2.6.1. Iogansen's Relationship

Based on the results obtained for various phenol complexes, in 1969 Iogansen and Rassadin proposed an empirical formula for the relationship between intermolecular hydrogen bond energy (the hydrogen bond enthalpy of formation) and the red-shift of the X-H stretching vibration frequency (Δ*ν*XH) that takes place upon the hydrogen bond formation [150,151]

$$E\_{\rm IIB} = 0.33\sqrt{\Delta\nu\_{\rm XH} - 40} > 0\tag{36}$$

The Δ*ν*XH red-shift can be obtained either from experimental measurements or theoretical computations. Unfortunately, although the Iogansen's relationship was derived for intermolecular hydrogen bonds [150,151], it is also used in order to estimate intramolecular hydrogen bond energies [119,152–154], where its applicability is at least unclear. One such use will be discussed in more detail here.

Using the formula (36), Nikolaienko et al. [119] estimated the energies of an impressively large number (However, it is highly doubtful that all of the more than 4000 conformers studied there actually correspond to true minima on the potential energy hypersurface and thus it is unclear what these conformers really mean.) of O-H··· O, O-H··· N, N-H··· O, and O-H··· C intramolecular hydrogen bonds in some biologically relevant DNA-related molecules. It is obvious that the use of this expression requires the knowledge of the reference vibration frequency *ν*free XH . Despite the fact that, in the case of an isolated molecule, obtaining such a quantity is, at least in terms of theoretical calculations, a fairly simple process; however, in the case of intramolecular interaction, it is controversial which frequency should be best taken as a reference [40,41]. It is enough to mention here a very common problem with coupling vibrations. Anyway, Nikolaienko et al. [119] stated that "*ν*free XH has been calculated as the simple average of stretching vibration frequencies for XH groups, such that: (i) their H atom does not participate in any XH··· Y bonding (i.e., no QTAIM bond path ends on it except for the one corresponding to the XH covalent bond), and (ii) unique normal vibration exists with *c*XH *j* > *<sup>c</sup>*th.", where *c*XH *j* = *<sup>∂</sup>l*XH/*∂<sup>x</sup>*j (*x*j is the *j*-th normal coordinate) and *c*th is the fixed treshold value (=0.92). Subsequently, based on the thus calculated energy values, these authors obtained, for each type of the hydrogen bond under consideration, a relationship with the determined value of the electron density at the bond critical point of a given hydrogen bond, *E*HB = *Aρ*BCP + *B*. The linear fit values for A and B thus obtained (the signs have been changed, so that the resulting *E*HB values are negative) by Nikolaienko et al. [119] for each of the types of hydrogen bonds considered by them are shown in Table 5.

In principle, one could complain that the *B* values should be exactly zeros, as a zero electron density value should result in no hydrogen bonding and, therefore, also zero energy value. On the other hand, however, a significant portion of the electron density in the BCP is only due to the mutual overlapping of atomic orbitals of various atoms, and not only H and Y [107]. This fact was completely ignored here. Most likely, a correction is required to at least subtract the electron density contributions from H and Y, and perhaps even X. A somewhat similar correction has recently been proposed by Scheiner for calculating corrected NMR chemical shift for a proton involved in an intramolecular hydrogen bonding [141].


**Table 5.** Linear fit parameters for the linear relation *E*HB = *Aρ*BCP + *B* between the hydrogen bond energy (in kcal/mol) and electron density at the BCP (au) of the indicated hydrogen bond [119].

## 2.6.2. Chemical Shift—Based Method

As mentioned earlier, one of the characteristic effects accompanying the formation of a hydrogen bond is the 1H NMR signal shift for the donor proton, i.e., the so-called downfield shift or proton magnetic deshilding, Δ*δ* [138–141]. As early as in 1961, Gränacher [155] noticed a linear correlation between the proton chemical shift and the shift of the infrared absorption band, announcing the possibility of obtaining values of intermolecular hydrogen bond energies via the following equation [145]

$$E\_{\rm IIB} = \Delta \delta + (0.4 \pm 0.2) > 0\tag{37}$$

Quite recently, this expression was used by Afonin et al. [121] in order to estimate energies of many different intramolecular hydrogen bonds, including improper, blue-shifting [156,157] ones, the energy of which cannot be estimated while using Equation (36). Importantly, their aim was to compare the energies that were obtained in this way with their counterparts obtained using other popular methods of estimating the energy of hydrogen bonds. Moreover, this approach allowed them to obtain new correcting parameters in the formulas combining the hydrogen bond energy with the local potential energy density (*V*BCP) and the electron density (*ρ*BCP) at the BCP of these interactions and determine the quality (However, the combination of experimental rather than theoretical 1H NMR data with theoretically determined QTAIM parameters for theoretically obtained geometries of the molecules under consideration is somewhat suspicious.) of the methods that are based on these parameters [121]. In this way, Afonin et al. showed that the estimates based on geometric parameters [158–160] are very poor (therefore, they are not discussed in this review). Nevertheless, even less reliable (2–3 times overestimation) values were obtained while using the uncorrected Espinosa's formula, whereas applying a multiplier of 0.31 significantly improved the results. Based on the values of the hydrogen bond energies that were obtained by Equation (37) and the calculated values of either *V*BCP or *ρ*BCP, Afonin et al. [121] obtained Equation (33) for the former parameter and *E*HB = −191.4 *ρ*BCP + 1.78 for the latter one, where the coefficients A and B are noticeably close to those that were obtained earlier by Nikolaienko et al. [119] (see last column in Table 5).
