Ab initio water hexamers and octamers: tool to study hydrogen-bonded pattern in liquid water

Fourteen different structures of water hexamer found ab initio within a 6- 311G** basis set in the interval of 1.7 kcal/mol above the global minimum represent an unprecedently wide range of conformational plasticity of liquid water. The present work also provides the first ab initio demonstration...

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Datum:	1998
1. Verfasser:	Kryachko, E.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Інститут фізики конденсованих систем НАН України 1998
Schriftenreihe:	Condensed Matter Physics
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/118939
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Zitieren:	Ab initio water hexamers and octamers: tool to study hydrogen-bonded pattern in liquid water / E. Kryachko // Condensed Matter Physics. — 1998. — Т. 1, № 2(14). — С. 213-238. — Бібліогр.: 28 назв. — англ.

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spelling	irk-123456789-1189392017-06-02T03:02:29Z Ab initio water hexamers and octamers: tool to study hydrogen-bonded pattern in liquid water Kryachko, E. Fourteen different structures of water hexamer found ab initio within a 6- 311G basis set in the interval of 1.7 kcal/mol above the global minimum represent an unprecedently wide range of conformational plasticity of liquid water. The present work also provides the first ab initio demonstration of the existence of pentacoordinated water clusters. Чотирнадцять різних структур гексамера води, які знайдені методом ab initio в базисі 6-311G в інтервалі 1.7 kcal/mol вище глобального мінімуму, є безпрецедентним прикладом високої конформаційної пластичності рідкої води. Дана робота також вперше демонструє існування пентакоординаційних кластерів води на рівні прецизійних ab initio розрахунків. 1998 Article Ab initio water hexamers and octamers: tool to study hydrogen-bonded pattern in liquid water / E. Kryachko // Condensed Matter Physics. — 1998. — Т. 1, № 2(14). — С. 213-238. — Бібліогр.: 28 назв. — англ. 1607-324X DOI:10.5488/CMP.1.2.213 PACS: 61.20.Ja, 61.25.Em, 61.20.Gy, 62.30.+d, 63.20.Pw, 63.90.+t, 64.70.Ja, 65.50.+m, 64.30.+t http://dspace.nbuv.gov.ua/handle/123456789/118939 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Fourteen different structures of water hexamer found ab initio within a 6- 311G** basis set in the interval of 1.7 kcal/mol above the global minimum represent an unprecedently wide range of conformational plasticity of liquid water. The present work also provides the first ab initio demonstration of the existence of pentacoordinated water clusters.
format	Article
author	Kryachko, E.
spellingShingle	Kryachko, E. Ab initio water hexamers and octamers: tool to study hydrogen-bonded pattern in liquid water Condensed Matter Physics
author_facet	Kryachko, E.
author_sort	Kryachko, E.
title	Ab initio water hexamers and octamers: tool to study hydrogen-bonded pattern in liquid water
title_short	Ab initio water hexamers and octamers: tool to study hydrogen-bonded pattern in liquid water
title_full	Ab initio water hexamers and octamers: tool to study hydrogen-bonded pattern in liquid water
title_fullStr	Ab initio water hexamers and octamers: tool to study hydrogen-bonded pattern in liquid water
title_full_unstemmed	Ab initio water hexamers and octamers: tool to study hydrogen-bonded pattern in liquid water
title_sort	ab initio water hexamers and octamers: tool to study hydrogen-bonded pattern in liquid water
publisher	Інститут фізики конденсованих систем НАН України
publishDate	1998
url	http://dspace.nbuv.gov.ua/handle/123456789/118939
citation_txt	Ab initio water hexamers and octamers: tool to study hydrogen-bonded pattern in liquid water / E. Kryachko // Condensed Matter Physics. — 1998. — Т. 1, № 2(14). — С. 213-238. — Бібліогр.: 28 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT kryachkoe abinitiowaterhexamersandoctamerstooltostudyhydrogenbondedpatterninliquidwater
first_indexed	2025-07-08T14:56:26Z
last_indexed	2025-07-08T14:56:26Z
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fulltext	Condensed Matter Physics, 1998, Vol. 1, No 2(14), p. 213–238 Ab initio water hexamers and octamers: tool to study hydrogen-bonded pattern in liquid water E.Kryachko 1,2,3 1 Bogolyubov Institute for Theoretical Physics, Kyiv-143, Ukraine 252143 2 Cherry L. Emerson Center for Scientific Computation and Department of Chemistry Emory University, Atlanta, GA 30322, U. S. A. 3 Department of Chemistry, The Johns Hopkins University, Baltimore, MD 21218, U. S. A. Received March 18, 1998 Fourteen different structures of water hexamer found ab initio within a 6- 311G basis set in the interval of 1.7 kcal/mol above the global minimum represent an unprecedently wide range of conformational plasticity of liquid water. The present work also provides the first ab initio demonstration of the existence of pentacoordinated water clusters. Key words: liquid water, H-bond pattern, orientational defect, water cluster, “Dangling” bond, Ab initio HF 6-311∗∗ calculation PACS: 61.20.Ja, 61.25.Em, 61.20.Gy, 62.30.+d, 63.20.Pw, 63.90.+t, 64.70.Ja, 65.50.+m, 64.30.+t 1. Introduction Liquid water is the most mysterious substance for natural scientists because of its great importance in the Universe and its versatile abnormal properties are far from being completely understood. It is formed due to hydrogen or, shortly, H- bonds and it is, in fact, its H-bonded pattern with interconnectivity and tortuosity that always plays the role of the key starting point in numerous studies aimed at resolving water paradigm [1] (see also [2]). It has been known for a long time that H-bonded patterns of liquid water and hexagonal ice Ih are tetrahedral [1(a)]. They were also believed to have much more similarities between them. That is why ice was often chosen as a reasonable reference model for the study of an H-bonded water pattern. The aforementioned tetrahedrality originates, in fact, from the tetrahedral charge distribution around c© E.Kryachko 213 E.Kryachko the oxygen atom in a water monomer possessing two positive partial charges at the positions of hydrogen atoms and two negative partial charges that refer to ear-like lone electron pair. This idea was expressed by Bernal-Fowler-Pauling and called ice rules [3]: (i) There are exactly two hydrogen atoms that belong to each oxygen atom with the O-H bond length of 1 Å; (ii) There is exactly one hydrogen atom that occupies each O· · ·O bond between any pair of neighbouring oxygen atoms. What might an H-bonded pattern of liquid water be like then? Well, obviously, like ice. Its “icelikeness” is manifested in the fact that each oxygen atom is involved in two covalent and two hydrogen bonds, or equivalently, that each water molecule is four-bonded and is surrounded by four nearest-neighbour water molecules. The latter ones compose its first coordination shell. Within the Bernal-Fowler-Pauling H-bonded pattern, the oxygen atoms are arranged in such a way that all the O- O-O bond angles between the nearest-neighbour water molecules are tetrahedral, i. e., equal to circa 109.47o. The hydrogen atoms decorate therein O-O “bonds”, establishing the so-called “hydrogen bigamy”. Let us ask then the following question. Does this “icelikeness” adequately de- scribe the whole H-bonded pattern of liquid water? Or, in other words. Does liquid water contain any “patches” in its pattern that in fact violate Bernal-Fowler- Pauling ice rules and are crucial for turning liquid water into what it actually is? The answers are certainly ‘no’ and ‘yes’, respectively. It is now apparently pretty hard to believe in resolving a liquid water paradigm denying the very existence of such non-icelikeness “patches”. To put it simpler, water can be roughly described by a “two-state” model with one “state” obeying Bernal- Fowler-Pauling rules and another one violating them. The “two-state” model satisfies either the energetical or geometrical criterion imposed on the H-bonded pattern [4-6]. The former sug- gests that two water molecules are H-bonded if their interaction energy V < VHB. The negative threshold H-binding energy VHB plays the role of a model cutoff pa- rameter. It is allowed to take on a sequence of values, VHB = −2nε with integer n from interval (10, 41) and ε = 0.07575 kcal/mol. VHB varies then from - 6.2115 to - 1.5150 kcal/mol. According to the geometrical criterion (for current references see [7]), two water molecules are H-bonded if the following three constraints are accomplished alto- gether. The first one is the constraint on interoxygen separations that must be less than Rthr = 3.5 - 3.7 Å [8]. Rthr determines the position of the first minimum of the gOO radial distribution function and defines, in fact, the boundary of the first coordination shell. The second constraint is imposed on the distance between the oxygen atom of the acceptor water molecule and the hydrogen atom of the donor one. It should be smaller than rthr = 2.45 Å, that is, the distance at which the first minimum of the radial distribution function gOH takes place [8]. The third constraint limits angle δH between the participating oxygen, the hydrogen of the donor molecule and the oxygen atom of the acceptor molecule: δH > 160o. This constraint seems to be quite fragile and is often omitted because a reasonable de- 214 H-bonded pattern in water viation of δH from 160o to smaller angles indicates a stronger nonlinearity of an H-bond. These constraints are sometimes supplemented by another one imposed on the values of the lone pair-oxygen-oxygen angle. Regarding the O-O distribution function gOO of liquid water at 25o C [8], its first maximum is sharply peaked at 2.86 Å and its integral from zero to Rthr, which is interpreted as the mean number of water molecules within the first co- ordination shell, is about 4.5. This implies that the H-bonded pattern of liquid water partly possesses five-fold coordinated “patches”. Their very existence un- ambiguously shows how far real liquid water is from that modelled by the Bernal- Fowler-Pauling rules. One may even insist that its natural perfection is, in fact, in its imperfection which makes water so anomalous and so mysterious for natural science. In terms of patterns, this imperfection might be explained by the fact that water possesses some sort of defects, namely, such “patches” in its H-bonded pattern where the Bernal-Fowler-Pauling rules are violated. It has been thought that these defects manifest themselves in many ways, for instance, in conductiv- ity, in water restructuring resulting in a flow of water, in interpreting the Raman [9,10] and infrared spectra [1]. They are usually divided into two types: ionic and orientational, depending on which the Bernal-Fowler-Pauling ice rule is violated. In the present work we focus only on orientational defects which represent the violation of the second Bernal-Fowler-Pauling rule. The simplest model of an orientational defect was a long time ago suggested by Bjerrum [11(a)] (see also [1]). The Bjerrum orientational defects are of two types. One corresponds to the “empty” O · · ·O bond and is called L-defect [11(b)]. Another one, D-defect [11(b)] is, in fact, a doubly occupied bond O - H · · · H - O. This definition of orientational defects is rather schematic and does not take into account the cooperative nature of H-bonding in liquid water. The progress towards understanding the nature of orientational defects was made due to the first and in some sense semi-empirical calculations by Dunitz [12(a)], Cohan et al. [12(b)], and Eisenberg and Coulson [12(c)] (see also [12(d-h)]). These calculations show that an orientational defect is actually some area of the H-bonded pattern around an “empty” or doubly oc- cupied H-bond that should include some distorted H-bonds around. Distortion is primarily thought in terms of δH angle whose value in the range of 160o − 130o describes a leaned or nonlinear H- bond that converts to the so-called bifurcated H-bond when this angle approaches 120o − 100o. A physical model of a solitonic- type orientational defect where the central part is, in fact, a sort of a bifurcated H-bond was elaborated in [13] (for recent review see [14]). A bifurcated H-bond is such a specific type of H-bond where the hydrogen atom simultaneously participates in or donates two hydrogen bonds. In other words, the hydrogen atom is shared by a couple of oxygen atoms simultaneously [9]. It is believed that these bonds contribute significantly to the Raman [10] and infrared [1] (see also [13(b)]) spectra of liquid water. It is also believed that they are important in mobility of water molecules. This view was supported experimentally [15] and by computer simulations [16,17]. However, it should be stressed that all these Bjerrum-type defects rely on the tacit assumption of preserving a four-fold 215 E.Kryachko coordination. It has been recently suggested [17] that a five-fold “patch” is a new type of an orientational defect with a bifurcated H-bond which appears as a result of approaching the fifth water molecule (see also [10(b)]). As suggested in [17] as well, this “patch” facilitates a transition from one Bernal-Fowler-Pauling pattern to another through a lowered energy barrier. The similar model of a five-fold “patch” was discussed in [18]. There is still another very specific feature of liquid water worth mentioning. It is the O-O-O bond angle distribution function that, according to the recent studies [19] (see also [17(b), 18]), reveals two maxima. One corresponds to the tetrahedral bond angle. Another one, peaked at ∠O-O-O = 60o, demonstrates that the corresponding water molecules are settled in nontetrahedral directions. In this context, pentacoordinated “patches” within an H-bonded pattern of liquid water are an appealing concept, since they constitute a simple and, to some extent, a rather universal principle of the underlying liquid water dynamics. The aim of the present paper is to reveal such “patches” in water clusters by means of performing ab initio search over their total potential energy surfaces and to study the properties of these defects. Section II deals with the computational GAUSSIAN-type methodology of this work. The next Section III starts with a brief description of the present status of ab initio water clusters and continues with the exhaustive search of the total potential energy surface of water hexamer cluster to reveal novel lower-lying water hexamer structures and in particular those of hexamers and octamers that possess pentacoordinated “patches”. This is, in fact, the first evidence of ab initio five-fold coordinated water clusters. Section VI discusses and summarizes the present work. 2. Computational methodology All the calculations were performed with the Gaussian-94 suite of programs [20] at the Cherry L. Emerson Center for Scientific Computation of Emory Uni- versity in Atlanta. The ground-state geometry of water clusters was optimized at the Hartree-Fock level of computation with the triply split valence 6-311G ba- sis set of 180 basis functions that includes polarization functions on the oxygen and hydrogen atoms in particular [21]. In all the computations, no constraints were imposed on the geometry of water clusters. Full geometrical optimization was performed for each water cluster structure, and the attainment of the energy minimum was verified by calculating the vibrational frequencies that result in the absence of negative eigenvalues. Default options were used for the SCF convergence and for the threshold limits determining the final changes in the maximum forces and displacements in the geometry optimization. Vibrational modes and the corre- sponding frequences are based on a harmonic force field. Empirical scaling factors were not used. For conciseness, the tables report only the most intensive infrared (IR) and Raman active bands for which IR intensity and Raman activity exceed 100 km/mol and 20 Å 4 /amu, respectively. 216 H-bonded pattern in water 3. Pentacoordinated water clusters 3.1. Prelude: hexamer and octamer Study of water clusters is one of promising ways to resolve the liquid water paradigm. It is trivial, on the one hand, that a larger cluster better mimics bulky water. On the other hand, it is also quite evident that a larger cluster possesses a richer potential energy surface picture. It is then a rather well-spread and rea- sonable belief that to grasp the water paradigm, it might be sufficient to study those water cluster structures that occupy lower-lying energy minima on the po- tential energy surface. Should we be confident with this belief? Apparently, ‘yes’, if among lower-lying water cluster structures there are, in particular, those with five-fold coordinated water molecules. Hexamer and octamer water clusters were the subject of a number of ab initio studies [22,23]. It has been recently shown [22(d)] that at the HF/6-311G level of the computational theory, the global minimum of the hexamer potential energy surface is attained by prism structure. There are also two lower-lying local minima, one of which, corresponding to the cyclic chair- type structure, is 0.84 kcal/mol. Another one occupied by the boat-like structure distances from the chair-type one by 1.08 kcal/mol. There is yet another global minimum found at the cage- like water hexamer cluster with 8 H-bonds [23(d-g)]. To resolve this controversy, we have done an exhaustive search of the landscape of the total potential energy surfaces of water hexamer. The result of this search is reported in table 1 for energy, zero-point vibration energy (ZPVE) enthalpy and entropy, and in table 2 – for rotational constants and the total dipole moment and are displayed in figure 1. As it is seen in figure 1, three prism structures of (H2O)6 are at the bottom of the total potential energy surface. Prism I occupies its global minimum, but two others lie very close, at 2.4 and 5.8 cm−1, taking the zero-point vibration energy into account. Five cage-type structures are followed by them. They are shown in figure 2. Table 2 demonstrates a good agreement of rotational constants for cages calculated in the present work and reported in [23(g)]. Four lower-lying cages I - IV possess very particular geometries. Their interoxygen distances between O3 and O4 are 3.577, 3.577, 3.466, and 3.570 Å, respectively. This implies that according to the first geometrical constraint these are precisely “dangling” bonds because of the absence of a hydrogen atom between these oxygens. Furthermore, some O-O- O interbond angles move away from the tetrahedral values to lower angles. For example, in cage I, ∠O3−O14−O4 = 75.86o; in cage II, ∠O3−O14−O4 = 75.85o; in cage III, ∠O3 − O15 − O4 = 70.99o; in cage IV, ∠O3 − O14 − O4 = 75.49o; and finally, in cage VI, ∠O1−O15−O4 = 75.52o. The cage structure of water hexamer mimics the basic unit of one of the high-density polymorphs of ice, ice VI (compare with [1(a)], figure 3.8). Proceding to the next structures, it is worth mentioning that prism VI is exactly the one found in [22(d)] at the global minimum. It is interesting that the open prism structure shown in figure 3 with a very high dipole moment of 4.38 D appears in the present list of lower-lying structures of water hexamer. Another cage structure, 217 E.Kryachko cage VI displayed in figure 4, is followed by an open prism and also possesses a rather high dipole moment equal to 3.45 D. Chair- or ring-type structure with Table 1. Energy, enthalpy H, entropy S, and zero-point vibrational energy ZPVE of lower-lying water hexamers. Structure - Energy ∆E ZPVE - Enthalpy ∆H S +456 hartrees kcal/mol kcal/mol +456 hartrees kcal/mol cal/mol K chair 0.356702756 2.3619426 99.85516 0.180874 1.708691 132.506 0.7666526 0.113401 cageVI 0.357035657 2.1530463 100.99548 0.180491 1.949024 121.942 1.6980763 1.494054 op. prism 0.357431391 1.9047220 100.74973 0.181046 1.600760 123.652 1.2040020 0.900040 prism IV 0.358047985 1.5178074 101.30789 0.181304 1.438864 119.300 1.3752474 1.296304 prism VI 0.358048203 1.5177980 101.30876 0.181303 1.439492 119.297 1.3761080 1.297757 prism V 0.358048107 1.5177308 101.30871 0.181303 1.439492 119.301 1.3759908 1.297752 cage V 0.358758406 1.0720161 101.01288 0.182218 0.865327 121.718 0.6344461 0.427757 cage IV 0.358944226 0.9554135 101.09977 0.182346 0.785006 121.111 0.6047335 0.434326 cage III 0.3590113236 0.9133095 101.10253 0.182410 0.744846 120.990 0.5653895 0.396926 cage II 0.359012959 0.9122833 101.10311 0.182409 0.745474 120.983 0.5649433 0.398134 cage I 0.359013050 0.9122262 101.09889 0.182410 0.744846 121.022 0.5606662 0.393286 prism III 0.360449222 0.0110239 101.45604 0.183583 0.008785 118.301 0.0166139 0.014375 prism II 0.360465772 0.0006388 101.45668 0.183591 0.003765 118.317 0.0068688 0.009995 prism I 0.360466790 0.0 101.45045 0.183597 0.0 118.347 0.0 0.0 a zero total dipole moment concludes the present list. Table 1 also reports the calculated thermodynamic properties of the aforementioned structures. As it is seen there, the chair structure is characterized by the highest entropy of 132.51 cal/mol T. Entropies of cage VI and the open prism are larger than those of the prism I structure. This implies that despite the fact that the latter is at the global minimum at T=0 K, the energy ordering might change with raising the temperature. In figure 5 we display a phase-like diagram of some water hexamer structures. It is seen that prism I remains at the global minimum at T ¡ 130 K. If T> 130 K, the ring structure becomes the lowest one by free energy. At room temperature, cages I-IV compete with prism I. It is interesting that at T> 300 K the open prism becomes more energetically favourable than prism I. 218 H-bonded pattern in water Figure 1. Lower-lying structures of water hexamer. In all figures O-H (solid line), H- (dashed), and “dangling” (dot-dashed) bonds. 219 E.Kryachko Table 2. Rotational constants and total dipole moment of lower-lying water hex- amer structures. Calculated rotational constants from [23(g)] are in parentheses. Structure Rotational constants, GHz Dipole moment, D chair 1.16607 1.16607 0.59509 0.0 cage VI 2.06059 1.08402 1.07861 3.4507 op. prism 1.57297 1.29319 1.07046 4.3841 prism IV 1.63253 1.28955 1.26493 2.8531 prism VI 1.63298 1.28948 1.26525 2.8512 prism V 1.63251 1.28955 1.26522 2.8523 cage V 2.11851 1.08603 1.02710 2.0095 cage IV 2.10894 1.09042 1.03060 2.0980 (2.1334) (1.1055) (1.0790) cage III 2.13939 1.07922 1.01612 1.8585 (2.1336) (1.1021) (1.0755) cage II 2.14024 1.07891 1.01597 1.8667 (2.1341) (1.1032) (1.0754) cage I 2.13849 1.07943 1.01628 1.8524 (2.1332) (1.1027) (1.0747) prism III 1.61272 1.32655 1.28996 2.9921 prism II 1.61462 1.32616 1.29057 2.9101 prism I 1.61452 1.32630 1.29033 2.9038 Table 3. Energy E, enthalpy H, entropy S, zero-point vibrational energy ZPVE, and free energy of defect water hexamer. Prisms I and VI and chair hexamers are chosen as the reference ones. ∆prismI ≡ ∆o,∆prismVI ≡ ∆p,∆chair ≡ ∆c. ∆xX in kcal/mol, X = E, H, and G (at room T=298.15 K); ∆x S in cal/mol K, x = o, p, c. E (hartree) H (hartree) S (cal/mol K) ZPVE (kcal/mol) ∆oE ∆oH ∆oS ∆oG ∆pE ∆pH ∆pS ∆pG ∆cE ∆cH ∆cS ∆cG -456.3534556 -456.1773545 127.64111 100.24399 4.40 3.92 9.29 1.49 2.88 2.48 8.34 - 0.01 2.04 2.21 - 4.86 3.66 With regard to water octamer, it is well known that the cubic structure Ca [23(a)] of D2d is the most stable one on the water octamer potential energy surface computed at the HF/6-311G level. The cubic structure of S4 symmetry occupies the local minimum that lies 3.49 kcal/mol above the Ca structure. In addition, there have been found another twenty five lower-lying octamer structures among which the ring structure R8a is energetically higher than the Ca one by 12.91 kcal/mol. At room temperature, their free energy difference diminishes to 0.19 kcal/mol [23(a)]. 220 H-bonded pattern in water Figure 2. Five lowest cage structures of six molecules of water. 221 E.Kryachko 3.2. “Dangling” bond in water hexamer Figure 3. Opened prism water hexamer. A novel lower-energy local-mini- mum structure of (H2 O)6 water cluster is revealed on the water hexamer poten- tial energy surface at the HF/6-311G level. It is shown in figure 6 and here- after is named Defect I. Its total dipole is equal to 2.47 D. It lies above prism VI and chair water hexamers by 2.88 and 2.04 kcal/mol, respectively, and the distances from the prism I structure by 4.40 kcal/mol which reduces to 3.19 kcal/mol with taking ZPVE into acc- count. The zero-point vibrational en- ergy, enthalpy, free energy of Defect I and their differences with respect to the corresponding quantities for the prism and chair structures tabulated in [22(d)] are listed in table 3. One readily figures out that at room temperature, the energy of formation of Defect I is 1.49 kcal/mol with respect to that of prism I. It is worth mentioning that Defect I and prism VI are almost isoenergetical at room temperature (see also figure 7). Figure 4. Cage VI cluster of six water molecules. Internal coordinates of Defect I are presented in table 4. One sees the- re that in Defect I, water molecule H2H3O1 distances from the nearest- neighbor water molecules with which it forms a covalent or H-bond rather different from the typical O-O separa- tion of 2.86 Å inherent for the tetrahe- dral pattern. To specify, R(O1 and O4) = 2.93 Å, R(O1 and O5) = 3.08 Å, and R(O1 and O15) = 2.97 Å. Due to this, the water molecule has a fifth nearest-neighbour which is settled on the oxygen atom O10 and character- ized by the distance R(O1 and O10) = 3.16 Å (see figure 6). The latter is less than Rthr and thus, one might think of these molecules as bonded to each other by the so-called “dangling” bond. One, therefore, concludes that this wa- ter cluster with a “dangling” bond is a five-folded “patch” that may appear among four-folded ones in an H-bonded 222 H-bonded pattern in water pattern of liquid water. Table 4. Internal coordinates of defect water hexamer. Interoxygen distance, Å R(O1 − O4) R(O1 − O5) R(O1 −O10) R(O1 −O15) R(O4 − O10) R(O4 −O14) R(O4 −O15) R(O5 − O10) R(O5 −O14) R(O10 − O14) 2.932 3.075 3.162 2.969 2.842 2.824 2.897 2.953 2.869 4.120 H-bond length, Å r(O1 −H9) r(O1 −H18) r(O4 −H3) r(O4 −H11) r(O5 −H13) r(O10 −H8) r(O14 −H6) r(O15 −H7) 2.337 2.171 1.997 1.960 1.949 2.070 1.931 2.074 H-bond angle δH , deg O1 −H3 −O4 O1 −H9 − O5 O1 −H18 − O15 O4 −H6 −O14 O4 −H7 −O15 O5 −H8 − O10 O10 −H11 − O4 O14 −H13 − O5 167.99 134.63 141.29 155.45 144.44 154.30 153.55 161.84 O-O-O bond angle, deg O4 −O1 −O15 O10 − O1 − O15 O5 −O1 −O15 O14 − O4 −O15 O1 − O4 − O15 O10 − O5 − O14 O1 −O5 −O10 O10 − O5 − O14 O4 −O14 − O5 O1 −O15 −O4 62.75 43.89 87.60 73.37 59.57 86.70 93.11 86.70 63.41 57.67 Formation of a “dangling” bond promotes the appearance of nonlinear H-bonds between O1, on the one hand, and O5 and O15, on the other one, with r (O1−H9) = 2.34 Å, ∠O1−H9−O5 = 134.63o, r (O1−H18) = 2.17 Å, and ∠O1−H18−O15 = 141.29o. This is clearly seen in figure 6. It follows from table 4 that the “dangling” bond causes distortion of the rest of an H-bonded pattern of Defect I water hexamer that remains four-folded. For instance, R(O5 and O10) = 2.95 Å, r (O10 − H8) = 2.07 Å, ∠O5 − H8 −O10 = 154.30o, and R(O4 and O15) = 2.90 Å, r (O4 − H18) = 2.07 Å, ∠O4−H7−O15 = 144.44o. It should be noted that some O-O-O bond angles are clustered around 60o. These are, in particular, the following: ∠O4−O1−O15 = 62.75o,∠O1−O4−O15 = 59.57o, and ∠O4−O14−O5 = 63.41o. One also sees from table 4 that the lone-pair distribution of O1 loses its tetrahedrality characterized by ∠O5 − O1 − O15 = 87.60o due to the five-fold coordination. We would like to conclude this paragraph by noting that due to the “dangling” bond, the separation between the oxygen atoms O1 and O14 considered as second-neighbours is 4.22 Å. 223 E.Kryachko The assignment of harmonic vibrations of the Defect I “patch” is shown in table 5. Harmonic frequencies computed for inter- and intra-molecular modes of the Defect I structure are listed in the second column of this table. Its third and fourth columns report the corresponding theoretical IR intensity and Raman activity. First of all, before studying table 5, it should be mentioned that the theoretical spectra of the chair and boat water hexamers [22(d)] do not have vibrations in the range 4000-4200 cm−1. It implies that the hydrogen atoms in these structures are solely of two sorts: the hydrogens involved in forming H-bond, and “free” hydrogens participating in unbonded OH groups. On the contrary, the prism hexamers and Defect I do have H-stretching vibrations in this region. The most intensive IR band of Defect I is the librational one centered at 663.5 (IR intensity 564 km/mol, Raman activity 0.5 Å 4 /amu) comparing with the most IR intensive ones for the prism, chair, and boat hexamers that fall into the region of H-stretching vibrations. The band at 663.5 cm−1 is assigned to the composed libration of the O4 − H6, O5−H8, and O5−H9 bonds. Other IR intensive bands of Defect I are the following: band 467.2 cm−1 (343.3, 2.2) associated with the composed librational vibration of O1 − H3, O5 − H8, O5 − H9, and O15 − H18 bonds and others belong to the H-stretching region. These are band-centered at 4001.96 cm−1 (511.8, 22.0) with the O4−H6, O14−H13 stretching vibration character and the band at 4111.4 cm−1 (369.8, 21.3) assigned merely to O4−H7 stretching. Regarding the O-H stretching vibrational modes with the frequencies νtheor 1 = 4141.9 cm−1 (17.7, 65.5; symmetric) and νtheor 3 = 4237.6 cm−1 (57.4, 32.4; asymmetric) computed for the Figure 5. Phase-like diagram of lower-lying water hexamer clusters. 224 H-bonded pattern in water Table 5. Most intensive bands of Defect I water hexamer: vibrational frequen- cies ν, cm−1, IR intensity (km/mol), Raman activity (Å 4 /amu), force constant (mdyne/Å), and reduced mass (amu). No. ν IR Raman Force Reduced cm−1 intensity activity constant mass Assignment 1 96.52 131.68 2.08 0.0068 1.2344 O15 −H17 translation 2 174.06 124.45 2.47 0.0251 1.4086 O14 −H16 translation 3 220.78 108.64 1.34 0.0329 1.1462 O10 −H12 translation 4 352.05 122.04 0.71 0.0761 1.0421 O1 −H2, O5 −H8 −H9, O10 −H12 libration 5 426.87 191.66 1.32 0.1163 1.0830 O1 −H2, O15 −H17 −H18 libration 6 450.08 103.90 0.47 0.1293 1.0830 O1 −H2 −H3, O5 −H9, O10 −H11 −H12, O15 −H18 libration 7 467.16 343.34 2.20 0.1357 1.0554 O1 −H3, O5 −H8 −H9, O15 −H18 libration 8 591.19 203.24 4.31 0.2162 1.0497 O1 −H3, O4 −H7 libration 9 663.51 564.12 0.47 0.2763 1.0651 O4 −H6, O5 −H8 −H9 libration 10 750.18 102.25 0.41 0.3462 1.0442 O4 −H7, O5 −H8, O10 −H11 libration 11 792.43 292.40 0.88 0.3940 1.0650 O4 −H6 −H7, O5 −H9, O14 −H13 libration 12 913.97 112.38 0.88 0.5155 1.0475 O1 −H3, O4 −H6 −H7 libration 13 1756.83 203.58 3.66 1.9633 1.0797 H2 −O1 −H3,H11 −O10 −H12, H17 −O15 −H18 scissor 14 1822.21 107.83 3.87 2.1091 1.0781 H2 −O1 −H3,H8 −O5 −H9 scissor 15 3967.47 79.66 162.57 9.7897 1.0566 O4 −H6, O14 −H13 stretch 16 4001.96 511.76 22.01 9.9429 1.0537 O4 −H6, O4 −H7, O14 −H13 stretch 17 4016.36 167.01 25.95 10.0461 1.0570 O10 −H11 stretch 225 E.Kryachko Table 5. Continued from the previous page. 18 4047.50 227.75 89.05 10.1860 1.0553 O1 −H3, O4 −H6, O10 −H11 stretch 19 4068.80 143.11 36.64 10.2569 1.0516 O5 −H8, O5 −H9 stretch 20 4085.39 211.26 48.75 10.3511 1.0526 O15 −H17, O15 −H18 stretch 21 4111.36 369.79 21.25 10.7142 1.0758 O4 −H7 stretch 22 4159.71 177.57 29.98 10.9754 1.0766 O5 −H9 stretch 23 4183.20 111.31 32.23 11.0601 1.0727 O1 −H2 stretch 24 4210.83 124.08 51.80 11.2187 1.0739 O10 −H12 stretch 25 4212.04 81.71 59.75 11.2060 1.0721 O14 −H16 stretch 26 4220.61 113.57 47.29 11.3126 1.0779 O15 −H17 stretch HF/6-311 G water monomer (for comments see [24]), one may divide the whole stretching region of Defect I into three groupings. The first one spans the range Figure 6. Pentacoordinated water hexamer cluster Defect I. 226 H-bonded pattern in water Figure 7. Phase-like diagram of Defect I and some lower-lying water hexamers. from 3967 to 4069 cm−1 and possesses the dominant character of stretching the H-bonded hydrogen atoms. The band 3967.5 cm−1 (79.7; 162.6) describes the com- posed symmetric stretching vibration of O4 −H6, O14 −H13 bonds and is, in fact, the strongest Raman active one. The aforementioned band 4002.0 cm−1 corre- sponds to asymmetric stretching vibration. Comparing with the stretching bands of a single water molecule, the latter are both redshifted, considerably enhanced, and become closer to each other by 60 cm−1. The second grouping of bands lies in the interval from 4085 to 4180 cm−1 and is primarily contributed by nonlinear or bent H-bonded hydrogens. The last one consists of bands from 4183 to 4221 cm−1. They are assigned to the stretching vibrations of “free” hydrogen atoms. These bands are slightly blueshifted and mainly twice enhanced comparing with the stretching mode ν3 of a water monomer. Regarding the intramolecular or scissor modes of our defect hexamer structure, it has to be mentioned first that the har- monic mode ν (monomer) 2 of a water monomer calculated at the HF/6- 311G level is 1750.3 cm−1. Its IR absorption and Raman activity constitute 79.2 km/mol and 6.4 Å 4 /amu, respectively. Two intra-molecular modes of Defect I are redshifted com- paring with ν (monomer) 2 and more pronounced. Their normal vibration assignment is also shown. For instance, the first intramolecular mode of Defect I with frequency 1756.8 cm−1 describes the composed scissor vibrations of water molecules H2H3O1 and H11H12O10 bonded to each other via a “dangling” bond. 227 E.Kryachko 3.3. Five-fold coordinated water octamers Figure 8 displays a pentacoordinated water octamer. Its distinction is that it has formed 3 H-bonds and looks identical to the structure R6c found in [23(a)], although a slight deviation in their properies is revealed [25]. Energy, enthalpy, Figure 8. Pentacoordinated water octamer cluster Defect II. zero-point vibrational energy and entropy calculated at the HF/6-311G level of the computational theory of the octamer are reported in table 6. Before going further, one more word about the studied water clusters should be added. As to the cages and Defect I, they serve as examples in lower-lying clusters. Water molecule may form a first coordination shell with five water molecules in such Table 6. Energy E, enthalpy H, entropy S, zero-point vibrational energy ZPVE, and free energy of defect water octamer. Cubic octamer Ca is chosen as the reference one. For other notations see caption of table 3. Defect E (hartree) H (hartree) S (cal/mol K) ZPVE (kcal/mol) ∆caE ∆caH ∆caS ∆caG Defect II -608.4765824 - 608.2403860 156.181 134.42923 12.291 10.802 17.905 5.47 228 H-bonded pattern in water a way that the approaching fifth water molecule is bonded to the central water molecule by a “dangling” bond. Defect II portrayed in figure 8 is referred to an absolutely different type of pentacoordinated water clusters. It is seen there that the oxygen atom O4 has three elongated and rather weak H-bonds with the oxygen atoms O1,O5, and O11 separated from it by circa 3 Å (table 7). Despite the fact that the angle ∠H6 − O5 − H7 = 107.13o remains unchanged comparing with the water monomer. However, the lone-pair angular distribution suffers drastic changes, which is revealed in the calculated values of the corresponding angles: ∠H3−O4−H9 = 69.19o, ∠H9−O4−H14 = 70.54o, and ∠H3−O4−O14 = 115.65o. Its arrangement relative to the H6 − O4 − H7 plane is determined by the anlges ∠H3−O4−H7 = 88.91o, ∠H6−O4−H14 = 79.56o, and ∠H6−O4−H9 = 125.61o. Table 7. Internal coordinates of defect water octamer. Interoxygen distance, Å R(O1 − O4) R(O1 − O10) R(O1 − O18) R(O1 − O21) R(O4 − O5) R(O4 − O11) R(O4 − O19) R(O4 − O21) R(O5 − O11) R(O5 − O18) R(O10 − O19) R(O11 −O19) 3.051 2.883 2.842 2.930 2.989 2.972 2.802 2.890 2.990 2.862 2.875 3.061 H-bond length, Å r(O1 −H13) r(O1 −H24) r(O4 −H3) r(O4 −H9) r(O4 −H14) r(O5 −H16) r(O10 −H17) r(O11 −H8) r(O11 −H22) r(O18 −H2) r(O19 −H6) r(O21 −H7) 1.965 2.086 2.164 2.120 2.207 1.948 1.955 2.238 2.370 1.941 1.892 2.051 H-bond angle δH , deg O1 −H3 − O4 O4 −H9 −O5 O5 −H8 − O11 O1 −H13 −O10 O4 −H14 − O10 O1 −H2 −O18 O5 −H16 −O18 O4 −H6 − O19 O10 −H17 − O19 O4 −H7 −O21 O11 −H22 −O19 O1 −H24 −O21 155.49 152.07 136.03 161.78 137.24 157.20 160.34 158.87 162.59 146.57 129.83 147.54 O-O-O bond angle, deg O1 − O18 − O5 O4 −O5 − O18 O10 −O1 − O18 O11 −O5 −O18 O5 − O4 − O19 O1 −O10 −O19 O4 − O19 − O10 O11 −O4 −O19 O5 −O4 − O21 O10 −O1 − O21 O11 −O4 −O21 O18 −O1 − O21 O19 −O4 −O21 77.73 104.14 122.58 138.43 103.22 84.15 99.21 63.95 113.46 112.69 169.30 122.74 111.81 229 E.Kryachko Being a Ca structure, Defect II possesses 12 H-bonds albeit 7 of which may be considered relatively weak because their lengths exceed 2 Å. What is most impressing in the H-bonded pattern of Defect II is that water molecule H14H15O11 is weakly bonded to its nearest neighbours. This is clearly seen in the fact that the inwarding H-bonds, such as O11 −H8 and O11 −H22 have the lengths 2.24 and 2.37 Å, respectively. The outwarding H-bond O4 − H14 is also strongly elongated to 2.21 Å. The angles ∠O5 − H8 − O11 = 136.03o, ∠O4 − H14 − O11 = 137.24o, and ∠O11 − H22 − O19 = 129.83o keep, therefore, a very low profile. The angle ∠O11 − O4 − O19 = 63.95o shows, in particular, a loss of tetrahedrality in this area of the pentacoordinated “patch”. It seems worth discussing right now the distribution of oxygen-oxygen separations beyond the first coordination shell. They are quite well clustered around 4.65 Å and take the values: R(O4−O10) = 4.32 Å, R(O5−O19) = 4.54 Å, R(O4−O18) = 4.62 Å, R(O19−O21) = 4.71 Å, R(O10−O21) = 4.84 Å, R(O5 −O21) = 4.92 Å, and R(O1 −O11) = 4.94 Å. The oxygen atom is placed in a particularly privileged position because it stays very close to O5 and O19, namely, by 3.58 and 3.86 Å, respectively. The dipole moment of the Defect II “patch” is very low and equal to 0.87 D. Energetically, it is situated 12.30 kcal/mol above the Ca structure. At room temperature, their difference in free energy is just 5.47 kcal/mol. Regarding the spectrum of this “patch”, it is worth noting that the spectrum of Defect II possesses rather strong IR intensive bands. Namely, its most IR intensive and Raman active bands fall only in the region of stretching vibrations. This is the band νH2O 28 = 4113.9 cm−1 with IR intensity 453.6 km/mol attributed chiefly to the stretching of O1−H3 bond. The second one, νH2O 20 = 3978 cm−1, with Raman activity 142.27 Å 4 /amu and IR intensity 429.7 km/mol, is assigned to the composed symmetric stretching vibration of O1−H2 and O18−H16 bonds. From table 8 one can easily recognize the grouping of bands in the region 4091- 4168 cm−1 corresponding to the stretching vibrations of three O-H bonds which establish the five-fold coordination of the H6H7O4 water molecule and of the other two directed to H14H15O11. Comparing with the stretching modes of a water monomer, these bands are slightly redshifted, which indicates their weakness. This grouping of bands borders another one falling to 4205-4215 cm−1 and describing the stretching vibrations of unbonded or “free” O-H groups. Table 8 also lists the most intensive vibrations of fully deuterated Defect II calculated at the HF/6-311G level of computation. Its zero-point vibrational energy is 87.961 kcal/mol and entropy – 178.571 cal/mol·K. 230 H-bonded pattern in water Table 8. Most intensive bands of defect water octamer. For notations see caption of table 5. Values for fully deuterated Defect II are given in parenthesis. No. ν IR Raman Force Reduced cm−1 intensity activity constant mass Assignment 1 167.50 ( 139.68) 134.88 ( 66.82) 2.49 ( 0.13) 0.0217 1.3153 (2.4140) O21 −H23 translation 2 186.88 ( 145.25) 188.76 ( 47.14) 0.82 ( 0.50) 0.0286 1.3893 (2.8709) O18 −H20, O11 −H15 translation 3 454.54 ( 329.81) 247.97 (119.39) 1.45 ( 0.69) 0.1290 1.0594 (2.2083) O19 −H22 libration 4 479.03 ( 347.03) 156.33 ( 94.76) 3.36 ( 1.69) 0.1426 1.0549 (2.1971) O1 −H3, O19 −H22 libration 5 492.42 ( 357.49) 217.73 (108.48) 0.69 ( 0.39) 0.1513 1.0592 (2.2157) O21 −H24, O19 −H22 libration 6 533.88 ( 387.10) 161.88 (114.83) 0.31 ( 0.43) 0.1781 1.0605 (2.1872) O5 −H8, O11 −H14, O5 −H9 libration 7 546.94 ( 395.37) 170.08 ( 50.61) 3.40 ( 1.52) 0.1849 1.0492 (2.1872) O4 −H7, O21 −H24 libration 8 601.84 ( 436.54) 202.77 (116.59) 1.35 ( 0.65) 0.2256 1.0571 (2.2097) O5 −H9, O18 −H16 libration 9 654.55 ( 474.21) 134.04 ( 68.85) 0.87 ( 0.46) 0.2673 1.0591 (2.1829) O5 −H9, O19 −H17 libration 10 668.36 ( 486.34) 332.08 (191.89) 1.49 ( 0.75) 0.2802 1.0646 (2.2347) O18 −H16, O19 −H22 libration 11 693.47 ( 501.19) 250.72 (127.85) 0.93 ( 0.42) 0.2975 1.0501 (2.1782) O19 −H17, O4 −H6 libration 12 757.18 ( 548.76) 149.08 ( 73.69) 1.94 ( 0.96) 0.3574 1.0582 (2.1919) O4 −H6, O4 −H7 libration 13 798.45 ( 570.25) 381.95 (234.52) 0.13 ( 0.10) 0.3851 1.0488 (2.1742) O1 −H2, O4 −H6, O10 −H13 libration 14 844.58 ( 615.41) 177.47 (112.25) 0.40 ( 0.13) 0.4484 1.0670 (2.2478) O4 −H7, O1 −H2, O1 −H3, O21 −H24 −H18 libration 15 908.64 ( 654.81) 128.66 ( 77.92) 0.64 ( 0.30) 0.5079 1.0441 (2.1547) O4 −H6, O19 −H17 libration 16 1754.08 (1282.64) 132.62 ( 67.93) 6.41 ( 3.41) 1.9589 1.0806 (2.2559) O11 −H14 −H15 scissor 17 1774.04 (1295.89) 116.20 ( 54.80) 5.05 ( 2.71) 1.9991 1.0781 (2.2477) O21 −H23 −H24 scissor 18 1843.89 (1344.61) 188.18 (100.98) 1.93 ( 1.11) 2.1503 1.0735 (2.2360) O1 −H2 −H3, O5 −H8 −H9 scissor 19 3927.49 (2842.71) 257.45 (111.19) 122.55 (74.23) 9.6031 1.0566 (2.1797) O4 −H6 stretch 20 3978.39 (2880.00) 429.69 (252.81) 142.27 (77.02) 9.8550 1.0568 (2.1812) O1 −H2, O18 −H16 stretch 21 4003.26 (2897.42) 438.73 (199.00) 20.23 (11.63) 9.9717 1.0561 (2.1801) O10 −H13, O18 −H16, O19 −H17 stretch 22 4015.09 (2903.59) 114.94 ( 48.68) 36.68 (19.66) 10.0101 1.0539 (2.1756) O1 −H2, O18 −H16, O19 −H17 stretch 23 4034.36 (2917.34) 345.32 (143.67) 38.65 (19.40) 10.1096 1.0542 (2.1738) O10 −H13, O19 −H17 stretch 231 E.Kryachko Table 8. Continuation from previous page. 24 4058.52 (2937.93) 61.32 ( 36.05) 96.31 (40.38) 10.2950 1.0608 (2.1743) O21 −H24, O4 −H7 stretch 25 4085.48 (2954.27) 426.88 ( 35.75) 35.90 ( 8.48) 10.5052 1.0682 (2.1656) O4 −H7, O21 −H24 stretch 26 4091.03 (2955.79) 41.88 ( 53.56) 21.64 (45.36) 10.3624 1.0509 (2.1632) O11 −H14, O5 −H9 stretch 27 4094.86 (2985.00) 48.83 (233.61) 67.66 (21.11) 10.3814 1.0508 (2.2772 O11 −H14, O5 −H9 stretch 28 4113.90 (3005.70) 453.56 (233.57) 37.92 (19.01) 10.6611 1.0692 (2.2742) O1 −H3 stretch 29 4161.77 (3046.49) 96.49 ( 53.20) 26.93 (15.75) 11.0057 1.0785 (2.2812) O5 −H8, O5 −H9, O19 −H22 stretch 30 4168.43 (3049.50) 256.07 (169.94) 23.34 (10.82) 11.0101 1.0755 (2.2797) O19 −H22 stretch 31 4205.02 (3072.61) 104.06 ( 88.09) 49.36 (23.54) 11.1776 1.0729 (2.2705) O10 −H12 stretch 32 4208.77 (3074.79) 100.10 ( 85.73) 52.66 (24.84) 11.1935 1.0725 (2.2691) O18 −H20 stretch 33 4213.47 (3082.78) 116.91 ( 76.78) 43.67 (20.41) 11.2567 1.0762 (2.2801) O21 −H23, O21 −H24 stretch 34 4214.21 (3087.05) 110.76 ( 88.40) 47.58 (19.72) 11.2964 1.0796 (2.2878) O11 −H15, O11 −H14 stretch Hence, the ratio ZPVEH2O/ZPVED2O = 1.53 [27]. For the deuterium Defect II, the band νD2O 20 = 2880.0 cm−1 becomes most IR and most Raman active, si- multaneously being contrary to the protium Defect II. We should also mention the behaviour of ν27 assigned to the coupled asymmetric stretching vibrations of O11−H14 and O5−H9 bonds under isotopic substitution. For the protium Defect II this band possesses IR intensity equal to 48.8 km/mol while it becomes five times more pronounced than its deuterated isotopomer. Analyzing the second and sixth columns of table 8 one finds that the calculated harmonic frequencies and reduced masses of the protium and deuterium Defect II do not show the known isotopic re- lationships connecting the ratio of frequencies of translational and bending modes νH2O/νD2O with the square root of the ratio of the corresponding moments of iner- tia or masses, respectively. This is perhaps the common situation with computing harmonic vibrational frequencies [27], although, in our case with larger water clus- ters this deviation appears more pronounced than with water dimer. From table 8 it follows that, first, for νH2O 1 = 167.5 cm−1, the ratio νH2O 1 /νD2O 1 = 1.20 whereas, according to this relationship, it should be √ 20/18 = 1.05. Second, one directly obtains further that νH2O 2 /νD2O 2 = 1.29 and √ µH2O 2 /µD2O 2 = 1.44. Third, for the rest of vibrations listed in table 8, the isotopic frequency ratios behave rather regularly around 1.37-1.38 with the exception for the 13th and 15th modes when they are 1.40 and 1.39, respectively. The isotopic reduced mass ratio does not reveal such a simple regularity, although it is largerly clustered near 1.44-1.45. Altogether, this isotopic analysis emphasizes a very anharmonic picture of the total potential energy surface of water clusters. 232 H-bonded pattern in water 4. Summary The fourteen different structures in the interval of 1.7 kcal/mol represent an unprecedentedly wide range of conformational excursions for water hexamer clus- ter. They illustrate how rich the picture of the total potential energy surface of liquid water might be. The present work also provides the first ab initio demonstration of the penta- coordinated water hexamer structure. This structure is not, in fact, a defective one in the common sense of defects in the H-bonded pattern that violates the Bernal- Fowler-Pauling rules nor those structures that involve a bifurcated H-bond. It ac- tually represents a novel structure of H-bonds where the approaching fifth water molecule forms a “dangling” bond with the central water molecule generating in such a way a pentacoordinated “patch” of higher density. This “patch”, being incorporated into an H-bonded pattern of liquid water, on the one hand, partly contributes to nontetrahedral configurations and to the known blurred maximum of the O-O-O bond angle distribution function in the interval 60o − 80o, in par- ticular. On the other hand, it contributes to tetrahedral configurations as well. Notwithstanding that such hexamer and octamer five-fold coordinated “patches” occupy lower- lying local energetic minima on the total potential energy surfaces, they become energetically accessible at room temperature. Acknowledgements The author thanks Joel Bowman, Keiji Morokuma, and Jamal Musaev for many helpful discussions and hospitality. GAUSSIAN computing resources were provided by the Cherry L. Emerson Center for Scientific Computation at Emory Univer- sity in Atlanta. The author also acknowledges stimulating discussions with En- rico Clementi, Gina Corongiu, Jens Peder Dahl, Ludwig Hofacker, Yves Marechal, Francesco Scortino, Georg Zundel, and Sotiris Xantheas. The author is grateful to Ihor Stasyuk for the inspiring scientific atmosphere and many useful discussions during his visits to Lviv. References 1. (a)Eisenberg D., KauzmannW. The structure and properties of water. Oxford, Claren- don, 1969; (b)F. Franks (Ed.) Water: a comprehensive treatise. New-York, Plenum, 1973. Vols.1-7, and References therein. 2. For Stasyuk’s contribution to the hydrogen bond theory consult with: (a)Stasyuk I. V., Levitsky R. R. On elementary excitations in ferroelectrics with hydrogen bond. //Ukr. Fiz. Zh. 1969, vol 14, No 7, p.1097-1105; (b)Stasyuk I. V., Levitsky R. R. Coupled vibrations of a proton-ion system in ferroelectrics with hydrogen bonds of KH2PO4 type. //Ibid., 1970, vol 15, No 3, p.460-469; (c)Stasyuk I. V., Levitsky R. R. The role of proton-phonon interaction in the phase transition of ferroelectrics with hydrogen bonds. //Phys. Stat.Sol.(b), 1970, vol 39, No 1, p.K35-K38; (d)Stasyuk I. V. Proton-phonon interaction in ferroelectrics with hydrogen bonds (strong coupling 233 E.Kryachko approximation). //Teor. Mat. Fiz., 1971, vol 9, No 3, p.431-439; (e)Stasyuk I. V., Levitsky R. R. Dynamical theory of NH2H2PO4-type antiferroelectrics with hydrogen bonds. //Izv. Acad. Nauk SSSR, Ser. Fiz., 1971, vol 35, No 9, p.1775-1778; (f)Stasyuk I. V., Ivankiv A. L. Reduced set model for explanation of the molecular complexes with chains of hydrogen bonds. //Ukr. Fiz. Zh., 1991, vol 36, No 6, p.817-823; g)Stasyuk I. V., Stetsiv R. Y. Electronic states and optical effects in KH2PO4 type crystals with hydrogen bonds. //Izv. Acad. Nauk SSSR, Ser. Fiz., 1991, vol 55, No 3, p.522-525; (h)Stasyuk I. V., Ivankiv A. L. Thermodynamics of the molecular complexes with chains of hydrogen bonds. //Mod. Phys. Lett. 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X-ray diffraction study of liquid water 234 H-bonded pattern in water in the temperature range 4-200o C. //Disc. Faraday Soc., 1967, vol. 43, p. 97-107; (b)Narten A. H., Levy H. A. Observed diffraction pattern and proposed models of liquid water. //Science, 1969, vol. 165, No 3892, p. 447-454; (c)Soper A. K., Phillips M. G. A new determination of the structure of water at 25o C. //Chem. Phys., 1986, vol. 107, No 1, p. 47-60; (d)Narten A. H., Levy H. A. Atom-pair distribution function of liquid water at 25o C from neutron diffraction. //Science, 1982, vol. 217, No 4564, p. 1033-1035; (e)Morgan J., Warren B. E. X-ray analysis of the structure of water. //J. Chem. Phys., 1938, vol. 6, No 9, p. 666-673. See also: Cho C. H., Singh S., Robinson G. W. An explanation of the density maximum in water. //Phys. Rev. Lett., 1996, vol. 76, No 10, p. 1651-1654. 9. Giguére P. A. Bifurcated hydrogen bonds in water. //J. Raman Spectrosc., 1984, vol. 15, No 5, p. 354-359; The bifurcated hydrogen-bond model of water and amorphous ice. //J. Chem. Phys., 1987, vol. 87, No 8, p. 4835-4839. 10. (a)Walrafen G. E., Hokmabadi M. S., Yang W.-H., Chu Y., Monosmith W. B. Collision-induced Raman scattering from water and aqueous solutions. //J. Phys. Chem., 1989, vol. 93, No 8, p. 2909-2917; (b)Walrafen G. E., Hokmabadi M. S., Yang W.-H. Raman investigation of the temperature dependence of the bending ν2 and com- bination ν2 + νL bends from liquid water. //Ibid., 1988, vol. 92, No 9, p. 2433-2438; (c)Walrafen G. E. Raman spectrum of water: transverse and longitudinal acoustic modes below ≈ 300cm−1 and optic modes above ≈ 300cm−1. //Ibid., 1990, vol. 94, No 6, p. 2237-2239. 11. (a)Bjerrum N. Structure and properties of ice. //K. Danske Vid. Selsk. Mat.-Fys. 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Application of ab initio molecular orbital calculations to the structural moieties of carbohydrates. 5. The geometry of the hydrogen bonds. //J. Am. Chem. Soc., 1979, vol. 101, No 18, p. 1997-2002; (g)Jeffrey J. A., Saenger W. Hydrogen bonding in biological structures. Berlin, Springer, 1991. p. 20ff; (h)Head- Gordon M., Head-Gordon T. Analytic MP2 frequencies without fifth-order storage. Theory and application to bifurcated hydrogen bonds in the water hexamer. //Chem. Phys. Lett., 1994, vol. 220, Nos 1,2, p. 122-128. 13. (a)Kryachko E. S. The cooperative model for orientational defects in ice: continuum approximation. //Chem. Phys. Lett., 1987, vol. 141, No 2, p. 346-349; Collective model of orientational defect in ice within the continuum approach. //Solid State Phys. (USSR), 1987, vol. 29, No 2, p. 345-350; (b)Yanovitskii O. E., Kryachko E. S. Model for orientational defects in quasi-one-dimensional ice crystals. //Phys. Stat. Sol. (b), 1988, vol. 147, No 1, p. 69-81; (c)Kryachko E. S. Recent developments in solitonic model of proton transfer in quasi-one-dimensional infinite hydrogen-bonded systems. 235 E.Kryachko In A. Müller, H. Ratajczak, W. Junge, and E. Diemann (Eds.), Electron and proton transfer in chemistry and biology. Amsterdam, Elsevier, 1992. pp. 363-385. 14. Devlin J. P. Vibrational spectra and point defect activities of icy solids and gas phase clusters. //Int. Rev. Phys. Chem., 1990, vol. 9, No 1, p. 29-65; Fisher M., Devlin J. P. Defect activity in amorphous ice from isotopic exchange data: insight into the glass transition. //J. Phys. Chem., 1995, vol. 99, No 29, p. 11584-11590. 15. Prielmeier F. X., Lang E. W., Lüdemann H. D., Speedy R. J. Diffusion in supercooled water to 300 MPa. //Phys. Rev. Lett., 1987, vol. 59, No 10, p. 1128-1131. 16. Geiger A., Mausbach P., Schnitker J. Computer simulation study of the hydrogen- bond network in metastable water. In G. W. Neilson and J. E. Enderby (Eds.), Water and aqueous solutions. Bristol, Adam Hilger, 1986; pp. 15-30. 17. (a)Sciortino F., Geiger A., Stanley H. E. Network defects and molecular mobility in liquid water. //J. Chem. Phys., 1992, vol. 96, No 5, p. 3857-3865; (b)Sciortino F., Geiger A., Stanley H. E. Isochoric differential scattering functions in liquid water: the fifth neighbor as a network defect. //Phys. Rev. Lett., 1990, vol. 65, No 27, p. 3452-3455. 18. Head-Gordon T., Stillinger F. H. An orientational perturbation theory for pure liquid water. //J. Chem. Phys., 1993, vol. 98, No 4, p. 3313-3327. See Figs. 5 and 20 in particular. 19. (a)Jedlovszky R., Bakó I., Pálinkás G., Radnai T., Soper A. K. Investigation of the uniqueness of the reverse Monte Carlo method: studies on liquid water. //J. Chem. Phys., 1996, vol. 105, No 1, p. 245-254; (b)Bruni F., Ricci M. A., Soper A. K. Unpre- dicted density dependence of hydrogen bonding in water found by neutron diffraction. //Phys. Rev., 1996, vol. B 54, No 17, p. 11876-11879. 20. GAUSSIAN 94, Revision B.3, M. J. Frisch, G. W. Trucks, H. B. Schlegel, P. M. W. Gill, B. G. Johnson, M. A. Robb, J. R. Cheeseman, T. Keith, G. A. Petersson, J. A. Montgomery, K. Raghavachari, M. A. Al-Laham, V. G. Zakrzewski, J. V. Ortiz, J. B. Foresman, C. Y. Peng, P. Y. Ayala, W. Chen, M. W. Wong, J. L. Andres, E. S. Replogle, R. Gomperts, R. L. Martin, D. J. Fox, J. S. Binkley, D. J. Defrees, J. Baker, J. P. Stewart, M. Head-Gordon, C. Gonzalez, and J. A. Pople, Gaussian, Inc., Pittsburgh, PA, 1995. 21. (a)Krishnan R., Binkley J. S., Seeger R., Pople J. A. Self-consistent molecular orbital methods. XX. A basis set for correlated wave functions. //J. Chem. Phys., 1980, vol. 72, No 1, p. 650-654; (b)Frisch M. J., Pople J. A., Binkley J. S. Self-consistent molecu- lar orbital methods 25. Supplementary functions for Gaussian basis sets. //Ibid., 1984, vol. 80, No 7, p. 3265-3269. 22. (a)Xantheas S. S., Dunning T. T., Jr. Ab initio studies of cyclic water clusters (H2 O)n, n=1-6. I. Optimal structures and vibrational spectra. //J. Chem. Phys., 1993, vol. 99, No 11, p. 8774-8792; (b)Xantheas S. S. Ab initio studies of cyclic water clusters (H2 O)n, n=1-6. II. Analysis of many-body interactions. //Ibid., 1994, vol. 100, No 10, p. 7523-7534; (c)Tsai C. J., Jordan K. D. Theoretical study of the (H2 O)6 cluster. //Chem. Phys. Lett., 1993, vol. 213, Nos 1, 2, p. 181-188; (d)Krishnan P. N., Jensen J. O., Burke L. A. Theoretical studies of water clusters. II. Hexamer. //Ibid., 1994, vol. 217, No 3, p. 311-318; (e)Estrin D. A., Paglieri L., Corongiu G., Clementi E. Small clusters of water molecules using density functional theory. //J. Phys. Chem., 1996, vol. 100, No 21, p. 8701-8711. 23. (a)Jensen J. O., Krishnan P. N., Burke L. A. Theoretical study of water clusters: 236 H-bonded pattern in water octamer. //Chem. Phys. Lett., 1995, vol. 246, Nos 1, 2, p. 13-19; (b)Kim J., Mhin B. J., Lee S. J., Kim K. S. Entropy-driven structure of the water octamer. //Ibid., 1994, vol. 219, Nos 3, 4, p. 243-246; (c)Knochenmuss, R., Leutwyler S. Structure and vibrational spectra of water clusters in the self-consistent-field approximation. //J. Chem. Phys., 1992, vol. 96, No 7, p. 5233-5244; (d)Kim K., Jordan K. D., Zwier T. S. Low-energy structures and vibrational frequencies of the water hexamer: comparison with benzene-(H2 O)6. //J. Am. Chem. Soc., 1994, vol. 116, No 25, p. 11568; (e)Liti K., Brown M. G., Carter C., Saykally R. J., Gregory J. K., Clary D. C. Characterization of a cage form of the water hexamer. //Nature, 1996, vol. 381, No 6582, p. 501- 503; (f)Gregory J. K., Clary D. C. Structure of water clusters. The contribution of many-body forces, monomer relaxation, and vibrational zero-point energy. //J. Phys. Chem., 1996, vol. 100, No 46, p. 18014-18022; (g)Liu, K., Brown M. G., Saykally R. J. Terahertz laser vibration-rotation tunneling spectroscopy and dipole moment of a cage form of the water hexamer. //Ibid., 1997, vol. A 101, No 27, p. 8995-9010. 24. (a)Mills I. M. Harmonic and anharmonic force field calculations. In R. N. Dixon (Ed.), Theoretical chemistry. London, Chemical Society, 1974. Vol. 1-Quantum Chem- istry, p. 110-159; (b)Honegger E., Leutwyler S. Intermolecular vibrations of small water clusters. //J. Chem. Phys., 1988, vol. 88, No 4, p. 2582-2595, Table II and Refs. therein. Experimental harmonic frequencies of water molecule are the follow- ing: νexpt1 = 3832 cm−1, νexpt2 = 1649 cm−1, and νexpt3 = 3942 cm−1; (c)νtheor2 = 1750.30 cm−1 at the HF/6-311G level; (d)Scaling factors fi = νtheori /νexpti , i = 1, 2, 3 are equal to 1.0809, 1.0614, 1.0750, respectively. The average scaling fac- tor < f >= (f1 + f2 + f3)/3 = 1.0724; (e)The scaled frequencies νsca i = νtheori / < f > (with the relative error, δ = (νsca i − νexpti )/νexpti ,%) are 3862.17 cm−1(+0.79%), 1632.09cm−1(−1.03%), 3951.43cm−1(+0.21%). 25. The relative properities of Defect II with respect to the R6c structure are: ∆E = 2 ·10−4 kcal/mol, ∆H = - 4 ·10−4 kcal/mol, and ∆S = - 21.74 cal/mol·K. 26. For instance, the frequencies ν and reduced masses µ of water isotopomers H2O and D2O calculated via STO-3G basis set are the following: ν H2O(D2O) 1 = 4142.29 ( 2992.49 ) cm−1, µ H2O(D2O) 1 = 1.0491 ( 2.1747 ) amu; ν H2O(D2O) 2 = 2169.80 ( 1584.65 ) cm−1, µ H2O(D2O) 2 = 1.0785 ( 2.2429 ) amu; and ν H2O(D2O) 3 = 4393.38 ( 3212.88 ) cm−1, µ H2O(D2O) 3 = 1.0774 ( 2.2696 ) amu. Further, ZPVEH2O = 15.30 kcal/mol, ZPVED2O = 9.94 kcal/mol, and their ratio ZPVEH2O/ZPVED2O = 1.54. Also, νH2O 1 /νD2O 1 = 1.38, νH2O 2 /νD2O 2 = νH2O 3 /νD2O 3 = 1.37, and √ µH2O 1 /µD2O 1 = √ µH2O 2 /µD2O 2 = 1.44, √ µH2O 3 /µD2O 3 = 1.45. 27. We analyze the ratio νH2O/νD2O for the harmonic vibrational frequencies of water dimer calculated in [28] at the HF/6-31G* and MP2 levels. Following Tables 1 and 2 of [28], one easily derives that this ratio spans the range 1.16-1.18 for all listed modes. 28. Scheiner S., C̆uma M. Relative stability of hydrogen and deuterium bonds. //J. Am. Chem. Soc., 1996, vol. 118, No 6, p. 1511-1521. 237 E.Kryachko Ab initio гексамери і октамери води: знаряддя до вивчення водневозв’язаних кластерів у рідкій воді Є.Крячко 1,2,3 1 Інститут теоретичної фізики ім. М.М.Боголюбова, 252143 м. Київ-143, вул. Метрологічна, 14б 2 Центр наукових обчислень Черрі Л. Емерсона і Відділення хімії Університету Еморі, Атланта, GA 30322, США 3 Університет Джона Хопкінса, Відділення хімії, Балтімор, MD 21218, США Отримано 18 березня 1998 р. Чотирнадцять різних структур гексамера води, які знайдені методом ab initio в базисі 6-311G** в інтервалі 1.7 kcal/mol вище глобально- го мінімуму, є безпрецедентним прикладом високої конформаційної пластичності рідкої води. Дана робота також вперше демонструє іс- нування пентакоординаційних кластерів води на рівні прецизійних ab initio розрахунків. Ключові слова: рідка вода, H-зв’язаний комплекс, орієнтаційний дефект, водний кластер, “гойдальний” зв’язок, ab initio HF 6-311∗∗ обчислення PACS: 61.20.Ja, 61.25.Em, 61.20.Gy, 62.30.+d, 63.20.Pw, 63.90.+t, 64.70.Ja, 65.50.+m, 64.30.+t 238

Ab initio water hexamers and octamers: tool to study hydrogen-bonded pattern in liquid water

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