J. Phys. Chem. 1991, 95,8726-8731
8726
( ~ 2 % The ) ~ratio ~ ~of vibrational partition functions was evaluated by considering only the new vibrations that arise in the complex, it being assumed that all other vibrations cancel in the ratio. The following metal-ligand vibrational frequencies (cm-') were used: CuCO, 363,200, and 200; CuC2H2,480 and 349; and CuC2H4,390,352, and 227. The 363-cm-' frequency for CuCO Conclusions is from an ab initio calculation;2othe 200-cm-' bending mode frequency is estimated. Frequencies for CuCZH2are based upon Our results show that copper atoms form diligand complexes ab initio calculations3' of the normal mode frequencies of the with CO, C2H2,and C2H4 in the gas phase near room temperature. aluminum/acetylene complex AIC2H2. Those for CuC2H4are The binding energies of the intermediate, monoligand complexes based upon a combination of experimental'* and the~retical'~ are in the range 25 f 5 kJ mol-' in all cases. A simple kinetic results for AIC2H4and experimental results"*18for Cu(C2H4) model that incorporates the effects of different third-body cdlision complexes. The magnitude of the quantity 6 in eq 18 depends eficiencies for the reactant gas and Ar has been shown to provide upon the average internal energies of Q and CuQ. Contributions a reasonable description of our results, from which binding energies from vibrations were obtained by a standard method32using the for the monoligand complexes and third-order rate constants for estimated metal-ligand vibrational frequencies noted above. the first ligand addition reactions may be derived. SimpMed Unimdecuhr RateCaladaWUw the Fonnrlism Acknowledgment. We thank Dr. H. van den Bergh and D. of Troe."~~~ Structural parameters and vibrational frequencies Braichotte of Institut de Chimie Physique, &ole Polytechnique of the monoligand complexes were obtained as described above. FgdCrale, Lausanne, Switzerland, for the preparation and puriThe frequencies for CuC2H2were assumed to be identical with fication of the C U ( C ~ H F ~ O ~ ) sample. ~-~H~O those found from an ab initio calculation for A1C2H2.'' For CuC2H4,the following frequencies (an-')were used: 3095,3090, Appendix 3003,2993,1443,1393,1198,1194,829,788,706,686,390,352, and 227. In the notation of T r ~ e ,we ~ used ~ * the ~ ~following for Partitha F"for M o a o Complexes. ~ Estimates of the Cu/CO (s = 4, m = 3), Cu/C2H2 (s = 9, m = 2), and the ratios [Zo~Q)Ivib/[Zo(CUQ)Iviband [ZO(Q)lmt/[Zo(fiQ)Imt Cu/C2H4 (s = 15, m = 3) association reactions at 295 K: Zu of vibrational and rotational partition functions are required in = 6.6 X lo-'' cm3 s-' in all cases, Fsnb= 1.73, 1.14, and 1.1 1; eq 17. The ratio of rotational partition functions follows directly from the molecular structures. For CuCO2Aa and C U C ~ Hwe~ , ~ p(E0 = 25 kJ mol-!) = 16.6, 8.65, and 31.0 (kJ mol-l)-I; FE = 1.21, 1.30, and 1.37; and Fa = 2.44,4.29, and 3.85, respectively. used structural parameters derived from ab initio calculations. Fra was calculated by using the method of ref 25. The results CuC2H2was assumed to have the same Cu-C bond distance and for W, are given in Table VI1 under the heading k(3)q(calc). C-C-H angle as CuC2H4. The C-C bond distance of CuC2H2 was assumed to have the same fractional increase relative to free R ~ s NO. Q CU,7440-50-8; CO, 630-08-0; CZH2, 74-86-2; C2H4, C2H2as found theoretically for CuC2H4compared with C2H4 74-85-1; CUCO,55979-21-0; CUCZHZ,65881-80-3; CUC~H~, 60203-82-9. at least -50 kJ mol-'. A larger binding energy for the second ligand compared with the first ligand is consistent with simple considerationsof bonding mechanisms.1s.'6 The diligand complexes may react further to add a third ligand, but our results provide no information on this process.
Integtai Cross Sections tor H 4- H2(0,0) --* H2(0,/') 4- H: Comparison of Quasic(aegical and Quantum Results M.E.Mandy**+ Department of Chemistry, Lash Miller Chemical Laboratories, University of Toronto, Toronto, Ontario, M5S 1Al Canada
and P.G . Martin Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, Ontario, M5S I AI Canada (Received: March 20, 1991; In Final Form: May 10, 1991)
-
State-testate integral cross sections for the exchange reaction H + H2(0,0) H2(0j') + H have been calculated with quasiclassical trajectories (QCT) on the LSTH potential energy surface. Microscopic reversibility permits an alternative, indirect, means of calculating cross sections based on trajectories starting in the (04') state in addition to the usual direct one based on trajectories starting in (0.0). Thaw QCT cross sections are compared with one another and with recent exact quantum calculations at relatively low total energy (- 1 eV). We identify some systematic effects relating to the exchange barrier in the potential, which is important for defining the threshold for low j'. Some differences between direct and indirect cross sections close to threshold arise from the binning procedure by which the outcomes of quasiclassical trajectories are assigned to quantum states. Overall we find that indirect QCT agrees best with the quantum results near threshold and that the agreement improves as j'increases. Therefore, at least for this particular reason, we recommend the use of indirect QCT when quantum calculations are unavailable or intractable. Some additional QCT cross sections were calculated by using the DMBE surface. The relative agreement between QCT and quantum calculations persists. Differences between the DMBE and LSTH results seen in both QCT and quantum methods indicate rather similar tunneling contributions in the energy range available for comparison. introduction For interactions involving a large total energy ( E >~2 ev), ~ the quasiclassicaltrajectory method (QCT) is presently the only Present address: Department of Chemistry, Dalhousie University, Halifax, Nova Scotia B3H 335 Canada.
tractable means of obtaining the cross sections for even small systems like H + H2; QCT does not increase in difficulty with increasing energy. BY contrast, it is not Yet possible to determine all desired state-testate cross sections with 3-D quantum calculations, because the number Of channels necessary for convergence It is therefore desirable to have a increases rapidly with &,,.I
0022-3654/91/2095-8726S02.50/00 1991 American Chemical Society
H
+ H2(0,0)
-
HZ(0j')
+H
The Journal of Physical Chemistry, Vol. 95, No. 22, 1991 8727
clear sense of how the results of QCT compare to those of quantum calculations. Agreement of QCT with quantum calculations at both the rate constant and cross section levels is sensitive to factors such as the potential energy surface and masses of the species invol~ed.~-~ Earlier work has included comparison of QCT and exact quantum results in restricted dimensionality with approximate potential energy surfaces64 and comparison of QCT rate constants with predictions of variational transition-state theorya9 Compared here are new fully three-dimensional QCT calculations of state-testate cross sections for H H2on a chemically accurate potential energy surface and corresponding recent exact quantum results. The H + H2 system has been well studied both theoretically and experimentally for over 50 years.l0 In recent years experiment and theory have converged to the point that theorists and experimentalists are comparing results directly with one another both for H + Hzll and its isotopic combinations.'"" A recent feature article in this journal15describes recent quantum calculations by various groups that have been carried out in effort to clarify whether it is possible to observe resonances in experimental ~tudies.'~J' These quantum calculations were carrried out on the chemically accurate LSTH'* and DMBEI9 potential energy surfaces and have included determination of fully converged three dimensional integral cross sections for state-testate transitions involving the para-to-ortho exchange reaction. Although most of the calculations have focused on particular product states that have been studied in the experiments, i.e., H H2(u=OJ=O) H2(u'= 1j'= 1,3) + H, cross sections for a limited number of other states in the u' = 0 manifold have been determined as We compare the QCT and exact quantum state-to-state cross sections for the reaction H + H2(0,0) H2(0j') H over the low-energy range (Eio,up to 1.35 eV). Our QCT calculations are part of a systematic study of state-testate integral cross sections and rate constants for a p plication to the study of the excitation and dissociation of warm (> 1000 K) interstellar H2. Transitions involving large energy changes as well as transitions from and among even highly excited ( u j ) states can be i m p ~ r t a n t . ~However, ~ . ~ ~ because of the low density of the gas and deexcitation by quadrupole emission, most of the population will be in low ( u j ) states. Collisional transitions among these low ( u j ) states will dominate the kinetics, particularly the exchange kinetics. In the absence of protons, transitions
+
-
+
-
+
(1)Chapman, S.;Green, S.J . Chem. Phys. 1977,67,2317. (2) Neuhauser, D.; Judson, R. S.; Jaffe, R. L.;Baer, M.; Kouri, D. J. Chem. Phys. Left. 1991, 176,546. (3) Schatz, G. C. J . Chem. Phys. 1985,83,5677. (4)Schatz, G. C.J. Chem. Phys. 1985,83,3441. ( 5 ) Schatz, G. C.; Kuppcrmann, A. J . Chem. Phys. 1976,65,4668. (6)Bowman, J. M.; Schatz, G. C.; Kuppermann, A. Chem. Phys. Lett. 1974,24,318. (7) Bowman, J. M.; Kuppcrmann, A. Chem. Phys. Left. 1973, 19, 166. (8) Schatz, G. C.J . Chem. Phys. 1983, 79,5386. (9)Blais, N. C.;Truhlar, D. G.; Garrett, B. C. J. Chem. Phys. 1983, 78, 2363. (10)Proceedings of the Farkas Symposium: 50 Ycars of H + H2 Kinetics, published as a spccial issue of Inr. J . Chem. K i m . 1986, 18. (11) Borman, S.Chem. Eng. News 1990,68,32. (12)Schatz. G. C.Chem. Phys. Lerr. 1984, 108, 532. (13)Blais, N . C.;Zhao, M.; Mladenovic, M.; Truhlar, D. G.; Schwenke, D. W.; Sun, Y.; Kouri, D. J. J . Chem. Phys. 1989,91,1038. (14)Zhao, M.; Truhlar, D. G.; Blais, N . C.; Schwenke, D. W.; Kouri, D. J. J . Phys. Chem. 1990, 94,6696. (15)Miller, W.H.;Zhang, J. 2. H. J . Phys. Chem. 1991, 95,12. (16)Nieh, J. C.;Valentini, J. J. Phys. Rev. Len. 1988,60, 519;J.Chem. Phys. 1990, 92,1083. (17)Kliner, D. A. V.;Adelman, D. E.; a r e , R. N. J . Chem. Phys. 1991, 94,1069. (18) (a) Liu, B. J . Chem. Phys. 1973,58, 1925. (b) Siegbahn, P.; Liu, B. J . Chem. Phys. 1978,68,2457.(c) Truhlar, D. G.; Horowitz, C. J. J. Chem. Phvs. 1978. 68. 2466: 1979. 71. 1514. 119) Varandas, A. J . C.; Brown,-F. B.; Mead, C. A,; Truhlar, D. G.; Blais, N . C. J . Chem. Phys. 1987,86,6258. (20)Zhang, J. 2. H. Private communication. (21)Manolopoulos, D. E.;Wyatt, R. E. J. Chem. Phys. 1990, 92,810. (22)Dove, J. E.;Mandy. M. E. Asrrophys. . . J . 1986.311. L93 and references therein. (23) Chang, C. A.; Martin, P. G. Asrrophys. J . 1991, 378,202.
TABLE I:
b
r
g
j
E:eV
0
0.269 0.284 0.351
1 3
bels ~ Of H,(OJ) AEb 0.015 0.081
AEb
j
E:eV
5 7
0.485 0.664 0.889
9
"Relative to potential energy minimum of
0.215 0.395 0.619
H1.bRelativc to (0,O).
between ortho and para states of the H2 molecule can still occur as the result of reactive collisions with H. The H H2 system is one for which the fundamental differences between quantum and quasiclassicalapproaches might be expected to be manifest. The atoms involved are light in mass and thus the system is further removed from the classical limit. Energy levels of the H2 molecule are widely spaced, too much so to be treated as a continuum. Although in the quasiclassical method trajectories are initiated in conditions corresponding precisely to a ( u j ) quantum state, all variables are treated as continuous throughout the trajectory. Trajectories leading to exchange must cross an energy barrier. On the LSTH'* potential energy surface, the height of this barrier changes from 0.424 eV in the collinear conformation to 2.75 eV in the T conformation. The energy range of the aforementioned experimental and theoretical studies is such that tunneling through the exchange barrier might be expected to make a contribution. The classical calculation by its very nature excludes tunneling. As described in the Discussion, there is also the question of zero-point energy leak in the QCT approach. Low-energy exchange transitions involving only the states (Oj
H2(0.0)
H2(0.1)
+
H
H
t " " ' " " ' 1
+
H2(0.0)
--> H2(0.3)
+
H
a
st 0
t
1 u
m " . M
0.6
0.4
H
+
H,(O,O)
+
-> H,(0.5)
0.8
H
H
+
H,(O.O)
1 .o
->
H.JO.7)
1.2
+
1.4
H
t " " " " " 5: a
5 E 2
"5
s .-6
3
t,
0, 0
0
0.4
0.6
0.8
Eto,
1.o (4
1.2
1.4
e 0
a .% E
.
-e s.-. k
k'l
6" c ._ . P 5
.
v)
I
;:I
.
M
2.: 0
0
0.4
0.6
0.8
1 .o
1.2
1.4
Figure 1. Integral cross sections as a function of total energy. Solid line: quantum results of Zhang and Miller" using the LSTH surfacc. Dashed line joining points with error bars: direct upward QCT results using the LSTH surface. Dotted line: indirect cross sections (using microscopic reversibility) from downward QCT (LSTH). Beaded line: quantum cross sections of Manolopoulos et aL*' using the DMBE surface. Filled squares: direct upward QCT using the DMBE surface. Filled triangles: indirect QCT (DMBE). Arrow: total energy required classically for the transition, the larger of the energy of the final state or the barrier in the collinear conformation.
Such contributions a t low energy cannot be eliminated simply by adjusting the binning of trajectory outcomes; this effect seems to be consistent with ZPE leak. As we have mentioned, removal of this leak is in general problematical (lack of tunneling would persist too). It might be that removal of this leak would reduce
the QCT cross sections to below the quantum results near threshold, pointing to an important tunneling contribution. At a somewhat higher energy (0.8 eV), the QCT results are below the quantum values; this might signal a tunneling contribution not cancelled by the effects of ZPE leak (note in Figure 2a that
8730 The Journal of Physical Chemistry, Vol. 95, No. 22, 1991 H
+
Hz(O,O)
->
+
H2(0.1)
Mandy and Martin
H
H ?
[
'
+
'
I
H2(0.0) I
->
HJ0.3) '
I
'
+
H
'
'
1; I
I
b '
H
+
Hz(O,O)
-> H2(0.5)
+
H
0
C
0
1
0.6
al4
.
1
.
Elol H
+ Hz(0,O)
--> H2(0,9)
+
1
1 .o
0.8
.
I
1 .?
.
I
J
1.4
(4
H
0
9
4
I L
.
!
.
!
by this energy (j') has approached the center of range for this histogram bin). At somewhat higher energies still, all results come into good agreement. For the transition (0,O) (0,3),both QCT estimates of the cross section are below the quantum calculation. The quantum dynamic thteshuld is about the same as for the transition (0.0) (O,l), which is expected if the barrier dominates the behavior, but the dynamic threshold in QCT is somewhat higher. This
-
-+
L
I
1 1 ,
I
,
,
!
'I_ .
!J
possibility indicates a greater tunneling contribution, not compensated by ZPE leak. For the states withi', 5, the energy of the final state is greater than that of the barrier in the cottinear conformation, and so ultimately it becomes the primary factor affecting the dynamic threshold The effects of ZPE leak and tunneling should become less important. For (0,O) (0,5), the direct and indirect QCT cross sections bracket the quantum values near threshold. This
-
8731
J . Phys. Chem. 1991,95, 8731-8737
-
evolution continues as j’increases, such that by (0,O) (0,9), the indirect estimates are in close agreement with the quantum ones, whereas the direct ones are systematically high. As we have discussed in connection with Figure 2, it appears empirically within the bin histogram method that when trajectories with low j ‘ (relative to the center of the bin) dominate the contributions to the direct upward cross section, we have a sufficient condition for QCT to overestimate the cross section near threshold. The other commonly used method of determining cross sections from quasiclassical trajectories is the smooth histogram method35of Truhlar et al., which in fact aggravates this type of overestimation of direct upward cross sections in the threshold region even further, since trajectories of even lower j’are allowed to contribute to the cross section to higher states. Where the barrier is not dynamically important, QCT estimation of upward cross sections indirectly appears to minimize the systematic effects due to binning. We have made the same observation for (0,O) (~’=l,j’=1,3,5).*~ Sometimes experimental and theoretical cross sections (or relative populations) are presented as a function ofj’for fixed values of E,,, and 0’. For example, see Figure 6 in Barg et al.,36 Figure 18 in Buchenau et al.,” or Figure 12 in Rinnen et which all compare upward direct QCT cross sections with quantum or experimental results. They show that the results from direct QCT are higher at (or extend to) higher j’. Examination of our sequence of Figure 1, parts a-e, in which j’increases from 1 to 9 will show the form of such a diagram for our results. Using E,,, = 1.1 eV as an example, we see that the direct QCT cross section is slightly lower than the quantum curve for the lower j’ but higher for larger j’, particularly for j’= 7,9. Thus our direct cross sections will also appear rotationally hot with respect to the quantum results. Note however that this can be alleviated by use of indirect cross sections which eliminates the overestimation at higher j ’ (fixed E,,,). DMBE Potential: QCT-Exact Quantum Comparison. Manolopoulos et aL2’ carried out the calculation of integral cross sections in the energy range 0.95-1.35 eV for (0,O) (0,j’= 1,3,5,7) on the DMBE surface. These are also plotted in Figure 1. For j ’ = 1, 3 their cross sections are consistently higher than
-
-
(35) Blais, N. C.; Truhlar, D.G. J . Chem. Phys. 1977, 65, 5335. (36) Barg, G.-D.;Mayne, H. R.; Toennies, J. P.J. Chem. Phys. 1981,74, 1017. (37) Buchenau, H.; Toennies, J. P.; Arnold, J.; Wolfrum, J. Ber. BunsenGes. Phys. Chem. 1990, 94, 1231. (38) Rinnen, K. D.; Kliner, D. A. V.; Zare, R. N. J . Chem. Phys. 1989, 91. 7514.
the corresponding quantum LSTH calculations. The interpretation is that since the collinear bamer on the DMBE surface is slightly lower than that on the LSTH surface, exchange occurs more easily. By j ’ l 5, their cross sections agree closely with the LSTH results. For these transitions the collinear barrier is less important dynamically and away from this barrier the two surfaces at low energy are very similar.39 The good agreement of exact quantum cross sections for (0,O) (v’=l, j’=l,3) bears this out.39 We have calculated QCT cross sections at a few energies using the DMBE surface (see Figure 1). These QCT results are shifted with respect to the corresponding QCT calculations using the LSTH surface by approximately the same amount that the two quantum results were. In this limited energy range somewhat above the threshold for exchange, it appears that tunneling contributions for the DMBE surface are not larger than those for the LSTH surface. Recommendations. Where the dynamic threshold is affected by the barrier height, both direct and indirect QCT appear to suffer from the effects of ZPE leak. For cases where the dynamic barrier is dominated by the energy difference between initial and final states, direct (upward) cross sections near threshold are overestimated by the histogram binning technique used, whereas the indirect ones are not. We therefore recommend that in the absence of quantum results the indirect cross sections should be adopted near threshold. This is consistent with the recommendations of Ashton et al.,27 arrived at through consideration of factors affecting reaction barriers, of Leasure et a1.28 in the consideration of dynamic thresholds, and of Blais et al.? arrived at through comparison of thermal rate constants with those from variational transition state theory. At higher energies where the indirect and direct cross sections agree within their statistical errors, a weighted average can be taken.
-
Acknowledgment. We thank J . Z. H. Zhang for kindly making available his unpublished quantum results, W. J. Keogh for help with the DMBE calculations, and P. W. Brumer and J. P. Valleau for useful discussions. We acknowledge the contributions of J. E. Dove, sadly now deceased, to the formative stages of this work. This work was supported by grants from the Natural Sciences and Engineering Research Council of Canada and the Connaught Fund of the University of Toronto. Registry NO. H,12385-13-6. (39) Auerbach, S. M.;Zhang, J. Faraday Tram. 1990,86, 1701.
Z. H.; Miller, W. H. J. Chem. Soc.,
Gas-Phase Aromatic Substitution. Reactivity of (Trifluoromethoxy)benzene toward Charged Eiecttophiies Fulvio Cacace,* Maria Elisa Crestoni, Annito Di Marzio, and Simonetta Fornarini University of Rome ‘La Sapienza”, P. le Aldo Moro, 5, 00185 Rome, Italy (Received: March 22, 1991: In Final Form: May 28, 1991) The reactivity of C6H50CF3toward gaseous cations, including C2H5+,i-C3H7+,(CH3)2F+,and CH30(H)N02+,has been investigated with mass spectrometric and radiolytic techniques. Two distinct kinetic patterns emerge, depending on whether formation of an early electrostatic adduct between the reactants is rate determining. In this case, which occurs in ethylation and isopropylation, the reactant displays positional but not substrate selectivity, e.g., i-C,H7+ is characterized by a kvpCF,/kw ratio of 1 .O and by a 77% ortho, 12% meta, and 11% para orientation. By contrast, methylation and nitration display both substrate and positional selectivity, e.g., CH30(H)N02+is characterized by a kCsH50CF3/kC6Hs ratio of 0.096 and by a 35% ortho, 5% meta, and 60% para orientation. Only the latter reactions lend themselves to construction of free-energycorrelations, e.&, a up+value of +0.12 has been calculated for the nitration of CsHSOCFoby CH30(H)N02+in CH, at 37.5 OC, compared with the +0.07 value from the nitration by N02+BF4-in solution. The gas-phase reactions are compared with the solution-chemistry counterparts, and their distinctive features are discussed. The mechanism of electrophilic aromatic substitution has been the focus of sustained interest for over a century and a cornerstone
in the development of physical organic chemistry. Far from declining, the interest in this class of reactions has recently ’been
0022-3654/91/2095-873 1%02.50/0 0 1991 American Chemical Society
