Choosing the Optimal Method and Basis Set. The method and basis set to be used for
the study of the dimerization of cyclopentadienone was done by comparing a high level computational method, G4 on a B3LYP/6-31G* optimized structe, against a series of low cost computational methods and basis sets. The results obtained from these calculations are summarized in Table 3.2, they show that M06/6-31+G** energetics are comparable to those obtained from G4.
Table 3.2 Computational methods and basis sets used to explored the energy surface for cyclopentadienone dimerization.
Method Basis Set Cope T.S Monomer Dimer
G4 / -8.0 -11.3 -35.8 B3LYP 6-31G* -6.3 -17.2 -32.1 B3PW91 6-31G* -9.3 -13.7 -37.1 6-31G** -9.1 -13.8 -36.7 6-31+G* -9.2 -14.6 -36.3 6-31+G** -9.0 -13.8 -36.7 M06 6-31G* -8.1 -9.5 -37.6 6-31G** -7.8 -9.6 -37.1 6-31+G* -8.0 -10.3 -36.5 6-31+G** -7.8 -10.4 -35.9 M062X 6-31G* -6.4 -11.2 -39.1
Four stationary points were located along the energy surface using M06/6-31+G** and their structures are shown in Figure 3.4. The located saddle points, b and c, show the symmetry of both structures in agreement with Caramella’s report29. Transition state
structures have only one imaginary frequency while the starting material and product have zero imaginary frequencies.
Figure 3.4 Stationary points located on the potential energy surface for cyclopentadienone dimerization in the gas phase using M06/6-31+G**. a. Optimized structure of the starting material. b. Saddle point for the cyclopentadienone dimerization. The structure illustrates the participation of two sets of [4+2] orbital interactions. c. Saddle point corresponding to the Cope re-arrangement for the interconversion between isotopomeric products. d. Optimized structure of the unsymmetrical cyclopentadienone dimer.
Dynamic Trajectories. Direct dynamic trajectories were initialized following an
analogous procedure to that employed on Chapter II. In order to over come the computational cost of running dynamic trajectories with a 13C in one of the
cyclopentadienone moieties we took advantage of our trick of starting dynamic trajectories with in silico, superheavy carbon atoms. Quasi-classical dynamic trajectories in the gas phase, on a M06/6-31G* energy surface were initialized from isotopomers of the cycloaddition transition state structure, containing a single 16C, 20C, 28C, 44C, 76C, or 140C(12+2N amu, N =2, 3,4, 5, 6, or 7). Each normal mode in the transition state structure was give its zero point energy (zpe), along with a Boltzman distribution of a sample at 78 or 25 °C, with a random phase. Because the lowest-energy real normal mode in the cycloaddition transition state structure, “mode 3” desymmetrizes the structure in a way that has a large effect on trajectory outcomes, an equal number of trajectories were given positive versus negative velocities in this mode. The transition vector was given a Boltzmann sampling of energy ‘forward’ from the col. The trajectories were integrated until either the product was formed (median time 80 fs) or the starting materials were reformed and the results are summarized in Table 3.3 and Table 3.4.
Table 3.3 Quasi-classical dynamic trajectories initialized from the cycloaddition transition state structure on a M06/6-31G* energy surface at 298 K.
Position /
mass Total Runs Total x Total x' Recrossed KIE, x/x'
a/a' 140 9016 3787 5229 252 0.72 ± 0.03 a/a' 76 6554 2801 3753 182 0.75 ± 0.04 a/a' 44 6493 2881 3612 193 0.80 ± 0.03 a/a' 28 8240 3819 4421 248 0.86 ± 0.03 a/a' 20 11215 5323 5892 289 0.90 ± 0.03 a/a' 16 19730 9654 10076 548 0.96 ± 0.03 extrapolated to 13C 0.984 b/b' 140 7459 3633 3826 151 0.95 ± 0.04 b/b' 76 7302 3625 3677 188 0.99 ± 0.04 b/b' 44 10137 5085 5052 257 1.01 ± 0.04 c/c' 140 8536 6070 2466 224 2.46 ± 0.05 c/c' 76 5161 3427 1734 115 1.98 ± 0.05 c/c' 44 5505 3381 2124 131 1.59 ± 0.03
Dynamic trajectories correctly predict the major product in each case, and the magnitude of the selectivity in the trajectories follows the same trend as in the experimental KIEs. As the mass of the “super heavy” carbon atoms is increased the selectivity of the reaction is also increased in all positions. Trajectories with a label on position b exhibit a small selectivity with a heavy carbon of mass 140 amu, but with 76C very small selectivity was observed. A minimal amount of recrossing is observed, however the Newtonian preference to place the lighter atoms at the newly formed sigma bond pre- dominates.
Table 3.4 Quasi-classical dynamic trajectories initialized from the cycloaddition transition state structure on a M06/6-31G* energy surface at 195 K.
Position /
mass Total Runs Total x Total x' Recrossed KIE, x/x'
a/a' 140 9278 3838 5440 294 0.71 ± 0.03 a/a' 76 6559 2794 3765 211 0.74 ± 0.04 a/a' 44 7433 3367 4066 219 0.83 ± 0.03 a/a' 28 8238 3853 4385 234 0.88 ± 0.03 a/a' 20 13085 6289 6796 355 0.93± 0.03 a/a' 16 21644 10562 11082 642 0.95 ± 0.02 extrapolated to 13C 0.985 b/b' 140 11556 5655 5901 328 0.96 ± 0.03 b/b' 76 6924 3423 3501 195 0.98 ± 0.04 c/c' 140 8628 6511 2117 242 3.08 ± 0.07 c/c' 76 6419 4449 1970 173 2.26 ± 0.05 c/c' 44 7074 4519 2555 202 1.77 ± 0.03
In order to evaluate whether dynamic trajectories accurately predict the experimental observations of a temperature independent KIEs, we initialized a second set of quasicalssical dynamic trajectories at 195 K. The computationally calculated KIEs at 195 K (Table 3.3) are statistically indistinguishable from those at 298 K (Table 3.4). Dynamic trajectories at both temperatures are in good agreement with experimental intramolecular KIEs for the dimerization of dicyclopentadienone.
The uncertainties in the trajectory ratios were calculated by setting up an Excel spreadsheet that would repeatedly simulate the complete set of trajectory runs with each choice of superheavy carbon. In the simulations, the outcome of each individual trajectory depended on a random number and on a weighting that corresponded to the ratio of outcomes actually observed in the set of trajectories. Because of the weighting,
the average outcome of an infinite number of Excel simulations would be equal to the outcome observed in the trajectories. Each individual simulation, however, departs from the observed outcome in a way that reflects the role of random chance on the results in a set of trajectories. From the results of 119 simulations, standard deviations was calculated: mass-140: 0.018; mass 76 0.022; mass 44 0.025; mass 28 0.018. The 95% confidence ranges would be twice these standard deviations.
Extrapolation of Trajectory Results to 13C. The process for extrapolation of the
trajectory results to 13C starts with the assumption that the additional isotope effect per additional mass unit decreases as the mass grows. For example, the effect of going from 13C to 14C would be larger than the effect of going from 140C to 141C. We further assumed that this decrease with mass is nonlinear with a decreasing slope, that is, that a plot (Figure 3.5) of the additional isotope effect per additional mass versus mass would have a greater slope at low masses (e.g., 13C to 14C) than at high masses (e.g. 140C to 141C). These simple assumptions exclude some extrapolation processes that would lead to physically unreasonable isotope effects at both high masses and at 13C.
Figure 3.5 13C extrapolated KIE for a/a’ at 195 K and 298 K from quasiclassical dynamic trajectories on Table 3.3 and Table 3.4. The corresponding equation that describes the trendline at each temperature are also shown.
Tables 3.3 and 3.4 and the associated graph show the data used in the extrapolation, the extrapolated results, and the complete linear regression analysis and statistics. The Nominal KIEs come from the Table 2.2 on the previous section. The “additional mass” is defined as the mass beyond 12 amu. The “isotope effect per mass” is defined as the geometric average. The extrapolation was then carried out by assuming a linear relationship between the log of the additional mass (the base is arbitrary but set conveniently here as 2) versus the isotope effect per mass. The extrapolation process used the Regression tool in Microsoft Excel 2011 for Mac,Version 14.3.9
y = 0.0019x + 0.9837 R² = 0.94015 y = 0.0017x + 0.9849 R² = 0.99566 0.982 0.984 0.986 0.988 0.990 0.992 0.994 0.996 0.998 1.000 0 1 2 3 4 5 6 7 8 KIE per added mass
log (base 2) of addiPonal mass