**2. Results and Discussion**

The geometric parameters (interatomic distances *r*1, *r*2, *r*<sup>3</sup> and *r*4) of hydrogen bonds in investigated complexes are collected in Table S1. The set of complexes covers a wide range of hydrogen bonds geometries—there are complexes without proton transfer in both hydrogen bonds (OH···O−PyrN···HO), complexes with proton transfer in one of the hydrogen bonds (O−···HOPyrN···HO or OH···OPyrNH···O−) and complexes with two hydrogen bonds with proton transfer (O−···HOPyrNH+···O−). The dependencies of the *q*2*<sup>a</sup>* = *r*<sup>1</sup> + *r*<sup>2</sup> coordinate on *q*1*<sup>a</sup>* = 0.5·(*r*<sup>1</sup> − *r*2) for the OHO hydrogen bond and the *q*2*<sup>b</sup>* = *r*<sup>4</sup> + *r*<sup>3</sup> coordinate on *q*1*<sup>b</sup>* = 0.5·(*r*<sup>4</sup> − *r*3) for the OHN hydrogen bond are shown in Figure 3. It is seen that the complexes cover a range of *q*<sup>1</sup> from ca. −0.4 to ca. 0.35 Å forming a parabolic curve.

**Figure 3.** Dependencies of the (**a**) *q*2*<sup>a</sup>* = *r*<sup>1</sup> + *r*<sup>2</sup> coordinate on *q*1*<sup>a</sup>* = 0.5·(*r*<sup>1</sup> − *r*2) for OHO hydrogen bond; dependencies of the (**b**) *q*2*<sup>b</sup>* = *r*<sup>4</sup> + *r*<sup>3</sup> coordinate on *q*1*<sup>b</sup>* = 0.5·(*r*<sup>4</sup> − *r*3) for OHN hydrogen bond. Dashed lines are guides for the eye.

Despite the fact that OHO and OHN hydrogen bonds are divided in space by 4 hydroxypyridine anion, they "feel" the presence of each other through the electronic system of the complex as a whole, the phenomenon is called cooperativity. It manifests as the changes of the proton position in one hydrogen bond depending on the proton position in another hydrogen bond. For example, if one fixes the substituents R1, R2 and

R3 in the proximity of the OHO hydrogen bond (see a set of points of the same shape and color in Figure 4), the variation of substituents in OHN hydrogen bond causes a change in *q*1a. Complexes with a "tail-to-tail" (OH···O−PyrN···HO) and a "head-to-head" (O−···HOPyrNH+···O−) configuration are anti-cooperative (blue areas in Figure 4): i.e., the strengthening of one hydrogen bond causes the weakening of another hydrogen bond. For complexes with "head-to-tail geometry" (O−···HOPyrN···HO and OH···OPyrNH···O−) (green areas in Figure 4) cooperative effects are observed: i.e., the strengthening of one hydrogen bond causes the strengthening of another. Thus, the character of mutual influence of geometries of OHO and OHN hydrogen bonds in such a system can be either cooperative or anti-cooperative.

**Figure 4.** The dependence of proton position in the NH···O hydrogen bond *q*1b on the proton position in the OH···O hydrogen bond *q*1a. The shape and color of a marker indicate a series of complexes with the same set of substituents *R*1, *R*<sup>2</sup> and *R*<sup>3</sup> (shown in legend) and varying set of substituents *R*4, *R*<sup>5</sup> and *R*6. Green areas correspond to cooperative hydrogen bonds and blue areas to anti-cooperative ones.

Among the NMR parameters of the 4-hydroxypyridine anion, which potentially are sensitive to the geometries of OH···O and OH···N hydrogen bonds, chemical shifts of the nuclei located in the proximity of hydrogen bonds, i.e., carbon (C1) and nitrogen (N4), are the first candidates. The changes of carbon and nitrogen chemical shifts upon complexation, Δ*δ*C1 and Δ*δ*N4, are shown in Figure 5.

In Figure 5, it is clearly seen that the C1 carbon chemical shift changes with the change of geometries of both hydrogen bonds—upon moving the bridging proton along OHO hydrogen bond (with fixed geometry of OHN hydrogen bond), Δ*δ*C1 changes by up to 8 ppm, and moving it along the OHN hydrogen bond causes a change of Δ*δ*C1 by up to 2.5 ppm. In other words, the steepness of the slope of the Δ*δ*C1(*q*1a, *q*1b) surface is larger in the direction of *q*1a than in the direction of *q*1b. Contrarily, the nitrogen chemical shift is more sensitive to the geometry of the closest (OHN) hydrogen bond. The shape of the isolines of the distributions of Δ*δ*C1 and Δ*δ*N4 makes the unequivocal solving of the inverse spectral problem possible for most cases (except for *q*1a ≈ −0.3 Å due to the shape of the Δ*δ*C1 isolines in this region caused by a presence of a hill with its maximal value). Together with the high sensitivity discussed above, this makes the pair Δ*δ*C1 and Δ*δ*N4 quite promising parameters. However, the experimental issues of measuring nitrogen chemical shift (low natural abundance of 15N and line broadening due to quadrupolar interactions in 14N NMR spectra) encourage us to discuss additional NMR parameters.

**Figure 5.** Distributions of the change upon complexation of chemical shift of (**a**) the C1 atom Δ*δ*C1 and (**b**) the N4 atom Δ*δ*N4 along *q*<sup>1</sup> coordinates for OHO (*q*1a) and OHN (*q*1b) hydrogen bonds. The coefficients *a*, *b*1, *c*1, *d*1, *b*2, *c*<sup>2</sup> and *d*<sup>2</sup> (Equation (1)) and *R*<sup>2</sup> are given in Table S2. Isolines are drawn with a step of 0.5 and 10 ppm, respectively.

The next pair of "promising" NMR parameters are the changes of chemical shifts of bridging protons in the OHO and OHN hydrogen bonds upon complexation, Δ*δ*Ha and Δ*δ*Hb; their dependences on *q*1a and *q*1b are shown in Figure 6. Along the OHO hydrogen bond, Δ*δ*Ha increases from 8 to 17 ppm for *q*1a < 0, reaches maximal value at *q*1a ≈ 0 and then decreases to 0 ppm. The change of the geometry of the OHN hydrogen bond (*q*1b) does not significantly influence Δ*δ*Ha. A similar situation is observed for Δ*δ*2b: moving along the OHN hydrogen bond (*q*1b) changes Δ*δ*Hb from 9 to 18 ppm for *q*1b ≈ 0 and then down to 0 ppm. It is clearly seen that both surfaces have a hill of maximal values of Δ*δ*<sup>H</sup> and slopes in the direction of one of *q*<sup>1</sup> axes (*q*1a for Δ*δ*Ha and *q*1b for Δ*δ*Hb). It means that the magnitude of the bridging proton chemical shift in a given hydrogen bond is determined almost exclusively by the geometry and electronic features of the three atoms forming this bond. In other words, the chemical shifts of the two bridging protons are not coupled. For the purposes of solving the reverse spectral problem, this is an advantage, because the

magnitude of each chemical shift can be used for the independent evaluation of hydrogen bond geometries.

**Figure 6.** Distributions of the change upon complexation of chemical shift of bridging protons (**a**) in the OH···O hydrogen bond Δ*δ*Ha and (**b**) in the OH···N hydrogen bond Δ*δ*Hb along *q*<sup>1</sup> coordinates for OHO (*q*1a) and OHN (*q*1b) hydrogen bonds. The coefficients *a*, *b*1, *c*1, *d*1, *b*2, *c*<sup>2</sup> and *d*<sup>2</sup> (Equation (1)) and *R*<sup>2</sup> are given in Table S2. Isolines are drawn with a step of 2 ppm.

However, there are two principal problems with such an approach. The first problem was mentioned in introduction and is caused by the fact that a particular value of one of the proton chemical shifts (Δ*δ*Ha or Δ*δ*Hb) corresponds to an isoline in distributions shown in Figure 6a,b that form a hairpin curve. Two hairpin isolines have four intersecting points (see Figure 7), i.e., each set of Δ*δ*Ha and Δ*δ*Hb values could correspond to four alternative hydrogen bond geometries. The second problem with using Δ*δ*Ha and Δ*δ*Hb for the evaluation of hydrogen bond geometries is due to the fact that, in the experimental spectrum, it is difficult to distinguish which signal will relate to the OHO and which to the OHN hydrogen bond. This issue demands the usage of additional NMR parameters, for example, chemical shifts of the nuclei of 4-hydroxypyridine.

**Figure 7.** Schematic representation of the non-unequivocal inverse spectral problem solving. The black dots correspond to the four possible solutions. NMR parameters of the 4-hydroxypyridine anion, the chemical shift of carbon *δ*C2, *δ*C3, *δ*C5, *δ*C6 and the hydrogen atoms *δ*H2, *δ*H3, *δ*H5, *δ*H6, were also analyzed for their sensitivity and applicability for the evaluation of the geometries of OHO and OHN hydrogen bonds. We found that the most promising parameters are arithmetically averaged chemical shifts of C2 and C6 atoms (*δ*C26), C3 and C5 atoms (*δ*C35) and H3 and H5 atoms (*δ*H35). The distributions of changes of these parameters are shown in Figure 8. Both carbon chemical shifts, Δ*δ*C26 and Δ*δ*C35, are more sensitive to the closest hydrogen bond (OHO and OHN, respectively). The range of values of Δ*δ*C26 was about 10 ppm, Δ*δ*C35—15 ppm, which makes both of them suitable for the accurate evaluation of hydrogen bond geometries. The change of *δ*H35 within the change of the OHO and OHN hydrogen bonds geometries slightly exceeded 1 ppm. The topology of these three surfaces is such that almost all combinations of pair of isolines of Δ*δ*C26, Δ*δ*C35 and Δ*δ*H35 have a single intersection point, thus making the solution of an inverse spectral problem unequivocal.

For an additional estimation of the mutual influence of OHO and OHN hydrogen bonds, calculations for two extra sets of systems with a single hydrogen bond were performed (4-hydroxypyridine anion with one substituted methanol from either the oxygen or nitrogen side, 10 complexes in each extra set). The results the of NMR calculations (*δ*C26 and *δ*C35) are presented in Figures S1 and S2 of the Supporting Information. The difference plot between data shown in Figure 8 (system with two coupled hydrogen bonds) and Figures S1 and S2 (hypothetical system with no coupling between hydrogen bond modeled as a sum of systems with a single hydrogen bond) is shown in Figure S3. It is clearly seen that the cooperativity effects on the NMR parameters are substantial (±2–4 ppm, making them non-negligible) and non-monotonous. The strongest effects are observed for the configuration O−···HOPyrNH+···O−.

In order to test the approach proposed in this work, we performed additional calculations of structure and NMR parameters for three arbitrarily chosen complexes of 4 hydroxypyridine anion with CH3CFHOH and CF3CFHOH molecules: **1** (CH3CFHOH···<sup>−</sup> OPyrN···HOCFHCF3), **2** (CF3CFHOH···−OPyrN···HOCFHCF3) and **3** (CF3CFHOH···<sup>−</sup> OPyrN···HOCFHCH3); the optimized structures of **1**–**3** are shown in Figure 9. The results of the comparison of the "predicted" geometry based on pairs of NMR chemical shifts and the "real" (calculated) geometries of the two hydrogen bonds are summarized in Table 2 and shown in Figures S4–S6 in the Supporting Information. It can be concluded that hydrogen bond geometries estimated using two-dimensional correlations differ from those directly calculated by quantum-chemical methods by not more than 0.04 Å. The only exception is complex **3** (CF3CFHOH and CF3CFHOH), for which the chemical shifts of carbons C2 and C6 differ significantly due to the presence of an additional hydrogen bond between the CH-group of the 4-hydroxypyridine anion and the fluorine atom of the substituted methanol (see Figure S7). In this case, using the chemical shift of carbon atom not involved

in additional hydrogen bonding instead of the arithmetically averaged chemical shift Δ*δ*C26 is more appropriate.

**Figure 8.** Distributions of the change upon complexation of average chemical shift of (**a**) C2 and C6 atoms Δ*δ*C26, (**b**) C3 and C5 atoms Δ*δ*C35 and (**c**) H3 and H5 atoms Δ*δ*H35 along *q*<sup>1</sup> coordinates for OHO (*q*1a) and OHN (*q*1b) hydrogen bonds. The coefficients *a*, *b*1, *c*1, *d*1, *b*2, *c*<sup>2</sup> and *d*<sup>2</sup> (Equation (1)) and *R*<sup>2</sup> are given in Table S2. Isolines are drawn with a step of 0.5 ppm for (**a**,**b**) and 0.05 ppm for (**c**), respectively.

**Figure 9.** Structures of additional complexes with two hydrogen bonds formed by 4-hydroxypyridine anion as a hydrogen bond acceptor and two substituted methanols as donors used for testing the proposed approach. From top to bottom: **1** (proton donors CH3CHFOH, CF3CHFOH), **2** (proton donors CH3CHFOH, CF3CHFOH), **3** (proton donors CF3CHFOH, CH3CHFOH).

**Table 2.** Results of testing the proposed approach for three additional complexes, shown in Figure 8: **1** (proton donors CH3CHFOH, CF3CHFOH), **2** (proton donors CH3CHFOH, CF3CHFOH), **3** (proton donors CF3CHFOH, CH3CHFOH).



**Table 2.** *Cont.*

## **3. Computational Details**

Geometry optimization was performed using second-order Moller–Plesset perturbation theory (MP2) [12,13] with Dunning' correlation-consistent polarized double-ζ basis set with diffuse functions aug-cc-pVDZ [14]. All calculated geometries were checked for the absence of imaginary frequencies. Chemical shieldings were calculated using DFT (B3LYP) with the augmented polarization consistent triple-ζ basis set aug-pcS-2, which is specially designed for the calculation of shieldings at the DFT level with a high accuracy [18].

Calculations were carried out using the Gaussian16 software. Computational resources were provided by the Computer Center of Saint Petersburg University Research Park (http://www.cc.spbu.ru/, accessed on 1 April 2020).

Visualization was performed in GaussView 6.0 and MATLAB 2021b software packages. For the description of the geometries of OHO and OHN hydrogen bonds, the parameters *q*<sup>1</sup> and *q*<sup>2</sup> were used [2,16,17]: *q*<sup>1</sup> = 0.5·(*rOH* − *rHY*), *q*<sup>2</sup> = *rOH* + *rHY*, where Y = O, N. The meaning of *q*<sup>1</sup> and *q*<sup>2</sup> coordinates is pretty clear for linear hydrogen bonds—the *q*<sup>1</sup> coordinate is the shift of the hydrogen atom from the hydrogen bond center, the *q*<sup>2</sup> coordinate is the total length of the hydrogen bond (for strictly linear hydrogen bonds, *q*<sup>2</sup> is the distance between the heavy atoms O...O or O...N; it should be noted that, as hydrogen bonds deviate from linearity, *q*<sup>2</sup> loses this geometrical meaning, because in this case *q*<sup>2</sup> is slightly longer than the distance O...O or O...N). For the OHO hydrogen bond, the following pair of coordinates was used *q*1*<sup>a</sup>* = 0.5·(*r*<sup>1</sup> − *r*2) and *q*2*<sup>a</sup>* = *r*<sup>1</sup> + *r*<sup>2</sup> and for OHN, they were *q*1*<sup>b</sup>* = 0.5·(*r*<sup>4</sup> − *r*3) and *q*2*<sup>b</sup>* = *r*<sup>4</sup> + *r*3, respectively.

The algorithm of construction for the distributions of spectral parameters (Δ*δ*C1, Δ*δ*N4 and others discussed in this work) along the *q*1*<sup>a</sup>* and *q*1*<sup>b</sup>* was as follows. For a set of complexes shown in Figure 1, NMR parameters were calculated. Let us denote the spectral parameter as *f*. The value of a change of a spectral parameter upon complexation is defined as Δ*f* = *fcomplex* − *f f ree* (where *f f ree* is the value of a given spectral parameter for an isolated 4-hydroxypyridine anion or methanol, and *fcomplex* is the value of the same parameter within a hydrogen-bonded complex). Calculated Δ*f* values were approximated as a function of *q*1*<sup>a</sup>* and *q*1*<sup>b</sup>* using the Curve Fitting Tool implemented in the MATLAB 2021b software package by a polynomial function of a third-degree along *q*1*<sup>a</sup>* and *q*1*<sup>b</sup>* without cross-terms:

$$
\Delta f(q\_{1a}, q\_{2b}) = a + b\_1 q\_{1a} + c\_1 q\_{1a}^2 + d\_1 q\_{1a}^3 + b\_2 q\_{1b} + c\_2 q\_{1b}^2 + d\_2 q\_{1b}^3 \tag{1}
$$

The coefficients *a*, *b*1, *c*1, *d*1, *b*2, *c*<sup>2</sup> and *d*<sup>2</sup> for all discussed parameters and the coefficient of determination *R*<sup>2</sup> for each approximation are given in Table S2 in the Supporting Information. The resulting functions were plotted as contour plots, in which each isoline

corresponds to the particular value of Δ*f* and is marked by a color and an isovalue for clarity in the following figures. The polynomials were taken as one of the simplest forms that could describe the strongly non-monotonous behavior of spectral parameters. There is no deeper reason for this choice, and for the fitting purposes, other functions could be selected if the data points would allow doing so.

## **4. Conclusions**

In this work, the possibility of solving the inverse spectral problem for a system with two coupled hydrogen bonds was demonstrated on the example of 61 complexes formed by 4-hydroxypyridine anion with two substituted methanols with OHO and OHN hydrogen bonds. The algorithm for using two-dimensional correlations is as follows: for measured or calculated change of two given spectral parameters, one should find isolines on their corresponding plots and overlap them (see Figure 2). The coordinates of the intersection point of the two isolines will give the proton positions in both hydrogen bonds. It was demonstrated that any pair of parameters Δ*δ*N4, Δ*δ*C1, Δ*δ*C26, Δ*δ*C35 and Δ*δ*H35 is suitable for the evaluation of hydrogen bond geometries. However, Δ*δ*C1, Δ*δ*C26, Δ*δ*C35 and Δ*δ*H35 are more easily available from an experiment.

Thus far, we have only tested our approach computationally on three examples of complexes. The applicability of the method "in practice" awaits experimental verification by future researchers. Indeed, the proposed approach has obvious limitations. Firstly, such 2D maps are constructed for each compound individually (in our case for 4-hydroxypyridine anion). Secondly, the experimental application of the approach requires the measurements of signals that are not averaged out by fast (in the NMR time scale) molecular exchange—a condition that is not necessarily satisfied for intermolecular complexes. The coexistence of several types of molecular complexes in a solution at the same time can also complicate the solving of the reverse spectral problem. Thirdly, the accuracy of the 2D chemical shift maps worsens in the presence of additional non-covalent interactions between the proton donors and the central proton-accepting molecule. Fourthly, for other systems, multiple extrema in the 2D maps of spectral parameters can make determination of hydrogen bonds geometries non-unequivocal, as shown in Figure 7. An unfortunate combination of all four factors could, of course, render the proposed approach totally unreliable. Nevertheless, we have shown in principle that, for systems where these limitations are absent or can be neglected, the NMR spectral data can be sufficient to solve the two-dimensional reverse spectral problem even with a significant entanglement of spectral parameters.

The accuracy of geometry estimations with this approach for systems with two OHO and OHN hydrogen bonds (and without additional interactions) is in the range ±0.04 Å.

**Supplementary Materials:** The following supporting information can be downloaded at: https:// www.mdpi.com/article/10.3390/molecules27123923/s1, Table S1:Geometric parameters of OHO and OHN hydrogen bonds; Table S2: Coefficients *a*, *b*1, *c*1, *d*1, *b*2, *c*<sup>2</sup> and *d*<sup>2</sup> for each discussed spectral parameter, *R*<sup>2</sup> for each approximation; Figure S1:Δ*δ*C35 for complexes of hydroxypyridine anion with a single substituted methanol molecule from an oxygen side (OHO hydrogen bond) Δ*δ*C35a and from a nitrogen site (OHN hydrogen bond) Δ*δ*C35b. Distributions of the sum Δ*δ*C35 = Δ*δ*C35a + Δ*δ*C25b along *q*1a and *q*1b hydrogen bonds; Figure S2: Δ*δ*C26 for complexes of hydroxypyridine with a single substituted methanol molecule from an oxygen side (OHO hydrogen bond) Δ*δ*C26a and from a nitrogen site (OHN hydrogen bond) Δ*δ*C26b. Distributions of the sum Δ*δ*C26\_ = Δ*δ*C26a + Δ*δ*C26b along *q*1a and *q*1b hydrogen bonds; Figure S3: Δ*δ*C26–Δ*δ*C26, Δ*δ*C35–Δ*δ*C35\_ along *q*1a and *q*1b; Figure S4: Solution of the inverse spectral problem for *R*<sup>1</sup> = H, *R*<sup>2</sup> = F, *R*<sup>3</sup> = CH3, *R*<sup>4</sup> = H, *R*<sup>5</sup> = F, *R*<sup>6</sup> = CF3; Figure S5: Solution of the inverse spectral problem for *R*<sup>1</sup> = H, *R*<sup>2</sup> = F, *R*<sup>3</sup> = CF3, *R*<sup>4</sup> = H, *R*<sup>5</sup> = F, *R*<sup>6</sup> = CF3; Figure S6: Solution of the inverse spectral problem for *R*<sup>1</sup> = H, *R*<sup>2</sup> = F, *R*<sup>3</sup> = CF3, *R*<sup>4</sup> = H, *R*<sup>5</sup> = F, *R*<sup>6</sup> = CH3; Figure S7: Optimized geometry of complex 3 with CF3CFOH and CH3CFHOH as hydrogen bond donors.

**Author Contributions:** Conceptualization, P.M.T. and M.V.S.; methodology, E.Y.T.; validation, P.M.T. and M.V.S.; formal analysis, E.Y.T. and M.V.S.; investigation, E.Y.T. and M.V.S.; data curation, E.Y.T.; writing—original draft preparation, E.Y.T.; writing—review and editing, P.M.T. and M.V.S.; visualization, E.Y.T.; funding acquisition, P.M.T. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Russian Science Foundation grant number 18-13-00050.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data is contained within the article or Supplementary Material.

**Acknowledgments:** This study was supported by the Russian Science Foundation.

**Conflicts of Interest:** The authors declare no conflict of interest.

**Sample Availability:** Samples of the compounds are not available from the authors.
