Microstructure and Hydrogen Bonding in Water−Acetonitrile Mixtures

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Microstructure and Hydrogen Bonding in Water-Acetonitrile Mixtures Raymond D. Mountain† Chemical and Biochemical Reference Data DiVision, Material Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-6320, United States ReceiVed: June 8, 2010; ReVised Manuscript ReceiVed: NoVember 8, 2010

The connection of hydrogen bonding between water and acetonitrile in determining the microheterogeneity of the liquid mixture is examined using NPT molecular dynamics simulations. Mixtures for six, rigid, threesite models for acetonitrile and one water model (SPC/E) were simulated to determine the amount of water-acetonitrile hydrogen bonding. Only one of the six acetonitrile models (TraPPE-UA) was able to reproduce both the liquid density and the expermental estimates of hydrogen bonding derived from Raman scattering of the CN stretch band or from NMR quadrupole relaxation measurements. A simple modification of the acetonitrile model parameters for the models that provided poor estimates produced hydrogen-bonding results consistent with experiments for two of the models. Of these, only one of the modified models also accurately determined the density of the mixtures. The self-diffusion coefficient of liquid acetonitrile provided a final winnowing of the modified model and the successful, unmodified model. The unmodified model is provisionally recommended for simulations of water-acetonitrile mixtures. I. Introduction Mixtures of water with molecules that can accept hydrogen bonds are often spatially inhomogeneous on the scale of a few molecular sizes but are homogeneous on longer length scales.1-5 This feature has come to be labeled microheterogeneity. Both molecular geometry and the ease in forming hydrogen bonds between water molecules and other molecules6 are important factors entering into the details of the microheterogeneity of a mixture. In this paper, we are concerned with microheterogeneity in water-acetonitrile mixtures as these liquids are used extensively as solvents and working fluids.7 Local variation in liquid composition can influence how solutes are solvated. In particular, we examine the interactions between water and acetonitrile since simulations of water-acetonitrile mixtures are of importance for simulation based studies of reversed phase liquid chromatography.8-10 It has been suggested recently that pressureinduced changes in the microstructure of water-acetonitrile mobile phases at high water content for reversed phase liquid chromatography may change the observed retention of solutes.11 There are several indications based on experimental measurements that water-acetonitrile mixtures are microheterogeneous on the scale of a few molecular diameters. Measured thermodynamic property data have been used to evaluate the Kirkwood-Buff integrals12 for the mixtures.13,14 The likemolecule integrals are positive and the unlike-molecule integrals are negative. This indicates that the near neighbor pairs are predominantly like molecules while the unlike molecule pairs tend to be spatially separated. This is an indication of microheterogeneity but does not provide the spatial extent of the water-rich and acetonitrile-rich regions. Small angle X-ray and neutron diffraction studies also support the view that the mixtures are spatially inhomogeneous on the scale of a few molecular sizes.15,16 † To whom correspondence [email protected].

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There are other experimental measurements,17 depolarized Raman scattering18 and NMR quadrupole relaxation,19 that provide more detailed information within the framework of a model used to analyze and interpret the data. The basic idea of the model is that acetonitrile molecules in water can be characterized as either “free” or “bound”. A bound acetonitrile molecule is hydrogen bonded to a water molecule while a free acetonitrile molecule is not. Of course, the “free” molecules still interact with their neighbors, so “free” has this specific meaning for acetonitrile molecules. The CN stretch band observed by Raman scattering in water-acetonitrile solutions is found to be split18 with a blueshifted component that varies with the concentration of acetonitrile. It follows from the analysis of the Raman scattering17 that the integrated intensity of the higher frequency band is proportional to the mole fraction of bound molecules and the intensity of the lower frequency band has the same proportionality to the mole fraction of free molecules. Since the composition of the mixture is known, the mole fraction of bound molecules can be extracted from the spectra. A similar approach produces the same estimated amount of water-acetonitrile bonding from NMR quadrupole relaxation data.19 Estimates of the amount of water-acetonitrile hydrogen bonding derived from infrared absorption measurements20 are significantly lower than the estimates for bonding amounts obtained by CN stretch and NMR quadrupole relaxation measurements. In the following we use the results from Raman scattering18 as the amount of water-acetonitrile hydrogen bonding. Having two independent measurement methods yield the same results builds confidence in those results relative to other estimates that do not have independent confirmation. Molecular dynamics and Monte Carlo simulations are a way to examine the spatial extent of the water-rich and acetontirilerich regions proVided the interaction models used are able to reliably describe the appropriate physics. In this case, the appropriate physics is the amount of water-acetonitrile hydrogen bonding. Rather different conclusions on the amount of spatial heterogeneity in water-acetonitrile mixtures have been

This article not subject to U.S. Copyright. Published 2010 by the American Chemical Society Published on Web 11/19/2010

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TABLE 1: Acetonitrile Interaction Model Parametersa εMe εC εN σMe σC σN qMe qC qN dMe-C dC-N

M1

M2

M3

M4

M5

M6

0.817 0.500 0.500 0.375 0.355 0.295 0.269 0.129 -0.398 0.154 0.116

1.589 0.416 0.416 0.360 0.340 0.330 0.269 0.129 -0.398 0.146 0.117

0.755 0.875 0.407 0.380 0.300 0.340 0.269 0.129 -0.398 0.146 0.117

1.366 0.350 0.350 0.348 0.329 0.319 0.287 0.138 -0.425 0.146 0.117

0.782 0.544 0.628 0.378 0.365 0.320 0.206 0.247 -0.453 0.146 0.115

0.866 0.628 0.711 0.378 0.365 0.320 0.15* 0.28* -0.43* 0.146 0.115

a

The units for the ε’s are kJ/mol, the units for the σ’s and bond lengths, d, are nm, and the units for charges, q, are e, the charge on the proton. Me refers to the methyl site represented as a united atom.

reached for different model interactions.5,21 An attempt to understand the source of the different conclusions about the amount of microheterogeneity in the simulated mixtures provided the motivation for the study reported here. In this paper, we employ molecular dynamics simulations to examine how reliably several three-site models for acetonitrile combined with the SPC/E model for water reproduce the amount of water-acetonitrile hydrogen bonding indicated by Raman scattering at ambient conditions. We also explore modifications of the models so that they can reproduce the experimentally determined amount of water-acetonitrile hydrogen bonding. A few simulations were repeated using the TIP4P/2005 model22 in place of the SPC/E model to determine the sensitivity of the results to the water model. The simulation details are described in section II and the results of the simulations of the model systems and of modified model systems are presented in section III. Some summary observations are presented in section IV. II. Simulations In this paper, we use molecular dynamics simulations to examine the fraction of bound and free acetonitrile molecules predicted for water-acetonitrile mixtures using six model potentials for acetonitrile and the SPC/E model for water. The acetonitrile models, labeled here as M1,23 M2,24 M3,25 M4,26 M5,27 and M628 are rigid, linear, three-site models with LennardJones and Coulomb interactions between all sites on interacting molecules. The methyl group of acetonitrile is treated as a united atom. The model parameters are listed in Table 1. The SPC/E water model used with each of the acetonitrile models is also a rigid three-site model.29 The Lennard-Jones interaction is only between oxygen sites with ε ) 0.65 kJ/mol and σ ) 0.31657 nm. The OH bond length is 0.1 nm, the HOH angle is 109.47°, and the charge on the oxygen site is -0.8476 e. All unlike-site Lennard-Jones interactions are obtained using the LorentzBerthelot mixing rules.30 As noted above, a few simulations were made using the TIP4P/2005 model for water. This is a planar, rigid, four-site model with the OH distance of 0.095 72 nm and the HOH angle of 104.52°. The M-site is located on the HOH bisector 0.015 46 nm from the oxygen site in the direction of the hydrogen sites. There is a Lennard-Jones interaction between oxygen sites with ε ) 0.7743 kJ/mol and σ ) 0.315 89 nm. The charge on the M-site is -1.1128 e with neutralizing charges on the hydrogen sites. In the simulations 500 molecules in a cubic simulation cell are subject to periodic boundary conditions in the NPT ensemble.31 Five compositions were examined for each model with

Figure 1. Distribution of the water-acetonitrile pair energy, Epair, for the equimolar composition for M1. A pair has a hydrogen bond if the pair energy is less than the minimum, Emin, at -13.0 kJ/mol. The distribution of pair energies is scaled so that the area under the curve to the left of Emin is unity.

mole fractions of acetonitrile, XAcn, equal to 0.1, 0.25, 0.5, 0.75, and 0.9. The equations of motion for the NPT ensemble32,33 are intergrated with a time step of 1 fs using a velocity-Verlet algorithm34 adapted to describe molecule orientation using quaternions.35 Direct interactions are truncated at half the simulation cell size at each time step and the long-range parts of the Coulomb interactions are determined using the Ewald method.36 The temperature of the system is maintained at 298 K using a separate thermostat for the rotational degrees of freedom.37 The pressure was set to 0.1 MPa to correspond to the available experimental values of the density.38 The coupling times for the thermostats regulating the translational and rotational degrees of freedom are 0.1 ps, the time for the barostat is 1.1 ps and the time for the thermosat chain for the barostat is 0.11 ps. These values provide stable temperature and pressure regulation. Once a system was stabilized for at least 1 ns, a production run of 2 ns was made. Such long runs are needed to adequately sample the configurations of the system when microheterogeneity extends for more than a few tenths of a nanometer. In some cases, a second 2 ns run was made to confirm the stability of the results for the state being examined. A water-acetonitrile pair share a hydrogen bond if the following three conditions are satisfied. The first condition is that rNH, the distance between the hydrogen and nitrogen sites is less than 0.25 nm. That is the position of the first minimum in the hydrogen-nitrogen site pair function. The second condition is that the intermolecular potential energy of a water-acetontrile pair, Epair, is less than the position of the local minimum, Emin, in the distribution of pair energies, P(Epair), as indicated in Figure 1. There the distribution of pair energies is shown for the equimolar case for model M1 with the distribution scaled so that the area under the curve to the left of the minimum, Emin, is unity. In this case, Emin is -13.0 kJ/mol and that condition was used for all cases examined. The third condition is that the angle between the OH vector of the water molecule and rON, the vector between the oxygen and the nitrogen sites, is less than 30°. The third condition is the same one used when the orientation of hydrogen-bonded pairs of water molecules is examined.39 It is important that the third condition be used as many of the pairs satisfying the first two conditions fail the third one. During the run, the number of water-acetonitrile hydrogen bonds, the number of acetontirile molecules with more than one hydrogen bond, and b(t), the number of acetonitrile molecules with at least one hydrogen bond were determined. This information was recorded in 1 ps block averages for postpro-

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Mountain

Figure 2. Mole fraction of bound acetonitrile molecules, XAcn-b, and the liquid density for the equimolar mixtures compared with the experimental values. The experimental point is the shaded circle. The calculated points for each of the models: M1 (square), M2 (diamond), M3 (triangle up), M4 (triangle left), M5 (triangle down), M6 (triangle right). The temperature is 298 K, and the pressure is 0.1 MPa.

cesssing. The Raman band integrated intensity for the CN stretch mode is proportional to the number of bonded molecules, not the number of bonds. The cumulative average per sample, 〈b(t)〉, was also written to a file at 1 ps intervals. The stability of the cumulative average for long times provides a check on the adequacy of the sampling. In all cases, the long time average of 〈b(t)〉, 〈b〉, is required to be stable to within (1% or less. Since 〈b〉 is the number of “bound” acetonitrile molecules, the mole fraction of “bound” acetonitrile molecules, XAcn-b, is 〈b〉/ N, where N is the total number of molecules. It is essential to distinguish between the number of hydrogen bonds and the number of acetonitrile molecules that are “bound” as we found in some cases that up to 10% of the bound molecules experience more than one hydrogen bond. The comparison of the simulation results with the experimental determination of the fraction of bound acetonitrile molecules in a mixture provides an important check on the adequacy of the model used in a simulation. The amount of nitrogen-hydrogen bonding is related in some way to the spatial extent of the inhomogeneity. An increasing number of nitrogen-hydrogen bonds is an indication of increasing uniformity of the mixing of the components. The view of inhomogeneity that emerges from this study will be presented below. III. Results The initial screening of the six acetonitrile models was made by determining the mole fraction of bonded molecules, XAcn-b, and the liquid density for the equimolar mixtures. The results are compared with the experimental results in Figure 2. Clearly, M1 produces the best estimates. Since only M1 of the six models predict hydrogen bonding in close agreement with the experimental results and simultaneously provide an accurate prediction of the liquid density, some tweaking of the model parameters for the other models was done to see if better agreement with the experimental results could be obtained. The Lennard-Jones parameter that has the greatest ability to modify the amount of hydrogen bonding per amount of change in the parameter is σNN. This was determined from a series of test cases and a decrease in σNN results in an increase in both XAcn-b and in the liquid density. For each models M2 through M6, the parameter σNN was shifted down in magnitude. Since M2 and M4 have densities larger than the experimental density, this shift resulted in an even larger density and these models are not considered further.

Figure 3. Calculated values of XAcn-b and the liquid densities for M1 (squares) and M5a (triangles-down) compared with the experimental values (circles) for the five compositions. The lines are intended only to guide the eye.

For M3, σNN was decreased from 0.340 to 0.3175 nm and the density matches the experimental. However, XAcn-b only increased from 0.102 to 0.193 and M3 will not be considered further. For M5, σNN was decreased from 0.320 to 0.300 nm and the density now matches the experimental value and XAcn-b increased from 0.180 to 0.257 in reasonable agreement with the value for M1. This revised model will be labeled M5a. For M6, σNN was decreased from 0.320 to 0.290 nm and the density now matches the experimental value. XAcn-b increased from 0.0883 to 0.213, and this model will not be considered further. This leaves us with M1 and M5a as candidate models. The simulation results as a function of the mole fraction of acetonitrile in the mixture, XAcn, are compared with the experimental results in Figure 3. The results for both models are in close agreement with each other and are within a few percent of the experimental values over the compositions examined. It is necessary to examine other physical properties of the fluids to decide between M1 and M5a. One possibility is the self-diffusion coefficient of the fluids. Experimental results are available for pure acetonitrile.40 For 298 K and 0.1 MPa, the experimental value of the self-diffusion coefficient of acetonitrile is 4.3 × 10-9 m2/s. The computed values of the self-diffusion coefficient are obtained using the Einstein relation between the self-diffusion coefficient and the mean-square displacement of the molecules.41 The value of the self-diffusion coefficient of liquid acetonitrile for M1 is 3.7 × 10-9 m2/s and for M5a is 3.1 × 10-9 m2/s. From this we tentatiVely conclude that M1 is the model to use in conjunction with SPC/E water for simulations of water-acetonitrile mixtures. The simulations for M1 were repeated using the TIP4P/2005 model in place of the SPC/E model. The results for the two sets of simulations are in good agreement. The values for XAcn-b for the TIP4P/2005 simulations are at most 0.01 lower than the values for the SPC/E simulations. The densities agree to within 1% for the five compositions. This indicates that the amount of water-acetonitrile hydrogen bonding is not sensitive to the water model used. A second series of simulations using model M1 for the five compositions were made with the pressure set to 50 MPa at

Water-Acetonitrile Mixtures

Figure 4. Oxygen-oxygen site-site pair functions, gOO(r), for M1 for the compositions examined here. The acetonitrile mole fraction is 0.9 (black line), 0.75 (red line), 0.5 (green line), 0.25 (blue line), and 0.1 (dashed cyan line).

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Figure 5. Hydrogen-nitrogen site-site pair functions, gHN(r), for M1 for the compositions examined here. The acetonitrile mole fraction is 0.9 (black line), 0.75 (red line), 0.5 (green line), 0.25 (blue line), and 0.1 (dashed cyan line).

298 K to see how the hydrogen bond structure is modified at elevated pressure. The result is that essentially no change in the amount of hydrogen bonding was found. This indicates that the changes in retention observed in liquid chromatography11 are not due to structural changes in the mobile phase. IV. Discussion Molecular dynamics simulation has been used to examine how well various three-site models for acetonitrile predict the observed hydrogen bonding of acetonitrile when mixed with SPC/E water at ambient conditions. The amount of hydrogen bonding between water and acetonitrile is closely related to the extent the mixture exhibits microheterogeneity. The more water acetonitrile hydrogen bonding present, the more uniform the mixing of water and acetonitrile at the molecular level. The different conclusions reached about the amount of microheterogeneity using models M121 and M25 are due to the different amounts of water-acetonitrile hydrogen bonding generated by those models. Since M2 does not yield the correct amount of hydrogen bonded acetonitrile molecules while M1 does, the results obtained using M1 are more correct. A rough indication of the size of the microheterogeneous regions can be obtained by examining the pair functions for the mixture.3,5 Here we use the functions for M1 to estimate the nature of microheterogeneity as a function of the acetonitrile mole fraction. First we examine the oxygen-oxygen pair function, gOO(r), in Figure 4. With the exception of the XAcn ) 0.1 state, there is enhanced short-range correlation of the water molecules extending over 0.7 nm or more. This indicates that there are regions with dimensions on the order of 1 nm or larger where the water molecules are concentrated and other regions where the water molecules are depleted relative to the system average distribution of water. This behavior of the pair functions has been observed in an earlier simulation using the SPC/E water model with M6.4 A related conclusion is reached when the water-hydrogen acetonitrile-nitrogen pair function, gHN(r), is examined. In Figure 5, these functions for M1 are shown for the compositions simulated. These functions indicate that there is, except for the XAcn ) 0.9 case, a depletion of unlike next neighbors for a range of up to about 0.7 nm. Since the edge of the cubic simulation cell varies from 3.4 nm for XAcn ) 0.9 to 2.6 nm for XAcn ) 0.1, the simulated systems are large enough to avoid serious size effects on the scale of the microheterogeneity. The spatial distribution of acetonitrile-acetonitrile pairs is different from the water-water and water-acetonitrile pairs,

Figure 6. Carbon-carbon site-site pair functions, gCC(r), for M1 for the compositions examined here. The acetonitrile mole fraction is 0.9 (black line), 0.75 (red line), 0.5 (green line), 0.25 (blue line), and 0.1 (dashed cyan line).

as shown in Figure 6 where the carbon-carbon pair functions, gCC(r), for M1 are displayed. As the amount of acetonitrile increases, the next-neighbor structure is modified, but any other indication of preferential aggregation of acetonitrile molecules, such as the approach to unity from above present in Figure 4, is absent. This indicates that the overall distribution of acetonitrile molecules and the local distribution for regions of size 0.5 nm are approximately the same. The microheterogeneity of the mixtures is due to the preference of water molecules to be near neighbors and not to “clustering” of acetonitrile molecules or to the presence of a significant amount of water-acetonitrile clusters. This has been found for other models for water and acetonitrile.3 These observations could be enhanced by performing some sort of cluster analysis5 but that would be beyond the scope of this study of interaction models. The use of pair functions to estimate the sort of microheterogeneity present works reasonably well for water-acetonitrile mixtures but does not appear to be useful for water-amine mixtures2 or water-methanol mixtures,1 as the simulated pair functions for those mixtures do not exhibit the approach to unity from above of Figure 4 or the approach to unity from below of Figure 5. The term “microheterogeneity” does not imply a specific form of local order, only that deviations from uniformity have dimensions larger than one or two molecular sizes. An alternative approach to characterizing microheterogeneity is to use mole-fraction enhancement/depletion as a measure of the local composition of a mixture. Conclusions similar to those based on pair functions are reached about the extent of the water rich regions in water-acetnitrile mixtures using that approach.21

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For now, model M1 would be the model of choice when water-acetonitrile mixtures are simulated using three-site models. This is not to say that the parameters in M1 are optimal or unique. Those parameters would be a reasonable starting point if further refinement of the model were undertaken. Acknowledgment. Katrice Lippa provided valuable assistance with the experimental literature. References and Notes (1) Dougan, L.; Bates, S. P.; Hargreaves, R.; Fox, J. P.; Finney, J. L.; Re´at, B.; Soper, A. K. J. Chem. Phys. 2001, 121, 6456. (2) Zoranic´, L.; Mazighi, R.; Sokolic´, F.; Perera, A. J. Phys. Chem. C 2007, 111, 15586. (3) Kovacs, H.; Laaksonen, A. J. Am. Chem. Soc. 1991, 113, 5596. (4) Bergman, D. L.; Laaksonen, A. Phys. ReV. E 1998, 58, 4706. (5) Mountain, R. D. J. Phys. Chem. A 1999, 103, 10744. (6) Vishnyakov, A.; Lyubartsev, A. P.; Laaksonen, A. J. Phys. Chem. A 2001, 105, 1702. (7) Gagliardi, L. G.; Castells, C. B.; Rose´s, M.; Ra`fols, C.; Bosch, E. J. Chem. Eng. Data 2001, 55, 85. (8) Sun, L.; Siepmann, J. I.; Schure, M. R. J. Chem. Theory Comput. 2007, 3, 350. (9) Mountain, R. D.; Lippa, K. A. J. Phys. Chem. B 2008, 112, 7785. (10) Rafferty, J. L.; Siepmann, J. I.; Schure, M. B. AdV. Chromatogr. 2010, 48, 1. (11) Fallas, M. M.; Neue, U. D.; Hadley, M. R.; McCalley, D. V. J. Chromatogr. A 2010, 1217, 276. (12) Kirkwood, J. G.; Buff, F. G. J. Chem. Phys. 1951, 19, 774. (13) Blandamer, M. J.; Blundell, N. J.; Burgess, J.; Cowles, H. J.; Horn, I. M. J. Chem. Soc., Faraday Trans. 1990, 86, 277. (14) Marcus, Y.; Migron, Y. J. Phys. Chem. 1991, 95, 400. (15) Takamuku, T.; Tabata, M.; Yamaguchi, A.; Nishimoto, J.; Kumamoto, M.; Wakita, H.; Yamaguchi, T. J. Phys. Chem. B 1998, 102, 8880. (16) Takamuku, T.; Noguchi, Y.; Matsugamai, M.; Iwase, H.; Otomo, T.; Nagao, M. J. Mol. Liq. 2007, 136, 147. (17) Gordon, R. G. J. Chem. Phys. 1965, 42, 3658. (18) Reimers, J. R.; Hall, L. E. J. Am. Chem. Soc. 1999, 121, 3730.

Mountain (19) Dawson, E. D.; Wallen, S. L. J. Am. Chem. Soc. 2002, 124, 14220. (20) Bertie, J. E.; Lan, Z. J. Phys. Chem. B 1997, 101, 4111. (21) Rafferty, J. L.; Sun, L.; Siepmann, J. I.; Schure, M. R. Fluid Phase Equilib. 2010, 290, 25. (22) Abascal, J. L. F.; Vega, C. J. Chem. Phys. 2005, 123, 234505. (23) Wick, C. D.; Stubbs, J. M.; Rai, N.; Siepmann, J. I. J. Phys. Chem. B 2005, 109, 18974. (24) Edwards, D. M. F.; Madden, P. A.; McDonald, I. R. Mol. Phys. 1984, 51, 1141. (25) Hirata, Y. J. Phys. Chem. A 2002, 106, 2187. (26) Gee, P. J.; van Gunsteren, W. F. Mol. Phys. 2006, 104, 477. (27) Gua`rdia, E.; Pinzo´n, R.; Casulleras, J.; Orozco, M.; Luque, J. F. Mol. Simulation 2001, 26, 287. (28) Jorgensen, W. L.; Briggs, J. M. Mol. Phys. 1988, 63, 547. (29) Berendsen, H. J. C.; Grigera, J. R.; Straatsma, T. P. J. Phys. Chem. 1987, 91, 6269. (30) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, U.K., 1987; p 21. (31) Frenkel, D.; Smit, B. Understanding Molecular Simulation: From Algorithms to Applica-tions, 2nd ed.; Academic Press: San Diego, CA, U.S., 2002; pp 115-118. (32) Martyna, G. J.; Tobias, D. J.; Klein, M. L. J. Chem. Phys. 1994, 101, 4177. (33) Tuckerman, M. E.; Liu, Y.; Ciccotti, G.; Martyna, G. J. J. Chem. Phys. 2001, 115, 1678. (34) Martys, N. S.; Mountain, R. D. Phys. ReV. E 1999, 59, 3733. (35) Evans, D. J.; Murad, S. Mol. Phys. 1977, 34, 327. (36) Frenkel, D.; Smit, B. Understanding Molecular Simulation: From Algorithms to Applications, 2nd ed.; Academic Press: San Diego, CA, U.S., 2002; pp 292-300. (37) Martyna, G. J.; Klein, M. L.; Tuckerman, M. J. Chem. Phys. 1992, 97, 2635. (38) Easteal, A. J.; Woolf, L. A. J. Chem. Thermodyn. 1988, 20, 693. (39) Swiatla-Wojcik, D. Chem. Phys. 2007, 342, 260. (40) Hurle, R. L.; Woolf, L. A. J. Chem. Soc., Faraday Trans. I 1982, 78, 2233. (41) Kushick, J.; Berne, B. J. In Statistical Mechanics Part B: TimeDependent Processes; Berne, B. J., Ed.; Plenum Press: New York, NY, U.S., 1977; p 55.

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