*2.1. Conformational Methods*

As noted in the Introduction, it is impossible (see, however, the further discussion on the QTAIM-based methods) to precisely define the energy of the intramolecular hydrogen bond XH··· Y (or more generally of the intramolecular interaction X··· Y), because it is impossible to create a reference system in which there would be no such interaction, but in which the configuration of all atoms would be preserved. In such a reference system, the interaction of interest would be simply "switched off". Crucially, this approach is eqivalent to the following partition of the total energy of the system (the so-called closed or chelate form) containing the interaction of interest (e.g., a hydrogen bond)

$$E(\text{closed}) = E^f(\text{closed}) + E\_{\text{HB}} \tag{2}$$

in which *<sup>E</sup>*(closed), *Ef*(closed), and *E*HB correspond successively to the total energy of the closed form, the total energy of a fictitious closed form with the interaction switched off, and the hydrogen bond interaction energy. Of course, such an exclusion is impossible, but, nevertheless, one may be tempted to find another system being very similar to the fictitious closed one. Due to the fact that total energy depends on the number and type of particles making up a given molecule, the phrase "another but very similar system" should be understood as a different conformer of the closed form of a molecule. This leads to so-called conformational methods, i.e., methods which use total energies of at least two conformers of a molecule having the intramolecular interaction.

## 2.1.1. The Open-Closed Method (OCM)

The simplest and the most frequently used method of estimating the energy of intramolecular interactions, including intramolecular hydrogen bonds, is the so-called open-closed method (OCM) [5,19]. Apart from the molecule that contains a given interaction (i.e., the closed or chelate form), OCM requires the use of one more reference form (the so-called open), in which this interaction is absent [20–45]. It is then assumed that

$$E^f(\text{closed}) \approx E(\text{open})\tag{3}$$

which means that the total energy of another conformer, i.e., the open form, can be used instead of the impossible to obtain total energy of the fictitious closed system. Substituting expression (3) to (2) leads to a simple expression for the intramolecular hydrogen bond energy in OCM:

$$E\_{\rm IIB}^{\rm OCM} = E(\text{closed}) - E(\text{open}) < 0 \tag{4}$$

It should be emphasized that this article adopts the convention according to which a negative value of the obtained interaction energy means local stabilization that results from H··· Y, while on the contrary, a positive value means local destabilization. Thus, of course, as being stabilizing interactions, hydrogen bonds should be characterized by negative values.

Equation (3) requires that the open form does not differ much from the (fictitious) closed form. Therefore, the open form is most often obtained by rotating the donor or acceptor group by 180◦, as shown in Figure 1.

**Figure 1.** Scheme showing two open forms obtained by rotation of either the hydrogen-acceptor (lhs) or the hydrogen-donor (rhs) group.

It is understood that, in general, these reference open forms give different values of *E*OCM HB [33]. In principle, one can also try to use a different open form. However, I will come back to this issue further. Although the expression (3) suggests that the open form should be fully optimized, i.e., it should correspond to a local minimum on the potential energy hypesurface, another possibility is to use an open form having the geometry (more precisely, geometrical parameters) of the closed form [5,19,33,38,39,42,44,45]

$$E^f(\text{closed}) \approx E^{\text{closed}}(\text{open})\tag{5}$$

Of course, this leads to a different energy value

$$E\_{\rm HB}^{\rm QCM} = E(\text{closed}) - E^{\rm closed}(\text{open}) \tag{6}$$

since *<sup>E</sup>*closed(open) = *<sup>E</sup>*(open). In fact, Schuster advocated this option, suggesting that the reference open form should have "the least changes in molecular geometry besides a cleavage of the H-bond" and proclaiming that it "need not be a local minimum of the energy surface" [5]. Moreover, in his opinion, the full optimization of the open form geometry is even inadvisable, because this approach mixes the energy of isomerization (resulting from the change of the conformer) into the determined

energy value [5]. In fact, both of these approaches introduce different definitions of the intramolecular interaction energy (cf. Equations (4) and (6)). This situation is somewhat similar to the one that occurs when determining the interaction energy from Equation (1). Namely, the use of the monomers A and B with their geometries taken from the AB dimer defines the interaction energy, while their full optimization leads to the binding energy. The latter quantity also takes into account the correction for geometry change that takes place during the transition from the isolated form to the bound form in the dimer. Because of the fact that, in OCM, the fictitious closed form is replaced by the open form obtained by some conformational change, Schuster stressed that any splitting of the energetical difference between both forms is artificial [5]. However, it seems that this opinion may be slightly weakened by some corrective approaches [39,44]. It is valuable to present both variants of the partition of the total energy of the closed form in one scheme, as shown in Figure 2, where more concise notations are used for the respective energies.

**Figure 2.** Scheme showing two variants of the closed form total energy partition (*E*c) to the interaction energy (*E*int) and the total energy of the fictitious closed form (*Ef*c ) obtained after 'excluding' this interaction.

It is worth noting that |*<sup>E</sup>*o| > |*E*f,c o |, which, in principle, should lead to the relationship |*E*OPT int | < |*E*SPint|. It seems that at present the variant based on full geometry optimization of the open form (leading to *E*o and then *E*OPT int ) is much more popular [37] than the variant based on single point calculations (leading to *E*f,c o and then *E*SPint). In this variant, the isomerization energy mentioned by Schuster [5] is 'absorbed' into the interaction energy. In other words, this variant assumes that the changes in geometrical parameters that take place during the open form → closed form transition are related to the continuous process of creating the interaction (e.g., an intramolecular hydrogen bond) in the closed form [42,45].

Although OCM seems to be the most frequently [20–45] used theoretical method of estimating the energy of an intramolecular interaction, it is not free from further problems. The rotation of the proton-donor or the proton-acceptor group quite often leads to a new, significant interaction (either repulsive or attractive) in an open form [24,27–29,31,33,39–46]. Unfortunately, this possibility is quite often ignored. Moreover, sometimes, one or even both of the open forms cannot be used due to

symmetry of these groups. Some simpe examples representing both cases are shown in Figure 3. Of course, similar examples can be easily invented endlessly.

**Figure 3.** Examples of problematic cases in the open-closed method: (**a**) malondialdehyde, (**b**) 3-aminoacrolein, (**c**) 1-amino-2-nitroethylene.

In the case of (a) relating to the intramolecular hydrogen bond O-H··· O in malondialdehyde, the rotation of the proton-donor group -OH leads to a new rather significant interaction O··· O, while the rotation of the proton-acceptor group -CHO leads to also rather significant new interaction H··· H. In the case of (b) (3-aminoacrolein), due to the symmetry of the amino group, its rotation leads to practically the same system, while the rotation of the aldehyde group leads to a new significant H··· H interaction, similar to the case of (a). The closed form of 1-amino-2-nitroethylene does not have any such simple open forms due to the symmetry of both groups, i.e., -NH2 and -NO2.

Another, but important, question is whether these new interactions can be completely ignored [24,27–29,33,39–46]. For example, in the case of malondialdehyde, geometry optimizations (B3LYP/aug-cc-pVTZ) of the open form shown on the left-hand side of Figure 3 gives 2.89 Å for the O··· O distance and 2.02 Å for H··· H in the open form shown in the right-hand side of this figure. In the case of the open form of 3-aminoacrolein, the distance H··· H is 2.18 Å. Therefore, it would seem that these distances are too large for the interaction energy to be uncertain. However, on the other hand, the comparison of the CCC angle values in both forms (119.6◦ vs. 126.8◦ and 125.2◦ in malondialdehyde and 122.0◦ vs. 125.1◦ in 3-aminoacrolein) shows that the closed form → open form transition leads to an opening of the molecular skeleton, which may sugges<sup>t</sup> significant repulsive actions of both these interactions. It seems that the O··· O contact, in particular, cannot be completely ignored here. It is worth mentioning that both forms, i.e., closed and open, may differ in some structural aspects, e.g., the amino group in 3-aminoacrolein (b) is flat in the closed form, whereas slightly pyramidal in the optimized open form. In this case, one would have to decide whether the pyramidalization energy of the amino group should be shelled out or included in the hydrogen bond energy value [47].

In such and similar cases, it may be tempting to find other reasonable open forms, obtained after the rotation of one of the groups around the CC double bond. On the one hand, such new interactions will be avoided, but on the other hand, the configuration of the carbon skeleton of the molecule will be changed. For example, Buemi et al. [33] rebuked the use of the most extended enol and enethiol tautomers of thiomalondialdehyde [48,49] as reference structures [24,50], because, in their opinion, the *trans* configuration of double bonds seems to be too different that the *cis* arrangemen<sup>t</sup> in the closed form (Figure 4).

**Figure 4.** Closed and the most extended enol and enethiol forms of thiomalondialdehyde.

It is also worth adding that the most extended conformers are very often the global minima of a given molecule. On the other hand, open systems with a changed configuration of backbone atoms can be more reasonable in many cases. In fact, the selection of the most reasonable reference system is an individual matter for the closed form of the molecule under consideration. Therefore, this issue should be carefully analyzed before starting the appropriate calculations while using OCM.

The fundamental issue for OCM is that the presence of a new significant interaction in the reference open form leads to either an overestimation or underestimation of the determined value of the interaction energy in the closed form [42,45]. Both of the situations are shown in Figure 5.

**Figure 5.** Scheme showing the presence of a new either repulsive or attractive interaction as a cause of either overestimating or underestimating the determined value of the intramolecular interaction energy.

The presence of a new significantly repulsive interaction in the reference open form leads to a less negative total energy of this form (*E*rep o ), and thus to an overestimation of the determined value of the interaction energy (*E*OPT int,r > *E*OPT int ). Conversely, the presence of a significant attractive (stabilizing) interaction, e.g., a new hydrogen bond, results in underestimating the determined energy value (*E*OPT int,a < *E*OPT int ). Moreover, because the most extended forms are often the most stable (as already mentioned), *E*ext o < *E*c, their frivolous use can underestimate the value of the interaction energy so much that this value can even change the sign (*E*OPT int,e ) [42]. As open forms with presence of new locally repulsive interactions X··· Y (e.g., O··· O, O··· S, S··· S, etc.) and, in particular, H··· H are often treated favorably, the resulting energies may often be overestimated. This, in turn, may lead to overinterpretations of the considerable strength of some intramolecular hydrogen bonds [39].

Given the fact that the full geometry optimization of the open form can lead to a new significant interaction (repulsive or attractive) or to a significant change in structure as compared to the closed form, a solution may be to perform a partial (i.e., constrained) geometry optimization [42]. In many cases, it is enough to 'freeze' one or two dihedral angles that define the spatial orientation of the proton-donor or proton-acceptor group, the optimization of which would lead to the previously mentioned undesirable effects. However, sometimes, it is also necessary to freeze other geometric parameters [42]. The approach that is based on partial geometry optimization of the open form is, in fact, another variant of OCM, leading to the interaction energy value between these described by Formulas (4) and (6).

This variant was first proposed [42] to estimate the energy of Si-H··· Al intramolecular charge-inverted hydrogen bonds [51,52] in ten model systems. The energy values of Si-H··· Al in these systems were determined while using seven variants of OCM. In addition to either the full optimization (OPT) or complete freeze (SP) of the open form geometry, five variants of the constrained optimizations of the open form geometry were also used: (P1) only bonds optimized, (P2) only bonds and plane angles optimized, (P3) all geometric parameters optimized but dihedral angles governing the positions of the Si atom and the -AlH2 group in relation to the carbon skeleton of the reference form, (P4) all geometric parameters optimized but dihedral angles governing the positions of the Si atom and both hydrogen atoms from the -AlH2 group, and (P5) all geometric parameters optimized, but dihedral angles governing the positions of both hydrogen atoms from -AlH2. Of course, the values of the non-optimized geometric parameters in the variants SP and P1–P5 were taken from the closed form. Therefore, it can be seen that the P1–P2 variants in a controlled manner increase the number of optimized parameters (degrees of freedom), which increases the flexibility of the approach. Because the obtained results [42] very well reflect the mentioned problems related to the use of OCM, these results are shown for three molecules (Figure 6) in Table 1.


**Table 1.** Determined (B3LYP/aug-cc-pVTZ) energy values (in kcal/mol) of Si-H··· Al interactions in **1c**, **2c** and **3c** (see Figure 6).

**Figure 6.** Closed and some open forms of (**1**) H3Si-CH2-CH2-CH2-AlH2, (**2**) H3Si-CH=CH-CH= CH-AlH2 and (**3**) H3Si-CH=CH-CH2-CH2-AlH2.

First of all, it can be seen that the determined values of the interaction energy vary widely, depending on the variant of the open-closed method used in the calculations. In the case of molecule **1**, it is from −7 to −1 kcal/mol and, in the case of **3**, from −10.6 to about −0.8 kcal/mol. The values decrease (i.e., become less negative) with an increased degree of flexibility regarding the geometric parameters optimized in a given variant. It can be seen that especially even a partial optimization of dihedral angles has a large influence on the determined interaction energy values. Moreover, the rotation of the -SiH3 group in general gives significantly different values from that when the −AlH2 group is rotated. This is especially visible for the least flexible variant SP, while on going from P1 to P5 these differences become smaller and smaller. It is instructive to analyze the results from the last column of Table 1, i.e., regarding the variant with full geometry optimization of the proposed open form. While in case of **1** one reasonable value was found (−1.08 kcal/mol), in the case of **6** two significantly different values were obtained (−4.97 and −0.75 kcal/mol). The latter results from the fact that two open reference forms (see **3o1** and **3o2** in Figure 6) with quite different characteristics were obtained. Despite the fact that both forms have identical carbon frame configuration (*cis*), the **3o2** form has two new H*δ*<sup>+</sup> ··· H*δ*<sup>+</sup> interactions. On the other hand, the **3o1** form has two pairs of probably less important H*δ*− ··· H*δ*<sup>+</sup> interactions. Case **2**, on the other hand, is an important example illustratating the significant impact of the presence of a completely new type of interaction in an open form on the quality of the estimation of the interaction energy is a closed form. Namely, in both open forms (**2o1** and **2o2**), the -AlH2 group (Al has an empty *p* orbital) takes a coplanar position to the CH=CH fragment with a formal C=C double bond. This arrangemen<sup>t</sup> allows for the *pπ* →Al coupling (highlighted in Figure 6 by drawing a C=Al double bond), which significantly lowers total energies of these forms. Consequently, the estimates of the interaction energy of Si-H··· Al in **2c** are highly unreliable.

The variant of OCM with partial geometry optimization of the open form was then used [45] to estimate energies of Si-H··· B contacts in some 1-silacyclopent-2-enes and 1- silacyclohex-2-enes and helped to successfully support the earlier Wrackmeyer's suggestion based on NMR spectroscopic data [53] that this contact is considerably stronger in the latter system than in the former one. Additionally, the energies of Ge-H··· Al and Ge-H··· H-N interactions in some alkenylhydrogermanes were estimated [46] in a similar way (see Figure 7).

**Figure 7.** Closed and open forms investigated in ref. [46].

The variant of OCM with partial geometry optimization of the open form should rather be treated as a certain, but probably not the only, possible solution when the full geometry optimization of this form gives (for the reasons discussed earlier) highly unreliable estimates of the interaction energy [42,45].

The results that are presented here are enough to show that OCM in which only one reference system is utilized must be used with grea<sup>t</sup> caution so as not to write with reserve. It should be so especially when its—nevertheless the most popular—variant with the full geometry optimization of the reference open form is used. Not as rare as it may seem at first, the occurrence of new interactions (whatever attractive or repulsive) or significant structural changes (e.g., changing the skeleton of a molecule) can lead to highly unreliable estimates of the energy value of the intramolecular interaction of interest in a closed form. Indeed, Rozas et al. [32] went so far as to say that the energy value obtained from OCM should scarcely be taken as the value of the energy of the interaction in a closed form. Simply, it should rather be treated as the energetical difference between the respective forms of a molecule. On the other hand, this criticism seems a bit exaggerated. If the open form is very similar to the closed form both in terms of structure and the interactions occurring in these forms, then it seems that OCM is a worthy method of choice. The substantial similarity that is referred to herein can be provided by the presence of some rigid part of the molecule to which both the donor and acceptor groups are attached. This is the case, for example, with a benzene ring, leading to the variant of OCM, described as the *ortho*-*para* method [38,54,55], which is described in more detail in the next subsection.
