Ab Initio Calculations of the Transition State Energy and Position for

predicted by the Marcus equation or Miller's equation. Interestingly, the Hammond hypothesis gives a reasonable fit to the data. The energy of the tra...
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J. Phys. Chem. 1996, 100, 10945-10951

10945

Ab Initio Calculations of the Transition State Energy and Position for the Reaction H + C2H5R f HH + C2H4R, with R ) H, CH3, NH2, CN, CF3, C5H6: Comparison to Marcus’ Theory, Miller’s Theory, and Bockris’ Model W. T. Lee and Richard I. Masel* UniVersity of Illinois, 600 South Mathews, Urbana, Illinois 61801 ReceiVed: February 26, 1996; In Final Form: April 24, 1996X

Some years ago, Murdoch proposed that one can use the Marcus equation to predict the position of the transition state for chemical reactions. A slightly different equation was proposed by Miller. In this paper, ab initio calculations are used to test Murdoch’s and Miller’s proposal for a series of hydrogen abstraction reactions: H + CH3CH2R f H2 + CH2CH2R, with R ) H, CH3, CN, CF3, C5H5. We find that in all cases the reactions have late or very late transition states. If we define χ as a dimensionless reaction coordinate which goes from 0 at the reactants to 1 at the products, we find that the χ of the transition state varies from 0.63 to 0.86, for the cases here. In contrast, the Marcus equation and Miller’s equation give quantitatively incorrect results, i.e., early to middle transition states (χ ) 0.44-0.51). There does not appear to be any correlation between the positions of the transition state estimated from the ab initio calculations and those predicted by the Marcus equation or Miller’s equation. Interestingly, the Hammond hypothesis gives a reasonable fit to the data. The energy of the transition state is reasonably well predicted by the Marcus equation, however. Detailed analysis of our results indicates that Pauli repulsions (i.e. electron-electron repulsions) play a key role in determining the position of the transition state. The Pauli repulsions are ignored in Marcus’ model.

Introduction

χq ) 0.5 +

The position of the transition state plays an important role in the theory of reactions. In 1978, Miller1 devised a formalism to relate the position of the transition state to the thermodynamics of the reaction. Miller defined a reaction coordinate, χ, which went from 0 at the position of the reactants to 1 at the position of the products. Miller then fit the potential energy surface to second-order splines in χ and asserted that χq, the value of χ at the position of the transition state, is given by

∆H 8E0a

(3)

where ∆Hr is the heat of reaction and Ea is the activation barrier to the reaction. Miller called χq “the position of the transition state”, and we will adopt that notation here. Miller had proposed that eq 1 was very general and did not depend on the form of the potential surfaces. However, Murdoch2 later showed that eq 1 arises from a “scaled symmetry” relationship where the two sides of the potential energy surface were equivalent. Murdoch suggested that the scaled symmetry relationship worked for a large series of reactions, including some such as

where E0a is the intrinsic activation barrier, could be used to quantitatively predict the position of the transition state. Equation 3 can actually be derived from the Marcus equation without assuming that the reaction obeys the scaled symmetry relationship. However, Murdoch demonstrated that there would be deviations from the Marcus equation in cases where the scaled symmetry relationship failed. Murdoch states that eqs 1 and 3 are in good agreement with ab initio calculations. However, Murdoch did not actually do SCF calculations of the transition state geometry. Instead, in his work with Chen,3 Murdoch fit other people’s SCF calculations of equilibrium geometries and vibrational frequencies to splines, calculated the position of the transition state from the maximum in the splines, and then compared to eq 3. No comparisons to ab initio transition state geometries were done. The only actual comparisons that we can find are the work of Yamataka et al.,4,5 who found that the transition state geometry calculated at the HF/3-21G level did not match that predicted from eq 1 for a number of hydrogen transfer reactions. No comparisons were done to the Marcus equation, eq 3. There has been considerable work trying to verify the standard Marcus equation, which can be written

F + H2 f HF + H

EA ) EA0 + γP∆Hr

χq )

1 ∆Hr 2Ea

(1)

(2)

where the environment of the reactants and products was changing greatly. Murdoch also asserted that a modification of the Marcus equation, * Send correspondence to this author. X Abstract published in AdVance ACS Abstracts, June 1, 1996.

S0022-3654(96)00578-3 CCC: $12.00

(4)

with γP, the transfer coefficient, given by

γP ) 0.5 +

∆Hr 8E0a

(5)

In a recent paper8 we examined the potential energy surface, the intrinsic barrier, and transfer coefficient for the reaction © 1996 American Chemical Society

10946 J. Phys. Chem., Vol. 100, No. 26, 1996

Lee and Masel

Figure 1. Geometries considered in this paper.

D + C2H6 f DCH3 + CH3

(6)

The calculations showed that reaction 6 had a slightly late transition state, χq ) 0.52. However, Murdoch’s extension of the Marcus formalism predicted an early transition state (χq ) 0.48). Miller’s formalism also predicts χq ) 0.48. When we wrote our earlier paper, we were surprised that the Marcus equation gave incorrect results. However, the deviations were small so we did not concentrate on them. Still there was the possibility that the Marcus equation and Miller’s formalism would give some unexpected results for other systems. The objective of this paper was to do a more careful test of the Marcus equation, Miller’s formalism, and the Hammond hypothesis, for the series of reactions:

D + CH3CH2R f DH + CH2CH2R with R ) H, CH3, NH2, CN, CF3, C5H6.

(7)

Calculational Methods In this paper, we have used ab initio molecular orbital calculations to determine the energetics of the molecules and transition states. All of the calculations were performed with the GAUSSIAN929 and GAUSSIAN9410 programs. In all systems, it was assumed that the deuterium attacks the R-trans H of CH3CH2R as shown in Figure 1. Equilibrium and transition state structures were optimized (frozen core) at the second-order Moller-Plesset perturbation theory (MP2) using various basis sets ranging from 6-31G to 6-311G(d,p). We have also done calculations at MP4 and QCISD(T) as well as using the G-2 technique to estimate the electron correlation energies. Spin projection was used to account for spin contamination for open shell structures. The transfer coefficient γP was determined from the slopes of the intrinsic reaction pathway as described in our earlier paper.8 This was done by stepping off from the transition state structure toward the reactants and products along the mass-

H + C2H5R f HH + C2H4R Reaction

J. Phys. Chem., Vol. 100, No. 26, 1996 10947 0 and 0.5 for an early transition state, and negative in the Marcus inverted region. In the case of the reaction in Figure 2, χLq works out to be 0.663, which is indicative of a fairly late transition state. One can also get a measure of the position of the transition state from a Mullikin population analysis. We define χMq, the position of the transition state as calculated from a Mullikin population analysis, by

χMq )

Figure 2. Potential energy surface for the reaction H + C2H6 f HH + C2H5 calculated for the geometry shown in Figure 1. The calculation was done at the PMP2/6-31G* level of theory. The contour lines are 5 kcal/mol apart.

weighted reaction coordinate, as described in ref 8. To be consistent, the maximum slope was used in the calculation. The potential energy surfaces reported in this paper were calculated by freezing the C-H and H-H distances and then optimizing all of the other coordinates in the molecule at the MP2/6-31G* level of theory. Results Figure 2 shows a potential energy surface (PES) for reaction 7 with R ) H calculated for a hydrogen atom approaching with an ethane molecule on a trajectory along the carbon-hydrogen bond axis of the ethane. The PES was generated using the 6-31G* basis set at the MP2 level. The potential energy surface looks quite standard for an SN2 reaction. The potential goes up as the hydrogen approaches and then down again as the products fly away. The saddle point energy in Figure 2 is 16.1 kcal/mol. This compares to an experimental activation barrier of 11.8 kcal/mol.11 One interesting feature in Figure 2 is that the transition state for reaction occurs late in the reaction coordinate. At the transition state the C-H bond is greatly extended, while the D-H bond length is almost at its equilibrium value. Therefore, we can conclude that the reaction has a late transition state. One can quantify this effect by defining a reaction parameter χL as described by Miller.

χL )

0 rC-H - rC-H 0 0 (rC-H - rC-H ) + (rD-H - rD-H )

(8)

where rD-H and rC-H are the D-H and C-H distances at any point along the intrinsic reaction coordinate (IRC), rD-H is the 0 is D-H bond length in an isolated D-H molecule, and rC-H the C-H bond length in an isolated ethane molecule. This definition is not unique. For example, any definition of the form

χn )

0 rC-H - rC-H 0 0 (rC-H - rC-H )n + Cn(rD-H - rD-H )n

(9)

with Cn > 0 would also meet Miller’s criterion. However, the function in eq 8 is the simplest function to meet Miller’s criterion. Note that χLq, the value of χ at the transition state, will be between 0.5 and 1.0 for a late transition state, between

%H-D %C-H + %H-D

(10)

where %C-H and %H-D are the fractional C-H and H-D bond orders calculated from a Mullikin population analysis. For the example in Figure 2, χMq works out to 0.67, which is again indicative of a late transition state. One can also use Pauling bond orders in eq 10. In that case χPq works out to be 0.66. We have also done calculations with R ) H, CH3, NH2, CN, and CF3 at the PMP2/6-31+G(d,p) level of theory. Table 1 summarizes the results of these calculations. In all cases the reactions show fairly late transition states. The calculations in Figure 2 and Table 1 were done with a modest basis set and a moderately accurate calculational procedure. We have also done calculations where we varied the basis set and the level of theory for three of the cases in Table 1, namely, R ) H, R ) CH3, and R ) CN. Tables 2, 3, and 4 show the results of those calculations. Generally we find that we need to go to the G-2 level of theory to get reasonably accurate heats of reaction. Accurate intrinsic activation barriers can be computed at the PMP2/6-311++G(d,p) level of theory. We have also calculated the position of the transition state using Miller’s formula (1) and Murdoch’s extension of the Marcus equation (3). The heats of reaction were estimated with G-2 calculations for everything except R ) C5H6. The intrinsic activation barriers were estimated both from the Marcus equation and Bockris’ formula, as described earlier.8 Numerical values of all the parameters are given in Table 1. The results of the calculations are given in Table 5. Generally, the Marcus equation and Miller’s approximation predict χq’s of 0.5 or less, which are indicative of early transition states. Discussion It is useful to compare the ab initio results for the position of the transition state to the predictions of the Marcus equation and Miller’s formula. The ab initio calculations show some very clear trends. First, all of the reactions have late transition states. The transition state lies far to the left of the 45° line in Figure 2. The position of the transition state corresponds to a χq of between 0.66 and 0.68, which is indicative of a very late transition state. The transition state positions (i.e. χq’s) calculated via a Mullikin population analysis are similar to those calculated from the transition state geometry. Interestingly our results generally agree with the Hammond postulate, γP ) χq, in that there is a reasonable agreement with γP calculated as described in our previous paper8 and the position of the transition state χq calculated from eq 8. This tells us that our definition of χ is appropriate for this system. The results, however, show that there is little agreement between the positions of the transition state (i.e. χq) in the ab initio calculations and the position of the transition state calculated from the Marcus equation or Miller’s equation. Figure 3 compares the position of the transition state calculated via our ab initio calculations and those predicted by the Marcus equation and Miller’s formula. Notice that Miller’s and Marcus’ methods predict early to middle transition states (χq ) 0.430.51), while the ab initio calculations show late or very late

10948 J. Phys. Chem., Vol. 100, No. 26, 1996

Lee and Masel

TABLE 1: Comparison of the Activation Barriers, Heats of Reaction, and Transition State Position Calculated for the Reaction H + C2H5R f HH + C2H4R at the PMP2/6-31+G*d,p) Level of Theory R

saddle point energy, kcal/mol

heat of reaction, kcal/mol

χLq (structure)

χMoq (Mullikin)

Ea0 (γP from the Marcus eq)

Ea0 (assuming γP by Bockris’ procedure)

H CH3 NH2 C6H5 CF3 CN

16.1 16.23 16.41 16.01 17.26 17.46

+2.54 +3.26 +3.29 +3.61 +4.11 +4.91

0.65 0.66 0.66 0.67 0.67 0.68

0.67 0.72 0.69 0.72 0.82 0.83

14.80 14.56 14.72 14.14 15.13 14.90

14.34 13.92 14.07 13.37 14.26 13.92

TABLE 2: Comparison of the Energies of Various Species, Activation Barriers, and Heats of Reaction Calculated for the Reaction H + C2H6 f HH + C2H5 via a Series of Different Basis Sets and Computational Procedures computational procedure

basis set

UMP2 UMP2 UMP2 UMP2 UMP4 QCISD QCISD(T) G-2 experiment

6-31G 6-31+G** 6-311G** 6-311++G** 6-311G** 6-311G** 6-311G**

ethane energy, hartrees

ethyl energy, hartrees

heat of reaction, kcal/mol

saddle point energy, kcal/mol

-79.385 60 -79.545 79 -79.570 89 -79.571 67 -79.614 52 -79.606 14 -79.615 79 -79.630 86

-78.734 76 -78.882 31 -78.905 07 -78.906 88 -78.944 59 -78.938 17 -78.946 63 -78.969 25

3.1 2.5 3.4 2.7 1.3 -0.3 0.4 -3.8

17.9 16.1 14.8 14.5 14.8 13.8 13.6 10.4 11.8

χLq 0.62 0.65 0.66 0.66

TABLE 3: Comparison of the Energies of Various Species, Activation Barriers, and Heats of Reaction Calculated for the Reaction H + C3H8 f HH + C3H7 via a Series of Different Basis Sets and Computational Procedures computational procedure UMP2 UMP2 UMP2 UMP2 UMP4 QCISD QCISD(T) G-2

basis

propane

propyl

∆H

Ea

χ

6-31G 6-31+G** 6-311G** 6-311++G** 6-311G** 6-311G** 6-311G**

-118.495 29 -118.729 10 -118.766 05 -118.767 43 -118.827 27 -118.813 47 -118.828 79 -118.843 96

-117.843 37 -118.064 47 -118.099 31 -118.101 57 -118.156 569 -118.144 81 -118.158 95 -118.183 92

3.8 3.3 3.9 3.4 1.8 0.1 0.8 -4.76

18.1 16.2 15.1 14.8 15.0 14.1 13.8 10.9

0.63 0.66 0.67 0.67

TABLE 4: Comparison of the Energies of Various Species, Activation Barriers, and Heats of Reaction Calculated for the Reaction H + C2H5CN f HH + C2H4CN via a Series of Different Basis Sets and Computational Proceduresa computational procedure UMP2 UMP2 UMP2 UMP4 QCISD QCISD(T) G-2

basis

propane nitrile

propyl nitrile

∆H

Ea

χ

6-31+G** 6-311G** 6-311++G** 6-311G** 6-311G** 6-311G**

-171.551 61 -171.601 82 -171.606 40 -171.660 86 -171.632 52 -171.657 59 -171.739 11

-170.884 36 -170.932 68 -170.937 75 -170.987 85 -170.961 60 -170.985 49 -171.071 98

4.9 5.4 5.1 3.2 1.5 2.2 -0.3

17.4 16.4 16.6 16.3 15.3 15.0 10.6

0.68 0.69 0.68

a The experimental values were calculated by adding the heats of formation of the various species from the JANAF tables to the sum of the atomic energies for the reactants.

TABLE 5: Comparison of the Activation Barriers, Heats of Reaction, and Transition State Position Calculated for the Reaction H + C2H5R f HH + C2H4Ra R

saddle point energy, kcal/mol

heat of reaction, kcal/mol

χLq

χMoq

χqMarcus

χqMiller

γP (Bockris)

H CH3 NH2 CF3 CN

14.54 14.81 14.90 15.98 16.57

-3.8 -3.7 -3.83 -3.3 +0.73

0.657 0.669 0.663 0.682 0.687

0.69 0.72 0.69 0.82 0.83

0.47 0.47 0.47 0.48 0.51

0.44 0.44 0.44 0.45 0.49

0.70 0.71 0.71 0.73 0.72

a The activation barriers of χ’s are calculated at the PMP2/6-311++G(d,p) level of theory. The heats of reaction are calculated at the G-2 level of theory

transition states (χq ) 0.66-0.83). There is little correlation between the χq’s in the ab initio calculations and those predicted by the Marcus equation or Miller’s formula. Interestingly, the Hammond postulate predicts χq’s that are only slightly larger than the χq’s calculated from the transition state geometry or the Pauling bond order. The Hammond postulate gives less good of a fit to the value of χq calculated from a Mullikin

population analysis. The differences between Marcus’ and Miller’s formulas and the calculation deviations are quite substantial, much more than one can explain via calculational error. The deviations are not simply small quantitative errors. Rather, the Marcus and Miller equations are giving qualitatively incorrect results (early transition states in systems with late transition states). Thus, it seems that the Marcus equation and

H + C2H5R f HH + C2H4R Reaction

J. Phys. Chem., Vol. 100, No. 26, 1996 10949

Figure 5. Plot of the ethyl-H and H-H potentials along the intrinsic reaction coordinate. The actual energy along the reaction path is included for comparison.

Figure 3. Plot of the transition state positions calculated by a variety of methods.

Murdoch derived eq 3 by assuming that the potential energy surface for a reaction of the form

R-H + X f R + H-X

(11)

is given as a combination of the R-H and H-X potentials for isolated R-H and H-X molecules and then fitting the potentials to parabolics to derive the Marcus equation. Our examples apparently do not follow the Marcus equation, so it is interesting to understand why they do not. Consider the reaction

ethyl-H + H f ethyl + H-H Figure 4. Plot of the MP2 transition state energy versus the MP2 heat of reaction. The points are the MP2 data. The line is the Marcus equation with Ea0 estimated from the curve crossing in Figure 5.

Miller’s approximation do not give reasonable predictions of the positions of the transition states for these examples. One can also use the Marcus equation to estimate the energy of the transition state for each of these reactions. Figure 4 compares the predictions of the Marcus equation for the activation barriers for the reaction to the MP2 values for the energy of the transition state. In this plot, E0a was calculated from MP2 calculations for the isolated reactants, as described at the end of this section, and ∆H was estimated from the MP2 energies. Notice that the Marcus equation gives a good fit to the transition state energies even though it does not fit transition state positions. This leads to the main result of this paper: The Marcus equation seems to give a good fit to the activation barriers estimated from ab initio calculations. However, neither the Marcus equation nor Miller’s formula gives a good fit to the transition state positions. It is useful to relate the findings in this paper to the findings of Murdoch.2 Murdoch showed that the Marcus equation would fail in cases that do not obey a “scaled symmetry relationship”. However, Murdoch asserted that the Marcus equation, eq 3, would quantitatively predict the transition state position for any reaction that follows a “scaled symmetry” relationship. Murdoch also asserted that most reactions would follow the scaled symmetry relationship.

(12)

Murdoch’s derivation makes the assumption that the potential for this reaction will be like that for the reactants near the beginning of the reaction and like that at the products near the end of the reaction. Murdoch makes the additional assumption that the position of the transition state corresponds to the intersection at the ethyl-H and H-H potentials. To see if this idea works, we need to have a plot of ethyl-H and H-H potential as a function of χ. We generated the plots by using Gaussian to calculate the energy of an ethane molecule as a function of rC-H, the length of the C-H bond. In this calculation we allowed the lengths of all of the other bonds in the ethane to vary. We added the energy of the ethane molecule to the energy of a hydrogen atom. That let us calculate a plot of the energy of an ethane molecule plus an isolated hydrogen atom as a function of rC-H. Next we converted that to a plot of the energy as a function of χ not rC-H. First we did a standard IRC calculation for reaction 12. The IRC calculation gave us rC-H and rH-H at each point along the minimum energy pathway for reaction 12. We then used eq 8 to calculate χ corresponding to each value of rC-H. At this point we had a plot of the energy of the reactants (i.e. an ethane molecule with a hydrogen atom) and χ as a function of rC-H. Therefore, one can make a cross plot of the energy of the reactants as a function of χ. The result is line A in Figure 5. We repeated the calculation for the products (H2 and an isolated ethyl group). The result is line B in Figure 5. Finally, we plot the total energy of the system

10950 J. Phys. Chem., Vol. 100, No. 26, 1996

Lee and Masel

E(χ1) EA

(14)

E(χ2) EA - ∆H

(15)

χ for χ < χq q χ

(16)

1-χ for χ > χq 1 - χq

(17)

G1(χ1) ) G2(χ2) ) χ1 ) χ2 )

Figure 6. Plot of the data in Figure 5 in mass-weighted coordinates.

from the IRC calculation as a function of χ. That is line C in Figure 5. Figure 6 shows the same data plotted in massweighted reaction coordinates. Notice that the ethyl-H and H-H potentials look very standard. The curves are nearly parabolic and fairly symmetric. The two potentials intersect at an energy that lies close to the calculated activation barrier. However, the position of the intersection of the two curves does not correspond to the position of the transition state, and the shapes of the curves show no correspondence to the actual potential surface. Interestingly, the position of the intersection of the two curves shows a close correspondence to the position calculated from the Marcus equation, eq 3. This result shows that the Marcus equation does correctly predict the position of the intersection of the C-H and H-H potentials. However, the point of intersection of the ethyl-H and H-H potentials does not correspond to the position of the transition state! Following Murdoch, we can also see if the system follows a scaled symmetry relationship:

G1(χ1) ) G2(χ2) where

(13)

We replot the data in Figure 4 in reduced coordinates and look for symmetries. Figure 7 shows the result. One finds that the C-H and H-H potentials do approximately follow a scaled symmetry relationship. There are some small deviations. However, these deviations are comparable to others that are said to follow the scaled symmetry relationships. Therefore, it seems that the deviations from the Marcus equation found in this paper are not caused by a failure of the scaled symmetry relationship. Rather, they seem to be associated with the fact that the position of the intersection of the ethyl-H and H-H potentials does not correspond to the position of the transition state. These results suggest that there is a fundamental flaw in Murdoch’s derivation. Murdoch assumes that the potential energy surface for the reaction can be approximated by the ethyl-H potential near the reactants and the H-H potential near the products. Figure 5, however, shows that the real potential does not look anything like the ethyl-H potential at the early part of the reaction, nor does the real potential look anything like the H-H potential near the end of the reaction. According to the ab initio calculations, when an incoming hydrogen atom first approaches the hydrogen, there is a Pauli repulsion between the incoming hydrogen and the hydrogen in the ethane. The C-H bond in the ethane is compressed. That compression is ignored when one calculates the potential of an isolated H-ethyl molecule, and so the shape of the actual potential energy contour near the reactants is quite different from the shape of the potential contour for an isolated C-H molecule. A similar thing happens near the products, but the effect is smaller. The net result is that the Pauli repulsions push the transition state toward the products. These Pauli repulsions are ignored in the Marcus model. There also is significant mixing

Figure 7. Test of the scaled symmetry relation for the reaction H + C2H6 f HH + C2H5.

H + C2H5R f HH + C2H4R Reaction of states near the transition state, which flattens the top of the potential energy surface near the transition state. The net effect of all of this is that the basic assumption in Murdoch’s derivation fails. The potential energy surface for the reaction does not ride up the ethyl-H for the reaction and down the H-H potentials for the products, as assumed in Murdoch’s derivation. Still, as noted previously, the Marcus equation does not do a bad job of predicting the energy of the transition state. Figure 4 compares the energy of the transition state estimated from the Marcus equation with E0a calculated from intersections of the curves in Figure 5 to the energies calculated from ab initio calculations. We find that the energy of the transition state is reasonably well predicted from the intersection of the two curves in Figure 5, even though the position of the transition state is not well predicted. Physically, what is happening is that there is a cancellation of errors. There is a Pauli repulsion when the reactants come together, which raises the energy of the transition state. There is also some mixing of states at the avoided crossing, which lowers the energy of the transition state. It happens that those two effects approximately cancel. As a result, the energy of the transition state is reasonably approximated by the energy of the intersection of the ethyl-H and H-H curves; even the actual potential energy surface looks nothing like the ethyl-H and H-H curves. It is unclear whether the terms will cancel in other examples. However, for the examples here, the Marcus equation does give reasonable predictions of the energy of the transition state, but not the position of the transition state. Conclusion In the work here, we have used ab initio calculations to estimate the positions and energies of the transition states for a number of hydrogen abstraction reactions. We find that the energies of the transition states are reasonably well represented by the Marcus equation. However, the positions of the transition states represented by the Marcus equation or Miller’s results are qualitatively incorrect. In all cases, the ab initio calculations predict late or very late transition states (χ ) 0.66-0.83), while the Marcus equation and Miller’s results predict early to middle

J. Phys. Chem., Vol. 100, No. 26, 1996 10951 transition states (χ ) 0.43-0.51). These results suggest that there may be a fundamental problem with using the Marcus equation to predict transition state positions, even though the Marcus equation gives reasonable transition state energies. In particular Pauli repulsions play a key role in determining the transition state positions. The Pauli repulsions are ignored in the Marcus model. Acknowledgment. This work was supported by the National Science Foundation under Grant CTS 94-03840. The work used the Silicon Graphius Power Challenge at the National Center for Supercomputing Applications, University of Illinois, Urbana-Champaign. References and Notes (1) Miller, A. R. J. Am. Chem. Soc. 1978, 100, 1984. (2) Murdoch, J. R. J. Am. Chem. Soc. 1983, 105, 2667. (3) Chen, M. Y.; Murdoch, J. R. J. Am. Chem. Soc. 1984, 106, 4735. (4) Yamataka, H.; Nagase, S.; Ando, T.; Hanafusa, T. J. Am. Chem. Soc. 1986, 108, 601. (5) Ymataka, H.; Nagase, Shigeru J. Org. Chem. 1988, 53, 3232. (6) Shaik, S. S.; Schlegel, H. B.; Wolfe, S. Theoretical Aspects of Physical Organic Chemistry; John Wiley & Sons, Inc.: 1992, and reference therein. (7) Marcus, R. A. J. Phys. Chem. 1968, 72, 891. (8) Lee, W. T.; Masel, R. I. J. Phys. Chem. 1995, 99, 9363. (9) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Gill, P. M. W.; Johnson, B. G.; Wong, M. W.; Foresman, J. B.; Robb, M. A.; Head-Gordon, M.; Replogle, E. S.; Gomperts, R.; Andres, J. L.; Raghavachari, K.; Binkley, J. S.; Gonzales, C.; Martin, R. L.; Fox, D. J.; Defrees, D. J.; Baker, J.; Stewart, J. J. P.; Pople, J. A. Gaussian 92/DFT, Revision G.2; Gaussian, Inc.: Pittsburgh, PA, 1993. (10) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Gill, P. M. W.; Johnson, B. G.; Robb, M. A.; Cheeseman, J. R.; Keith, T.; Peterson, G. A.; Montgomery, J. A.; Raghavachari, K.; Al-Laham, M. A.; Zakrzewski, V. G.; Ortiz, J. V.; Foresman, J. B.; Peng, C. Y.; Ayala, P. Y.; Chen, W.; Wong, M. W.; Andres, J. L.; Replogle, E. S.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Binkley, J. S.; Defrees, D. J.; Baker, J.; Stewart, J. P.; HeadGordon, M.; Gonzalez, C.; Pople, J. A. Gaussian 94, Revision B.3; Gaussian, Inc.: Pittsburgh, PA, 1995. (11) Nicholas, J. E.; Bayrakeen, F.; Fink, R. J. J. Chem. Phys. 1972, 56, 1008. Oldershaw, G. A.; Gould, R. L. J. Chem. Soc., Faraday Trans. 2, 1985, 81, 1507.

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