Nature of H-bonds in water vapor

    NATURE OF H-BONDS IN WATER VAPOR Nikolay P. Malomuzh, Igor V. Zhyganiuk, Mikhail V. Timofeev PII: DOI: Reference:

S0167-7322(17)31213-8 doi: 10.1016/j.molliq.2017.06.127 MOLLIQ 7573

To appear in:

Journal of Molecular Liquids

Received date: Revised date: Accepted date:

19 March 2017 19 May 2017 30 June 2017

Please cite this article as: Nikolay P. Malomuzh, Igor V. Zhyganiuk, Mikhail V. Timofeev, NATURE OF H-BONDS IN WATER VAPOR, Journal of Molecular Liquids (2017), doi: 10.1016/j.molliq.2017.06.127

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NATURE OF H-BONDS IN WATER VAPOR

I.I. Mechnikov National University of Odessa, 2, Dvoryans’ka Str., Odessa 65026, Ukraine b National Academy of Sciences of Ukraine, Institute of Environmental Geochemistry, 34a, Palladin Ave., Kyiv 03142, Ukraine

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Nikolay P. Malomuzha , Igor V. Zhyganiukb , Mikhail V. Timofeeva

Abstract

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The work is devoted to the discussion of the H-bond nature in rare enough water vapor. For achievement of this aim 1) the new inter-particle potential between water molecules is constructed and 2) the overlapping degree of electron shells for water molecules forming an isolated dimer is in details studied. In order to estimate the diameter of a molecule, the averaged interparticle potential between water molecules is used. It is shown that the overlapping degree does not exceed 0.03 that leads to the conclusion about the electrostatic nature of H-bonds in water vapor.

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Keywords: water, H-bond, dimer, interaction potential PACS: 82.30.Rs 1. Introduction

There is a vast literature on the nature of H-bonds in water in other substances [1, 2, 3, 4, 5, 6, 7, 8]. An H-bond is considered as a combination of electrostatic attraction and irreducible one caused by exchange effects arising due to overlapping of the electron shells of water molecules. At that, the electrostatic interaction manifests itself on all distances between molecules while the specific H-bond interaction is nonzero only on distances leading to the overlapping of electron shells. In fact, the main question is reduced to the following: how is significant the contribution of non-electrostatic nature? In the majority of works [9, 10, 11, 12, 13, 14, 15, 16, 17] used in computer simulations this irreducible contribution is at all ignored. There are also the attempts to take it into account [18].

Preprint submitted to Journal of Molecular Liquids

June 30, 2017

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In connection with this, we want to investigate this question in the present paper. Our main attention will be focused on the study of the overlapping degree for electron shells of water molecules forming dimers in a water vapor. It is necessary to note that in the framework of standard representations [1, 19] the formation of dimer is caused by the H-bonding of molecules. Note that the overlapping degree ∆(l, d) is the simplest characteristics of such a kind. It is immediately connected with the distance between oxygens l for the ground state of a water dimer and size d of a water molecules. The first value is well known [20]. The second one can be determined from the equation of state (EoS) for rare water vapor. More exactly, the EoS allows us to determine the second virial coefficient, after this we can find the parameters of the corresponding averaged interparticle potential. Here it is very important that the EoS is connected with the averaged interaction potential. Formally the last can be found with the help of known bare potentials: SPC [11], SPC/E [21], TIPS [13] and other. Unfortunately, their parameters are unsatisfactory for vapor states since they are determined from the analysis of liquid ones. These bare potentials have the effective character and cannot be used for accurate modeling of intermolecular interaction in gases. Therefore the construction of suitable bare and averaged interparticle potentials for rare water vapor becomes very actual. We will show that the overlapping degree for electron shells of water molecules forming a dimer is unessential that allows us to conclude that the properly H-bonding between water molecules is small. This conclusion is also supported by the following two facts: 1) the frequencies of normal vibrations of a dimer are reproduced within the electrostatic picture quite satisfactory [22] and 2) the frequency shift for valent vibrations of hydrogen in liquid water comparatively with that in water vapor is with good accuracy explained by the electrostatic model potential GSD [15] (see also [23]). Here we should stress that the adequate analysis of the H-bond nature is possible only for rare water vapor where mutual influence of neighboring molecules is negligible. Only in this case the formation of dimers can be attributed to H-bonding. The realization of our program includes: 1) the construction of the bare potential between water molecules; 2) the calculation of the averaged potential describing the interaction between water molecules; 3) the determination of d; 4) the definition and thorough calculation of ∆ for water molecules and 5) the Discussion of the results obtained and the nature of H-bonds.

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2. Bare intermolecular potential

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This Section is devoted to the construction of a potential of interparticle interaction between water molecules. We suppose that the potential is determined by the sum of repulsive, dispersive and electrostatic contributions. The electrostatic interaction is modeled by the multipole expansion including all terms up to dipole-octupole contribution. The dispersive interaction is approximated by the power expression applicable for spherical molecules. The repulsion between molecules is also modeled by simple power term with exponent determined from the special physical requirements (see below). Such simple shapes for dispersive and repulsive interactions are naturally explained by the continuous rotational motion of water molecules, that leads to the selfaveraging of these contributions to the full potential. The angular dependence for considerably stronger electrostatic interactions is taken into account since namely they generate the dimerization processes in water vapor and determine its EoS. The weak interaction, arising due to overlapping of the electron shells, is omitted. This assumption is justified by all reasons given in the Introduction as well as by consequent estimates. Consistently small values of the overlapping degree and the frequency shift for the valent vibrations of hydrogen in water molecules at increase of density are the main evidences for the conclusion about smallness of properly H-bonding. In the case of necessity, the influence of irreducible H-bonding can be taken into account with the help of perturbation theory [24]. Thus, the potential of interparticle interactions between water molecules is assumed to be equal Φ(1, 2) = ΦR (r12 ) + ΦD (r12 ) + ΦE (r12 , Ω1 , Ω2 ),

(1)

where ΦR and ΦD are the terms describing the repulsive and dispersive interactions correspondingly (they depend only upon the distance between centers of mass because of continuous rotation of water molecules), Φ E is the electrostatic contribution depending additionally on the angles Ω 1 and Ω2 which determine the orientations of molecules (Ω = (θ, ϕ), if we use the spherical coordinates). In accordance with said above we put ΦR (r) =

A B and ΦD (r) = − 6 . n r r

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(2)

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3 B = Iα2 4

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The constant B is estimated with the help of the London’s formula (see [25]) (3)

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where I is the ionization potential and α is the average electronic polarizability. The values of A and n will be determined from two following requirements: 1) reproduction of the ground state energy Ed for a water dimer as well as its dipole moment µd and 2) reproduction of the temperature dependence for the second virial coefficient of a rare water vapor [26, 27]. The electrostatic interaction is determined by the multipole expansion: ΦE = Φµµ + ΦµQ + ΦQQ + ΦµO + ...

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 ! ! " (1) (2) (1) (2) nβ µβ − µα µα Φµµ = − 13 3 nα µα r  ! ! (1) (2) 1 ΦµQ = 4 5 nα µα nβ nγ Qβγ − r ! !" (2) (1) nα nβ Qαβ + − n γ µγ !" (2) (1) (1) (2) +2 µα Qαβ nβ − µα Qαγ nγ  ! ! (1) (2) 1 ΦQQ = 5 35 nα Qαβ nβ nγ Qγλ nλ − 3r " (1) (2) (1) (2) −20nα Qαβ Qβλ nλ + 2Qαβ Qαβ   ! ! (1) (2) ΦµO = − 15 7 nα µα nβ nγ nλ Oβγλ + r! !" (2) (1) nα nβ nγ Oαβγ − + nλ µλ !" (2) (1) (1) (2) −3 nβ nγ Oβγα µα + nβ nγ Oβγα µα

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where

(4)

ˆ O ˆ denote the dipole moment and tensors of quadrupole Here, the symbols: µ $ , Q, and octupole moments correspondingly, $n is the unit vector directed along the bisectors of angle between OH 1 and OH2 . If indexes are repeating, the summation is carried out on them. In our calculations we will use the experimental values for dipole [28] and quadrupole [29] moments, which are close to these determined with the help of quantum chemical methods [30], and the quantum chemical value for octupole moment [30]. The values of A, n, σ are determined from the best reproduction of the ground state energy for an isolated dimer and its dipole moment as well as 4

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µd , D 2.62

r#OO 0.989

χ, ◦ 3.1

θ, ◦ 34.5

#d E -9.96

1.85 2.27 2.35 2.35 2.35

2.6 [31, 32] 2.81 2.81 3.58 2.76

1.0 [20] 0.924 0.906 0.956 0.913

6 [20] -2.5 -1.14 -1.135 -9.96

57 [20] 22.2 27.61 51.8 41.2

-9.96 [33] 12.18 -12.66 -12.7 -14.3

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µ, D 1.85

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Present work (n = 28) Exp. SPC [11] TIPS [13] ST2 [16] SSD [17]

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Table 1: Characteristics of isolated water molecule and a dimer: µ and µd are the dipole moments of isolated water molecule correspondingly, rOO — distance between oxygens, angles χ and θ are defined in [20, 31].

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temperature dependence of the second virial coefficient for the water vapor. The value n = 28 is the most unexpected result of this fitting procedure. The # = 1.506 for n = 28 corresponding value of the repulsive coefficient equals to A # = A/kB Ttr , Ttr is the triple point temperature). For standard n = 12 the (A agreement between theoretical and experimental data is essentially worse. The characteristics of an isolated dimer, calculated with the help of our potential and the known potentials: SPC, SPC/E, TIPS and ST, as well as their experimental values are placed in the Table 1. Note, that values of A [20] and E0 = kB Ttr lengths and energies are measured in units: r0 = 2.98 ˚ - correspondingly (˜ r = r/r0 , E˜ = E/E0 ), dipole moments are measured in Debye units (D). Note that the dipole moment and the ground state energy are the most important characteristics of a dimer. Their values are successfully reproduced only with the help of the presented multipole potential (MP). This potential is also consistent with the experimental value of dipole moment for an isolated water molecule and multipole moments of higher order. Values of dipole moments for an isolated molecule and dimer, corresponding to such potential as SPC, TIPS and etc, significantly exceed their experimental ones (more than 20%). These potentials unsatisfactory reproduce also experimental value of the dimer ground state energy. 3. Averaged interparticle potential for water molecules It is evident that the chief thermodynamic properties of a water vapor, such as the EoS and coexistence curve are determined by the averaged interparticle 5

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potentials. From the physical point of view, the averaging of potential is caused by rapid rotational motion of water molecules. It is not difficult to !1/2 I , is considerably verify that the characteristic rotational time, τr ∼ kB T lesser than the free path time for a molecule in rare enough water vapor. The averaged interparticle potential Ua (r) is defined by the relation (see details in [34]):

Ω1 =4π

Ω2 =4π

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exp (−βUa (r)) =< exp (−βΦ (r, Ω1 , Ω2 )) >≡ $ $ dΩ1 dΩ2 ≡ exp (−βΦ (r, Ω1 , Ω2 )) , 4π 4π

(5)

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where β = 1/kB T . Such a definition of the averaged potential is genetically connected with the physical requirement: the free energy and corresponding to it the configurational integral should not depend on choice of bare or averaged potentials in the pair approximation. It is necessary to note that Ua (r) differ essentially from the more standard average: < Φ (r, Ω1 , Ω2 ) exp (−βΦ (r, Ω1 , Ω2 )) > < exp (−βΦ (r, Ω1 , Ω2 )) >

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U (a) (r) =

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These averages are connected with each other by simple equation 1 U (a) ≈ Ua 2

only in the limit case:|−βΦ| ≪ 1. One can show that the average potential Ua (r), corresponding to the described bare potential, has the Lennard–Jones structure (see Fig. 1): & % ! σ 28 σ !6 ∼ Ua (r) = 4ε (6) − r r The values of ε and σ as well as min Ua (r) ≡ Umin are presented in the Table 2. The considerable distinction of the repulsive exponent n in (6) from that in the standard Lennard–Jones potential (n = 12) is not very surprising. Such a value for n is even possible for Argon [35]. More exactly, the value of n for argon takes different values in dependence of pressure and temperature on the argon phase diagram. 6

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σ #

0.849

0.896

0.919

4.36

3.072

2.593

Umin

-4.36

-4.72

-4.89

0.935

2.423 -5.0

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n

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Table 2: Numerical values of ε, σ and Umin vs. n at T = 308 K

1.0

1.2

1.4

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Figure 1: The comparative behavior of the averaged potential constructed according to (5) at T = 308 K (points) and its approximation with the help of (6) (solid line).

Parameters ε and σ as well as rmin , corresponding to min Ua (r), are weak functions of temperature (Table 3). Their temperature dependencies require the caution in the usage of such potentials [36]. 4. Overlapping of electron shells for water molecules forming dimer If two water molecules form a dimer, the interaction energy between them is considerably smaller than that for molecules with covalent and ionic type of bonding. Therefore, the overlapping degree for their electron shells is expected to be small. In order to estimate it we will use the simplest geometrical construction

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1.0032 -5.0

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0.937

2.311

1.0046 -4.77

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2.046

1.0082 -4.23

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σ #

1.0103 -3.96

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Table 3: Values of ε, σ and r#min for different temperatures at n = 28

mentioned in the Introduction (see Fig. 2):

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l ∆=1− , d

(7)

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where l is the inter-oxygen spacing. Its spectroscopic value for an isolated dimer equals to l = 2.98 ˚ A [20].

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d

Figure 2: The overlapping of electron shells

The characteristic values of ∆ for dumbbell-like molecules with covalent type of bonding between them are given in the Table 4. The effective diameter of a water molecule d should be determined with the help of bare potentials used in literature for the description of interaction between water molecules. In connection with this, we will consider several different cases. 4.1. Bare potential (MP) proposed in this work In this case l = 2.946˚ A is assumed to be equal to the equilibrium distance between oxygens for an isolated water dimer (this value is very close to l = 8

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Table 4: The characteristics of dumbbell-like molecules with covalent type of b onding

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B2 [37] N2 [37] O2 [37] F2 [37] Ne2 [38] 0.534

0.369

0.406

0.473

1.037

1.208

1.107

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1.06

∆

0.56

0.67

0.61

0.54

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# E

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199.33

62.54

64.07

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2.089

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1.058

1.186

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e∆

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2.98˚ A following from the spectroscopic analysis). This distance is evaluated from the minimum for the bare inter-particle potential for water molecules forming the dimer configuration. In order to define the diameter d of a water molecule we will use the analogous procedure applied to the averaged potential. The value d is defined as the intermolecular distance corresponding to the minimum of the averaged potential: These averages are connected with each other by simple equation Ua (d) = minUa (r)

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The values of d determined in such a way for several temperatures as well as the overlapping degree ∆ are presented in the Table 5. At that, the values d#1 and ∆1 correspond to the immediately calculated averaged potential (points in the Fig. 1), at the same time d#2 and ∆2 correspond to the approximating potential (6) with its parameters from the Table 3. Thus, the overlapping degree for two water molecules, forming a dimer, does not exceed ∆ ≈ 0.03, i.e. it is essentially lesser than for molecules formed by ion or covalent types of bonding (here we ignore the difference between the positions of center of mass and center of oxygen). Note, that we can use other definitions of the molecular diameter too. However, the definition used by us has an essential advantage – it is genetically connected with corresponding procedure for a bare potential. Now we complete the results obtained by taking into account the screening effects. It is evident that the lasts mainly modify the electrostatic contribu-

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Table 5: The effective diameter of a water molecule and the overlapping degree of electron shells for several temperatures

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0.017

1.0043

0.016

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1.0071

0.018

1.0065

0.018

473

1.0106

0.022

573

1.0128

0.024

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1.0153

0.026

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1.0142

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tion:

1 ΦE → ΦE , ζ

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where ζ is the dielectric permittivity on frequencies of rotational motion of water molecules. With suitable accuracy it is connected with the substance density ρ by the relation: ζ(ρ) − 1 4π ρ = α ζ(ρ) + 2 3 m

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where α is the electron polarizability and m is the mass of a water molecule. The corrected in such a way values of l, d and ∆ on the coexistence curve are presented in the Table 6. As it is should be the corrections for small densities are small. So, at T = 473 K the high frequency dielectric permittivity equals to ζ = 1.005, that leads to the necessity to raise the accuracy of calculations. Rigorously speaking the value l changes also because of excitations of vibration and rotational degrees of freedom. However, these excitations lead to the diminish of the overlapping degree. Therefore, details of this effect we omit. 4.2. SPC, TIPS and ST potentials The potentials SPC, SPC/E, TIPS and ST describe the interaction between hard molecules, which parameters remain invariable. For them we will estimate d and l in the analogous way, i. e. they are identified with the position 10

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Table 6: The effective diameter of a water molecule and the overlapping degree of electron shells for several temperatures

∆1

∆2

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0.017

1.0043

0.016

373

1.0071

0.018

1.0065

0.018

473

1.0106

0.022

573

1.0143

0.024

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Table 7: Values of d# and ∆ at T = 308K for several potentials

SPC, [11] TIPS, [13] ST2, [16] SPC/E, [21] SD, [14] GSD, [15] MPM, [3] 0.957

0.019

0.025

0.977

0.946

0.94

1.0067

0.87

0.02

0.035

-0.036

0.0607

-0.077

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d#

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of minimum for averaged and bare SPC, TIPS and ST potentials. These data are presented in the Table 7. 4.3. SD and GSD potentials There are also the potentials [14, 15], molecular parameters of which adapt to the inter-particle distances and configurations formed by interacting molecules. Their characteristic properties are determined by the screening functions. Namely they reflect the existence of electron shells and determine their diameters and ones for water molecules. The corresponding estimates for d and ∆ are also presented in the Table 7. Thus, all obtained results allow us to conclude that the H-bond between water molecules cannot be related to the standard chemical nature: ion or covalent. Ending this subsection we note that 1) the MP allows us to reproduce main properties of an isolated water dimer and the EoS for rare enough water vapor more successfully in comparison with other potentials whose parameters were determined by fitting to some characteristics of liquid water; 2) the 11

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0.3

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Figure 3: The values of e∆ as functions of ∆ for different chemical compounds (see in [39]). The arrow shows the direction of small shift if temperature increases.

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analysis of the overlapping degree of electron shells for water molecules forming an isolated dimer leads to the conclusion that the exchange effects are small, so the irreducible non-electrostatic contribution to an H-bond is also small and 3) the smallness of a properly H-bonding between water molecules is consistent with its very weak influence on the frequency shifts of valent vibrations for hydrogens in water molecules (they are fully explained by electrostatic forces). In connection with this we conclude that the special consideration of exchange effects is not reasonable for rare enough water vapor. 4.4. Density of the H-bonding energy on the overlapping interval Let ' ( Ed e∆ = 0.01 ∆ be the density of intermolecular interaction energy on the overlapping interval. As density characteristics, the value e∆ characterizes the interaction of water molecules more sufficiently than their interaction energy Ed . The values of e∆ for different chemical compounds with covalent and ionic bonding as well as for water dimer are presented in the Fig. 3. 12

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From the Fig. 3 it follows that the groups of chemical compounds with covalent and ionic type of bonding are essentially separated from each other. At that, the water dimer abuts on the ionic group. This circumstance also testifies that the water dimers arise due to electrostatic forces. Since the ground state energy for dimer is less than the ionic bonding energy approximately on one order of magnitude, it means that the formation of dimers is mainly caused by dipole-dipole forces. This conclusion together with all other cannot be occasional.

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5. Discussion of the results obtained

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The main result of our consideration is the receipt of the estimate for the overlapping degree of electron shells for water molecules forming a dimer. It is very essential since in the framework of the standard representations the dimerization is caused by the H-bonding of two molecules. Therefore, the careful analysis of the electrostatic contribution to H-bonding energy is very important. In this paper, we have shown that the overlapping degree is close to zero. It means that the irreducible part of H-bond energy should be also close to zero. The key role in our consideration plays the careful analysis of main parameters of a water dimer. This analysis is carried out within the electrostatic picture. In order to find the repulsive constant A and the considering exponent n as well as the diameter σ for water molecules we attract the experimental values for the ground state energy Ed of a dimer and its dipole moment as well as the second virial coefficient for a water vapor. At that, the equilibrium distance between oxygens and dimer angles are reproduced practically exactly. They are very important evidences in the favor of the proposed electrostatic model. For other potentials of type SPC, SPC/E, TIPS so consistent reproduction of main dimer parameters is impossible. In order to find the diameter of a water molecule we apply to the EoS for rare water vapor which properties are determined by the average interparticle potential. In this case, the diameter of electro-neutral water molecules is naturally identified with the inter-particle spacing corresponding to the minimum of the averaged potential. It is necessary to stress that such a definition of the molecular diameter d is genetically connected with the equilibrium inter-oxygen distance l for an isolated dimer.

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In such a way we get ∆ ≈ 0.03. This estimate testifies on the smallness of the overlapping degree. At that, the electrostatic interaction energy of water molecules forming a dimer at rOO = l coincides with its ground state energy practically exactly. From here it follows that the contribution of exchange forces to the inter-molecular interaction potential is negligibly small and we can ignore it in our calculations. In fact, all relevant parameters of a dimer are satisfactory reproduced within the electrostatic model proposed in this paper. It is necessary to note that our conclusions about the H-bonding nature is rigorously correct only for the rare water vapor. The situation becomes more complex in liquid water and dense enough vapor. In these cases the overlapping degree of electron shells increases especially for large enough pressures. It is possible that in such extreme situations it is necessary to apply to representations developed in the IUPAC Recommendations [5, 6]. We plan to discuss this problem in the separate paper. Here we would like to concern briefly two nontrivial examples discussed in the literature [40, 41]. We apply our approach to more complex compounds: − F− . . . HCO− 3 and (H2 PO4 )2 . Estimating ∆ and e∆ we will get additional criterion for identification of the bonding character for these compounds. In + the first case anions F− and HCO− 3 can weakly attract if H is situated on the line F− . . . C is equal to ∆ ≈ 0.18 and corresponding e∆ ≈ 1.25 that is characteristic for artificial complexes of type Ne 2 . From the Fig. 3 it follows that these parameters allow us to attribute this attraction to weak covalent bond. The compound (H 2 PO− 4 )2 of dimer type can be realized in two configurations: one of them corresponds two bonds O − H . . . O and the second — to three bonds O −H . . . O. At that corresponding H-bonds appear to be unequivalent. So in the configuration created by two H-bonds the values (1) of ∆ and e∆ for one of them are ∆1 ≈ 0.01 and e∆ ≈ 4.9 while ∆2 ≈ 0.46 (2) and e∆ ≈ 0.6 for another bond. It means that the first H-bond has the electrostatic (dipole-dipole) character and the second – the covalent one. The analogous situation is characteristic for the configuration with three H-bonds although our estimates in this case are not so reliable. We would like to thank Professor E. Arunan for the attention to our results and courteous acquaintance with IUPAC Recommendations of 2011. We remember always with gratitude Professor Galina Puchkovskaya stimulating our investigation of this question. We thank also Professor L. A. Bulavin, Professor L. Pettersson, Profes14

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sor V. Pogorelov and Dr. I. Doroshenko for discussions of touched questions.

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[2] F. Franks, Water: A Comprehensive Treatise, Plenum Press, New York, 1975.

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[3] A. Antonchenko, V.Ya. Davydov, V. Ilin, Fundamentals of Physics of Water, Naukova Dumka, Kyiv, 1991.

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[4] N. Sokolov, Usp. Fiz. Nauk 57 (1955) 205. [5] E. Arunan, G. Desiraju, R. Klein, J. Sadlej, S. Scheiner, I. Alkorta, D. Clary, R. Crabtree, J. Dannenberg, P. Hobza, H. Kjaergaard, A. Legon, B. Mennucci, D. Nesbitt, Pure Appl. Chem. 83 (2011) 1637.

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Highlights

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• A bare potential correctly describing the dimerization in water vapor is proposed.

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• Equation of state for water vapor is determined by the averaged potential. • The overlapping degree of electron shells for molecules in dimer is close to zero.

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• H-bond in water has the electrostatic nature.

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