**2. Molecular Tailoring Approach**

The molecular tailoring approach (MTA) is a fragmentation-based technique developed by Gadre et al. [68–82]. Within MTA, a spatially extended molecular system under consideration is partitioned into a set of overlapping fragments (called the "main fragments") on which ab initio calculations for one-electron property or the energy are carried out. The fragmentation may be carried out automatically or manually. The quality of the fragmentation scheme is gauged by a parameter called R-Goodness (Rg), which may be estimated as follows: Put a sphere of radius R centered on a reference atom i so that all the atoms of the parent system lying within this sphere belong to at least one of the main fragments. The maximum value of R obeying this condition is called the Rg value of atom i in the given fragmentation scheme. The minimum of such atomic Rg values is called the Rg value of the scheme. In general, the larger the Rg value of a fragmentation scheme, the better is the chemical environment of each atom mimicked [69]. After choosing an appropriate fragmentation scheme, molecular energy E, of the spatially extensive parent system is estimated approximately by patching those of the individual fragment energies [69] using Equation (1).

$$\mathcal{E} = \sum\_{\mathbf{i}} \mathcal{E}^{\mathcal{E}\_{\mathbf{i}}} - \sum\_{\mathbf{i} < \mathbf{j}} \mathcal{E}^{\mathcal{E}\_{\mathbf{i}} \cap \mathcal{F}\_{\mathbf{j}}} + \dots + (-1)^{\mathbf{k}} \sum\_{\mathbf{i} < \mathbf{j} < \dots < \mathbf{n}} \mathcal{E}^{\mathcal{F}\_{\mathbf{i}} \cap \mathcal{F}\_{\mathbf{j}} \cap \dots \cap \mathcal{F}\_{\mathbf{n}}} \tag{1}$$

where the energy E of the parent molecule is estimated as the sum of energies of primary fragments { *Fi*} minus the sum of energies of binary overlap fragments {Fi ∩ Fj plus the sum of energies of ternary overlap fragments, etc. Here, k stands for the degree of overlap., e.g., for binary overlap, k = 1, for ternary overlap, k = 2, etc. Equation (1) is generalized for estimating an electronic property of the molecule, such as the energy gradients, the Hessian matrix elements, etc.

We now illustrate the fragmentation procedure with a test example, viz., the αtocopherol molecule (shown as M) in Scheme 1, fragmented into three primary fragments F1, F2, and F3 (shown by appropriate circles). The fragments F4 and F5 are the binary overlaps of fragments F1, F2, and F2, F3, respectively. Here, the term binary overlap means the common structural part of two primary fragments. In fragmentation Scheme 1, the ternary fragment (overlap of three fragments F1, F2, and F3) is absent. The valencies of the cut regions are satisfied by placing the H-atoms at the appropriate C–H distance of

1 Å along the cut C–C bond. The calculations for the single point energy (or the property) of these fragments F1 to F5 are performed. For this test case, we report the MTA energy calculation of M at HF/6-31+G(d,p) level theory using Equation (1). It should be noted that the HF method employed here is only for illustrative purposes. MTA method works at any correlated level of theory. All the calculations are performed with the Gaussian 16 package [83].

**Scheme 1.** Illustration of the molecular tailoring approach (MTA)-based fragmentation procedure for α-tocopherol, shown as the parent molecule, M. See text for details.

In the present case, the energy (EMTA) of the parent molecule (M) is obtained by Equation (2) as

$$\mathbf{E\_{MTA}} = \{\mathbf{E\_{F1}} + \mathbf{E\_{F2}} + \mathbf{E\_{F3}}\} - \{\mathbf{E\_{F4}} + \mathbf{E\_{F5}}\} \tag{2}$$

In the present case of α-tocopherol this energy is calculated utilizing the energies of the fragments as EMTA = {(−769.56869) + (−469.57421) + (−391.51579)} − {(−118.26152) + (−235.36444)} = −1277.03273 hartrees (a.u.). The actual energy (EFC) at the HF/6-31+G(d,p) level of theory is EFC = −1277.03288 a.u. Thus, the error (EError) in the estimation of molecular energy is given EError = EFC − EM = −0.00016 a.u. This error can be further reduced using the so-called grafting procedure embedded in the current version of MTA [77,78].

Gadre et al. first proposed the MTA methodology for estimating the electrostatic properties of large, closed-shell molecules [68]. However, in the last 24 years, the scope of MTA was extended for the estimation of molecular energy [69], geometry optimization [70,71], the estimation of the Hessian matrix [72], the computation of vibrational infrared [73] and Raman spectra [74], and binding energies of large molecular clusters and complexes [75–82]. Since the fragment computations are independent of each other, MTA has the advantage that the energy computation of the parent molecule is intrinsically parallel. Further, MTA can currently work with Gaussian, GAMESS, and NWChem at the back-end, thereby becoming a powerful tool for ab initio treatment of large molecules and clusters, when used in conjunction with a high level of theory, such as MP2 or CCSD(T) employing a large basis set. One important application of MTA is in estimating the IHB energy [52,66,67,84–92]. This will be discussed in the next section.

## **3. Intramolecular Hydrogen Bond Energy Estimation by Molecular Tailoring Approach**

With the above brief introduction to MTA, we now discuss its application for estimating the IHB energy. As discussed in the introduction section, intermolecular X–H···Y HB energy in a complex A···B is estimated as EHB = E A···B − (E A + EB). This estimation is possible because the energies of the two monomers A and B can be separately calculated. In the case of intramolecular X–H···Y interaction, such a separation is, in general, difficult. However, the MTA procedure allows the generation of fragments so that the atoms/functional groups involved in the HB formation are parts of two different fragments.

The fragmentation procedure is illustrated in Scheme 2 for the test molecule of 1,2,4- butanetriol. In Scheme 2, the parent test molecule, denoted as M, is shown at the center. The geometry of 1,2,4-butanetriol was optimized at the MP2/6-31+G(d,p) (default option: frozen core) level of theory using the Gaussian package [83]. The energy of the optimized structure is −383.01926 a.u. at MP2/6-31+G(d,p) level. The three oxygen atoms are shown as O1, O2, and O3 (see Scheme 2), with the two HB's, viz., **HB1** (O2-H···O1) and **HB2** (O3-H···O2) whose energy is to be estimated. For this purpose, the parent molecule is cut into three primary fragments F1, F2, and F3, obtained by replacing −O1H, −O2H, and −O3H groups, respectively, with an H atom each. Dotted circles show these cut regions on the original molecule. The H-atoms are added along the respective C–O bonds (which are cut to form these primary fragments) so that the C–H distance is 1 Å. Hydrogen is the simplest monovalent atom that can be used for satisfying the valencies of cut regions. It is emphasized here that H-atoms placed at slightly different distances (say at 0.9 or 1.1 Å) from the C-atom do not change the results appreciably. This is because of the cancellation of errors in estimating the molecular energy using these fragments. Fragments F4, F5, and F6 are obtained by taking the binary intersection of these primary fragments, i.e., (F1 ∩F2), (F2 ∩F3), and (F1 ∩F3), respectively. Here, intersection means the common structural parts between two primary fragments apart from added H-atoms. For instance, in fragment F4, C1(H2)–C2(H2)–C3(H2)–C4(H2)O3(H) is the common structural part that is also present in fragments F1 and F2. Similarly, fragment F7 is the common intersection of three primary fragments F1, F2, and F3, i.e., (F1 ∩F2 ∩F3). A single point energy evaluation, at MP2/6-31+G(d,p) level of theory, is carried out on all seven fragments obtained by the above fragmentation procedure. The fragment geometries are not optimized to avoid the conformational changes in them so that they lead to reliable estimates of IHB energies. It is necessary first to provide a check on MTA application to the parent molecule, M. As discussed above, the actual energy of the original molecule (M) is *E*M = −383.01926 a.u. at the MP2 level of theory. Using the MP2 single point energies of these fragments, the estimated molecular energy of M is: E M = {EF1 + EF2 + EF3} − {EF4 + EF5 + EF6}+EF7 = {−307.97304 + ( −307.9642) + ( −307.97765)} − {−232.92410 + ( −232.92514) + ( −232.93273)} + (−157.88552) = −383.01844 a.u. The error, ΔE = |MTA energy - actual energy| in molecular energy indeed turns out to be very small, viz., 0.00082 a.u. This excellent agreemen<sup>t</sup> between the MTA-estimated and actual energy suggests that the present fragmentation scheme is reliable for evaluating HB energies.

Now we estimate the energy of two hydrogen bonds **HB1** and **HB2,** in the parent 1,2,4-butanetriol. Recall that the hydroxyl groups involved in the formation of hydrogen bond **HB1** are O1–H and O2–H. These hydroxyl groups were replaced in fragments F1 and F2, respectively, by H-atoms. Putting the geometry of fragment F1 over F2, we regenerate the parent molecule except following two things: (i) the O–H···O H-bond, i.e., the **HB1** interaction between O1–H and O2–H present in the parent molecule is missed out and (ii) there is double counting of common structural part between F1 and F2 (viz., the secondary fragment, F4). Upon addition of single-point energies of fragments F1 and F2, followed by subtraction of the energy of fragment F4 would give the energy of the parent molecule except that the energy of the HB, viz., **HB1** is missed out. If the energy of the parent 1,2,4-butanetriol E M is subtracted from (EF1 + EF2 – EF4), the HB energy E**HB1** is obtained as E**HB1** = (EF1 + EF2 − EF4) − E M = 3.84 kcal/mol. In a similar fashion, E**HB2** is obtained

as E**HB2** = (EF2 + EF3 − EF5) − EM = 1.60 kcal/mol. It should be noted here that these estimated HB energies are in the gas phase. However, the MTA-based method in principle can provide HB energies in the solvent phase, wherein the energies of the fragments in solvent (using continuum solvation model) could be employed.

**Scheme 2.** Fragmentation procedure for estimating the energies of the H-bonds, **HB1**, and **HB2** in 1,2,4-butanetriol (Parent M) molecule. See text for details.

We note that the two HBs, **HB1** and **HB2**, are interconnected, forming an H-bond network. Such networking of H-bonds leads to a phenomenon called cooperativity [67].In general, it is anticipated that the strengths of **HB1** and **HB2** are enhanced because of this networking effect. To estimate the contribution of cooperativity toward each of these two H-bonds, we reestimated the HB energy of these two HBs by isolating them from each other. The difference between the HB energy estimated earlier (in the presence of network) and the one when they are isolated (in the absence of a network) is the cooperativity contribution toward this HB. For example, consider fragment F3 in which only **HB1** is present and fragment F1 in which **HB2** is present. To estimate the energy of **HB1** in the absence of the networking effect of **HB2**, we consider fragment F3 as our parent molecule.In the present case, fragments F5 and F6 are the two primary fragments that, when placed over each other, would give us the parent fragment F3 except **HB1**, and fragment F7 is the binary overlap of F5 and F6. Therefore, utilizing these fragments' energies, the energy of **HB1** is obtained as EHB1 = (EF5 + EF6 − EF7) − EF3 = 3.32 kcal/mol. Similarly, the energy of **HB2** in the absence of the networking effect of **HB1** is obtained as EHB2 = (EF4 + EF6 − EF7) − EF1 = 1.09 kcal/mol. These reestimated HB energies are indeed smaller than those estimated in the presence of the networking effect. The difference in the energy is cooperativity contribution. The cooperativity contribution to **HB1** is Ecoop HB1 = 3.84 − 3.32 = 0.52 kcal/mol and that for **HB2** is Ecoop HB2 = 1.60 − 1.09 = 0.51 kcal/mol. In the present test case, the estimated cooperativity contributions are not large because only two HBs are present. The later sections will show that cooperativity values in some molecules can indeed be as large as a typical HB energy.

The HB energies obtained by applying the above procedure to some alkanetriol molecules are shown in Table 1 [66]. The estimated HB energies fall in a range between 1.50 and 4.97 kcal/mol (see Table 1). This is the expected energy range from chemical intuition. Further, these HB energies are in a qualitative agreemen<sup>t</sup> with those expected from the corresponding HB distances. For instance, the strongest HB in 1,2,5-pentanetriol has an energy of 4.97 kcal/mol, with the corresponding HB distance being the shortest (1.80 Å) among all the alkanetriols reported in Table 1. One of the noteworthy results in Table 1 is that the error in estimating molecular energies of all the alkanetriols is quite small, viz., between 0.40 to 0.65 kcal/mol. By considering this accuracy, we estimate the maximum error associated with our calculated HB energies to be 0.3 kcal/mol. The present method is thus capable of calculating accurately the IHB energies and cooperativity values of multiply H-bonded systems. This is a significant advantage of the current method over the other indirect approaches reported in the literature.

**Table 1.** The H-bond (HB) distances (in Å), HB energies (in kcal/mol), and the error in the molecular energy estimation for alkanetriols using similar fragments, ΔE=|EM − Ee|. The corresponding O–H stretching frequencies (cm−1) and the molecular electron density (MED) value at the (3, −1) bond critical point (BCP) (a.u.) are also shown. The calculations are performed at MP2 (full)/6-311++G(2d,2p) level theory.


a The MP2 (FC)/6-311++G(2d,2p) optimized geometries were employed. Table 1 is partially reproduced from our earlier study reported in Ref. [66]; *Copyright (2006) The American Chemical Society.* b The triols wherein three OH groups are present on the successive C-atoms show three H-bonds. See text for details.

#### **4. Critical Comparison of MTA with Other Methods**

We now compare the estimated HB strengths in these molecules with those qualitatively estimated by other indirect measures. These measures include the O–H stretching frequency, molecular electron density (MED) value at (3, −1) BCP, and the HB energy estimated using the isodesmic reaction approach (IDRA) and that by using Espinosa's equation. Both the MED value at (3, −1) BCP and the shift in the stretching frequency of the O–H involved in the HB show a good qualitative correlation with the estimated HB energies (see Table 1). For instance, the strongest HB found in 1,2,5-pentanetriol (4.97 kcal/mol) corresponds to the highest MED value (0.0334 au) at the (3, −1) BCP. The stretching frequency of the O–H bond involved in the HB is 3669 cm<sup>−</sup>1, showing a redshift. The weakest HB is seen in 2,3,4-pentanetriol (1.50 kcal/mol), with the (3, −1) BCP being absent and a large O–H stretch frequency of 3820 cm<sup>−</sup>1. In general, the calculated HB energies show a good qualitative rank–order relationship with the corresponding O–H stretching frequencies and the MED value at the respective (3, −1) BCP.

We now critically compare our MTA-based results with those obtained by ye<sup>t</sup> another indirect method, viz., the isodesmic reaction approach (IDRA) [33,48]. In IRDA, the IHB- making/breaking reaction is written such that, except for the HB under consideration, the number and type of other bonds are conserved on both sides of the reaction. Within IDRA, all the reactant and product geometries are optimized at the appropriate level of theory. The energy change for such a reaction is taken as the HB energy. The main disadvantage of IDRA is that there is no unique way of writing an isodesmic reaction for estimating the HB energy. For example, in Scheme 3, four possible reactions are shown for evaluating the HB energy E**HB2** in 1,2,5-pentanetriol, which retain **HB1** on both sides of these isodesmic reactions. See Ref. [52] for details of several other isodesmic reactions for estimating the HB energies of **HB1** and **HB2**.

**Scheme 3.** Some possible isodesmic reactions for the estimation of H-bond energy, E**HB2** in 1,2,5- Pentanetriol. See text and Ref. [52] for details.

Figure 1 shows a histogram of the estimated HB energy E**HB2** by these four reactions and also by the MTA-based method. The HB energies estimated by the isodesmic reactions vary significantly across the levels of theory and from each other. These estimated energy values are much smaller (30 to 40%) than the respective MTA-based ones. The reasons for these smaller HB energy values by IDRA can be understood as follows: In IDRA reactions presented in Scheme 3, on the reactant side, the **HB2** bond formation between the -OH groups at C2 and C5 positions leads to a seven-membered ring-like structure involving one O–H···O, two C–O, and three C–C bonds. This ring formation has the ring strain effect in the parent 1,2,5-pentanetriol molecule. Since the reactant and product geometries are optimized, this ring strain effect is not preserved on the product side of these reactions. Therefore, the molecules on the product side are more relaxed, losing most of their ring strain due to the loss of the **HB2** bond. It may be noted here that the ring strain may be small or canceled out when one estimates the energy of **HB1** using IDRA. This is because the formation of **HB1** leads to the formation of a five-membered ring-like structure involving one O-H···O, two C–O, and only one C–C bonds. This ring (five-membered) backbone is expected to be maintained (to some extent if not fully) on both reactant and product sides as it involves only one C-C bond. For more details about **HB1** energy by IDRA, see Ref. [52].

**Figure 1.** The H-bond energy, E**HB2**, in 1,2,5-pentanetriol calculated at different levels of theory using isodesmic reactions (see Scheme 3) and also by molecular tailoring approach (MTA). See text for details.

Thus, the finer bonding effects are not mimicked evenly on both sides of the reaction, resulting in poor estimates of HB energy by IDRA. On the contrary, in the MTA method, the geometry of the fragments is not optimized, and the fragment backbone structure is preserved, leading to accurate estimates of the HB energies. For instance, the evaluation of HB energy E**HB1** as (EF1 + EF2 − EF4) − EM in 1,2,4-butanetiols involves the energies of F1, F2, and F4 (see Scheme 2). Thus, the MTA-based method leads to unique isodesmic/homodesmic reaction, viz., M + F4 → F1 + F2. As seen in Scheme 2, the carbon backbone structure is retained on either side of this unique reaction. In other words, as advocated in the literature [49–52], IDRA does not yield the true HB energies. Another drawback of the IDRA is for estimating the HB strengths in multiply hydrogen-bonded systems. Here, one has to write different reactions for different HBs. In summary, the MTA-based method for the estimation of IHB energy is accurate and can be applied to the evaluation of multiple IHB energies in a given molecule. In the following sections, we present and discuss the results of IHB energies obtained by applying the MTA-based method to a variety of systems.

We now present a comparison of the HB energy estimated by the MTA-based method with those by Espinosa's empirical relation [59]. For this purpose, we consider two molecular systems having O–H···O=C HB, viz., 2-Hydroxyacetophenone, and methyl 2-hydroxybenzoate. The geometries and HB energies by Espinosa's method in these two molecules were taken from the Ref. [93]. The reported HB energies at B3LYP/6-311++G(d,p) level, by Espinosa's method, in 2-Hydroxyacetophenone and methyl 2-hydroxybenzoate are 15.3 and 11.4 kcal/mol, respectively. The HB energies for these molecules were calculated by us at the given geometries, at the same level of theory, by MTA-based method are 9.7 and 7.8 kcal/mol, respectively. As can be observed, the HB energies estimated by Espinosa's method are significantly overestimated, compared to the HB energies by the MTA method. These results are in agreemen<sup>t</sup> with an earlier report that the use of Espinosa's method significantly overestimated the energy of HBs [62,93]. Although our MTA-based method requires extensive calculations and more computational time, reliable direct estimates of the HB energies are thereby obtained. Further, the internal benchmarking of the total energy is possible for the MTA-based method. Such benchmarks are not available in Espinosa's method.

#### **5. Application to Large Molecules and Clusters with Multiple Hydrogen Bonds**

We now discuss the application of the MTA-based method for IHB energy estimation in several large systems. One such interesting system is a class of carbohydrates, viz., sugar molecules. These molecules play an important role in biological processes, which are mainly governed by weak interactions such as hydrogen bonding, hydrophobic effects, etc. Hence, understanding the interactions such as H-bonding is of utmost importance. Further, the O–H groups in carbohydrates form a network of interconnected O–H ···O HBs. It is suggested that the strengths of these individual O–H···O HBs are enhanced due to such networking of HBs.

In our earlier work [67], the IHB energies and the contribution to cooperativity were investigated in eight aldopyranose monosaccharides, which vary in the position of hydroxyl groups [axial (*ax*) or equatorial (*eq*)], as shown in Figure 2. The estimated HB energies are in the range of 1.2 to 4.1 kcal/mol at the MP2(full)/6-311++G(2d,2p) level of theory. It is found that the OH···O *eq-eq* interaction energies are between 1.8 and 2.5 kcal/mol, with axial-equatorial ones being stronger (2.0 to 3.5 kcal/mol). The strongest bonds involve nonvicinal *ax-ax* O–H groups (3.0 to 4.1 kcal/mol). The cooperativity contribution to the HBs is seen to fall between 0.1 and 0.6 kcal/mol for *eq-eq* HBs and is seen to be higher (0.5 to 1.1 kcal/mol) for *ax*–*ax* HBs. This work [67] was one of the first attempts for estimating the IHB energies and the respective cooperativity contributions in sugar molecules.


**Figure 2.** General structure of the aldopyranose sugar. In Table, *ax* represents axial, and *eq* represents equatorial orientations of the hydroxyl group at carbons C2-C4. Figure partially reproduced from Ref. [67] with the permission from American Chemical Society (ACS). *Copyright (2008) The American Chemical Society*.

> A similar class of compounds having no ring oxygen atom is hexahydroxy–cyclohexane,called inositol. Inositol derivatives function as intracellular signal transduction molecules, playing an important role in biological processes. It is hence important to understand the structure and stability of various inositol conformers [84]. On applying the MTA-based method, the estimated HB energies in the presence of a cooperative HB network are seen to be in the range of 2.2 to 3.8 kcal/mol at the MP2/6-311+G(d,p) level of theory. The sum of all the HB energies in these conformers of inositol falls between 7.2 to 18.1 kcal/mol. The total cooperativity contribution in these conformers is rather large, between 2 to 5 kcal/mol. It increases on going from isomers with more *eq* O-H groups to those with more *ax* ones. Importantly, the highest stability of the *scyllo* isomer in the solvent was attributed [84] to weaker intramolecular OH···O HBs between *eq* hydroxyl groups. It is suggested that these weak OH···O HBs in *scyllo* isomer may facilitate favorable intermolecular interactions with solvent molecules. In contrast, the inositol isomers with *ax* O-H groups are involved in the formation of relatively strong HBs. Therefore, they are less stable due to large steric factors in the gas phase and unfavorable intermolecular interaction in the solvent phase.

> The MTA-based method was also employed for understanding the conformational stability of fructose [85]. The experimental rotational spectroscopic studies sugges<sup>t</sup> that the molecules of fructose and ribose preferentially adopt the β-pyranose structure in the gas phase. It was noted that [85] the relative stability of different conformers of fructose in the gas phase could be explained in terms of three collective effects: (i) the sum of HB energies in a given conformer, (ii) the strain energy of a bare fructose ring, and (iii)

the sum of anomeric stabilization (*endo* + *exo*) energies. It was concluded [85] that the small ring strain, sufficiently large sum of the IHB energies, and the higher stabilization due to anomeric interactions in β-fructo-pyranose makes it a conformationally locked predominant structure in the gas phase.

Another molecular system wherein the D-glucose units are joined to each other by 1–4 linkage is cyclodextrins (CDs), which are macrocyclic oligosaccharides. They possess a unique ability to entrap gues<sup>t</sup> molecules in their cavities owing to their bucket/bowl shape. Such inclusion complexes are used in the pharmaceutical industry for a variety of formulations. The most commonly known CDs are α-, β- and γ-CDs which have six, seven, and eight glucose units, respectively (see Figure 3). The IHB's between the secondary O–H groups of glucose units result in the formation of a smaller rim of the CD bowls, whereas the primary O–H groups form the larger ones. The strength of the IHBs is suggested to be an important factor governing the stability of the CDs [94–96]. Moreover, one would expect a larger cooperativity contribution due to the formation of a more extended network of HBs between different types of O–H groups. The estimated IHB energies obtained by using the MTA-based method [86] belonged to a wide range of 1.1 to 8.3 kcal/mol at the B3LYP/6- 311++G(d,p) level, suggesting that strong HBs are seen in CDs. For the O6H···O6 HBs, HB energies fall between 6.7 to 8.3 kcal/mol, and for O3H···O2 HBs, they are between 3.3 to 5.5 kcal/mol and 1.9 to 2.8 kcal/mol for O2H···O3 HBs. The cooperativity contribution by the O6H···O6 HB is larger (1.3 to 2.7 kcal/mol) than that for O3H···O2 (0.3 to 1.0 kcal/mol) and O2H···O3 (0.25 to 1.10 kcal/mol) HBs. Note that the cooperativity contribution in glucopyranose (0.1 to 1.1 kcal/mol) is much smaller than that in CDs. The higher HB strength in CDs could be one of the possible reasons for the much lower aqueous solubility of the natural CDs than that of comparable acyclic polysaccharides [97–99].

**Figure 3.** (**a**) A schematic structure of β-cyclodextrin molecule (containing seven glucose units) indicating different types of hydroxyl groups and (**b**) a CD bowl, secondary hydroxyl groups at smaller and primary ones at the larger rim, respectively. See text for details.

We now discuss the HB energies and cooperativity contributions calculated by using MTA in ye<sup>t</sup> another macrocyclic system, viz., *p*-substituted Calix[n]arenes CX[n] (*n* = 4, 5), for exploring substitution effect on the strength of the HBs [87]. The estimated HB energies were between 4.6 to 8.2 kcal/mol, with cooperativity contribution being 0.2 to 2.6 kcal/mol, both calculated at B3LYP/6-311G(d,p) level of theory. The estimated HB energies showed [87] the following order: CX[n]-t-Bu ≈ CX[n]-NH2 > CX[n]-CH2Cl > CX[n] > CX[n]-SO3H > CX[n]-NO2 > S-CX[n]-t-Bu > S-CX[n]. It was also observed that the HB energies in CX[5] derivatives are larger than those in CX[4]. Additionally, as expected, large HB energy values and the respective cooperativity contributions were found in CX[n] hosts with electron-donating substituents.

Recently, we have widened the application of the MTA-based method for the estimation of individual O–H···O HB energy and their cooperativity contribution in water clusters [88]. Many works reported in the literature focus on the global minimum structure of water clusters of variable sizes [73,75,76,100–103]. In spite of a large number of studies, the nature of water clusters at the molecular level is not fully understood. For instance, the strengths of individual O-H···O HB interactions reported in the literature are mostly limited to the water–dimer. Larger clusters are expected to have a cooperativity contribution due to the networking of HBs therein. We recently applied the MTA-based method for the reliable estimation of individual O–H···O HB energies and their cooperativity contribution in small water clusters Wn, *n* = 3 to 8 (see Figure 4).

**Figure 4.** The MP2-optimized geometries of various water clusters (Wn). Figure reproduced from Ref. [88] with the permission from American Chemical Society (ACS). *Copyright (2020) The American Chemical Society*.

The fragmentation procedure for the estimation of O–H···O HB energy in Wn is similar to the one for molecules, discussed in the previous section. The difference is that no dummy atoms are needed here because no covalent bond is cut. In the present case, two water molecules involved in the formation of an O-H···O HB were removed for generating the two primary fragments keeping the other water molecules within the cluster intact (see Ref. [88]). The calculated HB energies in Wn, for n = 3 to 8, are in the wide range of 0.3 to 10.7 kcal/mol at the CCSD(T)/aug-cc-pVDZ level, with the respective cooperativity contribution being 1.2 to 7.0 kcal/mol. To check the reliability of the results, the sum of all the HB energies for a given cluster was added to the sum of monomers energies. The molecular energy of a water cluster thus estimated agreed well with the actual energy (typical error less than 8.3 mH), suggesting HB energies obtained by our MTA-based method [88] are reliable.
