2.2.1. General Problems
The first step in a quantum-chemical structure and energy analysis is the optimization of the molecular geometry. If more than one structural form (different conformers, tautomers) are to be considered, each of them has to be optimized. A very important point is the selection of a reliable theoretical method and the application of a satisfactorily large basis set. Clearly, one wants to obtain the best computational results possible within the technical limits of the given structural problem. When a seeming H-bonded system is under scrutiny, an additional problem emerges in that the system has to be identified whether it is really a hydrogen-bonding arrangement or not.
In the review [22], a reference list was provided for the theoretical approaches and basis sets most frequently used in the past 15–20 years for geometry optimizations and relative energy calculations for ground state, closed-shell systems. Geometries were mainly optimized at the ab initio Hartree-Fock (HF) and second-order Møller-Plesset perturbation theory (MP2) levels or using some DFT-based (density functional theory) method such as B3LYP or some more recent ones that account for the dispersion interaction like the B97D method of Grimme [33] or the M05 and M06 methods of Zhao and Truhlar [34]. For basis sets, the 6-311++G** Pople basis or correlation consistent basis sets like cc-pvXz or aug-cc-pvXz (X = d, t, q) [35] have been applied more often in the most recent studies. Relative energies of conformers and tautomers from ab initio calculations are sensitive to the applied level of theory (method + basis set). Accounting for the electron correlations beyond the MP2 approach has turned out to be very effective by the application of the coupled-cluster methods, CCSD and CCSD(T) (coupled-cluster singles and doubles and noniterative triples) [36,37]. Hobza proposed an extrapolation formula for calculating the molecular energy at the CBS (complete basis set level) utilizing the MP2 limit energy and the difference of the CCSD or CCSD(T) energy and the MP2 energy calculated with some smaller basis set [38]. Frequency analysis can verify local energy minimum geometries by finding all positive vibrational frequencies. Using classical partition functions for ideal-gas molecules, the free energies can be estimated at some temperature T and pressure p [39].
While many papers have proven that the results are very sensitive to the level of theory, calculations applying high-level theoretical methods in combination with large basis sets may not be practical even for the case of small molecules. This is true not only for individual geometric parameters and energy values where it is normal that energy decreases with a higher-level method and/or a larger basis set, but even for the relative energies between conformations. Changes in the relative values suggest that the energy differences have not reached a converged limit value yet. A disappointing example was presented by DePrince and Mazziotti [40], who compared two tautomers of the CH3NO molecule at the CCSD and CCSD(T) levels. Whereas the relative energies were calculated at 29.1 and 21.4 kJ/mol, respectively, utilizing the cc-pvdz basis set, the CBS values are 2.8 and −6.2 kJ/mol. These changes are dramatic. They indicate that the basis set effect on the relative energy is very large and the selected methods at the CBS limit even predict different relative stabilities. The CCSD(T)/CBS result is perhaps more reliable, yet it can not be ascertained in the absence of experimental information whether the obtained value is an acceptable limit or even higher-level methods should be considered. Note that this problem can be noticed even for a very small molecule. For a larger molecule (e.g., 15–20 heavy atoms and corresponding hydrogens) upgrading the level of theory is even less practical. The situation could be worse regarding the optimized geometries. For relative energies, we can ascertain at least that the computed limit is questionable while for geometries there is no clue about the correct bond lengths, angles, and torsion angles. If they do not vary monotonically in parallel with the increasing level of theory, one may have even less idea about the correct limit values in the absence of experimental information.
Why are the above, otherwise well-known computational experiences important in a review regarding H-bonds? As was shown above, the critical point can emerge after a small distortion of the optimized 1,2-dihydroxybenzene geometry. This finding can be interpreted to mean that the H-bond is disrupted in the optimal geometry. Indeed, it is quite possible that the existence of the BCP is very sensitive to the structure as can be seen by the earlier the quoted notes of Contreras-García et al. [11]. Thus, the suspicion may emerge that the level of theory is not high enough when small geometric changes can create or perturb a H-bond. Using again the paper of Mandado et al., [7] as an example, the (3, −1) BCP was found on the B3LYP/6-31+G** density map, but the BCP disappeared at the B3LYP/6-311++G** level, and did not appear either when the B3LYP/6-311++G (3d,3p) density was studied. The authors considered the B3LYP/6-31+G** result as an artifact and attributed it to the lack of diffuse functions on the hydrogen atoms. Disappearance of the BCP with larger basis sets clearly indicates, however, the basis set effect on some calculated topological indeces, and calls for studying a reliable electron density map. Obtainment of the latter is perhaps possible at a very high theoretical level, but such calculations are not practical for larger molecules. On the other hand, if gradually increasing basis sets provide different predictions with respect to the existence of a BCP, then conclusions based on medium-size basis sets remain uncertain.
Theoretical calculations are able to predict a shift in the X–H vibrational frequency if the bond is involved H-bond interaction. Gu et al., [41] studied the possible intramolecular H-bond for α-hydroxy acetic acid. The authors concluded that the red-shift of about 100 cm−1, based on former experimental results for the stretching frequency of the α-hydroxy group relative to that for a free O–H bond in methanol “can be attributed to internal OH…O= hydrogen bonding”.
The red-shift in the case of an intramolecular H-bond was also demonstrated, at least qualitatively, by HF/6-31G* calculations for the 1,2-ethanediol [42,43]. For the all-trans-OCCO conformer tTt (C2h symmetry, for the three-letter code see [44]), where the two oxygens are far from each other, the two O–H frequencies were calculated equally at 4124 cm−1. For the most stable OCCO gauche conformation, tG+g− (C1 symmetry) allowing for an intramolecular hydrogen bond, the two O–H frequencies differed in accord with the experimental finding (see below). The frequencies for the gauche form were calculated at 4095 and 4123 cm−1. The smaller value refers to the O–H vibration involved in intramolecular O–H…O bonding. The larger frequency is related to the free OH vibration in the tG+g− conformation, where “t” indicates the trans HOCC arrangement. As a free OH, its stretching frequency is practically not affected and is equal to that for the tTt conformer. The O…O and O…H distances are 277 and 236 pm, respectively, well within the structural parameter set accepted for a H-bond. Although the calculated high frequencies are generally overestimated by about 10% at the HF level in comparison with experimental values, the shift of the frequency for the O–H group involved in the intramolecular interaction has revealed. A similar conclusion can be drawn for the 2-OH benzoic acid, when the calculated phenolic O–H frequency in intramolecular interaction with the carbonyl oxygen is compared with the free OH stretching frequency, as 3952 vs. 4112 cm−1 [45].
Florio et al., [46] compared the OH stretching frequencies of the monomeric and dimeric forms of formic and benzoic acids. The experimental values showed a red-shift of 459 cm−1 upon formic acid dimerization. The calculated harmonic frequency differences were 556 and 435 cm−1 at the B3LYP/aug-cc-pvtz and MP2/aug-cc-pvtz levels, respectively. For formic acid and its dimer, the geometry optimizations by the two methods led to very similar structure parameters, generally also close to the experimental values. The predicted frequencies at the MP2/aug-cc-pvtz level showed consistent overestimations for the monomer and the formic acid dimer (FAD), resulting in a red-shift close to the experimentally observed value. B3LYP/aug-cc-pvtz calculations provided, however, a larger overestimation for the monomer than for the dimer, leading to an increased red-shift. The authors concluded that the results “provide strong evidence that the B3LYP method does not provide a quantitatively correct description of this aspect of the H-bonding in the FAD dimer”.
Upon the benzoic acid dimerization, the red-shift was 217 cm−1 experimentally as compared with the theoretical value of 616 cm−1 calculated at the B3LYP/6-311+G(2d,2p)/B3LYP/6-31+G(d) level. In this case the red-shift was even more strongly overestimated than that for FAD (see above) by the B3LYP method. These calculations utilized, however, a smaller basis set, B3LYP/6-311+G(2d,2p) for the carboxylic group and 6-31+G(d) for the atoms of the phenyl rings. For the benzoic acid systems, the calculated frequency was overestimated for the monomer and underestimated for the dimer. This latter result differs from that for the formic acid dimer. The calculated large red-shift for the benzoic acid systems may be explained by the interplay of the method and basis set. Since it was already qualified by the authors that the method does not quantitatively describe the H-bond for the dimer of a simple carboxylic acid, this likely applies to the benzoic acid dimer as well.
In intramolecular hydrogen bonds, the geometry for both the H…Y distance and the X–H…Y angle is primarily determined by the covalent structure of the molecule. While three-atom hydrogen-bonded rings are extremely rare, the four-atom substructures (e.g., carboxylic group, amides) deserve special consideration. In most cases, a H-bond can be expected if the system can form a five to seven-member ring, including arrangements utilizing a polar H. Prototypes are indicated in Figure 1 and typical representatives of five- and six-member rings are shown Figure 2 and Figure 3. Seven-member rings can be formed for γ-substituted carboxylic acids, 1,4-disubstituted butanes with OH and/or NH2 substituents.
Larger rings are probably not stable. Chen et al., [47] pointed out that no intramolecular H-bond exists in the prevailing conformer of 1,5-pentadiol and 1,6-hexadiol at room temperature. In these cases, formation of an intramolecular H-bond would require a ring conformation with eight and nine members, respectively. This is probably unfavorable due to entropy considerations even for seven-member rings. Nagy et al., [48] studied different conformers for γ-hydroxy-butyric acid. Although the lowest-energy conformer optimized at the MP2/6-311++G** level formed a seven-member ring with an O–H…O= H-bond, the free energy for this structure is higher by about 2 kJ/mol than that for the most stable gas-phase species where this bond is disrupted, as also found experimentally [49]. The results for the increased relative free energy suggest unfavorable entropy effects for the hydrogen-bonded seven-member ring. This explanation is supported by the argument of Blanco et al. [50], who investigated the gas-phase structure of γ-amino-butyric acid (GABA). Intramolecular H-bonds were noticed in both forms of N–H…O=C and N…H–O–C=O, although the two mostly populated species do not possess an intramolecular H-bond. In order to create such bonds, structures have to be formed which “contribute to decrease entropy and to increase the Gibbs energy” [50].