Kinetic Theory and Transition State Simulation of ... - ACS Publications

Chapter 23

Kinetic Theory and Transition State Simulation of Dynamics in Zeolites 1

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Chandra Saravanan , Fabien Jousse , and Scott M . Auerbach 1

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Downloaded by STANFORD UNIV GREEN LIBR on August 7, 2012 | http://pubs.acs.org Publication Date: April 8, 1999 | doi: 10.1021/bk-1999-0721.ch023

Departments of Chemistry and Chemical Engineering, University of Massachusetts, Amherst, MA 01003

We have developed and applied modeling techniques special ized for infrequent events to study the transport of benzene in the zeolite faujasite, focusing on the microscopic factors that control short and long range mobility. We describe the first exact site-to-site flux correlation function calculation for a non-spherical molecule inside a zeolite. We find that tran sition state theory is qualitatively correct for some but not all jumps. We outline a recently developed analytical theory indicating which transition states control diffusion in Na-Y zeolite. Our new theory predicts self-diffusion coefficients in qualitative agreement with pulsedfieldgradient NMR, and in qualitative disagreement with tracer zero-length column data.

The transport properties of adsorbed molecules play a central role in catalytic and separation processes that take place within zeolite cavities. Although significant effort has been devoted to understanding diffusion in zeolites (1,2), several basic questions persist: Can transition state theory and flux correlation function theory (3) be used to model complicated molecular jump processes in zeolites? Do these simulations need to allow for framework and molecular flexibility? How does the competition between energy and entropy modify the rates of these jump processes? What are the fundamental interactions that control the temperature and pressure dependence of diffusion in zeolites? In the present article, we begin to address these issues by modeling benzene jump diffusion in faujasite type zeolites. Significant effort has been devoted to modeling benzene adsorption and Corresponding author.

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© 1999 American Chemical Society

In Transition State Modeling for Catalysis; Truhlar, D., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1999.

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diffusion in the faujasites Na-X and Na-Y (4~20), motivated by persistent dis crepancies among different experimental probes of mobility (21-23). Pulsed field gradient N M R diffusivities for benzene in Na-X decrease monotonically with loading (21), while tracer zero-length column data increase monotoni cally with loading (23). Addressing this discrepancy with theory and simu lation will provide deep understanding of the microscopic physics essential to these transport phenomena. Our new theory predicts self-diffusion coefficients in qualitative agreement with pulsed field gradient N M R , and in qualitative disagreement with tracer zero-length column data. Benzene Jump Dynamics in Na-Y Zeolite We model benzene self-diffusion in Na-Y by replacing the zeolite with a three dimensional lattice of binding sites. Benzene has two predominant binding sites in Na-Y (Si:Al=2.0). In the primary site, denoted as Sn, benzene is fa cially coordinated to a zeolite 6-ring, ca. 2.7 A above a Na cation (24). In the secondary site, denoted as W , benzene is centered in the 12-ring window separating adjacent cages, ca. 5.3 A from the Sn site (24)- Figure 1 shows schematic adsorption sites and jumps for benzene in Na-Y. The lattice of ben zene binding sites in Na-Y contains four tetrahedrally arranged Sn sites and four tetrahedrally arranged, doubly shared W sites per cage. A saturation cov erage of ca. 6 molecules per cage is found for benzene in Na-Y, corresponding to occupation of all Sn and W sites.

Figure 1. Schematic sites and jumps for benzene in Na-Y. Adapted from Ref. (20). Since benzene becomes trapped with long residence times at cationic sites, transition state theory (TST) and flux correlation function (FCF) theory (3) must be used to calculate site-to-site jump rate coefficients. The exact rate coefficient for a jump from site i to site j takes the standard form: x fij(r),


(1)

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where kj]jj is the TST rate coefficient given by:

In Equation (1), fij(r) is the normalized reactive flux correlation function evaluated at the plateau time, i.e. the typical time required for relaxation from the transition state, which is usually much less than Please see Ref. (18) for computational details. Although previous TST and F C F calculations for jumps of spherical species in zeolites have been reported (25-28), modeling jumps of non-spherical species such as benzene is much more complicated due to angle-dependent dividing surfaces (16-18,29-31). We begin to address this by defining a planar dividing region of spatial width e, which can encompass some dividing surface curvature, and by using F C F dynamics to correct TST for an inaccurately defined dividing surface. Our calculations below provide the first numerically exact site-to-site rate coefficients for non-spherical molecules in zeolites. We applied Voter's Monte Carlo displacement vector approach (32) to the calculation of the partition function ratio in Equation (2). We generalized this method to the 6-dimensional case of a rigid, non-spherical molecule as described in Ref. (18). A l l T S T and F C F calculations were carried out with our forcefield for aromatics in zeolites (9), which includes electrostatic and Lennard-Jones interactions between benzene and Na-Y. For the case of a single benzene molecule in Na-Y, we found the fast multipole method (33) to be the most efficient for calculating electrostatic interactions; while for the other cases considered below, the standard Ewald approach was the best. The simulation cell consisted of 640 zeolite atoms and one benzene molecule, under cubic periodic boundary conditions with a repeat distance of 24.8 A. The Si:Al ratio of 2.0 requires 64 Na atoms in the simulation cell, assumed to occupy fully the I' sites in the (3 cages and the II sites in the supercages. We used the "average T-site" approach, wherein the 128 Si and 64 A l atoms are replaced by 192 identical "T-atoms," each with properties intermediate between that of Si and A l . A l l calculations reported below were carried out on an I B M RS/6000 with a 604e 200 MHz PowerPC processor, capable of calculating ca. 20 host-guest energies per C P U second. Rigid Benzene in a Rigid Framework. We begin by calculating the Sn—>Sn, Sn—>>W, W—>Sn and W - » W rate coefficients keeping both benzene and Na-Y rigid. A typical TST calculation for this system required 10 C P U hours, averaging over 10 —10 configurations. The corresponding F C F dy namical correction typically required 48 additional C P U hours, consisting of 2000 trajectories, each lasting ca. 2 ps. In addition to the usual checks for statistical convergence of kJ^J and fij(r), we confirmed that the TST rate coefficients obey detailed balance for the S n ^ W equilibrium, by calculating independently #eq(Sii-»W) using Voter's jump vector method. In addition, we 5

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299 confirmed explicitly that the TST rate coefficients depended upon the dividing surface location; while the F C F rates were independent of the dividing surface location. The results of these calculations are shown in Table I. Several remarks can be made about the data in Table I. First, the Suv^YV rate coefficients do indeed obey detailed balance, when compared to the simulated equilibrium coefficient. This equilibrium coefficient energetically favors the Sn site for its strong 7T—cation interaction, but entropically favors the W site for its greater flexibility; a trend that is mirrored by the rate coefficients. In all cases the activation energies from F C F calculations, denoted "Con*, function" in Table I, agree well with those from minimum energy path calculations, denoted " M E P " in Table I. TST overestimates the F C F rate coefficients by a factor of ca. 2 for all jumps beginning and/or ending at an Sn site, but gives a qualitatively wrong activation energy for the W - » W jump. The prefactor for this jump, which is reasonably well approximated by TST, gives us a clue why TST is so bad in this case. Since the F C F W—>W prefactor is nearly an order of magnitude smaller than that for the W—>S\\ jump, there is likely to be a strong entropic bottleneck in the former case. This can arise from either a tight transition state, which TST should be able to handle, or from other final states that lie close to the W—»W dividing surface, which TST cannot treat accurately because of its blindness to the eventual fate of dividing surface flux. Figure 1 shows that the W—>W path crosses right through the Sn—•Sn path, suggesting that most of the flux through the W - » W dividing surface relaxes at an Sn site. Most of these W—>W dividing surface configurations have nothing to do with actual W—•W jumps, but do have energies slightly higher than the W site energy, explaining the very small TST activation energy for this jump.

Table I. Activation Energies and Prefactors for Benzene in Na-Y Arrhenius prefactors Activation Energy (kJ mol ) MEP Corr. function TST TST Corr. function fc(W^S„) 1.1 10 s16 17.0 ± 0.1 16.4 ± 0.3 2.7 10 s" fc(W->W) 18 2.4 10 s" 1.1 ± 0.5 15.1 ± 4.0 6.0 10 s" 41 1.6 10 s" 0.8 10 s" 44.8 ± 0.1 44.4 ± 0.1 MSn^w) 0.8 10 s" 35 37.4 ± 0.1 36.8 ± 0.3 1.6 10 s" fc(Sii->S ) 25 28.0 ± 0.2 7.1 tfe (S„^W) 1

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SOURCE: Adapted from Ref. (18). Flexible Benzene in a Flexible Framework. In the previous section we held benzene and Na-Y rigid to test the feasibility of performing TST and F C F calculations, and to obtain initial estimates of the jump rate coefficients. Here we relax these constraints to determine how flexibility affects these rates. Since these new calculations can be rather computationally intensive, we attempted to diagnose which host-guest vibrational couplings would be most important.



300 Towards this end, we performed several constant—NVE molecular dynamics (MD) calculations for benzene in Na-Y, using our previously published forcefield for Na-Y framework dynamics (9). We used M D to calculate power spec tra, i.e. Fourier transforms of velocity autocorrelation functions (34), keeping various parts of the system rigid. Data were collected during 200,000 1 fs steps, after initial equilibration periods of 20,000 steps, all at approximately 300 K . Figure 2a shows the power spectrum of Na at site II in Na-Y (see Fig ure 1 for Na(II) position), in the absence (solid line) and presence (dashed line) of benzene. Figure 2a suggests that the vibrational frequency of Na(II) is blue-shifted upon adsorption of benzene, as benzene pulls Na(II) up an anharmonic vibrational potential. Figure 2b shows the power spectrum of benzene's center of mass (COM) at the Sn site in Na-Y, with full framework flexibility (solid line), with only the nearby Na(II) flexible (dashed line), and with a rigid framework (dots). Several remarks can be made about Figure 2b. The peaks near 13-17 c m correspond to C O M motion parallel to the local Na-Y sur face, whereas the higher frequency peaks arise from vibrations normal to the surface. Only the normal vibrations are perturbed from framework flexibility. These normal vibrations are significantly red-shifted by nearby Na(II) vibra tions, and are further slightly red-shifted by the remaining lattice vibrations. Indeed, roughly 80% of the total red shift arises from Na(II) motion. These power spectra suggest qualitatively which zeolite vibrations must be included in T S T / F C F calculations, namely the nearby Na(II); and suggest quantitatively how these coupled vibrations might affect the jump prefactor in a harmonic picture of adsorption. In particular, the counterbalancing blue-shifted Na(II) vibration and red-shifted benzene normal vibration foreshadows a partial can cellation, potentially giving a flexible-lattice jump prefactor nearly equal to the rigid lattice one. - 1

Wave number (cm" ) 1

Wave number (cm ) -1

F i g u r e 2. Power spectra for (a) Na(II) and (b) benzene's C O M .

In order to quantify these effects, we performed several T S T and F C F calcu lations, allowing for benzene internal flexibility and/or Na vibration in Na-Y zeolite. We focused on the Sri—•Sn, S11-+W and W—^Sn jumps, since TST



301 agreed rather well with F C F for these jumps as discussed above, allowing us to use T S T for these much more demanding calculations. Where possible, F C F calculations were performed for comparison with TST (not shown in Table II), generally giving agreement comparable to that shown in Table I. We consid ered three new combinations of flexibility: flexible benzene and rigid zeolite (denoted "Flex B-Rigid Z" in Table II), flexible benzene and vibrating nearby Na(II) ("Flex B-Na(II)"), and flexible benzene and all Na cations vibrating ("Flex B-Na(all)"). A typical "Flex B-Na(all)" T S T calculation required 35 C P U hours on an I B M PowerPC. The results in Table II demonstrate a remarkable insensitivity to the space of coordinates included. It is difficult to extract clear trends from Table II, except the fact that jumping from an Sn site is slightly faster when the nearby Na(II) is allowed to move. Even the W—^Sn barrier decreases when Na(II) is flexible, presumably because Na(II) approaches and stabilizes the W-»Sn transition state. The prefactors in Table II also appear relatively insensitive to the space of coordinates included, and show no clear trend. This presumably arises from the competition between the Na(II) and benzene C O M vibrational frequencies discussed in Figure 2. One conclusion from our work is that site-to-site jump dynamics in zeolites are well described by TST when the initial or final sites involve relatively deep potential mimina. Another conclusion is that molecular jump dynamics in a large pore zeolite is well described by including only a small number of degrees of freedom. Completely different conclusions might be drawn from modeling the jump dynamics of molecules in small pore zeolites, where window breathing modes can play a crucial role in transition state stabilization.

Table II. Effect of Benzene and Na Flexibility on Jumps in Na-Y Activation Energy (kJ mol" ) Prefactors (10 s ) SII->SH Sn-»W W - > " S ^ ~ Sir+Sn Sn-»W W->Sn RigidB-Z 37.4 ± 0 . 1 44.8 ± 0 . 1 17.0 ± 0 . 1 1.6 1.6 0.27 Flex B-Rigid Z 36.7 ± 0 . 1 44.6 ± 0 . 4 16.5 ± 0 . 3 1.4 2.8 0.28 FlexB-Na(II) 36.7 ± 0.4 43.3 ± 0.3 15.2 ± 0.5 2.6 1.9 0.17 Flex B-Na(all) — 44.0 ± 0.3 — — 2.1 — 1

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Benzene Diffusion Theory in Na-Y Zeolite In order to make contact with macroscopic transport measurements (1,2), we must relate our site-to-site jump rate coefficients with quantities such as the self-diffusion coefficient. In the following sections, we describe such a connec tion based on the mean field dynamics of cage-to-cage transition state motion. Transition State Picture of Diffusion. We can simplify diffusion in cagetype zeolites by imagining that—although hops really take place among actual potential minima—long range motion involves jumps from one "cage site" to


302 an adjacent "cage site" (14). A random walk through Na-Y reduces to hopping on the tetrahedral lattice of cages. In what follows, the W and Sn sites are denoted sites 1 and 2, respectively. We have previously shown that for self-diffusion in cage-type zeolites at finite fractional loadings, 0, the diffusion coefficient is given by: De = \kea , where a$ is the mean intercage jump length (a$ = 11 A for Na-Y) and l/ke is the mean cage residence time (14)- Furthermore, we have determined that k$ = K-kyPi, where P i = [\+K< (l-t2))~ is the probability of occupying a W site, ( T J ) = 1/fci is the mean W site residence time, and K is the transmission coefficient for cage-to-cage motion (14)- Our theory thus provides a picture of cage-to-cage motion involving transition state theory (k\-P\) with dynamical corrections («). This finding is significant since it shows that kinetics needs to be considered only for jumps originating at threshold sites between cages. It is reasonable to expect that K = | for all but the highest loadings. We also expect that Pi will increase with loading, and that k\ will decrease with loading. Below we determine the loading dependencies of «, k\ and P i using mean field theory (35) and kinetic Monte Carlo simulations. Our results elucidate how the balance between k\ and Pi controls the resulting concentration dependence of the self-diffusion coefficient. 2

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Parabolic Jump Model. We have determined P i , the probability of occupy ing a W site, using standard Ising lattice theory (35). In order to develop viable theory and simulation strategies for modeling k , the total rate of leaving a W site, we need to account for blocking of target sites and adsorbate-adsorbate interactions that modify jump activation energies. In order to account for such effects, we have generalized a model that relates binding energies to transition state energies used previously by Hood et al. (36) and also used by us for predicting mobilities in zeolites (10). We assume that the minimum energy hopping path connecting adjacent sorption sites is characterized by intersect ing parabolas, shown in Figure 3, with the site-to-site transition state located at the intersection point. The mathematical details of this model will be re ported in a forthcoming paper (20). We have performed several many-body reactive-flux correlation function calculations (35) that give barriers in quali tative agreement with the parabolic jump model. A more detailed test of this method will be reported shortly (37). x

Mean Field Ising Theory. As discussed above, K = | in mean field the ory, and standard Ising mean field theory can be used to obtain P i . The parabolic jump model for ki is also amenable to mean field theory. Assum ing that fluctuations in the pre-exponentials can be ignored and that activa tion energies are Gaussian-distributed, we have that (k^j) = i/i^j(e~ ^) = i / ^ e " ^ ^ ^ ' ^ ^ " ^ ^ ^ * ' ^ / , where al(i, j) is the variance of the Gaussian distribution of activation energies, i.e. crl(i,j) = {[E (i,j) — (E (i,j))] ) = ([E (i,j)] ) — (E (i,j)) . These quantities can all be obtained analytically using the parabolic jump model. 0Ea

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Kinetic Monte Carlo Simulation. To determine the accuracy of our mean field treatment, we perform kinetic Monte Carlo ( K M C ) simulations (15) on benzene in Na-Y, using the parabolic jump model described above. A hop is made every K M C step and the system clock is updated with variable time steps (38). The details of our implementation are reported elsewhere (15). We use time averages to calculate equilibrium values of Q\, 0 , AC, PI, fci, k$, a$ and DQ. Below we focus on comparing theory and simulation for ke, the cage-to-cage rate coefficient. For a given temperature and loading, the following parameters are required by our new model to calculate the self-diffusion coefficient: E^(i,j), v^j, a,ij and J{j for i,j = 1,2. These are the infinite dilution jump activation energies and pre-exponential factors (cf. Tables I and II), jump lengths and Ising nearest neighbor interactions for each site pair. The jump lengths can be deduced from structural data (24), and fall in the range ca. 5-9 A. The Ising coupling can be obtained from the second virial coefficient of the heat of adsorption (39), yielding ca. —3 kJ mol" . Since the diffusivity is especially sensitive to activation energies, we must recognize that our calculated barriers in Tables I and II may not be the most accurate of all available data. We first regard these barriers as flexible parameters, to determine in the most general sense what loading dependencies are consistent with our model. Figure 4a shows that three "diffusion isotherm" types emerge. We see in Figure 4a excellent qualitative agreement between theory (lines) and simulation (dots). Theory consistently overestimates simulated diffusivities because mean field theory neglects correlation effects that make K < \ for finite loadings. These diffusion isotherm types differ in the coverage that gives the maximum diffusivity: 0 = 0 is defined as type I, 0 6 (0,0.5] is type II, and 0 € (0.5,1] is type III. Defining the parameter = P[E ? (2,1) - E^(l, 2)], we find that type I typically arises from \ < 1> type II from \ ~ 1> d type III from x > 1. This suggests that when the Sn and W sites are nearly degenerate, i.e. x ~ 1, the coverage dependence of P\ is weak, and hence k$ and D$ are dominated by the decreasing coverage dependence of k\. Alternatively, when \ ~ 1) the enhancement of Pi at higher loadings dominates the diffusivity until 9\ ~ 6 , at 2

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which point the decreasing k\ begins to dominate. We have also compared the temperature dependencies generated from theory and simulation for various loadings. We have found again that our theory gives excellent qualitative agreement with simulation.

Loading (molec/cage)

Loading (molec/cage)

Figure 4. (a) Possible diffusion isotherms, (b) benzene in Na-Y. Adapted from Ref. (20). Barriers consistent with energetic data for benzene in Na-X (5), the system for which experiments disagree, are E (2-> 1) = 25 kJ m o l and E (l - » 2 ) = 10 kJ mol" . Figure 4b shows the resulting diffusion isotherms for various values of E (l —> 1) at T = 468 K , compared to pulsed field gradient N M R data (21) (uniformly scaled by a factor of 5) and tracer zero-length column data (23) (uniformly scaled by a factor of 100) at the same temperature. Our model predicts that benzene in Na-X has a type II diffusion isotherm, in qualitative agreement with the pulsed field gradient N M R results. Benzene in Na-Y has a significantly larger value of x than in Na-X (10), as seen from the activation energies in Tables I and II. Thus, our model predicts that benzene in Na-X will have a type II diffusion isotherm, while benzene in Na-Y will have a type III diffusion isotherm. - 1

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Conclusions. We have described a powerful combination of transition state simulation and analytical theory of activated self-diffusion in zeolites with adsorbate-adsorbate interactions. Our new theory, which is based on mean field dynamics of cage-to-cage motion, gives excellent qualitative agreement with kinetic Monte Carlo simulations for a wide variety of system parameters. Moreover, our theory provides deep understanding of the microscopic physics essential to these transport phenomena. Our new theory predicts self-diffusion coefficients for benzene in Na-X in qualitative agreement with pulsed field gra dient N M R , and in qualitative disagreement with tracer zero-length column data. We anticipate that this combination of transition state simulation and mean field theory can be used to complement experimental data for a wide variety of other host-guest transport systems.


305 Acknowledgments. We acknowledge support from NSF grants CHE-9625735 and CHE-9616019, and from Molecular Simulations, Inc. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, under grant A C S - P R F 30853-G5.


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