2.5.1. Espinosa's Method (EM)

Based on empirical data for many systems featuring intermolecular hydrogen bonds of the H··· O type, Espinosa et al. [115] proposed the following relationship between the energy of an intermolecular hydrogen bond and the local electronic potential energy density that is determined at the bond critical point of a given hydrogen bond

$$E\_{\rm HB}^{\rm EM} = \frac{1}{2} V\_{\rm BCP} \tag{32}$$

It should be noted that both the simplicity of the formula (32) and the easy availability of the *V*BCP value (QTAIM calculations) resulted in a rather uncritical acceptance of this expression for determining not only the energy of intermolecular hydrogen bonds, but also the energy of intramolecular interactions. The latter, however, must be firmly criticized [44]. Since *V*BCP is a negatively defined quantity [94], the energy of any hydrogen bond (or other interaction) determined by formula (32) will always give a negative *E*EMHB value. Therefore, according to EM, any interaction will be stabilizing. However, as shown [44], many C-H··· O contacts, which many would probably consider weak hydrogen bonds, are, in fact, destabilizing, i.e., repulsive in nature. Indeed, *V*BCP, which is crucial in formula (32), was interpreted [115] as the pressure exerted by the system on the electrons in the closest vicinity of BCP. Therefore, it is easy to imagine a situation that the short distance H··· Y is merely forced, e.g., by the stiffness of the molecular skeleton or some steric interactions, which lead to a high value of *V*BCP and, consequently, to a large value of *E*EMHB , suggesting strong hydrogen bond, though in reality the interaction may be locally repulsive in nature. For this reason, the use of EM in the cases of intramolecular interactions is not recommended [44].

It should be added that some concerns regarding EM have also been pointed out by Gatti et al. [118] and more recently by Nikolaienko et al. [119]. For example, the latter authors complained that the expression (32) was obtained while using data relating to crystallographic structures in which, as is known, the distances H··· Y are often much shorter due to lattice forces. Moreover, this expression was obtained for X-H··· O (X = C, N, O) hydrogen bonds only and its use for hydrogen bonds of other types is unreliable. They also refer to the example of hydrogen bonds of the H··· F type, for which the Espinosa formula (32) should rather have a factor of 0.31 [120]. To all of these allegations [119], it can be added that the hydrogen bond energies were obtained [115] with a real mixture of theoretical methods. Therefore, it seems necessary to revise the derivation of formula (32). In fact, some modifications to the orginal Espinosa's formula (Equation (32)) have been proposed [44,118,120,121]. For example, Afonin et al. [121] obtained the formula

$$E\_{\rm IIB} = 0.277V\_{\rm BCP} + 0.45\tag{33}$$

in which the slope value of 0.277 is very close to the mentioned 0.31 value that was obtained by Mata et al. [120]. Even a little earlier, Jabło ´nski and Monaco proposed correcting the Espinosa's formula (32) by adding a constant *k* of 3.4 kcal/mol, thus to use the expression *E*HB = *E*EMHB + 3.4 [44]. Importantly, this expression was specifically dedicated to intramolecular hydrogen bonds.

## 2.5.2. Interacting Quantum Atoms (IQA)

As already mentioned, the Interacting Quantum Atoms (IQA) approach [116,117], which is based on QTAIM, allows for the total energy of a system to be divided into mono- and polyatomic components. Among many energy terms available by means of IQA, the most important one in the context of this article is the interatomic interaction energy defined as follows

$$E\_{\rm int}^{\rm E\_1\rm E\_2} = V\_{\rm nn}^{\rm E\_1\rm E\_2} + V\_{\rm en}^{\rm E\_1\rm E\_2} + V\_{\rm ne}^{\rm E\_1\rm E\_2} + V\_{\rm ee}^{\rm E\_1\rm E\_2} \qquad (E\_1 \neq E\_2) \tag{34}$$

where *V*E1E2 nn is the repulsion energy between nuclei of atoms E1 and E2, *V*E1E2 en is the attraction energy between electrons of the atom E1 and the nucleus of the atom E2, *V*E1E2 ne is the attraction energy between the nucleus of the atom E1 and the electrons of the atom E2, and *V*E1E2 ee is the interatomic two-electron repulsion energy. Because the *E*1 and E2 atoms may be e.g., the H and Y atoms from the X-H··· Y hydrogen bridge, it is evident that IQA via Equation (34) can be a suitable tool for calculating the energy of inter- and, more importantly, intramolecular hydrogen bonds. In this case, Equation (34) takes the following form

$$E\_{\rm int}^{\rm H\cdots Y} = E\_{\rm HB}^{\rm IQ} = V\_{\rm nn}^{\rm HY} + V\_{\rm en}^{\rm HY} + V\_{\rm ree}^{\rm HY} + V\_{\rm ee}^{\rm HY} \tag{35}$$

It should be emphasized that the determination of the interaction energy of H··· Y using formula (35) does not require assuming any reference system or referring to empirical data, and from this point of view the IQA-based approach is absolutely unique and, therefore, also worth a wider study of its applicability. It should also be added that *E*1 and *E*2 of Equation (34) can be any atoms, and therefore the interaction energy of any interatomic contact, not just hydrogen bonding, can be determined in a similar way. Moreover, these atoms do not need to be linked to each other by a bond path, nor do they need be in a close proximity to each other.

Unfortunately, as compared to intermolecular hydrogen bonds [122–126], the IQA-based estimates of the energy of intramolecular hydrogen bonds are relatively rare [127–130]. It is worth noting here that the list of IQA applications given most recently by Guevara-Vela et al. [117] can be easily supplemented with various repulsive interactions [105–108,112–114], which are often related to the presence of an appropriate bond path. Therefore, these cases are important in the very discussion on the interpretation of a bond path and the earlier connection of its presence on molecular graphs with the stabilizing nature of interactions, as stated in the orthodox QTAIM [94].

It should be emphasized that, as it seems, the interatomic interaction energy itself (i.e., *E*E1E2 int ) is currently not as popular quantity as the exchange-correlation component (*E*E1E2 ee,xc) of the interelectron interaction energy (*E*E1E2 ee = *E*E1E2 ee,C + *E*E1E2 ee,xc). This, in turn, results from the fact that *E*E1E2 ee,xc was associated with the presence of bond path via the concept of so-called privileged exchange channels [131]. Moreover, even more importantly, at short distances [113] *E*E1E2 ee,xc is related to the strength of a given bond [123]. Therefore, it turns out that, despite the possibility of determining the interatomic interaction energy and thus also of an intramolecular hydrogen bond, which is significant for the theory of intramolecular interactions, this quantity in IQA has become less important than dimensionally much lower exchange energy.
