Chemical Kinetics of a Bipalladium Complex - The Journal of Physical

Article pubs.acs.org/JPCA

Chemical Kinetics of a Bipalladium Complex Edward W. Doddridge,*,†,‡,§ Larry K. Forbes,† and Brian F. Yates‡ †

School of Mathematics and Physics, University of Tasmania, Private Bag 37, Hobart, Tasmania 7001, Australia School of Chemistry, University of Tasmania, Private Bag 75, Hobart, Tasmania 7001, Australia

‡

ABSTRACT: A theoretical model is presented, for reductive elimination in a bipalladium complex, based on the model of Ariafard et al. (2011). This reaction is of particular interest due to the novel Pd(III) intermediate. A thermo-kinetic model is proposed for this reaction scheme, and the rate laws and energy balance are given as a system of ordinary differential equations. A simplified model is then derived that only involves two key variables, so that the system can be analyzed completely in a phase plane. It is shown that kinetic oscillations do not occur, but that there are multiple steady states for the reaction. These new features are confirmed by a numerical analysis of the full model scheme. The predictions provide a mechanism to test the model and the underlying computational chemistry.

I. INTRODUCTION The mathematical analysis of chemical reactions has been an active area of research since the early part of the 20th Century. Early work done by Lotka1,2 showed that sustained oscillations far from thermodynamic equilibrium were possible. At the time it had been assumed that chemical reactions must monotonically approach thermodynamic equilibrium. While this is still believed to be true close to equilibrium within the linearized domain, it has been demonstrated that far from equilibrium chemical systems may exhibit a range of exotic behaviors.3−12 This paper analyses a bipalladium catalytic system. Palladium (Pd) is commonly used in catalysis. As long ago as 1971 a Pd(II)/Pd(IV) redox catalytic cycle was proposed.13 Recent research has shown that bimetallic palladium complexes can act as catalysts for carbon−carbon and carbon-heteroatom bond formation.14−17 The mechanism of these catalytic cycles is an area of current research; Powers and Ritter16 give some evidence to support an intermediate with two Pd(III) centers rather than the traditional Pd(IV) intermediate. The possibility of two Pd(III) centers raises interesting questions regarding the mechanism of the redox catalytic cycle. Powers et al.18 provide evidence of reductive elimination from the two Pd(III) centers with redox synergy between the two metal centers. This is beneficial because “the presence of the second metal ... lowers the activation barrier of reductive elimination.”18 Lowering the activation barrier allows the reaction to proceed at a faster rate, or under less extreme conditions. Computational chemistry has been used to investigate the mechanism of these proposed bimetallic palladium redox cycles.18,19 Ariafard et al.19 investigated the reactions reported by Powers and Ritter16 and Powers et al.18 and proposed two © 2012 American Chemical Society

pathways for reductive elimination from a bimetallic Pd(III) complex. The reaction scheme is shown in Figure 1. The first pathway, the “direct” pathway, involves a reductive elimination step followed by the dissociation of a chloride ion from one of the palladium centers. Ariafard et al.19 computed that the reductive elimination occurred primarily at one palladium center, leaving the molecule as a Pd(I)−Pd(III) species. This is highly unusual. The second pathway, the “dissociative” pathway, involves the same two steps, but their order is reversed; the dissociation of a chloride from one of the palladium atoms occurs first, followed by reductive elimination. After these steps the two pathways form the same intermediate, at which point a chloride ion associates with the other palladium center. In Figure 1 the species designated “X” is either a chloride, as was present in the experimental work by Powers and Ritter,16 or a phenyl group, as was present in the experimental work by Deprez and Sanford.14 Ariafard et al.19 computed a potential energy surface for this reaction in two solvents, dichloromethane and acetonitrile, with the two options for “X”. Figure 2 shows the potential energy surface computed by Ariafard et al.19 for the chloride version of the bipalladium complex. The analysis presented in this paper focuses on the reaction where “X” is a chloride and dichloromethane is used as the solvent because the experimental work by Powers and Ritter16 includes rate measurements at a range of temperatures. These data will be used to estimate some of the intermediate rate parameters. Received: October 19, 2012 Revised: December 13, 2012 Published: December 17, 2012 541

dx.doi.org/10.1021/jp310368g | J. Phys. Chem. A 2013, 117, 541−549

The Journal of Physical Chemistry A

Article

Figure 1. Reaction scheme proposed by Ariafard et al.19 Transition states include “TS” in their name. The oxidation state of each Pd center is shown in Roman Numerals.

In section II, of this paper the computational work done by Ariafard et al.19 will be used to develop a system of coupled ordinary differential equations for the concentration of each of the chemical species. A number of simplifying assumptions will be used to reduce the complexity of the resulting mathematical model. Section IIIa presents a regression analysis of experimental data from Powers and Ritter16 that provides a method for estimating the rate parameters of the elementary reactions. In section IIIb, a two-dimensional approximation of the full model is derived and analyzed. No oscillatory solutions are found, but there exist regions in the phase-space with multiple steady states. Dulac’s theorem is used to rule out the possibility of oscillatory solutions in regions of the phase-space. Analysis of the full model is presented in section IIIc. In section IIId, an alternative model is investigated, in which the chloride dissociation is modeled as a fast thermodynamically controlled equilibrium; however, this is then shown to lack selfconsistency and must therefore be discounted.

5_X + + Cl− → 6_X_Cl + Cl− + Q 4 , reaction rate = k4

Direct Pathway:

reaction rate = k 2

2_TS_X + → 5_X + + Q 3 ,

reaction rate = k 3

1_TS _X_Cl → 4_X_Cl + Q 6 ,

reaction rate = k6

4_X_Cl → 5_X + + Cl− + Q 7 ,

reaction rate = k 7

d[3_X +] = k1[2_X_Cl] − k 2[3_X +] dt

(1)

The nondimensionalisation for temperature proposed by Burnell et al.20 and Gray et al.21 θ=

RT Ea

(2)

(in which Ea is the activation energy) will be used in this paper. This allows the ambient temperature to be used as a bifurcation parameter without affecting the values of other nondimensionalized constants. This nondimensionalisation, and assuming the reaction occurs under the approximation of a continuous-flow stirred tank reactor, leads to the following set of equations relating the concentration of the chemical species to their rate of change with time:

reaction rate = k1

3_X + → 2_TS_X + + Q 2 ,

reaction rate = k5

These elementary reactions can be used to construct ordinary differential equations for the concentration of each species in the reaction scheme. For example

II. DERIVATION OF A MATHEMATICAL MODEL Consideration of the full chemical reaction, as shown in Figure 1, leads to seven elementary reactions, in which Q represents the change in Gibbs free energy for the reaction. The reaction rates for these elementary reactions are assumed to follow the Arrhenius form. The elementary reactions are shown below. Dissociative Pathway: 2 _X_Cl→ 3_X + + Cl− + Q 1 ,

2_X_Cl → 1_TS _X_Cl + Q 5 ,

dx = γx0 − x(ρ5 e−1/ θ + ρ1 + γ ) dτ 542

(3a)



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Figure 2. Reaction potential energy surface from Ariafard et al.19 in (a) dichloromethane and (b) acetonitrile, with calculated free energies in kJ mol−1 (enthalpies are shown in parentheses).

dy = ρ5 e−1/ θ x − y(ρ7 + γ ) dτ

(3b)

dz = ρ7 y + ρ2 e−α / θ − z(ρ4 + γ ) dτ

(3c)

dw = ρ1x − w(ρ2 e−α / θ + γ ) dτ

(3d)

volume and thus represents the inverse of a mean residence time, τ is the nondimensionalized time, and θ represents nondimensionalized temperature. The constants ρ1 and so on represent dimensionless reaction rates; the reaction rates have been nondimensionalized by dividing by Z5. That is, ρ1 = k1/Z5 and equivalently for each reaction rate. This nondimensionalisation means that ρ5 is defined to be 1. The e−1/θ terms are the Arrhenius temperature dependencies of the reaction rates, and because the temperature has been nondimensionalized with respect to E5 the parameter α is the ratio of E2 to E5. The two quantities θa and θin in eq 3e are the dimensionless ambient temperature and the dimensionless temperature of the inflowing solution containing 2_X_Cl, respectively; for the rest of the analysis presented, these two temperatures are assumed to be equal. Analyzing continuous-flow stirred tank

dθ = xρ5 e−1/ θ + wρ2 e−α / θ − β(θ − θa) − γ(θ − θin) dτ (3e)

in which x, y, z, and w represent nondimensionalized concentrations of the four species 2_X_Cl, 4_X_Cl, 5_X+, and 3_X+ respectively, γ is the ratio of flow rate to reactor 543



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Table 1. Symbols Used in This Paper and Their Meanings symbol

meaning

w, x, y, z θ γ En α kn ρn Zn τ

nondimensional concentrations (subscript “e” refers to equilibrium values) nondimensional temperature (subscript “a” refers to ambient and “e” refers to equilibrium) nondimensional flow parameter (inverse of a mean residence time) dimensional activation energy (n = 2, 5) ratio of E2 to E5 dimensional reaction rate (n = 1,..., 7 nondimensional reaction rate (n = 1, ..., 7) dimensional pre-exponential in Arrhenius rate parameter (n = 2, 5) nondimensional time

rate at which the starting material reacts is equal to the rate at which the final product is formed. Interpreting this in terms of the model presented in Figure 3 gives the following set of equalities

reactor systems with inflow temperature as a bifurcation parameter can yield interesting results,22 but is beyond the scope of this paper. Table 1 contains a list of the symbols used in this paper and their meanings. The derivation of this model requires a number of simplifying assumptions. These assumptions lead to the simplified reaction scheme and potential energy surface shown in Figures 3

[2_X_Cl](k1 + Z5e−E5 / RT ) = k4[5_X +] =

d[P] = [2_X_Cl]exp kexp dt

(5)

where the subscript “exp” subscript refers to experimental values. The data collected by Powers et al.18 are shown in Table 2. These data are for the dichloro version of the bipalladium Table 2. Reaction Rates for 2_Cl_Cl in Dichloromethane with an Initial Concentration of 20 mM at Various Temperatures

Figure 3. Simplified model reaction scheme.

temperature (K)

and 4, respectively. For ease of comparison with experimental data, the dimensional reaction rates will be used for much of the following discussion. It is of particular importance to remember that the two reaction rates k2 and k5 in Figure 3 are assumed to be temperature dependent and to follow Arrhenius kinetics k 2(T ) = Z 2 exp( −E2 /RT )

(4a)

k5(T ) = Z5 exp( −E5/RT )

(4b)

278.37 286.50 292.46 299.50 308.03

reaction rate (s−1) 5.99 1.73 3.49 6.75 1.32

× × × × ×

10−4 10−3 10−3 10−3 10−2

complex with dichloromethane as the solvent. The rest of the analysis presented in this paper will be based around this combination of solvent and reagent. Combined with the theoretical work done by Ariafard et al.19 this gives enough information to estimate k1 and Z5. Equation 5 may be simplified, and [2_X_Cl] canceled, to give

in which E2 and E5 are the activation energies for the two reactions and R is the universal gas constant. The remaining reactions are assumed to occur isothermally, so that k1, k4, and k7 are all assumed to be constant, as indicated in Figure 4.

k1 + Z5e−E5 / RT = kexp

III. ANALYSIS AND RESULTS a. Estimating Intermediate Rate Constants. The experimental rate data collected by Powers et al.18 can be used to estimate intermediate rate coefficients through regression. If it is assumed that the intermediates in the actual reaction quickly reach their equilibrium concentrations then the

(6)

which allows the data presented in Table 2 to be used to estimate k1 and Z5. If the temperature data in Table 2 are transformed into a new variable e−E5/RT and the rate data are regressed against this new variable, then k1 will be the intercept and Z5 the slope of the regression equation.

Figure 4. Simplified potential energy surface for the model. 544



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roughly in dynamic equilibrium with respect to both x and θ over the time periods considered. Thus, a quasi-equilibrium simplification may be made in which w is replaced by the value ρ1x/(ρ2e−α/θ+γ). Replacing w with this quasi-equilibrium value results in the following planar system of equations:

The regression analysis fits the data well: the proportion of variance explained is high, R2 = 0.975, and the linear model fits significantly better than the null model, F(1,3) = 156.7, p = 0.001099; see Table 3 for coefficient estimates. Table 3. Summary from the Regression of Experimental Reaction Rates Against the Transformed Variable e−E5/RT intercept (k1) slope (Z5)

estimate

standard error

t value

p value

1.020E−03 8.655E+12

4.873E−04 6.914E+11

2.093 12.519

0.1274 0.0011

dx = γx0 − x(ρ5 e−1/ θ + ρ1 + γ ) ≡ f1 (x , θ ) dτ

⎛ ρx dθ = xρ5 e−1/ θ + ⎜⎜ −α1/ θ dτ + ⎝ ρ2 e

This regression analysis contains a number of flaws; the data display obvious nonlinearity, as shown in Figure 5, the estimate

(7a)

⎞ ⎟⎟ρ e−α / θ 2 γ⎠

− (β + γ )(θ − θa) ≡ f2 (x , θ )

(7b)

It can be shown from eq 7a that the equilibrium value for x is xe =

γx0 ρ5 e

−1/ θ

+ ρ1 + γ

(8)

Substituting this value into eq 7b gives a transcendental equation for the ambient temperature, θa in terms of the equilibrium temperature, θe, of the form θa = θe −

1 β+

⎛ ⎛ ρ1x ⎜x ρ e−1/ θe + ⎜ e 5 ⎜ ⎜ −α / θe γ⎝ + ⎝ ρ2 e

⎞ ⎞ ⎟⎟ρ e−α / θe⎟ 2 ⎟ γ⎠ ⎠ (9)

This equation provides a means of exploring the relationship between ambient temperature and equilibrium temperature. Figure 6 shows a plot of nondimensionalized equilibrium temperature, θe, against nondimensionalized ambient temperature, θa.

Figure 5. Experimental rate measurements plotted against the transformed variable e(−E5/RT) with regression model shown.

for k1 is not significantly different from zero and the error estimates associated with the rate constants are large. A number of other regressions were tried, but the results from the regressions using a forced estimate of zero for k1, an altered activation energy and a quadratic term were unsatisfactory and difficult to justify. For these reasons the estimates presented in Table 3 will be used. b. A Planar Approximation. Two-dimensional (or planar) systems are substantially simpler to analyze than higherdimensional systems. Plotting the variables against each other gives a phase-plane, rather than some higher-dimensional phase-space. Visualizing solution trajectories is much simpler on a plane, as is searching for limit cycles. The Poincaré and Bendixson criteria provide methods to refute the existence of limit cycles for regions of the parameter space In the full system of eqs 3a−3e, it can be seen that none of x, w, and θ depend on y or z. Thus, without any loss of accuracy or generalizability the equations for y and z may be dropped from the model and solved as forced differential equations. That is true, once x, w, and θ have been solved, for they may be considered as forcing functions when solving for y and z. This leaves a three-dimensional system. Analysis of the threedimensional system showed that the conversion of w into z occurs on a much faster time scale than the other reactions (not shown). This means that the concentration of w remains

Figure 6. Equilibrium temperature, θe, plotted against ambient temperature, θa. Otherwise known as a bifurcation diagram for the planar system, in which the ambient temperature, θa, has been used as the bifurcation parameter. Thick line segments represent stable equilibria and thin segments unstable equilibria.

Figure 6 shows a number of interesting features: four saddlenode bifurcations (the locations at which the curve is vertical), two sections of unstable equilibria (the thin line segments) and three of stable equilibria (the thick line segments). Examining the phase-plane of this two-dimensional system near one of the suspected saddle-node bifurcations provides 545



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Bendixon’s negative criterion and Dulac’s theorem each provide a means for excluding the presence of periodic solutions within a bounded domain (see Guckenheimer and Holmes,24 p 44). For Bendixon’s negative criterion the partial derivatives of interest are as follows:

further evidence to support this conclusion. A saddle-node bifurcation is suspected for θa ≈ 0.009. A portion of the x−θ phase-plane for an ambient temperature just above this critical value is shown in Figure 7, and just below in Figure 8. The

∂f1 /∂x = −(ρ5 e−1/ θ + ρ1 + γ )

(10a)

∂f2 /∂θ = xρ5 e−1/ θ /θ 2 + (ρ1ρ2 γαx e−α / θ ) /(θ 2(γ + ρ2 e−α / θ )2 ) − (β + γ )

(10b)

If the sum of these two terms does not change sign, then limit cycles (oscillatory behavior) are not possible. As shown in Figure 9, plotting the sum of these two partial derivatives on

Figure 7. Phase-plane portrait for the planar system at θa ≈ 0.0096, with separatrices and two solution trajectories shown.

Figure 9. Graphical representation of Bendixon’s negative criterion. The surface is color coded; blue indicates negative values and red positive.

the x−θ phase-plane shows that limit cycles are not possible for temperatures below θ ≈ 0.025 (≡263 K). This temperature is where the direct pathway is expected to begin out-competing the dissociative pathway. Dulac’s Theorem states that if there exists a function B(x,θ), such that

Figure 8. Phase-plane portrait for the planar system at θa ≈ 0.0091, with separatrices and three solution trajectories shown.

∂ ∂ (Bf ) + (Bf ) = S(x , θ ) ∂x 1 ∂θ 2

important feature to notice is the presence of three equilibria (the points at which the dotted lines intersect are equilibria) in Figure 7 and only one equilibrium in Figure 8. The effect of the dotted lines in Figure 7 is to separate the phase-plane. In combustion theory these lines are called ignition or explosion limits12,23 and mark out the boundary in the phase-plane between areas which undergo rapid selfheating and those which cannot. In Figure 7, two solution trajectories are plotted, one from either side of the separatrix. The difference in initial conditions for the two solutions is very small, yet after some time they diverge and move toward different steady-states. Below the critical ambient temperature all solutions converge to the single stable node, as shown in Figure 8. It can also be seen that the convergence in x is much faster than the convergence in θ. These predictions provide a method for testing the validity of the model and possibly the underlying computational chemistry.

(11)

does not change sign in a simply connected domain, then periodic solutions are not possible (see Strogatz,25 p 202). By carefully selecting the function B, it is possible to eliminate the possibility of limit cycles on sections of the phase-plane. Theorem: There cannot be limit cycles if α 1/4 Proof: Choose B(x,θ) = (1/x)e1/θ and then S < 0 always, when these two conditions are satisfied. c. The Full Model. The bifurcation diagram for the threevariable model is identical to the one shown in Figure 6. This occurs because the two-dimensional model was obtained by applying a quasi-equilibrium assumption to the full model. 546



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or limit cycles. One of the eigenvalues of the three-dimensional Jacobian, shown below, is always negative. The second and third change sign, and hence determine the stability of the system. The Jacobian for the three-variable system has the following form:

The three-variable model appears to possess the same bifurcations, and stability properties as the simpler two-variable model. A more sophisticated analysis with XPPAUT,26 MATLAB,27 and BIFOR228 did not reveal any Hopf bifurcations ⎛ ⎜− (ρ e−1/ θe + ρ + γ ) 0 5 1 ⎜ ⎜ ⎜ ⎜ ρ1 − (ρ2 e−α / θe + γ ) ⎜ ⎜ ⎜ ρ5 e−1/ θe ρ2 e−α / θe ⎜⎜ ⎝

and was obtained by taking the derivatives of the differential equations in the full model and evaluating them at the equilibria. One of the three eigenvalues becomes positive between each pair of saddle-node bifurcations, as can be seen in Figures 10 and 11.

⎞ ⎟ ⎟ θe 2 ⎟ ⎟ − weρ2 α e−α / θe ⎟ θe 2 ⎟ ⎟ ⎟ weρ2 α e−α / θe ⎞ ⎟ − β − γ⎟ + ⎟ 2 ⎟ θe ⎠ ⎠ − xeρ5 e−1/ θ

⎛ x ρ e−1/ θe ⎜ e5 ⎜ θ2 ⎝ e

In higher dimensional systems, there is no equivalent to Bendixon’s negative criterion, which makes testing for limit cycles more difficult. Brute force simulations that followed the solution trajectory from initial conditions spanning the phase space did not show any oscillatory behavior. This indicates that limit cycles are extremely unlikely to exist for the range of parameter values tested. d. Modeling the Chloride Dissociation as a Thermodynamic Equilibrium. The model potential energy surface shown in Figure 4 assumes that the dissociation of the chloride is an isothermal process. The potential energy surface from Ariarfard et al.19 shows that the dissociation is not an isothermal process. Another reason for testing whether the dissociation can be modeled as an equilibrium is the inclusion of a fast thermo-dynamically controlled equilibrium in the catalytic cycle proposed by Powers and Ritter.16 This modifies the model reaction scheme to the new scheme shown in Figure 12.

Figure 10. Second eigenvalue plotted against equilibrium temperature. Figure 12. Model reaction scheme with the chloride dissociation modeled as a thermodynamic equilibrium.

The two dissociation steps are no longer modeled as isothermal processes. The new model potential energy surface is shown in Figure 13. Assuming the species are in thermodynamic equilibrium allows the van’t Hoff equation29 to be used to estimate the fraction in each form. This leads to the following multivariate regression equation: °

kexp = Z5e−E5 / RT + Z 2

(e−Δr H / RT e−E2 / RT ) [Cl−]

(≡Z5ξ5 + Z 2ξ2)

(12) 18

The experimental data provided by Powers et al. showed that the reaction rate was not sensitive to the concentration of chloride over the range of concentrations tested. Therefore, when evaluating eq 12 the concentration of chloride will be assumed to be 10 mM, a value in the middle of the range tested by Powers et al.18 The results of this regression are summarized

Figure 11. Third eigenvalue plotted against equilibrium temperature. 547



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Figure 13. Potential energy surface for modeling the chloride dissociation as a thermodynamic equilibrium.

potential energy surface used to derive the model and the surface that must be invoked to explain the results indicates that the prediction should not be considered valid.

Table 4. Regression Summary for the Intermediate Rate Constants When the Dissociation Is Modeled as a Thermodynamic Equilibrium estimate slope (Z5) slope (Z2)

8.322 × 10 −7.647 × 1020 12

standard error

t value

p value

1.695 × 10 3.99 × 1019

49.10 −19.16

1.86 × 10−5 0.000 31

11

IV. DISCUSSION AND CONCLUSIONS This work has shown that the experimental data of Powers et al.18 can be fitted to a set of coupled ordinary differential equations. The original system of equations contains five ordinary differential equations, of which two decouple. That is, these two equations do not affect the other three. Hence, they may be disregarded without any loss of accuracy or generalizability. When the remaining three equations were analyzed, it became apparent that one of the concentrations, w, varied on a much shorter time scale than the others. This allowed for further simplification by assuming that this concentration stayed very close to its dynamic equilibrium value with respect to x and θ, yielding a two-dimensional system of equations. As well as fitting experimental data the analysis provided a range of testable predictions. These predictions include the conditions under which each pathway is predicted to be favored. At low temperatures the dissociative pathway is predicted to dominate, but as the temperature increases and the activation barrier for the direct pathway is overcome the reaction is predicted to favor the direct pathway. This model also predicts that there are ambient temperatures at which the reaction possesses multiple stable and unstable equilibrium temperatures. The ambient and equilibrium temperatures at which these multiple steady states occur are dependent on the activation energy, inflow concentration and mean residence time. Therefore, experimentally locating these equilibria would provide an excellent method for testing the validity of the model. Locating these multiple steady states would provide substantial evidence to support the model, and the computational chemistry upon which it is based. However, a failure of the model does not necessarily indicate that the computational work is invalid; the model may include too many simplifications in its derivation. The bifurcation diagrams presented also show a number of locations at which substantial increases in temperature are expected; these provide an additional means of testing the predictive validity of the model. Attempts to include a fast equilibrium, as suggested by Powers and Ritter16 resulted in unrealistic predictions for the reaction rate and required two potential energy surfaces: one to derive the model, and a completely different one to justify the result. This lack of consistency provides strong evidence to suggest that there is not a fast, thermodynamically controlled equilibrium.

in Table 4. The regression model fits the data very well: the proportion of variance explained is extremely high, R2 = 0.9998; and the model fits significantly better than the null model, F(1,3)= 1.034 × 104, p = 1.746 × 10−6. Unfortunately, these intermediate rate estimates lead to unrealistic behavior. The estimate for Z2 is negative, indicating that z is being formed through the direct pathway and being converted back into starting material through the dissociative pathway. This leads to the predicted reaction rate shown in Figure 14, in which the predicted rate goes through zero, and

Figure 14. Predicted reaction rate when the dissociation has been modeled as a thermodynamic equilibrium. The open circles represent the measurements taken by Powers et al.18

remains negative for all higher temperatures. For this to occur the potential energy surface must have two very specific features: first, the activation energy barrier must be smaller going backward from z to w than it is in the forward direction; second, there must be a way to form z without forming w as an intermediate. Since the proposed mechanism has two pathways the second of these conditions is satisfied. However, the first condition does not agree with the model potential energy surface shown in Figure 13. The inconsistency between the

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. 548



Article

Present Address §

Magdalen College, Oxford, OX1 4 AU, U.K.

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

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REFERENCES

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Chemical Kinetics of a Bipalladium Complex - The Journal of Physical

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