Strength of hydrogen bonds of water depends on local environment

Abstract

In-depth knowledge of water-water potential is important for devising and evaluating simple water models if they are to accurately describe water properties and reflect various solvation phenomena. Water-water potential depends upon inter-molecular distance, relative orientation of water molecules, and also local environment. When placed at a favorable distance in a favorable orientation, water molecules exhibit a particularly strong attractive interaction called hydrogen bond. Although hydrogen bond is very important for its effects on the elements of life, industrial applications, and bulk water properties, there is no scientific consensus on its true nature and origin. Using quantum-mechanical methods, hydrogen bond strength was calculated in different local environments. A simple empirical linear relationship was discovered between maximum hydrogen bond strength and the number of water molecules in the local environment. The local environment effect was shown to be considerable even on the second coordination shell. Additionally, a negative linear correlation was found between maximum hydrogen bond strength and the distance, at which it was observed. These results provide novel insights into the nature of hydrogen bonding.

INTRODUCTION

Water is one of the most abundant compounds in the part of Earth accessible to humans. At the same time it played a key role in the development of early life and sustaining all known life forms today, as well as in the technological processes and industry. Additionally, water bears many properties that differentiate it from other compounds;^{1, 2} the best known examples are density maximum at 4 °C, lower density of the solid phase as compared to the liquid phase, nearly constant heat capacity in the liquid phase, negative expansion coefficient below the temperature of density maximum, high surface tension, and viscosity. These properties are largely governed by the formation of hydrogen bonds (HB). To understand the behavior and properties of water and aqueous solutions is therefore crucial to understand hydrogen bonding and its molecular background.

In principle, this can be done with quantum-mechanical calculations.^{3, 4} While they offer the highest degree of exactness, a high computational cost of these approaches, however, limits their use to small- and medium-sized molecules and systems.⁵ Insights gleaned from employing them on small systems should allow for the development and fine tuning of simplified water models that make running simulations of larger systems with a couple hundreds water molecules feasible.^{6, 7, 8}

Usually, simplified water models are manipulated through employing them in molecular dynamics (MD) or Monte Carlo simulations (MC). First, simulations with adjusted parameters are run repeatedly to parameterize a model in such a way as to achieve the best agreement of an arbitrarily chosen bulk property with the experimental data. Later on, the accuracy of a model is validated by repeating the simulation to calculate other bulk properties and comparing them with experimental data. We argue, however, that it is vital that the model already accurately describe the behavior on the molecular scale if it is to accurately predict macroscopic properties of water and various solvation phenomena.

When two water molecules are placed at a favorable distance in a favorable orientation, they exhibit a particularly strong attractive interaction called hydrogen bond. Because of the small size of the two molecule dimer, experimental determination of the interaction strength is difficult and mostly avoided in lieu of ab initio electronic structure studies.^{9, 10, 11, 12, 13, 14, 15} Nevertheless, measurement of the water dimer binding energy in gaseous phase has been attempted by studying thermal conductivity of the vapor of H₂O and D₂O, (Ref. ¹⁶) while bond strength in the solid phase can be inferred from light and neutron scattering measurement^{17, 18} or from the heat of sublimation of ice.^{19, 20} It has been shown computationally that the HB strength in ice is greater than in liquid water.^{21, 22, 23} Dependence of HB strength upon the number of water molecules in the local environment, their orientation and placement can, however, only be calculated theoretically. There have only been few attempts to calculate the HB interaction in the quantum water dimer, mostly its dependence upon water molecule geometry²⁴ or upon basis sets used,⁹ with varying results. Xantheas also evaluated the effect of cooperativity on hydrogen bonding for trimers, tetramers, pentameters, and hexamers in various orientations.²⁵ In this work, we studied even larger clusters and calculated dependence of water-water interaction on the presence of additional water molecules in the first and second solvation shell.

METHODS

Theoretical methods

Standard density functional calculations were performed using the GAUSSIAN 09 program suite,²⁶ employing B3LYP density functional (DFT) method with 6-31++G(df) (Ref. ²⁷) and aug-cc-pVTZ (Refs. ^{28, 29}) basis sets. The B3LYP functional is a linear combination of Hartree-Fock exchange,^{30, 31} 1988 Becke exchange,^{32, 33} and LYP correlation.³⁴ Geometry of the water molecule was constructed to match the experimental data (OH bond length 0.9572 Å, ∠HOH angle 104.52°) (Ref. ³⁵) for an isolated water molecule in a vacuum and then kept constant throughout the computations. Use of these two basis sets is a reasonable assumption since geometric optimisation using B3LYP, CCSD,^{36, 37, 38, 39} and MP2 (Ref. ⁴⁰) chemistries with the aug-cc-pVTZ (Refs. ^{28, 29}) basis set yields very similar geometry (bond lengths 0.960, 0.959, and 0.961 Å; angles 105.2°, 104.5°, and 104.1°, respectively). Water-water potential for two molecules in a vacuum (see Figure 1) was also calculated with different methods used (maximum interaction 4.46, 3.77, and 3.76 kcal mol⁻¹; oxygen-oxygen distance 2.93, 3.05, and 3.06 Å, using B3LYP, CCSD, and MP2, respectively), and is in agreement with the literature data.^{9, 41} B3LYP/6-31++G(df) and B3LYP/aug-cc-pVTZ were ultimately chosen as the best compromise between the computational complexity and the accuracy of the results.

Figure 1.

Distance dependence of water-water potential in a vacuum calculated by different methods. Solid line: B3LYP/aug-cc-pVTZ; dotted line: MP2/aug-cc-pVTZ or CCSD/aug-cc-pVTZ. The MP2 and CCSD calculations do not differ.

Electronic energies are reported as difference between energy of the dimer and energy of two isolated water molecules at the infinite distance. Water molecules in the local environment are placed each at an oxygen-oxygen distance of 2.892 Å in a tetrahedral orientation.

Calculation details

Two water molecules were placed in an orientation that is particularly favorable for the formation of hydrogen bond (for clarity henceforth referred to as the dimer) (Refs. ^{42, 43, 44, 45}) HB donor molecule is placed relative to the HB acceptor molecule in such a way that it donates its hydrogen bond to lone pair electrons of the HB acceptor's oxygen. Assigning atoms of the first molecule labels H^1a, H^1b, and O¹ and those of the second molecule labels H^2a, H^2b, and O², the orientation can be described as having atoms O¹, H^1a, and O² on a straight line that forms the angle of 125° with the bisector of the angle ∠H^2aO²H^2b (see Figure 2). When using chemistry B3LYP/6-31++G(df), interaction in this configuration attains its maximum strength of 6.147 kcal mol⁻¹ at the oxygen-oxygen distance 2.892 Å. Consequently, we decided to keep water molecules in the local environment fixed at this distance.

Figure 2.

Geometry of the water dimer when hydrogen bond is the strongest.

When a hydrogen atom of a molecule participates in the hydrogen bond, that molecule is called a HB donor; conversely, the other molecule is known as a HB acceptor. In the tetrahedral orientation, each water molecule can form two bonds as a donor and two as an acceptor. Since one bond is already formed with the other partner in a dimer, each molecule can form three additional hydrogen bonds with the surrounding molecules that are called the local environment. The distances between each water molecule in the dimer and the three molecules forming its environment remain constant throughout the computation, while the distance between the two water molecules in the dimer is varied (see Figure 3).

Figure 3.

Local environment breaks into two pieces when the oxygen-oxygen distance in the dimer increases.

From zero to six additional water molecules were added in the first solvation shell of dimer in all possible variations, each time at an oxygen-oxygen distance of 2.892 Å and in the tetrahedral orientation. The final cluster consisting of eight water molecules resembles the structure as found in the ice (see Figure 4 for the geometry). We also calculated the effect of water in second solvation shell on strength of hydrogen bond by placing one water at all possible configurations in second solvation shell.

Figure 4.

Water dimer can form six additional hydrogen bonds with water molecules in the first coordination shell.

To ensure that only the interaction between the two molecules in the dimer is considered in calculation of HB strength and not additional undesired interactions between the environment of one molecule and the other molecule or between both environments, calculation is performed four times (see Figure 5): for the whole system (labeled I), for the system without water molecule 1 (II), for the system without water molecule 2 (III), and for the system without both water molecules (IV). HB interaction is obtained as follows:

$$\begin{array}{c}\hfill E_{HB}=E_I-E_{II}-E_{III}+E_{IV}.\end{array}$$ (1)

Figure 5.

Different configurations used in extraction of interactions of water dimer.

RESULTS AND DISCUSSION

Energy of the dimer as calculated from Eq. 1 was set to zero for the infinite distance between the molecules in the dimer. Relative distances and orientation of both parts of local environment to the each closer molecule of the dimer were kept constant (see Figure 3). Upon evaluating the distance dependence of potential, it was found that the maximum interaction between the water molecules in the dimer (HB strength) and the corresponding oxygen-oxygen distance (HB length) vary with the changes in the local environment. This manifests as a shorter and stronger hydrogen bond in highly ordered structure of ice as compared to an isolated dimer in vacuum (see Figure 6).

Figure 6.

Distance dependence of water-water potential for two water molecules in vacuum (solid B3LYP/6-31++G(df), dashed B3LYP/aug-cc-pVTZ), and in ice-like structure (dotted B3LYP/6-31++G(df), dot-dashed B3LYP/aug-cc-pVTZ).

Bond strength

Effect of introducing additional water molecules in the local environment was thoroughly investigated. It was found that the placement of each additional water molecule in the local environment of the dimer changes HB strength of the dimer. Changes are additive, increasing or decreasing the interaction each for 0.73 kcal mol⁻¹ when using B3LYP/6-31++G(df) (see Figure 7) and 0.78 kcal mol⁻¹ when using B3LYP/aug-cc-pVTZ (see Figure 8). HB strength decreases for each water molecule that is introduced into the local environment of the donor molecule in the dimer as an acceptor or into the local environment of the acceptor molecule as a donor. Conversely, introducing a donor water molecule to the local environment of a donor molecule in the dimer or introducing an acceptor water molecule to the local environment of an acceptor molecule increases HB. This relationship can be described as

$$\begin{array}{c}\hfill E_{HB}=A+n\times B,\end{array}$$ (2)

with parameters A_6-31++G(df) = 6.265 kcal mol⁻¹, B_6-31++G(df) = 0.73 kcal mol⁻¹ and A_aug-cc-pVTZ = 4.785 kcal mol⁻¹, B_aug-cc-pVTZ = 0.78 kcal mol⁻¹. The variable n describes the number of water molecules in the local environment and is defined as n = n_AA + n_DD − n_DA − n_AD. Indices AA and DD denote donor molecule in the environment bonded to the donor molecule of the dimer or acceptor molecules in the environment bonded to the acceptor molecule in the dimer, respectively (cooperative effect). Indices AD and DA designate the opposite relationship, i.e., donors in the environment bonded to the acceptors in the dimer and vice versa (anti-cooperative effect) (see Tables 1, 2).

Figure 7.

Energy of hydrogen bond depending on number of water molecules in first solvation shell for B3LYP/6-31++G(df).

Figure 8.

Energy of hydrogen bond depending on number of water molecules in first solvation shell for B3LYP/aug-cc-pVTZ.

Table 1.

Hydrogen bond and strength for different configurations of water molecules in the local environment. A and D denote a HB acceptor or donor, respectively, on the second coordination shell for B3LYP/6-31++G(df).

			On HB acceptor		On HB donor
	EHB	rOO
n	[kcal mol−1]	[Å]	Donors	Acceptors	Donors	Acceptors
0	6.15	2.89	0	0	0	0
1	5.52	2.92	0	0	0	1
1	6.84	2.87	0	1	0	0
1	6.90	2.86	0	0	1	0
1	5.46	2.92	1	0	0	0
2	6.15	2.90	0	1	0	1
2	7.59	2.84	0	2	0	0
2	6.28	2.89	0	0	1	1
2	7.78	2.83	0	1	1	0
2	7.59	2.84	0	0	2	0
2	4.96	2.96	1	0	0	1
2	6.15	2.89	1	1	0	0
2	6.21	2.89	1	0	1	0
3	6.84	2.87	0	2	0	1
3	7.09	2.86	0	1	1	1
3	8.60	2.81	0	2	1	0
3	6.90	2.86	0	0	2	1
3	8.60	2.81	0	1	2	0
3	5.59	2.92	1	1	0	1
3	6.90	2.87	1	2	0	0
3	5.65	2.92	1	0	1	1
3	7.03	2.86	1	1	1	0
3	6.84	2.86	1	0	2	0
4	7.78	2.84	0	2	1	1
4	7.72	2.84	0	1	2	1
4	6.21	2.90	1	2	0	1
4	6.34	2.89	1	1	1	1
4	9.41	2.78	0	2	2	0
4	7.78	2.83	1	2	1	0
4	6.21	2.89	1	0	2	1
4	7.66	2.83	1	1	2	0
5	8.60	2.81	0	2	2	1
5	7.09	2.86	1	2	1	1
5	7.09	2.86	1	1	2	1
5	8.60	2.80	1	2	2	0
6	7.84	2.83	1	2	2	1
6 + 1	8.06	2.83	1	2 + A	2	1
6 + 1	7.72	2.85	1	2 + D	2	1

Table 2.

Hydrogen bond and strength for different configurations of water molecules in the local environment. A and D denote a HB acceptor or donor, respectively, on the second coordination shell for B3LYP/aug-cc-pVTZ.

			On HB acceptor		On HB donor
	EHB	rOO
n	[kcal mol−1]	[Å]	Donors	Acceptors	Donors	Acceptors
0	4.46	2.93	0	0	0	0
1	3.82	2.98	0	0	0	1
1	5.22	2.90	0	1	0	0
1	5.37	2.88	0	0	1	0
1	3.93	2.97	1	0	0	0
2	4.48	2.95	0	1	0	1
2	6.15	2.86	0	2	0	0
2	4.66	2.93	0	0	1	1
2	6.38	2.84	0	1	1	0
2	6.19	2.85	0	0	2	0
2	3.38	3.03	1	0	0	1
2	4.63	2.93	1	1	0	0
2	4.76	2.92	1	0	1	0
3	5.32	2.90	0	2	0	1
3	5.55	2.89	0	1	1	1
3	7.30	2.81	0	2	1	0
3	5.42	2.90	0	0	2	1
3	7.34	2.81	0	1	2	0
3	4.10	2.98	1	1	0	1
3	5.49	2.89	1	2	0	0
3	4.15	2.97	1	0	1	1
3	5.69	2.87	1	1	1	0
3	5.50	2.88	1	0	2	0
4	6.39	2.85	0	2	1	1
4	6.29	2.85	0	1	2	1
4	4.76	2.94	1	2	0	1
4	4.35	2.91	1	1	1	1
4	8.33	2.78	0	2	2	0
4	6.56	2.84	1	2	1	0
4	4.82	2.92	1	0	2	1
4	6.41	2.84	1	1	2	0
5	7.36	2.81	0	2	2	1
5	5.77	2.89	1	2	1	1
5	5.82	2.88	1	1	2	1
5	7.52	2.80	1	2	2	0
6	6.66	2.84	1	2	2	1
6 + 1	6.89	2.84	1	2 + A	2	1
6 + 1	6.51	2.86	1	2 + D	2	1

We noticed that different methods and basis sets yield absolute energy difference, but the increase or decrease in strength is around 0.7 kcal mol⁻¹ per added molecule in all cases. Initial calculation using MP2/aug-cc-pVTZ chemistry also showed the HB strength change of 0.65 kcal mol⁻¹ per added molecule. This confirms that the calculated HB strength change per added molecule is consistent across different methods (B3LYP and MP2) and basis sets (6-31++G(df) and aug-cc-pVTZ) used in calculation.

The strongest hydrogen bond of 9.41 kcal mol⁻¹ (basis set 6-31++G(df)) or 8.33 kcal mol⁻¹ (basis set aug-cc-pVTZ) is observed for a highly unlikely coordination, where the donor molecule of the dimer forms bonds with two additional donor molecules while the acceptor molecule of the dimer forms bonds with two additional acceptor molecules. Conversely, the weakest hydrogen bond of 4.96 kcal mol⁻¹ (basis set 6-31++G(df)) or 3.38 kcal mol⁻¹ (basis set aug-cc-pVTZ) manifests for an equally unlikely scenario of each molecule in the dimer forming a single additional hydrogen bond of the opposite directionality (donor bonding with another acceptor and vice versa).

When all possible hydrogen bonds are formed, the structure of the cluster resembles a highly ordered structure of ice. In that instance HB strength was calculated to be 7.84 kcal mol⁻¹ (basis set 6-31++G(df)) or 6.66 kcal mol⁻¹ (basis set aug-cc-pVTZ). This is in agreement with theoretical calculations of HB strength, which range from 3 to 8 kcal mol⁻¹ with the best computational estimate of 5.0 ± 0.1 kcal mol⁻¹ (Ref. ⁹). Experimental results from measuring thermal conductivity of the vapor suggest the hydrogen bond strength to be 3.6 ± 0.5 kcal mol⁻¹ (Ref. ⁹).

Effects of varying the environment on the second coordination shell were also investigated. An additional water molecule was introduced as either a HB donor or acceptor on the second coordination shell of the dimer with six already existent water molecules in the local environment. HB strength changed for 0.13–0.23 kcal mol⁻¹. This shows that the local environment effect of the second coordination shell is ∼20%–30% as strong as that of the first coordination shell.

Observed cooperativity effect is in excellent agreement with known facts about the nature of hydrogen bond. Its strength depends upon the position of the electron density; the more the hydrogen atom is stripped off the electron density the stronger the hydrogen bond is. When the donor molecule of the dimer forms an additional bond with another acceptor molecule, the electron density is pushed further away from the dimer molecule, which in turn increases HB strength. Conversely, when the donor molecule of the dimer forms an additional bond with another donor molecule, electron density is pushed back towards the donor molecule, thereby decreasing electron density separation and HB strength. The effect also quickly falls off with distance, falling to less than a third at the second coordination shell.

Bond length

Interaction between two isolated water molecules in a vacuum is the strongest at the oxygen-oxygen distance 2.892 Å or 2.930 Å when it reaches 6.147 kcal mol⁻¹ (basis set 6-31++G(df)) or 4.455 kcal mol⁻¹ (basis set aug-cc-pVTZ). Placement of additional water molecules in the local environment was also found to change the length of hydrogen bond. Changes in the HB length are additive, increasing or decreasing for 0.028 Å (basis set 6-31++G(df)) or 0.040 Å (basis set aug-cc-pVTZ) per added molecule (see Figures 910).

Figure 9.

Dependence of hydrogen bond length on number of water molecules in first solvation shell for B3LYP/6-31++G(df).

Figure 10.

Dependence of hydrogen bond length on number of water molecules in first solvation shell for B3LYP/aug-cc-pVTZ.

HB length increases for each water molecule that is introduced into the local environment of the donor molecule in the dimer as an acceptor or into the local environment of the acceptor dimer molecule as a donor. Conversely, introducing a donor water molecule to the local environment of a donor molecule in the dimer or introducing an acceptor water molecule to the local environment of an acceptor molecule shortens the HB. This relationship can be described as

$$\begin{array}{c}\hfill r_{OO}=A+n\times B,\end{array}$$ (3)

with parameters A_6-31++G(df) = 2.891 Å, B_6-31++G(df) = −0.0280 Å and A_aug-cc-pVTZ = 2.930 Å, B_6-31++G(df) = −0.0403 Å. The variable n describes the number of water molecules in the local environment and is defined as n = n_AA + n_DD − n_DA − n_AD. Indices AA and DD denote donor molecule in the environment bonded to the donor molecule of the dimer or acceptor molecules in the environment bonded to the acceptor molecule in the dimer, respectively (cooperative effect). Indices AD and DA designate the opposite relationship, i.e., donors in the environment bonded to the acceptors in the dimer and vice versa (anti-cooperative effect) (see Tables 1, 2).

As with the HB strength, hydrogen bond is the shortest at 2.78 Å (both basis sets) for an improbable coordination of each molecule in the dimer forming a single additional hydrogen bond of the opposite directionality (donor bonding with another acceptor and vice versa). Conversely, the longest hydrogen bond at 2.96 Å (basis set 6-31++G(df)) or 3.03 Å (basis set aug-cc-pVTZ) is formed when the donor molecule of the dimer forms bonds with two additional donor molecules or when the acceptor molecule of the dimer forms bonds with two additional acceptor molecules.

It is hence evident that the relationship between the maximum HB strength and its length is approximately linear (see Figures 1112), described by

$$\begin{array}{c}\hfill E_{HB}=A\times r_{OO}+B,\end{array}$$ (4)

with parameters A_6-31++G(df) = −26.05 kcal mol⁻¹Å⁻¹, B_6-31++G(df) = 81.50 kcal mol⁻¹ and A_aug-cc-pVTZ = −19.86 kcal mol⁻¹Å⁻¹, B_aug-cc-pVTZ = 62.95 kcal mol⁻¹. This allows us to narrow any further research of local environment effects on HB strength only since bond length strongly correlates with it.

Figure 11.

Correlation between hydrogen bond strength and distance for B3LYP/6-31++G(df).

Figure 12.

Correlation between hydrogen bond strength and distance for B3LYP/aug-cc-pVTZ.

Second coordination shell effects on HB length were much less pronounced than on its strength. Introducing additional water molecules on the second coordination shell negligibly lengthened or shortened the hydrogen bond for less than 0.01 Å, which is less than the difference between the combinatorially equivalent configurations of the first coordination shell.

CONCLUSION

We have calculated dependence of HB strength of water on the local environment by performing quantum chemical calculation of different clusters of water molecules. We have varied the distance between two water molecules forming HB and the number of water molecules bonded to this dimer. Calculations have shown that the introduction of additional water molecules to the local environment linearly increases or decreases the HB strength for ∼0.7 kcal mol⁻¹ and its length for ∼0.03 Å per added molecule. Whether bond strength and length is increased or decreased depends on the donor or acceptor nature of the newly formed hydrogen bonds with the local environment. Changes on the second coordination shell were found to influence the HB strength for about a third as much as on the first shell, while not impacting its length. HB length was also found to correlate linearly with the HB strength.

This result provides new evidence and insight into the nature of hydrogen bond and general features of such bonding. Local environment appears to play far greater role in the HB strength and length than previously thought. It was found that its strength can vary for as much as 90% between extreme cases of cooperativity and anti-cooperativity nature of hydrogen bonding. Even on the second coordination shell, the local environment effect on HB strength is not negligible.

Simple water models take into account only pair potential between water molecules, which our calculations showed is not enough. Using only pairwise interactions is insufficient since the presence of additional molecules considerably changes the potential between two molecules. One way to address this is by using polarisable models of water where interactions are not pairwise. Alternatively, it is possible to construct water models that use pairwise and three-molecule interactions.⁴⁶ We believe that these results are useful in developing new water models and assessing the quality of the current ones.