the NSAF (Grant No.U1230201). Some calculations are performed on the ScGridof Supercomputing Center, Computer Network InformationCenter of Chinese Academy of Sciences.

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