Hydrogen-Bond Dissociation Energies from the Properties of Isolated Monomers

The strength of binding, as measured by the equilibrium dissociation energy De of an isolated hydrogen-bonded complex B···HX, where B is a simple Lewis base and X = F, Cl, Br, I, CN, CCH, or CP, can be determined from the properties of the infinitely separated components B and HX. The properties in question are the maximum and minimum values σmax(HX) and σmin(B) of the molecular electrostatic surface potentials on the 0.001 e/bohr3 iso-surfaces of HX and B, respectively, and two recently defined quantities: the reduced electrophilicity ΞHX of HX and the reduced nucleophilicity ИB of B. It is shown that De is given by the expression De = {σmax(HX)σmin(B)} ИB ΞHX. This is tested by comparing De calculated ab initio at the CCSD(T)(F12c)/cc-pVDZ-F12 level of theory with that obtained from the equation. A large number of complexes (203) falling into four categories involving different types of hydrogen-bonded complex B···HX are investigated: those in which the hydrogen-bond acceptor atom of B is either oxygen or nitrogen, or carbon or boron. The comparison reveals that the proposed equation leads to De values in good agreement in general with those calculated ab initio.


INTRODUCTION
It has long been an aim of those concerned with noncovalent interactions of molecules, particularly the hydrogen bond, to predict the properties of the complex so formed from those of the separate components. For example, Drago and co-workers proposed a relationship between the dissociation enthalpy of hydrogen-bonded complexes and two parameters associated with the separate components, one assigned to the hydrogenbond donor and the other assigned to the hydrogen-bond acceptor. 1,2 The approach favored by Taft and Abraham utilized experimental acidity and basicity scales of the hydrogen-bond donor and acceptor molecules, respectively, 3−7 while Platts employed acidities and basicities calculated theoretically. 8−11 Alternative approaches 12−14 discussed hydrogen bonding in complexes B···HX in terms of the distances r(B···H) and r(HX) and also via the relationship between electron density properties and r(B···H) distances. 15−20 From geometries determined in microwave spectroscopic studies of various hydrogen-bonded complexes B···HX (X is a halogen atom), rules for predicting angular geometries based on HX acting as a probe for the directions of nonbonding and π-bonding electron pairs were formulated and discussed in several articles. 21−23 The rules were electrostatic in character. Buckingham and Fowler 24,25 were able to predict successfully the angular geometries of hydrogenbonded complexes in terms of the electric-charge distribution of the separate molecules, each described by a distributed multipole analysis. In the present article, we propose a method of predicting two measures of the strength of the hydrogen bond, namely, the equilibrium dissociation energy D e for the process B···HX = B + HX and the intermolecular quadratic, stretching force constant k σ of the complex B···HX from the properties of the isolated molecules B and HX. The properties in question are the so-called molecular electrostatic surface potentials (MESP) of B and HX together with the reduced nucleophilicity of the acceptor molecule B and the reduced electrophilicity of the hydrogen-bond donor HX, as recently introduced.

THEORETICAL METHODS
The equilibrium dissociation energies D e of most of the hydrogen-bonded complexes used to produce the generalizations presented here are taken from recent publications. 26−28 For complexes not considered in earlier publications, their geometries and those of the isolated monomers were calculated by exactly same the approach as in refs 26−28. Thus, geometry optimizations were conducted at the CCSD(T)(F12c) computational level 29,30 in the frozen core approximation and with the choice of cc-pVDZ-F12 basis sets, 31 using the MOLPRO program. 32 D e values were taken as the difference of the electronic energies of the complex and those of the isolated monomers and, as previously, were corrected for basis set superposition error (BSSE) by means of the full counterpoise method of Boys and Bernadi. 33 The molecular electrostatic surface potentials (MESP) of the isolated Lewis bases B and acids HX were calculated with the GAUSSIAN program 34 by employing the MP2/aug-cc-pVTZ wavefunction and analyzed on the 0.001 e/bohr 3 electron density iso-surface with the Multiwfn program. 35

Background.
It was shown as long ago as 1987 36 that the intermolecular stretching force constant k σ for an isolated hydrogen-bonded complex B···HX (available from the spectroscopically determined centrifugal distortion constant) 37  An important molecular property in the present context is the quantity called the molecular electrostatic surface potential (MESP). 46 This property of a molecule is defined as the electrostatic potential energy of a unit positive charge on the isosurface at which the electron density has a constant value, in this case 0.001 e/bohr 3 . Given that noncovalent interactions have a substantial electrostatic component, that the axes of nonbonding electron pairs or π-bonding pairs are directions associated with most negative (minimum) electrostatic potential, and that the atoms acting as donors in hydrogen bond, halogen bond, etc. formation are associated with regions of maximum electrostatic potential, it seems reasonable that the MESP has a role when noncovalent interactions are discussed. This is made clear by while division of eq 2 by σ min (B) results in When D e /σ max (HX) was plotted against N B for complexes B··· HX (X = F, Cl, Br, I), the separate straight lines through the origin (generally having different gradients) that were previously obtained from the D e versus N B plots for different molecules B became conflated to a single straight line. This led to the definition of the quantity Ξ HX = E HX /σ max (HX) as the reduced electrophilicity, a property common to the HX molecules, independent of whether F, Cl, Br, or I was attached to H. Likewise, when D e /σ min (B) was plotted against E HX for a range of Lewis bases B, the separate straight lines through the origin obtained from the D e vs E HX plots (generally of different gradients) became conflated to a single straight line and suggested the definition of И B = N B /σ min (B) as the reduced nucleophilicity of the group of Lewis bases B involved. Analysis in ref 28 showed that values of И B determined from different types of Lewis base were not significantly different when the atom of B directly involved in the hydrogen bond in B···HX was the same and hence that И B is an intrinsic property of the atom, independent of the remainder of B. Moreover, И B vary little when that atom was any one of the first-row series B, C, N, or O. The starting point of the analysis presented in this article is to note that if eq 3 is divided by σ min (B) or eq 4 is divided by σ max (HX), the result is We note that the quantities on the right-hand side of eq 6 are properties of the isolated Lewis base B and the isolated Lewis acid HX. Hence, eq 6 suggests a route to the calculation of the dissociation energy of an isolated hydrogen-bonded complex B···HX from the properties of the individual molecules B and HX. Table 1 records values of σ max (HX) and σ min (B), as calculated at the MP2/aug-cc-pVTZ level, for the molecules B and HX to be considered here. Table 2 carries values of the reduced electrophilicities Ξ HX and the reduced nucleophilicities И B of the molecules of interest, as determined in refs 26 and 28, respectively. Groups of complexes B···HX in which the hydrogen bond to HX is to O are discussed first, followed by corresponding groups in which the bond is successively to N, C, and B atoms.

Comparison of D e Calculated from Eq 6 and Ab Initio Calculated Values.
3.2.1. Complexes B···HX in Which the Hydrogen-Bond Acceptor Atom Is O. Table 3 lists the complexes containing the  Table 2), while the final three columns show D e (eq 6), D e (ab initio), and the difference between these two values, respectively. The agreement between D e (eq 6) and D e (ab initio) is mainly good, except perhaps for the complexes B···HCN. In fact, it will be noted later that HCN is generally anomalous in this respect, for reasons presently unknown. An interesting feature of Table 3 is the group of complexes CO···HX, in which the hydrogen bond is to the oxygen atom of carbon monoxide. In fact, both ends of carbon monoxide are negative and capable of sustaining hydrogen bonds, that is CO is ambi-nucleophilic, as may be seen from the two values of σ min (CO) included in Table 1. This is also true of N 2 . It will be seen in Section 3.2.2 that the other isomer, OC···HX, is the more strongly bound. It is worth noting that, although the molecules B involved in Table 3 range from diatomics, through linear triatomics, to polyatomic asymmetric tops, the predictions of D e by eq 6 are all good, except for those involving O···HCN, as mentioned earlier. Figure 3a shows a graph of D e (eq 6) plotted against D e (ab initio) for the 42 O··· HX complexes listed in Table 3. The continuous straight line is the result of a linear regression fit to all of the points and has a gradient of 0.999 (29) and R 2 = 0.9683. The fit is slightly better if the O···HCN points are excluded, as can be seen in Figure 3b Table S1, which is available in the Supporting Information. The graph of all D e (eq 6) plotted against D e (ab initio) is shown in Figure 4a. The complexes consist of CH 3 CN···HX, HCN···HX, FCN···HX, N 2 ···HX, and  Figure 4a. When the points involving HCN either as proton donor or acceptor are removed, the scatter is reduced with the gradient and R 2 of the regression fit both closer to 1, as can be seen in Figure 4b.  Table S2, which is available in the Supporting Information. The graph of all D e (eq 6) plotted against D e (ab initio) is shown in Figure 5. The complexes consist of CH 3 NC···HX, HNC···HX, FNC···HX, OC···HX, and SC···HX. Guided by the anomalous behavior of HCN complexes when the H-bond acceptor atom is O or N, the five hydrogen bonds of the type C···HCN type (with HCN as Hbond donor) were removed from Figure 5, with the result shown in Figure S1. Again there is a small improvement in the scatter of points, as indicated by the movement of both the gradient and R 2 both closer to 1.0 than in Figure 5.

Complexes B···HX in Which the Hydrogen-Bond Acceptor Atom of the Lewis Base Is
Boron. The compound F− B has been characterized experimentally, 47 and theory has shown 48 that the predominant contribution to its valence-bond description is a Lewis structure having a single covalent bond, three equivalent nonbonding electron pairs on F, and one nonbonding pair on the axis of FB. Hence, F−B should form hydrogen bonds with the Lewis acids HX (X �F, Cl, Br, I, HCN, HCCH, and HCP) of the type F−B···HX. Complexes R−   Table S3 in the Supporting Information, along with their ab initio calculated counterparts D e (ab initio) and the differences between these two quantities. Figure 6 shows the graph of D e (eq 6) as the ordinate and D e (ab initio) as the abscissa for these complexes. The corresponding plot when the three complexes F−B···HCN, H−B···HCN, and H 3 C−B···HCN are removed is shown in Figure S2 in the Supporting Information. The fit quality is only slightly improved from that in Figure 6. Figures  3−6 and Tables 1 and S1−S3 are encouraging in that they show that the equilibrium dissociation energy D e of simple complexes of the type B···HX (X = F. Cl, Br, I, CN, CCH, and CP) can be predicted with reasonable accuracy from the properties of the isolated molecules B and HX, namely, σ min (B) and σ max (HX), the reduced nucleophilicity И B of B, and the reduced electrophilicity Ξ HX of HX. However, И B and Ξ HX were determined by use of the D e values of all of the complexes listed in Tables 1 and S1 (except the values for CO···HX), S2, and S3. It remains to apply a more stringent test, that is to calculate D e from eq 6 and compare those with a set calculated ab initio for complexes B···HX that were not used in generating И B and Ξ HX . The D e (eq 6) and D e (ab initio) values for the CO···HX complexes are in Table 1, while Table S4 carries these quantities for the complexes, ClCN···HX, ClNC···HX, and R− B···HX, where R = H 3 Si, Cl, Br, I, CN, NC, and F 3 C. None of these were used to determine И B and Ξ HX . The graph of D e (eq 6) vs D e (ab initio) is displayed in Figure 7 for the full set of these 70 complexes. The resulting points show some scatter with respect to the regression fit, with R 2 = 0.9669 and a gradient = 0.949 (21). When the 10 complexes that employ HCN as the Lewis acid are removed, the scatter is significantly reduced, with the result shown in Figure S3 in the Supporting Information. Thus, the value of R 2 and the gradient increase to 0.9833, and 0.963 (17), respectively. It is concluded from these observations that, for hydrogen-boned complexes formed between simple molecules of the type considered here, eq 6 can predict with reasonable accuracy the value of D e , even when these complexes were not involved in the determination of the reduced quantities И B and Ξ HX . Moreover, the range of D e values that can be   predicted from these properties of the separate molecules is significant, from about 4 to 40 kJ mol −1 .

Calculation of Intermolecular Stretching Force
Constants k σ from D e . It was shown in ref 38 that, for a wide range of hydrogen-bonded and halogen-bonded complexes, the quadratic intermolecular stretching force constant k σ is directly proportional to the dissociation energy D e . Experimental values of k σ can be determined from centrifugal distortion constants available from analysis of rotational spectra using the expressions due to Millen 37 or can be calculated ab initio by finding the second derivative of the potential energy with respect to the intermolecular distance evaluated at the equilibrium. 49 The former values have necessarily employed zero-point spectroscopic constants in the Millen formulae, so the latter are preferred here. It was shown in ref 49 that when ab initio values of D e calculated at the CCSD(T)/CBS level were plotted against k σ for hydrogen-bonded B···HX of type considered here, the appropriate form of eq 1 combined with eq 2 for complexes of the type considered here is Equation 7 therefore provides a means of predicting k σ from the properties of the isolated molecules B and HX.

CONCLUSIONS
The equilibrium dissociation energies D e of 203 hydrogenbonded complexes of the type B···HX, in which B is a simple Lewis base molecule and X is one of F, Cl, Br, I, CN, CCH, or CP, have been predicted by means of eq 6 and compared with the same quantities obtained by ab initio calculations at the CCSD(T)(F12c)/cc-pVDZ-F12 level after correction for basis set superposition error. Equation 6 involves only properties of the isolated molecules B and HX, namely, the minimum value σ min (B) of the MESP on the 0.001 e/bohr 3 iso-surface of B, the maximum value σ max (HX) of the MESP on the 0.001 e/bohr 3 iso-surface of HX, the reduced nucleophilicity И B of B, and the reduced electrophilicity Ξ HX of HX. The molecules HX are all linear and therefore in each case σ max (HX) lies on the molecular axis near the H atom. The molecules B were all chosen so that σ min (B) can be unambiguously associated with the direction of the axis of a nonbonding electron pair (as conventionally envisaged) and carried by the atom acting as the hydrogen-bond acceptor. И B has recently been identified as an intrinsic property of the atom acting as the hydrogen-bond acceptor, while the reduced electrophilicity Ξ HX is an intrinsic property of the H atom in HX when X is a halogen atom. When X = CN, CCH, or CP, Ξ HX has also a constant value but is different from that of the hydrogen halides. In view of these properties, the comparison of D e calculated with eq 6 with the ab initio value of D e was made in groups, in each of which the atom acting as the H-bond acceptor was the same, namely, O···HX, followed by N···HX, then C···HX and finally Boron···HX. The graph of D e (eq 6) vs D e (ab initio) for each group allowed a linear regression fit with a gradient near to 1, indicating that eq 6 does provide a reasonably accurate method for predicting D e from properties of the individual molecules forming the complex. It was noticed, however, that complexes in which HCN was either the H-bond donor or the H-bond acceptor molecule sometimes fell further from the fitted regression line than other complexes. When these were removed from the graph, the scatter was reduced. The reason for this behavior of HCN is not presently known.