3520
J. Phys. Chem. 1996, 100, 3520-3526
Lifetime of Trimethylene Calculated by Variational Unimolecular Rate Theory Charles Doubleday, Jr. Department of Chemistry, Columbia UniVersity, New York, New York 10027 ReceiVed: September 26, 1995; In Final Form: NoVember 16, 1995X
The calculation was undertaken to establish an RRKM prediction of the lifetime of trimethylene for comparison to the experimental result of Zewail and co-workers, 120 ( 20 fs. Nine stationary points on the C3H6 potential energy surface were located using 2,2-CASSCF or 4,4-CASSCF and a polarized valence triple-zeta basis set consisting of [4s3p2d] on C and [3s1p] on H. These included two minima and four saddle points associated with trimethylene previously reported by others. Two saddle points associated with propene formation were located, involving direct 1,2-H transfer and 1,3-H transfer to ethylcarbene, which was also optimized. Three intrinsic reaction coordinate paths were computed for cyclization to cyclopropane using 2,2-CASSCF/631G*, and microcanonical variational transition states were located with RRKM theory at seven values of the total energy. Under experimental conditions the lifetime is computed to be 118 fs. The match to experiment suggests that variational RRKM theory may be useful for biradicals even on this short time scale.
Recently Pedersen, Herek, and Zewail reported the first direct measurements of the lifetimes of trimethylene and tetramethylene biradicals.1 Two-photon excitation at 310 nm of cyclobutanone in a molecular beam followed by loss of CO was reported to yield trimethylene with a lifetime of 120 ( 20 fs.1 Our purpose is to find out how close transition state theory can come to computing the lifetime of trimethylene. The region of 10-13 s is likely to be near the edge of applicability of RRKM theory since energy randomization may not be complete. However, we were encouraged by Zewail’s observation that the decay fits reasonably well to a single exponential. In any case it is important to establish the RRKM prediction for this archetypal biradical. Previously, we reported canonical2a,c,d and microcanonical2b variational transition state theory calculations for tetramethylene and canonical variational results for trimethylene.3 Here we report the application of microcanonical variational unimolecular rate theory to the lifetime of trimethylene as a function of total energy. From the beginning, calculations of trimethylene predicted the potential energy surface (PES) to be nearly flat with respect to internal CH2 rotation. Notable early achievements were Hoffmann’s extended Hu¨ckel calculation4a and Salem’s location of the saddle point for single epimerization.4b The flatness of the PES was an important theme in the recent work of Getty, Davidson, and Borden5 (GDB) and Baldwin, Yamaguchi, and Schaefer 6 (BYS), who have clarified some subtle features of one- and two-center epimerizations, discussed below. GDB5 reported eight stationary points including two minima and four saddle points, and BYS6 reported many of these stationary points at a higher level of theory. These papers contain a complete list of references to the trimethylene literature spanning 25 years. Computational Methods Geometry optimizations and reaction paths were computed with a complete active space multiconfiguration self-consistentfield (CASSCF) wave function7 with either two electrons in two orbitals (2,2-CAS, three singlet configurations) or four electrons in four orbitals (4,4-CAS, 20 singlet configurations) using the MESA8 programs. Reaction paths for cyclization were computed with the 6-31G* basis set.9 Nine stationary points X
Abstract published in AdVance ACS Abstracts, January 15, 1996.
0022-3654/96/20100-3520$12.00/0
SCHEME 1
were optimized with a larger basis set consisting of [4s3p2d] on carbon obtained from Dunning’s [4s3p2d1f] cc-pVTZ set10 by removing the f functions and [3s1p] on hydrogen obtained from the three contracted s functions of cc-pVTZ plus a single set of p functions with ζ ) 0.75. This basis set contains 111 contracted functions (sets of six Cartesian d functions were used) and is designated VTZ(2d,p). Stationary points were characterized by analytical second derivatives (all positive eigenvalues for minima, one negative eigenvalue for saddle points), and all Cartesian gradient components were less than 10-5 au. At stationary points and at selected points on the reaction paths, the energies were corrected for dynamic electron correlation by multireference CI with all single and double excitations from the CASSCF reference (CISD), excluding excitations involving the lowest three occupied or highest three virtual orbitals, which were kept doubly occupied or unoccupied, respectively. With the VTZ(2d,p) basis set the number of singlet configurations in C1 symmetry was 1 031 945 for 2,2-CAS-CISD and 5 761 732 for 4,4-CAS-CISD. We define the lifetime of trimethylene as τ ) (kcyc + kpr)-1 (Scheme 1). Rate constants for cyclization and propene formation were computed with variational unimolecular rate theory,11 a method we recently applied to tetramethylene.2b The procedure is similar to canonical variational transition state theory,12 except that the rate constants are calculated microcanonically by RRKM13 theory. It requires a reaction path, for which we used the intrinsic reaction coordinate14 (IRC), the steepest descent path from the saddle point in mass-weighted coordinates. Progress along the IRC is characterized by the arc length s in units of bohr‚amu1/2, with s ) 0 at the saddle point, reactants at s < 0, and products at s > 0. We computed IRC paths for cyclization to cyclopropane using 2,2-CAS/631G* with the Page-McIver algorithm15 in the corrected local quadratic approximation.16 In this method analytical second derivatives at each point are combined with finite-difference © 1996 American Chemical Society
Lifetime of Trimethylene third derivatives along the path direction to predict the next step. After projecting out the gradient, rotations, and translations, each mass-weighted second-derivative matrix was diagonalized to yield 3N - 7 ) 20 eigenvalues. At the points on the IRC paths at which state sums were evaluated, the energies were recalculated with 2,2-CAS-CISD/6-31G*, and CH2 rotational potentials were also computed, described below. In comparing one path to another, their relative energies were adjusted according to the relative CISD/VTZ(2d,p) energies of their saddle points. By computing state sums17 at several points along the IRC, we obtained the RRKM rate constant k(E,J,s) as a function of arc length s along the IRC path for a given total energy E and total angular momentum J. The variational transition state was located by polynomial interpolation at the value s ) s‡ for which k(E,J,s) was a minimum The RRKM rate constant17 is given by k(E,J,s‡) ) h-1N(E,J,s‡)/F(E,J), where N(E,J,s‡) is the sum of states at s ) s‡, F(E,J) is the density of states of the trimethylene reactant, and h is Planck’s constant. Since the biradicals in the Zewail experiment were generated in a supersonic expansion1, they are expected to be rotationally cold, and we put J ) 0 in our calculations. To compute the sum of states, we modified the RRKM program of Zhu and Hase18 to include a Stein-Rabinovitch19 direct count of the energy levels for internal rotation as well as the standard Beyer-Swinehart20 count of the remaining (harmonic) modes. The internal rotation energy levels were obtained as described for tetramethylene:2a,b computation of the rotational potential at several points along the IRC path and fitting to a six-term Fourier series, calculation of the reduced moments of inertia,21 and calculation of the energy levels22 in a basis of 150 sines and 151 cosines. The rotational potential functions were obtained by constrained optimizations at several fixed values of φ, the dihedral angle between the CCC plane, and the plane perpendicular to and bisecting the rotating CH2. In these optimizations the CCC angle and both torsion angles of the other CH2 were also fixed. All other internal coordinates were optimized at each fixed value of φ, including the pyramidalization of the rotating CH2. The potential functions have a period of π with a dominant cos 2φ Fourier term. Calculation of the reduced moment of inertia for CH2 rotation was easier than for tetramethylene2a,b because the trimethylene rotors are attached to a rigid frame instead of a flexible frame. The original Pitzer-Gwinn formulas21a were adequate. In computing sums and densities of states, the harmonic levels associated with CH2 rotation were replaced by the anharmonic levels. For paths 1 and 2 (see below), the harmonic rotation mode involves both CH2 groups, and this was replaced by anharmonic levels for single CH2 rotation. Though not rigorously correct, the change in the sum of states for single rotation along the IRC is expected to be a reasonable parametrization of the changes using the full dimensionality. Results and Discussion Electronic Structure. In preliminary work using 2,2-CAS/ 6-31G*, we located the two trimethylene minima and four saddle points that GDB5 had reported using the identical GVB/6-31G* wave function. Our energies and geometries matched theirs exactly. We then reoptimized these six stationary points with 2,2-CAS/VTZ(2d,p) to give structures 1, 1‡, 2, 2‡, 3, and 4 shown in Figures 1 and 2. 1 and 1‡ are the local minimum and saddle point associated with conrotatory double CH2 rotation (C2 symmetry). 2 and 2‡ are the minimum and saddle point for disrotatory rotation (Cs symmetry). 3 is the saddle point for interconversion of 1 with 2. 4 is the saddle point for cistrans isomerization of cyclopropane via a single twist. The
J. Phys. Chem., Vol. 100, No. 9, 1996 3521
Figure 1. Conrotatory (1, 1‡) and disrotatory (2, 2‡) double-twist stationary points optimized with 2,2-CAS/VTZ(2d,p). Structures shown are the minima 1 and 2.
Figure 2. Point 3: saddle point for 1-2 interconversion. Point 4: saddle point for cis-trans isomerization of cyclopropane via a single twist. Both were optimized with 2,2-CAS/VTZ(2d,p).
geometries strongly resemble the 6-31G* results,5 and 4 is close to Salem’s STO-3G saddle point for single epimerization.4b Table 1 lists energies, harmonic zero point vibrational energies (ZPE), and the imaginary frequencies of saddle points, for all stationary points located with 2,2-CAS/VTZ(2d,p). Figure 3 shows the trimethylene stationary points on a 2-dimensional plot as a function of the torsion angles, following the format of the plot originally given by GDB.5 The con and dis paths are the diagonals, and cis-trans isomerization follows close to the axes, via 4. The con and dis paths intersect in the C2V edge-to-edge or (0,0) structure,4 which GDB5 and BYS6 report to have two imaginary eigenvalues of the force constant matrix. All important features of the trimethylene PES calculated by GDB5 and nearly all those reported by BYS6 are maintained on the 2,2-CAS/VTZ(2d,p) PES. Our structures 1, 1‡, 2, 2‡, 3,
3522 J. Phys. Chem., Vol. 100, No. 9, 1996
Doubleday
TABLE 1. Energies of Stationary Points Obtained with the VTZ(2d,p) Basis Set structure
CAS active space
CAS energy (au)
∆ECASa
ZPEb
1 1‡ 2 2‡ 3 4 7 2 5 6
2,2 2,2 2,2 2,2 2,2 2,2 2,2 4,4 4,4 4,4
-117.028 264 -117.028 258 -117.027 652 -117.026 723 -117.027 540 -117.025 868 -117.020 243 -117.043 606 -117.018 662 -117.012 876
0 0.004 0.38 0.97 0.46 1.50 5.03 0.38f 16.03 19.67
48.62 48.37 48.56 48.42 48.10 48.35 49.93 48.07 47.09 50.33
ν‡ c 124i 310i 163i 309i 1562i 993i
CISD energy (au)
∆ECId
∆ECI + ZPEe
-117.446 177 -117.446 386 -117.445 408 -117.444 220 -117.445 739 -117.443 209 -117.436 226 -117.450 596 -117.438 716 -117.435 159
0 -0.13 0.48 1.23 0.28 1.86 6.25 0.48f 7.94 10.17
0 -0.38 0.42 1.03 -0.24 1.59 7.56 0.42f 6.91 12.37
a Relative CAS energy in kcal/mol. b Harmonic zero-point vibrational energy in kcal/mol from CAS analytical second derivatives. c Imaginary frequency (cm-1) of transition vector. d Relative CISD energy in kcal/mol. e Relative CISD energy including CAS ZPE correction, kcal/mol. f Defined to have the same ∆E relative to 1 as 2,2-CAS in order to place them on the same scale. Other 4,4-CAS entries in this column are relative to this defined value for 2.
Figure 3. Trimethylene stationary points as a function of dihedral angles H1C1C2C3 and H5C3C2C1. Cyclopropanes (triangles in the corners) are shown deuterated to clarify stereomutation pathways. Conrotatory and disrotatory double epimerization (paths 1 and 2) and single epimerization (path 3) are shown. Only one path 3 is shown to minimize clutter.
and 4 correspond to GDB’s structures 6, 8, 5, 7, 9, and 10, respectively.5 Our 2, 2‡, and 3 correspond to BYS’s Cs(int), Cs(ts), and C1(ts).6 The shallow minima 1 and 2 disappear with inclusion of ZPE or CISD. 1 is barely distinguishable from 1‡ even at the CAS level (maximum gradient components 75 kcal, path 3 also has a bottleneck in the region s > +2. It is only at high energies that entropic restrictions begin to affect the rate constants.2 Under thermolytic conditions35 the transition states are probably well represented by saddle points on the PES. With three paths leading from the biradical region of the PES down to cyclopropane, the question is how to define the overall cyclization rate constant kcyc. It is formally correct to put kcyc ) kcyc1 + kcyc2 and ignore kcyc3, because path 3 involves neither 1 nor 2. However, the flat PES implies that path 3 is energetically as accessible as the other paths and may contribute to kcyc. This would cause a problem because there is no unique way of including path 3 in the computation. However, the following argument suggests that path 3 has no effect on kcyc, and the formal definition is correct. We pointed out above that path 3 approaches path 1 so that, for example, the two transition states at s ≈ 2.3 are very close structurally. The energy of s ) 2.3 on path 3 is 2.4 kcal/mol higher than that of s ) 2.4 on path 1 with either CAS or CISD, an indication that path 1 is better described as the minimum-energy path. We investigated the energy ridge separating these structures by computing a fivestep path of linear synchronous transit in the four CH2 dihedral angles, optimizing all other internal coordinates at each point except the CCC angle, which was identical in the two structures and was held constant. The largest of the dihedral step sizes was 4.4°. The energy decreased by 0.7 kcal/mol on the first step from s ) 2.3 of path 3 and continued monotonically down to s ) 2.4 of path 1. Evidently the energy ridge separating paths 1 and 3 is so small that even a 4.4° change steps over it. At this point in the cyclization process, path 3 is best described as a weakly defined shelf 2.4 kcal up on the side of the local reaction path valley belonging to path 1.36 Therefore, kcyc3 is ignored and kcyc ) kcyc1 + kcyc2. It is interesting to compare tetramethylene,2a,b which also has cyclization paths with C2 and C1 symmetry analogous to paths 1 and 3. There the situation is clearer because the two paths merge before the variational transition state, and no additional calculations have to be done to verify the proper transition state. Table 3 shows the rate constants for propene formation and cyclization via paths 1 and 2 for seven values of the total energy
3526 J. Phys. Chem., Vol. 100, No. 9, 1996 E relative to 1 and the lifetimes τ ) (kpr + kcyc1 + kcyc2)-1. For kpr, we computed a reaction path for propene formation29 and found that the transition state was negligibly different from 5, which we used in computing kpr at all values of E. The rate constants for ethylcarbene formation, computed using harmonic frequencies of 6, are 2 orders of magnitude smaller than kpr (k ) 2.0 × 1010 s-1 at E ) 99 kcal/mol) and are neglected. The value of τ goes through a maximum at intermediate energies because the energy dependence of kcyc1 is opposite to that of kcyc2 and kpr. Cyclization via path 1 encounters no barrier, only an entropic restriction at s ≈ 2.3, and exhibits the inverse energy dependence typical of entropy-controlled reactions.2,37 At high energies τ is dominated by kcyc2 and kpr, the reactions with barriers. At low energy the entropic restriction on kcyc1 fades, kcyc1 becomes dominant, and τ decreases. The energy dependence of τ is predicted to be small at high energies, similar to tetramethylene.2b However, the experimental lifetime of tetramethylene1 was reported to decrease by a factor of 2.5 from E ) 77 to E ) 100. This appears to be a feature for which RRKM is inadequate. In Table 3 the agreement with experiment at E ) 99 is probably fortuitous, and in any case τ has a wide margin of uncertainty associated with the flatness of the PES. For example, a referee asked how τ would change if a more accurate calculation found no local minimum for trimethylene. In that case the RRKM prediction is not clear because the existence and location of the trimethylene reactant is debatable (perhaps as a local maximum in the sum of states), but certainly τ would be expected to be much shorter. The computed lifetime would also change if trimethylene had only a single local minimum instead of two minima with an effective density of states33 Feff ∝ F1/σ1 + F2/σ2. Since F1 ≈ F2 in the present case, a single minimum with σ ) 1 or 2 would have a lifetime 33% greater or smaller, respectively, than those of Table 3. Subject to all these uncertainties, the agreement with experiment nevertheless suggests that variational RRKM theory may be useful for biradicals on this short time scale. Acknowledgment. We thank the National Science Foundation for support of this research and the Pittsburgh Supercomputing Center for a grant of time on the Cray C90. We also thank Professors A. H. Zewail and W. T. Borden for stimulating and helpful discussions and Professor W. L. Hase for his RRKM program and a preprint of ref 33. Supporting Information Available: Internal coordinates and vibrational frequencies of stationary points, kcyc(E,s), along paths 1, 2, and 3 (9 pages). Ordering information is given on any current masthead page. References and Notes (1) Pedersen, S.; Herek, J. L.; Zewail, A. H. Science 1994, 266, 1359. (2) (a) Doubleday, C. J. Am. Chem. Soc. 1993 115, 11968. (b) Doubleday, C. Chem. Phys. Lett. 1995, 233, 509. (c) Doubleday, C.; Camp, R. N.; King, H.; McIver, J., Jr.; Mullally, D.; Page, M. J. Am. Chem. Soc. 1984, 106, 447. (d) Doubleday, C.; Page, M.; McIver, J. W., Jr. J. Mol. Struct. (THEOCHEM) 1988, 163, 331. (3) Doubleday, C.; McIver, J. W., Jr.; Page, M. J. Phys. Chem. 1988, 92, 4367.
Doubleday (4) (a) Hoffmann, R. J. Am. Chem. Soc. 1968, 90, 1475. (b) Horsley, J. A.; Jean, Y.; Moser, C.; Salem, L.; Stevens, R. M.; Wright, J. S. Ibid. 1972, 94, 279. (5) Getty, S. J.; Davidson, E. R.; Borden, W. T. J. Am. Chem. Soc. 1992, 114, 2085. (6) (a) Baldwin, J. E.; Yamaguchi, Y.; Schaefer, H. F. III J. Phys. Chem. 1994, 98, 7513. (b) Yamaguchi, Y.; Schaefer, H. F. III; Baldwin, J. E. Chem. Phys. Lett. 1991, 185, 143. (7) (a) Roos, B.; Taylor, P.; Siegbahn, P. Chem. Phys. 1980, 48, 157. (b) Ruedenberg, K.; Schmidt, M. W.; Gilbert, M. M.; Elbert, S. T. Chem. Phys. 1982, 71, 41. (c) Lengsfield, B., III J. Chem. Phys. 1982, 77, 4073. (8) Molecular Electronic Structure Applications, written by P. Saxe, B. Lengsfield, III, R. Martin, and M. Page. (9) Hariharan, P. C.; Pople, J. A. Theor. Chim. Acta 1973, 28, 212. (10) Dunning, T. H. J. Chem. Phys. 1989, 90, 1007. (11) Hase, W. L. Acc. Chem. Res. 1983, 16, 258. (12) (a) Truhlar, D. G.; Garrett, B. C. Acc. Chem. Res. 1980, 13, 440. (b) Truhlar, D. G.; Isaacson, A.; Garrett, B. C. In Theory of Chemical Reaction Dynamics; Baer, M., Ed.; CRC Press: Boca Raton, FL, 1985; Vol. IV, p 65. (13) Marcus, R. A. J. Chem. Phys. 1952, 20, 359. (14) Fukui, K. J. Chem. Phys. 1970, 74, 4161. (15) Page, M.; McIver, J. W., Jr. J. Chem. Phys. 1988, 88, 922. (16) Page, M.; Doubleday, C., Jr.; McIver, J. W., Jr. J. Chem. Phys. 1990, 93, 5634. (17) Robinson, P. J.; Holbrook, K. A. Unimolecular Reactions; WileyInterscience: London, 1972. (18) Zhu, L.; Hase, W. L. RRKM, submitted to QCPE (modification of QCPE 234 by Hase and Bunker). (19) Stein, S. E.; Rabinovitch, B. S. J. Chem. Phys. 1973, 58, 2438. (20) Beyer, T.; Swinehart, D. F. Commun. ACM 1973, 16, 379. (21) (a) Pitzer, K. S.; Gwinn, W. D. J. Chem. Phys. 1942, 10, 428. (b) Pitzer, K. J. Chem. Phys. 1945, 14, 239. (22) Lewis, J.; Mallory, T., Jr.; Chao, T.; Laane, J. J. Mol. Struct. 1972, 12, 427. (23) Doering, W. von E.; Sachdev, K. J. Am. Chem. Soc. 1974, 96, 1168. (24) Marcus, R. A. J. Phys. Chem. 1991, 95, 8236. (25) Yamaguchi, Y.; Osamura, Y.; Schaefer, H. F. III J. Am. Chem. Soc. 1983, 105, 7506. (26) (a) Huzinaga, S. J. Chem. Phys. 1965, 42, 1293. (b) Dunning, T., Jr. J. Chem. Phys. 1970, 53, 2823. (27) (a) Evanseck, J. D.; Houk, K. N. J. Phys. Chem. 1990, 94, 5518. (b) Ma, B.; Schaefer, H. F. III J. Am. Chem. Soc. 1994, 116, 3539. (c) Schaefer, H. F. III Acc. Chem. Res. 1979, 12, 288. (28) (a) Modarelli, D. A.; Platz, M. S. J. Am. Chem. Soc. 1991, 113, 8985; 1993, 115, 470. (b) Nickon, A. Acc. Chem. Res. 1993, 26, 84. (29) We computed IRC paths passing through 5 and 6 using 4,4-CAS and a basis set consisting of 6-31G* except for the single transferred H, for which we used [3s2p] obtained from the [3s2p1d] cc-pVTZ set for H with the d functions removed. (30) Waage, E. V.; Rabinovitch, B. S. J. Phys. Chem. 1972, 76, 1965. (31) (a) Purvis, G. D.; Bartlett, R. J. J. Chem. Phys. 1982, 76, 1910. (b) Scuseria, G. E.; Janssen, C. L.; Schaefer, H. F. III Ibid. 1989, 89, 7382. (32) The energy along path 2 looks very similar to path 1 (Figure 5) and is not shown. (33) Peslherbe, G. H.; Hase, W. L. J. Chem. Phys. 1994, 101, 8535. (34) Benson, S. W. Thermochemical Kinetics, 2nd ed.; Wiley: New York, 1976; p 277. (35) (a) Berson, J. A.; Pedersen, L. D.; Carpenter, B. K. J. Am. Chem. Soc. 1976, 98, 122. (b) Berson, J. A. Annu. ReV. Phys. Chem. 1977, 28, 111. (c) Cianciosi, S. J.; Ragunathan, N.; Freedman, T. B.; Nafie, L. A.; Dewis, D. K.; Glenar, D. A.; Baldwin, J. E. J. Am. Chem. Soc. 1991, 113, 1864. (d) Baldwin, J. E.; Cianciosi, S. J.; Glenar, D. A.; Hoffnam, G. J.; Wu, I.; Lewis, D. K. J. Am. Chem. Soc. 1992, 114, 9408. (36) On the s < 0 side of path 3, the energy of the transition state at s ) -1.8 (Figure 10) is 5.5 kcal/mol higher than s ) 2.4 on path 1 with 2,2-CAS (6.1 kcal with CISD). (37) Houk, K. N.; Rondan, N. G.; Mareda, J. Tetrahedron 1985, 41, 1555.
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