Abstract
The concept of covalency is widely used to describe the nature of intermolecular bonds, to explain their spectroscopic features and to rationalize their chemical behaviour. Unfortunately, the degree of covalency of an intermolecular bond cannot be directly measured in an experiment. Here we established a simple quantitative relationship between the calculated covalency of hydrogen bonds in liquid water and the anisotropy of the proton magnetic shielding tensor that can be measured experimentally. This relationship enabled us to quantify the degree of covalency of hydrogen bonds in liquid water using the experimentally measured anisotropy. We estimated that the amount of electron density transferred between molecules is on the order of 10 m while the stabilization energy due to this charge transfer is ∼15 kJ mol^{−1}. The physical insight into the fundamental nature of hydrogen bonding provided in this work will facilitate new studies of intermolecular bonding in a variety of molecular systems.
Introduction
From its early days, NMR spectroscopy has been instrumental in the study of liquid water^{1,2}. The magnetic shielding tensor, which relates secondary induced electronic magnetic fields to an external static magnetic field, is at the heart of NMR spectroscopy. The shielding tensor is characterized by the three components in the principal axes frame σ_{X}≤σ_{Y}≤σ_{Z} or, alternatively, by the isotropic part σ_{iso}=(σ_{X}+σ_{Y}+σ_{Z}), the anisotropy Δ=σ_{Z}−σ_{iso}, and the asymmetry η=(σ_{Y}−σ_{X})^{3}. However, because of the random orientation and motion of molecules in the liquid phase, only the isotropic part of the tensor can be directly measured by experiment. Nevertheless, even the isotropic ^{1}H shielding can provide valuable information: it has been long established that σ_{iso} decreases on the formation of hydrogen bonds (HBs)^{4,5,6}.
A quantitative investigation between the other components of the shielding tensor and the HB geometry in liquid water^{7,8} and small cluster models^{4,9,10} has been made possible by advances in electronic structure theory^{11,12,13,14,15,16}. Moreover, a recently reported elegant experimental approach has allowed the indirect measurement of the ^{1}H shielding anisotropy in liquid water via its contribution to spin relaxation rates^{17}. Combined with first principles calculations, the new experimental data^{8} have underscored previous findings that some of the tensor components are more sensitive to HB interactions than the average σ_{iso} (refs 4, 18).
In this work, we reveal a quantitative relationship between the calculated components of the ^{1}H magnetic shielding tensor and the degree of covalency of HBs in liquid water^{19,20}. The covalent component of intermolecular bonds is commonly defined as the strength of donor–acceptor orbital interactions between molecules^{21}. Covalency, like many other individual fundamental mode of binding (for example, frozen electrostatics, polarization), cannot be measured, even in principle, by experiment; only the total strength of binding is measurable. Nevertheless, similar to various other unmeasurable auxiliary concepts (for example, wave function in quantum mechanics), covalency is a fundamental, theoretically well defined and physically meaningful quantity that is widely used by chemists to investigate the nature of intermolecular bonding. Furthermore, it is widely accepted that donor–acceptor interactions between molecules in liquid water affect its directly measurable structural, spectroscopic and chemical properties.
Results
Decomposition analysis and NMR shielding tensor
To quantify the strength of covalent interactions in a HB, we employed the decomposition analysis for condensed phase systems based on absolutely localized molecular orbitals (ALMO DA)^{21,22} within Kohn–Sham density functional theory^{23}. The ALMO DA makes this feat possible by first filtering out frozen electrostatic and polarization effects from the total manybody intermolecular binding energy, and then splitting the remaining covalent component into twobody terms, each corresponding to an individual HB. The ALMO approach allows to define two scales to measure the strength of covalent interactions: the amount of electron density transferred from the occupied orbitals on molecule A to the virtual orbitals on molecule B (ΔQ_{A→B}), and the energetic stabilization due to this charge transfer (ΔE_{A→B}). These twobody terms are obtained selfconsistently and include cooperativity effects, which are essential for a correct description of the HB network in liquid water^{24,25,26,27}.
All ensemble averaged components of the NMR shielding tensor (Table 1), geometric and ALMO DA descriptors of HBs were calculated from density functional theorybased ab initio molecular dynamics (MD) simulations performed to represent liquid water at ambient conditions (see Methods for details). The computed values are close to the experimentally measured absolute shielding values at 27 °C (σ_{iso}=25.7 p.p.m. and Δ=18.3 p.p.m., respectively)^{17}. We followed the common practice of denoting the average of σ_{X} and σ_{Y} components of the axially nearly symmetric NMR tensors (η≈0) as the perpendicular shielding component σ_{⊥}. It is easy to see that σ_{⊥} is directly related to the anisotropy: σ_{⊥}=σ_{iso}−.
Correlation between σ components, HB covalency and geometry
Figure 1a,d show that σ_{⊥} exhibits a striking correlation with the strength of electron density transfer as measured on both charge and energy scales. By comparison, the correlation between σ_{iso} and the strength of charge transfer (Fig. 1c,f) is weaker, because σ_{iso} is just the strongly correlated σ_{⊥} component plus the noise from σ_{Z} (Fig. 1b,e).
The presence of the scattered points in Fig. 1 that do not follow the general trend is a manifestation of the complex nature of the HB network, in which some hydrogen atoms do not form HBs and some interact strongly with more than one electron donor. Describing such configurations in detail is beyond the scope of the present work and will be presented elsewhere. Here we just would like to note that including or excluding the outliers does not change our statistical results, models or conclusions significantly (see Methods).
To obtain a more comprehensive view on HBs in liquid water we collected information on the correlation between electronic, NMR and geometric descriptors into a single matrix shown in Fig. 2. In addition to the key relationship established above, the correlation matrix clearly shows the wellknown relationship between the HB length and σ_{iso}. However, we found that the correlation of the HB length and σ_{⊥} is stronger. As in the case of the electronic descriptors, the correlation between the HB length and σ_{iso} is merely a manifestation of the underlying strong correlation with σ_{⊥} plus a noisy contribution from σ_{Z}.
ΔQ_{A→B} and ΔE_{A→B} are also strongly correlated with the HB length, even though this correlation is somewhat weaker than that with σ_{⊥} because the HB length alone is insufficient to characterize a HB (that is, other geometric descriptors, such as the HB angle, should be taken into account). We note that no correlations were found for the electron transfer terms where the water molecule bearing the shielding tensor acts as an electron donor. This shows that the ^{1}H shielding tensor is insensitive to electron transfer from the lone pairs of its own water molecule. There is no significant correlation of the HB angle with any individual shielding tensor component, except for a rather weak correlation with Δ. The HB angle alone does not appear to influence electronic terms either, in agreement with the insensitivity of ΔQ_{A→B} and ΔE_{A→B} to intermolecular rotations in water dimer^{28}.
In addition to studying the shape of the shielding ellipsoid, we also examined its orientation relative to the water molecule. As in previous studies^{29}, our simulations showed that the longest Z axis of the tensor points along the covalent O—H bond: the mean deviation angle between the longest axis and the covalent O—H bond is only 5±3° (Fig. 3, top). Moreover, despite the very low value of η, we find that the two short axes still exhibit a distinct spatial preference: the shortest X axis is normal to the plane of the water molecule, whereas the intermediate Y axis is coplanar with the molecule (Fig. 3, bottom). In other words, the shielding tensor is rigidly fixed to the water molecule. Previous reports have shown that the only nonzero offdiagonal elements of the shielding tensor are σ_{yz} and σ_{zy} in the Eckart frame^{7,30}. This is consistent with our finding, as the transformation from the principal axes frame to the Eckart frame is merely a rotation around the X axis.
Linear model that relates σ_{⊥} to the covalency of HBs
The physical basis underlying the correlation between the electronic and NMR terms (Fig. 1) can be understood by considering the dependence of both of these quantities on the HB length R. We found that σ_{x} and σ_{y} decrease linearly with R^{−3} over the entire HB length range in liquid water (see Supplementary Fig. 1). This strong negative correlation implies that the induced magnetic field from the electron donor molecule (a schematic depiction is given in Supplementary Fig. 2) is the major contributor to deshielding in the plane orthogonal to the HB—it accounts for 88% of the variance in σ_{⊥}. The inverse cubic dependence also suggests that this field can be accurately represented as that of a magnetic dipole. That is, higher order terms in the multipole expansion (Supplementary Equation 1) are negligible. Thus, the R^{−3} dependence of σ_{⊥} enables one to write:
where σ_{⊥}^{∞} characterizes an HBfree water molecule, and has a value of 33.7 p.p.m. as obtained by linearly extrapolating σ_{⊥} to R^{−3}=0. The strength of donor–acceptor interactions between molecules decreases with the tails of the orbitals, that is, exponentially with distance^{21}
The combination of the two equations predicts the following relationship between the chargetransfer term and perpendicular component of the NMR shielding tensor:
where γ_{Q}=−β_{Q}α^{1/3} is the dimensionless proportionality constant. A similar relationship can be written for the energy scale with β_{E}, γ_{E}, λ_{E} as parameters. Indeed, plotting both sides of equation (3) reveals clear linear trends (Fig. 4) with a Pearson coefficient of −0.94 for ΔE_{A→B} and −0.92 for ΔQ_{A→B}. The values of the parameters that represent the best fit of the simulation data are λ_{Q}=270.6 m, λ_{E}=579.4 mHa, γ_{Q}=− 19.86 and γ_{E}=−24.07, respectively. It is worth noting that the λ coefficients should not be interpreted as the limiting values for R→0, but rather as statistical coefficients that best fit the linear trend over its limited range of validity.
While σ_{⊥} shows the strong R^{−3} dependence, the weak correlation between σ_{Z} and R^{−3} (Supplementary Fig. 1) indicates that the change in σ_{Z} results from the factors other than the dipolar field of the electron donor molecule. We found that a satisfactory regression model capable of predicting the behaviour of σ_{Z} should include at least three predictor variables: R^{−3}, the HB angle θ and the covalent O—H bond length OHr. The standardized regression coefficients of this model (Supplementary Table 1) show that R^{−3} and OHr give opposite contributions to σ_{Z} largely cancelling each other out.
The established quantitative dependence between σ_{⊥} and the strength of intermolecular donor–acceptor bonding has one important implication: since σ_{⊥} can now be measured experimentally in liquids^{17}, our model represents an indirect method for probing the covalent component of HBs. Table 2 presents our estimates of ΔE_{A→B} and ΔQ_{A→B} in liquid water based on a linear approximation to the regression model established here and the σ_{⊥} measured experimentally at different temperatures^{17} (Supplementary Note 1 for full analysis). As expected, the strength of the covalent component in HBs decreases with increasing temperature as more water molecules form distorted or even broken HBs with their neighbours.
We would like to emphasize that, unlike population analysis methods (for example, Mulliken, Löwdin), which partition the total electron density between molecules, the ALMO approach measures the reorganization of electron clouds on the formation of a HB^{31}. This conceptual advantage of the ALMO method leads to the remarkable result that the fractional electron transfer ΔQ_{A→B} between water molecules in the liquid phase is only a few millielectrons (Table 2). The reorganization of charge has a welldefined energy given by the ΔE_{A→B} term. Whereas it may seem unusual that so little charge transfer (7–10 m) can stabilize a HB by 12–18 kJ mol^{−1} (equivalent to 19 eV per one transferred electron), it is consistent with simple theoretical estimates. Perturbation theory shows that the transfer energy per electron is equal to the energy gap between donating and accepting orbitals^{28}. Since the energy gap between the most important donating and accepting orbitals on two molecules lies between 10 and 40 eV (virtual orbitals form almost a continuum of states), a value of 19 eV for the effective donor–acceptor gap is entirely reasonable.
To put the estimates of ΔQ_{A→B} into a context, we calculated that 4 m is transferred between molecules in the water dimer if the two molecules are at their equilibrium separation of 2.0 Å. However, ΔQ_{A→B} becomes 7.6 m (that is, significantly closer to the value obtained for the ambient liquid) if the molecules in the dimer are separated only by 1.77 Å—the typical length of a HB in the liquid phase^{27}.
It is important to note that the timescale of fluctuations in ΔQ_{A→B} and ΔE_{A→B} is the same as the timescale of intermolecular vibrations—hundreds of femtoseconds^{22,32,33,34}, which is several orders of magnitude faster than the typical timescale of NMR spectroscopy. This implies that NMR measurements of covalency are capable of yielding only timeaveraged values of ΔQ_{A→B} and ΔE_{A→B}.
Discussion
To summarize, we established a simple quantitative relationship between the perpendicular component σ_{⊥} of the axially nearly symmetric ^{1}H magnetic shielding tensor and the degree of covalency of HBs in liquid water. Covalency was defined as the amount of electron density transfer between water molecules and the stabilization energy associated with it. The physical origin of this relationship is the field induced almost entirely by the magnetic dipole located on the neighbouring water molecule involved in the HB. The major implication of our findings is that this relationship provides the calibration necessary to quantify the covalency of HBs experimentally.
Recent advancements in measuring σ_{⊥} for liquid water^{17} enabled us to estimate that the average amount of charge transferred between the molecules on the formation of an average HB is on the order of 10 m, while the corresponding stabilization energy is estimated to be 15 kJ mol^{−1}. From the practical perspective, using σ_{⊥} rather than σ_{X} or σ_{Y} offers an experimental advantage because σ_{⊥} can be determined as a linear combination of σ_{iso} and Δ and it is technically easier to measure the latter.
In contrast to σ_{⊥}, the σ_{Z} component of the shielding tensor exhibits a complex dependence on the local environment around the proton. Its fluctuations cannot be explained only by the magnitude of the induced magnetic fields originating at the protonaccepting water molecule. Therefore, although it is trivial to measure σ_{iso} experimentally, the noisy contribution of σ_{Z} makes it unsuitable for predicting covalency.
Methods
Second generation Car–Parrinello ab initio MD simulations
Ab initio MD simulations of a periodic cubic cell with 128 light water molecules were performed at constant temperature (330 K to mimic nuclear quantum effects in liquid water at ambient conditions^{35}) and density (0.9966, g cm^{−3}) using the second generation Car–Parrinello method^{36,37}. In this approach, the fictitious Newtonian dynamics of the original Car–Parrinello scheme^{38} is replaced with an improved coupled electronion dynamics that keeps electrons close to the instantaneous electronic ground state without defining a fictitious mass parameter. The superior efficiency originates from the fact that the iterative wave function optimization is fully bypassed. Thus, not even a single diagonalization step is required, while at the same time allowing for integration time steps up to the ionic resonance limit.
The energy and forces were computed using the mixed Gaussianplane waves approach^{39}, where the Kohn–Sham orbitals were represented by an accurate tripleζ basis set with two sets of polarization functions (TZV2P)^{40}, while planewaves with cutoff of 400 Ry were used to represent the charge density. The BLYP exchangecorrelation functional plus a damped interatomic potential to account for van der Waals interactions^{41} was employed. Previous works have shown that this setup provides a realistic description of many important structural, dynamical and spectroscopic characteristics of liquid water, including the partial pair correlation functions, selfdiffusion and viscosity coefficients, HB lifetime, vibrational spectra and magnetic shielding tensor components^{14,15,27,42,43}. The system has been equilibrated at constant temperature and density for 30 ps before ten decorrelated snapshots separated by 1 ps were extracted.
Computational analysis
The ALMO DA was performed for each snapshot using the same computational setup as in ref. 22. The magnetic shielding tensors were computed using density functional perturbation theory^{14,44,45} that is based on the allelectron GAPW method^{46,47} and IGLOIII basis set^{48}. All computations were performed using the QUICKSTEP and ALMO modules of the CP2K suite of programmes^{49}. To keep our results consistent, we performed sampling over all protons including the outliers that are clearly visible in Figs 1 and 4. The only exception is the analysis presented in Fig. 2 (supported by Supplementary Fig. 1 and Supplementary Table 1). The data presented in Fig. 2 requires computing geometric characteristics of HBs. These characteristics can be computed only for well defined HBs. To define a HB, we used a geometric criterion that was derived from the twodimensional (R versus θ) potential of mean force^{43,50}.
Additional information
How to cite this article: Elgabarty, H. et al. Covalency of hydrogen bonds in liquid water can be probed by proton nuclear magnetic resonance experiments. Nat. Commun. 6:8318 doi: 10.1038/ncomms9318 (2015).
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Acknowledgements
R.Z.K. is grateful for financial support from the Swiss National Science Foundation; T.D.K. kindly acknowledges the Gauss Center for Supercomputing (GCS) for providing computing time through the John von Neumann Institute for Computing (NIC) on the GCS share of the supercomputer JUQUEEN at the Jülich Supercomputing Centre (JSC).
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H.E. and T.D.K. designed the research, H.E. and R.Z.K. performed the numerical calculations, R.Z.K. and T.D.K. developed and implemented the employed methodologies and computer codes, T.D.K. supervised the project. All authors contributed equally to writing the manuscript.
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Supplementary Figures 13, Supplementary Tables 12 and Supplementary Note 1 (PDF 1292 kb)
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Elgabarty, H., Khaliullin, R. & Kühne, T. Covalency of hydrogen bonds in liquid water can be probed by proton nuclear magnetic resonance experiments. Nat Commun 6, 8318 (2015). https://doi.org/10.1038/ncomms9318
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