Chapter 4
Isotope Effects in Addition Reactions of Importance in Combustion Theoretical Studies of the Reactions CH + H
CH *
2
3
CH + H 2
Albert F. Wagner and Lawrence B . Harding Chemistry Division, Argonne National Laboratory, Argonne, IL 60439
Ab initio electronic structure characterizations of the addition reaction path for the title reaction are described. Variational R R K M calculations employing the reaction path properties are then used to compute thermal rate constants for comparison to kinetics measurements on the title reactions and its isotopic variation. In combustion chemistry, both addition and abstraction reactions have important roles to play. However, addition reactions, and their reverse, dissociation reactions, often provide a more complex chemistry because of the presence of long-lived metastable adducts that can be stabilized in the presence of third-body collisions or eliminate to produce a variety of products. Addition reactions frequently involve barrier-less reaction paths, often involve close competition in adduct decay between atomic or molecular elimination, and often can involve electronic excitation in the reactive chemistry. The particular reaction CH(2n) + H
2
~
CH * ![M] CH 3
3
CH ( BO + H 2
(la) (lb)
3
where M is a third body and the * indicates metastability, demonstrates all of these features. Consequently this reaction has been the subject of several different types of experimental studies and a variety of theoretical studies as well. Experimental kinetics studies have measured the thermal rate constants for addition in both directions (1-14). Isotope effects on three variants of reaction (1) have been performed (6 -10): CD+D
CH+D
2
2
-
^
CD * 3
CHD * 2
->
CD + D 2
(2a)
->
CD
(2b)
->
CD + H
(3a)
->
CHD
(3b)
3
2
2
0097-6156/92/0502-0048$06.00/0 © 1992 American Chemical Society
4. WAGNER & HARDING
CH+D
CD+H
2
2
Isotope Effects in Addition Reactions in Combustion
-
—
CHD * 2
CHD * 2
->
CD + HD
(3c)
—>
CHD + D
(3d)
—>
CDH + H
(4a)
->
CDH C H + HD CH + D
(4c)
(4b)
2
—> —»
(4d)
2
The thermal dissociation rate constant (15) for reaction (1) has also been measured. Non-kinetics measurements using molecular beams (16-18) have characterized the state-resolved reactive and inelastic cross sections for reactions (1) and (3). Spectroscopy of CH3 and all its isotopic variants have been measured with special attention (19,20) to the out-of-plane umbrella motion which has quartic characteristics. Reaction (1) has also been the subject of several theoretical studies and is quite accessible to reasonably rigorous theory due to the few number of electrons involved. Several of the earliest electronic structure studies pointed out that reaction (1) is the simplest reaction with a non-least-motion pathway (21), i.e., C H does not insert into H along a C path. Several ab initio electronic structure studies have mapped out the general nature of either the CH + H (21-22) or the C H + H (23) reaction path. However, these pioneering studies employed a relatively small basis set and modestly correlated wave function, resulting in limited accuracy in the reaction path characterization. A l l previous dynamics studies have focused on the calculation of rate constants. The earliest study (6) did not have the benefit of electronic structure characterizations of the reaction path. The other (23) did employ computed reaction path characteristics but examined only the high pressure limit of the reverse reaction (la). Recently, a new series of theoretical studies (24, 25), of which this is the third, have begun to examine the reaction path characteristics (via electronic structure calculations) and the kinetics [via Rice-Ramsperger-Kassel-Marcus (RRKM) calculations] of reactions (1) - (4). The first study in this series, hereafter called Paper I, gave a description of the Multi-Reference Singles and Doubles Configuration Interaction (MRSDCI) electronic structure method used and the resulting characterization of the fragments C H , C H , C H 3 , H , and the planar C H + H addition reaction path at the harmonic level. The high-pressure limiting rate constant of reaction (-la) was also provided. The second study in this series, hereafter called Paper II, provided a preliminary harmonic description of the C H + H reaction path and of the anharrnonic out-of-plane motion along both reaction paths. In this paper, a brief review of the electronic structure theory method used in the work will be provided. A fuller description of the out-of-plane anharrnonic motion will be discussed. Variationally R R K M theory will then be applied to produce rate constants for comparison to the experimental results on reactions (1) - (4). 2
2 v
2
2
2
2
2
2
Details of Electronic Structure Calculations As fully described in Papers I and II, all MRSDCI electronic structure calculations in this work were performed with the COLUMBUS program system (26). The multireference wavefunction, the configuration interaction, and the basis set will now be briefly reviewed. The standard multi-reference wavefunction used in these studies is a FORS/CASSCF type (27) with 1 inactive orbital (the C(ls) orbital) and 7 active molecular orbitals correlating the remaining 7 electrons. This wave function, written
49
50
ISOTOPE EFFECTS IN GAS-PHASE CHEMISTRY
in abbreviated form as (7mo/7e), results in 784 configuration state functions (CSFs), an expansion which is well within the capabilities of modern M C S C F methodology. At planar geometries, the above reference space reduces to the direct-product wave function (6mo/6e)x(lmo/le), because the radical orbital must remain singly occupied in order for the total wave function to possess the required A symmetry. This roughly halves the reference expansion length to 364 CSFs. The singles and doubles configuration interaction calculations are based on either the (7mo/7e) or (6mo/6e)x(lmo/le) reference wavefunction. Configuration interaction calculations performed with both reference wavefunctions at selected geometries of CH+H2 showed differences that never exceeded 0.1 kcal/mole. A l l energy changes due to out-of-plane motion from planar geometries consistently use the (7mo/7e) reference in Papers I and II and also here. In papers I and II, a Qorrelation-£onsistent polarized yalence jriple zeta (cc-pVTZ) orbital basis set (28) was used to characterize the planar reaction path and equilibrium reactant and adduct properties at largely a harmonic level. In this paper, extensive non-planar, non-harmonic calculations along both reaction paths will be presented. Because of the expense of such calculations, a smaller basis set, cc-pVDZ, was used. This basis set has dissociation energies for CH3->CH+H2 and the CH3—>CH2+H about 3 and 5 kcal/mole smaller, respectively, than the energies for the cc-pVTZ basis set which in turn are very close to experiment (as will be discussed later). However, the computed reactant frequencies differ by no more than 50 cm" , the reactant geometries are essentially identical, and the variation from the asymptotic value of the potential energy along the reaction path differs by no more than a few tenths of a kcal/mole between the two basis sets. Consequently, previous cc-pVTZ and current cc-pVDZ MRSDCI calculations will be carefully mixed together in the reaction path characterizations used in the rate constant calculations. The absolute energetics separating reactants, products, and adducts will be taken from the cc-pVTZ calculations. In addition to the MRSDCI energies, the use of various multireference Davidson corrections (30) has also been examined. Such empirical corrections are used to estimate the contributions of higher-order excitations. However, since the reference space itself is quite large in the cc-pVTZ calculations, these corrections appear to overestimate the importance of higher-order corrections and were not used. In the ccpVDZ calculations, these corrections were used and generally gave comparable energies to cc-pVTZ where comparisons between the two calculations were made. M
1
Reaction Path Results In order to fully describe the kinetics, the reactants, products, adduct, and all reaction paths between them must be described. A l l reactant, product, and adduct theoretical descriptions are taken from Paper I and summarized in Table I. The frequencies listed in the table are harmonic frequencies. The harmonic representation will be used for all vibrational motion except out-of-plane motion along the reaction path. As indicated in Table I, the agreement between theory and experiment is quite satisfactory. The energetics separating adduct, reactants, and products are all taken from the cc-pVTZ calculations of Paper I and are summarized in Table II. The energies listed in the table all include zero-point corrections, i.e., they are enthalpies at 0 K. In computing the zero-point corrections for the theoretical entries to the table the harmonic frequencies in Table I are used. Two theoretical entries are listed: one labelled "ab initio" and the other "adjusted". The ab initio column is directly calculated by the electronic structure calculations described in Paper I. By comparison with the experimental J A N A F entries (31) in the table, the directly
4. WAGNER & HARDING
Isotope Effects in Addition Reactions in Combustion
computed values are one or two kcal/mole below the nominal experimental values and only outside the error bars for experiment in the case of the dissociation energy for CH3—»CH2+H. The adjusted column represents allowing slight changes in the computed energetics in optimizing the agreement in the computed and measured rate constants (to be described later). These show only one kcal/mole or less adjustments. The small variations between ab initio values and adjusted or experimental values testifies to the intrinsic accuracy of the electronic structure calculations and implies no more than a 1 % underestimation of the dissociation energies of CH3. Table I. Calculated and observed properties of H2, C H , CH2, and CH3 -1
H2: theory exp't
r (Â) 0.745 0.7412
C H : theory exp't"
1.125 1.120
C H : theory exp'c
1.082 1.0766
133.38 134.037
3153 2985
3361 3205
CH3: theory exp'td
1.0821 1.Q790 1.0790
120.000 120.000
3126 3270
3231,3231 3285-3297
e
a
2
O (deg.)
harmonic frequency (cm ) 4185 4401
e
2719 2859 1143 963 1446,1446 1436-1447
520.0 495-545
a
B. Rosen, Spectroscopic Data Relative to Diatomic Molecules (Pergamon Press, New York, 1970). G . Herzberg and J. W. C. Jones, Astrophys. J. 158, 399 (1969). Bunker, P. R.; Jensen,Per; Kraemer, W. P.; Beardsworth, / . Chem. Phys. 1986, 85, 3724. Schatz, G. C ; Wagner, A . R ; Dunning, Jr., T. H.; J. Physical Chem. 1984, 88, 221. b
c
d
Table II.
Calculated and observed reaction enthalpies at 0° Κ
process
3
theory (ab initio)
C H -> C H + H 3
2
C H -> C H + Η 3
2
C H + Η -> C H + H 2
2
3
theory (adjusted)
experiment (Ref. 31)
102.7
104.0
105.6 ± 4 . 4 kcal/mole
106.7
107.75
108.3 ± 1.2
4.0
3.75
2.7 ± 5.2
Zero-point corrections made using harmonic frequencies in Table I. The harmonic characterization of the planar reaction paths for CH+H2 or CH2+H addition have been described in Papers I and II and in paper II a cursory characterization of the out-of-plane motion in both paths was presented. Extensive characterizations of the out-of-plane motion have resulted in a somewhat more precise description for CH+H2 but a substantially revised description of the entire reaction path in the case of CH2+H. This channel is found to have a new-planar reaction path, even though CH3, both in experiment and in the calculations, has a planar equilibrium structure. The results of these new calculations will be presented below. A more detailed account of these results is in preparation.
51
52
ISOTOPE EFFECTS IN GAS-PHASE CHEMISTRY
CH + H
2
Reaction Path
As discussed in Paper II, the CH+H2 reaction path is planar in the kinetically important region but does not follow the least motion C pathway where the C H bond inserts along the perpendicular bisector of the H H bond. Rather, at large distance of separation along the reaction path, the C H bond is aligned nearly parallel with the H H bond but displaced in the parallel direction with the C atom approximately located on the bisector of the HH bond, as in ' — I where the line on the left is CH while the line on the right is HH. Unlike this schematic representation, the C H and H H bonds are not perfectly parallel to each other. The C H bond has a slightly acute and the H H bond has a slightly obtuse angle with respect to the vector R between C and the center-of-mass of H and the dotted line in the schematic. Only well down into the potential well does the reaction path incorporate the angular motion that leads to the symmetric configuration of CH3. The out-of-plane motion along the reaction path can be thought of as the dihedral motion of the H-C-(center-of-mass of H ) plane relative to the C - H plane, i.e., the twirling of the CH bond about the vector R. Schematically, a 180° variation in this dihedral angle would correspond to ' · · · I -> | · · · I. If all the other degrees of freedom perpendicular to the reaction path are allowed to relax during the much slower dihedral motion, then the barrier to this internal rotation would occur at the symmetric geometry where the dihedral angle is exactly at 90° and the H H bond is perpendicular to the R vector. Calculations of this relaxed barrier to hindered rotation have been carried out as a function of R at the cc-pVDZ level. Test calculations at one value of R suggest a simple constant plus cosine fit to potential change as a function of dihedral angle is reliable. The resulting sinusoidal fits in the vicinity of the reaction bottleneck (as discussed later) are displayed as a function of the dihedral angle in Figure 1 for various values of R - R where R is the value R assumed by CH3 at equilibrium, i.e., 1.02 aCH2+H, and stabilization of the complex, e.g., CH3*+M-»CH3+M (from which the pressure dependence arises). While empirical or adjustable stabilization rates must be provided, R R K M theory provides a statistical, transition-state-theory like expression for all the other rates in terms of a ratio of partition functions at the transition state and at the reactants times a rate of crossing through the transitition state. A variational R R K M approach is required for there are no potential energy barriers to addition along either pathway to locate the transition state and therefore its
4. WAGNER & HARDING
Isotope Effects in Addition Reactions in Combustion
location must be varied along the reaction path until the most constraining, and therefore optimum, location (i.e., the reaction bottleneck) is found. There are five features to the R R K M calculations reported here that are described below. First, a canonical, rather than micro-canonical, variational R R K M theory was carried out. This means that the reaction botdeneck was located for each reaction path (including all isotopic variations) as a function of temperature. A more rigorous location of the bottleneck as a function of total energy Ε and total angular momentum J has not yet been done. Second, explicit summation over the total angular momentum was included in the calculations in order to be sensitive to the changes in extension of the geometries of the reaction bottlenecks for the different reaction paths. The quantum number Κ for the projection of J on the principal axis of C H 3 * was treated as active. Third, the influence of the buffer gas M was treated in an analytic way (36) that approximates the rigorous Master Equation description of the effect of the buffer gas. In this approximate approach, an effective rate constant for buffer-gas collisioninduced stabilization of metastable CH3* to thermalized CH3 is derived from a gas kinetic rate constant (using approximate Lennard-Jones parameters (37)) modified by a scaling constant γ that is a function of AE t, the average energy lost per collision of CH3* with M . The scaling constant is derived from analytic solutions to a simplified Master Equation model in which only one unimolecular decay route (not two as in CH3) is available. There is no direct measurements of AEiot and in the calculations it is used as a temperature and isotopically independent adjustable constant. The significant approximations used in the treatment of the buffer gas are probably partially remedied by the adjustable value of Δ Ε ^ . Fourth, the treatment of the out-of-plane motion was determined by preliminary calculations of the high pressure limit to CH3 addition along both pathways. Since the high pressure limiting rate constant is reached only at pressures where every adduct formed is stabilized, it is sensitive only to the addition reaction path characteristics, not to any competing pathways for adduct unimolecular decay or to the properties of the adduct itself. For CH+H2, negligible variation in the rate constant (CD2+D. However such an increase would also push to higher temperatures the rate constant increase in CH+H2 to such an extent that the calculated rate constant would fall below the error bars at high temperature in the Becker et al. experiment (9). If one accepts that the theory is correctly determining small variations between isotopes, then the results in Figures 5 and 6 could be interpreted as indicating a small inconsistency between the experiments of Becker et al. and Stanton et al. A further interpretation would be that the experiments of Stanton et al. may slightly underestimate the rate constant at higher temperatures. This interpretation is relevant to the following isotopic variation. For CH+D2, the only rate constant measurements over a variety of temperatures is by Stanton et al. at a fixed pressure of 20 torr of argon. The results, along with the theoretical calculations, are displayed in Figure 6. As can be seen from the figure, the temperature dependence of the rate constant changes dramatically from that of the fully protonated or deuterated system. This is due to the fact that this isotopic variant has two additional product channels, as detailed in reaction (3). One of those products, CD+HD, is an exoergic isotope exchange that requires no pressure or temperature activation. The theoretical calculations show that at the lower temperatures in the figure, this process completely dominates. There is almost no influence of pressure on the results, and, consequently, almost no influence of AEt t, because this isotope exchange can proceed at zero pressure. The rate constant at low temperatures does not fall with temperature, as in CH+H2 or CD+D2, because adduct stabilization is not an important channel. In this lower temperature region, the theory and experiment agree. At higher temperatures, the theory predicts that H+CD2 becomes a product channel competitive with isotope exchange. (The other triatomic product channel, D+CHD, is 1.5 kcal/mole more endoergic and over the temperature range displayed is always a trace product.) This endoergic channel is activated by temperature giving rise to the increase of the rate at higher temperatures for the same reasons found in the other isotopic variants. The experimental results show no clear evidence of this increase in the rate constant and tends to grow somewhat noisier at the higher temperatures. As in the case of CD+D2, the atomic bond fission channel could be made more endoergic, delaying its appearance in the temperature dependence of the addition rate constant to much higher temperatures. However, even massive changes in the endoergicity would lower but not eliminate the disagreement between theory and experiment at high temperatures. Since the atomic bond fission channels and the isotope exchange channel compete with each other, the calculations indicate that elimination of the atomic bond fission channel by artificially increasing its endoergicity will also have the effect of increasing the rate constant for isotope exchange. While the overall rate will decline, the calculated rate constant will still lie above the measured rate constant at the higher temperatures. An interpretation of this discrepancy is that the Stanton et al. measurements tend to somewhat underestimate the rate at higher temperatures in both CH+D2 and in CD+D2. For CD+H2, the measured rate constant consists of one value at room temperature in 100 torr of argon (6). This result and the calculated rate constant over the full temperature range are displayed in Figure 7. The calculations fall below the measured to
0
4. WAGNER & HARDING
Isotope Effects in Addition Reactions in Combustion
10 0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
1000/T Figure 6. The thermal addition rate constant for CH+D2 as a function of temperature at a fixed 20 torr pressure of Ar buffer gas.
10
^ ίο"
•
11
Berman et al. 1
Theory (ΔΕ, ,= -50 cm" ) η
1
Theory (AE = -75 cm" ) tQt
10 0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
1000/T Figure 7. The thermal addition rate constant for CD+H2 as a function of temperature at a fixed 100 torr pressure of Ar buffer gas.
61
62
ISOTOPE EFFECTS IN GAS-PHASE CHEMISTRY
value by about 10% to 20%. Consideration of the scatter found in the measured data as a function of temperature in the other isotopic variants (Figures 4 - 6 ) suggests this small difference is probably not significant. The overall theoretical temperature dependence is very similar to that of CH+D2. Both CD+H2 and CH+D2 have four product channels, i.e., reactions (4) and (3), respectively, and have similar temperature dependences for the same reason. In CD+H2, the simple isotope exchange to CH+HD is slightly endoergic. At quite low temperatures then, the product is dominated by C D H 2 product formation. In CH+D2, at such low temperatures, the isotope exchange process dominates. The remaining two other isotopic variants of CH+H2, namely CH+HD and CD+HD, have never been examined experimentally. These two sets of reactants sample reactions (3) and (4) from a direction different from that written. No theoretical results will be presented, but the temperature dependence at fixed pressure is very similar to that of CH+D2 and CD+H2 for related reasons. Conclusion Isotopic variations in the addition thermal rate constants for CH+H2 have been calculated and compared to experiment using variational R R K M theory and an ab initio electronic structure characterization of the reaction paths CH+H2 and H+CH2 and their isotopic variants. In general the agreement is good. Both theory and experiment clearly distinguish the change in character between the mixed isotope additions, i.e., CH+D2, and the fully protonated or deuterated additions. The consistency of agreement over all the isotopic variations suggests several of the experimental measurements may have somewhat underestimated the rate constant. Acknowledgments. This work was performed under the auspices of the Office of Basic Energy Sciences, Division of Chemical Sciences, U.S. Department of Energy, under Contract W-31-109-Eng-38. The large scale computing resources required for this project were provided in part through a Grand Challenge grant under the auspices of the Office of Basic Energy Sciences, Office of Scientific Computing, U.S. Department of Energy. Literature Cited 1. 2. 3. 4. 5. 6. 7 8. 9. 10. 11. 12. 13. 14. 15. 16.
Bohland, T.; Temps, F. Ber. Bunsenges. Phys. Chem. 1984, 88, 459. Grebe, J.; Homann, Κ. H. Ber. Bunsenges. Phys. Chem. 1982, 86, 581. Frank, P.; Bhaskaran, Κ. Α.; Just, Th. J. Phys. Chem. 1986, 90, 2226. Lohr, R.; Roth, P. Ber. Bunsenges. Phys. Chem. 1981, 85, 153. Peeters, J.; Vinckier, C Comb. (International) Symp. 1974, 15, 969. Berman, M . R.; Lin, M . C. J. Chem. Phys. 1984, 81, 5743. Zabarnick, S.; Fleming, J. W.; Lin, M . C. J. Chem. Phys. 1986, 85, 4373. Becker, K.H.; Engelhardt, B.; Wiesen, P.; Bayes, Κ. D. Chem. Phys. Lerr.1989, 154, 342. Becker, K. H.; Kurtenbach, R.; Wiesen, P. J. Phys. Chem. 1991, 95, 2390. Stanton, C.T.;Garland, N.L.; Nelson, H.H. J. Phys. Chem. 1991, 95, 1277. Braun, W.; McNesby, J. R.; Bass, A. M. J. Chem. Phys. 1967, 46, 2071. Butler, J. E.; Goss, L.P.; Lin, M . C.; Hudgens, J.W. Chem. Phys. Lett. 1979, 63, 104. Anderson, S. M.; Freedman, Α.; Kolb, C. E. J. Phys. Chem. 1987, 91, 6272. Bosnali, M . W.; Perner, D. Z. Naturforsch. 1971, 26a, 1768. Roth, P.; Barner, U.; Lohr, R. Ber. Bunsenges. Phys. Chem. 1979, 83, 929. Liu, K.; Macdonal, G. J. Chem. Phys. 1988, 89, 4443.
4. WAGNER & HARDING Isotope Effects in Addition Reactions in Combustion 63
17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39.
Liu, K.; Macdonal, G. J. Chem. Phys. 1990, 93, 2431. Liu, K.; Macdonal, G. J. Chem. Phys. 1990, 93, 2443. Jacox, M. E. J. Mol. Spectrosc. 1977, 66, 272. Holt, P. L.; McCurdy, K. E.; Weisman, R. B.; Adams, J. S.; Engel, P.S. Ibid. 1984, 81, 3349. Brooks, B. R.; Schaefer, III, H.F. J. Chem. Phys. 1977, 67, 5146. Dunning, Jr., T. H.; Harding, L. B.; Bair, R. Α.; Eades, R. Α.; Shepard, R. L. J. Phys. Chem. 1986, 90, 344. Merkel, Α.; Zulicke, L. Molec. Phys.1987, 60, 1379. Aoyagi, M.; Shepard, R.; Wagner, A. F.; Dunning, Jr., T. H.; Brown, F. B. J. Phys. Chem. 1990, 94, 3236. Aoyagi, M.; Shepard, R.; Wagner, A. F. Intl. J. Suprecomp. Appl. 1991, 5, 72. Shepard, R.; Shavitt, L; Pitzer, D. C.; Pepper, M.; Lischka, H.; Szalay, P. G.; Ahlrichs, R.; Brown, F. B.; Zhao, J.-G. Int. J. Quantum Chem. 1988, S22, 149. Shepard, R. in Ab Initio Methods in Quantum Chemistry II, Advances in Chemical Physics, K. P. Lawley, Ed. (Wiley, New York, 1987), Vol. 69, pp. 63-200. Dunning, Jr., T. H. J. Chem. Phys. 1989, 90, 1007. Aoyagi, M.; Dunning, Jr., T. H.(to be published) Shavitt, I.; Brown, F. B.; Burton, P. G. Int J. Quantum Chem. 1987, 31, 507. Chase, Jr., M.W.; Davies, C. Α.; Downey, Jr., J. R.; Frurip, D. J.; McDonald, R. Α.; Syverus, A. N. J. Phys. Chem. Ref. Data 1985, 14, 1211. Cobos, C.J.; Troe, J. J. Chem. Phys. 1985, 83, 1010. Pitzer, K. S.; Gwinn, W. D. J. Chem. Phys. 1942, 10, 428. Robinson, P. J.; Holbrook, K. A. Unimolecular Reactions : Wiley-Interscience, New York, 1972. Truhlar, D. G.; Hase, W. L.; Hynes, J. T. J. Phys. Chem. 1983, 87, 2264. Troe, J. J. Chem. Phys. 1977, 66, 4745. Hippler, H.; Troe, J.;Wendelken, H. J. J. Chem. Phys. 1983, 78, 6709. Dove, J. E.; Hippler, H.; Troe, J. J. Chem. Phys. 1985, 82, 1907. Timonen, R. S.; Ratajczak, E.; Gutman, D; Wagner, A. F. J. Phys. Chem. 1987, 91, 5325.
RECEIVED September 4, 1991
