J . Phys. Chem. 1988, 92, 4367-4371
4367
Singlet Biradicais as Intermediates. Canonical Variational Transition-State Theory Results for Trimethylene Charles Doubleday, Jr.,* Department of Chemistry, Columbia University, New York, New York 10027
James W. McIver, Jr.,* Chemistry Departments, State University of New York a t Buffalo, Buffalo, New York 14214, and Canisius College, Buffalo, New York 14208
and Michael Page* Laboratory for Computational Physics and Fluid Dynamics, Naval Research Laboratory, Washington, D.C. 20375 (Received: September 28, 1987) Ab initio complete active space multiconfiguration self-consistent field (CAS-MCSCF) calculations together with canonical variational transition-state theory (CVT) are applied to the geometrical isomerization of cyclopropane. The geometries and force constants of the saddle point and six points along the intrinsic reaction coordinate were obtained at the two-electron two-active-orbital CAS-MCSCF level of theory with a 6-3 1G basis set. The energies of these geometries were recalculated by using the six-electron six-active-orbital wave function and a 6-31G* basis set. The aim of these calculations is to determine whether trimethylene can exist as a free-energy intermediate at accessible temperatures. The results indicate that it cannot, but the question remains open as to whether substituted trimethylenes can act as such intermediates.
Introduction Understanding the behavior of singlet biradicals as intermediates in chemical reactions is a difficult and challenging problem. This is exemplified by tetramethylene. On the one hand, the experimentally observed loss of stereochemistry in many thermal and photochemical ring-breaking reactions1q2as well as in olefin dimerization~~ favors the existence of some sort of quasistable intermediate common to both the recyclization and fragmentation reactions.’ This is normally identified with a local minimum on the potential energy surface. Early ab initio calculations on tetramethylene supported this view.4 On the other hand, Hoffmann suggested that the very broad and flat nature of the biradical region of the potential surface allowed the interpretation of biradicals as “twixtyls” which can wander about the potential energy surface enough to lose stereochemistry yet not be necessarily identified with local potential energy minima.5 More recent ab initio MCSCF calculations6 are consistent with this picture in predicting no minima6a or very shallow (less than 1 kcal deep) minima6b for gauche- and trans-tetramethylene. In attempts to resolve this question we suggested6athat tetramethylene corresponds to a free energy minimum,’ as defined in the context of canonical variational transition-state theory* (CVT). Our early MCSCF calculations on gauche-tetramethylene supported this.6a Recent, extensive calculations on trans-tetra(1) (a) Dervan, P.; Uyehara, T. J . Am. Chem. Soc. 1976, 98, 1262. (b) Dervan, P.; Uyehara, T. J . Am. Chem. SOC.1979,101,2076. (c) Dervan, P.; Uyehara, T.; Santilli, D. J . Am. Chem. SOC.1979, 101, 2069. (d) Dervan, P.; Santilli, D. J. J. Am. Chem. Soc. 1980,102, 3863. (e) Schultz, P.; Dervan, P. J . Am. Chem. SOC.1982, 104, 6660. (2) Gerberich, H.; Walters, W. J . A m . Chem. Soc. 1961, 83, 488, 3935. (3) Scacchi, G.; Richard, C.; Back, M. Int. J . Chem. Kinet. 1977,9, 513, 525. (4) Segal, G. J . Am. Chem. SOC.1974, 96, 7892. ( 5 ) Hoffman, R.; Swaminthan, S.; Odell, B.; Gleiter, R. J . Am. Chem. Soc. 1970, 92, 7091. (6) (a) Doubleday, C.; Camp, R.; King, H.; McIver, J.; Mullally, D.; Page, M. J . Am. Chem. SOC.1984, 106, 447. (b) Bernardi, F.; Bottoni, A.; Robb, M.; Schlegel, H. B.; Tonachini, G. J . Am. Chem. Soc. 1985, 107, 2260. (7) The first suggestion that biradicals might correspond to a free energy minimum in the absence of a potential energy minimum was apparently made by Hay concerning trimethylene. See p 74 (footnote 18) of the following: Jeffrey Hay, P. Ph.D. Thesis, California Institute of Technology, 1972. (8) (a) Truhlar, D.; Isaacson, A.; Garrett, B. In Theory of Chemical Reaction Dynamics, Baer, M., Ed.; CRC Press: Boco Raton, FL, 1985; Vol. IV, p 65. Truhlar, D.; Garrett, B. Ace. Chem. Res. 1980, 13, 440. (b) Hase, W. Arc. Chem. Res. 1983, 16, 258.
0022-365418812092-4367$01.50/0
methylene9 suggested that the free energy barrier for fragmentation into olefins is too small for this species to be a mechanistically significant intermediate. The gauche conformer therefore appears to correspond to the experimentally observed’ intermediate. The simplest variational transition-state model suggests that above some temperature any biradical will correspond to a free energy minimum. In this model one picks a degree of freedom, the reaction path, which connects the biradical saddle point with reactants or products and computes the vibrational enthalpy and entropy due to the remaining degrees of freedom. The “free energy” along this path is then constructed to be the total enthalpy (the potential energy plus zero-point corrections plus vibrational enthalpy) minus the absolute temperature times the vibrational entropy. Because of the nearly free torsional degrees of freedom in the biradical relative to the cyclic reactants and products, the biradical is expected to have the higher entropy. This creates a local minimum in the -TAS term in the biradical region of the reaction path. There must therefore exist a temperature beyond which this entropic effect dominates the opposite enthalpic effect, leaving at least one free energy minimum somewhere between the cyclic reactants and products, thus creating an “entropy locked” intermediate. The important question is whether this temperature is low enough to be experimentally accessible. In this paper we ask if singlet trimethylene can be entropy locked as an intermediate. Trimethylene appears to be a good candidate for this effect since, although smaller than tetramethylene, it too is considerably floppier than the cyclic reactant cyclopropane.’ Moreover, trimethylene has been studied extensively by a b initio calculations,1° the results of which show no (9) Doubleday, C.; Page, M.; McIver, J. J . Mol. Strucf. (THEOCHEM) 1988, 163, 331. (10) (a) Hoffmann, R. J . A m . Chem. SOC.1968.90, 1475. (b) Buenker, R. J.; Peyerimhoff, S . C. J . Phys. Chem. 1969, 73, 1299. (c) Siu, A. K. 0.; St. John, W. M.; Hayes, E. F. J . Am. Chem. SOC.1970, 92,7249. (d) Jean, Y.; Salem, L. Chem. Commun. 1971, 382. ( e ) Hayes, E. F.; Siu, A. K. 0. J . Am. Chem. SOC.1971, 93, 2090. (0 Hay, P. J.; Hunt, W. J.; Goddard, W. A., 111 Ibid. 1972, 894, 638. (g) Horsley, J. A,; Jean, Y.; Moser, C.; Salem, L.; Stevens, R. M.; Wright, J. S. Ibid. 1972, 94, 279. (h) Jean, Y.; Chapuisat, X. Ibid. 1974, 96, 6911. (i) Chapuisat, X . ; Jean, Y. Ibid. 1975, 97,6325. (j) Yamaguchi, K.; Fueno, T. Chem. Phys. Lett. 1973,22,471. (k) Yamaguchi, K.; Nishio, A.; Yabushita, S.; Fueno, T. Ibid. 1978, 53, 109. (1) Yamaguchi, K.; Ohta, K.; Yabushita, S.; Fueno, T. Ibid. 1977,49, 555-9. (m) Kato, S.; Morokuma, K. Ibid. 1979,65, 19-25. (n) Doubleday, C.; McIver, J.; Page, M. J . Am. Chem. SOC.1982, 104, 6533.
0 1988 American Chemical Society
4368
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988
1.0695
Hi
U" I
IL
n
120.39
Figure 1. Saddle point for reaction 1 calculated with a 2-in-2 CASSCF wave function and a 6-31G basis set. Bond lengths are in angstroms, and angles in degrees. evidence of a potential energy minimum in the biradical region. Cyciopropane can rearrange by either a single rotation of a methylene or simultaneous rotations of both terminal methylenes in the biradical. Although the latter appears to be marginally favored experimentally," in this paper we examine the single epimerization mechanism. We have previously located the saddle point for this mechanism12 using a CAS-MCSCF wave function and analytically calculated force constants.13 Experimentally, there is evidence that substituted trimethylenes act as intermediates in gas-phase therm01yses.l~ As for trimethylene itself, Berson's own results for the gas-phase thermal stereomutations of 1,2-dideuteriocyclopropanes(reaction 1) were
A=k D
D
D E
interpreted in terms of predominant simultaneous epimerization of two carbons without the intervention of an intermediate." The mechanism is apparently sensitive to changes in substituent, and one suspects that subtle features of the potential energy surface (PES) may have a profound effect on the product distribution, especially on the ratio of one- to two-center epimerizations. Qualitatively, one expects that the single epimerization mechanism involves breaking the C-C bond to form the biradical and then twisting about through the saddle point (Figure 1) followed by cyclization. In the saddle-point region, there is little possibility for bonding interactions between the terminal methylene groups, and the 42torsion is expected to be a nearly free internal rotation. As cyclization proceeds, incipient bonding between the radical sites is expected to tighten the 42 torsion considerably. Conventional transition-state theory (TST) cannot account for this effect since it assumes that the free energy barrier to the reaction lies at the single saddle point to the reaction. If trimethylene is to be a free energy minimum, then there must exist free energy barriers separating it from the reactant and product cyclopropane. At least one of these barriers must be displaced from the saddle point along some path connecting reactant and product. The canonical variational transition-state theor? (CVT) provides both a definition of this free energy and a means of determining the free energy barriers. As described in ref 8a, CVT can share the same quasithermodynamic interpretation as TST. In the latter case one first locates the lowest potential energy barrier separating reactants (11) Berson, J. A.; Pederson, L. D.; Carpenter, B. K. J . Am. Chem. Soc. 1976, 98, 122.
(12) Mullally, D. Ph.D. Thesis, State University of New York at Buffalo, 1984. (13) Camp, R.; King, H.; McIver, J.; Mullally, D. J. Chem. Phys. 1983, 79, 1088. (14) (a) Berson, J. A. Ann. Reu. Phys. Chem. 1977.28, 111. (b) Berson, 3. A. In Rearrangements in Ground and Excited States; Vol. 1, deMayo, P., Ed.; Academic: New York, 1980; Vol. 1, p 311.
Doubleday et al. and products (the saddle point) and calculates the free energy from the partition function evaluated at this structure. Excluded from this calculation is the "reaction path" degree of freedom, which is taken to be the eigenvector of the force constant matrix (in mass-weighed Cartesian coordinates) that corresponds to the single negative eigenvalue. In CVT one constructs the reaction path or intrinsic reaction coordinate (IRC)lS connecting reactants, saddle point, and products. The entropy and enthalpy along this path is then constructed from the partition function, the vibrational contributions to which are obtained from the eigenvalues of the projected mass-weighted force constant matrix.16 This matrix is constructed by projecting the reaction path degree of freedom, the three translations, and the three rotations out of the massweighted force constant matrix. One next constructs the free energy as the enthalpy minus the absolute temperature times the entropy. This free energy versus the arc length along the reaction path can be. plotted at different temperatures. From these plots one can identify the locations of the free energy barriers and the depths of the free energy wells. The temperature-dependent free energies of activation are obtained from these. Further refinements include tunnelling corrections and anharmonic contributions to the potential energy.8 With the exception of the treatment of the 42torsion described below, neither are included in the calculations reported here. Methodology CVT Theory. We have utilized the CVT theory outlined in the previous section and described in detail by Truhlar et alegawith two modifications given below. These modifications are the special treatment of the internal rotation degree of freedom, the 42torsion, and an improved method of following the reaction path. For the one-center epimerization, we observed that the projected mass-weighted force constant matrix at each point on the reaction path had an eigenvector that corresponded almost exclusively to the $2 torsion. That degree of freedom was treated as an internal rotation rather than a harmonic oscillator in the calculation of the partition function. We computed the potential energy barrier V, to rigid 92torsions at each point on the reaction path and used the formula V = 1/2V0(l- cos 242) and the method of Pitzer and Gwinn" to estimate the internal rotation contribution to the partition function. The remaining eigenvalues of the projected force constant matrix were used to compute the vibrational contribution to the partition function in the usual way by using the harmonic oscillator approximation.l 8 The reaction path is a curved line in mass-weighted configuration space. It can be represented by the column vector X(s), where the X are mass weighted Cartesian coordinate^'^ and s is the arc length parameter defined by 3N
ds2 =
CdX? i- 1
The reaction path is defined as the solution to
(3) which connects the saddle point and reactants and products.I5 Here, g is the mass-weighted gradient vector and c is the normalization constant: c = (g+g)'/Z
(4)
It can be shown that the reaction path must approach the saddle (15) Fukui, K. Acc. Chem. Res. 1981, 14, 363.
(16) Miller, W.; Handy, N.; Adams, J. J . Chem. Phys. 1980, 72, 99. (17) Pitzer, K.;Gwinn, W. J . Chem. Phys. 1942, 10, 428. (18) Herzberg, G. Molecular Spectra and Molecular Structure; Van Nostrand Reinhold: New York, 194s; Vol. 2. 19) Mass-weighted Cartesian coordinates X(s) are defined by X(s) = M l L l ( s ) , where q(s) is the Cartesian position vector of dimension 3N ( N = number of atoms) and M is the 3N X 3N diagonal matrix of triplets of atomic masses.
The Journal of Physical Chemistry, Vol. 92, No. 15. 1988 4369
Singlet Biradicals as Intermediates
TABLE I: Total Electronic Energy and Zero-Point Energy along the Reaction Path at Several Levels of Theory. Relative Energies (in kcal/mol) Are Referenced to the Saddle-Point Enerm 2-in-2/6-3 1G*
2-in-2/6-31G sa
-(E+ 116)b
-2.74 -1.91 -1.23 0.0 0.74 1.63 2.39 cyclopropane
0.9451 1 0.93999 0.93718 0.93490 0.93584 0.9 39 56 0.94482
Emf -6.42 -3.20 -1.43
-(E
(0.0) -0.59 -2.93 -6.24
+ 116)b
0.99477 0.98876 0.98548 0.98298 0.98396 0.98819 0.99340
E,{ -7.42 -3.64 -1.57 (0.0) -0.62 -3.28 -7.12
6-in-6/6-31G*
+
-(E 117)* 0.02933 0.02314 0.01986 0.01763 0.01884 0.02340 0.03044 0.11703
E,,f -7.36 -3.46 -1.45 (0.0) -0.76 -3.63 -8.06 -62.52
AG ZPEC
ZPEret
49.77 49.13 48.83 48.71 48.65 49.07 49.59 53.90
1.06 0.42 0.12 (0.0) -0.06 0.36 0.88 5.19
at 0 KCqd -6.30 -3.04 -1.33 (0.0) -0.82 -3.27 -7.18 -57.33
"Arc length distance along the reaction path. in bohr amul/*. bTotal electronic energy E in hartrees. cIn kcal/mol. dFrom the 6-in-6/6-31G* E,,, values, corrected by ZPE,,,.
point along the "transition vector" or eigenvector of F, the mass-weighted force constant matrix at the saddle point, corresponding to the single negative eigenvalue.20,21 Equation 3 is normally integrated by an Euler method using small steps. At the saddle point the first step is alont the transition vector. Subsequent steps require only the energy gradient g. The difficulty with this procedure is that very small steps must be taken to avoid oscillating about the path. Various methods based on constrained energy minimization have been proposed to correct this problem, permitting larger step sizes.22 In many calculations, such as the one reported here, higher energy derivatives are required in addition to the gradient. It is shown elsewhere how this additional information can be used to more closely follow the path.21 Here we briefly summarize the method that should be used when the force constants are available. W e begin by first integrating the equation: dX/dt = 3
0
- Fo(X - Xo)
(5)
where the right-hand side is the negative of the energy gradient expanded through the first order about the nonstationary point Xo, presumed to lie on the path. The vector go and the matrix Foare the padient and force constant matrix at Xo, and t is the path parameter. The relation o f t to the arc length s is given below. The solution to eq 5 is X(t) = xo + A(t)go
(6)
where the symmetric matrix A is given as A(t) = Uocr(t)UJ
(7)
Here U is the orthogonal matrix of column eigenvectors of Fo and a ( t ) is a diagonal matrix whose elements are aii =
(e+' - l ) / k i
(8)
where the Xi are the eigenvalues of Fo. The relationship between the parameter t and the arc length s is obtained from (9)
points along the path at which force constants were not calculated, the most recently calculated force constant matrix was used in solving eq 4-9 to determine the reaction path step. Wave Functions and Basis Sets. The quantum chemical calculations were carried out with the MESA23system of programs on a Cray X-MP/48 at the Pittsburgh Supercomputer Center. The energies and analytical first derivatives were computed with a complete active space MCSCF, or CASSCF, wave function,24 which has proven useful in describing potential energy surfaces of biradical~.'@~J~ A CASSCF wave function designated m-in-n ( m electrons in n orbitals) includes all configuration state functions with a specified spin state generated by m electrons occupying n valence MOs. We used two CASSCF wave functions, 2-in-2 and 6-in-6, giving rise to 3 and 175 singlet configurations, respectively. With these wave functions we used two atomic basis sets, 6-31G26aand 6-31G*.26b For calculating the reaction path and the force constant matrices, we used the 2-in-2 level that has tetrabeen applied successfully to trimethylene,6a~2Sa~b*e~27 m e t h ~ l e n e , ~and ~ " ~cyclopentane- 1,3-diylZ8and is qualitatively reasonable. Force constants were evaluated by single-displacement finite difference for seven geometries along the path, including the saddlepoint. The energies of these seven geometries were recalculated at two higher levels, 2-in-2/6-3 1G* and 6-in-6/631G*. In the 6-in-6 calculation, the natural orbitals involved the two C-C bonding-antibonding combinations and the terminal p orbitals. The & barriers were recalculated at the 6-in-6/6-31G* level because they depend strongly on long-range C1-C3 bonding, which is best described at the higher level. To calculate the overall activation energy of the cis-trans isomerization, we optimized cyclopropane in Djhsymmetry at both the 6-in-6/6-31G and 6-in-6/6-31G* levels. The force constant matrix was then evaluated by using 6-in-6/6-31G, and the 6-in6/6-31G* energy was used to calculate AG*.The choice of 6-in-6 was imposed by the threefold symmetry, and we verified that the natural orbitals in these calculations were essentially the W a l ~ h ~ ~ ring orbitals. Thus, both cyclopropane and the reaction path geometries were calculated with a wave function in which electron correlation is complete within the subspace of carbon framework MOs.
where g', is the gradient in normal coordinates: g'o = Uo'go
For any desired arc length step, eq 9 can be numerically integrated by using small steps in t until this arc length is reached. For trimethylene, we have examined the reaction path for cis-trans isomerization within 6-7 kcal of the saddle-point energy on either side of the saddle point. Force constants were calculated at seven points along the path, including the saddle point. The determination of the entire path required about 120 steps. At (20) Pechukas, P. J . Chem. Phys. 1976,64, 1516. (21) Page, M.; McIver, J. J. Chem. Phys. 1988, 88, 922. (22) (a) Ishida, K.; Morokuma, K.; Komornicki, A. J . Chem. Phys. 1977, 66, 2153. (b) Muller, K.;Brown, L. Thear. C h i p . Acra 1979, 53, 75. (c) Schmidt, M.; Gordon, M.; Dupuis, M. J. Am. Chem. SOC.1985,107,2585. (d) Schlegel, H. Adu. Chem. Phys. 1987, 249.
(23) MESA (molecular electronic structure applications) by Paul Saxe, Byron Lengsfield, 111, Richard Martin, and Michael Page. The MCSCF code in MESA is based on: Lengsfield, B., 111 J . Chem. Phys. 1982, 77, 4073. (24) (a) Row, B. 0.;Taylor, P. R.; Siegbahn, P. E. M. Chem. Phys. 1980, 48, 157. (b) Ruedenberg, K.; Schmidt, M. W.; Gilbert, M. M.; Elbert, S.T. Chem. Phys. 1982, 71, 41. (25) (a) Doubleday, C.; McIver, J.; Page, M.; Zielinski, T. J . Am. Chem. Soc. 1985,107,5800. (b) Doubleday, C.; McIver, J. W., Jr.; Page, M. J . Am. Chem. Soc. 1982, 104, 3768. (c) Doubleday, C.; McIver, J. W.; Page, M. J. Am. Chem. Soc. 1985, 107,7904. (d) Carlacci, L.; Doubleday, C.; Furlani, T.; King, H.; McIver, J. J . Am. Chem. Soc. 1987, 109, 5323. (e) Caldwell, R.; Carlacci, L.; Doubleday, C.; Furlani, T.; King, H.; Ladoy, J.; McIver, J.; Nalley, E., submitted for publication in J . Am. Chem. SOC. (26) (a) Hehre, W. J.; Ditchfield, R.;Pople, J. A. J . Chem. Phys. 1972, 56,2257. (b) Hariharan, P. C.; Pople, J. A. Theor. Chim. Acta 1973,28,213. (27) Goldberg, A. H.; Dougherty, D. A. J. Am. Chem. SOC.1983,105,284. (28) Conrad, M.; Pitzcr, R.; Schaefer, H. F. J . Am. Chem. SOC.1979,101, 2245. (29) Walsh, A. D. Trans. Faraday SOC.1949, 45, 179
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988
4370
4
Doubleday et al. TABLE 11: Changes in the Low-Frequency Modes along the Reaction Path
-
c
2-
01
E -
4000" K
-
3000'K
;0 -
0 1
-
F-2-
2000°K
al
( -4=I -6 -3
1
'
-2
"
1
-1
1
"
1
0
1
1
1
'
2
3
s (bohr -domu Figure 2. Relative free energy curves along the reaction path for cistrans isomerization of cyclopropane at four temperatures. Each curve is separately referenced to the free energy at s = 0. Points were calculated by CVT; curves were obtained by cubic spline interpolation.
Results The saddle point calculated at the 2-in-2/6-31G level is shown in Figure 1. It is almost identical with Salem'sm structure obtained with the STO-3G basis set3' and looks basically as one expects: a C-C bond is broken, and one of the CH2 groups is twisted about 90". Recalculation with 6-in-6/6-3 l G * yielded a nonstationary point with a small gradient in which some of the bond-stretching components were nonnegligible. We did not reoptimize the saddle-point geometry on the 6-in-6/6-3 1G* potential energy surface. Table I lists the energies at seven points along the reaction path as a function of wave function and basis set, as well as the zero-point energy (ZPE) corrected relative energies based on the 6-in-6/6-31G* values. Relative to the 2-in-2/6-3 1G level, recalculation at the seven geometries at the 2-in-2/6-31G* level makes the descent of the path from the saddle point steeper. This is expected from the better description of long-range C-C bonding produced by the polarization functions. Recalculation at the 6-in-6/6-31G* level further lowers the energy of the path geometries far from the saddle point. However, in the vicinity of the saddle point the energy changes at the 2-in-2/6-31G and 6-in6/631G* levels nearly match. This is the most biradical-like section of the path (fca. 20" of twist) where the overlap between the terminal methylene p orbitals is very small. From Table I, the ZPE-corrected potential energy barrier is 57.5 kcal/mol. Substitution of deuterium, corresponding to the experimentally observable reaction 1, changes this value to 58.1 kcal/mol. Our calculated Arrhenius parameters for reaction 1 at 695 K are E , = 60.2 kcal/mol and log A = 15.4, and the calculated rate constant is 2.5 X lo4 s-l. Berson" reported the s-I, a factor rate constant for reaction 1 at 695 K to be 6.7 X of 3.8 lower than our calculated value. Experimental activation parameters are E, = 60-61 kcal/mol and log A = 14.6-16.4.32 The CVT results are summarized in Figure 2. For ease of presentation, each AG curve is separately referenced to the free energy of the saddle point geometry, s = 0, at the stated temperature. Because only seven points were calculated, the detailed curve shapes are uncertain. However, the qualitative result is established t h a t CVT predicts t h e existence of a free energy
sa
-2.74 -1.91 -1.23 0.0 0.74 1.63 2.39
u ~ ~ ~ *2 6~ , 455 568 435 482 364 432 335 382 254 392 318 431 447 469
~
u,h'
641 607 575 572 571 612 638
6,barrier, kcal /mol 2-in-2/6-31G 6-in-6/6-31G* 7.8 3.8 1.9 0.6 1.o 2.9 7.5
10.2 4.0 1.7 0.6 1.9 3.7 11.0
"Arc length distance along reaction path in bohr amulI2. bFrequency in cm-l. C C H 2pyramidalization on C H 2 undergoing rotation. dCCC bend. C C H 2pyramidalization on CHI not undergoing rotation.
minimum corresponding to a biradical, but only at a very high temperature. The frequency changes mainly responsible for this AG minimum (the CCC bend and the two CH2 out-of-plane pyramidalization modes) are shown in Table 11, along with the 42 torsion barrier.
Discussion The chief cohclusion of this study is that trimethylene is not an intermediate at the temperatures normally accessible in isothermal experiments. Even at 1000 K there is no local minimum in the free energy curve in Figure 2. In fact, nowhere in Figure 2 is a AG minimum larger than RT. Conventional TST remains valid (Le., the transition state intersects the saddle point of the PES) well past the temperature at which local AG minima start to appear. This contrasts with tetramethylene, which has an entropic barrier to cyclization somewhat greater than RT at 700 K.6a Figure 2 shows a small additional minimum a t s C 0. Whether this is due to an artifact (i.e., straying from the reaction path) is irrelevant to the conclusion that trimethylene is not an intermediate below 1000 K. The possible existence of the free energy minimum in Figure 2 is probably a moot point with regard to trimethylene itself, because of the very high temperature required. However, Figure 2 raises interesting questions about the dynamics of derivatives of trimethylene. The low-frequency modes will all be affected by substitution on the radical sites, but whether substituents will deepen the AG minima or lower the temperature at which the AG minima begin to appear is not known. From a computational point of view, Tables I and I1 show that addition of polarization functions to the basis set does not significantly alter the energetic changes in the biradical as long as overlap between the terminal p orbitals is small (ca. f 2 0 " of 41 twist either side of the saddle point). Changing the wave function from 2-in-2 to 6-in-6 also produces a very small effect on energetic changes in this region. Thus a simple 2-in-2 wave function with a split-valence basis set is virtually equivalent to a 6-in-6/6-31G* calculation with regard to biradical energies. Other properties, such as absolute values of force constants, are expected to require a better treatment of electron correlation. The activation energy for eq 1 is successfully reproduced at the 6-in-6/6-31G* level. Here, polarization functions are necessary to describe the bonding in the strained cyclopropane ring.33 Summary
(30) Horsley, J.; Jean, Y.; Moser, C.; Salem, L.; Stevens, R.; Wright, J. J. Am. Chem. SOC.1972, 94, 279. (31) Hehre, W.; Stewart, R.; Pople, J. J . Chem. Phys. 1969, 51, 2657. (32) The experimental activation parameters for reaction 1 were obtained in the following way, since the high-pressure values reported by Schlag and Rabinovitch (Schlag, E. W.; Rabinovitch, B.S . J. Am. Chem. SOC.1960,82, 5996) were apparently slightly in error. E, for the isomerization to propene has been measured many times, the values clustered about 64-65 kcal/mol. (For examples and leading references see: Klein, I. E.;Rabinovitch, B. S . Chem. Phys. 1978,35,439. Rabinovitch, B. S. Chem. Phys. 1982,67,201.) The difference in activation energies between propene formation and cis-trans isomerization has been accurately measured to be 3.7 (Waage, E.V.; Rabinovitch, B. S . J. Phys. Chem. 1972, 76, 1965). Thus E, for reaction 1 probably lies in the range 60-61 kcal/mol. See also: Doering, W. Proc. Natl. Acad. Sci. U. S . A. 1982, 78, 5279.
We have presented computational evidence that singlet trimethylene is not a gas-phase intermediate below IO00 K. The importance of entropy in the dynamics of tetramethylene6 convinced us not to rely on conventional TST but to use the CVTS dynamical method to compute the rate constant. In fact, our calculations show that TST gives a good description of the cistrans isomerization of cyclopropane below 1000 K. However, in the general case CVT is the lowest level of dynamical theory that can address the question of free energy minima in biradicals. From (33) Hehre, W.; Radom, L.; Schleyer, P.; Pople, J. Ab-initfo Molecular Orbital Theory; Wiley: New York, 1986.
J. Phys. Chem. 1988, 92,4371-4374 a computational point of view, although the 6-31G* basis is necessary to reproduce the overall activation energy, the 2-in2/631G results are virtually the same as 6-in-6/6-31G1 in describing relative energy changes in the vicinity of the saddle point, where overlap between the termini is small. Acknowledgment. We are deeply indebted to Dr. Paul Saxe
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of Los Alamos National Laboratory for helpful comments and to the National Science Foundation for Grant CHE84-21140. M.P. acknowledges support from the Office of Naval Research through the Naval Research Laboratory. Registry No. Trimethylene biradical, 32458-33-6; cyclopropane, 7519-4.
Unimolecular Thermal Reaction of Formaldoxime at High Temperatures: Experiments and Calculations KO Saito,* Koso Makisbita, Terumitsu Kakumoto, Takaharu Sasaki, and Akira Imamura Department of Chemistry, Faculty of Science, Hiroshima University, Higashisenda, Naka- ku, Hiroshima 730, Japan (Received: October 19, 1987; In Final Form: February 9, 1988)
The thermal decompositionof formaldoxime diluted in Ar has been studied behind reflected shock waves over the temperature mol cm-'. The decomposition process was range between 1050 and 1300 K and the total density range of (0.6-2.4) X monitored using the vacuum-UV absorption 9f the reactant and the IR emission from HCN. It was found that the main thermal reaction proceeded via (1) H2CNOH H 2 0 + HCN and that the alternate isomerization (2) H,CNOH CH3N0 exp(-48 kcal mol-'/RT) was negligible. Over the temperature range of the study, the decomposition rate constant is kl = s-I. There is a very slight pressure dependence over the experimental region. Ab initio MO calculations were performed for the decomposition and the isomerization channels. The barrier height for the isomerization was higher than that for the molecular decomposition by 24 kcal mol-' at the MP2/3-21G level. Rate constants calculated for the dehydration step were in good agreement with those measured.
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Introduction Formaldoxime (AHfo= 0 kcal/mol) is an isomer of formamide (AHfo = -44.5 kcal/mol) and nitrosomethane (AH," = 16 kcal/mol). At high temperatures formamide decomposes to ammonia and carbon monoxide but it does not isomerize to its isomers since these processes require complicated bond rearrangements.' The thermal isomerization of nitrosomethane to formaldoxime was investigated by Batt and Gowenlock2 by a flow method at 4.5 Torr over a temperature range of 633-698 K. They reported exp(-29.1 kcal mol-'/RT) s-I, for this a rate constant, k-2 = isomerization. Later Benson and O N e a l estimated a value, k-2 = 10t2.9exp(-39.3 kcal mol-'/RT) s-I, on the basis of assumed parameters for the transition state of this reaction.' Taylor and Bender4 studied thermal decomposition of formaldoxime using a static method over the temperature range of 623-488 K and at 4.5 Torr. They found that the reaction produced mainly water and hydrogen cyanide with a rate constant kl = lo9' exp(-39 kcal mol-'/RT) s-I. On the basis of the abnormally low preexponential factor, Benson and O'Nea12 suggested that the experiments were affected by oxygen and wall effects. Thus, we have no reliable kinetic data for these elementary reactions in a homogeneous system. For the determination of an elementary reaction rate constants, it is necessary to avoid several problems arising from secondary and heterogeneous reactions. In the current states, we determine the rate constant for the primary thermal reaction of formaldoxime in Ar at high temperatures behind shock waves where complications of the type mentioned above are avoided. In addition, we have calculated transition-state structures for plausible reaction paths by means of a molecular orbital method. The results are compared with those from the experiments. (1) Kakumoto, T.; Saito, K.; Imamura, A. J. Phys. Chem. 1985,89,2286. (2) Batt, L.; Gowenlock, B. G. Trans. Faraday Soc. 1960, 56, 182. (3) Benson, S. W.; ONeal, H. E. Kinetic Data on Gas Phase Unimolecular Reacfions; NSRDS-NBS 2 1; US.Government Printing Office.: Washington, DC, 1970. (4) Taylor, H. A.; Bender, H. J . Chem. Phys. 1941, 9, 761.
0022-3654/88/2092-437 1$01.50/0
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Experimental Section The experimental measurements were conducted behind reflected shock waves at temperatures between 1050 and 1300 K and in the density range (0.6-2.4) X mol cm-' using mixtures containing 0.1, 0.5, and 1.O mol % formaldoxime diluted in Ar. Formaldoxime trimer was prepared by the reaction of a formaldehyde solution and hydroxylamine as described in ref 5. The product was tested by C H N elemental analysis. The monomer was generated by heating the trimer to about 135 "C and then diluted with Ar to about 1% in a 6-L glass flask. The decrease of the monomer concentration in the flask was checked several times according to a spectroscopic method. The results indicated that the polymerization was negligible over a period of a few days. The shock-tube equipment used in this work was the same as that used in a previous studya6 The test section was about 3.8 m long with a 9.4-cm i.d. It was evacuated to less than 2 X 10" Torr before each run. A pair of MgF2 windows was mounted on the tube walls 2 cm upstream from the end plate. Time-resolved optical measurements of the reaction were perfbrmed through these windows. Several chemical species were monitored during the course of the reaction by their IR emission, UV absorption, and vacuum-UV absorption. In the IR region, several fundamental bands of H C N ( v l and v3) and H 2 0 (vl and v3) were selected by means of interference filters with an average half-bandwidth of Ax = 0.2 pm. The IR emission intensity was detected by an AuGe element cooled at 77 K. The time constant of the detection system was determined to be about 15 ps by using the CO, fundamental bands. Measurements of UV and vacuum-UV absorptions were performed at the same position as the IR observation. The wavelengths observed were 121.6, 130.5, 216, and 308 nm for the detection of H , 0, CH3, and OH, respectively. A microwave discharge tube containing flowing He with a few percent of ad( 5 ) Scoll, R. Chem. Eer. 1891, 24, 573. (6) Saito, K.; Kakumoto, T.; Imamura, A. J . Phys. Chem. 1984, 88, 1182.
0 1988 American Chemical Society
