J. Phys. Chem. 1986, 90, 3517-3524
Mode Specificity In the Model Unlmolecular Reaction H-C-C
-
3517
H 4- C=Ct
Kandadai N. Swamy,* William L. Hase,* Department of Chemistry, Wayne State University, Detroit, Michigan 48202
Bruce C. Garrett, Chemical Dynamics Corporation, Columbus, Ohio 43220
C . William McCurdy, Department of Chemistry, Ohio State University, Columbus, Ohio 43210
and J. F. McNutt Shell Development Company, Houston, Texas 77057 (Received: January 21, 1986; In Final Form: March 24, 1986)
-
Semiclassical and quantum mechanical studies of bound and resonance states for the model H-C-C H + C=C Hamiltonian are compared. Excellent agreement is found between the semiclassical and quantum mechanical bound-state eigenvalues and resonance positions. The close-coupling and stabilization graph with analytic continuation methodologies are used for the quantum calculations, and they give resonance positions and widths which are in good agreement. Resonance lifetimes vary by 5 orders of magnitude within a 1 kcal/mol energy interval. A correspondence between quasi-periodic/chaotic classical motion and regular/irregular quantum states is found for both bound and resonance states.
Introduction It is impossible to discuss unimolecular rate theory without considering the immensely significant work of Rudy Marcus. In collaboration with Rice in the 195Os, he developed the statistical quantum mechanical unimolecular rate theory which is widely known as RRKM (Rice-Ramsperger-Kassel-Marcus) theory.’ The period from approximately 1950 to 1970 could be considered “the golden age of RRKM theory”, for the theory provided a quantitative interpretation of nearly all unimolecular experiments, which at the time were performed under thermal conditions or by the “more modern” chemical activation technique. However, with advances in technology followed by sophisticated experiments and computer simulations, it has become clear that situations may arise in which the RRKM theory does not correctly describe the unimolecular event. Interpreting and searching for such nonRRKM effects is currently an active research area, in which Rudy Marcus is one of the most prominent participank2 Certainly, much of the excitement in the field of unimolecular dynamics originates from the work and inspiration of Rudy Marcus. At the present time, there is considerable interest in the study of mode-specific non-RRKM behavior in the unimolecular dissociation of vibrationally/rotationally excited molecules with covalent intramolecular potentials3 The characteristics of mode-specific dissociation depend on the properties and nature of the excited state, which may be broadly classified as either of two different types. The first are compound state resonances: which are the analogues of bound vibrational/rotational eigenstates. At energies in excess of the unimolecular threshold the resonance line widths of these states are broadened by unimolecular dissociation. The second type of excited state may be viewed as a superposition of these states,$ for which energy is initially localized in particular molecular motions, e.g., a C-H stretch local mode. Theoretical and experimental studies of mode specificity have taken different directions. Most quantum mechanical studies have dealt with compound state resonances for model two-dimensional Hamiltonians such as coupled Morse Oscillatorsb8 and the Henon-Heiles system”’, while the major experimental effort, as in + A preliminary report of this study was presented at the 190th ACS National Meeting, Chicago, IL, 1985. *Current address: IBM Corporation, Data Systems Division, Department 48B 428, Kingston, NY 12401.
chemical activationl”I6 and overtone excitation,”-I9 has dealt with the dissociation of superposition states. Superposition states have been analyzed theoretically in quantum mechanical calculations of isomerization in a two-dimensional coupled sytem.20 However, (1) Marcus, R. A.;Rice, 0. K. J. Phys. Colloid Chem. 1951, 55, 894. Marcus, R. A. J. Chem. Phys. 1952,20, 359. (2)Noid, D. W.; Koszykowski, M. L.; Marcus, R. A. Annu. Rev. Phys. Chem. 1981,32, 267. (3)“Photoselective Chemistry”; Aduances in Chemical Physics; Jortner, J., Levine, R. D., Rice, S. A., Eds.; Wiley: New York, 1981;Vol. XLVII. (4) Marcus, R. A. Discuss. Faraday SOC.1973,55,34.Feshbach, H. Ann. Phys. (New York) 1962,19,287. (5) E. Merzbacher, Quanrum Mechanics; Wiley: New York, 1970;p 450. W. L. Hase, Chem. Phys. Lett. 1985,116,312. This classification scheme becomes somewhat arbitrary if line widths for the resonances overlap. (6) Eastes, W.; Marcus, R. A. J. Chem. Phys. 1973,59,4757.Christoffel, K. M.; Bowman, J. M. J. Chem. Phys. 1983,78, 3952. (7)Hedges, Jr., R. M.; Reinhardt, W. P. Chem. Phys. Lett. 1982,91,241, J . Chem. Phys. 1983,78,3964. (8) Bisseling, R. H.; Kosloff, R.; Manz, J.; Schor, H. H. R. Ber. Bunsenges. Phys. Chem. 1985,89,270.Benjamin, I.; Bisseling, R. H.; Kosloff, R.; Levine, R. D.; Manz, J.; Schor, H. H. R. Chem. Phys. Lett. 1985,116,255. (9)Waite, B. A.; Miller, W. H. J. Chem. Phys. 1980,73,3713; 1981,74, 3910. 110) Christoffel. K. M.: Bowman. J. M. J. Chem. Phvs. 1982.76. 5370. ( I lj Yan Bai, Y.; Hose, G.; McCurdy, C. W.; Taylor,‘H. S. &em: Phys. Lett. 1983,99,342. (12)Rynbrandt, J. D.; Rabinovitch, B. S. J. Phys. Chem. 1971. 75,2164. Meagher,J. F.; Chao, K. J.; Barker, J. R.; Rabinohtch, B. S. J. Phys. Chem. 1974,78,2535. Flowers, M. C.; Rabinovitch, B. S. J . Phys. Chem. 1985,89, 563. (13) Parson, J. M.; Lee, Y. T. J. Chem. Phys. 1972,56, 4658. Farrar, J. M.; Lee, Y. T. J. Chem. Phys. 1976,65,1414. (14) Moehlmann, J. G.;Gleaves, J. T.; Hudgens, J. W.; McDonald, J. D. J. Chem. Phys. 1974,60,4790. Moss, M. G.;Ensminger, M. D.; Steward, G. M..; Mordaunt, D.; McDonald, J. D. J. Chem. Phys. 1980,73, 1256. (15) Rogers, P.J.; Selco,J. I.; Rowland, F. S. Chem. Phys. Lett. 1983,97, 313. Wrigley, S . P.;Oswald, D. A,; Rabinovitch, B. S. Chem. Phys. Lett. 1984,104, 521. (16)Reddy, K. V.;Berry, M. J. Chem. Phys. Lett. 1977,52, 1 1 1 ; 1979, 66,223. (17) Cannon, B. D.; Crim, F. F. J. Chem. Phys. 1981,75, 1752. Rizzo, T.R.; Crim, F. F. J. Chem. Phys. 1982,76, 2754. (18) Chandler, D. W.; Farneth, W. E.; Zare, R. N. J . Chem. Phys. 1982, 77,4447. (19)Jasinski, J. M.; Frisoli, J. K.; Moore, C. B. J . Chem. Phys. 1982,79, 1312. (20) Bowman, J. M.;Christoffel, K.; Tobin, F. J. Phys. Chem. 1979,83, 905. Christoffel, K.M.;Bowman, J. M. J. Chem. Phys. 1981,74, 5057.
0022-3654/86/2090-3517S01.50/00 1986 American Chemical Society
3518 The Journal of Physical Chemistry, Vol. 90, No. 16, 1986
the most extensive theoretical analyses of superpasition states have been classical trajectory calculation^^^-^^ in which the full dimensionality of the molecular Hamiltonian can be retained. Mode specificity may be observed by preparing a superposition state in which energy is initially localized in the reaction coordinate.22 The unimolecular decay will then be mode-specific until the initial localized state dephases/relaxes. Superposition states which remain localized for times exceeding s have been suggested from experimental studies for methyl and allyl isocyanide,16 and for both tetrallyltin and -germanium.I5 In recent experiments, it has become possible to selectively excite individual compound state resonances for molecules with covalent potentials.” There is very strong evidence for mode-specific decay from these states in f ~ r m a l d e h y d e . ~This ~ experimental result is supported by quantum mechanical calculation^.^^ There is also evidence, both theoretical and experimental, that compound state resonances are important in the dissociation of electronically excited DCN34 and in the dissociation of DN,.35 Mode specificity has been observed in a comprehensive classical trajectory study of the model unimolecular dissociation reaction H + C=C.36337 A large fraction of the trajectories H-C-C above the unimolecular threshold are quasi-periodic and, thus, do not dissociate. By use of the EBK semiclassical method which requires quasi-periodic classical motion, vibrational levels may be quantized at energies in excess of the unimolecular threshThese levels are expected to correspond to compound state resonances. In this paper we report a quantum mechanical study of H-C-C H C=C dissociation. The calculations are performed with two different techniques, the stabilization method with analytic c o n t i n ~ a t i o nand ~ ~ the close coupling method.45 Classicalsem-
TABLE I: Parameters for the H-C-C
-
Swamy et al. H
+ C=C Hamiltonian
parameter
value
mH. amu
1.008
mc, amu DH,kcal/mol
12.011 103.00 1.84 1.08 109.00
PH, A-’ ro, A
Da,kcal/mol Dd, kcal/mol
172.00
Pc, A-’ Rest A Rod, A cc,A-‘ rc, A
2.02 1.51 1.33
1.940 1.486
c,, ‘4-1
1.56
rB, A 5 ,
1.08 I
1
I
-
-
+
(21) Bunker, D. L. J . Chem. Phys. 1964, 40, 1946. (22) Bunker, D. L.; Hase, W. L. J . Chem. Phys. 1973, 59, 4621. (23) McDonald, J. D.; Marcus, R. A. J. Chem. Phys. 1976, 65, 2180. (24) Sloane, C. S.; Hase, W. L. J. Chem. Phys. 1977, 66, 1523. (25) Wolf, R. J.; Hase, W. L. J . Chem. Phys. 1980, 72, 316. (26) Hase, W. L.; Buckowski, D. G.; Swamy, K. N. J . Phys. Chem. 1983, 87, 2754. (27) Swamy, K. N.; Hase, W. L. J. Chem. Phys. 1985,82, 123. (28) Uzer, T.; Hynes, J. T. Chem. Phys. Lett. 1985,113,483. Uzer, T.; Hynes, J. T.; Reinhardt, W. P. Chem. Phys. Lett. 1985, 117, 600. (29) Kato, S. J. Chem. Phys. 1985, 82, 3020. (30) Holme, T. A,; Hutchinson, J. S. J . Chem. Phys. 1985, 83, 2860. (31) Kittvell, C.; Abramson, E.; Kinsey, J. L.; McDonald, S. A.; Reisner, D. E.; Field, R. W.; Katayama, D. H. J. Chem. Phys. 1981, 75, 2056. (32) Dai, H.-L.; Field, R. W.; Kinsey, J. L. J. Chem. Phys. 1985,82, 1606. Apel, E. C.; Lee, E. K. C. J. Chem. Phys. 1986, 84, 1039. (33) Waite, B. A,; Gray, S. K.; Miller, W. H. J . Chem. Phys. 1983, 78, 259. Gray, S. K.; Child, M. S. Mol. Phys. 1984, 53, 961. (34) Chuljian, D. T.; Ozment, J.; Simons, J. J. Chem. Phys. 1984,80, 176. (35) Simpson, T. B.; Mazur, E.; Lehmann, K. K.; Burak, I.; Bloembergen, N. J. Chem. Phys. 1983, 79, 3373. (36) Wolf, R. J.; Hase, W. L . J. Chem. Phys. 1980, 73, 3779. Hase, W.
L.; Wolf, R. J. In Potential Energy Surfaces and Dynamics Calculations; Truhlar, D. G., Ed.; Plenum: New York, 1981; p 37. (37) Hase, W. L.; Duchovic, R. J.; Swamy, K. N.; Wolf, R. J J . Chem. Phys. 1984, 80, 714. (38) An attempt was made to quantize quasi-periodictrajectories for which there is a Fermi resonance between the C H and CC stretch internal coordinates with the Sorbie-Handy method [Sorbie, K. S.; Handy, N. C. Mol. Phys. 1977, 33, 13191 and the DeLeon-Heller-Miller method [DeLeon, N.; Heller, E. J. J. Chem. Phys. 1984,81, 35731. However, neither of these approximate methods worked; Lemon, W. J.; Hase, W. L., to be published. (39) Hase, W. L. J . Phys. Chem. 1982,86, 2873. (40) Hedges, Jr., R. M.; Skodje, R. T.; Borondo, F.; Reinhardt, W. P., In Resonances in Electron- Molecule Scattering, Van der Waals Complexes, and Reactive Chemical Dynamics; Truhlar, D. G., Ed.; American Chemical Society: Washington, DC, 1984; p 323. (41) Duchovic, R. J.; Swamy, K. N.; Hase, W. L. J. Chem. Phys. 1984, 80, 1462. (42) Ramaswamy, R. J. Chem. Phys. 1985,82, 147. (43) Swamy, K. N.; Hase, W. L. Chem. Phys. Lett. 1985, 114, 248.
4 h
a”
v
2 3
2 ‘ 1
1
I
I
2
3
4
-
RCH
+
Figure 1. Contour diagram of the H-C-C H C-C potential energy surface. The energy interval between the contour lines is 5 kcal/mol.
iclassical-quantal correspondences are identified by comparing those quantum mechanical results with the previous classical and semiclassical studies. These quantum calculations are also of special interest since the H-C-C Hamiltonian is similar to those for the hydroperoxyl H0246and formyl HC04’ radicals.
-.
Hamiltonian The two-dimensional H-C-C H + C=C unimolecular system is described by the classical H a m i l t ~ n i a n ~ ~ ” ~
Dc(r)Iexp(-28c[R - Ro(r)l) - 2 exp(-Pc[R - Ro(r)l)I
+ Dc, (1)
The C-C bond energy Dc and equilibrium bond length Ro are written as functions of the C-H bond length r, DC(r) = Dcs + [Dcd - Dcslsc(r)
(2)
Ro(r) = RoS+ [Rod - RoS]S,(r)
(3)
These functions allow the C-C bond energy and “equilibrium” (44) McCurdy, C. W.; McNutt, J. F . Chem. Phys. Left. 1983, 94, 306. McCurdy, C. W. In Autoionization Recent Developments and Applications; Temkin, A., Ed.; Plenum: New York, 1985; p 135. (45) Child, M. S. Molecular Collision Theory; Academic: London, 1974. Lester, Jr., W. A. In Modern Theoretical Chemistry; Miller, W . H., Ed.; Plenum: New York, 1976; Vol. 1, p 1. (46) Dunning, Jr., T. H.; Walch, S.P.; Goodgame, M. V . J. Chem. Phys. 1981, 74, 3482. (47) Adams, G. F.; Bent, G. D.; F’urvis, G. D.; Bartlett, R. J. J. Chem. Phys. 1979, 71, 3697. Dunning, Jr., T. H. J . Chem. Phys. 1980, 73, 2304.
The Journal of Physical Chemistry, Vol. 90, No. 16, 1986 3519
Mode Specificity in Unimolecular Dissociation bond length to smoothly vary from single-bond to double-bond values via the switching functions S,(r) and Sp(r). These switching functions are expressed as S,(r) = 11 - exp[-C,(r - rC)]j2 S,(r) = 0.0
r
S,(r) = 0.0
r
r 2 r,
< rc
S,(r) = 1 - exp[-C,(r - rJ2]
0.09
r 2 r8
< rg
k (5)
+ V(R;R 4.0 au Ere: widthb 45.18700 2.496 (-4) 16 0.02 0.15 45.18664 2.610 (-4) 20 0.02 0.15 45.18663 2.617 (-4) 24 0.02 0.15 45.18664 2.617 (-4) 28 0.02 0.15 45.18475 2.385 (-4) 16 0.01 0.05 45.184 19 2.378 (-4) 16 0.005 0.025
+
Number of C-C Morse oscillator basis functions. Both resonance position and width are given in kcal/mol.
From these convergence tests and similar tests at other resonance energies, Ar step sizes of 0.01 au for r < 3.5 and 0.05 au for r > 3.5 au were chosen, as well as 20 C-C Morse basis functions. This number of C-C Morse functions is the same as that for the stabilization calculations. Resonances were located and characterized by the close-coupling calculations in the 42.37-47.00 kcallmol energy range, and at 54.91 kcal/mol. The results are listed in Table V. Here we have tabulated the lifetime T of the resonance instead of the width r; Le., T = h/I" Three resonances, denoted by the quantum numbers, (1,9), (0,12), and (0,15), were studied by both the stabilization and close-coupling calculational methods. The resonance positions for these levels, as calculated by these two different methods, agree to within five significant figurcs. For the lifetimes (widths) the agreement is within a factor of 2. In a close-coupling calculation the assignment of quantum numbers for excitations in the C C and C H modes can only be accomplished rigorously be examining the nodal structure of the total wave function in the interaction region. The R-matrix propagation method does not solve directly for the wave function and this analysis was not carried out in the present calculation. As an aid to locating and assigning quantum numbers to resonances which the stabilization calculations did not identify, an approximate adiabatic calculation was performed. The coupled equations for the radial wave functions (23) are approximated by
-
TABLE V: H-C-C H + C = C Resonances energy, kcal/mol lifetime, s level nr,nR semiclass. quantum" stabilizationb close-coupling 3,4' 1,9 0,12 2,7'
46.31
1,lO 0,13 23
48.29 49.23
1,11
51.35 qPd
0,14 23 1,12 0,15 2,lO 1,13 0,16
44.263 45.184 45.308 45.977 46.305 46.344 46.960 48.303 49.236 49.627 51.359 52.103 52.663 54.350 54.909 55.857 57.273 57.659
45.18
54.35 9P 57.28 qP
3.69 (-1 1) 1.07 (-10) 1.36 (-10) 1.50 (-8) 2.13 (-13) 5.38 (-11) 1.19 (-8) 1.90 (-13) 1.33 (-10) 2.53 (-10) 1.55 (-13) 2.48 (-1 1 ) 4.03 (-10)
3.14 (-10) 6.10 (-1 1) 2.06 (-13)
