Cooperativity and hydrogen bonding network in water clusters

Transcription

1 Chemical Physics 258 (2000) 225±231 Cooperativity and hydrogen bonding network in water clusters Sotiris S. Xantheas Environmental Molecular Sciences Laboratory, Paci c Northwest National Laboratory, 906 Battelle Boulevard, P.O. Box 999, MS K8-91, Richland, WA 99352, USA Received 4 January 2000 Abstract Clusters of water molecules are held together by hydrogen bonding networks. These networks are di erentiated by the participation of the individual water molecules in the hydrogen bonds either as proton donors (d), proton acceptors (a) or their combinations. It has long been assumed that the stability of clusters is determined by the dominant twobody interactions between the water molecules. We have found that homodromic hydrogen bonding networks, i.e. those exhibiting donor±acceptor (da) arrangements between all water molecules, are associated with the largest nonadditivities among other networks present in low lying minima of small water clusters. This nding o ers a novel explanation for the stability of homodromic rings that are the global minima for the clusters trimer through pentamer. Among the non-additive terms, three-body terms are mainly responsible for determining the relative stabilities between the various trimer through pentamer isomers. This suggests that purely two-body pairwise additive potentials will result in errors exceeding 20% for clusters larger than the pentamer. Among all higher order components, the three-body term was found to be the most important. Ó 2000 Published by Elsevier Science B.V. All rights reserved. Since Pindar, the Greek philosopher who used to refer to it as ``Aqirsom lem tdxq'' (signi cant indeed water), water has received much attention because of its abundance in nature and its role in many biological and chemical processes [1,2]. A molecular level understanding of the structure of water has been sought for almost half a century, yet there exists no comprehensive model that can su ciently describe a wide range of both its microscopic and macroscopic properties [3]. address: sotiris.xantheas@pnl.gov (S.S. Xantheas). Small clusters of water molecules can be considered as prototypes for understanding the fundamental interactions that govern hydrogen bonding. A renewed interest in these systems and their relevance in probing the structure of liquid water has been recently fueled by the vibration± rotation±tunneling (VRT) microwave experiments of Saykally and co-workers [4±13]. These measurements have provided the rst experimental data on the low frequency anharmonic tunneling motions. The VRT experiments have con rmed earlier rst principles theoretical predictions [15± 20] suggesting that the global minima of the trimer through pentamer clusters have ``ring'' structures consisting of homodromic con gurations with one hydrogen bond per molecule in which all members act as both proton donors and acceptors (da) to /00/$ - see front matter Ó 2000 Published by Elsevier Science B.V. All rights reserved. PII: S (00)

2 226 S.S. Xantheas / Chemical Physics 258 (2000) 225±231 nearest neighbors. For the water hexamer, the experimentally obtained rotational constants for the global minimum were best t by a cage rather than a ring structure [13,14]. An apparent manifestation of the cooperative e ects in the rst few water clusters has been associated with a systematic contraction of the nearest-neighbor O±O separation [20] and a red shift of the hydrogen-bonded OH stretch frequency with increasing cluster size [20,21]. For the ring water tetramer and pentamer the band positions for these frequencies are at 3416 and 3360 cm 1 respectively [21], compared to cm 1 for liquid water [22]. The corresponding (vibrationally averaged) O±O separations obtained from the experimentally tted rotational constants are 2.78 [8±10] and 2.76 A [11,12] as compared to 2.84 A for liquid water at 4 C [23] and 2.74 A for normal ice at 223 K [24]. These results are rather unexpected considering the di erent environments experienced by each water molecule in the cluster when compared to liquid water. Although the cyclic structure of the water trimer could be rationalized in terms of maximizing hydrogen bonding between three water molecules, there is no obvious ``physically intuitive'' argument justifying the energetically preferential ring arrangements of the water tetramer and pentamer with respect to other possible structures that have a larger number of hydrogen bonds. The fact that the ring minima up to the pentamer have only one hydrogen bond per water molecule suggests that they sample part of the con guration space not likely probed by con gurations in bulk water in which every water molecule participates on the average in two hydrogen bonds. Two questions therefore arise: (i) what is the underlying physical e ect resulting in the ring structures of the rst few water clusters, if not the maximization of the number of hydrogen bonds and (ii) are the modeling requirements quantitatively di erent for these structures than for bulk water? A quantitative account of the cooperative effects can be achieved by decomposing the interaction energy DE n of a system of n-bodies (water molecules) see, e.g. [25] DE n ˆ E 1; 2; 3; 4;...; n ne w Xn E i ne w Xn 1 X n j>i Xn 2 X n 1 X n D 2 E ij j>i k>j Xn 3 X n 3 X n 1 X n j>i k>j l>k Relaxation D 3 E ijk two-body D 4 E ijkl three-body four-body D n E 1; 2; 3; 4;...; n n-body; 1 where E i ; E ij ; E ijk ;... are the energies of the various monomers, dimers, trimers, etc. in the cluster and E w is the energy of the isolated water molecule. The relaxation or deformation component represents the energy penalty for distorting the individual fragments in the equilibrium cluster geometries with respect to those in isolation. The pairwise-additive two-body interactions and the higher three- through n-body non-additive components are de ned as D 2 E ij ˆE ij fe i E j g; D 3 E ijk ˆE ijk fe i E j E k g fd 2 E ij D 2 E ik D 2 E jk g; 2 3 D 4 E ijkl ˆE ijkl fe i E j E k E l g fd 2 E ij D 2 E ik D 2 E il D 2 E jk D 2 E jl D 2 E kl g fd 3 E ijk D 3 E ijl D 3 E ikl D 3 E jkl g: 4 Since it is of interest to investigate the e ect of the local hydrogen bonding network on the cooperativity, we performed this analysis for the global and various low-lying networks of the (H 2 O) n clusters (n ˆ 3±6) that exhibit di erent hydrogen bonding environments. The latter are di erentiated by the bonding arrangement of the individual

3 S.S. Xantheas / Chemical Physics 258 (2000) 225± water molecules participating in neighboring hydrogen bonds either as proton donors (d), proton acceptors (a) and their combinations (dd,da,aa), as de ned in Fig. 1. The energy decomposition is performed at the minimum energy geometries optimized at the second order perturbation (MP2) level of theory [26] using DunningÕs aug-cc-pvdz basis set [27,28]. During the MP2 calculations all electrons were correlated for consistency with our earlier calculations for the minima [20]; this as opposed of ``freezing'' the O(1s) electrons has practically no e ect in the computed structures and relative energy terms. The individual energy terms include corrections for basis set superposition error (BSSE) estimated by the function counterpoise method [29] as outlined elsewhere [30]. The calculations are performed with the GAUSSIAN-94 [31] and MOLPRO-94 [32] suites of programs. It should be noted that the basis set used here is far from been able to yield converged absolute binding energies for the clusters. However it reproduces both the order and relative energies of, for instance, the closely spaced various hexamer isomers Fig. 1. De nition of the local hydrogen bonding network for a water molecule in terms of a proton donor (d) or a proton acceptor (a) with respect to the neighbors. to within 0.1 kcal mol 1 or better when compared to results with the aug-cc-pv5z set and/or the estimates of the MP2 complete basis set limit [33]. The energy separation for the trimer through pentamer networks considered here is much larger than the corresponding one for the hexamers, a fact that clearly justi es the use of the aug-ccpvdz set for the purpose of the present study. The various water trimer, tetramer, pentamer and hexamer networks considered in this study are shown in Fig. 2. Their binding energies relative to the isolated fragments and the magnitude of the three-body term (in parentheses) are also indicated in kcal mol 1. A breakdown of the total energy di erence of the various networks from the most stable isomer for each cluster in terms of di erences in the relaxation, two-, three- and higher order contributions is listed in Table 1. The global minimum of the water trimer is a cyclic structure in which all water molecules act as both proton donors and acceptors (da) to neighbors [4±7,17,19]. The two ``open'' local minima exhibiting (a,dd,a) and (d,aa,d) arrangements are almost isoenergetic with each other, lying 6.1 kcal mol 1 higher in energy [34]. Besides small di erences in the relaxation energies of the fragments due to the strain associated with the formation of the ring (0.3 kcal mol 1 in favor of the ``open'' isomers), the energy separation between them and the cyclic minimum can be almost equally attributed to di erences in the two- and three-body terms (cf. Table 1). The di erences in the two-body terms (3.4 and 3.6 kcal mol 1, respectively) are due to the fact that the ``open'' structures have slightly repulsive two-body components (0.5 and 0.7 kcal mol 1 ) between the end fragments. Of comparable signi cance are the di erences of the threebody term (3.0 and 2.8 kcal mol 1 ) between the cyclic and the open minima. The latter arise from the fact that the three-body term is large and attractive ( 2.5 kcal mol 1 ) for the cyclic but smaller and repulsive (0.3 and 0.5 kcal mol 1 ) for the ``open'' minima, as shown in Fig. 2. The water tetramer global minimum is also a cyclic structure of S 4 symmetry having a (da,da, da,da) arrangement [8±10,20]. A cage structure in which three water molecules act as (da) and the

4 228 S.S. Xantheas / Chemical Physics 258 (2000) 225±231 Fig. 2. Binding energies (MP2/aug-cc-pVDZ/BSSE corrected) for the various trimer, tetramer, pentamer and hexamer minima. Numbers in parentheses indicate the magnitude of the three-body term in kcal mol 1. fourth as a double donor and a single acceptor (dda) lies 4.8 kcal mol 1 higher. Even higher in energy (10.5 kcal mol 1 ) lies a (dd,aa,dd,aa) ring minimum of D 2h symmetry. The energy separation between these structures can be mainly attributed to the di erences in the three-body terms which contribute twice as much as the di erences in the corresponding two-body terms. For example, the contribution in the energy separation between the global and cage minima is 3.0 kcal mol 1 for the three-body terms and 1.6 kcal mol 1 for the two-body terms as can be seen from Table 1. The same is true for the energy di erence between the global and the D 2h minima: almost 2=3 comes from di erences in the three-body terms with the remaining 1=3 due to di erences in the corresponding two-body terms. The four-body term is negligible ( 0.5, 0.1 and 0.1 kcal mol 1 ) for all three networks found in the tetramer minima considered here.

5 S.S. Xantheas / Chemical Physics 258 (2000) 225± Table 1 Breakdown of the energy di erence for the various hydrogen bonding networks with respect to the global minima which have all (da) arrangements for the trimer through pentamer Cluster Network D(DE e ) Total D(DE e ) Relax D(DE e ) 2-B Trimer (a,dd,a) (d,aa,d) D(DE e ) 3-B D(DE e ) 4-B D(DE e ) 5-B Tetramer (da,da,da,dda) (dd,aa,dd,aa) Pentamer (da,daa,- da,aad,da) Hexamer Cage Book S The water hexamer energy di erences are computed with respect to the more stable ``prism'' isomer (note that the ``cage'' minimum becomes more stable among all hexamer isomers upon inclusion of zero-point energy corrections). Positive/negative numbers indicate contributions against/in favor of the network with respect to the global minimum. The cyclic (da,da,da,da,da) arrangement is also the global minimum for the water pentamer [11,12]. A cage structure having six hydrogen bonds and a (da,daa,da,aad,da) network lies 1.7 kcal mol 1 higher in energy. The sum of two-body interactions for the cage structure is 1.3 kcal mol 1 more attractive than for the global minimum. Here the relative energies are determined solely by the three-body term, which is 2.1 kcal mol 1 more attractive for the ring than the cage structure, therefore compensating for the less attractive twobody term. The four-body term also contributes 0.8 kcal mol 1 in favor of the ring structure whereas the ve-body term is negligible (<0.03 kcal mol 1 for both arrangements). For the water hexamer the number of isomers that are close (within 1 kcal mol 1 ) in energy is larger [35±37] as previously reported by Jordan and co-workers and recently by others [38±40]. These previous studies were mainly concerned with the relative stability of the various isomers while the many-body energy decomposition was performed for the ring S 6 structure [37]. Interest in this cluster stems from the observation that it represents the ``transition'' from ring to a more compact, cage-like structures as regards the global minimum. When considering minimum energy di erences alone the minimum energy con guration is that of the ``prism'' isomer (cf. Fig. 2) with at least three other isomers within 1 kcal mol 1 higher in energy. However, in agreement with previous results [13,14,35±40], inclusion of zeropoint energy corrections results in the con guration with the two capping water molecules (``cage'' isomer) being the global minimum. The ring (da) homodromic network of S 6 symmetry lies 0.9 kcal mol 1 above the ``prism'' minimum (cf. Table 1), a separation mainly arising from the di erence in the two-body term (5.0 kcal mol 1 ) in favor of the ``prism'' besides a signi cant contribution of the higher order (three- through ve-body) terms in favor of the ring network. This is due to the fact that, in accordance with the observation for the smaller clusters, the ring S 6 minimum exhibits the largest non-additive (three- through ve-body) components of the binding energy among all other hexamer minima. We, therefore, propose that the underlying physical e ect behind the formation of small rings of the trimer through pentamer clusters exhibiting homodromic hydrogen bonding networks is the result of the maximization of the non-additive components of their binding energies. Homodromic networks were found to be associated with the largest non-additive components among other networks that are present in low-lying minima of the clusters examined. These non-additive components and, in particular, the three-body term

6 230 S.S. Xantheas / Chemical Physics 258 (2000) 225±231 Table 2 Percentage contribution of the two- and three-body terms to the interaction energy for the various networks of the (H 2 O) n, n ˆ 3±6 clusters Cluster Network Two-body Three-body Four-body Trimer (da,da,da) (a,dd,a) (R) (d,aa,d) (R) Tetramer (da,da,da,da) Cage (R) (aa,dd,aa,dd) (R) 0.7 (aa,da,dd,da) (R) Pentamer (da,da,da,da,da) Cage Hexamer Prism Cage Book S 6 (da,da,da,da,da,da) Repulsive contributions are indicated by (R) and are responsible together with the relaxation terms in producing two-body components that exceed 100%. For all pentamer and hexamer con gurations terms higher than four-body have negligible (<0.5%) contributions. determine the stability of the ring structures relative to other isomers for the water tetramer and pentamer. The water hexamer is the ``turning point'' where the magnitude of the total two-body additive component ``takes over'' in determining the most stable structure. Even for this cluster, however, the homodromic network in the ring S 6 minimum has the largest three- and four-body components among all networks in the other local minima. This nding suggests that asymmetric and locally inhomogeneous cluster environments may require more burdensome theoretical treatment than the on-the-average homogeneous environment found in liquid water. Purely two-body pairwise-additive interaction potentials, such as the ones derived from the water dimer potential energy surface, will result in errors exceeding 20% for clusters larger than the pentamer (cf. Table 2). Although e ective two-body pairwise-additive potentials like TIP4P which can be considered as incorporating non-additive e ects implicitly can be quite successful in reproducing the binding energies in a limited range of cluster sizes [41] when compared to more accurate many-body potentials [41±43], these models are not transferable across di erent environments (clusters, liquid water, ice phases) that exhibit di erent relative magnitudes of non-additivities [44,45]. Transferable models should include accurate estimates of many-body components and especially the three-body term and its dependence upon the local hydrogen bonding network environment. Interaction potentials with explicit consideration of the threebody interaction terms have been, for instance, previously used to model the structure and properties of various ice phases [46±48]. An alternative way of incorporating parts of higher-order manybody interactions is via the induction scheme (see Refs. [44,45,49]) whereas every water molecule is represented by a dipole and/or higher [50] moments that are determined iteratively. A recent study [51] has suggested that the three-body potential in the water trimer is dominated by mainly the second-order and induction non-additivity. Furthermore, it suggested that if induction e ects are to be included by iteration of the induced dipole moments and the corresponding electric elds, then one should proceed with this iteration beyond the rst step. Acknowledgements This work was performed at Paci c Northwest National Laboratory under the auspices of the O ce of Basic Energy Sciences, Division of

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