J. Phys. Chem. 1993,97, 12737-12741
12737
Theoretical Investigation of the Ethene-Ethene Radical Cation Addition Reaction J. Rad Alvarez-Idaboy,+Leif A. Eriksson, Torbjiirn Fiingstriim, and Sten Lunell' Department of Quantum Chemistry, Uppsala University, Box 518, S-751 20 Uppsala, Sweden Received: March 9, 1 9 9 P t
The addition reaction of an ethene radical cation and a neutral ethene molecule, as previously observed in low-temperature matrix isolation ESR measurements, has been studied using a number of computational methods, ranging from the semiempirical AM1 and P M 3 methods to ob initio UHF, MP2, and MP4 calculations using different basis sets, with and without spin projection included. An intermediate addition complex is observed as a local minimum on the potential energy surface of the reaction. The transition state between this local minimum and the final product, the 1-butene radical cation, is identified as having a hydrogen partly transferred between the carbons in the 2- and 4-positions, the activation energy calculated at the PMP4/6-31G**// MP2/6-31GS* level being 6 kcal/mol. Spin projection is found to be important for the addition complex, which has one very long (1.9 A) carbonxarbon bond, but not at the other stationary points of the potential energy surface.
Introduction The ethene radical cation has over the years been the subject of several detailed investigations, theoretical (refs 1-4 and references therein) as well as e ~ p e r i m e n t a l .By ~ ~studying the vibrational structures in UV absorption and photoelectron spectra, Meerer and Shoonveld,s and later Kappel et deduced that the ethene cation has D2 symmetry with an H-C-C-H torsional angle of about 25O and a 2Bl ground state. The twisted structure of the ethene radical cation has been confirmed by extensive ab initio calculations,z3including electron correlation,in conjunction with large basis sets containing both diffuse and polarization functions. The largest torsional angle obtained by ab initio methods is, however, only about half of the experimental value. On the other hand, semiempiricalcalculationsusing the MNDO method10 have been found to giverather good estimatesoftorsional angles in alkene radical cations, including C2H4+.4JLThis can be attributed to the fact that this and related semiempirical methods are parametrized to reproduce experimental properties and therefore in a sense model at least part of the correlation energy. In the present study both ab initio and semiempirical methods have been used. Due to its small size and high reactivity, the ethene cation has been found to be a rather difficult system to study experimentally. By creating the ion in a freon matrix at 105 K, Shiotani et al.' obtained an ESR spectrum which was originally believed to arise from the CzH4 cation, but in a later communication by Fujisawa et a1.,*it was ascribed to the 1-buteneradical cation. The ethene cation has, indeed, later been possible to isolate and study9 by using a neon matrix and very low temperature (4 K), the results of these measurements confirming the reinterpretation of the old experiments. The formation of the 1-butene cation under the original experimentalconditions7was explained as being caused either by diffusion by the small ethene molecules and ions in the matrix or by the existence of dimeric aggregates even in dilute solutions. Warming the samplefrom 77 to 113 K resulted in a better resolved spectrum (Le.,more product), easily interpretable as the 1-butene cation. Heating thesample further led toa spectralpattern which could be ascribed to a propagating radical, Le., products formed by stepwise polymerization of ethene, whereas if the matrix was Permanent address: Facultad de Quimica, Universidad de la Habana, Habana ]Woo, Cuba. * To whom correspondence should be a d d r d . *Abstract published in Advonce ACS Abstrocts, November IS, 1993.
0022-3654193f 2097-12737$04.00/0
instead illuminated with visible light, the isomerization reaction from the 1-butene radical cation to the cis-2-butene cation was observed. In the present paper we have concentrated on the first step in the above-mentioned polymerization process: the addition reaction of the ethene cation and a neutral ethene molecule. We have studied this by means of ob initio and semiempirical methods in order to identify the different stationary points on the potential energy surface characterizing the reaction.
Geometry Optimizations The main part of the present paper has been carried out at different levels of ab initio theory, within the UHF and MP2 formalisms, using the 3-21G, 6-31G, and 6-31G** basis sets, utilizing the program systems GAUSSIAN 9012and GAMESS.13 The attention is focused on the stationary points on the potential energy surface (PES) governing the addition reaction between ethene and the ethene cation: in this case a local minimum, a transition state, and the global product minimum. Full geometry optimizations were performed using gradient techniques. Additional total energy calculations, using M~ller-Plessetperturbation theory up to fourth order (MP4), without and with spin projection, were performed at each of the stationary points of the PES. For comparative purposes, parallel studies were also made with the semiempirical AMl14 and PM3l5 methods, using the program package MOPAC 6.0.16 In a preliminary study of the optimal way in which the two molecules approach each other, and to identify the intermediate addition complex, an intuitive supermolecule (Figure 1) was considered as a starting point for the geometry optimization calculations. As a first step, ab initio calculations at the UHF/ 3-21G level were performed. In this set of calculations,the results of which are depicted in Figure 2, the torsional angle cp wasvaried from 90° to 180°, with the internuclear ( 2 2 x 3 distance kept fixed at 2.0A. All other geometrical parameters were optimized at each point of the calculation. As seen in Figure2, the minimum on the potential energy curve is observed at cp = 180°, although it must be stressed that the surface is very flat. The calculations were repeated with the ( 2 2 x 3 distance fixed at 1.8 A. The results are very similar to those obtained at R ( C 2 4 3 ) = 2.0 A, with a shift of the whole potential energy curve by some 3.9 kcal/mol toward higher energies. As a second step, the ( 2 2 x 3 distance was varied between 1.5 and 2.5 A. All other parameters, including the torsional angle cp, wereoptimizedat theUHF/3-21Glevel. Theresultsareshown 0 1993 American Chemical Society
Alvarez-Idaboy et al.
12738 The Journal of Physical Chemistry, Vol. 97, No. 49, 1993
Figure 1. Labeling of the atoms of the ethene
+ ethene cation
a
supermolecule. Also indicated is the torsional angle cp, referred to in the text. .154.8105-
.154.9125
__--
- -.. '?* -
b
I
-154.9146
H
b
.--.-I
I
I
Figure 4. Optimizedgeometriesof the additioncomplex at the (a) UHF/ 3-21G, (b) MP2/6-31G** (UHF/6-31Gb* values within parentheses), and (c) PM3 levels.
I
-154.918 1 .so
I
2.00
i 2.50
CM diaaact (A)
Figure 3. As Figure 2, but varying the C2-C3 distance instead of the torsional angle cp. All other geometrical parameters were optimized at each point of calculation.
in Figure 3. It is observed that, although the total energy increases when the C 2 4 3 distance decreases from 2.1 A, the surface is also here rather flat. For all the distances considered, the minimum energy is observed at a torsional angle rp = 180°, giving a ground-state structure of C2h symmetry. Starting with the geometry having the lowest energy as obtained from the above-mentioned calculations (Le.,R(C2-C3) = 2.1 A and cp = 180°), full geometry optimization calculations were performed on the supermoleculein order to locate the intermediate addition complex. Several sets of ab initio calculations were performed, both without and with electron correlation included, from UHF/3-21G to MP2/6-31G**. The geometry thus obtained has & symmetry with a C 2 4 3 distance of 1.93-2.09 A, depending on the method used (cf. Figure 4). The counterpoise method" was used to calculate the basis set superposition error (BSSE). In the smallest basis set used (321G), the BSSE was found to be relatively large (7 kcal/mol). Since it has been shown that counterpoise corrections to results obtained using a small basis set are of doubtful significance18Jg and that the BSSE is better avoided by increasing the basis, the size of the BSSE was primarily used to check the reliability of the further geometry optimizations using larger basis sets (see below). The same procedure as outlined above was used to locate the intermediate addition complex at the semiempirical AM 1 and PM3 levels. Except for slightly shorter C2-C3 bond distances
(0.03 A shorter than MP2/6-31G**, in the case of PM3), the semiempirically optimized geometries are very similar to those obtained from the ab initio calculations. In Figure4 we show the (a) UHF/3-21G, (b) MP2/6-3lG**, and (c) PM3 optimized geometries of the addition complex. The addition complex is observed to have a long C2-C3 bond and two slightly weakened C-C double bonds, similar in length to the C - C bonds in other alkene radical cation^.^ To reach the 1-butene radical cation from the intermediate addition complex, hydrogen migration from C2 to C4 is necessary. This rearrangement was first modeled with the AM1 and PM3 methods, using the reaction coordinate method implemented in the MOPAC program. Starting from the intermediate addition complex, in which the migrating proton H10 is linked to C2 (cf. Figure l), the distance between the proton and the accepting carbon atom C4 was stepwise decreased. The point with maximum energy and minimum reaction gradient in the reaction path was used as input for calculations to optimize the transitionstate structure, using Bartel's gradient norm minimization method (NLLSQ)also implemented in MOPAC. The optimized transition-state structure (Figure Sa) was checked by means of frequency calculations,showing that only one negative eigenvalue of the Hessian existed. In order to check whether the semiempirical hypersurfaces coincide with the ab initio ones, several calculations with fixed C4-HlO distances were performed in thevicinity of the transition state. The results were very similar, and the maximum in the reaction coordinate was obtained at the same distance as in the semiempirical calculations (1.6 A). This geometry was used as input geometry for a transition-state search method implemented in the GAUSSIAN 90 program, with an analytical calculation of second derivatives at the first step. This procedure yielded a transition-state structure very similar to the one obtained with the semiempirical methods. On the product side, the optimized AM1 structure was used as input for the ab initio geometry optimization calculations of the 1-butene radical cation. The final product, depicted in Figure Sb, has a gauche conformation of the carbon framework with a torsional angle cp = 95.2O at the MP2/6-31G** level. The carbon-carbon distances are, at the
The Journal of Physical Chemistry, Vol. 97, No. 49, 1993 12739
Ethene-Ethene Radical Cation Addition Reaction
same level of calculation, 1.41 A (Cl-C2), 1.44 A (C2-C3), and 1.59 A (C3-C4). Asa final stepin thecalculations, theresultsof MPperturbation theory up to fourth order, including spin projection, were obtained a t the MP2/6-31G** optimized geometries. At all levels of the ab initio calculations, the BSSE was calculated by thecounterpoise method. For the UHFcalculations using the 6-31G and 6-31G** basis sets, the BSSE was found to beabout 2 kcal/mol, whereas for the MP2/6-31G** calculations it was only about 0.2 kcal/mol and may thus be considered almost negligible.
In the transition state, the difference between the uncorrelated and correlated energies is even larger, approximately 20 kcal/ mol. This is to be anticipated, since the transition state contains several abnormally stretched bonds, and such bonds are wellknown to be less well described in the Hartree-Fock level. The energy difference between transition state and the final products is, a t all higher levels of approximation, consistently predicted to be 21-23 kcal/mol. For the energy difference between the addition complex and the transition state, which defines the energy barrier toward formation of the final 1-butene radical by means of a 2,4-proton transfer in the primary addition complex, thevariation is, however, larger. One can, in particular, observe that this barrier becomes very small or even vanishes in the MP2/6-3 1G**calculationson the MP2 optimized geometries. The explanation of this unexpected behavior is found in Table 11, which shows a breakdown of the MP2 energy into its different spin components. As seen in Table 11, the contamination of higher spin states in the addition complex is very large, which is shown by the significant deviation of (Sz) from the ideal value of 0.75 (being 0.9105 for the zeroth-order wave function and 0.8547 for the first-order wave function). In contrast, thespin contamination is relatively insignificant a t the other two stationary points of the PES, which in the first-order wave functions have (S2)values of only0.7531 and0.7512, respectively. As alsoseen fromthetable, almost all of this spin contamination is of quartet character, requiring at least three unpaired electrons, corresponding to an orbitally excited state of significantly higher energy than the ground state. As shown in Table 11, the energy increase caused by the quartet contamination is a factor of 5 larger in the addition complex than at the other stationary points. Annihilation of quartet and higher spin contaminations thus lowers the total energy of the addition complex by 6.3 kcal/mol, while the transition state and final product are lowered by only 1.2 and 0.7 kcal/mol, respectively, giving an energy barrier of 4.3 kcal/mol at the PMP2/6-31G** level. Also, the MP4/6-31G** energies are affected by spin contamination, albeit to a somewhat smaller extent. The corresponding energy lowerings obtained by projecting out the unwanted spin components are in this case 4.4,0.7, and 0.4 kcal/ mol, yielding a total energy barrier of 6.1 kcal/mol at the PMP4/ 6-3 lG**//MP2/6-31G** level. A comparison with the results of the two semiempirical methods used, AM1 and PM3, shows that both these methods predict about the same total energy of formation for the final product, the I-butene cation, and that both are in good agreement with the ab initio calculations with electron correlation included. The predictions of the two methods are similar also for the energy of formation of the primary addition complex, while they predict rather different energy barriers toward the 2,4-proton rearrangement. Here, the value 3.5 kcal/mol, predicted by the PM3 method, is in considerably better agreement with the PMP4/ 6-31G**//MP2/6-31G** result than the value 12.4 kcal/mol, predicted by AM 1.
Potential Energy Surface
Comparison with Experiment
In Table I, we list the energies of the three critical points of the potential surface relative to the energies of the isolated reactants-a neutral ethene molecule plus an ethene radical cation-calculated at various levels of theory and including the BSSE correction in the case of the ab initio methods. One can first note that the ab initio energies for the addition complex and for the final product roughly fall into two main groups, those without electron correlation and those with electron correlation included, where the energies in the latter group are approximately 10 kcal/mol lower than those in the first group. This reflects the fact that the number of bonds is increased by one in these molecules (or rather, one *-bond is replaced by a a-bond) compared tothe freereactants, withanassociated increase of the total correlation energy.
The energy profile of the reaction, depicted in Figure 6, shows that the energies of all stationarypoints are lower than the energy of the reactants, independent of the level of calculation. This result thus supports the idea that stable dimeric aggregates are formed in solution.8 Quantiatively, there are some differences between the results obtained with the different computational methods, as discussed above. In the discussion of the mechanism of the reaction, we will in the following use the results from the PMP4/6-31GS*//MP2/6-3lG** calculations. The energy of the intermediate addition complex at the PMP4/ 6-31G**//MP2/6-31G** level is 32 kcal/mol below that of the reactants, and thus its formation must be considered to be an irreversible process which proceeds easily even a t 77 K. The energy of the transition state is a t the same level of theory 26
a
I
b
H Figure 5. MP2/6-31G** optimized geometriesof (a) the transition state and (b) theriactionproduct, the 1-buteneradicalcation(UHF/6-31GSS valucs within parentheses). TABLE I: Relative Energies in (kcal/mol) of the Reaction Species with Respect to the Isolated Reactants Obtained at Different Levels of Calculation relative energies (kcal/mol) addition transition product complex state level of calculation UHF/3-2lG UHF/6-31G UHF/6-31G** MP2/6-31G**//UHF/6-3 lG** MP2/3-21G MP2/6-3 1GS*//MP2/3-21G MP2/6-31G** PMP2/6-31G**//MP2/6-3 1G** MP4/6-3 1GS*//MP2/6-31G** PMP4/6-3 1G**//MP2/6-3 1G** AM 1 PM3
-19.6 -18.7 -17.7 -33.2 -30.3 -28.0 -28.6 -34.55 -27.9 -32.1 -23.5 -22.5
-0.4 -2.2 -8.9 -28.2 -17.0 -27.5 -29.5 -30.25 -25.5 -26.0 -11.1 -19.0
-36.2 -38.55 -39.5 -51.1 -47.1 -50.0 -50.9 -51.1 -48.8 -48.9 -45.2 -48.0
Alvarez-Idaboy et al.
12740 The Journal of Physical Chemistry, Vol. 97, No. 49, 1993
TABLE 11: Total Energies (hartrees) and (9) at the Different Stationary Points of the PES,Obtained at Different Levels of Calculation level of calculation addition complex transition state product MP2/6-3 1G** (s2(2,0)) “1)) PMP2/6-31G**, (S 1) component annihilated (S*(2,1)), (S 1) component annihilated PMP2/6-31GS*, (S 1, S 2) annihilated (s2(2,1)), (S 1, S + 2) annihilated PMP2/6-31GS*, (S 1 -S 3) annihilated (s2(2,1)), (S 1 -S + 3) annihilated
+
+
+
+ + +
+
+
MP4/6-31G**//MP2/6-3lG** PMP4/6-31G**//MP2/6-3lG**, (S + 1) annihilated PMP4/6-31G**//MP2/6-31G**, (S + 1, S + 2) annihilated PMP4/6-31G**//MP2/6-31G**, (S + 1 -S + 3) annihilated Relative Energy (kcal/mole) Reactants
Add. Complex
T.S.
Product
0.0 -10.0
-
-20.0
-
-156.329241 0.9105 0.8547 -156.339535 0.7502 -156.339298 0.7500 -156.339299 0.7500 -156.401094 -156.408336 -1 56.408148 -1 56.408148
-156.330703 0.7615 0.7531 -156.332677 0.7500 -156.332668 0.7500 -156.332668 0.7500 -156.397359 -156.398523 -1 56.3985 14 -156.398514
-156.364876 0.7564 0.7512 -156.365963 0.7500 -156.365959 0.7500 -156.365959 0.7500 -156.434524 -1 56.435 136 -156.435136 -1 56.4351 36
According to the mechanism proposed in the present study, the formation of the 1-butene radical cation is a two-step process. The first step, the formation of the addition complex, proceeds very fast and without any activation energy. This result can thus explain why the isolated ethene radical cation has proven to be very difficult to observe in ESR spectra, even in frozen matrices a t low temperatures.8 The second step involves the C2 C4 hydrogen migration via a transition state to form the 1-butene radical cation. The calculated activation energy for this process is about 6 kcal/mol (PMP4/6-31G**//MP2/6-31G**), which may explain the differences in matrix isolation spectra recorded at 77 and 113 K. The reaction profile for the composite reaction was found to be qualitatively the same, irrespective of the choice of method or basis set. From a quantitative point of view, however, the energies of the different species are very sensitive to basis set, electron correlation, and spin contamination. The latter is true especially for the addition complex, which has a very long (1.9 A) carboncarbon bond. The choice of method could therefore be important when studying systems in which the energy differences between the reactants and the products are small (which is not the case in this reaction), in which case the reaction profile could change qualitatively depending on method used. In general, the present results emphasize the importance of checking the purity of the spin state in reactions involving radicals to avoid an imbalanced spin contamination at different points of the potential energy surface.
-
-30.0 40.0 - -AM1
- - PM3
._.._.
PMP4/6.31 GH//MP2/6-31 G”
-60.0 Reaction Coordinate
Figure 6. Relativeenergies of the addition complex, the transition state,
and the reaction product, calculated at the AM1, PM3, UHF/6-31G**, and PMP4/6-31G**//MP2/6-3lG** levels of theory. The energies of the free reactants are used as zero level. kcal/mol below that of the reactants, so that an energy barrier of 6 kcal/mol separates the addition complex from the reaction product. At very low temperatures the addition complex may thus, in principle, be frozen and thereby observable. This result may explain the observed behavior of the matrix isolation ESR spectra at 77 Ke8 At this temperature, the primary addition complex is trapped in the local energy minimum and very little of the final product can be observed, thus the poor resolution of the observed spectra. When the matrix is warmed to 113 K, the small barrier becomes less important, and the 1-butene radical cation is obtained with a large yield resulting in the correspondingly high resolution of the ESR spectra. The 1-butene radical cation is 49 kcal/mol more stable than the reactants and 17 kcal/mol more stable than the intermediate addition complex. The big difference in energy thus guarantees the total conversion of the reactants into the product, once the thermal energy is large enough to overcome the small barrier of the transition state.
Conclusions In the present study, the first step in the radiation-induced polymerization process of ethene-the formation of the 1-butene radical cation by the addition of a neutral ethene molecule and an ethene radical cation-has been investigated by means of semiempirical (AM1 and PM3) and ab initio UHF, MP2, and MP4 methods, without and with spin projection included. The geometries of the reactants, the addition complex, the transition state, and the product have been optimized at semiempirical and ab initio UHF/3-21G, UHF/6-31G, and UHF/6-31G** levelsoftheory, aswellas MP2/3-21Gand MP2/ 6-3 1G**.
Acknowledgment. This research was supported by the Swedish Natural Science Research Council (NFR) and by the Swedish Agency for Research Cooperation with Developing Countries (SAREC). A grant of computer time a t the National Supercomputing Centre (NSC) in LinkiSping is gratefully acknowledged. References and Notes (1) Mulliken. R. S. Tetrahedron 1959, 5, 253. (2) Handy, N. C.; Nobes, R. H.; Werner, H. J. Chem. Phys. Lcrt. 1984, 110, 459. (3) Lunell. S.:Huana, M.B. Chem. Phw. Lett. 1990. 168. 63. (4) Lunell, S.; Eriksson, L. A.; Huang, M. B. Mol. Srmcr. (THEOCHEM) 1991, 230, 263. ( 5 ) Meerer, A. J.; Schoonveld, L. Can. J . Phys. 1969, 47, 1731. (6) KBppel, H.; Domcke, W.; Cederbaum, L. S.;von Niessen, W. J. Chem. Phys. 1978, 669,4252. (7) Shiotani,M.; Nagata, Y.;Sohma, J. J . Am. Chem. Soc. 1984, 104, 4640. (8) Fujisawa, J.; Sato, S.; Shimokoshi, K.Chem. Phys. Lcrr. 1986, 224, 391. (9) Shiotani, M.; Sjdqvist, L. Private communication. (IO) Dewar, M.J. S.;Thiel, W.J . Am. Chem. Soc. 1977, 99, 4899. (11) Belville, D. J.; Bauld, N. L. J . Am. Chem. Soc. 1982, 104, 294. (12) Frisch, M. J.; Head-Gordon, M.; Trucks, G. W.;Foresman, J. B.; Schlegel,H. B.; Raghavachari, K.; Robb, M. A.; Binkley, J. S.;Gonzalez,C.; Defrees, D. J.; Fox, D. J.; Whiteside, R. A.; Seeger, R.; Melius, C. F.; Baker J.; Martin, R. L.; Kahn, L. R.; Stewart, J. J. P.;Topiol, S.; Pople, J. A: Gaussian 90,Gaussian, Inc.: Pittsburgh, PA, 1990.
i.
Ethene-Ethene Radical Cation Addition Reaction (13) Dupuis, M.;Spangler, D.; Wendoloski, J. J. Narional Resourcefor Computations in Chcmfsrry S o f w a n Catalog; University of California: Berkeley, CA, 1980; Program QGO1. Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Kmki,S.;Jensen,J. H.;Gordon, M.S.;Nguyen, K.A.; Windus, T. L.; Elbert, S . T. QCPE Bull. 1990,10, 52. (14) Dewar, M.J. S.;Zoebisch, E.G.;Healy, E.F.; Stewart, J. J. P. J . Am. Chem. SOC.1985,107,3902.
The Journal of Physical Chemistry, Vol. 97, No. 49, 1993 12741 (15) Stewart, J. J. P.J . Comput. Chem. 1989,10, 221. (16) Stewart, J. J. P.Program No. 581, Quantum Chemistry Program Exchange, Indiana University, Bloomington, IN. (17) Boys, S. F.; Bemardi, F. Mol. Phys. 1970, 19, 553. (18) Schwenke, D. W.; Truhlar, D. G. J . Chem. Phys. 1985,82, 2418. (19) F r k h , M.J.; DclBcne, J. E.; Binkley, J. S.; Schaefer 111, H. F. J . Chem. Phys. 1986,84,2279.
