Hydrogen Bond Binding of Water to Two Cholic Acid Residues †

: Cholic acid is a trihydroxy bile acid with three hydroxy groups at C-3, C-7 and C-12 carbon atoms; two methyl groups at C-10 and C-13 carbon atoms of the steroid nucleus; and a carboxylic group at C24 of the side alkyl chain. The distance between the oxygen atoms linked to C-7 and C-12 (~4.5 Å) perfectly matches with the edge distance between oxygen atoms in ice. This leads to the design of a cholic acid dimer in which one water molecule is encapsulated between two cholic residues, resembling an ice-like structure. The water molecule participates in four hydrogen bonds, the water simultaneously being acceptor from the O12-H hydroxy groups (two bonds with lengths of 2.177 Å and 2.114 Å) and the donor towards the O-7-H groups (two bonds with lengths of 1.866 Å and 1.920 Å). Regarding this communication, we present the application of the “atoms in molecules” (AIM) theory to the tetrahedral structure. The analysis of the calculated electron density, ρ , is performed using its gradient vector, ∇ ρ , and the Laplacian, ∇ 2 ρ . The calculation of the complexation energy used correction of the basis set superposition error (BSSE) and the counterpoise method. As expected, four critical (3, − 1) points located in the H ... O bond paths were identiﬁed. All calculated parameters are in concordance with those of similar systems and obey the proposed criteria for hydrogen bonds. The total energy for the interaction is − 12.67 kcal/mol and is analysed using proposed energy/electron density equations.


Introduction
Bile acids have a bifacial polarity since the hydroxy groups (up to three at C-3, C-7 and C-12 carbon atoms) lie beneath the plane of the steroid nucleus (hydrophilic α-side).
In what follows, these oxygen atoms are identified with a superscript (for instance, O 7 or O 12 ). Two methyl groups are at the other side of the nucleus at the C-10 and C-13 carbon atoms (hydrophobic β-side). As an example, Figure 1 shows the structure of cholic acid (CA). This explains their amphiphilic behaviour in water-forming aggregates above a critical concentration, commonly referred to as critical micelle concentration (cmc). The biological importance of bile acids has been well described in the literature.
The characterization of the crystal structures of BA and their derivatives using X-ray analysis has been a topic of interest for years [1][2][3][4][5][6][7][8][9]. A common factor of all crystal structures is that the hydroxy groups are always involved in the formation of hydrogen bonds (HB) with another BA monomer, the solvent or both species. Nowadays, interesting and promising applications have emerged, mainly related to the ability of BA to form inclusion compounds in the solid state. This may be used for the resolution of racemates [10].
When analysing BA crystals, for accepting the formation of a hydrogen bond, the geometric criteria (bond lengths and angle) [11] have been used on an exclusively basis. Although enough for most cases, additional criteria may be required in special ones. α-and β-sides of cholic acid The characterization of the crystal structures of BA and their derivatives using X-ray analysis has been a topic of interest for years [1][2][3][4][5][6][7][8][9]. A common factor of all crystal structures is that the hydroxy groups are always involved in the formation of hydrogen bonds (HB) with another BA monomer, the solvent or both species. Nowadays, interesting and promising applications have emerged, mainly related to the ability of BA to form inclusion compounds in the solid state. This may be used for the resolution of racemates [10].
When analysing BA crystals, for accepting the formation of a hydrogen bond, the geometric criteria (bond lengths and angle) [11] have been used on an exclusively basis. Although enough for most cases, additional criteria may be required in special ones.
With the aim of illustrating the application of the "atoms in molecules" (AIM) theory [12,13] for the analysis of hydrogen bonds in crystal structures of BAs, we have chosen a BA crystal in which a single water molecule is encapsulated between two cholic residues in an ice-like structure [14].

Crystal Structure
The crystal structure of the reference system was published previously [14]. Cif files (CCDC 867499) contain the supplementary crystallographic data for the C-suc-C crystal (the acronym provided in that paper). These data can be obtained free of charge from The Cambridge Crystallographic Data Centre via www.ccdc.cam.ac.uk/data_request/cif (accessed on 13 November 2022).
It was shown that water accommodates in a hydrophilic region, establishing hydrogen bonds with the hydroxy groups O 7 -H and O 12 -H of two steroid residues belonging to two CA dimers. These four oxygen atoms form a distorted tetrahedral structure with a water molecule at its centre ( Figure 2) as evidenced by the fact that their centroid is exactly located on the oxygen of water molecule (O w ). The average tetrahedral angle is 110 ± 12°. The edge length values are shown in the figure, the average value being 4.61 ± 0.39 Å, which favourably compares with the one in ice [15]. Table 1 shows the values for the four hydrogen bonds in which water is involved since there are other hydrogen bonds in the crystal involving different groups, but they are not considered here. All the values fulfil the geometric criteria of the existence of a hydrogen bond [11]. In ice, the hydrogen bond distance is 2.76 Å [16]. From the Cambridge Structural Database, Steiner [17] obtained average values of 1.880(2) Å for the water dimer, HO-H … OH2, and 2.825(2) Å for H … OH2 and O … O distances, respectively. With the aim of illustrating the application of the "atoms in molecules" (AIM) theory [12,13] for the analysis of hydrogen bonds in crystal structures of BAs, we have chosen a BA crystal in which a single water molecule is encapsulated between two cholic residues in an ice-like structure [14].

Crystal Structure
The crystal structure of the reference system was published previously [14]. Cif files (CCDC 867499) contain the supplementary crystallographic data for the C-suc-C crystal (the acronym provided in that paper). These data can be obtained free of charge from The Cambridge Crystallographic Data Centre via www.ccdc.cam.ac.uk/data_request/cif (accessed on 13 November 2022).
It was shown that water accommodates in a hydrophilic region, establishing hydrogen bonds with the hydroxy groups O 7 -H and O 12 -H of two steroid residues belonging to two CA dimers. These four oxygen atoms form a distorted tetrahedral structure with a water molecule at its centre ( Figure 2) as evidenced by the fact that their centroid is exactly located on the oxygen of water molecule (O w ). The average tetrahedral angle is 110 ± 12 • . The edge length values are shown in the figure, the average value being 4.61 ± 0.39 Å, which favourably compares with the one in ice [15]. Table 1 shows the values for the four hydrogen bonds in which water is involved since there are other hydrogen bonds in the crystal involving different groups, but they are not considered here. All the values fulfil the geometric criteria of the existence of a hydrogen bond [11]. In ice, the hydrogen bond distance is 2.76 Å [16]. From the Cambridge Structural Database, Steiner [17]

Computational Details
Given the high number of atoms involved in the two bile acid dimers, to analyse the interaction with the water molecule, we simplified the system by reducing the number of atoms in the bile acid unit whilst keeping the same geometric parameters of the remaining atoms. Thus, A and D rings were suppressed, and their carbon atoms linked to B and C rings were replaced by hydrogen atoms. To keep the original interatomic distances obtained from the X-ray resolution of the complex, no minimization of the energy of the complex was carried out. Oxygen-oxygen distances (green lines; data in Å) of the tetrahedron formed by the O 7 -H and O 12 -H hydroxy atoms of the two steroid residues encapsulating a water molecule located at their centroid [14]. The four hydrogen bonds are indicated with blue lines.

Computational Details
Given the high number of atoms involved in the two bile acid dimers, to analyse the interaction with the water molecule, we simplified the system by reducing the number of atoms in the bile acid unit whilst keeping the same geometric parameters of the remaining atoms. Thus, A and D rings were suppressed, and their carbon atoms linked to B and C rings were replaced by hydrogen atoms. To keep the original interatomic distances obtained from the X-ray resolution of the complex, no minimization of the energy of the complex was carried out.
Calculations of the complexation energy used correction of the basis set superposition error (BSSE) with the counterpoise method implemented in Gaussian 19 [18]. Laplacian of electronic density and critical points (AIM) were calculated using Multi-wfn_3.8_dev software [19]. Oxygen-oxygen distances (green lines; data in Å) of the tetrahedron formed by the O 7 -H and O 12 -H hydroxy atoms of the two steroid residues encapsulating a water molecule located at their centroid [14]. The four hydrogen bonds are indicated with blue lines.
Calculations of the complexation energy used correction of the basis set superposition error (BSSE) with the counterpoise method implemented in Gaussian 19 [18]. Laplacian of electronic density and critical points (AIM) were calculated using Multiwfn_3.8_dev software [19].

Analysis of Electron Density
Electron density, ρ, is the starting point of the AIM theory. Its topology is easily deduced from the gradient vector, ∇ρ, and the Laplacian, ∇ 2 ρ. The electron density is usually visualized by drawing contour lines that connect electron density points with an identical value. Figures 3 and 4 show two examples for the present system. In Figure 3, the plane is defined by the nuclei of O57 and O25 (which are O 7 ), and O1 (O w ) oxygen atoms, while in Figure 4 the plane is defined by O24 and O56 (which are O 12 ) and O w oxygen atoms. The thin grey lines are defined by infinitesimal gradient vectors, thus describing gradient paths. duced from the gradient vector, , and the Laplacian,  2 . The electron density is usually visualized by drawing contour lines that connect electron density points with an identical value. Figures 3 and 4 show two examples for the present system. In Figure 3, the plane is defined by the nuclei of O57 and O25 (which are O 7 ), and O1 (O w ) oxygen atoms, while in Figure 4 the plane is defined by O24 and O56 (which are O 12 ) and O w oxygen atoms. The thin grey lines are defined by infinitesimal gradient vectors, thus describing gradient paths.  duced from the gradient vector, , and the Laplacian,  2 . The electron density is usually visualized by drawing contour lines that connect electron density points with an identical value. Figures 3 and 4 show two examples for the present system. In Figure 3, the plane is defined by the nuclei of O57 and O25 (which are O 7 ), and O1 (O w ) oxygen atoms, while in Figure 4 the plane is defined by O24 and O56 (which are O 12 ) and O w oxygen atoms. The thin grey lines are defined by infinitesimal gradient vectors, thus describing gradient paths.  When ∇ 2 ρ < 0, the electronic charge is locally concentrated, this being the case in covalent bonds [20]. When ∇ 2 ρ > 0, the electronic charge is locally depleted [13], the interactions are called closed-shell. This occurs in hydrogen bonds (HB) in which the charge concentrations are separately localized in the basin of the neighbouring atoms [21]. This kind of analysis leads to the proposition of criteria for characterizing hydrogen bonds (see below). Figure 5 shows bond critical points of covalent bonds (located, for instance, between two carbon atoms), while those with numbers 17, 43, 50 and 59 (located between hydrogen and oxygen atoms) correspond to hydrogen bond critical points (HBCP) where the gradient ∇ρ vanishes, being (3,−1) saddle points.
When  2  < 0, the electronic charge is locally concentrated, this being the case in covalent bonds [20]. When  2  > 0, the electronic charge is locally depleted [13], the interactions are called closed-shell. This occurs in hydrogen bonds (HB) in which the charge concentrations are separately localized in the basin of the neighbouring atoms [21]. This kind of analysis leads to the proposition of criteria for characterizing hydrogen bonds (see below). Figure 5 shows bond critical points of covalent bonds (located, for instance, between two carbon atoms), while those with numbers 17, 43, 50 and 59 (located between hydrogen and oxygen atoms) correspond to hydrogen bond critical points (HBCP) where the gradient  vanishes, being (3,−1) saddle points. In Figure 3, the contour lines of the electronic density around the water oxygen (O w ) basin resembles a Micky Mouse profile. This is a consequence in which the two hydrogen atoms of water (H w ) form covalent bonds with O w . In other words, O w behaves as an HB donor. The basins of O 7 of the hydroxy O 7 -H groups clearly have a circle shape. Similarly, Figure 4 shows the contour lines of the hydrogen bonds between water and the two O 12 -H hydroxy groups. The plane in the figure is defined by these three oxygen atoms. Now, the contour around the O w shows a basin while the profiles around the O 12 -H groups resemble those of peanuts. Now, the two O 12 are donors and O w is the acceptor. Furthermore, the bond paths of the four hydrogen bonds link the expected two atoms, the hydrogen and the acceptor. It is evident that the first condition of the criteria for characterizing a hydrogen bond is fulfilled. These criteria have been published by Koch and Popelier [22] and resumed by Popelier [20] as a table.
The second condition refers to the electron density at HBCP, b. According to Popelier [20], the b values should be in the range 0.002-0.035 au. Table 1 shows the calculated values for the four HB, all of them being within the expected range. These values are about one order of magnitude smaller than those found for a covalent bond (b = 0.391 au, for O-H in H2O) [23]. On the other hand, it may be noticed that the values when water is the donor are almost double than when it is the acceptor. In Figure 3, the contour lines of the electronic density around the water oxygen (O w ) basin resembles a Micky Mouse profile. This is a consequence in which the two hydrogen atoms of water (H w ) form covalent bonds with O w . In other words, O w behaves as an HB donor. The basins of O 7 of the hydroxy O 7 -H groups clearly have a circle shape. Similarly, Figure 4 shows the contour lines of the hydrogen bonds between water and the two O 12 -H hydroxy groups. The plane in the figure is defined by these three oxygen atoms. Now, the contour around the O w shows a basin while the profiles around the O 12 -H groups resemble those of peanuts. Now, the two O 12 are donors and O w is the acceptor. Furthermore, the bond paths of the four hydrogen bonds link the expected two atoms, the hydrogen and the acceptor. It is evident that the first condition of the criteria for characterizing a hydrogen bond is fulfilled. These criteria have been published by Koch and Popelier [22] and resumed by Popelier [20] as a table.
The second condition refers to the electron density at HBCP, ρ b . According to Popelier [20], the ρ b values should be in the range 0.002-0.035 au. Table 1 shows the calculated values for the four HB, all of them being within the expected range. These values are about one order of magnitude smaller than those found for a covalent bond (ρ b = 0.391 au, for O-H in H 2 O) [23]. On the other hand, it may be noticed that the values when water is the donor are almost double than when it is the acceptor.
There is a correlation between the O-O length and ρ b : the shorter the former, the higher the latter. Relationships between the hydrogen bond length and electron density have been published [24,25]. The values obtained here differ by less than ±0.004 au from those obtained using the equation [25] (r O...H in Å, from crystal data).
A third criterion refers to the Laplacian of the charge density evaluated at the bond critical point, where charge density is a local minimum along the bond path, i.e., ρ b is locally depleted with respect to neighbouring points along the bond path. The range values ( Table 2)   The water dimer is a system of two water molecules bound by a single hydrogen bond. Because of this, it is often used as the paradigmatic system, being the object of many experimental and theoretical investigations [26,27]. The equilibrium geometry and dissociation energy of the water dimer are well known. The dimer has a "trans-linear" structure, and the O-O distance was first measured from the microwave spectrum by Dyke et al. [28][29][30], the value being r OO = 2.98 ± 0.04 Å. The O-H distances depend on the role of the water molecules, whether they are the donor (r OH = 95.8 pm Å) or the acceptor (r OH = 0.95 Å) [26].
Previous ρ b and ∇ 2 ρ b values may be compared with those for the water dimer, H-O-H . . . OH 2 . Bader et al. [23] ascertained that ρ b (au) and ∇ 2 ρ b are 0.0199 and 0.0624 (data in au), respectively (Table 3). From the four values of ρ b obtained for the pseudo C-H 2 O-C crystal structure (Table 2), an average value of 0.020 au was obtained, which perfectly matches the one for the water dimer. The ∇ 2 ρ b value for H-O-H . . . OH 2 is closer to those in which the oxygens (O 12 ) of hydroxy groups are donors and O w is the acceptor. It should be noted that, in these two cases, the r OO lengths are also closer to the one of the H-O-H . . . OH 2 dimer. Other published values for the water dimer are shown in Table 3.
Another criterion consists in the estimation of the mutual penetration of the hydrogen (H) and acceptor atom (B, oxygen) upon hydrogen bond formation. In the literature, this criterion is often considered as a necessary and sufficient condition for classifying the intermolecular interaction as hydrogen bonding [36]. It is estimated as ∆r i = r i − r o i (i are the atoms involved in the hydrogen bond, B or H), the superscript referring to nonbonded radii, r o i , of these atoms and its absence the bonded radii, r i [22]. The nonbonded radius is the distance from a nucleus to a given electron density contour (usually 0.001 au) in the absence of interaction. This value is used because this yields atomic diameters in good agreement with gas-phase van der Waals radii [22]. The bonded radius is the distance from a nucleus to the bond critical point in question.  0.0199 0.0624 −5.5 [23] 1.949 −4.45 [31] 1.825 −10.97 [31] 0.023 0.091 1.950 −4.45 [32] 0.017 0.075 2.056 −4.25 [32] 0.0259 −4.93 [33] −4.96 [33] −4.32 ± 0.31 [34] 0.0219 0.0396 1.949 −4.33 [35] It should be noted that Isaev defined [36]  Remaining criteria proposed by Popelier [20] are not analysed here. They involve the analysis of properties in which the hydrogen atom is the focus of the analysis: increased net charge, energetic destabilization, decrease in dipolar polarization, and decrease in the atomic volume.

Energy of Hydrogen Bonds
The energy of hydrogen bond interactions lie between covalent bonds and weak van der Waals interactions, although there are no sharp borders between the three types of interactions [11]. Jeffrey [38] defined three categories of hydrogen bonds according to the energy involved: weak 0.1-1, moderate 5-15 and strong 20-60 (data in kcal/mol), but other classifications have been published. For instance, Emamian et al. [33] proposed four ranges for the interactions: very weak (>−2.5), weak to medium (−2.5 to −14), medium (−11.0 to −15) and strong (<15.0)-all data in kcal mol −1 .
Ruscic [39], from a new partition function for water, obtained dissociation enthalpy values for the water dimer, the values being 13.220 ± 0.096 kJ mol −1 and 15.454 ± 0.074 kJ mol −1 (3.693 ± 0.018 kcal mol −1 ) at 0 K and 298.15 K, respectively. The experimental result determined from the thermal conductivity of the vapor was −3.59 ± 0.5 kcal/mol [40], and Rocher-Casterline et al. [41] used state-to-state vibrational predissociation measurements following excitation of the bound OH stretch fundamental of the donor unit of the dimer to obtain an accurate value of 13.2 ± 0.12 kJ/mol for the dissociation energy. This value was compared with the theoretical one of 13.2 ± 0.05 kJ/mol, obtained by Shank et al. [42].
Some other values are shown in Table 3. As the complexing moieties of the water molecule are hydroxy groups, we also recompiled the values of the methanol-water system that was theoretically studied by Moin et al. [43] For a series of hydrogen-bonded complexes between nitrites and hydrogen chloride, Boyd and Choi [44] noticed a correlation between the electron density at the HBCP ρ(r b ) and the energy of the hydrogen bond. The energies ranged from 10 kJ/mol (NCCN . . . HC1) to 38 kJ/mol (LiCN . . . HCl) while the range of ρ(ρ(r b ) was 0.01103-0.02391. Many other equations have been proposed, and Rozenberg [45] reviewed the subject. After analysing 24 equations, he obtained the following relationship: The application of this equation to the present system provides the results shown in Table 4. Once E HB has been calculated for each HB, the total energy of the four hydrogen bonds was obtained by summation, the value being −16.951 kcal mol −1 , although its standard deviation was very high. By using different equations, the range of calculated values was from −11.8 to −28.5 kcal/mol. Ganthy et al. [46] carried out an analysis of the hydrogen bond of the water dimer but with the geometry found in ice, and compared it with the one in the gas phase. Let us recall that the nearest neighbour O-O distance in ice is 2.75 Å compared to 2.98 Å in a gas phase dimer. As we previously noticed, these two values are close to the values observed in our C-suc-C crystal (  The value calculated for the C-H 2 O-C system is −12.67 kcal/mol, although it cannot be exclusively ascribed to hydrogen bond interactions. However, it is the most important contribution to the interactions existing in the crystal. The value is higher than that predicted using the average in Equation (1), which probably is too low, as the value calculated for H-OH . . . OH 2 suggests, when compared with the most accepted experimental and theoretical ones.

Conclusions
There are two main oxygen-oxygen (r OO ) distances when a hydrogen bond is formed between water molecules, the one observed in the gas phase in the formation of a dimer (r OO = 2.98 Å) and the one in ice (r OO = 2.75 Å). Both lengths are observed in the C-suc-C crystal, in which a water molecule is encapsulated by four hydroxy groups belonging to two cholic acid dimers. The shorter one corresponds to hydrogen bonds in which the water oxygen is donor and the larger one when it is acceptor. The application of the AIM theory to a simplified system (C-H 2 O-C) confirms the existence of saddle critical points (HBCP) in all these four hydrogen bonds. The estimated interaction energy in the formation of the complex (−12.67 kcal/mol) is in adequate agreement with the summation of the energies of each hydrogen bond (E HB ) estimated from the electron density (ρ b ) of HBCP and published equations for E HB -ρ b linear relationships.