Intermolecular interactions in water

Journal of Molecular Structure, 189 (1988) 89-103 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

INTERMOLECULAR

INTERACTIONS

89

IN WATER*

MAREK J. WdJCIK Faculty of Chemistry, Jagiellonian University, 30-060 Cracow, Karasia 3 (Poland) (Received 22 January 1988)

ABSTRACT A review of different intermolecular interactions in water is reported in connection with the vibrational spectra of its hydrogen bonds.

INTRODUCTION

Interactions between water molecules are important for its properties. They determine the topology of water and ices, influence thermodynamic properties of the liquid and solid phases, density, characteristic constants, spectra, etc. Therefore, it is understandable that much work performed in water research in recent years has been connected with either determination or application of the intermolecular potentials of water [l-5]. The function of the potential energy V, for a many-body system, can be represented by a series:

(1) Analytical forms of the contributions V (3) and V (4) for water are not well known, and therefore one usually represents the total potential energy V as a sum of “effective” pair potentials V @), assuming that they include on the average many-body contributions. The intermolecular potentials for water fall into one of the categories: (a) empirical potentials fitted to chosen properties of water, such as the virial coefficients of the steam, static lattice energies or bulk moduli of ices, or (b) ab initio potentials, with or without correlation effects. Vibrational couplings in water are inseparably connected with the intermolecular potentials. The couplings are responsible for detailed structure of spectra of water, relaxation phenomena, proton movement, etc. Recently, considerable interest has been shown in studying cooperative phe*Dedicated to Professor D.J. Millen on the occasion of his retirement.

0022-2860/88/$03.50

0 1988 Elsevier Science Publishers B.V.

90

nomena in water [6-lo], which are probably responsible for its important thermodynamic properties, especially the anomalous properties. Some computer simulations suggested that certain patterns (rings) in water convert collectively during the motions of the molecules [ 61. There is no certainty what rings size is specifically characteristic for water - &membered or 6-membered. The available computational techniques and potentials do not yet allow us to determine exactly which topological structures and changes occur in the different phases of water. A study of the Hamiltonian for liquid water should lead to a determination of the transition energies for the collective modes of motion, which, in turn, should be related to the network geometry. Thermodynamic properties related to those collective motions and their temperature dependence are complex and, at the present state of knowledge, unknown. This article briefly describes the present state of knowledge of the intermolecular interactions, which’are important in dynamic processes in water, and of topology and cooperative phenomena, which are probably of great importance in the description of the anomalous properties of water. I feel that in these two fields sufficient progress has not yet been made. INTERMOLECULAR

POTENTIALS FOR WATER

Recent progress in studying the condensed phases of water is based on computer simulations, in which the knowledge of a proper analytical intermolecular potential is essential. Many such potentials have been proposed in recent years; it seems however that they are still unsatisfactory in the description of a broad range of the properties of water. They were used in molecular dynamics or Monte Carlo simulations of thermodynamic, dielectric, structural or spectroscopic properties of water [l--5], but none of them gave a precise enough description of a range of properties and none described its anomalous behaviour. One can expect that these properties are cooperative and require a manybody treatment. Pair potentials for water were reviewed in refs. l-3. This section recalls some of them, the most commonly used and successful in particular areas of water simulation. The functional forms of the potentials discussed are presented in Table 1. Probably the most commonly used potential in water simulations is the ST2, proposed by Stillinger and Rahman [ 111. This is based on a rigid four-pointcharge model in which the positive charges (protons) and the negative ones (lone electron pairs) are located tetrahedrally, though the distance from the oxygen to the negative charge is smaller than the distance from the oxygen to the positive charge. The potential function consists of a Lennard-Jones central potential acting between the oxygens, plus a modulated Coulomb term for the 16 pairs of point charges. Two further models of the pair potential for water were subsequently devel-

91 TABLE 1 Pair potentials

for water”

(A) Atom-atom portions STZb V,; =0.30589( (2.82/Roe)“VoH = VHH = 0.0

(2.82/R0o)~)

RCF ’

V,, = 26758.2/Rdos85g1-0.25exp[ -4(Roo -3.4)‘] -0.25exp[ Vo, =6.23403/R;$“= -10/{l+exp[40(Boo-1.05)]}-4/{1+ y[5.49305(RoH -2.2) ]} ~H~18/{1+exp[40(R,,-2.05)]}-17exp[-7.62177(R~~-1.45251)2]

PM6d

Voo=24.779exp[ -5.113(Roo-2.45)]+33.445/{l+exp[ll.739Roo]}+ 3.660/{l+exp3.975(Rc,o-3.77)]} V OH=l0exp[-3(Ro,-1.6)2]-198.0722802(Ro~-0.9854)exp[-l6(Ro~o.9854)2] Vnn =o.o

TIPSS”

V,, =6.95x lo5/R& V on =Vnn =o.o

RWKB’

Voo= -F[625.45(g~/Rsc)6+3390(g,/Rsc)8+1.5x21200(~~~/Rsc)‘0l Rsc = Roe x 0.94834673 F= 1 -3.8845Rgiz6exp( - 1.7921Roo) g, = 1 -exp( - 1.8817Roo/n-0.2475n-“2R&o) VOH =2.0736{exp[ -7.3615(Roo -1.637810)] -l}*-2.0736 V nn =631.918exp( -3.28059RHH)

MCYg

Voo =1.169848X 106exp( -5.202625Roo) -423.7835exp( V OH=168452exp( -2.720804Ron) Vnn =415.858exp(-2.456611Rnn)

- 1.5(Roo -4.5)*]

-6OO/R6,,

-2.157112Ron)

(B) Charges interacting through Coulomb’s Law ST2 q= 0.2357 e on each H atom q= -0.2357 e on points 0.8 A from the 0 atom, making tetrahedral O-H bonds RCF

q= 0.32983 e on each H atom q = - 0.65966 e on the 0 atom

PM6

q= 1.0 eon each H atom q=-2.0eontheOatom

TIPS2

q= 0.535 e on each H atom q= - 1.07 e on the 0 atom

RWK2

q= 0.6 e on each H atom q= - 1.2 e on H-O-H bisector 0.26 A from 0 atom

MCY

q= 0.75174 eon each H atom q= - 1.50348 e on H-O-H bisector 0.2581 a from 0 atom

(C) Shielding of Coulomb charges In the ST2 potential the electrostatic energy of an interacting function S (Roe ) which multiplies the electrostatic energy.

angles with the

pair of molecules is shielded by a

92 It is given by: SW,,)

0

=o

S(Roo)=(Roo-

2.0160)2(7.3701-2Roo)/1.377635

5 Roo 52.0160

2.0160 s RoolssT 3.1287 I Roe

S(Roo)=l

In the PM6 potential the electrostatic energy of an interacting pair oxygen-hydrogen by a function S (Roe ) which multiplies the electrostatic energy. It is given by: S(Roo) = 13.59449911exp( -4.050595693Roo)

-2

(D) Additional terms inpotential energy In the PM6 potential an additional nonadditive by:

The induced atomic moments

polarization

,& on oxygens are determined

energy q$ appears, which is given

through linear relations:

The dimensionless factors 1 -K and 1 -L account for the spatial extension electron cloud about each oxygen: l-K(r)=r3/[r3+2.116045(r-0.9854)2exp[

is shielded

of the polarizable

-8(r -0.9854)‘]

+20.335843exp(

-2.892496r)]

l-L(r)=l-[0.15/r3+0.30]+0.209485r3exp[-5.685254(r-0.967813)2]-0.5exp (-3.160792r)

[ 1+3.160792r+4.995304r2-24.597930r3+30.719350r4]

The SM potential h is a sum of the stretching potential ( Vpair) , bending potential ( V, ) , conformational energy ( Vs), nonbonded energy ( VNB), electrostatic energy ( V,,) and nonadditive contribution ( V,,). They are given by: V,,,,(R,,)

= -4.685-15.320(R0;;52;751)

v,=1.934[1-c0s(3eoon)C09(3e~ou)] Vg=1.171sin(6) V,, = -2.796(2.76/R00)6+3.996(2.76/Roo)9 V,, = - 0.6405 + l.9216(R0;;;;751)

VNA= -4.4+26.4(R0;$751) L-O denotes the lone pair of the acceptor. 6 is the conformational about the hydrogen bond. “Energies are given in kcal mol-I, distances modified by the authors. gRef. 20. hRef. 19.

angle describing

rotation

in A. bRef. 11. ‘Ref. 13. dRef. 15. ‘Ref. 16. Ref. 17,

93

oped by Stillinger and collaborators. The simplicity of a three-charge microscopic picture of the water molecule was captured by Lemberg and Stillinger in their central force potential [ 121 (I present here the revised version, RCF [ 131). This potential, which includes two important effects, i.e. internal degrees of freedom and self-association of the water molecules, is represented by the point-charge-point-charge interactions, encompassing Coulomb and nonCoulomb contributions. Another potential, called polarization potential, was proposed by Stillinger and David [ 141 (Table 1 gives the new version, PM6 [ 151); in which water is described as a bunch of bare protons and doublycharged oxygen ions polarizing electron clouds on oxygens. The total potential energy is a sum of two parts: one contains additive radial potentials for each pair of particles (including Coulomb charge-charge interactions), and the second is a many-body polarization term. The potential retains the feature of linear polarization response, but modulates electric fields on account of a spatial delocalization of the electron cloud of oxygen ions. A somewhat similar potential to the ST2, but simpler, was developed by Jorgensen (TIPSB) [ 161. This is a four-site model, with the charge on oxygen moved off the nucleus and towards the hydrogens on the HOH bisector. The potential is composed of one Lennard-Jones term acting between oxygens and of 9 Coulomb terms describing interactions between the charged sites. The RWK2 potential, recently developed by Reimers, Watts and Klein [ 17]* employs a similar four-site model for the water molecule. The interaction terms are, however, more sophisticated, including the dispersion term acting between oxygens, exponential repulsion between hydrogens and a Morse-type term acting between oxygen and hydrogen, plus a Coulomb term for each pair of the charged sites. This potential is frequently combined with an intramolecular potential function [ 181, which contains essentially Morse terms, giving rise to the pair potential RWKB-M for the relaxed water. Sceats and Rice [ 191 proposed a pair potential in water (SR) near the equilibrium configuration. They considered various contributions to this potential: stretching, bending and conformational ones, resulting from the hydrogen bonding, long-range non-hydrogen-bond contributions, including an electrostatic dipole-dipole interaction, a repulsive overlap interaction and a dispersive interaction, and nonadditive contributions. Finally, one should also note the ab initio CI pair potential for water (MCY), published by Matsuoka, Clementi and Yoshimine [20]. This potential was generated by fitting an analytical function, containing Coulomb and exponential terms, to the calculated energies of the dimer. The charge of oxygen is, in this model, shifted from the oxygen position towards the HOH bisector. Some of the pair potentials described in this section were recently tested in order to determine either the topology of different ice lattices [21], the lattice *A modified version supplied by the authors is presented

here.

94

mode spectra of the ices [ 221orthe infrared spectra of the water dimer and of complexes of water with ions [ 231. The results show that though none of the studied potentials gives quantitative agreement with a broad range of the properties of water, nevertheless the ab initio MCY potential gives more successful predictions than any other potential. COUPLING BETWEEN INTRAMOLECULAR

AND INTERMOLECULAR

VIBRATIONS

The idea of a coupling between the high-frequency intramolecular X-H stretching mode (Qi) and the low-frequency intermolecular X- *lY stretching mode (Q3) has been, for some time, considered essential to explain the structure and width of vibrational X-H bands in hydrogen-bonded systems [24301.It is, for the most part, related to the anharmonic force constant kl13. The X-H stretching vibration, with wavenumbers usually between 3000and 3500 cm-‘, in weak or moderately strong hydrogen bonds, is adiabatically coupled to the intermolecular mode X*.*Y, the wavenumber of which usually ranges between 100 and 200 cm-‘. The main reason for the adiabatic separation is a different time-scale of the motions; however, as recently shown by Romanowski and Bowman [31] for the bending and stretching vibrations of the water molecule, the adiabatic separation of vibrations is more general and is not limited only to vibrations of very different frequencies. The adiabatic coupling, considered in hydrogen-bonded systems, gives rise to the Franck-Condon progressions in the X-H vibrational transition, the components of which can be further split because of inter and intramolecular interactions. The adiabatic coupling between the O-H stretching and low-frequency intermolecular motions was never fully employed in explaining the spectroscopic properties of water. Fischer et al. [32,33] developed a quasiparticle model for ice crystals in which a strong interaction between the O-H stretching in ionic defects ( H30+ ) and optical phonons (0-s -0) vibration and wagging vibrations) was considered. An additional assumption was a symmetric double-minimum potential for the proton movement. They calculated a vibrational absorption profile with a phonon fine structure and predicted that the peak positions are almost temperature independent. Deuterium substitution causes large isotopic effect in the spectrum. Unfortunately, this work lacks a comparison with the actual spectra of ices. Rice and his group [ 34-401 emphasized the necessity to include the coupling between the OH stretching motion and the translational motions of the water molecules in condensed phases of water. However, they predicted that this coupling will alter only slightly the OH stretching region in their calculated Raman and infrared spectra of ice Ih, liquid water and amorphous solid water. Vernon et al. 1411 and Page et al. [42] argued that certain features of infrared photodissociation spectra of water clusters in supersonic molecular beams, observed by them, should be assigned to combination transitions in-

95

volving the hydrogen bond stretching and the hydrogen bond bending modes (in and out-of-plane). Watts and collaborators [ 43,441 found, from their potential for water [ 17,181, that the intensity of combination bands involving the 0.. -0 stretching was negligible. Recently, Wojcik and Rice [45] tested some potentials [13,15,17,18] for water, estimating, between others, coupling constants between the intramolecular and intermolecular vibrations. They found that the couplings, as estimated from these potentials, are very small. They claimed that existing pair potentials underestimate the interaction between the intra and intermolecular modes which is important in the dynamic processes in water. They also formulated a model for the infrared spectrum of the OH stretching region in the water dimer in which the intra-intermolecular coupling is included. FERMI RESONANCE

Fermi resonance is an anharmonic interaction between a fundamental state ( Q1 ) and an overtone (Qz) or a combination vibration ( Qz Q2, ) of a similar energy and symmetry [ 46,471. This interaction leads to a shift of energy levels and re-distribution of intensities in vibrational spectra. The importance of Fermi resonance in vibrational spectra of hydrogenbonded systems has been recognized widely [25,29,48-571. Fermi resonance in these systems is associated with the interaction of the fundamental X-H stretching vibration with the overtones or combination vibrations, involving the X-H bending, the C=O stretching, the C-H stretching, etc. The combination vibrations involved in Fermi resonance carry much lower intensity than the X-H stretching and the result of the reasonance is a borrowing of intensity, which leads to a broad and complex-featured vibrational spectrum, sometimes with deep holes, called “Evans holes” [ 581. In water, Fermi resonance is expected to occur between the overtone of the HOH bending motion and the OH stretching fundamental (thus being an intramolecular interaction). This is because the energy of the HOH bending overtone is close to the energy of the OH stretching fundamental. It is related to the anharmonic force constant KlZ2, the value of which was estimated by Kuchitsu and Morino [59] as 255.4 cm-’ and by Smith and Overend [60] as 71.71 cm-‘. Sceats et al. [61] estimated the value of this parameter as N 100 cm-’ for D,O and N 140 cm-l for HZ0 in isotopically diluted ice Ih. There are three major contributions to ItlZZ:the harmonic stretching and bending force constants k,, and lz,,, and the anharmonic force constant iz,,. These three terms account for about 95% of the magnitude of k,,, in the gas phase. For Hz0 in D,O ice Ih, klz2 is well accounted for assuming k,,,F 0, which means its reduction compared with the gaseous value (0.15 mdyn A-’ [59] or -0.4198 mdyn A-’ [60]).Th’ is c h ange was interpreted by &eats et al. as a result of change

96

of the character of the electronic wave functions in the OH bond upon hydrogen bonding (reduction of the sp3 hybridization and increase in p character). Further studies of Fermi resonance by Rice and his group [38-40,62-641 showed that it is a second-order effect in determining vibrational spectra of condensed phases of water (ice Ih, liquid water, amorphous solid water). It has more influence on the vibrational spectrum of DzO ice Ih than on that of HZ0 ice Ih. The inclusion of Fermi resonance in the description of the OD stretching dynamics materially improves the agreement between the predicted and observed vibrational spectra, however the major impact on the spectra is caused by the interaction between OH (OD ) oscillators on different molecules. Recently Wojcik and Rice [ 451 showed that there are significant differences in the Fermi resonance parameters for the bonded and non-bonded OH groups. The values of 12iZ2 estimated from different model pair potentials for water are widely scattered.

INTERMOLECULAR RESONANCE COUPLING

This type of coupling between neighbouring OH oscillators (also called Davydov coupling) has its origin in the multipole-multipole interactions. Combined with an adiabatic coupling between the O-H and O.*.O stretching vibrations, it was first employed by Mar&ha1 and Witkowski, in their theoretical model of infrared spectra of hydrogen-bonded dimers [ 261. Subsequently, the model was developed to reproduce infrared spectra of a number of hydrogen-bonded systems, with weak or moderately strong hydrogen bonds, both in the gaseous and crystalline state [ 26,27,65-681. This interpretation has been challenged by the evidence of Fermi resonance in many systems [ 51-531, and the prevailing thought in the vibrational theory of hydrogen-bonded systems is now that two cooperative mechanisms are responsible for vibrational spectra of these systems, namely coupling between intra and intermolecular vibrations and Fermi resonance between the O-H stretching and suitable binary combinations of vibrations in the system. It is the feeling of the present author that the three mechanisms are equally important, though acting differently in different systems. The presence of Fermi resonance does not preclude that of the intermolecular resonance coupling. The model calculations including all these three mechanisms were performed for the carboxylic acid dimer [ 691. In the case of water the presence of the coupling between OH oscillators of different water molecules has strong evidence in the isotopic studies of Haas and Hornig [ 701. They examined a series of isotopic mixtures of ice Ih. At the lowest concentration the vibrational spectrum has only a single peak corresponding to isolated guest oscillators. As isotope concentration is increased, two distinct sidebands are observed in the vibrational spectrum, indicating

97

coupling between neighbouring guest oscillators. From the positions of these sidebands Haas and Hornig calculated an intermolecular coupling constant of -0.123 mdyn A-‘. Whalley [ 711 viewed this mechanism as major in shaping detailed features of both IR and Raman spectra of water. He considered that effects such as Fermi resonance, coupling of OH stretches to the lower frequency lattice modes or “defects” in hydrogen bonding change the spectra only slightly. The same view was expressed by Rice and his group [ 34-40,62-64,721 in a series of calculations of vibrational spectra of condensed phases of water. They found that the strong harmonic coupling between nearest-neighbour OH oscillators in the network of hydrogen bonds (separated by only one hydrogen bond and one oxygen), resulting from the transition dipole-transition dipole interaction, is responsible for the major features of the vibrational spectrum, namely its breadth and gross distribution of intensity. They estimated the value of the coupling constant as -0.15 mdyn A-‘. This view was recently challenged by Reimers and Watts [ 731 who considered local anharmonicity, affected by hydrogen bonding, a major contributor to the vibrational spectrum of water, rather than the harmonic intermolecular coupling. FIELD EFFECTS

The interaction of proton vibration with an external electric field has been considered essential to explain broad and continuous infrared spectra of some charged hydrogen-bonded systems, such as H5 O,+ or H3 0; in liquids. Zundel et al. [ 74-771 argued that such bands appear in easily polarizable hydrogen bonds in which O-H dipole moment interacts strongly with the electric field of the ions. This mechanism is the most important in charged hydrogen-bonded systems. Zundel et al. also considered additional mechanisms complementing the dipole-field interaction, such as coupling between proton motions in different bonds, proton-hydrogen-bond coupling, inhomogeneous distribution of the 0. *-0 distances and double-minimum or flat and broad potential for proton vibration. Librovich et al. [ 781 explained the continuous bands by a mechanism analogous to the impurity-phonon coupling in doped crystals. Rijsch and Ratner, in their treatment of hydrogen-bonded crystals [ 79,801, also considered the interaction of the proton vibration with the dynamically fluctuating random local field. The coupling leads to broadening of individual lines and determines the bandshape of the O-H spectrum. Robertson and Yarwood [ 811 and Sakun [ 821 developed models describing band shapes of the OH stretching in liquids, assuming that the OH vibration is coupled to the low-frequency hydrogen-bond mode, which, in turn, interacts strongly with the “heat bath” of the medium. Strong infrared absorption in ice Ih implies the existence of dipole moment

changes associated with normal modes of the crystal. These changes imply that the fluctuating electric field will accompany the corresponding motion. The long range fields should therefore account for at least some features of the spectra of condensed phases of water. The long range fields modify the frequencies of longitudinal polarized modes relative to the transverse polarized modes. They also mix zone center modes having transition moment along the field wave vector. In the case of a single isolated mode the Lyddane-SachsTeller relation applies [ 831:

0: ---

E(W)

(2)

co:- E(0)

This relation can be derived on a very general basis, model independent, starting from the Kramers-Krijnig relation [ 841. It is therefore applicable to the spectra of disordered materials, such as ice, as well as to regular crystals. Klug and Whalley [85,86] assigned specific translational Raman bands in ices Ic and Ih on the basis of the Lyddane-Sachs-Teller relation. A similar assignment for ice Ih was done by Faure and Chosson [ 871. A tentative assignment of the longitudinal modes was also reported in amorphous silica [ 881. There is evidence in a recent study of the lattice mode spectra of ice Ih [89] which supports the contention of Klug and Whalley. In ice Ih the electric field effects in the mixing of translational states are important. However, the macroscopic fields are less important in the vibrational spectra of the OH stretching region. Calculations by Bergren and Rice [39] show that the long range electric field plays an important role in altering the states of the system; however, it does not influence the distribution of intensity in the Raman spectrum. The structure of the Raman spectrum of the OH cannot be assigned using only the Lyddane-Sachs-Teller relation, i.e. cannot be related to system modes arising under the influence of the longitudinal field. EFFECTIVE INTRAMOLECULAR

POTENTIAL

Sceats and Rice [90] have devised an effective intramolecular potential for HZ0 in condensed phases of water. They have introduced a single parameter q which characterizes the intermolecular influence (mostly hydrogen bonding) on the force constants of water. They concluded that only harmonic stretching force constants k,, and k,,. depend on the value of II: k,, =8.4452(1-v)‘mdyn

A-’

(3)

k,,, = -0.132+2.119r mdyn A-’ for ?j> 0.012. The value of q is related to the observed stretching frequency of mechanically isolated OH oscillators in DzO, or OD oscillators in HZ0 [391:

99

qon = - 0.8569 + (2.0344 - 1.3007&)

1’2


(4)

qoo = - 1.1183 + (3.0675 - 1.8169LD) 1’2 5on = (PO,, + 5) /2729.33 For ice Ih, where q= 0.087, k,,, is changed from the gas value - 0.1098 [ 601 to 0.055 mdyn A-’ in the crystal. The remaining harmonic constants Iz,, and k,, are independent of q and preserve their gaseous values. The same concerns the diagonal stretching anharmonicities k,, and k,,,,. All the other terms in the potential energy are negligible. The approximation introduced by Sceats and Rice is a direct result of the dipole-dipole or Coulomb interactions exerted on water in its condensed media. Higher derivatives of these potentials are small relative to the anharmonic force constants of the free water molecule. It can however be valid only in the small domain of configurational space, corresponding to an oxygen-oxygen distance from 2.7 to 3.0 A. This is the domain which is most important in the determination of the properties of the phases of water. Despite this, the influence of the intermolecular interaction on the normal mode anharmonicities is very large, through the increased vibrational amplitude of the stretching motion caused by the decrease in the OH stretching harmonic force constant induced by hydrogen bonding eqn. (3). The intramolecular potential of Sceats and Rice was used in successful theoretical interpretations of infrared and Raman spectra of condensed phases of water [ 38-40, 61-641, of the overtone spectra of ice Ih [ 901, and also in the interpretation of anharmonic constants of a series of complexes of methanol with various organic bases [90] and of the relation ~on/~on versus eon in isotopically isolated water molecules in solid hydrates [ 911. WATER TOPOLOGY AND COOPERATIVE EFFECTS

In a recent paper on structures present in water, Speedy [ 91 recalls Kepler’s botanical observations [ 921 (“five-sided pattern of flowers with five petals is common in almost all trees and bushes”), to convince a reader that unusual properties of water at low temperatures are caused by correlated pentagonal structures, which have properties of (a) self-replication, and (b) are associated with cavities. In an approximately tetrahedral network of hydrogen-bonded water molecules, the pentagon is planar and its sides are in an eclipsed conformation, which favours a pentagonal structure around it. Speedy admits, however, that a similar situation can also work for the advantage of hexagons, when the initial geometry is a chair-form hexagon. He believes however that the pentagonal arrangement prevents additional bonding above and below the

100

pentagonal plane, thus forming cavities of a size comparable with the volume of a small molecule. His further thermodynamic considerations suggest that this type of growing network could possibly account for the anomalous properties of water at low temperatures. Speedy and Mezei also performed a computer simulation study of water [lo], in which they evaluated the concentration of pentagonal rings in water and their spatial correlation function. This function g,, (r ) has a peak around r - 3.2 A, which is close to the distance between pentagons in the tetrahedral network. The concentration of pentagons vs. temperature showed an increasing tendency of forming pentagonal rings when temperature was lowered. These results seems rather suggestive and not conclusive because all simulations were performed at temperatures much above the freezing point and the concentration of hexagons was not counted. Another recent study by Mezei and Speedy [93], for the distribution g(S), where 6 denotes that dihedral angle for the hydrogen-bonded dimers and tetramers in liquid water, shows a significant excess of the staggered conformation, characteristic of ice-like rather than clathrate-like structures in water. This seems to contradict the pentagon hypothesis outlined in previous papers. The temperature changes are at this moment unknown and the conclusions drawn from the simulation studies of normal water need not be valid in the low-temperature limit. The work of Speedy and Mezei is a continuation of a long discussion concerning geometrical structures and cooperativity in water. A few years ago Speedy and Angel1 [94] gave evidence for anomalous density fluctuations in supercooled water. An X-ray scattering study by Bosio et al. [ 951 showed that the density fluctuations in water at - 20” C have a correlation length of about 8 A. Stanley and Teixeira [96,97] developed a model in which the role of the low-density species was attributed to clusters of four-bonded water molecules. In another model, Stillinger [98, 991 saw this low-density structure as polyhedral. The controversy is still unresolved. Quite recently Wojcik and Rice [ 1001 reported model calculations of the OH stretching region infrared spectra of 5,6 and 7-membered rings of water molecules. Although there are minor differences between the spectra of different configurations (boat, chair and planar) and of different rings, they are however not large enough to permit use of infrared spectroscopy to study ring distributions in real water.

ACKNOWLEDGEMENT

Part of this work was done in the James Franck Institute, University of Chicago. The author thanks Professor Stuart A. Rice for support and critical discussions.

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