2.1.4. Geometry-Corrected Method (GCM)

All of the methods for estimating the energy of intramolecular interactions (e.g., hydrogen bonds) discussed so far do not take into account changes in the values of geometric parameters upon considering an open reference form of a molecule. However, the presence of a conjugated system of double bonds, which is characteristic for 3-aminoacrolein and, thus, the existence of its four conformers (Figure 9) allowed for proposing a method to estimate the energy of the N-H··· O intramolecular hydrogen bond in the ZZ conformer with simultaneous partial consideration of geometric factors [39]. This method initially functioned under the name "Scheme A" [39–41,43], but later its meaningless name was changed to the Geometry-Corrected Method (GCM) [44].

Very helpful in understaning the idea of GCM and how to derive it are the diagrams presented in Figure 10, which show the energy relationships between the respective forms of 3-aminoacrolein.

**Figure 10.** Energy dependencies between the respective forms of 3-aminoacrolein used in deriving the formula for the energy of the intramolecular N-H··· O hydrogen bond in the ZZ form, according to Geometry-Corrected Method (GCM).

As in Equation (2), in the first step, it is assumed that the hydrogen bond in the ZZ form of 3-aminoacrolein can be simply 'turned off' without any changes in the electron density distribution of the system, therefore also without any changes in the geometrical parameters of this form. By introducing an approximation of the energy additivity, we obtain:

$$E\_{\rm HB} = E^{\rm ZZ} - E^{\rm ZZ,f} < 0 \tag{12}$$

where *E*ZZ,f is simply the total energy of the fictitious form of ZZ with the hydrogen bond just 'turned off'. The rotation of the aldehyde group around the C=C double bond, i.e., the transition ZZ →EZ leads not only to breaking the hydrogen bond, but also to some changes in the geometrical parameters. If the energy associated with these changes in geometric parameters is ΔZZ →EZ g , then

$$\rm E^{\rm EZ} \approx E^{\rm ZZ} + E\_{\rm IIB} + \Delta\_{\rm g}^{\rm ZZ \rightarrow \rm EZ} = E^{\rm ZZ, \rm f} + \Delta\_{\rm g}^{\rm ZZ \rightarrow \rm EZ} \tag{13}$$

and quite similarly for the ZZ →EE transition

$$E^{\rm EE} \approx E^{\rm ZZ} + E\_{\rm IIB} + \Lambda\_{\rm g}^{\rm ZZ \rightarrow \rm EE} = E^{\rm ZZ, \rm f} + \Lambda\_{\rm g}^{\rm ZZ \rightarrow \rm EE} \tag{14}$$

*Molecules* **2020**, *25*, 5512

Dividing the sum of Equations (13) and (14) by two, one obtains an expression that can be interpreted as the averaged energy that is related to the configuration change Z →E:

$$
\Delta\_{\mathbb{R}}^{\rm av} = \frac{1}{2} (\Delta\_{\mathbb{R}}^{\rm ZZ \to \rm EZ} + \Delta\_{\mathbb{R}}^{\rm ZZ \to \rm EE}) = \frac{1}{2} (E^{\rm EZ} + E^{\rm EE}) - E^{\rm ZZ, \rm f} \tag{15}
$$

Combining this equation with (12) gives the expression for the hydrogen bond energy in the conformer ZZ

$$E\_{\rm IIB} = E^{ZZ} - \frac{1}{2}(E^{\rm EZ} + E^{\rm EE}) + \Delta\_{\rm g}^{\rm av} \tag{16}$$

but in which there is (so far) the unknown quantity Δav g . In fact, the hydrogen bond energy, *E*HB, and the averaged contribution to the configuration change Z →E, Δav g , are formally non-separable quantities. However, the existence of conformers allowed for determining the unknown contribution Δav g from ye<sup>t</sup> another source. Let us introduce the fictitious equivalents of the conformers EZ and EE (EZ<sup>f</sup> and EEf, respectively), having the same values of all (of course, except the dihedral angle(s) changing the conformation) geometrical parameters as the conformer ZZ (see Figure 10). The energy that is associated with the transition ZZ<sup>f</sup> →EZ can then be assumed in the form

$$
\Delta\_{\text{g}}^{ZZ \to \to \to} = \Delta^{Z \to \to} + \Delta\_{\text{rel}} \tag{17}
$$

where ΔZ→<sup>E</sup> is the energy resulting from the change of the Z →E configuration while maintaining the constant values of all geometrical parameters, while Δrel is the relaxation energy of the fictitious EZ<sup>f</sup> form to its fully relaxed equivalent obtained after the full geometry optimization. The energy associated with the transition ZZ<sup>f</sup> →EE can be presented quite similarly

$$
\Delta\_{\mathbb{R}}^{ZZ \to \to \to} = \mathbb{\bar{\Lambda}}^{Z \to \to} + \Delta\_{\text{rel}} \tag{18}
$$

Changing the conformation from ZZ<sup>f</sup> to either EZ<sup>f</sup> or EE<sup>f</sup> (i.e., maintaining the same values of bond lengths and angles) should not have a significant influence on the energy change. With the neglect of changing the interactions between 'unbound' atoms, it can therefore be assumed that ΔZ→<sup>E</sup> ≈ Δ˜ Z→E ≈ 0. This approximation gives, after adding Equations (17) and (18) to each other, another expression for Δav g

$$
\Delta\_{\text{g}}^{\text{av}} = \frac{1}{2} (\Delta\_{\text{g}}^{\text{ZZ} \to \text{EZ}} + \Delta\_{\text{g}}^{\text{ZZ} \to \text{EE}}) \approx \frac{1}{2} [(E^{\text{EZ}, \text{f}} - E^{\text{EZ}}) + (E^{\text{EE}, \text{f}} - E^{\text{EE}})] \tag{19}
$$

which, inserted into Equation (16), gives the final formula for the value of the hydrogen bond energy in ZZ-3-aminoacrolein [39]

$$E\_{\rm HB}^{\rm CCM} = E^{ZZ} - \frac{1}{2} (E^{\rm EZ,f} + E^{\rm EE,f}) < 0\tag{20}$$

Thus, it can be seen that, to determine *E*GCM HB , only the total energy of the fully optimized ZZ conformer and the total energies of the fictitious EZ and EE conformers with the values of geometric parameters (except for the dihedral angle O=C–C=O) from the ZZ conformer are needed. It is worth repeating at this point that GCM, i.e., formula (20), to some extent takes into account the changes in geometric parameters when moving from the ZZ form to the reference forms.

At this point, it is instructive to compare GCM with the OCM variant, in which the open reference form is the ZE conformer, i.e., *E*OCM HB = *E*ZZ − *E*ZE (cf. with Equation (4)). As already discussed, the assumption of OCM is that the reference open system does not differ significantly from the closed form, whereas, in the case of ZZ-3-aminoacrolein, the rotation of the aldehyde group around the C–C bond leading to the ZE conformer introduces a new rather significant interaction of the H··· H type (see Figure 9). This interaction is practically not present in the closed form ZZ. Any estimate that refers to the ZE conformer as the reference form should take this change into account. Suppose (similar to the hydrogen bond in the ZZ conformer) that this H··· H interaction in the fictitious ZE<sup>f</sup> form can be

'turned off', which gives ZEf (note the prime sign in the superscript). The energy associated with the rotation of the aldehyde group (ΔZ→<sup>E</sup> s ) at the transition ZZ<sup>f</sup> → ZEf can be assumed to be negligible due to both the conservation of the same geometric parameters as in the conformer ZZ and also due to the neglect of additional H··· H repulsion at this stage (additionally, the negligible influence of changes in the interactions of unbound atoms other than H··· H is also assumed). This repulsion leads to an energy increase of ΔZErep and to the form ZEf, which still maintains the geometry of ZZ. Only full relaxation of the ZE<sup>f</sup> geometry leads to the optimized ZE conformer. The energy that is associated with this relaxation has been designated as ˜ Δ ˜ rel (see Figure 11).

**Figure 11.** Diagram showing the way of obtaining the ZE conformer from the ZZ one through various fictitious forms.

Therefore, the energy that is associated with the transition from the ZZ<sup>f</sup> conformer to the ZE conformer can be expressed as:

$$
\Delta\_{\rm g}^{\rm ZZ \rightarrow \rm ZE} \approx \Delta\_{\rm rep}^{\rm ZE} + \tilde{\Delta}\_{\rm rel} \tag{21}
$$

Given the assumption (12) and by the similarity to the previously defined changes in energies ΔZZ→EZ g (13) and ΔZZ→EE g (14), one gets

$$
\Delta\_{\rm g}^{\rm ZZ \to \rm ZE} = E^{\rm ZE} - E^{\rm ZZ,f} \tag{22}
$$

Inserting this expression together with Equation (21) into Equation (12) gives a relationship between the estimation of the hydrogen bond energy in ZZ-3-aminoacrolein that is obtained by GCM and that obtained by OCM with the ZE conformer as the reference open form

$$E\_{\rm IIB}^{\rm CCM} = E^{\rm ZZ} - E^{\rm ZE} + \left(\Delta\_{\rm rep}^{\rm ZE} + \tilde{\Delta}\_{\rm rel}\right) = E\_{\rm IIB}^{\rm OCM} + \left(\Delta\_{\rm rep}^{\rm ZE} + \tilde{\Delta}\_{\rm rel}\right) \tag{23}$$

Equation (23) shows that, when compared to OCM, the estimation that is based on GCM takes into account two terms with opposite signs. The repulsive term ΔZErep is positive, whereas the relaxation term ˜ Δ ˜ rel is negative. The mutual weights of these two terms cause that the value of the intramolecular hydrogen bond energy determined by GCM is either below or above the value obtained by OCM. Strong hydrogen bonds should cause significant changes within the X-H··· Y bridge and, thus, both a small distance H··· Y in the conformer ZZ and a small distance H··· H in the fictitious form ZEf (or ZEf) obtained after rotation of the proton-acceptor group while maintaining the geometrical parameters from the conformer ZZ (except for the dihedral angle O=C–C=C). As a consequence, in molecules with a strong intramolecular hydrogen bond, the role of H··· H repulsion at the ZEf → ZE<sup>f</sup> should be significant. On the other hand, the significance of the relaxation term ˜ Δ ˜ rel should be

dominant in the case of relatively small distances H··· H in the ZE<sup>f</sup> form (H··· Y in ZZ) and, which seems more important, in the case of bulky proton-acceptors.

At this point, it is worth comparing the hydrogen bond energy values that were obtained with GCM with those obtained with the traditional variant of OCM. The first comparison of this type was made for the ZZ-3-aminopropenal (ZZ-3-aminoacrolein) [47] discussed here and for the related ZZ-3-aminopropential [39], where sulfur atom replaces the oxygen atom. The energy values of hydrogen bonds *E*GCM HB and *E*OCM HB are shown in Table 2. Additionally, this table also shows the relative energies (in relation to ZZ) of the respective conformers, the H··· H distances in ZE<sup>f</sup> and ZE forms, and the values of ˜ Δ˜ rel, which will be used in the current discussion. All of these values are limited to the best method used (MP2/6-311++G\*\*), so as not to increase the amount of numerical data [39].

**Table 2.** Some energetic (in kcal/mol) and geometric (in Å) parameters computed (MP2/6-311++G\*\*) for different forms of 3-aminoacrolein (Y = O) and 3-aminopropential (Y = S).


In the case of 3-aminoacrolein, the following order of relative energies of conformers was obtained: *E*ZZ < *E*EE < *E*EZ < *E*ZE. It suggests a significant relaxation of the most extended EE conformer and a significant role of the H··· H repulsion in the ZE conformer. In the case of 3-aminopropential (Y = S), the relative energy of the EZ conformer, on the other hand, is significantly lifted up, so that it equals that of the ZE conformer. In turn, this result suggests a greater role of S··· H valence repulsion in 3-aminopropential than O··· H in the EZ conformer (a complementary explanation may also be the greater role of the attractive component in the O··· H interaction than S··· H, which lowers the relative energy of EZ-3-aminoacrolein in relation to EZ-3-aminopropential). It is also manifested by the values of the angle CCY, which is 128.3◦ and only 125.4◦ for Y = S and O, respectively. As for the estimated values of the hydrogen bond energy, interestingly, OCM suggests a somewhat stronger N-H··· O hydrogen bond in ZZ-3-aminoacrolein ( −6.50 kcal/mol) than N-H··· S in ZZ-3-aminopropential (−6.02 kcal/mol), whereas, in the case of GCM, the opposite is obtained, i.e., this method suggests that the latter bond is stronger ( −6.96 kcal/mol) than the former one ( −5.28 kcal/mol). Table 2 also presents the values of the distances H··· H in the ZE<sup>f</sup> and ZE forms of both molecules, as well as the changes of these distances at the ZE<sup>f</sup> → ZE transition, i.e., upon relaxation of this conformer. The much higher Δ*d*ZE,f →ZE H··· H value for the ZE-3-aminoacrolein ( −0.301 Å) than for ZE-3-aminopropential (−0.159 Å) suggests a much stronger H··· H repulsion in the former of these systems, which is most likely due to the much shorter initial distance (1.840 Å vs. 1.968 Å). This suggestion is actually confirmed by the obtained results. Namely, as can be seen from the last column of Table 2, 3-aminoacrolein and 3-aminopropential are characterized by the same value ( −1.87 kcal/mol) of ˜ Δ˜ rel, i.e., the relaxation term ZE<sup>f</sup> → ZE. Therefore, the greater change in the H··· H distance at the transition ZE<sup>f</sup> → ZE for the former of these molecules must result primarily from the greater repulsion ΔZErep, which, as a consequence, should significantly exceed the relaxation component ˜ Δ˜ rel. In turn, this should lead to a significantly lower *E*GCM HB when compared to *E*OCM HB . As can be seen from Table 2, such a relationship for ZZ-3-aminoacrolein does indeed take place since *E*GCM HB and *E*OCM HB amount to −5.28 and −6.50 kcal/mol, respectively. This result shows that the hydrogen bond energies obtained within GCM are consistent with the observable geometric changes.

In addition to the case of 3-aminoacrolein [47] and 3-aminopropential [39] discussed here, GCM was later used to estimate the energy of intramolecular C-H··· O/S interactions in few systems featuring a similar *quasi*-ring structure (Figure 12) [40,41].

**Figure 12.** Energy values (in kcal/mol) of intramolecular C-H··· O/S interactions obtained [41] (B3LYP/aug-cc-pVTZ) by either OCM (**black**) or GCM (**red**).

Importantly, contrary to popular belief, these calculations showed that the C-H··· O/S contacts in these systems are actually destabilizing. Therefore, no hydrogen bond in the usual sense is formed between the proton-donating C-H bond and proton-acceptor O or S atoms. This result was interpreted [40,41] in terms of the steric compression, which leads to the dominance of the valence repulsion contribution in the C-H··· O contact and it was further supported by observing both the increase in contact destabilization and the corresponding geometric changes during the flattening of some systems. Further detailed studies on an even larger group of systems (*vide infra*) showed, however, that intramolecular C-H··· O interactions may be destabilizing in some systems, while stabilizing in others [44]. The fact that the large number of X··· O (X = F, Cl, Br, I), O··· O and F··· F interactions, which some consider stabilizing due to the presence of a bond path tracing these contacts are, in fact, destabilizing in many molecules was also shown [43] by means of the energy values obtained, *inter alia*, by GCM and OCM. An example is shown in Figure 13.

**Figure 13.** Interaction energies (in kcal/mol) of the X··· O (X = F, Cl, Br, I) contact obtained [43] by either OCM (**black**) or GCM (**red**). The MP2/aug-cc-pVTZ level of theory was used for all systems but that with Y = I, for which MP2/aug-cc-pVTZ-PP was used instead.

Theoretical studies [39–41,43,44] show that GCM can be considered to be a reliable method of estimating the energy of both intramolecular hydrogen bonds as well as intramolecular non-bonding interactions. As this method takes into account changes in geometric parameters that occur when passing to reference systems, it is a more reliable approach than the standard OCM, which does not take into account these changes at all. Of course, the applicability of GCM, like most other methods, is limited. For example, the presence of bulky substituents can significantly reduce the reliability of this method. Moreover, of course, the analyzed molecule must have appropriate conformers, which is not always the case. However, OCM also has to deal with similar requirements. Nevertheless, OCM is less tricky.

It is obvious that obtaining the individual conformers needed while using conformational methods requires a grea<sup>t</sup> deal of care and attention. Unfortunately, this is not always the case. In their study of the N-H··· O and N-H··· S intramolecular hydrogen bonds in *β*-aminoacrolein, *β*-thioaminoacrolein, and their halogenated derivatives, Nowroozi and Masumian claimed that GCM performs worse than RBM and RRM, in particular [63]. However, it is enough to look at their Scheme 3 to realize that they used wrong conformers labeled as EZ and EE. Briefly, both of these conformers should have H and R3 at reversed positions! (Starting with the ZZ conformer, rotation of the -NHR3 group around the C=C double bond obviously leaves the H atom rotated with this group on the "inside" of the molecule, i.e., at the R3 site and close to R1.) Because EZ and EE conformers (either real or fictitious) are used in RRM and GCM, it is obvious that the results that are presented by Nowroozi and Masumian [63] are completely wrong (as evidenced, e.g., by low *R*<sup>2</sup> values). Moreover, these authors ignored the fact that some of the conformers they used experience new significant interactions, such as O··· Br, which, of course, significantly affect the total energy of a given conformer.

#### 2.1.5. Geometry-Corrected Related Rotamer Method (GCRRM)

It is worth noting that, when compared to GCM, RRM should give too negative values of interaction energy, because the total energy of the ZE conformer, *E*ZE, which is not present in the formula for *E*GCM HB (Equation (20)), appears with a negative sign. At first glance, it would seem difficult to further directly compare the two methods, as they do not use the same EZ and EE conformer structures; GCM overlays them with the values of the geometric parameters from the closed ZZ form, while RRM uses fully relaxed geometries. Nevertheless, the difference in estimations of the two methods can be written, as follows [44]:

$$E\_{\rm HB}^{\rm CCM} - E\_{\rm HB}^{\rm RBM} = \frac{1}{2}(E^{\rm EZ} - E^{\rm EZ,f}) + \frac{1}{2}(E^{\rm EE} - E^{\rm EE,f}) + \frac{1}{2}(E^{\rm EZ} - E^{\rm EE}) + (E^{\rm EE} - E^{\rm EE})\tag{24}$$

This expression shows that the difference between the estimates that were obtained with GCM and RRM results from the balance of the relaxation terms (first two terms) and the conformational changes (EZ →EE and ZE →EE). Importantly, both of these contributions have opposite signs and the latter ones are larger than the former. Moreover, the last term contributes without the factor 1/2. As a consequence, the difference (24) is positive. In the case of 13 molecules containing intramolecular C-H··· O contacts considered in the reference [44], the energies of the consecutive terms were, as follows: −1.3 ± 0.3, −2.0 ± 0.6, 2.5 ± 0.8, and 2.7 ± 0.3 kcal/mol, so that the difference (24) was 2.3 ± 0.3 kcal/mol. Because the first three values almost cancel themselves ( −0.4 kcal/mol), it can be assumed that the difference (24) comes mainly from the configurational change EE →ZE. This configurational change can then be considered as a two-step process: EE →EE<sup>f</sup> →ZE, where EE<sup>f</sup> is a fictitious conformer EE having the geometric parameters of the ZE conformer. Hence, the energy of the EE →ZE process can be written as the sum of the preparation energy of the ZE conformer and the E →Z isomerization energy:

$$E^{\rm ZE} - E^{\rm EE} = (E^{\rm EE, \rm f} - E^{\rm EE}) + (E^{\rm ZE} - E^{\rm EE, \rm f})\tag{25}$$

In the considered systems with the intramolecular C-H··· O interactions, the first term was 1.0 ± 0.2 kcal/mol. The second term is related to the H··· H repulsion that appears in the conformer ZE, the value of which was estimated at 0.60 ± 0.17 kcal/mol (median value) [44]. Together with the preparation energy, this energy suggests that the EE →ZE process is affected by close H··· H contact by roughly 1.6 kcal/mol, which is close to the actual value of 2.7 kcal/mol as well as the E → Z isomerization energy in 2-butene (1.04 kcal/mol). This fairly good agreemen<sup>t</sup> led to the proposition of a corrected RRM known as the Geometry Corrected Related Rotamers Method (GCRRM) [44]. According to GCRRM, the estimated value of an intramolecular hydrogen bond (or other interaction) can be obtained from the following formula

$$E\_{\rm IIB}^{\rm CGRRM} = E\_{\rm IIB}^{\rm RRM} + \left(E^{\rm EE,f'} - E^{\rm EE}\right) + E\_{\rm HH} \tag{26}$$

where the *E*HH value is 0.6 kcal/mol. The values obtained with GCRRM are between the values obtained with GCM and RRM and closer to the former, as shown in Figure 14.

**Figure 14.** Interaction energies (in kcal/mol) of the intramolecular C-H··· O contacts in the molecules investigated in ref. [44].

It is noteworthy that all four lines that are shown in Figure 14 have similar slopes; therefore, the methods differ by their intercepts that can be seen as "zeros of the interaction energy" [44]. Indeed, *E*RRM HB = *E*EMHB + 1.7 kcal/mol, *E*GCRRM HB = *E*EMHB +3.4 kcal/mol and *E*GCM HB = *E*EMHB +4.0 kcal/mol. At the same time, Figure 14 is a wonderful illustration displaying that a given intramolecular interaction in a certain system may be suggested to be much or less stabilizing according to one estimating method while another method may sugges<sup>t</sup> its rather repulsive nature.

#### *2.2. Rotation Barriers Method (RBM)*

A strong alternative to OCM with its various variants is the Rotation Barriers Method (RBM) [33,65–72] first used by Buemi et al. in order to estimate the energy of the O-H··· O intramolecular hydrogen bond in malonaldehyde [66] and a bit later of N-H··· N in formazan [67]. Quite rightly, this method assumes that an intramolecular hydrogen bond in the closed (chelate) form raises the height of the energy barrier that is associated with either the proton-donor or proton-acceptor group rotation by 180◦ to form an open form. Hence, when assuming the additivity of the respective energy terms, it can be written that

$$E\_{\rm RB} = E\_{\rm PB}^{\rm KBM} + E\_{\rm ARB} \tag{27}$$

where *E*RB is the rotation barrier and *E*ARB is (to use Buemi's terminology) the actual rotation barrier of the considered group [33]. The actual rotation barrier introduced as a result of the above additivity scheme is obviously related to a fictitious equivalent of a closed system in which the intramolecular hydrogen bond is 'turned off', and, therefore, it is not possible to calculate its value exactly. Nevertheless, *E*ARB can be estimated while using a certain reference system (see Figure 15).

**Figure 15.** Scheme showing the way of estimating the energy of an intramolecular hydrogen bond according to RBM.

Hence,

$$E\_{\rm HB}^{\rm RBM} = E\_{\rm ARB}^{\rm ref} - E\_{\rm RB} = (E\_{\rm 90}^{\rm ref} - E\_0^{\rm ref}) - (E\_{\rm 90} - E\_{\rm C}) < 0 \tag{28}$$

where the expressions in the former and in the latter brackets are rotation energy barriers for either the proton-donor or the proton-acceptor group in the reference and the closed form, respectively. In fact, the transition states for the rotations in both systems do not have to correspond exactly to the perpendicular orientation of the group. Nevertheless, the symbols denoting total energies of the transition states are given the subscript 90 in order to emphasize that often the transition state, that is associated with the rotation of a given group, roughly corresponds to its perpendicular orientation with respect to the molecular framework. Importantly, just like in the case of OCM, in RBM it is assumed that the reference system retains the earlier described significant similarity to the bound, i.e., closed form. This condition is not always easy to meet. On the other hand, the use of RBM is a reasonable method of choice in many of those cases where the energy estimate based on OCM is unreliable due to the presence of some bulky or highly electronegative substituents leading to new important interactions in the open reference form [33].

As already mentioned, this method was first used by Buemi et al. [67] in order to estimate the energy of the N-H··· N intramolecular hydrogen bond in one of the conformers of formazan (Figure 16).

**Figure 16.** The open and closed forms of formazan and the two reference systems (**A1** and **A2**) used by Buemi et al. [67] in RBM.

The abandonment of the traditional OCM and the need to use a different method, which led to RBM, resulted from the inability to find a reliable reference form. Because of the symmetry of the amino group, its rotation is useless, and the rotation of the N=N–H group leads to a close H··· H contact. On the other hand, the use of other conformers was considered [67] impractical, because it led to too large structural change. As a consequence of these problems, Buemi et al. proposed using, in RBM, two reference systems shown in Figure 16 as A1 and A2. Buemi et al. emphasized that they had previously successfully used this method to determine the interaction energy of the O-H··· O intramolecular hydrogen bond in malonaldehyde (using vinyl alcohol as a reference), obtaining (MP2/6-31G\*\*) a value similar to that of the traditional open-closed method ( −14.07 and −14.01 kcal/mol, respectively) [66]. Depending on the reference molecule A1 or A2 and on more subtle conditions concerning the structure of the amino group (planar vs. pyramidal), Buemi et al. obtained energies that ranged from −9.38 to −4.85 kcal/mol. Subsequently, however, the value close to the middle, i.e., −7.17 kcal/mol, was considered as the most reliable. Nevertheless, quite reasonably, Buemi and Zuccarello pointed out that such wide range of the obtained estimates does not allow for stating that the estimate of the N-H··· N hydrogen bond energy in formazan is as good as O-H··· O in malonaldehyde [66].

Buemi and Zuccarello then used RBM to estimate interaction energies of various intramolecular hydrogen bonds (O-H··· O, O-H··· halogen, O-H··· N, N-H··· O, N-H··· N, S-H··· O, O-H··· S, and S-H··· S) in many molecules (e.g., malondialdehyde, acetylacetone, and their variously substituted derivatives, formazan, 3-aminoacrolein, some *β*-thioxo- and *β*-dithioketones, 2-halophenols, 2-nitrophenol) [33]. From the many data shown there, I will only mention those obtained for malondialdehyde, acetylacetone, and 3-aminoacrolein. The closed form, the two open forms, and the two reference molecules used in RBM are shown in Figure 17, and the quoted values of the respective estimates are listed in Table 3.

**Figure 17.** The closed form, the two open forms and the two reference molecules used in RBM for malondialdehyde and acetylacetone (R = CH3) [33].

**Table 3.** Estimated values (MP2/6-31G\*\*) of intramolecular O-H··· O hydrogen bond energies (in kcal/mol) in malondialdehyde and acetylacetone [33].


As can be seen from Table 3, in the RBM calculations, Buemi and Zuccarello used four reference structures, two for the ARB for the -OH proton-donor group, and two for the ARB for the proton-acceptor -CHO group. In the former case, these systems were the open form A and the reference D obtained by replacing the group -CHO by the H atom. In the latter case, these were the open form D and the reference A obtained by replacing the OH group by H. Buemi and Zuccarello emphasized the very good agreemen<sup>t</sup> of the estimates based on OCM and RBM, whenever these methods use the proton-donor group rotation [33]. This result is especially obvious in the case of malondialdehyde (ca. −14 kcal/mol), whereas slightly less in the case of acetylacetone (from ca. −17 kcal/mol to ca. −15 kcal/mol), which was attributed to the new, probably quite significant, interaction between the methyl group and the hydrogen atom from the hydroxyl group in the open form A. On the contrary, worse agreemen<sup>t</sup> of the OCM and RBM results was noted for the estimates that are based on the rotation of the proton-acceptor group. However, it is noted that, in general, the estimates that are based on RBM (no matter whether it is a rotation of the proton-donor or the

proton-acceptor group) are closer to OCM estimates based on the proton-donor group rotation than the corresponding OCM based on the proton-acceptor group rotation [33].

As already mentioned, for ZZ-3-aminoacrolein (see Figure 9), it seems that the most reasonable reference form is EZ (although Buemi and Zuccarello also admitted the conformer ZE, this form experiences a new significant H··· H interaction). In the case of malonaldehyde, the EZ conformer gives a value of −9.7 kcal/mol, thus approximately 5.3 kcal/mol lower than the classic value of the O-H··· O hydrogen bond energy in malondialdehyde. Assuming that a similar underestimation would also act for 3-aminoacrolein, Buemi and Zuccarello renormalized the obtained value ( −5.2 kcal/mol), finally obtaining a value of about −10.5 kcal/mol [33]. The main model problem in the estimation of the N-H··· O hydrogen bond energy in ZZ-3-aminoacrolein using RBM is the change in the degree of amino group pyramidalization during rotation [33,47]. Because of the presence of the hydrogen bond, this group is planar in the ZZ conformer, whereas slightly pyramidal when rotating around the C-N bond. Depending on the constraint put on the rotating amino group and the reference system utilized (Figure 18), the estimated value of the N-H··· O hydrogen bond energy in ZZ-3-aminoacrolein is between −11.7 and −8.4 kcal/mol (MP2/6-31G\*\*).

**Figure 18.** Two reference molecules used in RBM for 3-aminoacrolein [33].

Unfortunately, this example shows quite a lot of freedom in terms of the possible choice of reference systems. On the one hand, the reference molecule A was obtained for ZZ-3-aminoacrolein by replacing the amino group with a hydrogen atom, whereas molecule D by replacing the aldehyde group with a methyl group (and not only with hydrogen). On the other hand, both of these reference molecules have the same number of heavy framework atoms. However, unlike D, molecule A features the presence of a conjugated system of two double bonds. Hence, it should be expected that the *π*-electron structure in both of these reference molecules is quite different.

In summary, RBM is a reasonable approach for estimating intramolecular hydrogen bond energy in many simple molecules and it can be successfully used as a replacement or supplement to the estimation based on OCM. However, like OCM, this method should also be used with grea<sup>t</sup> caution, because the presence of new interactions during rotation of a group in the parent or reference molecule may significantly reduce the reliability of the estimation. Moreover, in this method, both the problem of choosing a reasonable reference form and a certain freedom of this choice are noticeable. As noted by Buemi and Zuccarello [33], RBM is much more computationally expensive than OCM, as it requires calculating the rotation barriers for two systems, the bound, i.e., the closed one, and the reference molecule (Equation (28)). It is worth reminding here that the maxima of these barriers do not have to correspond exactly to the perpendicular arrangemen<sup>t</sup> of the rotated groups.
