J. Phys. Chem. 1996, 100, 15083-15086
15083
Tetramethylene Optimized by MRCI and by CASSCF with a Multiply Polarized Basis Set Charles Doubleday, Jr. Department of Chemistry, Columbia UniVersity, New York, New York 10027 ReceiVed: April 8, 1996X
Six stationary points of tetramethylene were optimized with a 44CAS-MCSCF wave function and a basis set consisting of [4s3p2d1f] on C and [2s1p] on H. Energies were corrected by multireference CI with all single and double excitations (CISD). Three stationary points were also optimized with CISD from a 22CAS reference and a basis set consisting of [3s2p1d] on C and [2s] on H. The current results predict a much flatter potential energy surface than previous calculations. The results suggest that several stationary points previously located at lower levels of theory effectively coalesce into a broad plateau from which both cyclization and fragmentation are accessible along paths of monotonically decreasing energy. The dynamical significance of the plateau region is discussed. On the whole the results support the use of single-point CISD energy corrections at CASSCF optimized geometries of tetramethylene.
Small biradicals offer a rare opportunity to compare theoretical dynamics with experimental lifetimes1 of an important class of reactive intermediates on a very short time scale. In our recent studies of tetramethylene,2 we used canonical variational transition state theory3 and variational RRKM theory4 to examine the dynamics as a function of temperature and total energy. Our results were in basic agreement with experimental product distributions5 and lifetimes,1 but significant differences from experiment persist. This suggests that small biradicals provide a stringent test of dynamical theories and that much remains to be learned from these systems. For this reason we decided to reexamine several of the stationary points of tetramethylene at a more reliable level of theory than before, prior to an examination of the classical dynamics of tetramethylene. In this paper we report six stationary points optimized with a much larger basis set than the 6-31G6a and 6-31G*6b used previously, followed by multireference CI energy corrections. We also report three of these stationary points optimized with multireference CI using a basis set of quality similar to 6-31G*. The results show a much flatter potential energy surface (PES) than previously,2a with some significant changes in the optimized geometries. Computational Methods. Stationary points were optimized with a four-electron-four-orbital complete active space MCSCF wave function (44CAS, 20 singlet state configurations) and a basis set consisting of Dunning’s cc-pVTZ,7 [4s3p2d1f], on carbon and cc-pVDZ, [2s1p], on hydrogen, designated ccpVTZ,cc-pVDZ. With Cartesian d and f orbitals the number of contracted basis functions was 180. Single-point energy corrections were computed with multireference CI consisting of all single and double excitations from the 44CAS reference (CISD), designated 44CAS-CISD/cc-pVTZ,cc-pVDZ. In these calculations the lowest four orbitals were kept doubly occupied and the highest four virtual orbitals were kept unoccupied. The number of configurations was 7 676 222 in C2h symmetry and 15 275 814 in C2 symmetry. In addition, three stationary points were reoptimized using analytical first derivatives of a CISD wave function whose reference space was 22CAS (three configurations) and whose orbitals were defined by 44CAS. The basis set for this optimization was cc-pVDZ,7 [3s2p1d], on carbon and [2s] on hydrogen, obtained from cc-pVDZ by removing the p functions. This basis set contains the same X
Abstract published in AdVance ACS Abstracts, August 15, 1996.
S0022-3654(96)01043-X CCC: $12.00
number of contracted functions (76) as 6-31G*. The calculation is designated 22CAS-CISD/cc-pVDZ,DZ. The lowest four orbitals were kept doubly occupied, and the highest four orbitals were unoccupied; there were 633 966 configurations in C1 symmetry. MESA code was used throughout.8 Results and Discussion. The six stationary points optimized with 44CAS/cc-pVTZ,cc-pVDZ are (in our previous nomenclature2a) the gauche minimum GM, the trans minimum TM, the gauche fragmentation saddle point GF, the trans fragmentation saddle point TF, the cyclization saddle point G1 (with simultaneous CH2 twist), and the saddle point for gauche-trans interconversion GT. The structures are shown in Figures 1-3, with comparisons of important geometric parameters optimized with 44CAS/6-31G and 44CAS/6-31G*. Each figure also includes the energy relative to that of GM in kcal/mol.9 G1 was optimized in C1 symmetry; GM, GF, and GT in C2 symmetry; and TM and TF in C2h symmetry. Maximum Cartesian derivative components ranged from 7 × 10-6 to 5 × 10-5 au. Analytical second derivatives were not computed with this large basis set, but the geometries and relative energies are very close to the previous values which were characterized by analytical second derivatives.2a The energy of each stationary point except G1 was recalculated with 44CAS-CISD/ccpVTZ,cc-pVDZ. GM, G1, and TM were also optimized with 22CAS-CISD/cc-pVDZ,DZ in C2, C1, and C2h symmetry, respectively. The energies are shown in Table 1. Comparing the 44CAS calculations with the three basis sets 6-31G, 6-31G*, and cc-pVTZ,cc-pVDZ, one finds that neither the geometries nor relative energies are sensitive to the basis set. A minor exception is that 6-31G places GF and TF lower relative to GM than do the larger basis sets, but the geometries are practically unchanged among all basis sets. The only large change is that the 22CAS-CISD optimized structure of G1 (Figure 2, lower structure) is close to that of GM and significantly different from the structure optimized with 44CAS (Figure 2, upper structure). On the 22CAS-CISD optimized PES, a slight twist of one of the CH2 groups of GM is enough to push it over the 0.04 kcal G1 saddle point leading to cyclobutane. An expected feature of 22CAS-CISD optimization is that the CH bonds are longer than with 44CAS because much of the CH electron correlation is included. For example, in TM the C1H1 and C2H3 bond lengths are 1.078 and 1.086 Å, respectively, with 44CAS/cc-pVTZ,cc-pVDZ and 1.096 and 1.102 Å with 22CAS-CISD/cc-pVDZ,DZ.9 © 1996 American Chemical Society
15084 J. Phys. Chem., Vol. 100, No. 37, 1996
Figure 1. Structures of gauche and trans minima, GM and TM, and trans fragmentation saddle point TF. Tables list important structural parameters for geometries optimized with the four wavefunctions shown at upper left and also include energies (kcal/mol) relative to GM for the given wave function. If the table contains three rows, the first three wave functions are implied. Definitions of the five torsion angles are enclosed in the rectangle.
The last two columns of Table 1 compare the 44CAS-CISD/ cc-pVTZ,cc-pVDZ relative energies to the previously calculated 44CAS-CISD/6-31G* energies, all with inclusion of zero-point vibrational energy (ZPE) previously computed2a with 44CAS/ 6-31G*. The two columns are qualitatively similar, but there is an interesting quantitative change. cc-pVTZ,cc-pVDZ lowers both fragmentation barriers by 0.71-0.79 kcal/mol compared to the 6-31G* barriers. Application of the ZPE correction decreases the barriers further. At the 44CAS-CISD level of theory, it appears that if GM and TM are local minima at all, they must be extremely shallow. In addition, the GT barrier is slightly lower with the cc-pVTZ,cc-pVDZ basis set, and on the whole the biradical is much floppier than with 6-31G*. Bernardi et al.10 optimized GM, TM, GF, TF, and GT with multireference second-order perturbation theory, 44CAS-MP2. Their geometries are close to ours, but their barriers are much higher. For example, with 6-31G* their ZPE-corrected trans fragmentation barrier is +0.90 kcal/mol compared to -0.99 and -0.28 in the last two columns of Table 1. In our 22CAS-CISD optimizations we examined mainly the cyclization of GM via G1. We briefly examined fragmentation
Doubleday
Figure 2. Structures of the cyclization saddle point G1 obtained by 44CAS (upper) and by 22CAS-CISD (lower). Table shows important geometric parameters and relative energies (see Figure 1 for definitions).
Figure 3. Structures of saddle points for gauche fragmentation (GF) and gauche-trans interconversion (GT). Tables show important geometric parameters and relative energies (see Figure 1 for definitions).
via GF and TF but found that 22CAS-CISD gave much too high an energy. Using 22CAS-CISD/6-31G*//44CAS/6-31G*
Tetramethylene Optimized by MRCI
J. Phys. Chem., Vol. 100, No. 37, 1996 15085
TABLE 1: Energies of Stationary Points Optimized with 44CAS/cc-pVTZ,cc-pVDZ and 22CAS-CISD/cc-pVDZ,DZ geom GM GF G1 GT TM TF GMg G1g TMg
44CAS -(E + 156)/au
CISD -(E + 156)/aua
0.085 602 0.082 736 0.085 438 0.083 263 0.088 828 0.086 877
0.656 071 0.655 941
0.0 0.08
0.654 360 0.659 595 0.660 197 0.473 293 0.473 228 0.476 409
1.07 -2.21 -2.59 0.0 0.04 -1.96
∆ECISDb
ZPEc
∆ECISD+ZPEd
68.58 68.04 67.96 68.70 68.99 68.38 68.58 67.96 68.99
0.0 -0.46 -0.52f 1.19 -1.80 -2.79 0.0 -0.58 -1.55
6-31G* ∆ECISD+ZPEe 0.0 0.33 -0.34 1.40 -1.77 -2.05
a 44CAS-CISD/cc-pVTZ,cc-pVDZ//44CAS/cc-pVTZ,cc-pVDZ for all entries except the last two, which are 22CAS-CISD/cc-pVDZ,DZ. b CISD energy relative to GM in kcal/mol. c Harmonic zero-point vibrational energy in kcal/mol from 44CAS/6-31G*.2a d ZPE-corrected CISD energy relative to GM in kcal/mol. e ZPE-corrected 44CAS-CISD/6-31G*//44CAS/6-31G* energy relative to GM in kcal/mol.2a f Based on ZPE-corrected 44CAS, not CISD. g Optimized with 22CAS-CISD/cc-pVDZ,DZ.
TABLE 2: Eigenvalues (10-2 hartree/radian2) of the 5 × 5 Torsional Second-Derivative Matrices for GM and G1 Computed with 22CAS-CISD/cc-pVDZ,DZ GM 0.13 0.24 2.94 3.12 3.64
disrotatory CH2 torsion conrotatory CH2 torsion asymmetric CH2 wag symmetric CH2 wag CCCC torsion
G1 -0.23 0.27 1.40 2.62 4.30
C4H7H8 torsion C1H1H2 torsion C4H7H8 wag C1H1H2 wag CCCC torsion
(very similar to the cc-pVDZ,DZ basis set) to compute relative energies of GM vs GF and TM vs TF, we found fragmentation barriers of 4.4 and 3.4 kcal/mol for gauche and trans, respectively. These are much higher than corresponding 44CASCISD/6-31G* barriers (Table 1). In the CI it is important to choose a space of reference configurations, with respect to which single and double excitations are defined, that encompasses the nondynamic electron correlation. For fragmentation this meant optimizing with 44CAS-CISD, which required too much computer time. Therefore we confined our 22CAS-CISD optimization to gauche structures and to the trans minimum. Optimization with 22CAS-CISD/cc-pVDZ,DZ yielded a very shallow GM minimum, optimized in C2 symmetry, and a singletwist cyclization saddle point G1 only 0.04 kcal/mol higher with a geometry barely displaced from GM (Figure 2). Because of the high expense and extremely flat PES in this region, we stopped the G1 optimization with a maximum gradient component of 3 × 10-4 au. Therefore the geometry of G1 is approximate. For GM and G1 we computed 5 × 5 torsional second-derivative matrices by single-point finite difference in the space of the four terminal HCCC dihedral angles and the CCCC torsion. Diagonalization yielded positive eigenvalues for GM and one negative eigenvalue for G1 (Table 2). The transition vector of G1 was mainly the C4H7H8 torsion. For GM, the eigenvector belonging to the lowest eigenvalue was the symmetry-breaking disrotatory double CH2 twist leading toward G1, similar to the result obtained from the full force constant matrix using 44CAS/6-31G*.2a To verify that G1 leads to cyclobutane, we displaced G1 along the transition vector followed by full minimization. However, the PES is so flat in this region that the minimization was unacceptably slow. Therefore we computed a cyclization path from G1 (Figure 4) consisting of constrained minimizations in which φ7 was changed in small increments while all other variables were optimized through 0-4 optimization cycles. The path basically follows the IRC previously computed with 44CAS/6-31G,2a and for 11 of the points optimization was not necessary. Because of the method of construction, the path is not completely smooth, but the energy decreases from G1 and the final geometry is clearly poised for cyclization. We also briefly examined a cyclization path without CH2 twisting and
Figure 4. Cyclization path via G1 obtained by constrained minimizations with 22CAS-CISD (see text). Horizontal axis lists the calculated points; 0 indicates G1, leftmost point is GM. Upper: 22CAS-CISD energy relative to G1, with structures of G1 and the final point on the way to cyclization. H7, which defines φ7, is shaded. Lower: torsion angles θ, φ1, φ21, φ7, φ87 along the path.
found that the energy rose above G1. Thus the small Woodward-Hoffmann preference for 2s + 2a cyclization of tetramethylene, predicted by Dewar and Kirschner11 and confirmed by our 44CAS calculations,2a is retained on the optimized 22CAS-CISD surface. Optimization of TM with 22CAS-CISD yielded a minimum, but this is likely an artifact of the inadequate CI reference space discussed above. Even so, we felt that TM was worth including in order to compare the gauche and trans regions of the PES. The 22CAS-CISD optimizations reveal some of the strengths and weaknesses of using a single-point CISD energy correction at a CASSCF optimized geometry. On the one hand, the 44CAS-CISD//44CAS relative energy of TM vs GM is close to the 22CAS-CISD optimized relative energy (Table 1). On the other hand, although 44CAS-CISD//44CAS correctly predicts the cyclization barrier to be lower than with 44CAS alone, it does not reveal whether GM is a minimum on the CISD surface. Overall, the 22CAS-CISD optimizations basically
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Figure 5. Qualitative representation of the gauche region of the PES as computed by 44CAS, which has clearly identifiable stationary points, vs CISD, in which they basically merge into a broad flat region.
support the use of CISD single-point energies at CASSCF optimized geometries to get an approximate CISD surface of tetramethylene. The 22CAS-CISD optimizations are appropriate mainly for cyclization, and for an overall picture of the PES we need to turn to the 44CAS-CISD results. The largest change in the gauche region brought about by the cc-pVTZ,cc-pVDZ basis set is that the energy of GF is below that of GM at the ZPEcorrected 44CAS-CISD level. At the CISD level, GM is evidently a mere dimple on the PES: a slight CH2 twist leads to cyclization and a slight C2C3 stretch leads to fragmentation. The gauche region of the PES is simpler than we reported before:2a GM, G1, and GF effectively coalesce into a broad flat region centered about GM that mediates fragmentation and cyclization and allows nearly free CH2 rotation. The situation is shown in Figure 5, which qualitatively compares the gauche region of the PES as computed by 44CAS (dotted energy curve) and by CISD (solid curve). Dynamical Significance of Shallow Biradical Minima. At the highest levels of theory there is effectively no minimum on the PES corresponding to a biradical, yet Zewail’s experiments revealed lifetimes approaching 1 ps at a total energy of ca. 80 kcal/mol.1 Previously we computed an entropy-dominated free energy minimum2a and microcanonical lifetimes2b of several hundred femtoseconds, but these predictions depended on the implicit assumption that a shallow minimum (GM or TM) optimized at the 44CAS level is dynamically significant even if it disappears at the CISD level or with inclusion of ZPE. The question is whether one is justified in making this assumption in order to compute rate constants for species with no barriers to product formation. As a practical matter there must be something right about this assumption, because it leads to reasonable lifetimes and product ratios for tetramethylene2 and without it the computed lifetimes would have been far too short. Similarly, our microcanonical lifetime calculated for trimethylene,12 118 fs, is close to the experimental value of 120 ( 20 fs1 and critically depends on the assumption of 3N - 6 degrees of freedom for some extremely shallow minima that disappear at the CISD level. Probably the thing that is right about assuming 3N - 6 modes for these phantom minima is that the mode lost on going from biradical reactant to the transition state (the reaction coordinate) is an internal rotation that is nearly decoupled from product formation at the reactant geometry. In the reactant region of the PES, motion along this mode does not lead immediately to product formation for the majority of
Doubleday biradicals. It is an entropically bound mode even though it supports no vibrational levels, and this justifies counting it as a bound reactant mode rather than the translational degree of freedom one would ordinarily assume from the lack of a local minimum. This idea is consistent with preliminary classical trajectory results for trimethylene,13 in which cyclization requires a relatively rare alignment of the two independently moving CH2 torsions. Conclusion. At the 44CAS level the changes in optimized geometry and relative energy on going from 6-31G* to the ccpVTZ,cc-pVDZ basis set are minor. However, at the ZPEcorrected 44CAS-CISD/cc-pVTZ,cc-pVDZ level the energies of GF and TF are computed to be well below those of their respective minima GM and TM. Optimization of GM and G1 with 22CAS-CISD/cc-pVDZ,DZ yields a cyclization barrier of only 0.04 kcal/mol and reveals an extremely flat PES (Figure 4). TM was also optimized with 22CAS-CISD, and on the whole the results appear to justify the use of single-point CISD energy corrections at CASSCF optimized geometries of tetramethylene. The picture that emerges from these calculations is that gauche tetramethylene resides in a broad, flat plateau from which both cyclization and fragmentation are accessible along paths of monotonically decreasing energy (Figure 5) and in which CH2 torsion is nearly free. This is consistent with our previously proposed model of stereomutations in tetramethylene,2a in which the flat PES allows the cyclization and fragmentation transition states each to mediate two or more stereochemically distinct processes. Acknowledgment. We gratefully acknowledge support from the National Science Foundation and a grant of time on the Cray C90 at the Pittsburgh Supercomputing Center. Supporting Information Available: Internal coordinates of stationary points (3 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. (3) (a) Truhlar, D.; Isaacson, A.; Garrett, B. In Theory of Chemical Reaction Dynamics; Baer, M., Ed., CRC Press: Boca Raton, 1985; Vol. IV, p 65. (b) Truhlar, D.; Garrett, B. Acc. Chem. Res. 1980, 13, 440. (4) Hase, W. L. Acc. Chem. Res. 1983, 16, 258. (5) (a) Dervan, P.; Santilli, D. J. Am. Chem. Soc. 1979, 101, 3663; 1980, 102, 3863. (b) Goldstein, M.; Cannarsa, M.; Kinoshita, T.; Koniz, R. Stud. Org. Chem. (Amsterdam) 1987, 31, 121. (c) Lewis, D.; Glenar, D.; Kalra, B.; Baldwin, J.; Cianciosi, S. J. Am. Chem. Soc. 1987, 109, 7225. (d) Chickos, J.; Annamalia, A.; Keiderling, T. J. Am. Chem. Soc. 1986, 108, 4398. (6) (a) Hehre, W.; Ditchfield, R.; Pople, J. A. J. Chem. Phys. 1972, 56, 2257. (b) Hariharan, P. C.; Pople, J. A. Theor. Chim. Acta 1973, 28, 212. (7) Dunning, T. H. J. Chem. Phys. 1989, 90, 1007. (8) Saxe, P.; Lengsfield, B., III; Martin, R., in press. (9) Complete geometries are listed in the supporting information. (10) Bernardi, F.; Bottoni, A.; Celani, P.; Olivucci, M.; Robb, M. A.; Venturini, A. Chem. Phys. Lett. 1992, 192, 220. (11) (a) Dewar, M. J. S.; Kirschner, S.; Kollmar, H.; Wade, L. J. Am. Chem. Soc. 1974, 96, 5242. (b) Dewar, M. J. S.; Kirschner, S. J. Am. Chem. Soc. 1974, 96, 5246. (c) Dewar, M. J. S.; Kirschner, S.; Kollmar, H. J. Am. Chem. Soc. 1974, 96, 5240. (12) Doubleday, C. J. Phys. Chem. 1996, 100, 3520. (13) Doubleday, C.; Bolton, K.; Peslherbe, G.; Hase, W. L. J. Am. Chem. Soc., submitted for publication.
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