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The Hierarchy of the Collective Effects in Water Clusters Imre Bako, and Istvan Mayer J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.5b10053 • Publication Date (Web): 15 Jan 2016 Downloaded from http://pubs.acs.org on January 20, 2016

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The Journal of Physical Chemistry

The Hierarchy of the Collective Effects in Water Clusters ´ ∗ and István MAYER Imre BAKO Institute of Organic Chemistry, Research Centre for Natural Sciences, Hungarian Academy of Sciences, H-1519 Budapest, P.O.Box 286, Hungary

Abstract The results of dipole moment as well as of intra- and intermolecular bond order calculations indicate the big importance of collective electrostatic effects caused by the non-immediate environment in liquid water models. It is also discussed how these collective effects are build up as consequences of the electrostatic and quantum chemical interactions in water clusters.

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Corresponding author. E-mail: [email protected]

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1. Introduction Liquid water is a very complex system, and its theoretical studies always reveal new and new aspects of its structure. Water shows many anomalous thermodynamic and dynamical properties, such as the decrease of density and increase of diffusivity upon isothermal compression. A comprehensive collection of these anomalies can be found in the literature1-4 . There are several speculations in the literature concerning that the cooperativity among different hydrogen bonds (static and dynamical range) play an important effect in determining the anomalous properties of water. Hydrogen bond cooperativity, for example, affects the vibrational spectrum of the OH stretches and the interaction energy between pairs of H-bonded water molecules in the condensed phase or water clusters. An additional well known manifestation of the cooperative effects in liquid water and water clusters has been associated with a contraction of the nearest-neighbor O-H separation as compared to the distance in the gas phase water dimer. It is also well-known that the dipole moment of an isolated water molecule is different from that in bulk water. The dipole moment of a water molecule in the gas phase was obtained to be 1.8546 Debyes (D) from Stark effect measurements5 , whereas the molecular dipole moment of liquid water was estimated to be 2.9 ± 0.6 D from X-ray measurements6,7 . Many theoretical studies8,9 indicated a significant (∼0.6-1.2 D) enhancement of the dipole moment of the individual water molecules in the clusters and in the condensed phase. The cooperative effects of hydrogen bonding in water clusters are not yet fully understood: many-body induction as well as charge transfer have been proposed to be their main reasons, when studied by using quantum chemical descriptors like atoms in molecule (QTAIM) analysis and perturbation theory based on locally projected atomic orbitals10,11 . The attention has mostly been concentrated on the identification of the dominating hydrogen-bonded structures12-25 . At the same time, it is obvious that other effects—in particular the electrostatic ones that are of longer range—should also have significant influence on the observed properties. More than two decades ago one of us called the attention26 to the fact that the presence of a point charge in the vicinity of a hydrogen-bonded water dimer can cause dramatic changes in the strength of the hydrogen bond, as measured either by the equilibrium H-bond distance or by the bond order index27,28 between the oxygen and hydrogen atoms forming the hydrogen bond. It is widely known that the hydrogen bonds in water clusters exhibit pronounced collectivity effects. Collective effects manifests already for the smallest clusters: one observes a systematic contraction of the nearest-neighbour 1


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O. . . O separation29 and a red shift of the hydrogen-bonded OH frequency30 with increasing cluster size. The aim of the present note is to consider the role and the importance of the polar and polarizable environment and quantum chemical (QM) bonding interactions in determining the resulting properties of liquid water. For that reason we have started from three standard clusters of 20, 24 and 30 water molecules used to model properties of liquid water31-40 and investigated how the environment influences the properties of the different individual water molecules and their hydrogen bonds. (We shall call “central” for distinction the molecule actually being considered.) Three types of systems have been investigated, increasing in a hierarchical manner,: — (Case A) The individual water molecules deformed according to the geometry they have in the “large” cluster, in order to reflect the purely geometrical effect. — (Case B) Different “small” clusters consisting of a “central” water molecule and its first solvation shell (those neighbours by which it forms hydrogen bonds), reflecting the local polarization and charge transfer effects (QM effects). — (Case C) The full “large” cluster containing all the effects of interest. This is illustrated by Figure 1. For sake of comparisons, as an alternative of extending the cluster, the effect of introducing a polarizable medium by standard solvation model of Marenich, Cramer and Truhlar41 (SMD) has also been considered.

2. Computational details The clusters considered contain 20, 24 and 30 water molecules. They have water-like dodecahedral or cage structures, as described in detail in Table 1 and shown on Figure 2. The structures were taken from Refs.32, 33, and were re-optimized42 by using the M05-2x DFT functional of Truhlar et al.43 and the standard 6-311G** basis. Li et al.44 have already showed that the M052x functional is one of the best in describing binding energies and geometric parameter’s of water clusters. The quantities computed were the dipole moments of the individual monomers as well as the intra- and intermolecular OH bond orders27,28 . The dipole moments of the individual monomers within the overall system have been computed as suggested by Bader and Matta45 with the aid of the AIMALL program46 , followed by a small local code, in the following manner. The atomic dipoles of the three atoms forming a water molecule, outputted

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by AIMALL were summed up, and the dipole moments of the atomic point charges with respect to the oxygen atom have been also added. (So, the dipole moments were calculated with respect of the oxygens, as origins.47 ) The bond orders were calculated by the program48 , using the “formatted checkpoint files” of the Gaussian-09. The calculations of energetic quantities would be better with adding also diffuse functions; the use of diffuse function is, however, known to be inadequate when doing population analyzes or studying bond orders49 , so we have not included them. The wave functions have been computed by using the Gaussian-09 program-suite50 .

3. Results and discussion The characteristic structural and electronic properties of the water monomer and dimer are shown in Table 2. The dipole moment of the free monomer calculated by the DFT functional and basis set applied is 2.14 D, which is somewhat larger than the experimental 1.86 D. This difference is simply coming from the lack of the diffuse function in the calculations: the dipole moment of water using the M05-2x/aug-cc-pVTZ level of theory is 1.92 D. However, we are interested in the tendencies and their reasons, so this discrepancy seems not to be of any importance. To check that, we have also investigated the effect of the significantly larger aug-cc-pVTZ basis set on the dipole moment of the monomers in water dimer, hexamer and decamer. (The data are presented in the Supplementary Material.) The correlation coefficient between the dipole moments calculated for 6-311G** and aug-cc-pVTZ basis sets, respectively, is 0.996. Characterization of the H-bond network in the investigated clusters The investigated hydrogen bonded clusters were characterized according to the hydrogen bond (HB) patterns existing in them. In this work we apply the following HB definition: the OO distance is less than 3.0 ˚ A and the OOH ◦ angle is smaller than 20 . We can classify the topology of the HB network by the number of H-bonds, and the classification of local HB patterns according to the number of proton-acceptor(A) and proton-donor(D) sites in a certain molecule. These graph-theoretical descriptors have already been applied for predicting to the local energy minimum structure of water clusters and to explore their possible geometries51,52 . These quantities are shown in Table 1 and the H-bond pattern is displayed on Figure 1.

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Cooperativity In order to study cooperativity effects, we used in our work the intra- and intermolecular bond orders and the dipole moments as quantum chemical descriptors, and investigated the variations of their values in the individual water monomers and dimers. The calculated average values of the intra- and intramolecular bond orders and of dipole moments, together with the average intramonomer and hydrogen bonded OH distances are presented in Table 2. It is to be noted that the values of physically relevant quantities depend sensitively on the type of water molecule in question, i.e., on the number of hydrogen bonds in which it plays the role of proton acceptor (A) or proton donor (D). Table 1 shows the average values of dipole moments for the three types of water molecules (1A:2D, 2A:1D and 2A:2D) together with the number of their occurrence in our clusters. One may conclude that the dipole moment of the water molecules of type 2A:2D forming four hydrogen bonds is, in average, larger that those forming only three H-bonds; the difference between the two types of the latter (1A:2D and 2A:1D) is less pronounced. The intramonomer and hydrogen bonded OH distances are useful parameters for describing the cooperative effect and they were already used for that purpose.53-55,56,57 Both of these parameters change significantly—and in the opposite directions—as compared to their values in monomer and dimer. The probability distribution of our QM descriptors calculated for the investigated clusters are shown on Fig 3. It is clear from these distributions that these parameters indicate significant cooperative effects in water clusters. Hierarchy the cooperative effects The changes of the monomer geometries (case A) within the complexes are significant; nevertheless the intramolecular bond orders for the free monomers calculated in the geometries they have in the complexes scatter only in the narrow range of 0.947 to 0.959. (The bond order in the optimized geometry of the free monomer is 0.961—all the OH bonds are stretched in the complexes.) Analogously, the dipole moments of the free monomers, calculated in the manner described above, do not change too much either: they scatter between 2.08 and 2.16 Debyes. When turning to the “small” clusters containing a “central” water molecule and its hydrogen-bonded nearest environment (case B), the dipole moments of the “central” water molecules increase considerably, to the values between 2.41 and 2.74 Debyes, showing the large polarizability of the water. This indicates that the critical factor determining the dipole moments of the

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given water molecule is its hydrogen bonded environment, not the variations of the individual monomer geometry.58 The situation is changing when the transition to the “large” clusters (case C) is considered. There is again some non-negligible overall increase of the dipole moments, but this is more-or-less uniform, leading to a wellpronounced correlation between the values calculated for the “small” and the “large” clusters, respectively, as indicated on Figure 4a. This figure also displays the data obtained in the following manner. The “small” clusters were recalculated assuming that they are placed in a polarizable continuous medium, using the model of Ref. 43. The dipole moments obtained in this manner are in an unexpectedly good general agreement with those calculated for the “large” clusters, indicating this solvation model to be pretty successful. The larger scatter in the data referring to the “large” clusters indicates that besides the homogeneous effects of the polarizable environment, the details of the actual neighbouring geometry of the other water molecules in the cluster have also a non-negligible role. Thus both the first neighbour shell and, to a smaller extent, the farther environment increase of the dipole moments of the water molecules: the farther polarizable environment enhances the effects of forming a hydrogen-bonded shell. There is only a rather weak correlation between the bond orders calculated for the “small” complexes and for the individual monomers having the same geometries. This is despite the fact that the intramolecular bond orders themselves undergo a significant—in some cases even drastic—reduction in the hydrogen bonded complexes (Figure 4b). This is in overall accord with the increases of bond lengths, but these increases are not the primary effects: as noted above, the bond orders do not change too much if individual molecules with the geometries in the complex are considered (single molecules calculated one at a time separately). The primary effect is the decrease of the intramolecular bond order during the hydrogen bond formation. In fact, the hydrogen atoms forming the hydrogen bonds have significant bond orders also with the oxygen atoms of the partner molecules (Figure 4c). At the same time, there is practically a single valence orbital on the hydrogen atom, and therefore the hydrogen atom has a valence close to one (c.f. the analysis in59 ). And the valence of the atom—if a closed shell wave function is used—is equal to the sum of the bond orders of that atom27,28 . Therefore formation of a hydrogen bond necessarily leads to the decrease of the bond order of the original O-H bond, indicating its some weakening. The main effect of the bond order change is observed when turning from a single molecule to the “small” complexes containing the first hydration shell of the individual molecules. When considering the full complex, then—as a tendency—a further, more-or-less uniform minor decrease of the intramolec5


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ular bond orders is observed (by some 0.03–0.04), and an analogous increase of the intermolecular ones can be seen. This is again basically an electrostatic effect of the dielectric medium, as it is observed also for the calculations applying the polarizable continuum model. For the actual complexes the effect is bigger than for the continuum model, but the scatter is also much more pronounced. In order to investigate the effect of the above factors on the water molecules in different hydrogen bonded environment, we have collected in Table 3 the data of intra- and intermolecular bond orders and dipole moments, treating separately the water molecules of the 1A:2D, 2A:1D and 2A:2D type—distinguishing also the H-bonded and “dangling” O–H bonds for 2A:1D waters having both of them. It can be seen that the inclusion of the polarizable medium changes the different quantities in a direction similar to that in the “big” clusters, but, in most cases, with reduced effects. That can be understand as a manifestation of the quantum chemical (probably mainly electron transfer) effects of the second and further neighbour molecules. An interesting exception is represented by the intramolecular bond orders and dipole moments in the systems having “dangling” O-H bonds, for which the continuum model overestimates the effect: this may be attributed to the fact that for the “dangling bonds” the idealized polarizable medium is assumed to approach too close to the actual system. It can also seen from these data that the bond orders reflect mainly the local effects connected with the next neighbour quantum mechanical interactions, while dipole moments change as a consequence of the more long-ranged electrostatic polarization. Looking at these data one should keep in mind that the presence of an electrostatic field can change significantly the strength and behaviour of a water-water hydrogen bond. This has been observed both experimentally and theoretically when the effect of a localized positive charge has been studied26 . Very recently Ishibasi et al.56 showed that the hydrogen bond strengths depend significantly on the next neighbouring hydrogen bonds and this cooperative effect can be explained by charge transfer processes. In order to characterize the effect of wider environment on the individual hydrogen bonds, we have calculated their “Ohno numbers” defined in Ref. 57: MOH = −d ′ + a ′ + d ′ ′ − a ′ ′ . In this formula d ′ and a ′ represent the “outside” donor and acceptor numbers of the proton donor molecule forming the hydrogen bond, while d ′ ′ and a ′ ′ represent those for the proton acceptor molecule. (“Outside” means that only the donor and acceptor numbers corresponding to the bonds other than the given hydrogen bond should be counted.) This is a number characterizing 6



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in an empirical fashion the farther bonding environment around the given hydrogen bond; it can change between -2 and 4. Considering relative small water clusters, Ohno et al. found that these Ohno numbers can be related to the O–H stretching frequencies and to the O–H bond lengths of the hydrogenbonded water molecule. We used the intermolecular bond order as a more direct parameter characterizing the strength of the hydrogen bonds. Considering our larger clusters than those used by Ohno et al. we also have observed the overall tendencies described by Ohno et al., but they are rather weak ones—see Figure 5 for our largest cluster of 30 water molecules. As this figure also shows, there is a large scattering in every case, indicating that the quantum effects of the wider environment—which cannot be described simply by counting the donor/acceptor numbers of the neighbouring water molecules—are of significant importance. Moreover, for the cluster of 24 water molecules the data for Ohno number equal 3 are very far from the tendency described by the other points, indicating that the use of Ohno numbers is of limited applicability to describe environmental effects.60 One could, probably, expect a tighter correlation with the Ohno numbers if all the hydrogen bonds were equivalent—this is not the case, however.

4. Conclusions We may conclude, therefore, that the results of dipole moment as well as of intra- and intermolecular bond order calculations indicate the big importance of collective electrostatic and quntum mechanical effects caused by the non-immediate environment in liquid water models. These longer-range effects may be basically attributed to the properties of water as a polarizable medium, but the peculiarities of the wave functions of the actual systems do also have some measurable role even for the farther environment. Acknowledgment The authors are much grateful to both Referees for their very helpful remarks that permitted to improve the manuscript. Thanks are also due to Dr. Pedro Salvador and Dr. András Stirling for reading the manuscript and useful comments. The authors also acknowledge the partial financial support of the Hungarian Scientific Research Fund (grant OTKA K108721).

Supporting information — Optimised structure of water clusters. — Dipole moment of water molecules (Debye, dimer,hexamer and decamer) i calculated using Bader technique with 6-311G** and aug-cc-pVTZ basis sets.

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52. McDonald, S.; Ojamae L.; Singer, S.J. Graph Theoretical Generation and Analysis of Hydrogen-Bonded Structures with Applications to the Neutral and Protonated Water Cube and Dodecahedral Clusters.J. Phys. Chem. A 1998,102, 2824–2832; 53. Iwata S. Analysis of Hydrogen Bond Energies and Hydrogen Bonded Networks in Water Clusters (H2O)20 and (H2O)25 Using the ChargeTransfer and Dispersion Terms. Phys. Chem. Chem. Phys. 2014, 16, 11310-11317. 54. Miliordos, E,; Aprià, E.; Xantheas S.S 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. 2013, 139, 114302-114313. 55. Tainer, C.J.; Ni, Y.; Shi, L.; Skinner J.L. Hydrogen Bonding and OH– Stretch Spectroscopy in Water: Hexamer (Cage), Liquid Surface, Liquid, and Ice. J. Phys. Chem. Letter 2013, 4, 12–17. 56. Ishibashi,I.; Iwata,S.; Onoe,J.; and Matsuwaza, K. Hydrogen Bonded Networks in Hydride Water Clusters, F-(H2O)n and Cl-(H2O)n: cubic form of F-(H2O)7 and Cl-(H2O)7. J.Phys. Chem A. 2015, 119, 1024110253. 57. Ohno,K.; Okimura,M.; Akai, N. and Katsumoto, Y. The Effect of Cooperative Hydrogen Bonding on the OH Stretching-Band Shift for Water Clusters Studied by Matrix-Isolation Infrared Spectroscopy and Density Functional Theory. Phys . Chem. Chem. Phys. 2005, 7, 30053014. 58. No significant correlation could be observed between the dipole moments of the water molecules in the “small” cluster and the free monomer values calculated at the cluster geometry: the monomer dipole moments scatter around a horizontal line on Figure 4a. 59. Mayer, I. Bond Orders in Three–Centre Bonds: an Analytical Investigation into the Electronic Structure of Diborane and the Three– Centre Four–Electron Bonds of Hypervalent Sulphur. J. Mol. Struct. (Theochem), 1989, 186, 43–52. 60. That point breaks the tendency also if one considers the O–H distances, originally used by Ohno et al.57

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TABLE 1. Topological characterisation of the water clusters Number of mol.s

Number of HB

Water type

Average dipole (D)

20

30

10× 1A:2D 10× 2A:1D

2.67 (0.06) 2.70 (0.07)

24

42

7× 1A:2D 7× 2A:1D 10× 2A:2D

2.69 (0.05) 2.65 (0.06) 2.72 (0.11)

30

49

11× 1A:2D 10× 2A:1D 9× 2A:2D

2.64 (0.09) 2.65 (0.12) 2.73 (0.12)

(Standard deviations are shown in brackets.)

TABLE 2. The average value of intra- and intermolecular distances, bond orders and dipole moments of individual water molecules in the investigated clusters. (Standard deviations are shown in brackets.)

R(O−H) (˚ A) Monomer 0.957 Dimer 0.964, 0.956, 0.958(2×) Wa20 0.973(0.010) Wa24 0.974(0.008) Wa30 0.975(0.011)

R(O. . .H) (˚ A) 1.912 1.74(0.10) 1.80(0.05) 1.76(0.08)

B(O−H)

B(O. . .H)

0.961 0.9, 0.96, 0.93(2×) 0.091 0.843(0.05) 0.121(0.02) 0.845(0.06) 0.139(0.03) 0.843(0.06) 0.129(0.2)

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Dipole(D) 2.14 2.24, 2.32 2.69(0.08) 2.69(0.09) 2.67(0.10)

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TABLE 3. The average values of intra- and intermolecular bond orders and dipole moments for different types of water molecules in the investigated clusters. (Standard deviations are shown in brackets.) Average intramolecular bond orders small clu. 20

1A:2D, 2A:2D 0.865(0.011) 2A:1D (bonded) 0.785(0.025) 2A:1D (free) 0.929(0.001) 24 1A:2D, 2A:2D 0.884(0.021) 2A:1D (bonded) 0.848(0.007) 2A:1D (free) 0.939(0.001) 30 1A:2D, 2A:2D 0.858(0.023) 2A:1D (bonded) 0.803(0.022) 2A:1D (free) 0.932(0.003)

small clu.+pol.

full clu.

monomer

0.837(0.022) 0.760(0.028) 0.898(0.002) 0.863(0.022) 0.831(0.001) 0.909(0.002) 0.831(0.023) 0.778(0.034) 0.900(0.001)

0.840(0.022) 0.738(0.043) 0.930(0.002) 0.843(0.060) 0.768(0.021) 0.930(0.001) 0.830(0.021) 0.762(0.049) 0.931(0.001)

0.954(0.001) 0.950(0.001) 0.955(0.001) 0.959(0.001) 0.958(0.001) 0.958(0.001) 0.953(0.001) 0.956(0.001) 0.955(0.001)

Average intermolecular bond orders small clu.

small clu.+pol.

full clu.

0.131(0.026) 0.127(0.018) 0.101(0.011) 0.104(0.014) 0.103(0.013) 0.124(0.025) 0.120(0.020) 0.123(0.025)

0.140(0.040) 0.139(0.037) 0.124(0.031) 0.127(0.026) 0.119(0.021) 0.130(0.024) 0.128(0.030) 0.132(0.031)

20 1A:2D 0.113(0.018) 2A:1D 0.108(0.025) 24 1A:2D 0.089(0.012) 2A:1D 0.093(0.015) 2A:2D 0.093(0.015) 30 1A:2D 0.106(0.011) 2A:1D 0.102(0.109) 2A:2D 0.105(0.011)

Average dipole moment (Debye)

20

1A:2D 2A:1D 24 1A:2D 2A:1D 2A:2D 30 1A:2D 2A:1D 2A:2D

small clu.

small clu.+pol.

2.525(0.016) 2.561(0.073) 2.478(0.016) 2.468(0.033) 2.557(0.061) 2.499(0.047) 2.499(0.076) 2.553(0.121)

2.708(0.021) 2.758(0.070) 2.663(0.020) 2.655(0.024) 2.698(0.066) 2.713(0.051) 2.720(0.071) 2.739(0.103)

full clu.

monomer

2.669(0.059) 2.123(0.012) 2.701(0.066) 2.136(0.010) 2.687(0.050) 2.135(0.011) 2.651(0.062) 2.1246(0.012) 2.720(0.106) 2.122(0.020) 2.640(0.085) 2.125(0.012) 2.654(0.120) 2.123(0.011) 2.737(0.118) 2.126(0.010)

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FIGURE CAPTIONS Figure 1 The three types of systems studied: water monomer, two “small” clusters consisting of a central monomer and its hydrogen-bonded environment and one of the “large” clusters. (Here the cage-type one is depicted) Figure 2 Water complexes of 20, 24 and 30 molecules in dodecahedral (a) and cage-like (b, c) arrangements. Figure 3 Distribution of intramolecular (a), intermolcular (b) bond orders and of dipole moments (c) calculated for the investigated water clusters (Case C). The corresponding values for the water monomer and dimer are also shown. Figure 4 Dependence of the dipole moment (a), intra- and intermolecular (b and c) O–H bond orders on the environment of the “small” cluster consisting of the “central” water molecule and its hydrogen-bonded nearest neighbours. On abscissae the values calculated for the small cluster, on ordinatae those for the “full” cluster or if a polarizable medium is added. (The dotted lines correspond to y = x.) The values corresponding to the free monomers at the geometry in complex are also shown. Figure 5 Intermolecular bond orders for the hydrogen bonded water pairs as functions of the Ohno number of the proton donor molecule. (The dashed line connects the mean values.)

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Monomer

small H-bonded cluster

Figure 1


cage like water cluster


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 2a


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Figure 2b



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 2c


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

a) 0,20

P(BO)

0,15

0,10

0,05

monomer dimer 0,00

0,65

0,70

0,75

0,80

0,85

0,90

0,95

intramolecular bond order (BO)

b) 0.35

0.30

P(BO)

0.25

0.20

0.15

0.10

0.05

0.00 0.05

0.10

0.15

0.20

0.25

0.30

intermolecular bond order

c) 0.25

P(dipole)

0.20

0.15

0.10

0.05

monomer

Dimer

0.00

2.0

2.1

2.2

2.3

2.4

2.5

2.6

dipole moment(D)

Figure 3


2.7

2.8

2.9


2,9

2,8

dipole moment (Debye)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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2,7

2,6

2,5 small cluster + pol. medium(SMD) full cluster monomer

2,4 2,2

2,1

2,0 2,4

2,5

2,6

2,7

2,8

dipole moment (Debye)

Figure 4a


2,9

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0,96

intramolecular OH bond order

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0,92

small cluster+pol. medium (SMD) full

0,88

cluster

monomer

0,84

0,80

0,76

0,72

0,68

0,64 0,64

0,68

0,72

0,76

0,80

0,84

0,88

0,92

intramolecular OH bond order

Figure 4b


0,96


0,25

intermolecular OH bond order

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0,20

0,15

0,10

small cluster + pol.medium (SMD) full cluster

0,10

0,15

0,20

intermolecular OH bond order

Figure 4c


0,25

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0.28 0.26 0.24 0.22

bond order

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.20 0.18 0.16 0.14 0.12 0.10 0.08

0

1

2 Ohno number

Figure 5


3

4


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Monomer

small H-bonded cluster

TOC graphics Hierarchical levels of systems considered


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cage like water cluster

Hierarchy of the Collective Effects in Water Clusters - ACS Publications

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