Interpretation of experimental hydrogen-bond enthalpies and entropies from COSMO polarisation charge densities †

In this work, experimental hydrogen-bond (HB) enthalpies measured in previous works for a wide range of acceptor molecules in dilute mixtures of 4-fluorophenol in non-polar solvents are quantified from COSMO polarisation charge densities s of HB acceptors (HBA). As well as previously demonstrated for quantum chemically calculated HB enthalpies, a good correlation of the experimental data with the polarisation charge densities is observed, covering an extended range of HBA (O, N, S, p systems and halogens) ranging from very weak to strong hydrogen bonds. Furthermore, for the first time, a quantitative analysis of experimental HB entropies is performed for such a chemical diversity of HBA. A good quantification of these entropies is achieved using the polarisation charge density s as a descriptor in combination with the logarithm of a directional partition function OHB. This partition function covers the directional and multiplicity entropy of HBA and is based on the s-proportional HB enthalpy expression taken from COSMO-RS. As a result, the experimental HB enthalpies and free energies of the B300 HB complexes are quantified with an accuracy of B2 kJ mol 1 based on COSMO polarisation charge densities.


Introduction
The hydrogen bond (HB) is recognised as the most important specific interaction between a molecule and its local environment.However, there is a tendency to view hydrogen-bond acceptors (HBA) and donors (HBD) as atomic sites and to consider them equivalent despite that the effects of organic functions and substituents that define the local molecular environment can have huge impacts on their HB properties.Indeed, although HBs are qualitatively well understood, it is generally admitted that quantitative data are needed.From an experimental point of view, the quantification of HB energy is complicated because a hydrogen bond can never be studied individually, its formation going unavoidably along with the concomitant dispersion and electrostatic contributions.
The situation is even worse in solution, since the formation of a hydrogen bond in this case additionally requires the removal of the solvent molecules in interaction with the acceptor prior to the HB formation.If the solvent molecules are not completely isotropic with respect to their electrostatics, or if they even have their own polar hydrogen atoms, i.e. their own prospective HBD site, the estimation of the free energy needed for removing the solvent at the position of the acceptor (desolvation) even requires statistical thermodynamics for getting appropriate averages.
The importance of hydrogen-bonding and the need of quantitative parameters for a comprehensive understanding and description of the interaction has led several research groups to concentrate their efforts in the construction of HB basicity scales, called pK HB , 1 log K b , 2 Sb 2 H , 3,4 SC a 5,6 or B. 7 Following the pioneering work of Taft and coworkers, 1 Laurence et al. 8 reported recently the development of the pK BHX database, which contains an unprecedented set of experimental values related to HB basicity.More precisely, HB free energies have been determined in tetrachloromethane (CCl 4 ) for a large number of chemically diverse HBA molecules using p-fluorophenol (pFP) as the reference HBD, chosen for historical and technical reasons. 1,9he measurements are made following the standard procedure based on the absorbance decrease in the OH absorption of pFP (n(OH) = 3614 cm À1 ; e = 237 dm 3 mol À1 cm À1 ) in dilute ternary pFP-base-CCl 4 solutions at 25 1C, leading to pK BHX values.More precisely, typical concentrations of 3-4 mM are prepared for our reference donor (pFP) and of 10-300 mM for the acceptors, depending on their HB basicity.Although contaminated to some degree with local desolvation and electrostatic contributions, the HB free energy is estimated from the equilibrium constants of complex association as -RT ln K, or equivalently as -RT pK BHX /ln 10.Using this notation, a strong HB acceptor forming a hydrogen-bonded complex with a large association constant (K), that is a low dissociation constant (K BHX = 1/K), has a large positive value of pK BHX .This can be reasonably justified based on the aprotic nature of CCl 4 , its high degree of isotropy with respect to polarity, and the use of a reference HBD.In its present version, the pK BHX database contains about 1340 values corresponding to nearly 1200 HBA.Furthermore, the HB enthalpies towards pFP for about 310 varied organic bases, ranging over 40 kJ mol À1 , have been published, 10 providing, in addition to the HB free energies, the quantitative establishment of the order of HB enthalpies according to a given atomic acceptor site.More exactly, this order of HB enthalpies has even been established according to the chemical function to which belongs the studied atomic site.These values have been measured from the temperature dependence determinations of the complex equilibrium constants.For this energetically extended and chemically diversified set of acceptors, the HB free energy was split into its enthalpic and entropic contributions.To our knowledge, this work is therefore the most extensive homogeneous experimental study of the thermodynamics of hydrogen bonding that has been carried out so far.From a theoretical perspective various attempts have been presented aiming for a predictive quantification of hydrogenbond strength.][4] Beyond more empirical increment schemes 11 quantum chemical descriptors have been used for a correlation of HB strength.Kenny 12 used the electrostatic potential as a quantum chemical descriptor for HBA strength of nitrogen bases.Schwo ¨bel and coworkers [13][14][15][16] presented several studies, in which they demonstrate the suitability of local frontier orbital descriptors 17 for the predictive quantification of Abraham's HBA and HBD parameters of a chemically much broader range of donor and acceptor sites.
Klamt et al. 18 recently presented a detailed study on HB enthalpies calculated quantum chemically for 2465 donor acceptor pairs.For reflecting the situation in a solvent, solvation effects were considered in the extreme limit of a perfect conductor employing the COSMO continuum solvation model. 19n a very large and chemically very diverse set of B2500 donoracceptor complexes, they demonstrated that the quantum chemically calculated HB enthalpies can be quantified with an accuracy of B1.3 kJ mol À1 assuming a bilinear dependence on the COSMO polarisation charge densities s don and s acc .The COSMO polarisation charge density is the local surface density of the conductor screening charges, i.e. the screening charge per unit area of the cavity surface, as resulting from the conductorboundary conditions in the continuum solvation model (COSMO).1][22] For those not familiar with the concept of COSMO, a more detailed description and discussion of the polarisation charge density s is given in appendix SI1 in the ESI.† The study on the quantum chemically calculated HB complexes confirmed the bilinear s-dependence of HB bond energies assumed within the COSMO-RS.
In this paper we demonstrate that the described linearity of HB energies with respect to s acc does not only hold for the calculated HB energies, but also for the experimental HB enthalpies reported by Laurence et al. 10 An analysis of the HB free energy with respect to s acc clearly discloses that the directional HB entropy has an important influence.Furthermore, we introduce a directional HB partition function which allows for the quantification of this directional entropy without any additional adjusted parameter.

Data sets and calculation methods
For the sake of consistency, we will only consider in this paper the 309 HBA molecules for which HB enthalpies DH o HB and HB free energies DG o HB with pFP in CCl 4 or tetrachloroethylene (C 2 Cl 4 ) have been reported by Laurence et al. 10 in the core and in the supplementary data of their paper.Some compounds (e.g.amines) are known to react with CCl 4 , and the corresponding measurements were therefore made in C 2 Cl 4 , which is very similar to CCl 4 , in particular with respect to polarity.The differences between the HB data obtained in the two solvents have indeed been shown to be small and no systematic differences were observed. 23or all compounds, DFT/COSMO calculations have been performed with the TURBOMOLE program 24 at the BP-TZVPD/ COSMO level, [25][26][27][28][29][30] as described in ref. 18.A conformational analysis following the concepts of COSMOconf workflow 31 was performed for each of the acceptor molecules.For each atom the maximum of the locally averaged polarisation charge density s was calculated with the COSMOtherm program 32 according to eqn (11) of ref. 21.Since negatively polar HBAs need to have positive s-values, the atom with the maximum s-value, i.e. the most polar acceptor, was selected as the HBA of interest in the case of polyfunctional molecules, bearing several HBAs.Actually, these theoretical assignments disagreed from those assumed in the experimental work only in one case, which will be discussed below.
While in the previous study on the s-dependence of QM/COSMO calculated HB enthalpies the molecular conformation was known for each donor and acceptor under consideration, the comparison with experimental HB enthalpies and free energies bears the complication that for some of the more flexible acceptors, the exact or predominant conformation in the solvent is not known from experiment.The most relevant conformation for hydrogen bonding may even be, in some cases, different from the predominant conformation in the solvent.To appropriately deal with such conformational ambiguities, we carried out a COSMO-RS study of the complex formation constants by calculating the contact probability of the dilute acceptor molecules and dilute pFP molecules in CCl 4 with the COSMOtherm program. 32At first, that calculation was performed for the entire conformational set, yielding, aside of the conformationally averaged contact probability, the population of the individual conformations in CCl 4 .Then we repeated the calculation of the complex formation constant only taking into account the most populated conformation in CCl 4 .The ratio f K,conf of the calculated complex formation constants without and with conformational multiplicity was calculated and is reported in Table SI1 in the ESI.† As shown in Fig. 1, for most of the considered acceptors the complex formation constant is rather insensitive to the conformational multiplicity.Only for seven compounds we found log(f K,conf ) to be lower than À0.4,all of them being benzylamines, except 1,2,3,4-tetrahydroisoquinoline, the extreme value being À3.2 log-units for 3-(trifluoromethyl)benzylamine.For these molecules, our calculations show that the lowest energy conformation in CCl 4 is much less involved in HB interactions than the other conformations.A closer investigation showed that this behaviour is due to the formation of a 5-membered ring stabilized by a weak intramolecular interaction of the slightly positive ortho-hydrogen with the amine lone pair, shown in the right inset in Fig. 1.On that basis, the 18 amines having the same 5-membered ring structure in their lowest energy conformer in CCl 4 have been removed from our sample.Being a single exception, we kept 1,2,3,4-tetrahydroisoquinoline in the data set.For all compounds with |log(f K,conf )| less than 0.4, we estimate that the potential error resulting from the neglect of additional conformations on the free energies of HB formation should be smaller than B2 kJ mol À1 .
For two very bulky tertiary amine compounds (N,N-diisopropyl-3-pentylamine and N,N-diisopropylisobutylamine) a more serious problem occurred.In all conformations generated by COSMOconf the amine lone-pair was entirely hidden by the bulky substituents.No surface area with sufficiently high s for hydrogen bonding was found.Even manual attempts to generate the conformations with the accessible amine lonepair failed, since the quantum chemical energies of these conformations were too high in order to be relevant.These theoretical trends agree well with the experimental behaviour observed for tertiary amines 33 for which the nitrogen is hidden by long and/or branched alkyl chains.Consistently, these two compounds have also been removed from our analysis dataset, which finally consisted of 289 acceptor molecules.

Results and discussion
As the first step of our analysis, we plotted the experimental HB enthalpy DH o HB vs. the polarisation charge density s acc of the strongest acceptor position, as shown in Fig. 2. A linear dependence can clearly be observed.This relationship is characterised by a correlation coefficient r 2 of 0.935, and a standard deviation of 2.4 kJ mol À1 with a rather homogeneous error distribution.Hence, we may conclude as a first result that the HB enthalpies measured in CCl 4 confirm the assumption of s-proportionality of the HB enthalpy as suggested in our previous theoretical study. 18ext, we considered the analogous plot for the experimental HB free energy DG o HB , as shown in Fig. 3.The DG o HB data are the primary results of the equilibrium constant determination while the enthalpies are derived from the van't Hoff plot established with the equilibrium constants measured on a range of 60 1C, from À5 1C to 55 1C.For this reason, the experimental error is found to be smaller for DG o HB (AE0.2 kJ mol À1 ) than for DH o HB (AE1 kJ mol À1 ).However, despite the higher experimental accuracy for DG o HB , the correlation with the polarisation charge density s acc , with r 2 = 0.729 and s = 3.0 kJ mol À1 , is much worse than the corresponding one found for DH o HB .This is not really unexpected since the entropic contribution arising from site and lone-pair multiplicity, from the degree of directional softness and from the vibrational changes arising from the deep and narrow potential energy minima typical for HB cannot be assumed to be simply proportional to s acc .After a closer analysis of the deviations from the regression line, it turns out that the s-regression strongly underestimates DG o HB for exposed oxygen atoms, the extreme case corresponding to hexamethylphosphoramide, while on the upper side of the regression line the DG o HB obtained are in contrast overestimated for sterically hindered acceptor atoms such as tertiary amine nitrogens.This observation highlights the need of a quantitative measure of the directional entropy, i.e. of the number of almost equivalent positions on the molecular surface at which the hydrogen bond can be formed, in the HB free energy.
The entropic contribution arising from the multiplicity is to some extent already taken into account in the experimental data, The 309 compounds are ordered with respect to the ratio f K,conf of the donoracceptor contact probability calculated without and with conformational multiplicity.The red symbols mark the compounds removed from the sample (see the text).The inset illustrates the preferred conformation of benzylamine in polar solvents with an exposed nitrogen lone-pair, and the preferred conformation in a non-polar solvent as CCl 4 , in which the nitrogen lone-pair interacts with an orthohydrogen atom and thus is less available for hydrogen bonding.
since the entropic values ÀDS o are statistically corrected by a ÀR ln(n) term, where n is the number of (quasi)-equivalent sites, in order to put these values on a per acceptor atom basis.However, some cases have been trickier to handle.Thus, the aromatic bases have been treated considering one benzene ring as a single HBA.In the same vein, for molecular systems bearing two HBAs in close spatial proximity (e.g.syn-2,4-difluoroadamantane) the experimental values refer to one HBA site, based on the experimental and theoretical evidence of bifurcated (three-centre) HB interactions in these cases. 34Lastly, in the case of polyfunctional bases, experimental arguments have been used as guides for the statistical correction to apply.For example,  in the case of triethylthiophosphate, which possesses three oxygen and one sulphur atoms as potential HBAs, we have considered the sulphur atom as the only HBA site, based on a previous experimental investigation. 35This last assignment is in contradiction with the trends obtained from the computed polarisation charge densities s acc , since the three oxygen atoms are found to be the strongest HBA sites.These various situations have led us to the decision to return to the unbiased original DG o values for the entire sample of molecules, and to calculate the full HB directional and multiplicity entropic correction by a HB partition function, evaluated over the entire molecular surface.Based on the assumption of s-proportional HB enthalpy, and using the slope of the regression line in Fig. 1 as a reasonable guess for the dependence of the local HB enthalpy on s acc , we thus constructed a HB partition function O HB and the respective entropy descriptor S HB dirmult = R ln(O HB ), which is expected to capture this missing entropic contributions.For details of the calculation of O HB see Appendix 1.As a side product, this partition function also gives us an expectation value hs acc i which should be a better measure for the enthalpy than the maximum value.
As can be seen in Fig. 4, the correlation between the experimentally reported DG o HB values, corrected by -RT ln(O HB ), and s acc is strongly improved (r 2 = 0.923 and s = 2.1 kJ mol À1 ).This goes along with a significant change in the slope of the regression line.We attribute this to the fact that the directional entropy is to some degree correlated with s acc .This is because the nitrogen acceptors, and especially the tertiary amines, have the highest values of s acc , but very small exposed surface area available for the single lone-pair, while oxygen acceptors typically have smaller values of s acc and two lone-pairs, i.e. larger HB area.Finally, the very weak halogen and p-acceptors have small s acc values but a large accessible surface for complexation of quite similar s, i.e. a large directional entropy.
If we replace the maximum of the acceptor polarisation charge density, so far denoted s acc , by the expectation value hs acc i, and replace the reference s in the partition function accordingly, then the statistics does not change, i.e. we still have r 2 = 0.923 and s = 2.1 kJ mol À1 .But if we replace the descriptor s acc by hs acc i in the enthalpy correlation, i.e. in an analogue of Fig. 1, the correlation improves to r 2 = 0.945 and s = 2.2 kJ mol À1 , while the slope of the regression is almost unaffected.Hence, overall, the hydrogen-bond expectation value hs acc i seems to be the slightly better descriptor, but from a practical perspective the local maximum value s acc is more readily available.
Going a next step toward thermodynamic consistency, we can calculate the partition functions and expectation values taking into account multiple conformations of the acceptor.This requires the knowledge of the conformational population w i of each conformation i.If we use the COSMO-RS model in its COSMOtherm implementation, we get estimates of these conformational populations in the solvent.Based on these we can easily get the respective HB partition functions O HB,conf and expectation values hs acc i conf .Inserting these values we end up with a tiny improvement for the HB enthalpy regression (r 2 = 0.946, s = 2.2 kJ mol À1 ; see Fig. 4) and also for the HB free energy regression (r 2 = 0.924, s = 2.1 kJ mol À1 ; see Fig. 5).Since we had already excluded the 18 compounds showing conformational preferences with specific intramolecular interactions, it is not surprising that the improvement in this dataset is small.But, as shown in Fig. 5 and 6, these 18 HBAs are now described with the same quality as the other compounds using the conformationally averaged quantities.
Based on the two regressions (in units of e nm À2 for s and kJ mol À1 for energy) DH o HB = À17.9hs acc i + 5.4 Hence the HB entropy consists of an acceptor independent negative constant, a directional and multiplicity contribution and a major s-proportional entropy loss.The latter can be easily interpreted by the deeper and thus more narrow minimum on the molecular potential energy surface with increasing HB enthalpy, and hence with increasing s acc .
It might be worth noting that the s-slope of the HB enthalpies found here, i.e. the value of À17.9 kJ mol À1 nm 2 e À1 , is only 63% of the s-slope of the regression line found in eqn (1) in the study on quantum chemically calculated HB enthalpies in a conductor environment 18 for a donor of s don = 2.0, i.e. using the value found on the donor hydrogen atom of pFP.Hence, in a conductor, the hydrogen bonds would be roughly 50% stronger than in CCl 4 .This is not surprising given the stronger polarisation of the donor and acceptor molecules in a conductor compared with the non-polar CCl 4 environment used here.Indeed, this is in quite good agreement with the fact that the polarisation charge densities in a non-polar solvent typically are by a factor of 0.8 smaller than in a conductor, due to the almost complete lack of electronic back-polarisation in non-polar solvents. 36Since we are using as descriptors s values calculated in a conductor environment here, and since this reduction applies to the donor and acceptor, we would expect a reduction of the slope of DH o HB , vs. s by a factor of B0.8 2 = 64%.

Summary and conclusions
Experimental HB free energies and enthalpies measured in CCl 4 for a wide and varied selection of HB bases have been analysed from COSMO polarisation charge densities s acc of the acceptor atoms.The HB enthalpies show a strong correlation with s acc with r 2 = 0.94 and a standard deviation of B2 kJ mol À1 .Based on the observed s-proportionality of the HB enthalpies, partition functions have been constructed for the molecular free energy of hydrogen bonding.Using these partition functions for the description of the directional and multiplicity entropy of hydrogen bonding, the HB free energy can as well be described as a linear function of s acc with almost the same correlation coefficient and standard deviation.The lower slope of the HB free energy s-regression compared to the HB enthalpy regression corresponds to a s-proportional part of the HB entropy.The relations presented herein between experimental HB enthalpies or free energies and the COSMO polarisation charge From a theoretical point of view the results present for the first time a quantitative model for the HB entropy in solution.This model demonstrates that entropy contributions arising from the directional flexibility are of comparable importance as the entropy loss which arises from the vibrational restriction of the HB length, which increases proportionally to the HB strength.This quantitative insight into the different contributions of the HB entropy can be useful for improvements of the HB expressions in solvation models as COSMO-RS, but also for force field improvements.

Appendix 1: construction of the HB partition function
If we assume that every surface segment n with surface area a n and polarisation charge density s n can form a hydrogen bond of enthalpy where the n summation is over all surface segments of a molecule a, and s ref can be either the maximum value of s acc on the acceptor molecule, or its expectation value according to eqn (A3).The i summation is over the conformations of molecule a, where w a i is the relative conformational population of conformation i.In the case of just one conformation it is unity.a hb is the effective hydrogen bond contact area, which has a value of 4.57 Å in the parameterisation used here.
In our case of hydrogen bonds with a fixed donor (pFP) in a fixed solvent (CCl 4 ) the value of c hb s don can be taken from the slope of the HB enthalpy with respect to s acc , i.e. from the slope of the regression line in Fig. 1.
The expectation value of the hydrogen bond s acc arising from the multiple directional choices of the HB and potentially from multiple HB sites on the acceptor surface consequently is given as

Fig. 1
Fig.1Conformational sensitivity of the calculated complex formation constant.The 309 compounds are ordered with respect to the ratio f K,conf of the donoracceptor contact probability calculated without and with conformational multiplicity.The red symbols mark the compounds removed from the sample (see the text).The inset illustrates the preferred conformation of benzylamine in polar solvents with an exposed nitrogen lone-pair, and the preferred conformation in a non-polar solvent as CCl 4 , in which the nitrogen lone-pair interacts with an orthohydrogen atom and thus is less available for hydrogen bonding.

Fig. 3
Fig. 3 Experimental HB free energies, DG o HB , of 289 acceptor molecules plotted vs. the COSMO polarisation charge density s acc of the most polar acceptor position.

Fig. 2
Fig. 2 Experimental HB enthalpies, DH o HB , of 289 acceptor molecules plotted vs. the COSMO polarisation charge density s acc of the most polar acceptor position.

Fig. 5
Fig. 5 Experimental HB enthalpies, DH o HB , of the 307 HBA molecules (289 + 18) plotted vs. the HB expectation value of the COSMO polarisation charge density hs acc i conf .The full green symbols mark the 18 conformational problematic cases (mainly benzylamines).For comparison, open green symbols mark these compounds before conformational averaging.
h n hb = c hb s don s n (A1)with a hydrogen bond donor of polarisation charge density s don , then the relative partition function of hydrogen bonding compared to a reference isO a hb;conf ¼ a hb À1 Àc hb s don ðs n À s ref Þ RT (A2)