Hydrogen bonded configurations in liquid water and their correlations with local tetrahedral structures

Hydrogen-bonded structures in liquid water lies at the root of its many anomalous properties. The ability of water to form hydrogen bonds is perceived to be the reason for the formation of locally favored tetrahedral structures (LFTS) in liquid water. These tetrahedral environments also lay the foundation for the two-state theories of water. In molecular dynamics simulations, hydrogen bonds are often assigned between molecules if they satisfy certain geometric or energetic criteria, or a combination of both. Potential of Mean Force landscapes may be used to identify the geometric criteria for identifying hydrogen bonds, and distinguishing ‘ice-like’ hydrogen bonds. In this work, we report the statistical information of hydrogen-bonded structures in liquid water from molecular simulations of the iAMOEBA water model. The fraction of different hydrogen-bonded structures is also compared with the fraction of LFTS molecules to gain insights into the hydrogen-bonded configuration of LFTS molecules. The structure of different hydrogen-bonded configurations, using various structural descriptors is also reported. We find that tetrahedral environments in liquid water (as modeled by the iAMOEBA force-field) are largely, but not exclusively constituted by the molecules that donate and accept two hydrogens each when ‘ice-like’ hydrogen bonds are considered. GRAPHICAL ABSTRACT


Introduction
Water exhibits numerous anomalies, when compared to 'normal' liquids [1].The structural fluctuations between a bond-ordered (low-density) molecular environment and a density-ordered (high-density) molecular environment in liquid water, have been used to explain water's anomalies [2][3][4].Jagla showed that most of the anomalous thermodynamic properties of water can be qualitatively explained by the existence of two competing equilibrium values for the inter-particle distance [5].There has been a lot of work in the literature to model the properties of liquid water based on the structural fluctuations between two local structural environments -one of them being tetrahedral [2][3][4][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23].Recent experimental evidence has suggested that at deeply super-cooled conditions, water has a liquid-liquid phase transition line, which ends in a liquid-liquid critical point [11,12].This implies that liquid water is a supercritical liquid, with strong structural fluctuations.However, authors like Soper have propounded that characterising water as a heterogeneous liquid is flawed [24][25][26].He proposes that ambient liquid water, even though possibly a super-critical liquid, has no distinct phases, the low-density water and high-density liquid, based on the distribution of number fluctuations [25].
The fraction of the molecules in the low-density environments (or locally favored tetrahedral structures, LFTS) is an important parameter in many theories to predict the anomalous properties of water [27][28][29].The widom line in water is also often characterised by maximization of these structural fluctuations (fraction of molecules in LFTS environments = 0.5).Hydrogen bonds play an important role in the formation of the LFTS in liquid water [29].There are also reports in the literature where anomalous properties of water are interpreted in terms of hydrogen-bonded networks [30].
Different experimental approaches have been reported to study the local hydrogen-bonded structures in liquid water [31][32][33][34][35][36][37][38].The formation of a hydrogen bond between two molecules involves three atoms: (i) a donor Oxygen (O) atom of a molecule (D) that donates a hydrogen atom to take part in the hydrogen bond, (ii) The Hydrogen (H) atom that is covalently bonded to the donor D, and (iii) an acceptor Oxygen atom (A) belonging to the second water molecule that accepts the H atom to form a hydrogen bond.A pictorial representation of the atoms involved in the formation of a hydrogen bond is shown in Figure 1.The relative positioning of these atoms near each other gives rise to the geometric variables, which may be used to identify hydrogen bonds in simulations [39].
Molecular simulation is an excellent tool that has been used extensively to study the structure and properties of water [40].It gives a molecular perspective on the underlying physical phenomena on time and length scales that are difficult to obtain from experiments.There have been a lot of models that were developed to study the properties of water, ranging from simple pairwise additive models to more complex and expensive explicit many-body potential models [40][41][42][43][44][45][46].
Usually, a hydrogen bond is identified in molecular simulations based on a combination of these relevant geometries and/or energy of interaction between the molecules [47].In a recent work [51], we estimated statistically favorable configurations from simulations using the iAMOEBA water model and demonstrated how to identify hydrogen bonds based on Potential of Mean Force (PMF) landscapes along the O-H distance and the O-H-O angles (r and α in Figure 1).The method may also be used to distinguish 'ice-like' hydrogen bonds in water.A water molecule can donate a maximum of two hydrogen atoms to take part in hydrogen bonds.Therefore, it can form either double-donor (DD), single-donor (SD) or non-donor (ND) configurations in a system.Similarly, a water molecule can accept a maximum of two hydrogen atoms through the lone pairs on its oxygen atom.Therefore a water molecule may belong to double-acceptor (DA), single-acceptor (SA), or non-acceptor (NA) configurations.Molecular dynamics simulations enable us to identify such configurations and study them individually.In this work, we report the statistics of the different hydrogen configurations in liquid water.We would like to emphasise that in the remainder of the article, we refer to the results from the liquid water simulations using the iAMOEBA water model when we discuss our results about the structure of 'liquid water'.The work tries to gain insights into the hydrogen-bonded structures, and its correlation with the structural fluctuations in water as mentioned by the two-state theory [4,[20][21][22]29,52].
We also report the structural characteristics of different hydrogen-bonded configurations in liquid water using different structural descriptors that are reported in the literature, especially to characterise local tetrahedral environments in liquid water.Structural descriptors such as ζ [29], tetrahedral order parameter (q) [53,54], local structure index (LSI) [55], and the number of neighbors have been used in the literature to explore the two-state theories of water and characterise local structural environments in water.Some of these descriptors have been studied by others, to explore the correlation between them [56].We also report the structure of hydrogen-bonded configurations using a new structural descriptor (θ avg ) that we proposed recently to characterise tetrahedral environments in liquid water simulations.It is based on the angles the oxygen atom of a water molecule makes with its neighbors [57].We study the two-dimensional probability distributions using different structural descriptors to understand which combinations can visually represent the different structural environments in liquid water.Further, we report the structure of different hydrogen-bonded configurations using these descriptors to analyze which hydrogen-bonded configurations majorly contribute to the local tetrahedral environments.The analysis throws insight into the role of hydrogen bonds in the formation of local tetrahedral environments in water, and the significance of identifying 'ice-like' hydrogen bonds in water.

Molecular simulations
Molecular simulations were performed using the iAMOEBA water model [58].The water model belongs to the AMOEBA (Atomic Multipole Optimized Energetics for Biomolecular Applications) forcefield class.The iAMOEBA water model is a polarizable water model, and it is parameterised by both experimental and abinitio data [58].This water model has been reviewed extensively and is found to be able to reproduce liquidphase properties of water like density, dielectric constant, self-diffusion coefficient, vapor-liquid equilibrium curve, and many more [27,40,45,59].It has been also used to explore the two-state theories of water [27,57,60].We performed NPT simulations of 2094 water molecules using the OpenMM 7.5 [61] simulation package.The system size was chosen so that the size of the simulation box is considerably larger (nearly 40 Å) than the distance beyond which the O-O pair correlation function out (around 10-12 Å) in water.The time evolution of energies and densities of the system was tracked in super-cooled simulations to ensure that we were sampling liquid configurations of the system.A crystallization event would often be indicated by a sudden drop in energy and density of the system.Trajectories generated ranged from 10 ns to 100 ns -at lower temperatures, longer trajectories were used.See table S1 in the supporting information for details of the lengths of trajectories and the number of configurations analyzed at different temperatures.Langevin integrator with a time-step of 1 femtosecond was used to perform the time integration, and to maintain the temperature of the system.Pressure was maintained using the Monte-Carlo barostat [62,63] with a coupling time of 25 time-steps.Non-bonded interactions were cut-off at 9 Å and correction terms were applied.Long-range electrostatic contributions were calculated using the Particle Mesh Ewald method [64].Periodic boundary conditions were applied in all directions.The molecular trajectories were read and analysed using the MDAnalysis library [65].

Identification of hydrogen bonds
As discussed earlier, hydrogen bonds are assigned between water molecules, based on geometric criteria, energetic criteria, or a combination of both.In a recent work, we proposed to identify hydrogen bonds based on PMF landscapes along r and α (see Figure 1).We identified two water molecules to be hydrogen-bonded, if their mutual orientation (specified by r and α) belonged to the region where PMF ≤ 0 kT [51].In this work, it was also reported that the hydrogen bond geometries of a vast majority of ice Ih crystals, may be described by the condition PMF ≤ −2 kT.Hydrogen bonds identified by this criterion, are therefore both stronger, and have 'ice-like' geometries.Therefore, 'ice-like' hydrogen bonds were identified if mutual orientation between two molecules (r-α combination) lie in the region PMF ≤ −2 kT.We used these definitions to identify hydrogen bonds between molecules for the current study.Interested readers are referred to our previous work for more details [51].There have been instances in the literature about the relevance of distinguishing hydrogen bonds.For example, Stanley and colleagues studied the relation between thermodynamic properties and hydrogen bonds, through the study of lattice-gas models which were modified to include the local geometries of hydrogen bonds [66].In that work, they used an energy-based distinction of strong hydrogen bonds, and report that water molecules might be involved in forming strong hydrogen bonds, when specific relative mutual orientations are similar to that of ice I h .The PMF-based definition of strong hydrogen bonds proposed by us also shows that stronger hydrogen bonds have 'ice-like' geometries [51].

Structural descriptors
We report the structure of local tetrahedral structures in liquid water using some of the most used structural descriptors from the literature.

Zeta (ζ ) ζ :
ζ is a structural descriptor that can capture local structures in liquid water, and takes into account hydrogen bond formation between water molecules.It was first proposed by Russo and Tanaka [29] to identify locally favored tetrahedral structures in water.For any given water molecule i, ζ is defined as the difference between the distance d ij to the closest neighbor that is not hydrogen-bonded to i and the distance d ij to the farthest neighbor that is hydrogen-bonded to it.

Local structure index (LSI):
LSI was first proposed by Shiratani and Sasai [55], who showed that the distribution of LSI in liquid water may be decomposed into two individual distributions corresponding to structured and de-structured populations.Later, Wikefeldt et al. showed that LSI is capable of capturing the bimodality of inherent structure of water in molecular simulations [67].It is defined by Equation ( 1) where, where

Tetrahedral order parameter (q):
The tetrahedral parameter was proposed to characterise tetrahedrality [53,54].It is calculated based on the angles oxygen atoms make with each other and is defined by Equation ( 3) where θ jk is the angle formed by a molecule with its jth and kth neighbors.A higher value of q indicates a higher extent of tetrahedrality.

Average O-O-O angle (θ avg )
θ avg : We recently proposed a structural descriptor (θ avg ) to characterise local tetrahedral environments in liquid water, also based on the angles oxygen atoms make with their neighbors [57].It is defined as the average angle an oxygen atom makes with its neighboring molecules and can be calculated using Equation ( 4) where θ ij is the angle formed with neighbors i and j, n is the number of neighbors, and consequently, n C 2 is the number of angles that a molecule forms with its neighbors.The neighbors were identified using a distance cut-off on the distance between the oxygen atoms.
A cut-off value of 3.7 Å was used to identify the nearest neighbors.We reported that the distribution of θ avg is bimodal, with one peak at 109.5 • (internal angle of a regular tetrahedron).This indicates that θ avg is a promising structural descriptor to characterise local tetrahedral configurations in water.

Hydrogen bond statistics
The two-state theory of water postulates structural fluctuations in liquid water between two local structural environments.One of them is predominantly tetrahedral, and is formed by maximizing hydrogen bonds.Such structures are low in entropy because of the limited structural degeneracy imposed by the structure.There are evidence in the literature to show a hidden 'ice-like' environment in liquid water [68].Identifying 'ice-like' environments is therefore relevant in gaining insights into water's structure and properties.In our earlier work, we reported that the fraction of molecules forming four 'ice-like' hydrogen bonds closely follows the trend of locally favoured tetrahedral structures (LFTS) estimated from simulations [57].The reader may note that some authors use the term low-density liquid (LDL) fraction to report the fraction of these tetrahedral environments in water [3,27].As discussed previously, it is possible to identify different configurations like Double-Donors/Acceptors (DD/DA), Single Donors/Acceptors (SD/SA), Non-Donors/Acceptors (ND/NA) from molecular simulations.The fraction of molecules that belong to these configurations at 300 K and 1 bar is shown in Figure 2. The error bars shown in the plot are the standard deviation of the data.
It is clear that when all hydrogen bonds (PMF ≤ 0 kT) are considered (red bars in Figure 2), the majority of the water molecules (nearly half of them) belong to the category of DD-DA.That is, they donate two hydrogen atoms, and accept two hydrogen atoms from neighbors simultaneously, forming 4 hydrogen bonds.However, the distribution into different configurations is more even when only 'ice-like' hydrogen bonds (PMF ≤ −2 kT) are considered (blue bars in Figure 2), and the most common configuration is SD-SA where the water molecule donates one hydrogen atom and accepts one hydrogen atom from a neighbor forming 2 'ice-like' hydrogen bonds.Recently, Paesani and colleagues [69] also reported the fraction of these hydrogen-bonded configurations using many-body molecular simulations with the MB-POL water model [70][71][72].Our results (when all hydrogen bonds are considered) agree with the statistics of hydrogen-bonded configurations reported in the literature (see fig.S1 in the supporting information).
To gain a better understanding of the structures as defined by donor and acceptor configurations of molecules, we also looked at the distribution of the number of hydrogen bonds formed by molecules when they belong to DD/SD/ND or DA/SA/NA configurations.The results are shown in Figure 3.
Considering all hydrogen bonds, when a molecule donates or accepts 2 hydrogen atoms (DD or DA respectively), there is about 70% chance that they form 4 hydrogen bonds (See red bars in Figure 3(a-d)) at 300 K and 1 bar.There is also ∼ 20% chance that DD/DA molecules accept or donate just one hydrogen atom, resulting in 3 hydrogen bonds.Almost 60% of the single donors tend to form 3 hydrogen bonds, indicating that they accept two hydrogen atoms from their neighbors.Also, about 35 % of single donors accept just one hydrogen atom, to form 2 hydrogen bonds.ND or NA molecules tend to accept or donate (respectively) either two or one hydrogen atom(s).Now, we discuss these statistics when only 'ice-like' hydrogen bonds are considered.We find that most double donors form 3 hydrogen bonds ( ∼ 50%) -meaning that they are single acceptors (blue bars in Figure 3(a)).It should also be noted that a considerable fraction of molecules form 4 hydrogen bonds when they are double donors ( ∼ 30%).For double acceptors as well, a similar trend is observed (see blue bars in Figure 3(d)).For single donors, ∼ 55% molecules form 2 hydrogen bonds (or accept one hydrogen atom), and ∼ 30% molecules form 3 hydrogen bonds (or accept two hydrogen atoms).This can be seen from the blue bars in Figure 3(b).A similar trend is also observed for single acceptors (see blue bars in Figure 3(e)) -about 50% form two hydrogen bonds (donate two hydrogens) and about 30% form three hydrogen bonds (donate two hydrogen atoms).Looking at non-donor molecules (see blue bars in Figure 3(c)), almost 50% form one hydrogen bond (accept one hydrogen), and about 25% form 2 hydrogen bonds (accept two hydrogens).Non-acceptor molecules (see blue bars in Figure 3(f)), ∼ 55% molecules form 1 hydrogen atom (by donating one hydrogen atom).The rest of the nonacceptors form either 1 or 2 hydrogen bonds with almost equal probabilities.
The reader might note that DD-DA configurations are synonymous with molecules that are four times hydrogen-bonded in real water.However in simulations, in some cases, a molecule may donate/accept hydrogen atoms more than twice, even though this happens rarely.For instance, the presence of three or more hydrogen atoms close to an oxygen atom might be identified as a triple acceptor.If such a molecule also donates a hydrogen atom to a neighboring water molecule, the molecule gets identified as four times hydrogen-bonded, but they are not in the DD-DA configuration.Therefore, even though all DD-DA molecules are four times hydrogen-bonded, all molecules that are four times hydrogen-bonded may not belong to a DD-DA configuration.This is the reason we have shown the results of four times hydrogen-bonded molecules, and DD-DA molecules separately.
Given the breakdown of molecules into different hydrogen bond configurations, it is of interest to investigate their correlation with the locally favored tetrahedral environments in liquid water.The fraction of LFTS molecules, is often an important parameter in modeling water's anomalies based on the structural fluctuations between two local structural environments [27][28][29].For this purpose, first, we compare the fraction of different hydrogen-bonded structures in liquid water with the fraction of LFTS molecules we estimated using θ avg [57].The fraction of molecules forming DD/SD/ND and DA/SA/NA configurations, and the fraction of molecule k times (k ∈ 0, 1, . . .4) are shown in Figure 4. We find that DD and DA configurations qualitatively match the temperature trend of LFTS fraction (Figure 4(a-c)) when all hydrogen bonds are considered.However, even the nature of the curve is similar, the fractions of DD and DA molecules are much higher than that of the LFTS molecules.But, it is interesting to note that when only 'ice-like' hydrogen bonds are considered, the fractions of DD and DA molecules are closer to the fraction of LFTS molecules (see Figure 4(b-d)).The results are more quantitatively similar at lower temperatures, especially at temperatures ≤ 230 K. Tetrahedral environments are often perceived to be stabilised by the formation of hydrogen bonds with four neighbors.Therefore, we also compare the fraction of LFTS molecules with the fraction of molecules that form k hydrogen bonds (k ∈ 0, 1, . . .4).The results are shown in Figure 4(e-f).It is observed that the fraction of water molecules forming 4 hydrogen bonds is much higher than the fraction of LFTS molecules, indicating that these two configurations may not point to the same structural environments in water.When only 'ice-like' hydrogen bonds are considered, we find that the fraction of molecules that form four hydrogen bonds more closely follows the fraction of LFTS molecules.However, the deviation between the two is observed to grow at lower temperatures.
A molecule in a double donor configuration may also accept 0, 1 or 2 hydrogen bonds.Therefore, a double donor molecule may belong to DD-DA, DD-SA or DD-NA configurations depending on the number of hydrogen atoms it accepts.Similarly, a DA molecule may belong to DD-DA, SD-DA, or ND-DA, depending on the number of hydrogen atoms it donates.We also estimated the fraction of these configurations as a function of temperature, and they are shown in Figure 5.It can be observed that DD-DA molecules qualitatively follow the temperature trend of LFTS molecules.There are many more molecules in DD-DA configurations than there are LFTS molecules when all hydrogen bonds are considered (see Figure 5(a)).However, a quantitatively closer trend is observed between the fraction of LFTS molecules and DD-DA molecules when only 'ice-like' hydrogen bonds are considered (see Figure 5(b)).The temperature trends of the fractions of different hydrogen-bonded configurations (when all hydrogen bonds are considered) shown in Figures 4 and 5 also agree qualitatively with the values reported in the literature by other authors like Stanley, Soper and others using molecular simulations of different water models [69,73,74].A comparison with the temperature trends of hydrogen-bonded configurations reported using the MB-POL water model [69] is shown in Figure S1 in the supporting information.

Structural characteristics of hydrogen-bonded configurations
Given the understanding of the different hydrogenbonded configurations, and their temperature trends, and comparison with fraction of LFTS molecules, it is beneficial to study their structural characteristics.For this purpose, we studied the structure of these hydrogenbonded configurations using different structural descriptors that have been discussed in the literature.In a seminal work, Soper and colleagues reported the structure of hydrogen-bonded configurations in liquid water over a wide range of temperatures [74].They reported the O-H, H-H and O-O pair correlation functions, and the distribution of hydrogen bond angles in water.In the current work, instead of providing the overall structural characteristics, we report the structural characteristics of specific hydrogen-bonded configurations.
From the previous results discussed in the manuscript, it is observed that the tetrahedral environments are more predominant at lower temperatures.Therefore, in the following discussions, we show the structure of different hydrogen-bonded configurations at super-cooled conditions.We studied the distribution of ζ , θ avg , q and LSI for molecules in different hydrogen-bonded configurations.For these order parameters, we find that molecules that belong to either DD, DA, or DD-DA configurations have the highest probability of belonging to tetrahedral environments (see Section S2 in supporting information).However, we find that the tetrahedrality is more pronounced when only 'ice-like' hydrogen bonds are considered.The results indicate the relevance of identifying such 'ice-like' hydrogen bond interactions in liquid water.
There is also an interesting observation of which the reader might take note.Even though the probability of belonging to tetrahedral environments is higher when only 'ice-like' hydrogen bonds are considered (for DD, DA, and 4 times H-bonded molecules), the same is true for molecules that are in other configurations.For example, molecules in SD, SA, ND, NA, and k times hydrogen-bonded molecules have higher probabilities of belonging to tetrahedral environments when only 'ice-like' hydrogen bonds are considered when compared to the cases where all hydrogen bonds are considered.This is because of the fact that when we consider molecules in SD, SA, ND, and NA (considering 'ice-like' hydrogen bonds), they may also be a DD or DA or 4 times hydrogen-bonded, when all hydrogen bonds are considered.For example, a molecule that is SD when only 'ice-like' hydrogen bonds are considered, may belong to DD when all hydrogen bonds are considered.The apparent increase in tetrahedrality for molecules in SD/SA 'ice-like' hydrogen-bonded configurations is because of the fact that they may belong to DD/DA configurations when looking at all hydrogen bonds.
We also looked at the distribution of pairs of these structural descriptors to see which combinations can best describe the two local structural environments in liquid water.The two-dimensional probability density distributions of different combinations of the structural descriptors (at 230 K and 1 bar) are shown in Figure 6.The probability densities were calculated by dividing the probability of occurrence of an instance by the area of the bin.That is, if x and y are two order parameters, the probability density for x = x i and y = y i is calculated as P(x i , y i )/(dx × dy), where P(x i , y i ) is the probability that x = x i and y = y i simultaneously, and dx, dy are the sizes of the bins while discretizing x and y respectively.The reader should take note that we report the probability densities to normalise the effect of the bin size while studying any given structural descriptor.However, while comparing different structural descriptors (whose magnitudes are very different), probability densities may have differences in magnitudes as seen in Figure 6.This is a natural consequence of the difference in the sizes of bins (dx and dy in the equation above) for different structural descriptors.Here, we have shown the results considering all water molecules in the system -not just a particular hydrogen-bonded configuration.The intention behind the analysis is to understand which pair of structural descriptors may be able to visually represent the fluctuations between two local structural environments in liquid water.Figure 6(c) shows that fluctuations between two structural environments can be observed using a combination of LSI and θ avg .Two distinct peaks are observed in the two-dimensional probability distribution of LSI and ζ as well (Figure 6(b)), but they are very close to each other making it difficult for distinguishing the two structures easily.We had also previously shown that the combination of LSI and θ avg can show a distinct region of 'ice-like' structures in liquid water [57].The high probability region near θ avg = 109.5• corresponds to these 'ice-like' structures in water.The visualizations in Figure 6 show that there are patterns in the underlying molecular structure of water that indicate two prominent local structural environments, with intermediary structures between them.This also indicates the extent of structural degeneracy within these local structural environments in water.For example, the high probability region close to θ avg ∼ 109.5 • , is narrow along the θ avg axis, indicating a limited range of tetrahedrality.However, this structural cluster is wider along the LSI axis.
Given the understanding that a combination of LSI and θ avg can represent a tetrahedral environment in liquid water, we also report the LSI-θ avg two-dimensional probability distributions for different hydrogen-bonded configurations.The results for DD, DA, DD-DA, and 4 hydrogen-bonded molecules, which are the configurations that tend to correlate with tetrahedral structures based on our previous analysis, are shown in Figure 7.We find that the majority of these molecules lie in the tetrahedral structural configurations.The molecules in these hydrogen-bonded configurations may also belong to the other cluster that lies at lower values of θ avg .The distinction is very clearly visible when only 'icelike' hydrogen bonds are considered (see figures on the right panes in Figure 7).However, it is interesting to note that molecules in these hydrogen-bonded do not exclusively belong to tetrahedral clusters.That is, even 'icelike' DD-DA molecules can belong to the non-tetrahedral environments in water, but with a very low probability.

Conclusions
We reported the statistics of different hydrogen-bonded configurations in liquid water using molecular simulations of the iAMOEBA water model.Different configurations we reported are based on the number of hydrogen atoms a molecule donates or accepts (DD, DA, SD, DA, ND, NA) and the total number of hydrogen bonds a molecule forms (k times hydrogen-bonded molecules).A good qualitative agreement was observed with the statistics of hydrogen-bonded configurations reported in the literature.We also try to understand the link between some of these hydrogen-bonded structures and the local tetrahedral structural environments in liquid water.We find that the fraction of the DD, DA, and 4 times hydrogen-bonded molecules correlate closely with the fraction of tetrahedral molecules in liquid water.The results are closer when we consider 'ice-like' hydrogen bonds.
We also report the structural characteristics of these configurations using different structural descriptors.Different structural descriptors that have been reported in the literature to study the local structures in water -LSI, tetrahedral order parameter (q), ζ , and θ avg have been used to study the structure of these hydrogenbonded structures.We find that the hydrogen-bonded configurations which correlate well with the fraction of LFTS in water (DD, DA, DD-DA, and 4 times h-bonded molecules) have the highest degree of tetrahedrality in water, when we only consider 'ice-like' hydrogen bonds.
A combination of LSI and θ avg can help in the visual identification of the two local structural environments in liquid water.The existence of such local structural environments is proposed by the various two-state theories of water.In a previous work, we also showed that they can be used to identify hidden ice-resembling structures in liquid water.We find that tetrahedral environments are more prominent in DD, DA, and 4 times hydrogenbonded molecules, especially when only 'ice-like' hydrogen bonds are considered.
The results give insights into the hydrogen-bonded configurations in liquid water that lead to the formation of local tetrahedral structures.Results from the combination of different structural descriptors also reveal the underlying local structural environments in liquid water, as predicted by various two-state theories.Structural descriptors also suggest that there are differences in structural degeneracy between the two local structural environments, with the presence of intermediate local structures transitioning from tetrahedral to totally random structures.
The results discussed also indicate that the molecules which donate two hydrogens (DD) or accept two hydrogens (DA) or donate and accept two hydrogens at the same time (DD-DA) have the highest correlation with local tetrahedral structures in liquid water, when ice-like hydrogen bonds are considered.We show that the classical idea of viewing tetrahedral environments in water as four-times hydrogen-bonded molecules is not very accurate unless we consider 'ice-like' hydrogen bonds.The results also indicate the relevance of identifying and distinguishing 'ice-like' hydrogen bonds in water.

Figure 1 .
Figure 1.Pictorial representation of hydrogen bond formation between two water molecules.The relevant geometric variables involved in the formation are shown in the figure: O-O distance (R), O-H distance (r), O-O-H angle (β) and O-H-O angles (θ and α).Often a combination of these variables are used to identify hydrogen bonds in simulations [37,47-50].

Figure 2 .
Figure 2. Fraction of molecules when they fall to different donor/acceptor configurations.All results are at 300 K and 1 bar, using the iAMOEBA water model.The different configurations shown in the graph are DD-DA (double donor & double acceptor), DD-SA (double donor & single acceptor), DD-NA (double donor & non-acceptor), SD-DA (single donor & double acceptor), SD-SA (single donor & single acceptor), SD-NA (single donor & nonacceptor), ND-DA (Non-donor & double acceptor), ND-SA (nondonor & single acceptor) and ND-NA (non-donor & non-acceptor).The error bars shown are standard deviations of the data.

Figure 3 .
Figure 3. Fraction of molecules forming k hydrogen bonds when they belong to DD/SD/ND or DA/SA/NA configurations.The red bars show fractions when all hydrogen bonds are considered and the blue bars show them when only 'ice-like' hydrogen bonds are considered.The results shown are at 300K and 1 bar.Error bars are the standard deviations of the data.

Figure 4 .
Figure 4. Fraction of different configurations at different temperatures at 1 bar.The different configurations include Double Donor, Double Acceptor, and k-times molecules (k ∈ 0, 1, . . .4).The results when considering all hydrogen bonds are shown on the left side and those when only considering 'ice-like' hydrogen bonds are shown on the right side.

Figure 5 .
Figure 5. Fraction of different donor-acceptor configurations at different temperatures and 1 bar.The results when considering all hydrogen bonds is shown on the left side and those when only considering 'ice-like' hydrogen bonds are shown on the right side.

Figure 6 .
Figure 6.Two-dimensional probability density distributions of combination of different order parameters in liquid water at 230 K and 1 bar.The colours on the heat map indicate the probability density of the occurrence of two structural descriptors simultaneously.Most structural descriptors we reported have been used to study two-state theory of water.But a combination of LSI and θ avg is observed to demonstrate two distinct clusters in liquid water, as is seen in subplot (c).

Figure 7 .
Figure 7. LSI-θ avg two-dimensional probability density distributions for different hydrogen-bonded configurations (T = 230 K, P = 1 bar).On the left side are results considering all hydrogen bonds, and on the right side are corresponding results, when only 'icelike' hydrogen bonds are considered.The different configurations shown include DD (a,b), DA (c,d), DD-DA (e,f), and molecules forming four hydrogen bonds (g,h).