Article pubs.acs.org/JPCA

Composite Method for Implicit Representation of Solvent in Dimethyl Sulfoxide and Acetonitrile Anna Pomogaeva Inorganic Chemistry Group, Department of Chemistry, St. Petersburg State University, University Pr. 26, Old Peterhof 198504, Russia

Daniel M. Chipman* Radiation Laboratory, University of Notre Dame, Notre Dame, Indiana 46556-5674, United States S Supporting Information *

ABSTRACT: A composite method for implicit representation of solvent previously developed to compute aqueous free energies of solvation is extended to accommodate the polar aprotic solvents dimethyl sulfoxide and acetonitrile. The method combines quantum mechanical calculation of the solute electronic structure with a modern dielectric continuum model for long-range electrostatic interactions with solvent and individual models for short-range interactions arising from dispersion, exchange, and hydrogen bonding. The few parameters involved are optimized to fit a standard data set of experimental solvation energies for neutrals and ions. Results are better than other models in the literature, with average errors for ions comparable to or smaller than the estimated experimental errors. Some circumstantial evidence is also obtained to support one of the competing extrathermodynamic arguments recently used to determine the solvation energies of the proton, which are needed to separate measurements of paired cation plus anion solvation energies into absolute single ion solvation energies in these solvents.



INTRODUCTION The solvation free energy of a molecule can be efficiently estimated with an implicit solvation model. Due to its simplicitly and effectiveness, this general approach has become quite popular and has undergone considerable development in many diverse directions, as discussed in detail in several excellent reviews.1−5 One direction of particular interest has been to parametrize implicit solvation models for use with nonaqueous solvents. For example, the MST model6 has specific parametrizations for several solvents;7−11 the IEFPCM model12−14 also has several specific parametrizations,15−18 and the SMx series of models19 has univeral parametrizations that cover many different solvents.20−38 We have recently developed the CMIRS1.0 (composite method for implicit representation of solvent, version 1.0) model for aqueous solvation.39 This model is based on combining quantum mechanical calculation of the solute electronic structure with a modern dielectric continuum model for long-range electrostatic interactions with solvent and additional models for short-range dispersion, exchange, and hydrogen bonding interactions. In the present work, the CMIRS1.0 model is extended by developing specific parametrizations for use with the important polar aprotic solvents © XXXX American Chemical Society

dimethyl sulfoxide (DMSO) and acetonitrile (ACN) that can support ionic as well as neutral solutes.



METHODS Description of the CMIRS1.0 Model. The CMIRS1.0 model for solvation energy combines contributions from several different sources into two broad categories * ΔGCMIRS1.0 = ΔGSS(V)PE + ΔG DEFESR

Each contribution is obtained as a functional of the quantum mechanically calculated electronic charge density ρ(r) of the solute interacting with an isotropic homogeneous structureless polarizable solvent. The contributions are described only briefly here, with references to where complete details can be found. Long-range electrostatic interactions with bulk solvent are obtained with the SS(V)PE (surface and simulation of volume polarization for electrostatics) dielectric continuum model.40−42 This is an approximation to the SVPE (surface and volume Special Issue: Jacopo Tomasi Festschrift Received: September 29, 2014 Revised: December 1, 2014

A

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polarization for electrostatics) model,40,43−46 which, in contrast to most other formulations, provides an exact solution to the governing Poisson equation by taking account of the electrostatic effect arising from the solute charge density that quantum mechanically penetrates into the volume outside the cavity nominally containing the solute. The SS(V)PE model furnishes a reaction field that polarizes the solvent, which is allowed to back-polarize the electronic structure of the solute in an iterative procedure until mutual self-consistency is reached. If implemented with the same cavity, the SS(V)PE method becomes formally equivalent42,47 to the modified version of the IEFPCM method that is often currently used.13,14 Short-range interactions between solute and solvent are obtained with the DEFESR (dispersion, exchange, and fieldextremum short-range) model which has several components.

range interactions) method.54,55 They depend on the minimum (Fmin) and maximum (Fmax) values of the outgoing normal electric field produced by the solute on the cavity surface. These describe donation of a hydrogen bond by solvent and acceptance of a hydrogen bond by solvent, respectively. If a solute has no negative values of Fmin anywhere on the cavity surface then its Fmin is assigned the value zero. Similarly, if a solute has no positive values of Fmax then its Fmax is assigned the value zero. The dependence is generally found to be nonlinear, such that the Fmin and Fmax values are raised to some power γ. The working equation is γ ΔG FESR = C |Fmin|γ + DFmax

where the linear proportionality constants C and D as well as the exponent γ are parameters to be optimized. We note the limitation that the present version of the model only allows for treatment of the single strongest donor and single strongest acceptor sites on the solute, as discussed previously.54,55 For cyclohexane and benzene solvents, it was found48 that ΔGFESR made negligible contributions, as expected since those solvents have little propensity for hydrogen bonding interactions. The implicit solvation model described as SS(V)PE+disp+exch that was presented for cyclohexane and benzene48 may therefore be regarded as the CMIRS1.0 model for those solvents where the C and D parameters both happen to have values of zero. The aprotic solvents DMSO and ACN considered in this work can be expected to have very little ability to donate hydrogen bonds, and we therefore assume here that C = 0. They may, however, each have some ability to accept hydrogen bonds, so the D and γ parameters will be optimized for description of hydrogen-bonding interactions in DMSO and ACN. The validity of both assumptions, that C = 0 and D ≠ 0, will also be tested. In summary, the SS(V)PE method by itself has only one adjustable parameter, the value chosen for the isodensity contour that fixes the size of the cavity, while the CMIRS1.0 model for DMSO and ACN adds five more adjustable parameters, being A, B, D, γ, and δ. Computational Methods. Calculations on the internal electronic structure of the solutes were carried out with the B3LYP density functional method56,57 together with the 631+G* basis set,58−60 which combination gave better results than other electronic structure methods examined in previous studies on cyclohexane, benzene, and water solvents.39,48,55 The solute cavity was determined as an isodensity contour of ρ(r), considering the three particular values of 0.0005, 0001, and 0.002 au for the fixed contour value. All the computations were made using a locally modified version of the GAMESS software package.61 Experimental values for the dielectric constants were taken as 46.45 for DMSO and 35.94 for ACN. Values for the average solvent electron densities ρ̅solvent were taken to be 0.05279 au for DMSO and 0.03764 au for ACN, as determined from their experimental mass densities, atomic numbers, and molecular weights. Experimental Comparison. The small number of parameters in the CMIRS1.0 model were determined by least-squares fitting of the difference between calculated and experimental values for the free energies of solvation of all solutes in DMSO and ACN reported in the Minnesota Solvation Database, version 200962 that has been collected and kindly provided as an open source by the Minnesota group.

ΔG DEFESR = ΔG Disp + ΔG Exch + ΔG FESR

The DEFESR components are evaluated only once from the final solute charge density determined by the SS(V)PE treatment. Contributions from generic dispersion interactions are adapted48 from the VV09 nonlocal density functional originally suggested by Vydrov and Van Voorhis49 to account for dispersion interactions between atoms and molecules in the gas phase. The working formula in atomic units (where ℏ, e, me, and 1/4πϵ0 each have a value of unity) is ΔGdisp = A

∫solute d3r ρ(r) +

3c 4π

∇ ρ(r ) ρ(r )

4

ρ(r) ⎡ ⎢ ρ(r) + ⎣

3c 4π

∇ ρ(r ) ρ(r )

4

+

⎤ ⎥ ρsolvent ̅ ⎦

I(r; δ)

where I (r ; δ ) =

∫solvent d3r′ |r − r′|16 + δ 6

Here ρ̅solvent is the average electron density of the solvent, as taken from experiment, J is a constant taken over unchanged from the value 0.0089 that was optimized to fit a test set of gasphase intermolecular interaction energies,50 while the linear parameter A and nonlinear damping parameter δ introduced to prevent dispersion catastrophe at very short distances are to be optimized. The previously given formula48 for partial analytic evaluation of I(r; δ) in the case of a single-center treatment of the cavity has a few minor typographical errors, so a corrected verison is given here in the Appendix. Contributions from generic exchange interactions are adapted48 from an approach that represents the asymptotic exchange energy between two gas phase one-electron atoms as a surface integral over the flux of exchanging electrons.51−53 The working formula is ΔGexch = B

∫solvent d3r|∇ρ(r)|

where the linear proportionality constant B is a parameter to be optimized. We note that unconstrained quantum mechanical calculation of the solute charge density will necessarily produce a tail that penetrates into the solvent region, thereby allowing ΔGexch to be nonzero. Contributions from specific hydrogen-bonding interactions are obtained with the FESR (field-extremum model for shortB

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In light of the significant discrepancies in different approaches to obtain ΔGsolv * (H+) for use in establishing single ion solvation energies, we will consider in this work two alternative sets of experimental solvation energies for ions in DMSO and ACN, these being based on ΔG*solv(H+) values either from Kelly et al.65 or from Fawcett.70 Full details of the experimental and calculated results for all solutes in each solvent obtained with each of the three cavity contours are reported in the Supporting Information.

This database provides experimental free energies of solvation for 7 neutrals, 4 cations, and 67 anions for a total of 78 solutes in DMSO and for 7 neutrals, 39 cations, and 30 anions for a total of 76 solutes in ACN. The standard state assumed there and here (denoted by an asterisk superscript) corresponds to the Ben-Naim convention63 of identical concentrations in gas and in solution. Uncertainties in the data are stated62 to be about 0.2 kcal/mol for neutrals and about 3 kcal/mol for ions. The database also includes optimized gas-phase geometries for all solutes. We adopted these geometries in our work as unchanged upon solvation. The solutes are generally all fairly rigid molecules, except for free methyl rotors, so it is believed that changes in geometry and zero-point vibration energy due to solvation will not significantly affect the results.64 The singlecenter integration scheme used to evaluate the SS(V)PE solvation energy failed for the tri-n-butylamine cation solute in ACN with the 0.002 isodensity contour, leaving 75 instead of 76 solutes calculated for that case. It is believed that this has negligible influence on the statistical evaluation of that case. Experiments can determine the sums of paired cation plus anion solvation energies but not the separate single ion solvation energies. The experimental solvation energies of single ions can therefore only be given on a relative basis. They are often given relative to the solvation energy of the proton. Some extrathermodynamic assumption is then necessary to determine ΔG*solv(H+) and so place single ion solvation energies on an absolute scale. The experimental solvation free energies for ions in the Minnesota Solvation Database, version 2009 are based on taking ΔG*solv (H+) to be −273.3 kcal/mol for DMSO and −260.2 kcal/mol for ACN. These values for the proton solvation were obtained by Kelly et al.65 from an extrathermodynamic assumption based on cluster-pair approximations applied directly to solutes in these solvents, which were estimated there to have errors of 3−4 kcal/mol for DMSO and 2−3 kcal/mol for ACN. A hybrid cluster/continuum computational study by Westphal and Pliego66 on Li+ and Na+ has suggested that ΔG*solv(H+) is −273.2 kcal/mol in DMSO. This value differs by only 0.1 kcal/mol from the value obtained with the cluster-pair approximation.65 A value of ΔG*solv(H+) in DMSO, after conversion to the Ben-Naim standard state convention, of −270.5 kcal/mol has been obtained by Pliego and Riveros.67 This determination invoked extrathermodynamic assumptions of acceptance of the value for ΔG*solv(H+) in water obtained from a cluster-pair approximation applied directly to solutes in water68 and the TATB assumption that the large tetraphenylarsonium cation and tetraphenylborate anion have identical free energies of transfer from water to other solvents.69 With the use of this same procedure, a value for ΔG*solv(H+) in ACN of −255.4 kcal/mol can be obtained, although the value for the Gibbs free energy of transfer from water to ACN obtained under the TATB assumption is described69 as being doubtful. Using a parallel approach that included additional smoothing over analogous information derived from transfer energies of other ions, Fawcett70 obtained ΔGsolv * (H+), after conversion to the Ben-Naim standard state convention, similar results of −270.5 kcal/mol in DMSO and −256.2 kcal/mol in ACN. No error estimates were provided for this determination, which values are within the estimated error of the cluster-pair approximation65 value for DMSO but are outside that estimated error for ACN.



RESULTS AND DISCUSSION It should be noted that the experimental data sets used here for comparison are heavily weighted toward ions, with less than 10% of the data corresponding to neutrals. For DMSO there is a further heavy weighting toward just anions, with only about 4% of the data corresponding to cations. This situation may produce a bias in the parametrization to some unknown extent in favor of the well-represented categories of solutes. Optimum Parameters. After some experimentation, it was found that in all cases, the nonlinear parameter δ was optimal or near optimal at the value of 7 au, as was also found previously for cyclohexane, benzene, and water solvents.39,48 Therefore, the value of δ is fixed throughout here at that value. Table 1 presents optimized CMIRS1.0 values for the linear Table 1. Optimum Values of CMIRS1.0 Parameters A, B, D, and γa ρ0b

A

B

D

γ

DMSOc 0.0005 0.001 0.002

−0.003758 −0.014401 −0.023587

0.0005 0.001 0.002

0.001392 −0.009872 −0.019410

0.0005 0.001 0.002

−0.000463 −0.004984 −0.012247

0.0005 0.001 0.002

0.003232 −0.001832 −0.009912

0.017585 0.059515 0.061249 DMSOd −0.025825 0.036192 0.048330 ACNc 0.006345 0.028574 0.038322 ACNd −0.035571 0.006349 0.027819

−1041.53 −253.58 −211.38

4.4 4.3 5.0

−2989.37 −696.65 −69.58

5.0 5.0 5.0

−0.43405 −0.32009 −0.17084

1.2 1.3 1.4

−1.48289 −1.54873 −0.31097

2.0 2.5 3.0

a The nonlinear parameter δ is fixed at 7 au. The dispersion proportionality parameter A, the exchange proportionality parameter B, and the Fγmax proportionality parameter D are all given in au. bCavity size isodensity contour in au. cOn the basis of experimental energies with values for ions obtained from the proton solvation energies of Kelly et al.65 dOn the basis of experimental energies with values for ions obtained from the proton solvation energies of Fawcett.70

parameters A, B, and D and the nonlinear parameter γ for each cavity contour. Results are reported separately for using experimental energies with single ion values based on the Kelly et al.65 or on the Fawcett70 determinations of the proton solvation energies. The optimum values of the parameters A and B governing the strengths of dispersion and exchange, respectively, are generally in the same range as found previously for cyclohexane, benzene, and water solvents.39,48 A notable exception occurs in both solvents at the 0.0005 au contour when fitting to C

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Using experimental values for ions in DMSO based on the Fawcett determination SS(V)PE provides at the 0.0005 au contour level reasonable results for neutrals but large errors for all other categories. At the 0.001 au contour level it provides reasonable results for all categories of solutes except cations. At the 0.002 au contour level it provides reasonable results for neutrals and cations but large errors for all solutes, all ions, and anions. With the use of experimental values for ions in ACN based on the Kelly et al. determination65 SS(V)PE provides at the 0.0005 au contour level reasonable MUE results for neutrals and anions but large errors for all solutes, all ions, cations. At the 0.001 au contour level, it provides reasonable results for neutrals but large errors for all other categories. At the 0.002 au contour level it provides reasonable results for neutrals and cations but large errors for all solutes, all ions, and anions. Using experimental values for ions in ACN based on the Fawcett determination SS(V)PE provides at the 0.0005 au contour level reasonable results for neutrals and anions but large errors for all solutes, all ions, and cations. At the 0.001 au contour level it provides reasonable results for all categories of solutes except cations. At the 0.002 au contour level it provides reasonable results for all solutes, neutrals, and cations but large errors for all ions and anions. Overall, it is concluded that SS(V)PE alone is not capable of providing good results across the board in either of these solvents at any one contour level when using experimental values for ions based on the Kelly et al. determination or at the 0.005 or 0.002 au contour level when using experimental energies with values for ions based on the Fawcett determination. Reasonable results can be obtained in both solvents for most categories of solutes at the 0.001 au contour level when using experimental energies with values for ions based on the Fawcett determination, although even there large errors remain for cations in both solvents. CMIRS1.0 Mean Unsigned Errors. Table 3 presents the mean unsigned errors from experiment for free energy of solvation results in DMSO and ACN as obtained from the CMIRS1.0 model. Results are given separately for using experimental energies with single ion values based either on the Kelly et al.65 or on the Fawcett70 determinations of the proton solvation energies. The results for all solutes are again further broken down into the separate categories of neutrals, all ions, cations, and anions. The CMIRS1.0 model fares significantly better than SS(V)PE alone. Indeed, reasonable results with MUEs of less than 4.0 kcal/mol are found throughout with the single exception of neutrals in DMSO at the 0.002 au contour level when using experimental energies with values for ions based on the Kelly et al. proton determination.65 When using experimental energies with values for ions based on the Kelly et al. proton determination,65 the best overall performance is found at the 0.0005 au contour for both DMSO and ACN, although the 0.001 and 0.002 au contour levels perform a little better for the categories of all ions, cations, and anions in DMSO. When using experimental energies with values for ions based on the Fawcett proton determination, the best overall performance is found at the 0.001 au contour for both DMSO and ACN, although slightly better performance is found in DMSO at the 0.002 au contour level for all ions and anions and in ACN at the 0.0005 au contour level for cations. The best overall performance of the CMIRS1.0 model in both solvents occurs at the 0.001 au contour level while using

experimental energies with single ion values based on the Fawcett70 determinations of the proton solvation energies, where the mathematical optimum is not physically acceptable because it causes the A and B parameters to acquire unphysical signs. The optimum values for the parameter γ governing the nonlinearity of the hydrogen-bonding interaction are generally somewhat more sensitive to the cavity size in DMSO and ACN than previously found in water,39 where the common value of 3.6 was taken for all cavity sizes considered. The optimum γ values are somewhat higher in DMSO than in water, while in ACN they are somewhat lower. The D parameter governing the strength of the hydrogen-bonding interaction is strongly dependent on the value of the γ parameter, so it is not very meaningful to compare its values in DMSO and ACN to those in water. SS(V)PE Mean Unsigned Errors. Table 2 presents the mean unsigned errors (MUE) from experiment for free energy Table 2. Mean Unsigned Differences in kilocalories per mol between SS(V)PE Calculation and Experiment for Solutes in DMSO and ACN ρ0a

all solutes

neutrals

all ions

cations

anions

3.4 3.9 6.9

14.4 8.6 1.1

2.8 3.6 7.2

4.9 2.8 4.6

11.6 5.8 1.7

4.5 2.7 4.7

9.3 7.2 5.2

14.9 9.2 2.6

2.1 4.6 8.5

7.8 3.8 4.0

10.9 5.2 3.6

3.8 2.0 4.5

b

0.0005 0.001 0.002

3.3 3.7 6.5

0.0005 0.001 0.002

4.6 2.7 4.4

0.0005 0.001 0.002

8.6 6.7 5.0

0.0005 0.001 0.002

7.2 3.6 3.9

DMSO 1.3 1.2 2.8 DMSOc 1.3 1.2 2.8 ACNb 1.5 1.4 2.6 ACNc 1.5 1.4 2.6

a Cavity size isodensity contour in au. bOn the basis of experimental energies with values for ions obtained from the proton solvation energies of Kelly et al.65 cOn the basis of experimental energies with values for ions obtained from the proton solvation energies of Fawcett.70

of solvation results in DMSO and ACN as obtained from the SS(V)PE dielectric continuum method alone. Results are given separately for using experimental energies with single ion values based either on the Kelly et al.65 or on the Fawcett70 determinations of the proton solvation energies. The results for all solutes are further broken down into the separate categories of neutrals, all ions, cations, and anions. For the sake of concrete discussion, MUEs are somewhat arbitrarily described as being reasonable or large depending on whether they are below or above 4.0 kcal/mol, respectively. With the use of experimental values for ions in DSMO based on the Kelly et al. determination65, SS(V)PE provides reasonable MUE results at the 0.0005 and 0.001 au contour levels for all solutes together, neutrals, all ions, and anions but large errors for cations. At the 0.002 au contour level it provides reasonable results for neutrals and cations but large errors for all solutes, all ions, and anions. D

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and from 5.5 to 6.1 kcal/mol for ions in ACN, the ranges depending on the method used to determine atomic charges and on the electronic structure method employed. Compared to that, the CMIRS1.0 model gives much better results, with MUEs about half as large. It should be noted, however, that SM12 results would be improved if the parameters involved were to be specifically optimized for DMSO and ACN solvents instead of relying on universal parametrizations. Further Explorations. Several further avenues were explored to gain additional insight into the workings of the CMIRS1.0 model. In the interest of conciseness, illustrative results for them will be given only for the case of the 0.001 cavity contour and fitting to experimental energies with values for ions based on the Fawcett proton determination. Similar results are obtained in all other cases. Tests were made to determine the possible importance of additional contributions more strictly describing cavitation work by adding three more terms corresponding to a constant, a term linear in the cavity surface area, and a term linear in the cavity volume, with full reoptimization of all other parameters except δ. This gave negligible changes of 0.1 kcal/mol or less in the overall MUE for both DMSO and ACN. Furthermore, for ACN that mathematical optimum is not physically acceptable because it causes the A parameter to acquire an unphysical sign. This exercise suggests that whatever contribution cavitation work may make, it is already largely contained in the CMIRS1.0 model. The same conclusion was also previously reached for cyclohexane, benzene, and water solvents.39,48 This insensitivity to inclusion of terms commonly used to describe cavitation may in part be due to the fact that there is not a great range in sizes of the solutes in the database used. It was assumed above in this work that the CMIRS1.0 model for both DMSO and ACN has the parameter C = 0 because those aprotic solvents should have little ability to donate hydrogen bonds to solutes. This has been investigated by allowing C to take on nonzero values along with full optimization of all other parameters of the model except δ. Inclusion of this additional degree of freedom in the optimization is found to lower the MUE by a negligible amount of less than 0.1 kcal/mol in both DMSO and ACN. Furthermore, for ACN that mathematical optimum is not physically acceptable because it causes the C parameter to acquire an unphysical sign. These explorations verify that the assumption of C = 0 is appropriate for both DMSO and ACN.

Table 3. Mean Unsigned Differences in kilocalories per mol between CMIRS1.0 Calculation and Experiment for Solutes in DMSO and ACN ρ0a

all solutes

0.0005 0.001 0.002

2.3 2.4 2.7

0.0005 0.001 0.002

2.4 2.2 2.4

0.0005 0.001 0.002

2.7 3.0 3.3

0.0005 0.001 0.002

2.2 2.1 2.4

neutrals DMSOb 0.5 3.0 5.4 DMSOc 2.1 0.9 3.1 ACNb 2.7 3.1 3.9 ACNc 2.1 1.5 2.1

all ions

cations

anions

2.5 2.4 2.4

1.0 0.7 0.7

2.6 2.5 2.5

2.5 2.4 2.3

0.8 0.7 0.7

2.6 2.5 2.4

2.7 3.0 3.2

2.8 3.0 3.2

2.6 3.0 3.3

2.2 2.2 2.4

2.2 2.3 2.6

2.2 2.0 2.2

a Cavity size isodensity contour in au. bOn the basis of experimental values with ions obtained from the proton solvation energies of Kelly et al.65 cOn the basis of experimental values with ions obtained from the proton solvation energies of Fawcett.70

experimental results with values for ions from the Fawcett determination, and thereby provides some circumstantial evidence to prefer that determination to the one from Kelly et al., particularly for ACN. Interestingly, this same preference for the Fawcett determination was also found with the SM8 and SMD methods34 that compared to these same data sets. Figure 1 plots the experimental solvation energies, with values for ions based on the Fawcett proton determination, against the calculated CMIRS1.0 solvation free energies at the 0.001 au contour level for the two solvents. The MUE values in that case are 2.5 kcal/mol or less in DMSO and 2.3 kcal/mol or less in ACN for all categories of solutes. These CMIRS1.0 results can be compared to recent literature results from the SM1237 universal solvation model that utilized the same experimental data sets as used here and used ion values based on the Kelly et al. determination65. That reference reports SM12 MUE results for ions but not for neutrals. The SM12 MUEs range from 5.7 to 6.8 kcal/mol for ions in DMSO

Figure 1. Experimental free energies of solvation, with values for ions based on the Fawcett proton determination vs CMIRS1.0 results computed with the B3LYP/6-31+G* method at the 0.001 au cavity contour level. E

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APPENDIX Evaluation of the function I(r;δ) involved in the dispersion energy contribution was discussed previously48 for the case of a single-center treatment of the cavity, where it is useful to express the result as

The ability of each of these two solvents to accept hydrogen bonds was provided for above by permitting D ≠ 0. We have tested the possibility that this ability may be so weak that little would be lost by assuming that D = 0, meaning that there is no FESR contribution at all. However, it is found that removal of this degree of freedom in the optimization significantly raises the overall MUE by 1.0 kcal/mol in DMSO and by 0.9 kcal/ mol in ACN and furthermore causes the A and B parameters to acquire unphysical signs in ACN. It is concluded that providing for accepting of hydrogen bonds by solvent is significant, thereby verifying the importance of the assumption that D ≠ 0 for both DMSO and ACN.



Article

I (r ; δ ) =

∫ d2Ω′D(r, Ω′; δ)

with 2



D(r, Ω′; δ) =

=

CONCLUSION

The CMIRS1.0 model has been extended to determine free energies of solvation for neutral and ionic solutes in DMSO and ACN solvents. This method combines a modern dielectric continuum model for long-range electrostatic interactions with solvent and individual models for short-range interactions arising from dispersion, exchange, and acceptance of hydrogen bonds from solute. All solute−solvent interactions are determined as functionals of the quantum mechanically calculated electronic charge density of the solute. Optimal or near optimal results are found with cavities determined by either the 0.005 or 0.001 au solute isodensity contour, consistent with previous studies on water, cyclohexane, and benzene solvents.39,48 Using experimental solvation energies with ion values based on the proton solvation determination of Kelly et al.,65 the mean unsigned errors from experiment at the 0.0005 au cavity contour level are 2.6 kcal/mol or less in DMSO and 2.8 kcal/ mol or less in ACN for all solutes taken together and also for each category of neutral, all ion, cation, and anion solutes. Only slightly inferior MUE results are obtained at the 0.001 au cavity contour level, being 3.0 kcal/mol or less in DMSO and 3.1 kcal/mol or less in ACN for all categories of solutes. For ions, these MUEs are comparable to or lower than the estimated experimental errors and are much better than those from the SM12 method37 that compared to the same data sets with the same experimental values. With the use of experimental solvation energies with ion values based on the proton solvation determination of Fawcett,70 the MUEs at the 0.001 au cavity contour level are 2.5 kcal/mol or less in DMSO and 2.3 kcal/mol or less in ACN for all categories of solutes. This better performance provides some circumstantial evidence to prefer the Fawcett determinations over the Kelly et al. determinations of the proton solvation energy, particularly for ACN. In summary, very good results can be obtained with parameters optimized for either the 0.0005 or 0.001 au isodensity contours using either the Kelly et al.65 or Fawcett70 experimental values for ions. Some slight preference can be given to the best performing combination of the 0.001 au contour and the Fawcett determination, at least until the uncertainty in the proton solvation energies is more fully resolved. It is concluded that the CMIRS1.0 model is successful in this application and should provide a valuable and efficient means for predicting solvation energies of solutes in DMSO and ACN solvents.

′ ∫s′(Ω′) (r 2 + r′2 − 2r′rrd′ rcos θ′)3 + δ 6

T4 + T5 ⎞ 1 ⎛ T T T ⎟. 4⎜ 1 + 2 + 3 + q ⎠ 6δ ⎝

However, the analytic expressions given previously for the result of radial integration that produces the intermediate quantity D(r, Ω′;δ) contain some typographical errors. Errors occurred in the published expressions for T1, T2, T3, χ, and η±. The complete list of correct expressions for all terms is T1 = −2r cos θ′ln(p2 + δ 2 + r 2 sin 2 θ′), T2 = −2

(δ 2 − u 2 ) δ 2 + r 2 sin 2 θ′

arctan(p ,

δ 2 + r 2 sin 2 θ′ ),

T3 = r cos θ′[ξ − + ξ + + 2 3 (η− − η+)], T4 =

(ξ − − ξ +) 2 [(u − δ 2)cos χ + 2

3 (u 2 + δ 2)sin χ ],

T5 = (η− + η+)[ 3 (u 2 + δ 2)cos χ − (u 2 − δ 2)sin χ ]

where p = s′(Ω′) − r cos θ′, q = (δ 4 + r 4 sin 4 θ′ − r 2δ 2 sin 2 θ′)1/4 , u 2 = r 2(1 − 2 sin 2 θ′), χ=

1 arctan(δ 2 − 2r 2 sin 2 θ′, 2

3 δ 2),

ξ ± = ln(p2 + q2 ± 2pq cos χ ), η± = arctan(p ± q cos χ , q sin χ )

Note that the arctan functions in T2, χ, and η± need to be evaluated in the correct quadrant, so their arguments should not be expressed as a simple ratio of the second argument (numerator) to the first argument (denominator) as was originally indicated in the previously published work.



ASSOCIATED CONTENT

S Supporting Information *

Tables of all experimental and calculated solvation energies for neutral and ionic solutes in DMSO and ACN. This material is available free of charge via the Internet at http://pubs.acs.org.



AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. F

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The Journal of Physical Chemistry A



ACKNOWLEDGMENTS



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The generosity of the Minnesota group in making available their extensive solvation database is gratefully acknowledged. The research described herein was supported by the Division of Chemical Sciences, Geosciences and Biosciences, Basic Energy Sciences, Office of Science, United States Department of Energy through Grant DE-FC02-04ER15533. This is contribution number NDRL-5038 from the Notre Dame Radiation Laboratory.

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dx.doi.org/10.1021/jp5098519 | J. Phys. Chem. A XXXX, XXX, XXX−XXX

Composite method for implicit representation of solvent in dimethyl sulfoxide and acetonitrile.

A composite method for implicit representation of solvent previously developed to compute aqueous free energies of solvation is extended to accommodat...
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