Calculation of Molecular Volumes and Volumes of Activation Using

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Calculation of Molecular Volumes and Volumes of Activation Using Molecular Dynamics Simulations H. Wiebe,†,‡ J. Spooner,‡ N. Boon,† E. Deglint,† E. Edwards,† P. Dance,† and N. Weinberg*,†,‡ † ‡

Department of Chemistry, University of the Fraser Valley, Abbotsford, BC V2S 7M8, Canada Department of Chemistry, Simon Fraser University, Burnaby, BC V5S 1A6, Canada ABSTRACT: Traditionally, the effects of pressure on the reaction rates are expressed in terms of their pressure derivatives, known as volumes of activation. Within the framework of transition-state theory, these volumes of activation can be identified with the difference in volumes of transition states and reactants. Because the volumes of reactants are readily available experimentally, the volumes of activation provide a direct measure of the transition-state volumes and thus a unique glimpse of these transient species. We propose a new approach toward calculating molecular volumes based on molecular dynamics simulations. Molecular volumes and volumes of activation for nonpolar systems calculated using this approach agree well with the experimental data.

1. INTRODUCTION Comparison of experimental and calculated activation energies is one of the most important instruments for judging the viability of a proposed reaction mechanism. Activation energies determine temperature dependence of the rate constants and are found experimentally from analysis of such dependences. Along with temperature, pressure has a profound effect on the rates of chemical reactions.1 Typically, its effects are expressed in terms of the experimental logarithmic derivatives of the rate constants, known as volumes of activation. According to the transition state theory (TST),2 the volume of activation can be found as the difference between the volumes of transition state (TS) and reactant species 6¼

6¼


RTð∂ln k=∂PÞT ¼ ΔV ¼ V
VR

type of reaction homolysis

ΔV6¼ (cm3/mol) 5 to 20

radical polymerization

≈
20

Diels
Alder cycloadditions


25 to
40

intramolecular cycloadditions dipolar cycloadditions


25 to
30
40 to
50

ester hydrolysis (basic)


10 to
15

ester hydrolysis (acidic)

>
10

epoxide ring-opening


15 to
20

Wittig reactions


20 to
30

ð1Þ

The sign and magnitude of this difference dictates the scale and direction of the pressure effect. Activation volumes are also frequently used in qualitative discussions of reaction mechanisms,1
3 the most powerful example of which is application of activation volumes for the discrimination between concerted and stepwise mechanisms of pericyclic reactions.4 Different types of reactions are characterized by typical ranges of activation volumes, as shown in Table 1. These ranges reflect the mechanistic features specific to these reactions.5 It has also been argued that loss or gain of an active degree of freedom contributes ca. ( 6 cm3/mol to the total value of the activation volume.6 However, the usefulness of such predictions is severely limited by the imprecision of the existing models.7 As a result, activation volumes are not as widely used as activation energies in the context of mechanistic discussions. With a reliable computational r 2011 American Chemical Society

Table 1. Typical Ranges of Activation Volumes for Different Types of Reactions (Adapted From ref 3g)

method for relating the volumes of molecular systems to their geometrical parameters, the vast amount of experimental data on activation volumes available in the literature1 could become an invaluable source for elucidating reaction mechanisms. We show below that a model based on molecular dynamics simulations8 provides accuracy sufficient for this and thus restores the symmetry between the roles of activation energies and activation volumes in the discussion of reaction mechanisms. Special Issue: Chemistry and Materials Science at High Pressures Symposium Received: September 20, 2011 Revised: November 18, 2011 Published: November 21, 2011 2240

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2. MODEL The partial molar volume, VX, of a solute, X, in a given solvent, S, is the derivative of the overall volume, V, with respect to the number of moles of solute, nX.   ∂V VX ¼ ð2Þ ∂nX P, T For dilute solutions, this can be replaced with a finite-difference expression V ðnS , nX Þ
V ðnS , 0Þ ð3Þ VX ¼ nX

Figure 1. Hydrocarbons immersed in a model solvent (clockwise from the top left corner): hexane, cyclopentadiene, benzene, and toluene. The solvent trajectory is represented by an overlay of solvent configurations acquired at different instants of time. Solvent particles (white) avoid the solute, thus forming a cavity of the matching size and shape.

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That is, VX is defined by the difference between the volume of solution and the volume of pure solvent divided by nX. We propose to use constant-pressure molecular dynamics (MD) simulations to obtain this difference for a single solute molecule X (nX = 1/NA; NA is the Avogadro constant) and an N-particle solvent N 3 S (nS = N/NA). The volume of a single particle V(X) is then given by V ðXÞ ¼ V ðX þ N 3 SÞ
V ðN 3 SÞ

ð4Þ

and VX = NA 3 V(X). This method mimics the experimental procedure of determining partial molar volumes of stable compounds and is equally applicable to calculation of partial molar volumes of short-lived TS species, for which such experimental procedure is not feasible. Figure 1 shows four hydrocarbons immersed in a model solvent. In each case, the solvent trajectory demonstrates a clear pattern of avoidance due to the short-range solvent
solute repulsion, which results in the formation of a cavity around the solute. The incremental increase in the volume of the overall system due to cavity formation is described by eq 4. The size and the shape of the cavity depend on the geometry of the solute, the strength and type of the solute
solvent interactions, as well as temperature and pressure. Volume fluctuations in a constant-pressure MD run are quite significant and exceed the value of an incremental volume increase due to a single molecule. (See Figure 2.) However, the average volume fluctuates much less, and the standard error of the cumulative average decreases as the square root of the length of the trajectory. As can be seen in Figure 3, the errors can be reduced to an acceptable level if an MD run is sufficiently long.

3. COMPUTATIONAL DETAILS MD calculations were performed using the GROMACS package9 for a system of 256 solvent molecules in a cubic box with periodic boundary conditions. The system was maintained at a constant pressure of 0.1 MPa and a constant temperature matching experimental conditions using Berendsen temperature and pressure coupling.10 The MD trajectories were obtained using leapfrog integration with 1 fs time step with interaction cutoff radius

Figure 2. Instantaneous and average MD volumes of a system of 256 cyclopentadiene molecules. The incremental contribution from a single molecule estimated by the experimental molar volume of 82 cm3/mol is ca. 0.14 nm3. Large amplitude fluctuations of the instantaneous volume are somewhat stabilized by averaging over 10 ps intervals. Further improvement is reached by using a cumulative average. 2241

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Figure 3. Partial molar volume of cyclopentadiene calculated using eq 4 as a difference between volumes of 257 and 256 molecule systems. Note the difference in the time scales between this Figure and Figure 2.

of 0.9 nm. Monoatomic Lennard-Jones particles (m = 40 amu, σ = 0.35 nm, ε = 2.0 kJ mol
1) were used as a model solvent. Solutes and realistic solvents were described using the OPLS force field.11 Because OPLS was designed to model stable species, its standard set does not include parameters to describe partial bonds and transient geometries that inevitably occur in TSs. Missing force-field parameters12 were obtained by interpolation or by fitting to quantum mechanics.13 All quantum mechanical calculations were performed with the Gaussian 03 suite14 at either the B3LYP/6-31++G** or B3LYP/6-31++G level depending on the reaction considered. Structures of all TSs were optimized and verified with frequency calculations.15 Graphic rendering of MD trajectories shown in Figures 1 and 4 was produced using the VMD package.16

4. MOLECULAR VOLUMES OF HYDROCARBONS To verify the suitability of OPLS parameters for the volume calculations, we performed MD simulations and obtained the molar volumes of various cyclic and acyclic hydrocarbons by dividing the calculated average volume of a system of 256 molecules by the number of molecules. The results listed in Table 2 indicate that the experimental (Vexp) and calculated (Vbulk) values match with sufficient accuracy over a wide range of molar volumes spread between 80 and 230 cm3/mol. The accuracy of eq 4 for generating partial molar volumes was tested by the calculation of the volume difference between systems of 257 and 256 molecules. The resultant differences (Vdiff) constitute the volume of single solvent molecules and thus can be directly compared with Vexp and Vbulk. The data listed in Table 2 show that Vdiff and Vbulk are comparable in accuracy in reproducing experimental volumes. 5. VOLUMES OF TRANSITION STATES AND ACTIVATION VOLUMES FOR NONPOLAR REACTIONS The proposed approach was next applied to the calculation of activation volumes for three nonpolar reactions: Diels
Alder dimerization of cyclopentadiene (neat and in 1-chlorobutane), Diels
Alder dimerization of isoprene (in 1-bromobutane), and hydrogen transfer from methane to methyl radical (in hexane). The partial molar volumes of the reactants and TSs, necessary for

Figure 4. Solvated reactant state (top) and TS (bottom) of the hydrogen transfer reaction from methane to a methyl radical. In the reactant state, the solute generates a larger cavity than in TS, which results in a negative activation volume for this reaction. As in Figure 1, a model solvent was used here for illustrative purposes. The activation volume reported for this reaction in Table 3 was obtained in hexane solvent.

Table 2. Comparison of Experimental (Vexp) and Calculated (Vcalc) Molar Volumes (cm3/mol) of Some Hydrocarbons (Arranged in Order of Increasing Molar Mass)a Vcalc hydrocarbon

Vexpb

Vbulkc

Vdiff

T (°C)

cyclopentadiene

82.4

83.5

82.6

20

cyclopentene

88.2

90.9

90.0

20

cyclopentane

94.0

97.8

98.6

20

benzene

89.1

89.2

91.5

20

1,3-cyclohexadiene

95.3

96.4

95.9

20

cyclohexene

101.4

103.8

104.8

20

cyclohexane n-hexane

108.1 130.4

109.8 129.3

109.4

20 25

toluene

106.9

105.0

25

methylcyclohexane

127.6

127.8

20

n-octane

163.5

156.6

0

n-dodecane

227.3

225.8

25

a

Vbulk refers to the values obtained by dividing the volume of 256 molecules by their number; Vdiff refers to the values obtained as a difference between volumes of 257 and 256 molecules. b Ref 17. c Ref 8.

the calculations, were obtained from MD simulations of the appropriate systems using eq 4. Figure 4 visualizes these simulations for the reactant and TS of the hydrogen transfer reaction. The reaction system is longer in the reactant state and creates a 2242

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Table 3. Comparison of Experimental (ΔV6¼exp) and Calculated (ΔV6¼calc) Activation Volumes (in cm3/mol) for Three Nonpolar Reactions

a Experimental activation volumes were calculated from kinetic data using the El’yanov-Gonikberg equation (refs 18a and 18b) with parameters α = 0.170 and ß = 4.94 kbar
1. b Kinetic data from ref 19. c Kinetic data from ref 20. d Kinetic data from ref 21. e Estimated from experimental data for similar reactions (ref 1).

Figure 5. Volume profile (green triangles) for mutation of cyclohexane into hexane-like open-chain diradical. Horizontal red lines mark experimental volumes of cyclohexane and hexane. Torsional dispersion for the mutated cyclohexane (blue circles) was calculated as a standard deviation of the MD-generated probability distribution function for torsion angles.

larger cavity in the solvent, thus generating a greater displacement volume. The theoretical activation volumes for all reactions were calculated as the volume differences between TSs and reactants. The experimental activation volumes were evaluated from the experimental kinetic data as ΔV6¼ 0 parameters of El’yanov-Gonikberg equation18

  kp RT ln ¼
ΔV6¼0 ½1:17P
0:0344ð1 þ 4:94PÞ lnð1 þ 4:94PÞ k0

ð5Þ The results listed in Table 3 show good agreement between calculated and experimental values. The activation volumes for the cyclopentadiene dimerization reaction calculated for two different solvents (cyclopentadiene and 1-chlorobutane) are 5.7 cm3/mol apart, which matches closely the experimental difference of 5.2 cm3/mol observed for these solvents.

6. CONCLUDING REMARKS The results reported in this work demonstrate that the proposed displacement volume model is sufficiently accurate to be used for theoretical calculation of molecular volumes and that the standard OPLS force field is adequate for such calculations. The experimental activation volumes and their solvent dependencies for nonpolar reactions seem to be well-reproduced by the method. Because the calculated activation volumes are sensitive to the geometry of the postulated TS, they can be used, in a combination with the experimental kinetic data, for the analysis of reaction mechanisms. The necessary optimized geometries can be obtained using conventional quantum-mechanical methods22 for each TS of the reaction network that includes all plausible reaction channels. A comparison of the calculated and experimental activation volumes will rule out the reaction channels for which these volumes do not match within the allowed error threshold. Another interesting application of the proposed technique lies in its extension to the calculation of socalled volume profiles,23 ΔV(x), describing how the volume of a reaction system changes along its reaction coordinate x. This concept generalizes the idea of activation volume and is often used in mechanistic analysis.1,24 So far, however, it has been used only qualitatively23,24 and semiquantitatively25 because of the lack of a quantitative model of molecular volume. Our MD-based approach brings this concept to a quantitative level. Although intended to recreate reality as close as possible, MD simulations allow for a much greater degree of control over system parameters than an experiment. As a result, systems and their properties can be probed by MD in greater detail and under physical conditions unattainable in experiment.26 This opens a wide opportunity for the method described in this Article to be used for analysis of important factors and contributions to the observed molecular and activation volumes. For example, the fact that the packing coefficient,27 defined as a ratio of van der Waals volume7a
f to the total volume, is greater for cyclohexane than for hexane has been discussed3c,d,7i,28 in the context of mechanisms of the cycloaddition of butadiene and chloroprene7i,29 and was attributed to the difference in the percent contributions of the thermal expansion volumes30 to the total volumes of these compounds. Our method not only correctly reproduces the 2243

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The Journal of Physical Chemistry C experimental volumes of hexane and cyclohexane (Table 2) but also offers a way of testing this conjecture. Figure 5 displays an MD-generated volume profile for a hypothetical process of mutation of cyclohexane into hexane-like diradical (CH2)6 achieved by stretching the C1
C6 bond length from its value in cyclohexane to the average distance between C1 and C6 in hexane. The calculated volumes of mutated cyclohexane vary between the experimental volumes of cyclohexane and hexane, growing monotonously with the increase in the C1
C6 distance. Plotted on the same graph is the dispersion of the torsion angles in the mutated molecule measured by the standard deviation of the probability distribution function obtained for this molecule in the MD simulation. The increase in torsional dispersion reflects the increasing entropy of the mutated cyclohexane, primarily responsible for the thermal expansion volume. Positive correlation between the two curves supports the idea that the thermal expansion volume is the primary reason for the difference in the molar volumes of hexane and cyclohexane. This is also consistent with the finding28 that the entropies of formation and packing coefficients of cyclic hydrocarbons are anticorrelated.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT Financial support by the Natural Sciences and Engineering Council of Canada (NSERC) and the University of the Fraser Valley is gratefully acknowledged. This research has been enabled by the use of WestGrid computing resources, which are partially funded by the Canada Foundation for Innovation, Alberta Innovation and Science, BC Advanced Education, and the participating research institutions. WestGrid equipment is provided by IBM, Hewlett-Packard, and SGI. ’ REFERENCES (1) (a) le Noble, W. J. Prog. Phys. Org. Chem. 1967, 5, 207. (b) Asano, T.; le Noble, W. J. Chem. Rev. 1978, 78, 407. (c) van Eldik, R.; Asano, T.; le Noble, W. J. Chem. Rev. 1989, 89, 549. (d) Drljaca, A.; Hubbard, C. D.; van Eldik, R.; Asano, T.; Basilevsky, M. V.; le Noble, W. J. Chem. Rev. 1998, 98, 2167. (2) (a) Evans, M. J.; Polanyi, M. Trans. Faraday Soc. 1935, 31, 875. (b) Stearn, A. E.; Eyring, H. Chem. Rev. 1941, 29, 509.(c) Hamann, S. D. Physico-Chemical Effects of Pressure; Academic Press: New York, 1957. (d) Gonikberg, M. G. Chemical Equilibria and Reaction Rates at High Pressures; National Science Foundation: Washington, DC, 1963. (e) Isaacs, N. S. Liquid Phase High Pressure Chemsitry; Wiley: New York, 1981. (3) For recent reviews see, for example: (a) Swaddle, T. W.; Tregloan, P. A. Coord. Chem. Rev. 1999, 187, 255. (b) Stochel, G.; van Eldik, R. Coord. Chem. Rev. 1999, 187, 329. (c) Klarner, F.-G.; Brietkopf, V. Eur. J. Org. Chem. 1999, 2757. (d) Klarner, F.-G.; Wurche, F. J. Prakt. Chem. 2000, 342, 609. (e) Swaddle, T. W. Chem. Rev. 2005, 105, 2573. (f) Hubbard, C. D.; van Eldik, R. J. Coord. Chem. 2007, 60, 1. (g) Schettino, V.; Bini, R. Chem. Soc. Rev 2007, 36, 869. (4) (a) Houk, K. N.; Gonz
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