**3. Results and Discussion**

*3.1. Spectroscopic and Metric Parameters Associated with Intermolecular Hydrogen Bonds: Gas Phase vs. Crystalline Phase in the Light of Car–Parrinello Molecular Dynamics (CPMD) and Path Integral Molecular Dynamics (PIMD)*

The gas phase as well as the crystalline phase simulations were carried out for 2,6 difluorobenzamide, 5-hydroxyquinoline and 4-hydroxybenzoic acid to investigate the intermolecular hydrogen bonds. Car–Parrinello molecular dynamics and Path Integral molecular dynamics allowed for the quantitative and qualitative description of the spectroscopic and geometric features of the studied compounds. In Figure 1, the models used for the crystalline phase (based on crystal structures (A) 919101, (B) 908102 and (C) 908103 [51]) and gas phase molecular dynamics simulations are presented. In order to obtain a full spectroscopic description of the aforementioned models, all simulations were carried out in the gas and in the crystalline phases. It is worth emphasizing that the results obtained in vacuo served as a reference for the O-H and N-H stretching discussion. On the basis of the Fourier transform of the atomic velocity autocorrelation function, the classical vibrational spectrum was obtained (see Figure 3). A major advantage of the methodology used is the ability to estimate the individual contributions of selected atoms to the entire spectrum—these are shown in the third column of the Figure 3. Due to the characteristics of the CPMD method (nuclei dynamics is inherently classical), the Fermi resonance, and thus the splitting of the bands of nearly identical energies and symmetries as well as tunneling phenomena, cannot be observed. Moreover, the classical amplitudes of motion at 297 K allow for sampling a narrower part of the potential energy surface than the true quantum particle with its nuclear wavefunction delocalization. This leads in many cases to the underestimation of the anharmonicity and is another factor of deviation between the CPMD-derived X-H stretching position and the experimental spectrum. It is therefore a good idea to concentrate not on the absolute wavenumbers, but on the shifts between the hydrogen-bonded protons and free protons. In the case of the results shown in Figure 3 (see the first and the second panels), the presence of the two regions characterized by the increased intensities can be seen: the region of the deformation vibrations, from ca. 500 to 1800 cm−<sup>1</sup> in the solid and in the gas phase, as well as the region of stretching vibrations that extend from 2800 to 3700 cm−1. The former can be attributed to the heavy atom oscillations, whereas the latter is the signature region for the protons, including those involved in the hydrogen bond formation. The third panel corresponds to the particular protons involved in the HBs in the crystalline phase. In addition, the gas phase results presented in the third panel show the characteristic, sharp stretching modes of O-H, N-H and C-H not involved in the hydrogen bonding. For (A), one may observe the blueshift of the C-H stretching vibration (from ca. 3050 to 3150 cm−1) and thus CD-H. . . OA can be regarded as an anti-hydrogen bond [6,94]. In the case of ND-H. . . OA, it is visible that the N-H band for the solid phase is shifted towards lower wavenumber values (redshift). This is evidence for the ND-H weakening and the accompanying contraction and strengthening of the ND-H. . . OA bond, which can be interpreted as the charge transfer from the proton acceptor to the antibonding

orbital of the ND-H and as a sign of the hydrogen bond formation. In the case of (B) and (C), similar observations can be made—the O-H stretching is redshifted in the solid phase, when compared to the gas phase. In summary, the spectroscopic features obtained allow us to conclude that the hydrogen bonds were formed in each of the compounds studied in the solid state.

**Figure 3.** Atomic velocity power spectra obtained from the CPMD simulations. Left panel: the whole atomic spectra in the gas phase. Middle panel: the whole atomic spectra in the crystalline phase. Right panel: the contribution of the bridged protons in the crystalline and in the gaseous phases. (**A**) 2,6-difluorobenzamide, (**B**) 5-hydroxyquinoline and (**C**) 4-hydroxybenzoic acid.

The analysis of the Car–Parrinello (CPMD) and path integral molecular dynamics (PIMD) trajectories enables one to obtain a probability distribution of bridge proton positions in the studied compounds (see Figure 4). The characteristics of proton motion in all of hydrogen bridges highlighted in Figure 1 were obtained. Considering the impact of the NQEs on the proton behavior, one can observe that, in comparison with CPMD, the distance between the proton donor and its acceptor is insignificantly shortened for (A) (decrease of ca. 0.1 Å in each case). The same observation can be made for the ND-H and CD-H values—in these cases, the quantization of the nuclei does not change the proton dynamics. Indeed, both aforementioned hydrogen bonds cannot be regarded as strong hydrogen bonds—in the first case (A, ND), the proton donor is an amide group, with a behavior strongly altered by the presence of two fluorine substituents in -ortho positions. In the second case, the proton donor is the aromatic carbon atom and thus the HB can be considered weak (this is an example of the blue-shifting hydrogen bond, as we argued in the previous section). Different dynamical characteristics of the bridged proton were noticed for 5-hydroxyquinoline as well as the OD1 of the 4-hydroxybenzoic acid and quinoxaline co-crystal. For (B), one can observe that the proton mobility increased significantly, when the dynamics was performed in the PIMD scheme. On the contrary, the NQEs for the (C, OD1) provided further stabilization of the hydrogen atom at the donor side; in this case, the CPMD simulations indicated that the proton is more delocalized. Here, proton transfer, from the donor to the acceptor site in the hydrogen bond, has occurred, which can be easily discerned by the presence of the distinctive "tail" that extends towards larger r(OD-H) values. In the case of OD2-H. . . OA hydrogen bond (C, OD2), the only difference concerns the

negligible shortening of OD2. . . OA distance in PIMD, when compared to the CPMD. For (C, OD3), one may observe that the NQEs induce minimal r(OD3-H) covalent bond elongation and the accompanying contraction of the r(OD3. . . NA) distance. Summarizing, it can be noted that the inclusion of the NQEs for the presented set of compounds results in changes in the quantitative as well as qualitative nature. The latter are especially pronounced for (B), where the free energy surface was sampled by the proton more efficiently and for (C, OD1) case, where the NQEs inclusion resulted in a more localized behaviour of the proton in the hydrogen bridge. In all studied cases, with the exception of the blue-shifting (A, CD) bond, a decrease in the donor–acceptor distance is accompanied by increased proton delocalization and sharing.

**Figure 4.** Histograms presenting the relationship between the length of the corresponding donor– proton covalent bonds as well as the distance between the proton and its acceptor. CPMD vs. PIMD in the crystalline phase. (**A**) 2,6-difluorobenzamide, (**B**) 5-hydroxyquinoline and (**C**) 4 hydroxybenzoic acid. ND, OD and CD indicate proton donors. Probability density isocontours drawn at the 1 Å−<sup>2</sup> value.

The structures of the dimers discussed in this section were extracted from the 919,101 and 908,103 deposits in the CCDC database [51]. CPMD, as well as PIMD calculations, were performed to shed light on the dynamic features of the intermolecular hydrogen bridges

present in these structures (see Figure 2). The time evolution of the metric parameters of the hydrogen bonds is presented in Figure 5. Let us start the discussion with the dimer (D). In this case, the amide nitrogen is a proton donor, whereas the oxygen from the amide group serves as the proton acceptor. The distances between the donor and the acceptor of the hydrogen atom varied between ca. 2.6 and 3.9 Å, whereas the H. . . OA distance changes were within 1.5–3.0 Å. Throughout the whole simulation time, the hydrogen atom is located at the proton donor side. In the case of (E), where the oxygen atoms belonging to the carboxylic groups are the proton donors and acceptors, the observations that can be made are strikingly different. Here, the bridged proton freely switches its donor from one to another; proton transfer actually happened 3 ps after the production run of the dynamics started. Moreover, the proton was on both the donor and acceptor sides for a similar amount of time during the MD run. This substantial difference in the behaviors of the hydrogen bridges between these two dimers can be attributed to the differences between the amide and carboxylic groups, the electronegativity of the donor atoms as well as the impact of fluorine substituents in the structure of (D). We will come back to this issue later, when the electronic structure of both dimers will be studied in a more detailed way via the static approaches. In particular, the apparent large difference in strength of the hydrogen bonds in (D) and (E), leading to such different dynamic characteristics, will be confirmed by the QTAIM descriptors.

**Figure 5.** Time evolution of the metric parameters between the atoms involved in the hydrogen bond formation in the studied dimers. (**D**) 2,6-difluorobenzamide, (**E**) 4-hydroxybenzoic acid.

The estimation of the impact of the nuclear quantum effects (NQEs) on the proton was possible based on PIMD—the obtained results are presented in Figure 6. It can be seen that, in the case of dimer (D), using the quantum-classical isomorphism to impose the quantization on the nuclei does not significantly change the proton behavior. The PMF profile corresponding to the PIMD simulation resembles the classical harmonic one. The observations made for the dimer (E) show that the NQEs lower the barrier for the proton transfer (from ca. 3.1 kcal \* mol−<sup>1</sup> to 2.0 kcal \* mol−1) and predict two proton minima at the donor side, roughly at 1.55 Å and 1.70 Å of the H. . . OA distance. In the case of classical-quantum dynamics, we can observe two minima: one that occurs at ca. 1.05 Å, which is in accordance with the PIMD result, and the other one, which corresponds to 1.65 Å of the H. . . OA distance (proton at the donor side).
