*2.3. Dimer Model (DM)*

Importantly, in the context of the present considerations, all of the methods of estimating the energy of intramolecular hydrogen bond (or more generally, interaction) in the closed form that were discussed so far, were based on the assumed model of energy additivity, which leads to quite a lot of freedom in choosing a reasonable reference system. This, in turn, leads to the known problem that the resulting hydrogen bond energy value can be quite dependent on this reference system. Moreover, even within the adopted estimation method there are often many possible variants (e.g., in OCM with only partial optimization of the open reference form). Hence, the

idea was born to abandon the assumed total energy partition of the closed form and refer to the strictly defined interaction energy of the *inter*molecular contact (Equation (1)). This idea leads to the Dimer Model (DM) [73,74]. This model was most likely used for the first time by Palusiak and Krygowski in order to estimate the interaction energy of the intramolecular *π* ··· *π* contact in 1,3,5,7-cyclooctatetraene [73]. Subsequently, DM was used by Jabło ´nski and Palusiak [74] to support the previously obtained result [43] that the intramolecular Cl··· O interaction in 3-chloropropenal is, in fact, destabilizing (repulsive) and not stabilizing [75].

To relate to the results that were obtained for the 3-halogenopropenal previously presented in Figure 13, details on the ideas of DM will be discussed on the basis of this molecule [74]. The first step in this model is to build a reasonable dimer in which the fragment of the considered interaction from the bound molecule, i.e., its closed form, is preserved. In the case of 3-halogenopropenal, this is obviously the C–X··· O=CH fragment in the ZZ conformer (see Figure 13). In order to test the reliability of the model, two dimers, namely ZE··· EZ and ZE··· EE, were constructed where, fundamentally, the C–X··· O=CH fragment was taken from the ZZ-3-halogenopropenal and then built into these dimers, as shown by green ovals in Figure 19.

**Figure 19.** Spatial arrangemen<sup>t</sup> of reference dimers and their 'open' forms used in estimating the interaction energies of intramolecular X··· O contacts in ZZ-3-halogenopropenal [74].

Unfortunately, as clearly seen (red ovals) in Figure 19, the new very short C-H··· H-C contacts appear in the dimers thus constructed. However, they can be accounted for (and 'subtracted') by using appropriate rotated (inverted) forms of the proposed dimers, in which, importantly, the relative arrangemen<sup>t</sup> of all atoms in the C-H··· H-C fragment is conserved. Consequently, the formula for the interaction energy of the intramolecular X··· O contact in ZZ-3-halogenopropenal has the following form:

$$E\_{\chi\cdots\chi}^{\rm LJM} = E\_{\rm int}(\rm dimer) - E\_{\rm int}(\rm rotated \,\rm dimer) \tag{29}$$

Table 4 presents the results obtained using this formula.

**Table 4.** Interaction energies (kcal/mol) of the X··· O intramolecular contact in ZZ-3-halogenopropenal estimated by means of several dimers utilized in Dimer Model (DM) [74].


First, it should be noted that the thus obtained estimates are positive, not negative. This result confirmed the previously [43] obtained conclusion that the intramolecular X··· O interactions in ZZ-3-halogenopropenal are in fact locally destabilizing, i.e., repulsive. As expected, the obtained repulsion values increase in the order F < Cl < Br, and those obtained for Cl and Br are similar to each other, whereas the values for F differ from them. Due to a probably slight (2.44 Å) contamination of the rotated dimers with weak C-H··· O hydrogen bond (orange ovals), simplification of DM was then applied. Namely, halogenomethane··· formaldehyde (XMe··· Fa) and halogenacetylene··· formaldehyde (XAc··· Fa) dimers were then designed (Figure 19) with imposed restrictions on the structural requirements discussed earlier [74]. Although these simplified variants introduce some subtle problems [74] and the resulting estimates are clearly lower, the values are still positive, which supports the earlier conclusion regarding the repulsive nature of the X··· O contact in ZZ-3-halogenopropenal.

#### *2.4. Isodesmic Reactions Method (IRM)*

In many areas of physical organic and theoretical chemistry the so-called isodesmic reactions are used [63,76–93]. These are more or less hypothetical reactions, in which the same numbers of single and multiple bonds of the same type are present on both sides of this reaction, i.e., of the reagents and of the products. If, in addition, the relevant atoms conserve their hybridization, then these reactions are called the homodesmotic reactions [79–83,87–92]. The conservation of the atomic hybridizations makes the homodesmotic reaction a more reliable description of a given phenomenon than the less demanding isodesmic reaction. The use of isodesmic and homodesmotic reactions allowed for a more detailed theoretical description of many physical processes and effects, such as the extra stability due to cyclic *π*-electron delocalization [88]. Homodesmotic reactions are also often used in order to estimate the energy of intramolecular hydrogen bonds [32,38,39,44,63,84,86,87,89,91] or some other interactions of interest [42–44,55,87,93].

The reliability of the Isodesmic Reactions Method (IRM) is based on the assumption that the total energy of a molecule I can be partitioned into energies of chemically recognizable fragments, such as bond energies, and that those energies are transferable among various molecules which, however, involve similar chemical units. A general scheme of a simple homodesmotic (also isodesmic) reaction for a model system featuring an intramolecular X-H··· Y hydrogen bond is shown in Figure 20.

**Figure 20.** General scheme showing a homodesmotic reaction for a model molecule featuring an intramolecular X-H··· Y hydrogen bond.

In this figure, the molecular framework, which, of course, may vary from molecule to molecule, is drawn as a box for simplicity and, moreover, those C-H bonds in molecules II, III, and IV, which are not present in the parent molecule I are marked by a zigzag bond line. When comparing both sides of the homodesmotic reaction shown in Figure 20, it can be easily seen that all the bonds, except the only one denoting the H··· Y contact in the parent molecule I, on the left side of this reaction are also present on the right side of this reaction. Accordingly, the only missing 'bond' on the right side of this reaction equation is the intramolecular H··· Y contact in the parent molecule I. Thus, the interaction energy of this contact can be obtained by the following expression

$$E\_{\rm IIB}^{\rm IRM} = E(\rm I) - E^{\rm f}(\rm I) < 0 \tag{30}$$

where

$$E^{\rm f}(\rm I) = E(\rm III) + E(\rm IV) - E(\rm II) \tag{31}$$

In these two equations, *E*(I) is the total energy of the fully optimized parent molecule I, whereas *E*f(I) can be regarded to as the total energy of its fictitious counterpart featuring no H··· Y contact.

As with conformational methods, the question now arises as to what geometries to use for the auxiliary molecules II, III, and IV [42–44,91]. The vast majority of calculations that are related to isodesmic or homodesmotic reactions use fully optimized geometries, so that the total energies in Equations (30) and (31) are total energies of fully optimized molecules. For this reason, such an approach can lead to considerable doubts regarding the reliability of *E*IRMHB if only full geometry optimization of at least one of the molecules leads to new significant interaction(s) or to a significant change in molecular structure compared to I [42]. On the other hand, if the structural fragments in II, III, and IV do not differ significantly from those in I, then IRM can give reasonable estimates of interaction energies of the hydrogen bond (or any other contact of interest) in I. It seems that rigid ring molecules should be privileged here [44]. Another possibility that is very rarely considered [42–44] is that the geometry of the parent molecule I is transferred to the auxiliary molecules II, III, and IV. However, then, the question arises, what to do with the C-H bonds that the molecule I does not have (they are indicated in Figure 20 by a zigzag line). Hence, a field for different IRM variants arises here. For example, these bonds can be optimized, or they can be given the length of either the C-X or C-Y bond of molecule I, or any other reasonable value as, e.g., 1 Å. It is worth mentioning that the use of not fully optimized molecules II, III, and IV leads to an overestimation of the hydrogen bond energy value in I, i.e., *E*IRMHB .

Despite the fact that, as mentioned above, the method of estimating the interaction energy either of some hydrogen bonds or some other kinds of intramolecular interactions in I that is based on isodesmic/homodesmotic reactions is quite popular [32,38,39,42–44,55,63,84,86,87,89,91,93], its reliability, in general, may raise some doubts. For example, it has been shown that the comparison of the interaction energy in closely related systems disqualifies IRM (Figure 21) [44].

**Figure 21.** The values (in kcal/mol) of the interaction energies of the C-H··· O contacts estimated by OCM (black values) and IRM (red values) [44].

As one can see, the values of the interaction energies of C-H··· O contacts in both of these similar molecules are very close to each other (0.30 and 0.80 kcal/mol) when OCM is used, while IRM gives values very different from each other, also in terms of sign ( −2.97 and 0.86 kcal/mol). Moreover, the failure of IRM also manifested itself in the unphysical positive slope of the dependence of *E*IRMHB on the electron density at the critical point (*ρb*) of the C-H··· O interaction featuring a sp<sup>3</sup> hybridized carbon atom (see Figure 22) [44].

In contrast, RRM and GCM gave physically justifiable negative slopes.

**Figure 22.** Correlations between RRM-, GCM-, and IRM-based interaction energies of C-H··· O contacts featuring either sp<sup>2</sup> (open circle) or sp<sup>3</sup> (full circle) hybridized carbon atoms and the electron density at the bond critical points of these contacts [44].
