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Article

Nonlinear Hydrogen Bond Network in Small Water Clusters: Combining NMR, DFT, FT-IR, and EIS Research

by
Ignat Ignatov
1,
Yordan G. Marinov
2,
Paunka Vassileva
3,
Georgi Gluhchev
4,
Ludmila A. Pesotskaya
5,
Ivan P. Jordanov
6,7 and
Mario T. Iliev
8,*
1
Scientific Research Center of Medical Biophysics (SRCMB), 1111 Sofia, Bulgaria
2
Georgi Nadjakov Institute of Solid State Physics, Bulgarian Academy of Sciences, 72 Tzarigradsko Chaussee Blvd., 1784 Sofia, Bulgaria
3
Institute of General and Inorganic Chemistry, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
4
Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
5
Department of Ecology, Technology and Environmental Protection, Dnipro University of Technology, 49005 Dnipro, Ukraine
6
Department of Mathematics, University of National and World Economics, 1700 Sofia, Bulgaria
7
Institute of Mechanics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
8
Faculty of Physics, Sofia University, St. Kliment Ohridski, 1000 Sofia, Bulgaria
*
Author to whom correspondence should be addressed.
Symmetry 2025, 17(7), 1062; https://doi.org/10.3390/sym17071062
Submission received: 29 May 2025 / Revised: 23 June 2025 / Accepted: 2 July 2025 / Published: 4 July 2025
(This article belongs to the Section D: Chemistry: Symmetry/Asymmetry)
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Abstract

Water’s unique physicochemical properties arise from its dynamic hydrogen-bonding network, yet the precise molecular threshold at which these cooperative behaviors emerge remains a key question. This study employed nuclear magnetic resonance (NMR) spectroscopy and density functional theory (DFT) calculations to investigate the evolution of hydrogen bonding strength in small water clusters, ranging from dimers to pentamers. The observed exponential increase in NMR chemical shift up to the pentamer reflects growing hydrogen bond cooperativity, identifying the (H2O)5 cluster as a critical structural and energetic threshold. At this size, the network achieves sufficient connectivity to support key bulk-like phenomena such as proton transfer and dielectric relaxation. These conclusions were corroborated by complementary FT-IR and electrochemical impedance spectroscopy (EIS) measurements of bulk water. Our results position the water pentamer as the molecular onset of emergent solvent behavior, effectively bridging the divide between discrete clusters and the macroscopic properties of liquid water.

1. Introduction

Numerous recent studies have demonstrated that water clusters exhibit a highly cooperative hydrogen bonding behavior, which directly influences their chemical, electronic, and dielectric properties. Water’s anomalous nature is largely attributed to hydrogen bonding [1]. One such anomaly is the Mpemba effect, where, under certain conditions, hot water freezes faster than cold water, a phenomenon that remains only partially understood [2].
Encapsulating a single water molecule within a fullerene (C60) isolates it from external hydrogen bonding, allowing the observation of its free quantum rotational motion with discrete energy levels around 0.5–10 cm−1 [3]. This isolation underscores the crucial role of intermolecular interactions in defining water’s properties. Similarly, studies using electric nuclear quadrupole coupling have shown that hydrochloric acid (HCl) does not dissociate unless surrounded by at least five water molecules. Only with five or more water molecules does HCl begin to dissociate, indicating that this is the minimum number required to form a hydrogen-bonded network strong enough to exhibit solvent-like behavior [4]. Clusters with fewer than five water molecules retain a covalent H–Cl bond, while clusters of five and seven molecules form spontaneous ion pairs (H3O+…Cl−) in “book” and “cube” hydrogen-bonded structures [4].
Quantum chemical calculations (MP2/CBS-e) further support the idea that the water pentamer (H2O)5 represents a critical thermodynamic threshold; beyond this point, the decrease in free energy (ΔG) becomes minimal. The system achieves sufficient electrostatic stabilization at this cluster size to support proton transfer and enable acid dissociation [5]. Recent spectroscopic and computational investigations have emphasized the structural and energetic differences between smaller clusters and pentamers. Dimers and trimers adopt linear or cyclic geometries with limited hydrogen bonding cooperativity, while the tetramer shows partial ring closure and network stabilization [6].
The water pentamer marks a structural threshold where small water clusters transition from simple, locally constrained geometries to more dynamic and cooperative networks characteristic of the liquid phase [7]. The (H2O)5 cluster adopts a three-dimensional, cage-like structure in which each water molecule can function simultaneously as both a hydrogen bond donor and acceptor, forming a highly cooperative hydrogen-bonded network [8].
Measurements of electric dipole moments in nanosolvated acid–water clusters show a sharp increase at n = 5–6, signaling a structural reorganization that enhances the overall polarity [9]. Thermodynamic analyses of confined water clusters reveal a significant decrease in binding energy per molecule (ΔU/N) and entropy per molecule (ΔS/N) up to n = 4, stabilizing both values at n = 5 and beyond. This stabilization indicates the onset of structural saturation and a transition toward more organized, solvent-like behavior [10]. These observations support the hypothesis that the water pentamer represents a critical point between molecular clustering and emergent bulk-like behavior.
In this study, we aimed to identify the molecular threshold at which cooperative hydrogen bonding emerges in small water clusters, resulting in bulk-like properties. By applying nuclear magnetic resonance (NMR) spectroscopy and density functional theory (DFT) calculations to clusters ranging from dimers to pentamers, we analyzed exponential trends in chemical shifts as indicators of increasing hydrogen bond cooperativity. Our objective was to determine the critical point at which these localized interactions began to drive macroscopic phenomena such as solvation, proton transfer, and dielectric response.

2. Materials and Methods

2.1. Physicochemical Composition

Chemically pure distilled water with an electrical conductivity of 4.8 µS·cm−1, pH 7.1, and an oxidation-reducing potential (ORP) of +170 mV was analyzed by the accredited laboratory Eurotest Control, Sofia, Bulgaria, according to EU standards [11,12,13].

2.2. Nuclear Magnetic Resonance (NMR)

Nuclear magnetic resonance (NMR) spectra were recorded on a Bruker Avance II+ 600 MHz NMR spectrometer equipped with a 5 mm direct detection dual-broadband probe. The experiments were conducted at 298 K. 1H NMR spectra were acquired with 128 K time domain points, a spectral width of 9600 Hz, 16 scans, and a relaxation delay of 60 s. The chemical shifts were referenced to the residual DMSO-d6 resonance (2.5 ppm), used as an external reference. DMSO-d6 was placed in a coaxial capillary within the sample tube and served as the lock signal [14]. All measurements were conducted at the Institute of Organic Chemistry with Center of Phytochemistry, Bulgarian Academy of Sciences.

2.3. Hydrogen Bond Network and Water Clusters

Water clusters were studied using NMR and density functional theory (DFT), following the methodology described in [15]. The investigation focused on two specific types: (iii) the water trimer in the form of a cooperative ring and (iv) the tetramer forming a ring structure (S4) [16].

2.4. Theoretical Calculations

Theoretical calculations corresponding to the NMR results were performed using the Gaussian 16 software package [17,18]. The structural optimization of the water clusters was performed using several methods: MP2/CBS-e [5], M06-2X/aug-cc-pVDZ [19], B3LYP/6-31++G** [20], and MP2/aug-cc-pVTZ [21]. Additionally, we employed the MPW1PW91 hybrid DFT method combined with the 6-311+G(2d,p) basis set [22]. This combination has been shown to provide reliable NMR and geometric results for moderately sized hydrogen-bonded systems while maintaining reasonable computational costs [23,24]. Chemical shielding tensors for NMR were calculated using the Gaussian 16 software package, applying the gauge-including atomic orbital (GIAO) approach. The GIAO method, originally developed by Ditchfield and Hameka [25], is implemented within the DFT framework in Gaussian. The solvent effects were modeled implicitly using the SMD solvation model with built-in water parameters. For comparison with MP2/aug-cc-pVTZ, additional calculations were performed using the MP2/6-31G* level of theory to assess the effect of the basis set on the predicted NMR chemical shifts [15].
The formula was derived using nonlinear regression on δ values computed at the GIAO/MPW1PW91/6-311+G(2d,p)//MP2/CGS-e level of theory. The resulting model successfully captures the saturation trend of cooperative hydrogen bonding in (H2O)n clusters for n = 2–5 and provides a superior statistical fit, reflected by a lower residual sum of squares (RSS) compared to linear and logarithmic alternatives. As such, this formulation offers a novel yet physically grounded framework for describing cooperative behavior in hydrogen-bonded systems, building upon the conceptual foundation laid by earlier computational studies.

2.5. Fourier Transform Infrared (FT-IR) Spectroscpy

The water samples were analyzed using Fourier-transform infrared (FT-IR) spectrometry. The FT-IR spectra in our study were recorded with a “Thermo Nicolet Avatar 360 FT-IR” spectrometer (Waltham, MA, USA), equipped with a DTGS detector in the region 400–4000 cm−1, accumulating 64 scans at a spectral resolution of 2 cm−1 [26]. The experiments were performed at 298.15 K. All measurements were conducted at the Institute of General and Inorganic Chemistry, Bulgarian Academy of Sciences (BAS), Sofia, Bulgaria.

2.6. Electrochemical Impedance Spectroscopy (EIS)

The electrical conductivity studies were conducted at 298.15 K using BioLogic potentiostat/galvanostat (SP-200) (Claix, France). Complex electrical impedance spectroscopy [27] was applied in the 0.1 Hz–10 kHz frequency range, similarly to in our previous studies [13,28,29]. For the ionic conductivity and permittivity measurements, 20 µL distilled water was placed between two parallel flat round copper plates, constituting a plane electric capacitance with a gap of 1 mm. The plate diameter was 10 mm.

3. Results

3.1. Nuclear Magnetic Resonance (NMR)

Figure 1 and Table 1 present the nuclear magnetic resonance (NMR) results for the distilled water samples.

3.2. GIAO-DFT Calculations of NMR Chemical Shifts Based on MP2 Geometries

Table 2 presents the computed average chemical shifts for modeled water clusters (assuming fast exchange) at the GIAO/SMD/MPW1PW91/6-311+G-(2d,p)//MP2/CBS-e level of theory. The MP2/CBS-e geometries were obtained from references [15,16].
Figure 2 shows the relationship between the number of water molecules in clusters (H2O)n (n = 2–5) and their corresponding chemical shift (δ). To model water clusters, the geometries were optimized using various levels of theory, including MP2/aug-cc-pVTZ and MP2/6-31G*. As the number of water molecules increases, the chemical shifts (in ppm) also increase due to enhanced hydrogen bonding in small clusters such as (H2O)2, (H2O)3, (H2O)4, and (H2O)5. Both the MP2/aug-cc-pVTZ and MP2/6-31G* methods predict similar chemical shifts, although MP2/aug-cc-pVTZ yields slightly higher values, suggesting a more accurate description of hydrogen bonding in these systems.
The chemical shift continues to rise exponentially with the inclusion of pentamers (H2O)5. This supports the analysis that n = 5 marks a critical point in water clusters’ structural and energetic cooperativity [4,7]. Both computational methods exhibit a consistent exponential increase up to this point, rather than a plateau, indicating that cooperative hydrogen bonding continues to strengthen through the fifth molecule.
The relationship between chemical shifts (ppm) and the number of hydrogen bonds in (H2O)n clusters (n = 2–4) calculated for distilled water using the MP2/aug-cc-pVTZ method demonstrates an exponential trend. The exponential fit yields a residual sum of squares (RSS) value of 0.0617, compared to 0.1207 for the linear fit and 0.2655 for the logarithmic fit. A similar exponential dependence is confirmed using the MP2/6-31G* method for (H2O)n (n = 2–4), where the exponential model achieves a lower RSS value of 0.1656, compared to 0.2579 for the linear model and 0.4687 for the logarithmic model.
These results consistently support the exponential model as the best fit for chemical shift data from the dimer to pentamer in water. The coordinates obtained from the MP2/aug-cc-pVTZ and MP2/6-31G* calculations show a strong positive correlation for (H2O)n, n = 2–4, with a Pearson correlation coefficient of r = 0.997, respectively, demonstrating consistency between the high-level computational approaches.
For (H2O)n, n = 2–5, with RSS, the exponential model is most prominent. For MP2/aug-cc-pVTZ, the value for exponential dependence is 0.0839; for linear, it is 0.1247; and for logarithmic, it is 0.4277. For MP2/6-31G*, the value for exponential dependence is 0.1739; for linear, it is 0.2829; and for logarithmic, it is 0.7626. The coordinates obtained from MP2/aug-cc-pVTZ and MP2/6-31G* calculations show a strong positive correlation for (H2O)n (n = 2–4), with a Pearson correlation coefficient of r = 0.997, highlighting the consistency between these high-level computational approaches.
Experimentally observed chemical shifts (δ, ppm) in hydrogen-rich water (HRW) show a clear dependence on the number of hydrogen bonds, particularly in small symmetrical clusters such as trimers and tetramers. This combined interpretation of DFT calculations and NMR measurements confirms that the experimentally observed chemical shifts in small water clusters arise from the cooperative effect of hydrogen bonding, wherein water structures exhibit an exponential stabilization trend to 10 water molecules.
The formation of primary hydrogen bonds in dimers leads to significant changes in electronic structure, while trimers begin to form cooperative hydrogen bond networks. Tetramers emerge as stable configurations in which the first signs of exponential hydrogen bond stabilization become evident. Overall, the exponential trend highlights that the stability of the hydrogen bonding network increases nonlinearly, reaching a saturation point at n = 4 water molecules.
In this study, we focused on the most energetically favorable and frequently reported structures. These low-energy configurations are also the most likely to occur under experimental conditions relevant to our NMR and FT-IR measurements. In Table 3, we present the global minima and dominant low-energy structures for clusters up to n = 5, optimized at the MP2 level. Although alternative isomers—such as star-shaped or figure-eight structures—are theoretically possible, they are significantly higher in energy and less relevant to the cooperative hydrogen-bonding effects under investigation.
The computed average (fast exchange) chemical shifts in modeled water clusters were analyzed using the RSS method at the GIAO/SMD/MPW1PW91/6-311+G(2d,p)//MP2/CBS-e level of theory, based on MP2/CBS-e geometries (Table 3).
Table 4 presents the results achieved by fitting various mathematical models—linear, logarithmic, and exponential—for the relationship between the number of water molecules in clusters, from (H2O)2 to (H2O)n (n = 4–7), and the corresponding NMR chemical shifts (δ, ppm), based on the data in Table 3. These fits allow us to identify the most appropriate trend describing how cooperative hydrogen bonding within small clusters influences the observed chemical shifts. Theoretical analysis based on the data from Table 4 shows that symmetric cyclic trimers and tetramers exhibit an exponential dependence of chemical shifts (δ, ppm) on the number of hydrogen bonds. This trend results from the cooperative effect in symmetric structures, where nearly equivalent hydrogen bonds lead to the uniform strengthening of the hydrogen bond network. The theoretical model for the (H2O)2–(H2O)4 range in Table 4 demonstrates the best fit with an exponential trend, as indicated by the lowest residual sum of squares (RSS) value of 0.0005. For the (H2O)2–(H2O)5 range, the linear model yields an RSS of 0.0521. These results align closely with MP2/aug-cc-pVTZ and MP2/6-31G* calculations, which produce RSS values of 0.00153 and 0.0000347, respectively, for the exponential model. The notably lower RSS, compared to MP2/aug-cc-pVTZ, underscores the strong exponential dependence observed in the experimental data.
Although MP2/6-31G* achieves an even smaller RSS, the consistency across all methods and data sources confirms that an exponential stabilization model best describes cooperative hydrogen bonding in small symmetric clusters. However, this exponential trend diminishes when extending the range to (H2O)n, n = 2–28. In such larger clusters, logarithmic and linear models provide RSS values of 1.692 and 5.510, respectively, indicating that exponential behavior is primarily characteristic of smaller, symmetric systems.
A comparative analysis of MP2/aug-cc-pVTZ and MP2/6-31G* for modeling chemical shifts in small water clusters reveals that MP2/aug-cc-pVTZ consistently provides higher accuracy and reliability, particularly in systems with three or more hydrogen bonds. At n = 3, where stable, symmetric configurations begin to form, the difference in δ (ppm) between the two methods is minimal. However, for n = 4–5, MP2/6-31G* systematically overestimates the chemical shift, leading to a growing deviation and a higher mean absolute error (MAE) of 0.460 ppm, compared to 0.296 ppm with MP2/aug-cc-pVTZ. This trend is visualized in Figure 3, which plots the difference in chemical shifts as a function of the number of hydrogen bonds.
Figure 2 presents the GIAO-DFT-calculated chemical shift values (δ, ppm) a function of the cluster size (n = 2–5), with n plotted along the X-axis and δ along the Y-axis. Table 2 lists the corresponding δ values and the corresponding number of hydrogen bonds for each cluster. An exponential increase in δ is observed for n = 2–4, with a trend toward saturation at n = 5, indicating nonlinear and cooperative interactions within the hydrogen bond network. Table 4 compares different fitting models. The exponential model provides an exponential trend for n = 2–4. Linear and logarithmic approximations applied to larger clusters and hydrogen bond counts showed lower accuracy.
This behavior is consistent with previous studies, which showed that larger and more flexible basis sets, such as MP2/aug-cc-pVTZ, provided improved accuracy in molecular electronic properties compared to smaller sets, such as MP2/6-31G* [30].
We calculated the difference in calculated chemical shifts (δ, ppm) between MP2/aug-cc-pVTZ and MP2/6-31G* as a function of the number of hydrogen bonds in small water clusters. A minimum difference occurs at n = 3 hydrogen bonds, corresponding to the formation of symmetric cluster structures. Beyond this point (n = 4–5), the MP2/6-31G* level increasingly overestimates the chemical shift compared to MP2/aug-cc-pVTZ, resulting in larger deviations.
In 1996, Luzar and Chandler demonstrated that the dynamics of hydrogen bonds in liquid water are governed by cooperative effects, leading to logarithmic or power-law dependencies in the formation and breakage of the hydrogen bond network. Our computed chemical shifts for water clusters exhibit a similar logarithmic trend, with clusters ranging from 8 to 24 molecules, indicating the onset of dynamic equilibrium within the hydrogen bond network. These results confirm that, after the initial rapid stabilization phase observed in smaller clusters, the further growth of the system results in progressively minor changes, characteristic of a well-established cooperative hydrogen-bonded network [31]. The cooperative effects lead to the water clusters having a longer lifetime [32].
In previous studies, the cooperative hydrogen bonding behavior of small water clusters was analyzed using quantum chemical methods based on density functional theory (DFT) [5,7]. These analyses revealed a clear nonlinear trend in the energetic and structural properties of water clusters, particularly from the dimer to the pentamer, with a distinct saturation effect emerging beyond the fifth molecule. For instance, it has been demonstrated that both the free energy and internal energy per water molecule decrease rapidly up to n = 5, after which they stabilize, indicating a threshold in hydrogen bond cooperativity [5]. Similarly, a pronounced nonlinear increase in hydrogen bond interactions and NMR chemical shifts was observed from the trimer to the pentamer, transitioning toward liquid-like behavior at larger cluster sizes [7]. In the present study, we introduced an empirical exponential model to qualitatively describe the variation in chemical shifts resulting from cooperative hydrogen bonding:
δ (n) = δo + A (1 − e−kn),
where δo is the baseline chemical shift, A is the change amplitude due to hydrogen bonding, and k represents the exponential growth rate.
The formula was derived using nonlinear regression on δ values computed at the GIAO/MPW1PW91/6-311+G(2d,p)//MP2/CGS-e level of theory. The resulting model successfully captures the saturation trend of cooperative hydrogen bonding in (H2O)n clusters for n = 2–5 and provides a superior statistical fit, reflected by a lower residual sum of squares (RSS) compared to linear and logarithmic alternatives. This model fits both experimental and theoretical data with high accuracy, reflecting the rapid increase in, and saturation of, δ values up to n = 4. As such, this formulation offers a novel yet physically grounded framework for describing cooperative behavior in hydrogen-bonded systems, building upon the conceptual foundation laid by earlier computational studies.
Figure 4 shows the exponential model of chemical shifts and hydrogen bonds from the formula, confirming cooperative hydrogen bond enhancement up to the water pentamer.
Table 5 summarizes the correlation between the number of hydrogen bonds in small water clusters and their corresponding NMR chemical shifts and compares the model predictions with the calculated data.
The exponential formulas are as follows:
  • Theoretical model: δ (n) = 1.94 + 3.20 (1 − e−0.605n);
  • Model-MP2/6-31G*: δ (n) = 1.89 + 3.10 (1 − e−0.678n);
  • Model-MP2/aug-cc-pVTZ: δ (n) = 1.62 + 3.50 (1 − e−0.760n).
A comparative analysis between the theoretical and MP2/6-31G* experimental models yields a coefficient of determination R2 = 0.9987, indicating excellent agreement. In contrast, the MP2/aug-cc-pVTZ model yields a lower R2 = 0.7685, indicating moderate alignment with theoretical predictions. The difference in R2 values highlights the superior accuracy of the CBS-extrapolated method, which better captures long-range cooperative effects in hydrogen bonding. Despite variations in the absolute δ values, both experimental models follow the exponential trends observed for hydrogen bonds in cyclic water clusters with the formula (H2O)n, n = 2–4, affirming the physical consistency of the hydrogen bond saturation effect. These findings demonstrate that the proposed exponential model is mathematically robust, physically meaningful, and broadly applicable for the characterization of chemical shifts in hydrogen-bonded water clusters.
The corrected MP2/aug-cc-pVTZ model is more accurate, as its predicted chemical shifts are close to the experimentally measured values for small water clusters, with the lowest mean absolute error (below 0.5 ppm). Unlike the other two models, it uses a high-quality aug-cc-pVTZ basis set that better captures the electronic structure and cooperative hydrogen bonding interactions. Moreover, comparison with experimental NMR data shows that the MP2/aug-cc-pVTZ model provides quantitative and qualitative agreement, outperforming the purely theoretical model and the MP2/6-31G* approximation.
In this section, we present a quantum chemical analysis of small water clusters (H2O)n (n = 1–5), combining second-order Møller–Plesset perturbation theory (MP2) with density functional theory (DFT) to evaluate structural stability and NMR chemical shifts (δ). The geometries of the water clusters were optimized using the MP2/6-31G* and MP2/aug-cc-pVTZ levels of theory, followed by harmonic frequency analysis to ensure vibrational stability. The results show that all clusters are vibrationally stable, as imaginary frequencies were not detected in any of the structures. This confirms that the optimized geometries correspond to the true local minima on the potential surface.
The hydrogen bond network formed in each cluster is characterized by intermolecular O···H distances between 1.7 and 2.0 Å and O···O separations of approximately 2.7 Å. These distances indicate strong hydrogen bonding and are consistent with the expected chemical shift trends. These geometrical parameters align with the increasing chemical shift trends observed in the GIAO-DFT calculations. Subsequent DFT/GIAO calculations of NMR chemical shifts were performed at the B3LYP/6-31G* and MPW1PW91/6-311+G(2d,p) levels, applied to the MP2-optimized geometries. This combined approach is considered cost-effective and reliable in accurately predicting magnetic shielding in small molecular systems [15].

3.3. Fourier-Transform Infrared (FT-IR) Spectroscopy

Fourier-transform infrared (FT-IR) was performed on distilled water (Figure 5).
The FT-IR spectrum reveals characteristic absorption bands corresponding to the vibrational dynamics of O–H bonds and the underlying hydrogen-bonding network. In liquid water, the broad hydroxyl-stretching region observed between 3201 and 3504 cm−1 reflects the dynamic and extensive hydrogen bonding among water molecules. DFT calculations for isolated small water clusters—including dimers, trimers, and tetramers—predict the OH stretching frequency as the number of hydrogen bonds increases. FT-IR studies have shown that free OH groups not involved in hydrogen bonding exhibit stretching vibrations near 3700 cm−1 [33]. This spectral feature is characteristic of isolated water molecules (monomers) or environments where hydrogen bonding is partially disrupted. The red shift in the OH stretching band observed in small hydrogen-bonded clusters, such as dimers through tetramers, has been well documented in both experimental and theoretical studies. High-level ab initio calculations (e.g., MP2 and CCSD(T)) confirm that the OH stretching frequency shifts from ~3700 cm−1 in free water molecules to the 3200–3500 cm−1 range in small clusters, where cooperative hydrogen bonding stabilizes the network [28]. This trend is further supported by FT-IR measurements at hydrophobic interfaces, where free OH groups are detected around 3700 cm−1, while hydrogen-bonded OH groups exhibit lower stretching frequencies due to the weakening of the OH bond due to hydrogen bonding [30].
The experimentally observed broad, asymmetric band in the 3201–3504 cm−1 region (Figure 4) reflects the strengthening and cooperative nature of the hydrogen-bonding network.
This spectral profile aligns with DFT predictions for OH stretching frequencies in small clusters, where a distribution of hydrogen bond strengths results in a wide range of vibrational shifts [33]. The combined interpretation of DFT calculations and FT-IR measurements confirms that this broad band arises from water clusters of varying sizes and hydrogen-bonding configurations [34]. This approach, which links theoretical and experimental data, has been successfully applied in previous studies on hydrogen-bonding dynamics in bulk water and water at interfaces, with the role of free and hydrogen-bonded OH groups being analyzed [35,36,37].
In addition to the OH stretching region, the absorption band around 1650 cm−1, attributed to the H–O–H bending (scissoring) mode, is also indicative of hydrogen bonding. In our experimental data, this band appears at 1658 cm−1 [38]. DFT calculations predict that as water forms larger clusters, the scissor mode undergoes subtle frequency shifts and increases in intensity. This is consistent with the increased coupling between bending and stretching vibrations in hydrogen-bonded systems.
Finally, the combination band observed at 2130 cm−1, commonly seen in liquid water, arises from coupled bending and librational motions within the hydrogen bond network [39].
Furthermore, the broad OH stretching band observed in the FT-IR spectrum spanning approximately 3200 to 3500 cm−1 is indicative of threshold behavior at the size level of the water pentamer (H2O)5. The water pentamer is considered a critical threshold in the development of hydrogen-bonded aggregates, as it is the smallest cluster capable of forming a closed, cyclic network of five hydrogen bonds. This configuration enables the emergence of cooperative effects, vibrational redshifts, and geometries that closely resemble those found in bulk water [40,41].
Significant experimental work by Saykally and co-workers has shown, through high-resolution FT-IR and microwave spectroscopy, that, starting from the pentamer, water clusters exhibit broadened and redshifted O–H stretching bands akin to those in the liquid phase [42]. These features are not observed in smaller clusters and point to the pentamer as the smallest structure with liquid-like spectral characteristics.
Recent findings further reinforce this view. Xie et al. [4] reported that five water molecules are sufficient to completely dissociate a single HCl molecule, providing direct evidence that the pentamer exhibits incipient solvent behavior, including dielectric screening and ion stabilization. These results build on decades of computational studies that have identified the pentamer as the onset of three-dimensional hydrogen bond networks with enhanced cooperativity [4,40].
Thus, the distinct broadening and asymmetry of the OH stretching band observed in our FTIR measurements may be seen as spectroscopic confirmation of this structural threshold. The appearance of vibrational modes in the 3500–3600 cm−1 range, which are absent from smaller clusters, further supports this assignment [43].

3.4. Electrochemical Impedance Spectroscopy (EIS) Results

Complex electrical impedance spectroscopy is a powerful method of investigating the ionic conductivity of condensed matter layers. Figure 6a shows the real (Z′) and imaginary (Z″) parts of the complex electrical impedance in the Nyquist plot of a thin layer of distilled water. It can be seen that the parallel arrangement of bulk resistance and bulk capacitance can explain the typical contour of a semicircle in the frequency range studied. In contrast with the Nyquist diagram, the Bode plot (Figure 6b) presents the impedance modulus and phase dependencies on the frequency. This diagram allows the accurate localization of the relaxation processes and their relationship with the cooperative dynamics of the hydrogen network areas. Figure 7 illustrates the frequency spectra of the real and imaginary parts of dielectric permittivity for a thin distilled water layer.
In this study, the frequency dependence of the complex dielectric permittivity of distilled water was analyzed, revealing a decrease in both the real (ε1) and imaginary (ε2) components of the dielectric permittivity with increasing frequency. High dielectric permittivity values are observed in the low-frequency range (0.1–1 Hz), indicating the occurrence of ionic conductivity and hydrogen bond interactions. At a frequency of 10 kHz, both components reach low values, suggesting a weakening in dipole relaxation. The results confirm that, although distilled water is chemically pure, it exhibits structural dynamics associated with hydrogen bonding and ionic effects.
The results with RSS in the 0.1–1 Hz range show an exponential trend. There is cooperative hydrogen bonding and cluster formation, where the exponential trend is connected with relaxation processes associated with the hydrogen bond network. The results with RSS in the 1 Hz–10 kHz range show a logarithmic trend.
Complex electrical impedance spectroscopy (EIS) provides valuable information on distilled water’s ionic conductivity and polarization dynamics. The Nyquist plot (Figure 6a) reveals a characteristic semicircle, indicating that the water layer can be further modeled as a parallel combination of bulk resistance and capacitance.
The frequency dependence of the real (ε1) and imaginary (ε2) parts of the dielectric permittivity (Figure 7) exhibits a dispersion effect at the frequency range (0.1–1 Hz), where both real (ε1) and imaginary (ε2) reach exponential trends. This behavior reflects the contributions of cooperative dipole orientation processes and ionic conductivity effects. The two methods are closely related to the dynamic formation and reorganization of hydrogen bond networks.
For the frequency range (1 Hz–10 kHz), real (ε1) and imaginary (ε2), the sharps decrease and indicate transition regimes dominated by local dipole fluctuations and a reduced ability of larger cooperative structures to follow the applied field. The high dielectric permittivity at low frequencies is particularly notable, as it correlates with larger hydrogen-bonded clusters and extended cooperative networks. DFT calculations show that small cyclic clusters (especially trimers and tetramers) exhibit enhanced cooperative hydrogen bonding, which can be expected to contribute to the observed dielectric response. The exponential increase in hydrogen bonding strength and dipole alignment observed in DFT-calculated water clusters correlates with the enhanced dielectric response at low frequencies in the EIS measurements. The high values for real (ε1) and imaginary (ε2) permittivity below 1 Hz suggest dynamic reorientation and cooperative dipole behavior, which is structurally supported by DFT models of (H2O)2 and (H2O)5 clusters. Thus, the molecular-level cooperative effects are reflected in the macroscopic dielectric permittivity of bulk water.
The exponential trend of cooperative enhancement of hydrogen bonding in (H2O)n, n = 2–4, established through NMR and DFT, is also reflected in electrochemical impedance spectroscopy (EIS) within the frequency range of 0.1–1 Hz, where an exponential decrease in dielectric permittivity is observed, associated with the dynamics of the hydrogen bond network. This analysis is further supported by FT-IR data, where the broad OH stretching band (3200–3550 cm−1) and the bending mode at 1658 cm−1 provide evidence of a heterogeneous hydrogen bond network containing both weak and strong hydrogen bonds. The low-frequency dielectric response (high ε1 and ε1) is thus a macroscopic reflection of this structural heterogeneity, capturing the combined polarization response of various small clusters in the dynamic hydrogen bond network.
This study demonstrates that the dielectric properties of distilled water arise from the collective dynamics of small water clusters stabilized by a cooperative hydrogen bond network. By applying Bode plots, which present both the impedance magnitude and phase angle as a function of the frequency, it was possible to localize relaxation processes with frequency resolution. Bode analysis revealed an exponential decrease in impedance |Z| and a characteristic phase transition within the 0.1–10 Hz range, typical of the coordinated dipolar relaxation associated with the hydrogen bond network.
The experimentally observed exponential trend corresponds to DFT models of chemical shifts and the number of hydrogen bonds in water clusters. The exponential models are found in the range n = 2–4. This reflects phase-intensifying cooperativity, while at n = 5, a structural threshold is reached, which is the water pentamer (H2O)5. At this point, the hydrogen bond network acquires three-dimensional stability and begins to exhibit properties characteristic of bulk liquid water.
To conclude, the combination of EIS, FT-IR, and DFT data reveals that the dielectric properties of distilled water cannot be attributed to isolated water molecules. Instead, they emerge from the collective polarization and reorientation dynamics of small, cooperative water clusters, whose structural and vibrational properties have been independently confirmed using DFT and FT-IR. These findings are further supported by recent EIS measurements of distilled water at temperatures near freezing point (−0.1 °C and +0.1 °C), which revealed the formation of ordered ice Ih-type hydrogen-bonded clusters. These results show that cooperative dipolar polarization persists even in quasi-solid water states, highlighting the continuity of dielectric response [13].

4. Conclusions

This study combined experimental (NMR, FT-IR, and EIS) and theoretical (DFT) approaches to explore the cooperative behavior of hydrogen bonding in small water clusters. By analyzing trends in chemical shift data obtained from NMR spectroscopy, we observed an exponential increase in hydrogen bond strength with cluster size from dimers to pentamers. This exponential trend was validated through high-level DFT calculations (MP2/6-31G* and MP2/aug-cc-pVTZ), demonstrating excellent agreement up to the tetramer level and indicating saturation effects beyond this point.
The pentamer (H2O)5 emerges as a structural and energetic threshold at which water clusters gain sufficient connectivity to support bulk-like properties. These include critical phenomena such as proton transfer and dielectric relaxation, which do not manifest in smaller, more localized structures such as dimers or trimers. FT-IR spectra reinforce this interpretation by revealing broad OH stretching bands and redshifts that indicate a diverse, cooperative hydrogen-bonding network. Cyclic structures, particularly in trimers and tetramers, display spectral features consistent with enhanced stability.
Furthermore, EIS measurements revealed frequency-dependent dielectric behavior indicative of cooperative dipolar dynamics. Low-frequency exponential trends in dielectric permittivity correspond to large-scale dipole reorientation within clusters, while high-frequency logarithmic behavior reflects localized molecular fluctuations. These findings confirm that dielectric responses in water originate not from isolated molecules but from collective polarization effects within transient cluster networks.
In conclusion, our results position the water pentamer as the molecular onset of solvent-like behavior. This transition marks a shift from discrete molecular assemblies to a cooperative hydrogen-bonding network that supports the macroscopic phenomena that define liquid water. The integrated methodology presented here offers a valuable framework for studying emergent behavior in molecular systems and will inform ongoing research into solvation, transport phenomena, and aqueous-phase reactivity.

Author Contributions

I.I. and Y.G.M.; methodology, I.I. and Y.G.M.; software, G.G.; validation, I.I., Y.G.M., and G.G.; formal analysis, P.V. and L.A.P.; investigation, I.I., M.T.I., and Y.G.M.; resources, P.V. and L.A.P.; data curation, P.V.; writing—original draft preparation, I.I., Y.G.M., and P.V.; writing—review and editing, I.P.J. and M.T.I.; visualization, P.V.; supervision, G.G.; project administration, I.I.; funding acquisition, Y.G.M. and L.A.P. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

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Symmetry 17 01062 g001
Figure 1. The nuclear magnetic resonance (NMR) results for distilled water.
Figure 1. The nuclear magnetic resonance (NMR) results for distilled water.
Symmetry 17 01062 g001
Symmetry 17 01062 g002
Figure 2. Correlation between a number of water molecules in (H2O)n clusters and their NMR chemical shifts (δ, ppm).
Figure 2. Correlation between a number of water molecules in (H2O)n clusters and their NMR chemical shifts (δ, ppm).
Symmetry 17 01062 g002
Symmetry 17 01062 g003
Figure 3. Variation in chemical shift differences between MP2/aug-cc-pVTZ and MP2/6-31G* across hydrogen-bonded water clusters.
Figure 3. Variation in chemical shift differences between MP2/aug-cc-pVTZ and MP2/6-31G* across hydrogen-bonded water clusters.
Symmetry 17 01062 g003
Symmetry 17 01062 g004
Figure 4. Exponential model of chemical shifts δ (n) = δo + A (1 − e−kn).
Figure 4. Exponential model of chemical shifts δ (n) = δo + A (1 − e−kn).
Symmetry 17 01062 g004
Symmetry 17 01062 g005
Figure 5. FT-IR spectrum of distilled water.
Figure 5. FT-IR spectrum of distilled water.
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Symmetry 17 01062 g006
Figure 6. Nyquist complex impedance plot of real (Z′) and imaginary (Z″) parts of electric impedance (a) and Bode plot (b) of a thin layer of distilled water at room temperature.
Figure 6. Nyquist complex impedance plot of real (Z′) and imaginary (Z″) parts of electric impedance (a) and Bode plot (b) of a thin layer of distilled water at room temperature.
Symmetry 17 01062 g006
Symmetry 17 01062 g007
Figure 7. Frequency spectra of real (Re) and imaginary (Im) parts of dielectric permittivity ε for a thin layer of distilled water.
Figure 7. Frequency spectra of real (Re) and imaginary (Im) parts of dielectric permittivity ε for a thin layer of distilled water.
Symmetry 17 01062 g007
Table 1. The nuclear magnetic resonance (NMR) results for distilled water.
Table 1. The nuclear magnetic resonance (NMR) results for distilled water.
Sampleδ, ppmΔ1/2, Hz
Distilled water4.3587.3
Table 2. The computed averaged (fast exchange) chemical shifts in modeled water clusters at the GIAO/SMD/MPW1PW91/6-311+G-(2d,p)//MP2/CBS-e level of theory (MP2/CBS-e geometries).
Table 2. The computed averaged (fast exchange) chemical shifts in modeled water clusters at the GIAO/SMD/MPW1PW91/6-311+G-(2d,p)//MP2/CBS-e level of theory (MP2/CBS-e geometries).
Water MoleculeNumber of Hydrogen Bondsδ, ppmComment
H2O01.83
Cluster
(H2O)212.56
(H2O)333.69Cyclic
(H2O)444.38Cyclic
(H2O)554.60Cyclic
Table 3. The computed averaged (fast exchange) chemical shifts in modeled water clusters at the GIAO/SMD/MPW1PW91/6-311+G-(2d,p)//MP2/CBS-e level of theory (MP2/CBS-e geometries) [19].
Table 3. The computed averaged (fast exchange) chemical shifts in modeled water clusters at the GIAO/SMD/MPW1PW91/6-311+G-(2d,p)//MP2/CBS-e level of theory (MP2/CBS-e geometries) [19].
ClusterNumber of Hydrogen Bondsδ, ppmComment
H2O01.83
(H2O)212.56
(H2O)333.69Cyclic
(H2O)444.38Cyclic
(H2O)554.60Cyclic
(H2O)664.58Cyclic
(H2O)784.83CH1
(H2O)8125.42D2d
(H2O)9135.45D2dDDh
(H2O)10155.57PP1
(H2O)20305.60[19]
(H2O)24365.29[19]
(H2O)28425.33[19]
H3O+ (H2O)20346.02[20]
Table 4. Residual sum of squares (RSS) for fitting chemical shifts (δ, ppm) using different models across water cluster ranges (H2O)2–(H2O)n, where n = 4, 5, 6, and 7.
Table 4. Residual sum of squares (RSS) for fitting chemical shifts (δ, ppm) using different models across water cluster ranges (H2O)2–(H2O)n, where n = 4, 5, 6, and 7.
RSS Model/
Number of Hydrogen Bonds for Water Clusters		(H2O)2–(H2O)4(H2O)2–(H2O)5(H2O)2–(H2O)6(H2O)2–(H2O)7
Linear0.00450.05210.27710.03732
Logarithmic0.05840.06290.08800.08993
Exponential0.00050.11710.43090.08532
Fitted modelExponentialLinearLogarithmicLogarithmic
Table 5. Chemical shifts (δ, ppm) in small water clusters as a function of hydrogen bonding: theoretical model vs. experimental values.
Table 5. Chemical shifts (δ, ppm) in small water clusters as a function of hydrogen bonding: theoretical model vs. experimental values.
Water ClusterNumber of Hydrogen Bondsδ, ppm
Theoretical Model		δ, ppm
MP2/6-31G*
Calculated		δ, ppm
MP2/aug-cc-pVTZ
Calculated
(H2O)212.562.652.30
(H2O)333.693.353.10
(H2O)444.384.654.15
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Ignatov, I.; Marinov, Y.G.; Vassileva, P.; Gluhchev, G.; Pesotskaya, L.A.; Jordanov, I.P.; Iliev, M.T. Nonlinear Hydrogen Bond Network in Small Water Clusters: Combining NMR, DFT, FT-IR, and EIS Research. Symmetry 2025, 17, 1062. https://doi.org/10.3390/sym17071062

AMA Style

Ignatov I, Marinov YG, Vassileva P, Gluhchev G, Pesotskaya LA, Jordanov IP, Iliev MT. Nonlinear Hydrogen Bond Network in Small Water Clusters: Combining NMR, DFT, FT-IR, and EIS Research. Symmetry. 2025; 17(7):1062. https://doi.org/10.3390/sym17071062

Chicago/Turabian Style

Ignatov, Ignat, Yordan G. Marinov, Paunka Vassileva, Georgi Gluhchev, Ludmila A. Pesotskaya, Ivan P. Jordanov, and Mario T. Iliev. 2025. "Nonlinear Hydrogen Bond Network in Small Water Clusters: Combining NMR, DFT, FT-IR, and EIS Research" Symmetry 17, no. 7: 1062. https://doi.org/10.3390/sym17071062

APA Style

Ignatov, I., Marinov, Y. G., Vassileva, P., Gluhchev, G., Pesotskaya, L. A., Jordanov, I. P., & Iliev, M. T. (2025). Nonlinear Hydrogen Bond Network in Small Water Clusters: Combining NMR, DFT, FT-IR, and EIS Research. Symmetry, 17(7), 1062. https://doi.org/10.3390/sym17071062

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