PMC:4264129 / 23102-50966 JSONTXT

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Geometry Optimization\n\n2.2.1. General Problems\nThe 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.\nIn 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].\nWhile 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.\nWhy 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.\nTheoretical 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”.\nThe 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].\nFlorio 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”.\nUpon 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.\nIn 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.\nLarger 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].\n\n2.2.2. Special Problems\nIn general, for the past twenty years optimized molecular geometries in solution have been obtained by applying a continuum solvent model. The idea was introduced and subsequently developed by the Tomasi group as the PCM model [51,52]. Since the 1990’s, different continuum solvent models [53,54,55,56,57,58,59] and extension beyond the dielectric approximation [60] have been developed to account for the solvent effects on the geometry and energy/free energy of dissolved molecules. Several reviews summarize these models and compare the results obtained from different approaches [52,61,62,63].\nThe basic idea in the widely used PCM method [51] is that the solute is placed in a cavity carved in the continuum dielectric solvent, and the solute and the solvent mutually polarize each other. As a consequence, the solute’s geometry changes slightly and its internal energy increases when compared to its optimized gas-phase energy. The energy-increase is balanced by the developing solute-solvent electrostatic interaction energy. The final results are obtained through an iterative self-consistent-field (SCF) process that finds the total energy minimum and its related geometry. For the geometry optimization and energy/free energy calculations, all methods can be utilized, which were mentioned in relation to gas-phase calculations [33,34,35,36,37,38]. Thus, geometry optimizations can be performed by means of the HF, MP2 and DFT methods, and higher level energy calculations can be performed up to the CCSD(T)/CBS level. The customary basis set for geometry optimization and frequency analysis is 6-311++G**, but even the aug-cc-pvtz set has been applied [15,64].\nWhen a molecule dissolves, a close molecular environment is encountered that is in contrast to the most frequently applied ideal-gas model, where no potential energy interaction is considered even through the collisions of the molecules. Although the solute-solvent interactions are substantial, the effect of a non-polar or only slightly polar solvent (CCl4, CHCl3) on the molecular geometry is generally small [15,64]. The geometric effect could be, however, large when a solute with an intramolecular H-bond in the gas phase dissolves in a protic solvent such as water or methanol, which have both proton donor and acceptor sites. In this case, the X–H…Y intramolecular H-bond may collapse while solute-solvent H-bonds are formed using the free XH and Y sites.\nThe weakest point of the continuum dielectric solvent model is that the above solute-solvent H-bond(s) are only implicitly mimicked by polarization of the solvent and concomitant appearance of surface charges on the inner surface of the cavity: Negative surface charges opposite to a polar hydrogen and positive ones in the lone-pair regions of the solute’s oxygens and nitrogens. Although this response is qualitatively correct, the calculated solute-solvent stabilization energy is underestimated [65,66]. Thus, for proper calculation of the free energy changes when a polar solute with or without internal H-bond(s) dissolves in a protic solvent, explicit consideration of the solute-solvent intermolecular H-bonds becomes necessary.\nThis requirement can be largely satisfied by adopting the supermolecule + continuum approach, where the solute is surrounded by a number of explicit solvent molecules. The solute and the explicit solvent molecules mimic the H-bonds in the first solvation shell within the cavity carved in the continuum solvent. The critical question then becomes, how many explicit solvent molecules are to be considered.\nFor constructing the starting geometry of a supermolecule, knowledge of microsolvated solute structures is very helpful. In these systems, the central, polar molecule with or without an intramolecular H-bond is solvated by a few solvent molecules. Locations of the solvent molecules (water, methanol) indicate the most preferable solvation sites of the solute with a hydrogen donor/acceptor solvent. Useful information can be obtained from experimental gas-phase hydration/solvation studies augmented with theoretical calculations [12,13,14,16,23,67,68,69,70] or specific theoretical calculations for hydrated amino acid side chains, nucleotid base and sugar models [71,72,73,74,75].\nRecent calculations proved [15,64] that application of at least the aug-cc-pvtz basis set is required for reliable estimation of the relative solute free energies. If the solute has 6–10 C, N, O atoms and connected hydrogens, 500 basis functions could easily be required. If such a solute has to be surrounded by at least 5–6 water molecules, the number of basis functions increases to about 1000. The number of basis functions could be somewhat reduced by considering the solvent molecules with a lower basis set, with, e.g., 6-31+G**. While the supermolecule + continuum approach can be useful theoretically, it suffers from several technical challenges.\n(1) The geometry optimization for a system with 500–1000 basis functions is very slow in solution. If one wants to prove the local-energy-minimum character of the supermolecule and calculate thermal corrections, very small remaining forces should be allowed only at the end of the optimization. It is almost unreachable for a number of systems (or only by the application of the very time-consuming analytical second-derivative methods), in cases when torsion or intermolecular vibration frequencies could be as low as a few cm−1.\n(2) The number of explicit solvent molecules to be considered can become be critical. In a real, dilute solution the solute is surrounded by solvent molecules all around. Except for the simplest modeling cases like the partial solvation of γ-amino-butyric acid with 2–5 water molecules [76] or consideration of 3–8 water molecules during the HOCl catalyzed tautomerization of β-cyclopentadione [77], a considerably larger number of water molecules is generally required for reasonable modeling of the solvation sphere even for a small organic molecule. An impressive example was provided by Lu [78], who optimized the geometry of the Al(H2O)63+·12 H2O hydrate at the B3LYP/6-31+G(d,p) level in a water continuum by the PCM method. The resulting structure was of nearly spherical symmetry, easily allowing for the formation of the water network. Consideration of eighteen solvent molecules was necessary for mimicking the first and second hydration shells in a dilute solution.\n(3) The results of Lu and coworkers call attention to the need for the supermolecule to reasonably mimic the immediate in-solution environment of the solute. With a relatively small number of explicit solvent molecules within the supermolecule (for example, 3–4 water molecules, originally each of them facing a polar site), the water-water interactions may dominate over the solute-water interactions. Instead of forming 3–4 solute-water hydrogen bonds, a water cluster is then formed on some side of the solute and the number of water-solute hydrogen bonds would be smaller than expected in a water box with hundreds of water molecules. A successful tetrahydrate model in a continuum solvent was developed by Nagy for the syn-anti transformation of the acetic acid carboxylic group [79], whereas three waters in hydrogen bonds to the solute were not enough for modeling the immediate solvation environment of the transition state for 2F-phenol [15].\n(4) In general, only the first solvation shell around the polar sites can be modeled. Moreover, even in these cases, the explicit-solvent/continuum interface suffers from neglecting the consideration of the solvent-solvent hydrogen bonds. For methanol or acetonitrile solvents, the problem is not dramatic since the polar site of the solvent molecules should point toward the solute while the methyl group is located mainly on the outer surface of the supermolecule. Then the first-sphere solvent molecules can create a non-polar surface toward its continuum representation. This is surely not the case for explicit water molecules and is likewise questionable for a solvent like acetic acid with two stericly separated polar sites.\n(5) Geometry optimization for a supermolecule leads to an overly ordered structure, which is not maintained due to thermal disordering in a real solution.\n(6) If one wants to study the structure of a dilute or moderately concentrated (1 molar) solution as well as solute dimerization, boxes of a large number of explicit solvent molecules should be considered. These studies typically then require Monte Carlo (MC) [80] or molecular dynamics (MD) [81] simulations.\nDuring MC calculations, the solution model is a large solvent box with hundreds of solvent molecules and one or a few solute molecules embedded in the solvent. Atoms are represented by point-like centers characterized with atomic charges and assigned van der Waals parameters. The interaction energy of the atoms in different molecules is calculated by pair potentials and the total energy is the sum of these pair-interaction energies. Macroscopic thermodynamic quantities are estimated by averaging the individual values calculated for a large number of consecutive configurations. A configuration means a specific geometric arrangement of the elements in the solution box. The method is a probability method, meaning that a new configuration with modified geometric arrangement of the elements is considered for the above averaging upon the probability of the acceptance of the total energy change. If the new configuration is rejected, the old one is considered one more time in the averaging process. The most frequently applied sampling procedures are the Metropolis procedure [82] or some suitable procedure [83,84], which can accelerate the convergence of the calculated averages for thermodynamic quantities or help to more quickly reach an equilibrium solution structure by applying a biased energy calculation and probability for the acceptance of a new configuration. The goal is to generate a series of configurations corresponding to the Boltzmann distribution. For constant temperature simulations, the temperature is a parameter of the expression determining the acceptance probability. After having generated the required set of configurations, the average energy, enthalpy, volume, etc., can be calculated as an arithmetic mean of the individual values obtained with each configuration. If a biasing sampling was used, the probability of the acceptance has to be corrected before calculating averages.\nThe MD simulation is a deterministic process. The solution box is established as described above, but the atomic masses are also considered. A force field is used for calculating the total energy of the system with a given geometric arrangement (with Cartesian coordinates for each atom) at a reference time “t”. The force field contains terms accounting for the energy contributions by atoms bound along a 1–2–3–4 path, as well as for interactions of more remote atoms within the molecule and with all atoms in other molecules. The system generally is not in energy minimum, thus there are forces acting on the atoms. Using the gradient of the total energy, the forces acting on each atom can be determined. Applying Newton’s law, the position of the atoms at t + Δt can be calculated by means of the determined instantaneous velocities. Δt must be small, generally being chosen between 0.5 and 2 fs. In the latter case, the X–H distances are kept at a constant value. The temperature is related to the sum of the atomic kinetic energies. The simplest way to keep the temperature at a constant value is by scaling the determined atomic velocities or by coupling a thermostat to the system. Letting the simulation run long enough, sometimes for tens of nanoseconds, the average of the thermodynamic quantities can be obtained for a simulation period, where some structural characteristics, e.g., the solution density, have reached an equilibrium value already. By examining the trajectories calculated for geometric parameters of the solute, structural changes can be followed.\nUsing intermolecular pair-potentials such as OPLS-AA [85,86], Amber [87] or CHARMM22 [88], the largest problem is the development of the relevant atomic charges for the molecule. For example, Amber was originally parameterized for biopolymers and DNA, and no special charge parameters were available for, e.g., the HO–CHx–CHx–Y (Y = OH, NH2, NH3+, x = 1, 2) substructures. Furthermore, the OH and Y charges (and the CHx values, as well) should be conformation dependent, since, e.g., there is an intramolecular H-bond for the tG+g− conformer of 1,2-ethanediol, which is missing in the tTt form (Figure 2). Also, conformation dependent charges have to be used with Y = NH2, and NH3+ for the OCCN gauche and trans structures [89]. Recent developers of force fields suggest using molecular electrostatic potential (MEP) fitted charges, where the MEP should be obtained for the in-solution optimized solute. Since the solute and the continuum solvent mutually polarize each other, the MEP obtained at the end of the SCF procedure for the geometry optimization and energy minimization, reflects the electrostatic potential of a polarized solute in a polarized solvent environment. Charge fitting is a working tool, although different fitting methods (ES [90], RESP, [91], CHELPG [92]) lead to different results. Nonetheless, the problem is again of how to optimize the solute in a solvent environment. The MEP for the supermolecule is not relevant, since the solvent molecules are not so strictly bound to the solute due to thermal disorientation in a large solvent box as would follow from the structure of an optimized supermolecule. Furthermore, there is generally some charge transfer between the elements of the supermolecule. Although the programs force the charge of the total supermolecule to zero or some +/− integer for ionic solutes, the individual charges for the solute and the surrounding solvent molecules generally differ from 0, +/−1 etc. Thus, the MEP- fitted charges for a supermolecule should not be directly accepted for the atoms of the solute and applied in the MC of MD intermolecular pair-potentials. A possibility is that the solute geometry is accepted from the optimized supermolecule, and the MEP is fitted for the pure solute in a single-point calculation. This is, however, not a consistent procedure. The author has not found a good solution for this problem when surveying the literature.\nDespite the listed potential problems, the continuum solvent approach has been one of the most frequently used theoretical methods for characterizing the geometry and the energy/free energy for dissolved molecules. Since chemical equilibria depend on relative rather than absolute free energies, the problems mentioned above may not emerge in every case so harshly, and the errors could be partially cancelled. For example, the energy minimization in the supermolecule approach leads to too tightly bound water molecules. In a model, where the thermal disordering effect is also taken into consideration (MC and MD), a more loosely bound first solvation shell is expected. Nonetheless, since relative energy data are to be compared for the supermolecules with different solute conformations, the error is probably decreased. Also a more or less cancelled error may be expected regarding the interaction energy between the explicit solvent molecules at the outer surface of the supermolecules and the continuum. In a study for a series of compounds, it is a good practice to compare the computational results with available experimental values. Unfortunately, however, good-quality experimental results generally not are available in the literature for equilibria, where a number of conformers have been detected in solution.\n"}

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Geometry Optimization\n\n2.2.1. General Problems\nThe 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.\nIn 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].\nWhile 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.\nWhy 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.\nTheoretical 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”.\nThe 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].\nFlorio 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”.\nUpon 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.\nIn 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.\nLarger 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].\n\n2.2.2. Special Problems\nIn general, for the past twenty years optimized molecular geometries in solution have been obtained by applying a continuum solvent model. The idea was introduced and subsequently developed by the Tomasi group as the PCM model [51,52]. Since the 1990’s, different continuum solvent models [53,54,55,56,57,58,59] and extension beyond the dielectric approximation [60] have been developed to account for the solvent effects on the geometry and energy/free energy of dissolved molecules. Several reviews summarize these models and compare the results obtained from different approaches [52,61,62,63].\nThe basic idea in the widely used PCM method [51] is that the solute is placed in a cavity carved in the continuum dielectric solvent, and the solute and the solvent mutually polarize each other. As a consequence, the solute’s geometry changes slightly and its internal energy increases when compared to its optimized gas-phase energy. The energy-increase is balanced by the developing solute-solvent electrostatic interaction energy. The final results are obtained through an iterative self-consistent-field (SCF) process that finds the total energy minimum and its related geometry. For the geometry optimization and energy/free energy calculations, all methods can be utilized, which were mentioned in relation to gas-phase calculations [33,34,35,36,37,38]. Thus, geometry optimizations can be performed by means of the HF, MP2 and DFT methods, and higher level energy calculations can be performed up to the CCSD(T)/CBS level. The customary basis set for geometry optimization and frequency analysis is 6-311++G**, but even the aug-cc-pvtz set has been applied [15,64].\nWhen a molecule dissolves, a close molecular environment is encountered that is in contrast to the most frequently applied ideal-gas model, where no potential energy interaction is considered even through the collisions of the molecules. Although the solute-solvent interactions are substantial, the effect of a non-polar or only slightly polar solvent (CCl4, CHCl3) on the molecular geometry is generally small [15,64]. The geometric effect could be, however, large when a solute with an intramolecular H-bond in the gas phase dissolves in a protic solvent such as water or methanol, which have both proton donor and acceptor sites. In this case, the X–H…Y intramolecular H-bond may collapse while solute-solvent H-bonds are formed using the free XH and Y sites.\nThe weakest point of the continuum dielectric solvent model is that the above solute-solvent H-bond(s) are only implicitly mimicked by polarization of the solvent and concomitant appearance of surface charges on the inner surface of the cavity: Negative surface charges opposite to a polar hydrogen and positive ones in the lone-pair regions of the solute’s oxygens and nitrogens. Although this response is qualitatively correct, the calculated solute-solvent stabilization energy is underestimated [65,66]. Thus, for proper calculation of the free energy changes when a polar solute with or without internal H-bond(s) dissolves in a protic solvent, explicit consideration of the solute-solvent intermolecular H-bonds becomes necessary.\nThis requirement can be largely satisfied by adopting the supermolecule + continuum approach, where the solute is surrounded by a number of explicit solvent molecules. The solute and the explicit solvent molecules mimic the H-bonds in the first solvation shell within the cavity carved in the continuum solvent. The critical question then becomes, how many explicit solvent molecules are to be considered.\nFor constructing the starting geometry of a supermolecule, knowledge of microsolvated solute structures is very helpful. In these systems, the central, polar molecule with or without an intramolecular H-bond is solvated by a few solvent molecules. Locations of the solvent molecules (water, methanol) indicate the most preferable solvation sites of the solute with a hydrogen donor/acceptor solvent. Useful information can be obtained from experimental gas-phase hydration/solvation studies augmented with theoretical calculations [12,13,14,16,23,67,68,69,70] or specific theoretical calculations for hydrated amino acid side chains, nucleotid base and sugar models [71,72,73,74,75].\nRecent calculations proved [15,64] that application of at least the aug-cc-pvtz basis set is required for reliable estimation of the relative solute free energies. If the solute has 6–10 C, N, O atoms and connected hydrogens, 500 basis functions could easily be required. If such a solute has to be surrounded by at least 5–6 water molecules, the number of basis functions increases to about 1000. The number of basis functions could be somewhat reduced by considering the solvent molecules with a lower basis set, with, e.g., 6-31+G**. While the supermolecule + continuum approach can be useful theoretically, it suffers from several technical challenges.\n(1) The geometry optimization for a system with 500–1000 basis functions is very slow in solution. If one wants to prove the local-energy-minimum character of the supermolecule and calculate thermal corrections, very small remaining forces should be allowed only at the end of the optimization. It is almost unreachable for a number of systems (or only by the application of the very time-consuming analytical second-derivative methods), in cases when torsion or intermolecular vibration frequencies could be as low as a few cm−1.\n(2) The number of explicit solvent molecules to be considered can become be critical. In a real, dilute solution the solute is surrounded by solvent molecules all around. Except for the simplest modeling cases like the partial solvation of γ-amino-butyric acid with 2–5 water molecules [76] or consideration of 3–8 water molecules during the HOCl catalyzed tautomerization of β-cyclopentadione [77], a considerably larger number of water molecules is generally required for reasonable modeling of the solvation sphere even for a small organic molecule. An impressive example was provided by Lu [78], who optimized the geometry of the Al(H2O)63+·12 H2O hydrate at the B3LYP/6-31+G(d,p) level in a water continuum by the PCM method. The resulting structure was of nearly spherical symmetry, easily allowing for the formation of the water network. Consideration of eighteen solvent molecules was necessary for mimicking the first and second hydration shells in a dilute solution.\n(3) The results of Lu and coworkers call attention to the need for the supermolecule to reasonably mimic the immediate in-solution environment of the solute. With a relatively small number of explicit solvent molecules within the supermolecule (for example, 3–4 water molecules, originally each of them facing a polar site), the water-water interactions may dominate over the solute-water interactions. Instead of forming 3–4 solute-water hydrogen bonds, a water cluster is then formed on some side of the solute and the number of water-solute hydrogen bonds would be smaller than expected in a water box with hundreds of water molecules. A successful tetrahydrate model in a continuum solvent was developed by Nagy for the syn-anti transformation of the acetic acid carboxylic group [79], whereas three waters in hydrogen bonds to the solute were not enough for modeling the immediate solvation environment of the transition state for 2F-phenol [15].\n(4) In general, only the first solvation shell around the polar sites can be modeled. Moreover, even in these cases, the explicit-solvent/continuum interface suffers from neglecting the consideration of the solvent-solvent hydrogen bonds. For methanol or acetonitrile solvents, the problem is not dramatic since the polar site of the solvent molecules should point toward the solute while the methyl group is located mainly on the outer surface of the supermolecule. Then the first-sphere solvent molecules can create a non-polar surface toward its continuum representation. This is surely not the case for explicit water molecules and is likewise questionable for a solvent like acetic acid with two stericly separated polar sites.\n(5) Geometry optimization for a supermolecule leads to an overly ordered structure, which is not maintained due to thermal disordering in a real solution.\n(6) If one wants to study the structure of a dilute or moderately concentrated (1 molar) solution as well as solute dimerization, boxes of a large number of explicit solvent molecules should be considered. These studies typically then require Monte Carlo (MC) [80] or molecular dynamics (MD) [81] simulations.\nDuring MC calculations, the solution model is a large solvent box with hundreds of solvent molecules and one or a few solute molecules embedded in the solvent. Atoms are represented by point-like centers characterized with atomic charges and assigned van der Waals parameters. The interaction energy of the atoms in different molecules is calculated by pair potentials and the total energy is the sum of these pair-interaction energies. Macroscopic thermodynamic quantities are estimated by averaging the individual values calculated for a large number of consecutive configurations. A configuration means a specific geometric arrangement of the elements in the solution box. The method is a probability method, meaning that a new configuration with modified geometric arrangement of the elements is considered for the above averaging upon the probability of the acceptance of the total energy change. If the new configuration is rejected, the old one is considered one more time in the averaging process. The most frequently applied sampling procedures are the Metropolis procedure [82] or some suitable procedure [83,84], which can accelerate the convergence of the calculated averages for thermodynamic quantities or help to more quickly reach an equilibrium solution structure by applying a biased energy calculation and probability for the acceptance of a new configuration. The goal is to generate a series of configurations corresponding to the Boltzmann distribution. For constant temperature simulations, the temperature is a parameter of the expression determining the acceptance probability. After having generated the required set of configurations, the average energy, enthalpy, volume, etc., can be calculated as an arithmetic mean of the individual values obtained with each configuration. If a biasing sampling was used, the probability of the acceptance has to be corrected before calculating averages.\nThe MD simulation is a deterministic process. The solution box is established as described above, but the atomic masses are also considered. A force field is used for calculating the total energy of the system with a given geometric arrangement (with Cartesian coordinates for each atom) at a reference time “t”. The force field contains terms accounting for the energy contributions by atoms bound along a 1–2–3–4 path, as well as for interactions of more remote atoms within the molecule and with all atoms in other molecules. The system generally is not in energy minimum, thus there are forces acting on the atoms. Using the gradient of the total energy, the forces acting on each atom can be determined. Applying Newton’s law, the position of the atoms at t + Δt can be calculated by means of the determined instantaneous velocities. Δt must be small, generally being chosen between 0.5 and 2 fs. In the latter case, the X–H distances are kept at a constant value. The temperature is related to the sum of the atomic kinetic energies. The simplest way to keep the temperature at a constant value is by scaling the determined atomic velocities or by coupling a thermostat to the system. Letting the simulation run long enough, sometimes for tens of nanoseconds, the average of the thermodynamic quantities can be obtained for a simulation period, where some structural characteristics, e.g., the solution density, have reached an equilibrium value already. By examining the trajectories calculated for geometric parameters of the solute, structural changes can be followed.\nUsing intermolecular pair-potentials such as OPLS-AA [85,86], Amber [87] or CHARMM22 [88], the largest problem is the development of the relevant atomic charges for the molecule. For example, Amber was originally parameterized for biopolymers and DNA, and no special charge parameters were available for, e.g., the HO–CHx–CHx–Y (Y = OH, NH2, NH3+, x = 1, 2) substructures. Furthermore, the OH and Y charges (and the CHx values, as well) should be conformation dependent, since, e.g., there is an intramolecular H-bond for the tG+g− conformer of 1,2-ethanediol, which is missing in the tTt form (Figure 2). Also, conformation dependent charges have to be used with Y = NH2, and NH3+ for the OCCN gauche and trans structures [89]. Recent developers of force fields suggest using molecular electrostatic potential (MEP) fitted charges, where the MEP should be obtained for the in-solution optimized solute. Since the solute and the continuum solvent mutually polarize each other, the MEP obtained at the end of the SCF procedure for the geometry optimization and energy minimization, reflects the electrostatic potential of a polarized solute in a polarized solvent environment. Charge fitting is a working tool, although different fitting methods (ES [90], RESP, [91], CHELPG [92]) lead to different results. Nonetheless, the problem is again of how to optimize the solute in a solvent environment. The MEP for the supermolecule is not relevant, since the solvent molecules are not so strictly bound to the solute due to thermal disorientation in a large solvent box as would follow from the structure of an optimized supermolecule. Furthermore, there is generally some charge transfer between the elements of the supermolecule. Although the programs force the charge of the total supermolecule to zero or some +/− integer for ionic solutes, the individual charges for the solute and the surrounding solvent molecules generally differ from 0, +/−1 etc. Thus, the MEP- fitted charges for a supermolecule should not be directly accepted for the atoms of the solute and applied in the MC of MD intermolecular pair-potentials. A possibility is that the solute geometry is accepted from the optimized supermolecule, and the MEP is fitted for the pure solute in a single-point calculation. This is, however, not a consistent procedure. The author has not found a good solution for this problem when surveying the literature.\nDespite the listed potential problems, the continuum solvent approach has been one of the most frequently used theoretical methods for characterizing the geometry and the energy/free energy for dissolved molecules. Since chemical equilibria depend on relative rather than absolute free energies, the problems mentioned above may not emerge in every case so harshly, and the errors could be partially cancelled. For example, the energy minimization in the supermolecule approach leads to too tightly bound water molecules. In a model, where the thermal disordering effect is also taken into consideration (MC and MD), a more loosely bound first solvation shell is expected. Nonetheless, since relative energy data are to be compared for the supermolecules with different solute conformations, the error is probably decreased. Also a more or less cancelled error may be expected regarding the interaction energy between the explicit solvent molecules at the outer surface of the supermolecules and the continuum. In a study for a series of compounds, it is a good practice to compare the computational results with available experimental values. Unfortunately, however, good-quality experimental results generally not are available in the literature for equilibria, where a number of conformers have been detected in solution.\n"}