Revised Manuscript Quantification of Hydrogen Bond Strength

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Quantification of Hydrogen Bond Strength Based on Interaction Coordinates: A New Approach Sarvesh Kumar Kumar Pandey, Dhivya Manogaran, Sadasivam Manogaran, and Henry F. Schaefer J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b04752 • Publication Date (Web): 18 Jul 2017 Downloaded from http://pubs.acs.org on July 20, 2017

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The Journal of Physical Chemistry

Revised Manuscript (Manuscript ID: jp-2017-04752h.R2)

Quantification of Hydrogen Bond Strength Based on Interaction Coordinates: A New Approach Sarvesh Kumar Pandey1, Dhivya Manogaran1 Sadasivam Manogaran*,1 and Henry F. Schaefer III*,2

1

Department of Chemistry, Indian Institute of Technology, Kanpur, 208 016, India

2

Center for Computational Quantum Chemistry, University of Georgia, Athens, Georgia, 30602, USA

Corresponding Authors E-mail: [email protected]. Tel.: +91 512 259 7700. E-mail: [email protected].

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Abstract A new approach to quantify hydrogen bond strengths based on interaction coordinates (HBSBIC) is proposed and is very promising. In this research, it is assumed that the projected force field of the fictitious three atoms fragment (DHA) where D is the proton donor and A is the proton acceptor from the full molecular force field of the H-bonded complex characterizes the hydrogen bond. The "interaction coordinate (IC)" derived from the internal compliance matrix elements of this three atoms fragment measures how the DH covalent bond (its electron density) responds to constrained optimization when the HA hydrogen bond is stretched by a known amount (its electron density is perturbed by a specified amount). This response of the DH bond, based on how the IC depends on the electron density along the HA bond, is a measure of the hydrogen bond strength. The interand intramolecular hydrogen bond strengths for a variety of chemical and biological systems are reported. When defined and evaluated using the IC approach, the HBSBIC index leads to satisfactory results. Since this involves only a three atoms fragment for each hydrogen bond, the approach should open up new directions in the study of 'appropriate small fragments' in large biomolecules.

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1. Introduction "Noncovalent interactions" play a vital role in deciding the structure and properties of chemical and biological systems.1 One of the important non-covalent interactions having a profound effect is the "Hydrogen Bond (HB)". A vast number of papers, books, and reviews have been published on this topic in the last hundred years, exploring different aspects of HB, including quantification of hydrogen bond strength (HBS).1–9 The International Union of Pure and Applied Chemistry (IUPAC) adopted a broad definition of HB in 2011.10,11 The existence of life depends on bulk water and biomolecular aggregates. The functioning of these assemblies are decided by the making and breaking of the inter- and intramolecular HBs involved in their structures. Hence quantification of HBs would give a better understanding of the mechanism of the biological processes which in turn will provide us better ability to control them.12,13 Hydrogen bonding ability is also an important feature in drug-design.14 There are several concepts in chemistry like valency, bond order, aromaticity, and HBS which are used qualitatively by many chemists but difficult to precisely quantify. Very recently, we have proposed a method for the quantification of aromaticity based on interaction coordinates (AIBICs).15 Here, we describe a method for the quantification of "Hydrogen Bond Strength Based on Interaction Coordinates (HBSBIC)" and show its validity for several H-bonded systems. The assessment of HBS is addressed in the literature using both experimental and theoretical methods.5 Results from several experimental methods have been used, including rotational spectroscopy of gas phase H-bonded complexes,16 infrared,17,18 and NMR spectral techniques19–21 to investigate selected H-

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bonded systems. A number of systematic theoretical approaches at different levels of approximation from semi-empirical to ab initio methods have been used to study the HBS.3–6 The theory based quantifications of HB are largely based on geometry, energy, wave function, and relaxed force constant methods, which are briefly discussed here. Geometry Based Methods: The geometry criteria for the presence and approximate relative strengths of HBs may be developed empirically, and is based on the two bond lengths (DH and HA) and the internal angle (DHA). Such criteria are used to classify a HB based on HA length, DHA angle, and HB dissociation energy.3 Jeffery has defined a strong hydrogen bond as one with an H-A distance of 1.2 - 1.5 Å, a D-H-A angle of 170° - 180°, and a dissociation energy of 15 - 40 kcal/mole. A moderate hydrogen bond has r (H-A) = 1.5 - 2.2 Å, θ (D-H-A) = ~ 130°, and dissociation energy 4 - 15 kcal/mole and a weak hydrogen bond as having r(H-A) > 2.2 Å, θ (D-H-A) > 90°, and the dissociation energy less than 4 kcal/mole.3 Usually, if the distance between the heavy atoms DA is less than the sum of their van der Waals' radii, it is taken as a criterion for the presence of the HB.22–24 However, critical analysis of such results based on small molecules to large protein structures indicates that the geometric criteria of locating HBs may be error prone and requires additional physicochemical criteria.25,26 Energy Based Methods: The binding energy (BE) is obtained as the difference in energy between the H-bonded complex and the sum of the energies of the component systems in the case of an intermolecular HB. The zero point energy (ZPE) and the basis set superposition error (BSSE) may be taken into account. This BE is the sum of HB energy and the deformation energies of the monomers forming the H-bonded complex. When more

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than one HB exists between the two monomer units, the BE only estimates the total interaction energies of all HBs. Calculation of individual HBS will help in understanding the factors contributing to the total BE. Hence several approaches have been used to estimate the individual HBS.27-30 These involve building suitable model systems which closely resemble the original H-bonded complex with only one HB and estimating its BE.27–30 Each of these models involves several approximations and is likely to give only a good estimate for the HBs. Intramolecular HBs are trickier and are considered in a latter section. Wave Function Based Methods: Bader's theory of "Atoms in Molecules (AIM)" has been used extensively to study the nature of bonds including the HBs. AIM exploits the topological features of electron densities and gives satisfactory results for HBS in several cases. However, there are cases where AIM results are not very satisfactory.31–34 Weinhold’s Natural Bond Orbital (NBO) analysis is also widely used to study the nature of H-bonding. NBO analysis stresses the orbital interactions in the H-bonded complex and can give information about the origin of the HBs.35Good correlations have been observed between QTAIM and NBO results, implying that they complement each other.36 Relaxed Force Constant (RFC) Method: For a given system, the internal coordinate force constant matrix (F) elements depend on what other coordinates are used to describe the molecular force field. Its inverse, the compliance constant matrix (C = F-1) is independent of the other coordinates in the basis set.37,38 Accordingly, Jones defined the reciprocal diagonal compliance matrix element as the relaxed force constant (RFC).38–40 The RFC was recognized as a “more chemically meaningful bond strength parameter than the regular (force) constant”.40 Grunenberg and coworkers explored this idea for several covalent bonds

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in a series of papers and also extended this idea to noncovalent interactions like HBs, proposing that the RFC values of HBs could be used as a measure of HB strength. In the literature, there are several reports that this method works satisfactorily.41-49 This approach is easy to apply for larger systems, since the RFC of any one bond may easily be obtained from energy only calculations without the need to calculate the full Hessian.44 2. Methodology There are several reasonably successful cases in the literature where only the fragment DHA of the H-bonded complex is used in modeling the HB.50–54 Encouraged by the results, here we treat the three atoms fragment (DHA) of the H-bonded complex as a fictitious isolated system to study the features of the HB. The essential idea here is to obtain the projected force field of this fictitious three atoms fragment from the full molecular force field of the H-bonded complex15 and calculate the HBS from this projected force field using "interaction coordinates (ICs)". When the entire molecular system containing the DHA fragment is geometry optimized for minimum energy, we get its equilibrium structure. If we calculate the Hessian in Cartesian displacement coordinates (X) at the equilibrium geometry of the system, the computed force field (Fx) corresponds to the H-bonded complex at the equilibrium. Now, if we assume that the HB is characterized by the three atoms fragment (DHA) only, we project the force field of this fragment using its internal coordinates (R), two bonds (DH, HA), and one internal angle (DHA) from the Cartesian force field (Fx) of the complex.15 The potential energy in Cartesian displacement (X) and internal coordinates (R) is given by 2V = XTFxX = RTFRR

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The R and X are related by R = BX where B is Wilson's B Matrix.55 Rearranging gives X = B-1R and substituting in the potential energy expression yields 2V = (B-1R)TFx (B-1R) = RT (B-1)TFx (B-1R) = RTFR R which gives FR = (B-1)TFxB-1

(1)

Since B is a rectangular matrix, B-1 is obtained indirectly from Wilson's internal G matrix defined as G = BM-1BT.55 Inverting both sides and rearranging, one obtains the equation56 B-1 = M-1BTG-1

(2)

Using equations 1 and 2, we compute FR using Fx obtained from the Gaussian 09 program.57 Since there is no redundancy in R, the matrix CR = FR-1 could easily be obtained from FR. Although the FR and CR matrices incorporate the same information, the IC is defined in terms of the compliance matrix elements as (i)k = Cik/Ckk38 where the coordinates i and k correspond to the DH and the HA bonds, respectively. Although the procedure described here was used in evaluating the ICs in the present work, we propose the direct calculation of the compliance matrix elements from the energies without the need to calculate the full Hessian44. This provides a better understanding of the physical meaning of the ICs. In the H-bonded complex, at equilibrium the electron density along the bonds communicates and adjusts in such a way that the equilibrium bond lengths and angles of this structure will have zero forces on all internal coordinates. Now, suppose we pretend that we only have the DHA fragment and its projected force field. If we stretch the HA bond by a small amount, say 0.01 Å, we have a non-equilibrium structure, and nonzero forces appear on all three internal coordinates DH, HA bonds, and DHA internal angle. The electron

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densities along the bonds adjust with respect to the 0.01 Å extension of the HA bond (change in the electron density by a known amount). Keeping the extended HA bond fixed, if we do a partial optimization, the forces on the DH bond go to zero, and the DH bond responds by changing its bond length (electron density along the bond). Let the changes in energy and the bond lengths after partial optimization be described as V, Rk (= 0.01 Å) (HA), and Ri (DH). The compliance matrix elements are computed using

and

Ckk = Rk2/2V38

(3)

Cik = RiRk/2V38

(4)

In the H-bonded DHA fragment, the HA hydrogen bond is coordinate k and the D-H covalent bond is the coordinate i. Using equations 3 and 4 the IC can be defined as (i)k = Cik/Ckk = Ri/Rk 38

(5)

If Rk = 1, (i)k = Ri. When the HA hydrogen bond (Rk) is stretched by one unit (measured perturbation) and (keeping it fixed) if we do a constrained optimization, the response of the D-H covalent bond (Ri) corresponds to the IC, (i)k = Ri . This response of the DH bond, based on how the IC depends on the electron density along the HA bond, is a measure of the hydrogen bond strength. Thus the IC, (i)k, the covalent bond response during constrained optimization after constraining the HB by one unit, is a measure of HBS based on IC (HBSBIC). We define HBSBIC = (i)k = Cik/Ckk = Ri (when Rk = 1)

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HBSBIC is a ratio (Cik/Ckk or Ri/Rk when Rk = 1) and is a dimensionless number. Since Ri (response of the covalent bond) is expected to be very small, because the perturbed HB

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involves a relatively low electron density region, it is always less than 1.0 and its limiting values are 0.0 (low electron density, very weak or no bond) and 1.0 (high electron density, strong bond). The F-H-Ne complex is an example for a very weak H-bond system [MP2/aug-cc-pVTZ, HBSBIC = 0.0004] and [F-H-F]- ion is an example for a strong H-bond system [MP2/aug-cc-pVTZ, HBSBIC = 0.803]. The HBSBIC index is satisfactory in quantifying the HBS for the range of systems studied in the present work. In principle, the IC method could be used to quantify any bonding interaction, including covalent and noncovalent. Interaction Coordinate (IC) vs Relaxed Force Constant (RFC) The diatomic oscillator equation in the harmonic approximation is V= (1/2) kx2. So the restoring force on the oscillator -∂V/∂x = f = -kx takes the matrix form f = -FRR for polyatomic molecules where f is the general restoring force column matrix. Rearranging, we get R = -FR-1f = -Cf. Writing explicitly, Ri = -ΣkCikfk. When we have the nonzero force fk only on the coordinate Rk and forces on all other coordinates are zero (fi = 0 when i ≠ k), this takes the form Ri = -Cikfk and Rk = -Ckkfk. Substituting the value of fk = -Rk/Ckk in Ri = -Cikfk, we get Ri = CikRk/Ckk indicating that Cik/Ckk is independent of the force fk and depends only on Rk. When Rk = 1, the quantity (i)k = Cik/Ckk is invariant and likely to be a better diagnostic tool for the assessment of bond strengths. All interaction coordinates reported here, were computed using computer programs developed in IIT-Kanpur.

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3. Results and Discussion To assess the usefulness of the HBSBIC methodology, we examined several sets of interand intramolecular H-bonded systems. In the intermolecular case, we have quantified the HBSs for some small homo- and hetero-water and methanol clusters. DNA base pairs and their model systems, carboxylic acid dimers with symmetric and asymmetric substituents have also been studied. Finally, we considered various sulfur substituted formic acid dimers. Furthermore, in the intramolecular case, we explored oxalic acid, a few diols, and several interesting systems containing resonance assisted HBs. Intermolecular H-bonded Systems A. Water and Methanol Clusters Since our goal is to quantify the HBS, we selected the water and methanol dimers, and their higher cyclic n-mers [(H2O)n and (MeOH)n, n = 2 - 6]. We considered only the cyclic structures because extensive experimental and theoretical results are available in the literature for these small clusters. We optimized the (H2O)n (n = 2 - 6) and (MeOH)n (n = 2 - 4) structures using the MP2/aug-cc-pVTZ level of theory. Additionally, we performed MP2/6-311++G** computations for n = 2 - 6 in the case of (MeOH)n. While Figures 1 and 2 contain the structures of the optimized clusters, the optimized geometrical parameters for DH, HA, and DHA are given in Tables 1 (for water clusters) and 2 (for methanol clusters). The calculated HBSBIC values are also included in Tables 1 and 2 along with the reported BEs wherever available. In the case of the water clusters, the average HB length varies from 1.947 Å (dimer) to 1.723 Å (hexamer) and the average internal angle increases from 150.3° (trimer) to 179.1° (hexamer). Correspondingly, the HBSBIC values increase from 0.119 for the dimer to 0.205 [10]


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for the hexamer. There is a good correlation between the BEs and the HBSBIC values obtained using MP2/aug-cc-pVTZ. The reported BEs for MP2/aug-cc-pVTZ58 and CCSD(T)/CBS59 methods show a similar trend and are in agreement with the HBSBIC values reported here. It has to be noted that the HBSBIC value is a direct reflection of the HBS while the BE is the sum of the HBS and the deformation energy. In the case of the water trimer, the average internal angle 150° deviates most from the linear 180° and hence, based on geometry, we expect a weaker HB compared to the dimer. Due to the cooperativity effects of multiple HBs, the average HBSBIC in the trimer is more than the dimer (0.130 > 0.119). As observed from Table 2 for methanol clusters, the HB length decreases from 1.886 Å for the dimer to 1.687 Å for the hexamer, while the internal angle increases from 150° (trimer) to 177° (hexamer) at the MP2/6-311++G** level of theory. Our HBSBIC values are consistent with the calculated BEs reported in the literature.58,59 The tricyclic ring with three methyl substitutions has an unfavorable geometry for H-bonding and hence, the BE per HB decreases compared to the dimer [HBSBIC values 0.138 (trimer) < 0.165 (dimer)]. The BE of a cluster obtained from n-monomers can be decomposed to pairwise additive two-body interactions and higher non-additive n-body components (n > 2). The interaction energy due to these non-additive components with n > 2 is due to cooperativity effects. Thus the cooperativity effect of an n-mer system is defined as the interaction energy of n-mer system minus sum of the interaction energies of all its possible dimer sub-systems.60-62 In reality, this means that the formation of one HB leads to alteration of neighbouring bond polarities and subsequently the multiple HBs stabilizing each other. The cooperativity effect

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is quantified by computing the changes in BE between successive 'n' values (En-1 → En, n = 3 - 6). The values calculated for water clusters based on HBSBIC are -0.011 (2 → 3), -0.043 (3 → 4), -0.024 (4 → 5), and -0.008 (5 → 6). The negative values are a measure of the cooperativity effects in each (H2O)n, for n = 3 - 6. These cooperativity effects increase from trimer to tetramer, reach a maximum and then fall off through pentamer and hexamer, tending towards a limiting value, in agreement with the reported values in the literature.63 Similar to the water clusters, the cooperativity effects strengthen the HB in methanol clusters. The computed values based on HBSBIC are +0.032 (2 → 3), -0.047 (3 → 4), -0.017 (4 → 5), and -0.012 (5 → 6). The positive value for the trimer is due to the unfavorable geometry of the HBs. As in the case of water clusters, the cooperativity effect based on HBSBIC is maximum for the tetramer and smaller for the pentamer and hexamer. The cooperativity effects of small methanol clusters have been studied earlier.64,65 In addition to water and methanol clusters, mixed-water dimer complexes have attracted much attention because of their thermodynamic and physicochemical properties.66–68 Hence, we studied a few mixed-water dimers to probe the effect of different substituents on Hbonding. To understand the substituent effect, we looked at hetero water dimers with the other component being MeOH, Me2O, NH3, MeNH2, or Me3N. The optimized structures and the geometrical parameters are displayed in Figure 3 and in Table 3, respectively. In all these systems, water acts as a proton donor in the HB and the computed HBSBIC values show an excellent correlation with the BEs reported.69 If we compare, the HBSBIC values of 0.119, 0.159, and 0.144 for (H2O)2, (MeOH)2, and MeOH-HOH, respectively, they have the same trend as seen for the BEs. The mixing of MeOH and H2O has been studied [12]


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[(MeOH)2 + (H2O)2 → (MeOH-H2O) + (H2O-MeOH)] using the BEs of the corresponding homo and hetero dimers.70 Unlike BEs, the HBSBIC values are free from the deformation energies and hence, they are likely to provide better quantitative information in such cases. It is worthy to note that when the H atoms on the acceptor water molecules are substituted by methyl groups in succession, the HBSBIC values change from 0.119 [(H2O)2](1) to 0.144 (MeOH-HOH)(2), and 0.142 [(Me2O-HOH)(3)], implying that the first methyl substitution has a dominant effect on the computed HBS. A similar conclusion could be reached based on the ammonia-water dimer and methyl substituted ammonia-water dimers 4, 5, and 6 (see Table 3). In general Tables 1-3 show a good correlation between the computed BEs and the HBSBIC values. The linear least squares statistical analysis for BE(yi) Vs HBSBIC(xi), gave R2 values 0.980, 0.936 and 0.934. In Tables 2 and 3, the BEs and HBSBIC are obtained using different methods and hence the value of 0.93 for R2 is reasonable. The BE of different HBs contain small arbitrary deformation energies in addition to HBS while HBSBIC estimates only the HBS. So we do not expect R2 to be close to 1.0. The details of the analysis are given in Tables S1-S3 and in Figure S1 in the supplementary information. B. Difluoromethane Dimer (DFMD) Recently, there is considerable interest in blue shifting HBs, where the C-H stretching frequency shifts in the opposite direction compared to the normal HBs.18 The C-H---F HB is a known example of this type and has been studied in several dimers of fluoromethanes.71–74 Since, the difluoromethane dimer (DFMD) has been studied experimentally,73,74 we considered this system. In this molecule, there are three C-H---F HBs. Since they are of the

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same type, determining their HBS will be of interest to test the HBSBIC methodology. In general, the HBs involving C-H donor systems are relatively weak and here also, the studied C-H---F HBs are weak. In the present research, using the MP2 method, we have used several basis sets 6-311G, 6-31+G**, 6-311+G**, 6-311++G**, aug-cc-pVDZ, and aug-ccpVTZ to see the basis set effects on the computed values. The computed geometrical parameters and HBSBIC values are reported in Table 4. The optimized structure of the DFMD system may be seen in Figure 4. In all the basis sets, the H-bonded C-H covalent bond is shorter than that for the monomer and hence, all predict blue shifting HBs. The calculated C-H bond lengths are in agreement with the earlier reported distances.72,73 Two available experimental structures73,74 suggest that the HBs A and B are similar and weaker than C. The computed values indicate that C > B > A, and the differences between A and B are much smaller in comparison with their differences with C. C. DNA Base Pairs and Their Model Systems H-bonding plays a crucial role in several biological systems such as DNA and RNA base pairs. We extended our HBS calculations to these base pairs and their representative multi(triple and quadruple) H-bonded complexes to illustrate the applicability of the present approach based on the interaction coordinates. In systems containing multiple HBs, it is not easy to assess the contribution of individual HBs to the BE. Several approaches were tried to achieve this goal and some of the previously reported results are given in Table 5.29 The equilibrium structures of the systems studied are shown in Figure 5 and the optimized geometrical parameters along with HBSBIC values are given in Table 6.

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We considered the three base pairs Guanine-Cytosine (G-C), Adenine-Thymine (A-T), and Adenine-Uracil (A-U) for characterizing their individual HBSs. The G-C (C1) base pair contains three HBs A (N-H--O), B (N-H--N), and C (N-H--O), while A-T and A-U have a C-H---O HB for C instead of N-H--O as shown in Figure 5. In the case of A-T, the differences between HBSs A, B, and C are relatively larger. The HBSBIC values obtained are A: 0.309, B: 0.409, and C: 0.137 at B3LYP/6-311++G** level of theory and this trend is in good agreement with the reported results which follow the order B > A > C as shown in Tables 5 and 6.29,30,41 Since A-T and A-U differ only by a methyl substitution, the results are similar, as expected. As Tables 5 and 6 indicate for the HBs in the G-C base pair, the relative ordering of all three HBSs appear to be close to each other, and the final outcome appears to be largely dependent on the method and the basis set used in the computation.29 Since these molecules have a relatively larger number of atoms, performing higher level theoretical calculations with modest basis sets requires large computational resources. Also, there are reports for and against the aromaticity and resonance assistance to the HBs in this system.75 The computed HBSBIC values for the models containing multiple HBs are shown in Table 6 along with their geometrical parameters. The outcomes are consistent with those for the G-C, A-T, and A-U systems. D. Acid Dimers It is known that carboxylic acids form cyclic dimers with two equivalent HBs in the gas phase and in nonpolar solvents. Their dimerization constants are of similar magnitudes

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regardless of the side chains.76 Carboxylic acids serve as useful model compounds for a quantitative understanding of H-bonding, hydrophobic effects, and entropy factors.77–80 Such dimer systems have two COOH groups involved in H-bonding - one as a proton donor and other as a proton acceptor. When the two groups are identical, there are cooperativity effects between the carboxyl groups. As noted earlier, the presence of one HB polarizes the other HB, resulting in stabilization of both HBs. Several dicarboxylic acid dimers with symmetric and asymmetric substituents were considered in order to understand the contributions of substituent groups in determining the HBS and further validate the proposed methodology for quantifying the HBS of various H-bonded systems. The results for homo and hetero dicarboxylic acid dimers are reported in Tables 7 and 8, respectively. It is interesting to observe that when we add and divide by two, the proton donor and acceptor values corresponding to the HBs from the homo dimers, there is a good match with the HBSBIC values of the mixed dimer. When we sum the donor and acceptor values of HBSBIC for formic and acetic homo dimers, we get 0.292 + 0.306 = 0.598 at the MP2/augcc-pVTZ level of theory (Table 7). In the case of the formic acetic hetero dimer (F-AD, 1), the sum of the HBSBIC values of the two HBs is 0.313 + 0.285 = 0.598 (Table 8). From these matching values, we can easily conclude that the substituent effects are additive and as noted by others, the HBS in dicarboxylic acid dimers are largely independent of the substituents.76 A similar agreement is seen for other cases. From the homo and hetero acid dimers studied in this work, it appears that we could predict the HBS in any symmetric dicarboxylic acid by calculating the HBSBIC values of their hetero analogue with different substituents and vice versa. It has been reported in the literature that the proton donor plays

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a more important role than the proton acceptor in deciding the HBS.81 This may be observed in Table 8, with the HBSBIC values being higher when the donors contain electron withdrawing groups [3 (sp hybridized C≡C group) as well as (4-7)] and lower when donors contain electron releasing groups [(1, 2) (sp3 hybridized C-C group)]. E. Hydrogen Bonds Involving Oxygen and/or Sulfur in Formic Acid Dimers The HBs having oxygen/sulfur as donor and/or acceptor groups are important as model systems for several important pharmaceutical and biological molecules.82 The available literature indicates that the different kinds of sulfur HBs are not studied as widely as is the case of oxygen.7 The results given here are based on the projected force field of the three isolated atoms fragment (DHA) model for the HB. They can form four types of HBs; OHO, OHS, SHO, and SHS. Since the mass, size and electronegativity of oxygen and sulfur vary considerably, the HBs containing sulfur are likely to be different compared to the HBs containing oxygen. In order to investigate these differences, we studied ten differently substituted formic acid dimers where the sulfur acts as both proton donor and/or acceptor. The optimized geometrical parameters and the HBSBIC values are given in Table 9 at the PBE1PBE/6-31G**83 and MP2/aug-cc-pVTZ levels of theory. A representative system is shown in Figure 6. Since the proton donor is expected to play a dominant role compared to the proton acceptor,81 we expect the sulfur proton donors will have stronger HBs than the sulfur proton acceptors. The ab initio results also indicate that O-H---S HB is relatively weaker than O-H--O HB which is in agreement with the earlier report.84 Similar to substituents in carboxylic acid dimers, sulfur substituted formic acid dimers obey an additivity rule with

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approximately 10% errors. The electronegativity difference between H and S is very small (0.38) compared to the difference between hydrogen (2.20) and F (1.78), N (0.84), and O(1.24) in the Pauling scale, indicating that S is intrinsically not a good acceptor for hydrogen bonding. Also, the lone pair on O corresponds to 2p while that on S is 3p, making the overlap of H with O more effective than with S. Taking this difference into account the error is reasonable, assuming the validity of an additivity rule. For example, in Table 10, we get the HBSBIC values for symmetric dimers as 0.291 [OHO-OHO(D1)], 0.448 [SHOSHO(D5)], 0.245 [OHS-OHS(D6)], and 0.186 [SHS-SHS(D10)] at the MP2/aug-cc-pVTZ level. If we consider OHO-SHO(D2), the sum of HBSBIC values for both HBs is 0.303 + 0.393 = 0.696(D2) ≃ 0.70. This is within error limits equal to the sum 0.292 [OHOOHO(D1)] + 0.448 [SHO-SHO(D5)] = 0.740 ≃ 0.74. The error if we compare with 0.70, is 0.04, or about 6%. Similarly, we can validate this relation for other cases.

Intramolecular H-bonded Systems

When the HB donor and acceptor happen to be within the same molecule, it is called an intramolecular hydrogen bond (IMHB). Since this HB stabilizes a given conformation, our ability to understand and characterize this interaction will enhance our potential to control the functioning of a molecular system, because the function and conformation are intimately related. Although the quantification of intermolecular HBS using the supramolecular approach gives reasonable estimates, the evaluation of the intramolecular HBS presents difficulties. One has to choose a reference compound without the HB which closely resembles the structure of the system having the IMHB and calculate the energy difference between the two. Choosing a proper reference system may have problems, as more than one reference is possible. Bader's QTAIM approach has been extensively used in the study of [18]


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intramolecular HBs, although there are instances where the results are not very satisfactory. For example, the bond path for IMHB in oxalic acid is not observed in QTAIM.85 A bond path is observed in between the two hydrogens in phenanthrene and cis-2-butene though such a bond path is not realistic.86-88 We considered some simple intramolecular H-bonded systems, namely oxalic acid (OA), 1,4-butanediol (BD), and 1,2-ethanediol (ED). In addition, we also studied a set of typical intramolecular H-bonded systems where the HBS varies from weak to medium to strong. Oxalic Acid, a Few Diols, and Some Typical Intramolecular H-bonded Systems In oxalic acid, the intramolecular H-bonding is very well known and has been studied by energy, wave function, and compliance constant methods.49,85,89 The earlier reports using MP2 and B3LYP calculations with 6-31G** basis sets yield values of 7.3 - 7.6 kcal/mole.89 The present method gives an HBSBIC value of 0.060 for oxalic acid. There is a general consensus regarding the existence of IMHB in 1,4-butanediol90 and its calculated HBSBIC value is 0.102. However, in the case of 1,2-ethanediol (ED), the presence of HB is not a settled issue.49,90,91 The computed HBSBIC values are 0.036 (ED1), 0.031 (ED2), and 0.009 (ED3) for the three conformers containing IMHBs (Figure 7), respectively. Since there are 10 different conformers with small energy differences in 1,2-ethanediol,49 out of which only three have IMHBs, it is difficult to predict whether the HBs in these conformers could be observed experimentally. The HBSBIC values indicate the possibility of a very weak IMHB in two conformers of 1,2-ethanediol (ED1 and ED2). To assess the ability of HBSBIC to quantify the HBS we considered few systems having IMHBs known to vary from weak to medium to strong. These systems are shown in Figure 7 and their optimized geometrical parameters and HBSBIC values are reported in Table 10. [19]



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Based on the MP2/aug-cc-pVTZ level of theory the HBs in systems 11 and 12 with HBSBIC values 0.631 and 0.790, respectively, belong to the stronger category. Systems 6, 10, and 13 with HBSBIC values 0.155, 0.129, and 0.186, respectively, fall in the medium group while systems 7, 8, and 9 with HBSBIC values 0.099, 0.100, and 0.084, respectively correspond to weak hydrogen bonding. There is an overall agreement with the earlier reports34,92 in the cases of 6, 9, 11, 12, and 13. In other cases, there are minor differences which could be attributed to the different methods with different basis sets and the procedures used in the analyses.34 When we compare the B3LYP/6-311++G** results available in the literature92 with our HBSBIC values using the same basis set, the agreement is reasonable. There are reports for and against the dependency of HBS calculated based on energy and based on concepts of wave functions concerning resonance assistance.75 To test whether the HBS based on IC depends on the resonance assistance, we computed the HBSBIC values by including all the internal coordinates involved in the π-carbon framework for structures 6-9 (Table 10). We then used its projected force field to get the HBS. The results are indicated in bold font in Table 10. The minor differences between the HBSBIC values with and without a π-carbon framework appear to indicate that the projected force field based on three atoms DHA fragment model includes the resonance effect.

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5. Conclusions A method is proposed for assessing hydrogen bond strengths (HBSs) by considering the projected force field of the three atoms fragment (DHA) of the hydrogen bond (HB) from the full molecular force field based on the "interaction coordinate (IC)". This method "hydrogen bond strength based on interaction coordinates (HBSBIC)" is applied to the intermolecular HBs in small water, methanol, and mixed-water clusters to assess both their relative strengths and cooperativity effects. Additionally, the HBSBIC values were also computed for DNA and RNA base pairs, homodimer of formylformamide, and some quadruply hydrogen-bonded dimers, as well as the difluoromethane dimer for further validating our approach. Furthermore, we investigated the HBSBICs of the HBs in a series of dicarboxylic acid dimers with symmetric and asymmetric substituents. Differently sulfur substituted formic acid dimers and the additivity of HBSBIC values for them were also studied. For intramolecular hydrogen bonds (IMHBs), we studied the HBs in oxalic acid, a few diols, and some interesting typical systems containing resonance assisted HBs. Based on the computed results for a variety of chemical as well as biological systems, we conclude that the proposed approach for quantifying HBSs is promising and may open up new directions in characterizing a chosen large biomolecule by the consideration of a collection of appropriate small fragments.

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Acknowledgments The authors acknowledge the Department of Science and Technology (DST), Government of India, New Delhi for supporting this project in part and the Computational Facilities in the Computer Center, Indian Institute of Technology, Kanpur - 208 016, India. S.K.P. thanks, UGC and DST, Government of India, for Research Fellowships. D.M. thanks IITK for supporting postdoctoral research and Professor S. Yashonath for encouragement. The authors thank Dr. D.L.V.K. Prasad for helpful discussions. H.F.S. was supported by the U. S. Department of Energy, Office of Basic Energy Sciences, Theoretical and Computational Chemistry Program, Grant DE-FG02-97ER14748. Supplementary Information Results of Linear Least Squares Fit for BE Vs HBSBIC for water clusters, methanol clusters and mixed water dimer complexes are given as Tables S1-S3 and Figure S1 in the supplementary information.

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Biswal, H. S.; Shirhatti, P. R.; Wategaonkar, S. O-H---O versus O-H---S Hydrogen Bonding I: Experimental and Computational Studies on the p-Cresol.H2O and p-Cresol.H2S Complexes. J. Phys. Chem. A 2009, 113, 5633-5643.

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Hermida-Ramon, J. M.; Mosquera, R. A. Do Small Carboxylic Acids Present Intramolecular Hydrogen Bond?. Chem. Phys. 2006, 323, 211-217.

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Poater, J.; Visser, R.; Sola, M.; Bickelhaupt, F. M. Polycyclic Benenoids: Why Kinked is More Stable than Straight. J. Org. Chem. 2007, 72, 1134-1142.

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Grimme, S.; Muck-Lichtenfeld, C.; Erker, G.; Kehr, G.; Wang, H.; Beckers, H.; Willner, H. When Do Interacting Atoms Form a Chemical Bond? Spectroscopic Measurements and Theoretical Analyses of Dideuteriophenanthrene. Angew. Chem. Int. Ed. 2009, 48, 2592 –2595.

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Weinhold, F.; Schleyer, P. v. R.; McKee, W. C. Bay-Type HH “Bonding” in cis-2-Butene and Related Species: QTAIM Versus NBO Description. J. Comput. Chem. 2014, 35, 1499-1508.

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Chen, C.; Shyu, S. F. Conformers and Intramolecular Hydrogen Bonding of the Oxalic Acid Monomer and Its Anions. Int. J. Quant. Chem 2000, 76, 541-551.

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Lane, J. R.; Contreras-García, J.; Piquemal, J. P.; Miller, B. J.; Kjaergaard, H. G. Are Bond Critical Points Really Critical for Hydrogen Bonding?. J. Chem. Theory Comput. 2013, 9, 3263-3266.

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Mariam, Y. H.; Musin, R. N. Transition from Moderate to Strong Hydrogen Bonds: Its Identification and Physical Bases in the Case of O-H—O Intramolecular Hydrogen Bonds. J. Phys. Chem. A 2008, 112, 134-145.

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Table 1. Hydrogen Bond Strengths Based on Interaction Coordinates (HBSBIC), optimized bond distances (RO-H and RH---O in Å), and angles in degrees for small water clusters at the MP2/aug-cc-pVTZ level of theory. The structures are presented in Figure 1.

no.

1 2

3 4

5

a

species

water dimer and small cyclic water clusters

covalent bond (O-H)

H-bond (H--O)

angle

Total BEa

HBSBIC

151.2 148.7 151.2

-16.29b (-15.82, -15.70)c

0.124 0.137 0.127

172.9 176.3 175.5 175.6 176.4

-37.60b (-36.31, -36.01)c

(H2O)2

O-H (0.969)

H---O (1.947)

170.5

average

O-H (0.974)

H---O (1.899)

150.3

(H2O)3

(H2O)4 (H2O)5

average (H2O)6

O1-H2 (0.975) O4-H6 (0.974) O7-H9 (0.975) O-H (0.982)

O5-H15 (0.983) O1-H11 (0.984) O2-H12 (0.984) O3-H13 (0.984) O4-H14 (0.984) O-H (0.984) O-H (0.984)

H2---O4 (1.891) H6---O7 (1.913) H9---O1 (1.893) H---O (1.765)

H15---O1 (1.747) H11---O2 (1.733) H12---O3 (1.731) H13---O4 (1.730) H14---O5 (1.732) H---O (1.735) H---O (1.723)

-5.18b (-4.97, -5.03)c

—

167.4

-28.59b (-27.63, -27.43)c

175.4

—

179.1

— (-44.86, -44.60)c

0.119

0.130 0.173 0.219 0.190 0.194 0.194 0.191 0.197 0.205

BE refers to the dissociation energy for complete separation to isolated H2O monomers.

b

Reference 58, MP2/aug-cc-pVTZ; cReference 59, the values in parentheses correspond to (MP2/CBS, CCSD(T)/CBS).

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Table 2. Hydrogen Bond Strengths Based on Interaction Coordinates (HBSBIC), optimized bond distances (RO-H and RH---O in Å), and angles in degrees for small methanol clustersa. The structures are given in Figure 2.

no.

1 2

3 4

5

species

(MeOH)2 (MeOH)3

average

(MeOH)4 (MeOH)5

average

(MeOH)6

methanol dimer and small cyclic methanol clusters

covalent bond (O-H)

O-H (0.966, 0.969)

O14-H13 (0.972, 0.977) O2-H1 (0.973, 0.977) O8-H7 (0.972, 0.976) O-H (0.972, 0.977) O-H (0.980, 0.985)

O5-H10 (0.982) O1-H6 (0.983) O2-H7 (0.983) O3-H8 (0.983) O4-H9 (0.983) O-H (0.983)

O-H (0.984)

H-bond (H--O)

(1.886, 1.877) H13---O8 (1.879, 1.845) H1--O14 (1.870, 1.841) H7---O2 (1.899, 1.863) H---O (1.883, 1.849)

(1.736, 1.714)

H10---O1 (1.714) H6---O2 (1.699) H7---O3 (1.697) H8---O4 (1.696) H9---O5 (1.698) H---O (1.701)

H---O (1.687)

angle

172.3, 168.3 149.9, 152.8 151.7, 153.3

Energy per H-bond -5.023b

-4.820c -4.936b

-4.033c

149.4, 152.0 150.3, 152.4 167.6, 169.1 173.1 176.8 175.6 174.7 176.1

175.3

176.9

a

HBSBIC

0.165, 0.159 0.133, 0.136 0.138, 0.138

0.144, 0.144

-6.508b

-6.798c —

-7.200c

— —

0.138, 0.139

0.185, 0.185 0.201 0.203 0.202 0.201 0.201

0.202

0.214

Values given in normal font were computed at the MP2/6-311++G** (n = 2 - 6) while those in bold font were obtained at the more complete MP2/aug-cc-pVTZ (n = 2 - 4) level of theory. b Reference 65, HF/6-31G**//MP2/aug-cc-pVDZ; cReference 64, B3LYP/6-31+G*.

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Table 3. Hydrogen Bond Strengths Based on Interaction Coordinates (HBSBIC), optimized bond distances (RO-H and RH---X in Å where X = O and N), and angles in degrees for mixedwater dimer complexes at the MP2/aug-cc-pVTZ level of theory. The structures are given in Figure 3. no. 1 2 3 4 5 6 a

species

some mixed-water dimer complexes

(H2O)2 MeOH-HOH Me2O-HOH NH3-HOH MeNH2-HOH Me3N-HOH

covalent bond (O-H)

O-H (0.969) O-H (0.971) O-H (0.972) O-H (0.975) O-H (0.979) O-H (0.983)

H-bond (H--X)

H---O (1.947) H---O (1.888) H---O (1.860) H---N (1.957) H---N (1.900) H---N (1.857)

Reference 69, CCSD(T)/aug-cc-pVQZ.

angle

170.5 166.1 164.0 169.9 163.3 166.4

Total BEa

HBSBIC

-5.13 -6.13 -6.12 -6.53 -7.74 -7.95

0.119 0.144 0.142 0.188 0.212 0.232

Table 4. Hydrogen Bond Strengths Based on Interaction Coordinates (HBSBIC), optimized bond distances (RC-H and RH---F in Å), and angles in degrees for difluoromethane dimer (DFMD)a using the MP2 functional with different basis sets. [A], [B], and [C] are the three C-H---F interactions in the same system, as shown in Figure 4. basis set

6-311G

6-31+G**b,c 6-311+G**

6-311++G**d aug-cc-pVDZ

a

aug-cc-pVTZ

H-bond (H---F) [A]

2.8071, 91.9

DFMD

H-bond (H---F) [B]

2.5245, 109.8

H-bond (H---F) [C]

2.6070, 123.3

[1.0831, 0.007]

[1.0818, 0.032]

[1.0825, 0.080]

2.7956, 96.8

2.5818, 105.9

2.6893, 115.5

2.7813, 100.4

[1.0848, 0.026] [1.0886, 0.018] 2.7842, 97.0

[1.0886, 0.019] 2.7288, 98.3

[1.0962, 0.021] 2.8073, 92.2

[1.0860, 0.008]

2.5881, 108.3

[1.0842, 0.030] [1.0878, 0.024] 2.5863, 105.6

[1.0878, 0.024] 2.5416, 109.9

[1.0956, 0.032] 2.5524, 108.3

[1.0853, 0.030]

2.6933, 114.9

[1.0847, 0.061]

[1.0884, 0.065] 2.6893, 115.3

[1.0884, 0.064] 2.6348, 118.2

[1.0960, 0.070] 2.5922, 122.3

[1.0856, 0.084]

DFMc

(C-H)

1.0845 1.0859 1.0899 1.0899 1.0977 1.0869

Values given for DFMD are H---F bond distances and internal angles, while those in parentheses are C-H bond lengths and HBSBIC values.

b

Reference 73, MP2/6-31+G**; cReference 72, MP2/6-31+G**; dReference 71, MP2/6311++G**. c C-H bond lengths given for the difluoromethane monomer using the MP2 method with different basis sets.

[33]



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Table 5. The Relative Order of the Individual Hydrogen Bond Strengths in the G-C, A-T, and A-U Base Pairs from Earlier Researcha and Our HBSBIC Results. method used for quantification of H-bonds

rotationa compliance constanta atom replacementa EML equationa for experimental geometry EH---B vs FCP relationa NBOa HBSBIC (present work)

calculation level

G-C

A-T

A-U

B>A>C B>A>C

— —

B3LYP/D95** B3LYP/6-311++G** B3LYP/6-311++G** B3LYP/6-311++G**

C>B>A B>A>C A>C>B A>B>C

A>B>C B>A>C B>A>C A>B>C

B3LYP/6-311++G**

B≃A>C

B>A>C

B3LYP/6-311++G** B3P86/6-311++G** B3P86/6-311++G**

PBE1PBE/6-311++G(3df,3pd)

A>C>B A>B>C

A≃B>C A≃B>C

a

See reference 29 and references therein for the other methods.

[34]


B>A>C B>A>C

— — — —

B>A>C B>A>C B>A>C

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Table 6. Hydrogen Bond Strengths Based on Interaction Coordinates (HBSBIC), optimized bond distances (RX-H and RX---H in Å where X = O, N, C), and angles in degrees for DNA base pairs and their analogsa. The structures are given in Figure 5. no. 1

species

C1 (G-C)

type of H-bonds

covalent bond (X-H)

B

N-H (1.032)

A

C 2

C2 (A-T)

A B C

3

C3 (A-U)

A B C

4

C4

5

C5

— A B C D

6

C6

A B C D

7

C7

A B C D

N-H (1.034)

H-bond (H---X)

H---O (1.773)

angle

(178.4)

[1.041, 1.037]

[1.706, 1.710]

[179.2, 179.5]

N-H (1.021)

H---O (1.919)

177.6

[1.037, 1.032]

[1.023, 1.019] N-H (1.020)

[1.023, 1.019] N-H (1.045)

[1.054, 1.048] C-H (1.086)

H---N (1.921)

[1.860, 1.873]

[1.861, 1.872] H---O (1.927)

[1.869, 1.880]

H---N (1.841)

[1.760, 1.780] H---O (2.889)

176.8

[177.0, 177.1]

[178.3, 179.1] (173.5)

[173.8, 174.1] (178.8)

[179.4, 179.4] (132.3)

[1.087, 1.086]

[2.786, 2.803]

[132.8, 132.5]

N-H (1.046)

H---N (1.837)

(178.6)

N-H (1.020)

[1.023, 1.019]

H---O (1.926)

[1.869, 1.876]

[1.055, 1.049]

[1.757, 1.775]

N-H (1.013)

H---O (2.058)

C-H (1.086)

[1.087, 1.086] N-H (1.016) N-H (1.045) N-H (1.018) N-H (1.017) N-H (1.031) N-H (1.032) N-H (1.032) N-H (1.031) O-H (1.004)

N-H (1.025) N-H (1.025) O-H (1.004)

H---O (2.892)

(173.4)

[173.7, 174.1]

[179.3, 179.3] (132.3)

[2.788, 2.805]

[132.7, 132.4]

H---N (1.978)

(176.8)

H---O (1.959)

H---N (2.090) H---O (1.993)

H---O (1.749)

H---N (1.988) H---N (1.988) H---O (1.749) H---O (1.615)

H---N (2.039) H---N (2.039) H---O (1.615)

EHBa

-10.89 -8.21

-8.71 -5.58 -8.46

-1.75 — — —

(161.2)

-2.99

(173.3)

-3.02

(176.2) (177.7)

-1.59

(178.8)

-12.05

(172.3)

-3.63

(170.7) (172.3) (170.7)

[0.386, 0.386] (0.366)

[0.385, 0.380] (0.314)

[0.325, 0.323] (0.309)

[0.320, 0.318] (0.409)

[0.442, 0.434] (0.137)

[0.139, 0.139] (0.309)

[0.319, 0.319] (0.411)

[0.445, 0.436]

-8.83 -8.83

-12.09 -3.63

-12.09

The values given in parentheses are from the [6-311++G**,29 6-311++G(3df, 3pd)] and (6-311++G**)30 basis sets using the B3P86, PBE1PBE, and B3LYP methods, respectively. A, B, C, and D are the HBs as shown in Figure 5. a

[35]


(0.137)

[0.139, 0.138]

-6.07

-12.05

(179.0)

(0.363)

-8.28

(178.8) (179.0)

HBSBIC

(0.303) (0.248) (0.381) (0.291) (0.268) (0.370) (0.367) (0.367) (0.370) (0.365) (0.337) (0.337) (0.365)


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Page 36 of 46

Table 7. Hydrogen Bond Strengths Based on Interaction Coordinates (HBSBIC), optimized bond distances (RO-H and RH---O in Å), and angles in degrees for some symmetric acid dimersa (XCOOH)2 [X = H (1), CH3 (2), n-butyl (3), C≡CH (4), F (5), CF3 (6), CCl3 (7)] and (HNO3)2 (8). no. 1 2 3 4 5 6 7 8

species

covalent bond (O-H)

H-bond (H---O)

Angle

HBSBIC

formic acid dimer (FD)

1.000 (0.991) [0.995] {0.990}

1.659 (1.690) [1.725] {1.727}

179.9 (179.7) [177.7] {178.0}

0.292 (0.273) [0.262] {0.258}

butanoic acid dimer (ButD)

— (0.993) [0.998] {0.992}

— (1.670) [1.694] {1.694}

— (177.8) [179.6] {179.3}

— (0.292) [0.284] {0.281}

acetic acid dimer (AD)

Propiolic acid dimer (PropD)

F-formic F-formic acid dimer (FF-FFD) Trifluoroacetic acid dimer (TfaD)

Trichloroacetic acid dimer (TclaD) Nitric acid dimer (ND)b

1.000 (0.992) [0.996] {0.991} — (0.992) [0.997] {0.991} 0.997 (0.989) [0.992] {0.987} — (0.992) [0.995] {0.990} — (0.992) [0.997] {0.992} — (0.981)

1.649 (1.678) [1.704] {1.704} — (1.681) [1.708] {1.711} 1.642 (1.666) [1.713] {1.708}

— (1.678) [1.717] {1.712} — (1.674) [1.690] {1.684} — (1.784)

178.4 (177.9) [179.9] {179.9} — (179.0) [179.4] {178.6}

177.0 (177.1) [173.9] {174.0} — (178.6) [175.9] {176.4} — (179.1) [177.3] {177.5} — (172.3)

a

0.306 (0.288) [0.280] {0.275} — (0.284) [0.274] {0.270} 0.296 (0.282) [0.260] {0.260} — (0.278) [0.261] {0.260}

— (0.283) [0.277] {0.275} — (0.267)

Values given were computed using the MP2/aug-cc-pVTZ method, while those in parentheses are from the (6-311G(2d,2p)), [6-31++G**], and {6-311++G**} basis sets using the same MP2 method. b

ND gives transition state at MP2/6-31++G** and MP2/6-311++G** levels of theory.

[36]


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Table 8. Hydrogen Bond Strengths Based on Interaction Coordinates (HBSBIC), optimized bond distances (RO---H and RO-H in Å), and angles in degrees for some asymmetric acid dimersa XCOOH-X'COOH [therein X = H and X' = CH3 (1), n-butyl (2), C≡CH (3), F (4), CF3 (5), and CCl3 (6)] and HCOOH-HNO3 (7). no. 1

species

Formic Acetic (F-AD) D A A

2

3

4

5

Formic Butanoic (F-ButD) D

A

A

D

Formic Propiolic (F-ProD) D

A

A

D

Formic F-formic (F-FFD) D

A

A

D

Formic Tfa (F-TfaD) D A

6

7

a

D

A

D

Formic Tcla (F-TclaD) D

A

A

D

Formic Nitric (F-ND) D

A

A

D

covalent bond (O-H)

1.003 (0.994)

H-bond (H---O)

1.634 (1.665)

Angle 179.4 (179.0)

HBSBIC 0.313 (0.293)

[0.999] {0.993}

[1.692] {1.693}

[178.8] {179.0}

[0.284] {0.279}

1.004 (0.995)

1.625 (1.660)

179.2 (178.9)

0.319 (0.296)

0.997 (0.989)

[0.993] {0.988}

1.674 (1.702)

[1.737] {1.736}

178.9 (178.5)

[179.1] {179.2}

0.285 (0.269)

[0.258] {0.255}

[1.000] {0.994}

[1.683] {1.684}

[179.2] {179.4}

[0.288] {0.283}

0.999 (0.990)

1.663 (1.696)

179.5 (179.8)

0.289 (0.270)

0.997 (0.990)

[0.993] {0.988}

1.674 (1.700)

[1.736] {1.736}

178.9 (178.5)

[179.2] {178.8}

0.284 (0.270)

[0.258] {0.256}

[0.995] {0.989}

[1.728] {1.728}

[177.6] {177.5}

[0.260] {0.256}

0.995 (0.987)

1.687 (1.712)

177.7 (178.0)

0.270 (0.256)

1.002 (0.993)

[0.997] {0.992} [0.990] {0.986} 1.004 (0.996)

[1.000] {0.993} 0.995 (0.987)

[0.990] {0.986} 1.007 (0.997)

1.646 (1.676)

[1.711] {1.709} [1.765] {1.762} 1.607 (1.634)

[1.659] {1.662} 1.698 (1.726)

178.9 (178.5)

[179.3] {178.9} [174.8] {175.0} 179.4 (179.8)

[177.5] {177.5} 178.1 (178.7)

0.305 (0.287)

[0.274] {0.272} [0.235] {0.236} 0.325 (0.306)

[0.297] {0.291} 0.263 (0.249)

[1.774]{1.768}

[175.6] {176.3}

[0.231] {0.232}

1.688 (1.726)

178.2 (178.4)

0.270 (0.251)

1.611 (1.640)

179.9 (179.7)

[1.658] {1.659}

[0.992] {0.987}

[1.751] {1.743}

[176.0] {176.6}

[0.243] {0.244}

0.984 (0.979)

1.795 (1.801)

175.0 (176.2)

0.234 (0.232)

0.995 (0.987)

1.006 (0.998)

[1.001] {0.996} [0.982] {0.978} 1.003 (0.993)

[1.000] {0.993}

1.614 (1.638)

[1.662] {1.663} [1.880] {1.879} 1.645 (1.684)

[1.696] {1.699}

[178.2] {178.3}

0.327 (0.306)

[1.003] {0.997}

179.5 (179.0)

[178.9] {178.8}

[171.8] {172.3} 170.9 (170.2)

[172.6] {173.0}

[0.301] {0.296}

0.325 (0.310)

[0.299] {0.293}

[0.208] {0.210} 0.327 (0.302)

[0.309] {0.304}

Values given were computed using the MP2/aug-cc-pVTZ method, while those in parentheses are from the (6-311G(2d,2p)), [6-31++G**], and {6-311++G**} basis sets using the same MP2 method. [37]



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Table 9. Hydrogen Bond Strengths Based on Interaction Coordinates (HBSBIC), optimized bond distances (RX-H and RH---X in Å where X = O and S), and angles in degrees for oxygen/sulfur substituted formic acid dimersa (see Figure 6). no.

species

1

D1

2

D2

3

D3

4

D4

5

D5

6

D6

7

D7

8

D8

9

D9

10

D10

covalent bond (X-H)

H-bond (H---X)

angle

(Eb)b

HBSBIC

O-H (1.008, 1.000) O-H (1.008, 1.000)

H---O (1.604, 1.659) H---O (1.604, 1.659)

(179.2, 179.9) (179.2, 179.9)

-8.5360 -8.5360

(0.322, 0.292) (0.322, 0.292)

O-H (1.001, 0.996) O-H (1.008, 0.999)

H---S (2.146, 2.178) H---O (1.593, 1.657)

(179.6, 178.0) (172.0, 170.5)

-6.2998 -8.4872

(0.213, 0.197) (0.377, 0.336)

O-H (0.994, 0.990) S-H (1.373, 1.356) O-H (0.992, 0.989) S-H (1.371, 1.354) S-H (1.361, 1.351) S-H (1.361, 1.351)

O-H (1.000, 0.995) O-H (1.000, 0.995) S-H (1.379, 1.358) O-H (0.996, 0.990) S-H (1.362, 1.350) S-H (1.368, 1.353)

O-H (0.994, 0.989) S-H (1.376, 1.356) S-H (1.367, 1.352) S-H (1.367, 1.352)

H---O (1.697, 1.730) H---O (1.825, 1.923) H---S (2.236, 2.257) H---O (1.840, 1.941) H---O (1.944, 2.007) H---O (1.944, 2.007) H---S (2.152, 2.188) H---S (2.152, 2.188)

H---S (2.300, 2.411) H---O (1.674, 1.729) H---O (1.946, 2.025) H---S (2.429, 2.507) H---S (2.222, 2.263) H---S (2.338, 2.446) H---S (2.444, 2.542) H---S (2.444, 2.542)

(173.5, 172.7) (177.5, 175.9) (171.3, 169.4) (172.9, 172.7) (173.7, 175.4) (173.7, 175.4) (168.6, 166.6) (168.6, 166.6)

(179.1, 178.6) (165.3, 163.7) (164.6, 163.9) (172.6, 172.1) (161.1, 158.0) (171.1, 169.5) (163.5, 160.2) (163.5, 160.2)

a

-5.3832 -5.6029 -4.4186 -5.2485 -3.2651 -3.2651 -6.2125 -6.2125 -4.7779 -5.7655 -3.1804 -2.9844 -4.6131 -4.2349 -2.8866 -2.8866

(0.321, 0.303) (0.438, 0.393) (0.219, 0.208) (0.480, 0.432) (0.466, 0.448) (0.466, 0.448)

(0.271, 0.245) (0.271, 0.245)

(0.172, 0.136) (0.347, 0.313) (0.435, 0.391) (0.180, 0.161) (0.260, 0.230) (0.203, 0.165) (0.215, 0.186) (0.215, 0.186)

The values given in parenthesis were obtained from the [6-31G*, aug-cc-pVTZ] basis sets using the PBE1PBE and MP2 methods, respectively, for all. b Reference 83, PBE1PBE/6-31G**.

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Table 10. Hydrogen Bond Strengths Based on Interaction Coordinates (HBSBIC), optimized bond distances (RO-H and RH---O in Å), and angles in degrees for some intramolecular Hbonded systemsa. The structures are reported in Figure 7. no. 1 2 3 4 5 6 7 8 9 10 11 12 13

oxalic acid, 1,4-butanediol, and conformers of 1,2-ethanediol

speciesb

oxalic acid (OA)

1,4-butanediol (BD)

1,2-ethanediol (ED1) 1,2-ethanediol (ED2) 1,2-ethanediol (ED3) IMHB1

IMHB2d IMHB3d IMHB4 IMHB5 IMHB6 IMHB7 IMHB8

covalent bond (O-H)

H-bond (H---O) 2.096

118.1

0.963

2.317

108.6

0.977 0.971 0.964 0.960

1.834 2.311 2.703

angle

HBSBIC

156.3

0.102

112.0 89.8

some typical intramolecular H-bonded systemsc

0.060 0.036 0.031 0.009

1.001 (0.997, 1.000)

1.645 (1.698, 1.673)

149.0 (146.0, 146.9)

0.155, 0.151c (0.134, 0.149)

0.986 (1.041, 1.054)

1.775 (1.510, 1.473)

144.6 (150.9, 152.0)

0.100, 0.093c (0.295, 0.353)

0.986 (0.983, 0.985) 0.982 (0.980, 0.982) 0.966 (0.966, 0.965) 1.116 (1.082, 1.085) 1.142 (1.104, 1.106) 1.009 (1.000, 1.002)

1.778 (1.839, 1.808) 1.829 (1.878, 1.850) 1.878 (1.911, 1.910) 1.309 (1.378, 1.375) 1.258 (1.320, 1.321)

1.647 (1.710, 1.697)

144.6 (141.4, 142.5) 142.9 (139.9, 141.0) 145.4 (143.5, 143.8) 164.7 (162.0, 162.1) 179.5 (176.7, 177.0)

134.8 (131.5, 132.1)

0.099, 0.093c (0.086, 0.096) 0.084, 0.077c (0.076, 0.084) 0.129 (0.121, 0.123) 0.631 (0.485, 0.501) 0.790 (0.652, 0.659) 0.186 (0.149, 0.160)

a

Values given are from the MP2/aug-cc-pVTZ level of theory, while those in parentheses are from B3LYP/6-311++G** and B3LYP/aug-cc-pVTZ.

b

Reference 49, MP2/aug-cc-pVTZ. cReference 34 and reference 40 therein, B3LYP/6-311++G**.

c

Values given in bold font are the resonance assisted HBS.

d

The optimized structures are same for IMHB2 and IMHB3 systems using MP2/aug-cc-pVTZ method.

[39]



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1 [(H2O)2]

2 [(H2O)3]

4 [(H2O)5]

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3 [(H2O)4]

5 [(H2O)6]

Figure 1. Qualitative structures for the water dimer and four small cyclic water clusters.

[40]


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1 [(MeOH)2]

2 [(MeOH)3]

3 [(MeOH)4]

4 [(MeOH)5]

5 [(MeOH)6]

Figure 2. Qualitative structures for the methanol dimer and four small cyclic methanol clusters. [41]



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1 [(H2O)2]

4 [NH3-HOH]

2 [MeOH –HOH]

5 [MeNH2-HOH]

Page 42 of 46

3 [Me2O –HOH]

6 [Me3N-HOH]

Figure 3. Qualitative structures for the mixed-water dimer complexes.

Figure 4. Qualitative molecular structure of the difluoromethane dimer.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


C1 (G-C)

C2(A-T)

C3 (A-U)

C4

C5

C6

C7

Figure 5. Structures of the DNA base pairs and the model systems. [43]



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 6. Sketch of the doubly H-bonded sulfur/oxygen substituted formic acid dimer complex. The proton donor and proton acceptor pairs (DA) are shown in Table 9 where D and A are either sulfur or oxygen.

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1 (OA)

3 (ED1)

6 (IMHB1)

10 (IMHB5)

2 (BD)

4 (ED2)

7 (IMHB2)

5 (ED3)

8 (IMHB3)

11 [IMHB6 (-)]

12 [IMHB7 (-)]

9 (IMHB4)

13 [IMHB8 (-)]

Figure 7. Optimized structures of oxalic acid, 1,2-butanediol, conformers of 1,2-ethanediol and some typical intramolecular H-bonded systems.

[45]



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TOC graphic

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Revised Manuscript Quantification of Hydrogen Bond Strength

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