**1. Introduction**

The hydrogen bond (HB) is a dominant noncovalent interaction found in chemical and biological systems [1–4]. The term "hydrogen bond" seems to have emerged around 1930, from the works of Pauling [1] and Huggins [5,6]. However, the mention of weak, ye<sup>t</sup> specific interactions involving the hydrogen atom is much older. The dimeric association of molecules with hydroxyl groups was suggested by Nernst in 1892 [7]. The term "Nebenvalenz" by Werner [8] and "weak union" by Moore and Winmill [9] are other early stipulations of this noncovalent interaction. In 1920, Latimer and Rodebush [10] suggested that the hydrogen nucleus in an aqueous solution of amines is held jointly by two octets, constituting a weak bond. Barnes, while studying the structure of ice [11], suggested that the hydrogen atoms were midway between the two oxygen atoms, though he did not explicitly mention the hydrogen bond. Huggins [12] claimed that he was the first to propose the term "H-bond" in 1919. His later usage of the term "hydrogen-bridge" may have led to the German word "Wasserstoffbrücke." The concept of HB gained popularity after Pauling published his classic book, *The Nature of the Chemical Bond* in 1939 [1]. Pimentel and McClellan [13] suggested that an HB exists when (i) there is evidence of a bond and (ii) there is evidence that this bond involves a hydrogen atom already bonded to another atom. The recent definition of HB by IUPAC [14] is similar in spirit to that in Ref. [13]. The former [14] states that "the hydrogen bond is an attractive interaction between a hydrogen atom from a molecule or a molecular fragment X–H in which X is more electronegative than H, and an atom or a group of atoms in the same or a different molecule, in which there is evidence of bond formation."

An HB may be generally represented as X–H···Y, where X–H is the proton donor and Y is a proton acceptor. The X–H···Y interactions such as O–H···O, N–H···O, N–H···N,

**Citation:** Deshmukh, M.M.; Gadre, S.R. Molecular Tailoring Approach for the Estimation of Intramolecular Hydrogen Bond Energy. *Molecules* **2021**, *26*, 2928. https://doi.org/ 10.3390/molecules26102928

Academic Editor: Mirosław Jabło ´nski

Received: 17 April 2021 Accepted: 12 May 2021 Published: 14 May 2021

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S–H···O, etc., in neutral molecular systems, exhibit interaction energies lying between ~1 to 20 kcal/mol. Typical H···Y distances suggested in the literature fall in the range of ~1.2–3.0 Å and X–H···Y angles lie between 100 and 180◦ [2–4]. The HB's in liquid water are central to water's life-providing properties [1,15]. It is stipulated in the literature [16] that if HBs in water were 7% stronger or 29% weaker, water would not be a liquid at room temperature. HBs provide a significant driving force for the native structures and functions of biomolecules [17,18]. Hence, it is of grea<sup>t</sup> importance to reliably estimate these X − H··· Y HB strengths for shedding light on several physicochemical phenomena and life processes.

The theoretical estimation of intermolecular X–H···Y HB strength in a complex A···B is routinely performed using a supermolecular approach, in which the HB energy (EHB) is estimated as EHB = E A···B − (E A + EB). Several methods for estimating the intermolecular interaction energies are reported in the literature (see, e.g., Refs. [19–22]). On the other hand, quantifying an intramolecular hydrogen bond (IHB) strength is not as straightforward as the intermolecular one. The main difficulty lies in isolating the X–H···Y interaction present within a molecule than in a dimer or a complex. Many studies for gauging the strength of the IHB in the literature are based on spectroscopic- [23–26] and electron density topological approaches [27–30]. Nevertheless, some empirical, semiempirical, and ab initio procedures [31–39] have also been reported in the literature for estimating the IHB energy. These have been nicely summarized by Jablonski [40] in his article in this Special Issue, and we shall discuss only the aspects of these methods (e.g., their merits and demerits) that are not explicitly covered in Ref. [40].

One of the early approaches is the conformational analysis (CA), in which two different conformers of the reference molecule are considered. These conformers are chosen such that the HB is kept intact in one of the conformers and is broken in another. The energy difference between these two conformers is then taken as the measure of IHB energy [32,40,41]. A significant disadvantage of this method is that the estimated IHB energy is erroneous due to the incorporation of attractive (*syn-anti*) or repulsive (*anti-anti*) additional interaction in one of the conformers [42]. In another similar procedure, viz., the *ortho-para* method [43], the IHB energy of the X-H···Y bond formed by two substituents, which are *ortho* to each other, is taken as the energy difference between the *ortho* and *para* forms of the reference molecule. However, this method applies only to aromatic systems in which an HB is present in two substituents, which are *ortho* to each other. The main drawback of this method is that it assumes that the electronic effects caused by the substituent at different positions are similar between the *ortho*- and *para*-conformers [44–46].

Yet, another indirect approach is the isodesmic/homodesmic reactions. In the former, the IHB making/breaking reaction is written so that the number and type of bonds on either side of the reaction are equal, except the HB, which is retained in one of the reactants [33,47,48]. A further assumption is that the atomic hybridization is conserved on both sides of the reaction. In that case, the method is called homodesmic reaction, which is supposed to give more reliable energy estimates than the isodesmic reaction [49–51]. The main drawback of the isodesmic/homodesmic reaction approach is that it does not give HB energy but includes strain energy due to the formation of a ring structure [52]. Another major disadvantage of these indirect methods is that they are applicable only to the evaluation of energy of a single HB present in the system and cannot be employed in a system containing multiple HBs.

Another popular but indirect method is based on the quantum theory of atoms in molecules (QTAIM) [53]. In this method, the presence of a (3, −1) bond critical point (BCP) of the molecular electron density (MED) between H···Y, is considered as the signature of an HB. The large/small value of the MED at the BCP is seen to correlate with strong/weak X–H···Y interaction [54–58]. Espinosa et al. [59] proposed an empirical relation, *E*HB = 0.5 V(**r**cp), where V(**r**cp) is the potential energy density at BCP. Interacting quantum atoms (IQA) [60] framework, leading to QTAIM-compatible energy partition, is another indirect approach wherein the HB energy is calculated as the sum of the classical Coulombic interaction between groups involved in the HB and the exchange-correlation energy. It has

been pointed out that there is no check on the reliability of HB energy provided by both of these methods [61,62]. Further, these empirical equations are applicable only when a (3, −1) BCP is present. For instance, a (3, −1) BCP at O–H···O bond is conspicuous by its absence in all the polyols having an O–H···O interactions between the vicinal -OH groups. However, the weak O–H···O HB was confirmed in ethylene glycol-based on the vapor phase OH-stretching overtone spectroscopy [63–65]. Some other variants and indirect empirical equations proposed in the literature for estimating the strength of IHBs are summarized in Ref. [40]. Hence, we skip the discussion of these methods.

A brief review of the above literature suggests that these empirical and indirect methods for estimating IHB energy are generally limited to singly H-bonded systems. These cannot be readily extended to a system containing multiple HBs and hence also to the estimation of HB cooperativity due to an interconnected network of HBs. Most importantly, there is no check on the reliability of the estimated HB energies. Thus, it was felt necessary to come up with a direct theoretical method for a reliable estimation of IHB energy. Deshmukh and Gadre [66,67] proposed such a procedure, based on the in-house developed molecular tailoring approach (MTA) for the ab initio treatment of large molecular systems [68–82]. The MTA currently enables the calculation of one-electron properties, geometry optimization, and the calculation of vibrational infrared and Raman spectra of large molecules/clusters using DFT or correlated methods. Before discussing the application of MTA for the IHB energy estimation, we explain the working principle of MTA, with an illustrative example, in Section 2.
