J. Phys. Chem. 1988, 92, 7193-7204
7193
Interaction of Molecular Rotation with Large-Amplitude Internal Motions: A Rigid Twister Model of Hydrogen Peroxide Bobby G. Sumpter, Craig C. Martens:.: and Gregory S. Ezra*,$ Department of Chemistry, Baker Laboratory, Cornell University, Ithaca, New York 14853 (Received: March 9, 1988; In Final Form: June 7 , 1988)
The classical dynamics of interaction between molecular rotation and large-amplitude internal rotation is studied for a rigid twister model of hydrogen peroxide, obtained by assuming an adiabatic separation of the torsional degree of freedom from the remaining 3N - 7 vibrational modes. Use of the Augustin-Miller canonical transformation to express the molecule-fixed components of the total anguler momentum j in terms of the magnitude, j , the component along the body-fixed z axis, k, and x, the angle conjugate to k,results in a two degree of freedom rotation-torsion Hamiltonian, whose phase space structure can be characterized by using surfaces of section. Rigid twister surfaces of section are presented for several values of angular momentum and energy. Quasi-periodictrapping and crossing tori are found, together with regions of large-scale rotation-torsion chaos. The effects of deuteriation and variation of torsional barrier heights on phase space structure are investigated. Removing either the centrifugal or Coriolis coupling terms from the rigid twister Hamiltonian leads to a significant increase in stochasticity; we infer that there is a cancellation of coupling terms in the full Hamiltonian.
I. Introduction
Rotational effects on resonant interactions of normal modes in rotating molecules have been studied by using classical meThe importance of molecular rotation in the dynamics of polyatomic molecules has been illustrated in a number of recent theoretical studies.1*2 Rotation can either increase3-I4 or decrease'+ the rate and extent of intramolecular vibrational energy (1) McClelland, G. M.; Nathanson, G. M.; Frederick, J. H.; Farley, F. w. In ExcitedStates; Lim, E. C., Innes, K. K., Eds.; Academic: London, 1987; transfer. Rates of unimolecular decay and isomerization have Vol. 7. been shown to be strongly dependent on the total angular mo(2) Orr, B. J.; Smith, I. W. M. J. Phys. Chem. 1987, 91, 6106. mentum j.lS2I Energy transfer between vibrational and rotational (3) Bunker, D. L. J. Chem. Phys. 1972, 40, 3911. degrees of freedomlm-N has been suggested as an important factor (4) Hung, N. C.; Wilson, D. J. J . Chem. Phys. 1963,38,828. Hung, N. in determining rates of collisionally induced vibrational relaxaC. J. Chem. Phys. 1972,57, 332. (5) Parr, C. A,; Kupperman, A,; Porter, R. N. J. Chem. Phys. 1977, 66, tion,25v26and is probed in the fluorescence depolarizationz7and 2914. electric deflection2*experiments of McClelland and co-workers. (6) Rolfe, T. J.; Rice, S. A. J . Chem. Phys. 1983, 79, 4863. Although the mechanisms responsible for rotational effects are (7) Clodius, W. B.; Shirts, R. B. J. Chem. Phys. 1984, 81, 6244. not yet understood in complete detail, it has been established that (8) Shirts, R. B. J . Chem. Phys. 1986, 85, 4949. centrifugal coupling is primarily responsible for extensive rota(9) Shirts, R. B. Int. J . Quantum Chem. 1987, 31, 119. tion-vibration energy f10w,12922-23 while Coriolis interaction^^^ (10) Combs, J. A.; Hoover, W. G. J. Chem. Phys. 1984, 80, 2243. (11) Uzer, T.; Natanson, G. A.; Hynes, J. T. Chem. Phys. Lett. 1985,122, provide an additional coupling between vibrational modes not 12. present in the rotationless case.l0*ll (12) Frederick, J. H.; McClelland, G. M.; Brumer, P. J. Chem. Phys. 1985, Quantum calculations of rotation-vibration states of polyatomic 83, 190. molecules have so far been restricted to states with relatively low (13) Sumpter, B. G.; Thompson, D. L., submitted for publication in J . angular m o m e n t ~ m , ' ~since - ~ ~the , ~ size ~ of the rotation-vibration Chem. Phys. (14) Chen, C. L.; Maessen, B.; Wolfsberg, M. J. Chem. Phys. 1985, 83, Hamiltonian matrix increases linearly with j (cf., however, ref 1795. 32). Classical and semiclassical methods33 therefore continue (1 5) Stace, A. J. Ber. Bunsen-Ges. Phys. Chem. 1977, 81, 200. to be essential for the study of highly excited rotation-vibration (16) Quack, M.; Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1974, 78, 240. states. Work has been done on the semiclassical quantization of (17) Miller, J. A,; Brown, N. J. J. Phys. Chem. 1982, 86, 772. the asymmetric rigid rotor,the rigid bender$I and the triatomic (18) Grant, E. R.; Santamaria, J. J. J. Phys. Chem. 1981, 85, 2426. H 2 0 with inclusion of all three vibrational degrees of f r e e d ~ m . ~ ~ * ~ (19) ~ Viswanathan, R.; Raff, L. M.; Thompson, D. L. J. Chem. Phys. 1985, 82, 3083. The classical trajectory method has been widely used to in(20) Schlier, Ch. G. Mol. Phys. 1987, 62, 1009. vestigate the influence of molecular rotation on the internal dy(21) Wolf, R. J.; Hase, W. L. J. Chem. Phys. 1980, 73, 3779; 1981, 75, namics of polyatomic molecules. The pioneering studies of Bunker3 3809. (see also ref 4) established the activity of rotational degrees of (22) Frederick, J. H.; McClelland, G. M. J. Chem. Phys. 1986,84,4347. freedom at high internal energies. Parr, Kupperman, and Porter5 (23) Ezra, G. S. Chem. Phys. Lett. 1986, 127, 492. (24) Sumpter, B. G.; Ezra, G. S., to be submitted for publication. showed for several triatomic molecules that rotation can signif(25) Hippler, H.; Schranz, H. W.; Troe,J. J . Phys. Chem. 1986,90, 6158. icantly increase the rate of intramolecular energy redistribution. Schranz, H. W.; Troe, J. J. Phys. Chem. 1986,90, 6168. More recently, Clodius and Shirts7 investigated the effect of (26) Parson, R., preprint. rotational coupling on intramolecular vibrational energy flow for (27) Nathanson, G. M.; McClelland, G. M. J . Chem. Phys. 1986, 85, H 2 0 and D20and found that rotation can either increase or 4311; 1984,81,629; 1986,84,3170; Chem. Phys. Lett. 1985,114,441; Chem. Phys. Lett. 1985, 118, 228. decrease the amount of bond-bond energy transfer, depending (28) Farley, F. W.; Novakoski, L. V.; Dubey, M.; Nathanson, G. M.; on the relative signs and magnitudes of the various contributions McClelland, G. M. J . Chem. Phys. 1988,88, 1460. to the 1:1 resonant bond-bond coupling. Subsequently, Shirts (29) Jahn, H. A. Phys. Reu. A. 1939, 56, 680. has studied both the classical and quantum mechanics of rotational (30) Tennyson, J. Comput. Phys. Rep. 1986, 4, 1. decoupling of the local stretch modes in model H 2 0 8and HD09 (31) Natanson, G. A.; Ezra, G. S.; Delagado-Barrio. G.; Berry, R. S. J. Chem. Phys. 1986,84, 2035. rotating in the plane. Mathematical Sciences Institute Fellow. f Present address: Department of Chemistry, University of Pennsylvania, Philadelphia, PA 19104. Alfred P. Sloan Fellow; Camille and Henry Dreyfus Teacher-Scholar.
0022-3654/88/2092-7193$01.50/0
(32) Sutcliffe, B. T.; Tennyson, J.; Miller, S. Theor. Chim. Acta 1987, 72, 265. (33) Ezra, G. S.; Martens, C. C.; Fried, L. E. J. Phys. Chem. 1987, 91, 3721. (34) King, G. W. J . Chem. Phys. 1947, 15, 820.
0 1988 American Chemical Society
7194
The Journal of Physical Chemistry, Vol. 92, No. 26, 1988
chanics. Combs and Hooverlo investigated a model for rotating planar benzene and showed that Coriolis coupling leads to extensive energy flow between two degenerate harmonic modes. Uzer, Nathanson, and Hynes” investigated Coriolis-induced energy flow between anharmonic bend and asymmetric stretch modes for a collinear model ABA triatomic. Recent trajectory studies have established that energy flow between vibration and rotation (see, for example, ref 6) is significantly enhanced when there are low-order vibration-rotation resonances. Thus, Frederick, McClelland, and BrumerI2 have studied the onset of intramolecular vibration-rotation energy transfer in models for SO2and “bent OCS” (see also ref 43). At high energies, intramolecular vibrational-rotational energy transfer induces transitions between the two types of stable asymmetric top motions (A and C type, see The mechanism for rotation-vibration energy flow was determined to be strong centrifugal coupling between molecular rotation and bending vibration through a 2:1 resonance. Frederick and McClellandZ2 have studied the classical dynamics of rotation-vibration interaction for a rigid bender model of water and have found that bend-rotation resonances play an important role in the overall dynamics. Ezra23used the surface of section method to characterize quasi-periodic, chaotic, and resonant regions of phase space for the rigid bender model, and again noted significant rotation-vibration energy flow associated with 2:1 bend-rotation resonances. Long-time correlations, associated with trajectory segments corresponding to A- or C-type rotational motion, were also observed in the chaotic regime. A key feature of this work was the use of the Augustin-Miller t r a n s f ~ r m a t i o nto~ ~reduce ,~~ the rotational dynamics to a single degree of freedom. In the present paper we extend these studies of rotation-vibration dynamics in rigid bender models to consider the interaction of molecular rotation with large-amplitude internal rotation. We call the torsion-rotation Hamiltonian investigated here a “rigid twister” model, by analogy with the rigid bender.22,23 The parameters for the rigid twister model studied are appropriate for hydrogen peroxide, H202.46 One motivation for our work derives from intense current e ~ p e r i m e n t a l ~ ’and ,~~ t h e o r e t i ~ a l ’ ~interest , ~ ~ - ~ ~in the overtone-induced dissociation of H202. Although the classical trajectory calculations of Uzer et al. indicate that the torsional mode plays an insignificant role in (35) Colwell, S. M.; Handy, N. C.; Miller, W. H. J . Chem. Phys. 1978, 68, 745. (36) Harter, W. G.; Patterson, C. W. J . Chem. Phys. 1984, 80,4241. (37) Duchovic, R. J.; Schatz, G. C. J. Chem. Phys. 1986, 84, 2239. (38) Schatz, G. C. Comput. Phys. Commun., in press. (39) Patterson, C. W.; Smith, R. S.; Shirts, R. B. J . Chem. Phys. 1985, 85, 724. (40) Huber, D.; Heller, E. J.; Harter, W. G. J . Chem. Phys. 1987, 87, 1116. (41) Frederick, J. H.; McClelland, G . M. J . Chem. Phys. 1986, 84, 876; Frederick, J. H. Chem. Phys. Lett. 1986, 131, 60. (42) Sumpter, B. G.; Ezra, G. S. Chem. Phys. Lett. 1987, 144, 144. (43) Carter, D.; Brumer, P. J . Chem. Phys. 1982, 77, 4208. (44) Augustin, S. D.; Miller, W. H. J . Chem. Phys. 1974, 61, 3155. (45) Deprit, A. Am. J . Phys. 1967, 35, 424. (46) (a) Hunt, R. H.; Leacock, A.; Peters, C. W.; Hecht, K. T. J . Chem. Phys. 1965,42, 1931. (b) Olsen, W. B.; Hunt, R. H.; Young, B. W.; Maki, A. G.; Brault, J. W. J . Mol. Spectrosc. 1988, 127, 12. (47) (a) Rizzo, T. R.; Hayden, C. C.; Crim, F. F. Faraday Discuss. Chem. SOC.1983, 75, 112. (b) Rizzo, T. R.; Hayden, C. C.; Crim, F. F. J . Chem. Phys. 1984, 81, 4501. (c) Ticich, T. M.; Rizzo, T. R.; Dubal, H.-R.; Crim, F. F. J . Chem. Phys. 1986, 84, 1508. (d) Butler, L. J.; Ticich, T. M.; Likar, M. D.; Crim, F. F. J . Chem. Phys. 1986, 85, 2331. (e) Brouwer, L.; C o b s , C . J.; Troe, J.; Dubal, H.-R.; Crim, F. F. 1.Chem. Phys. 1987, 86,6171. ( f ) Crim, F. F. Annu. Rev. Phys. Chem. 1984.35, 657. (9) Likar, M. D.; Baggot,
J. E.; Ticich, T. M.; Vander Wal, R. L.; Crim, F. F. J . Chem. SOC.,Faraday Trans. 2, in press. (48) Scherer, N. F.; Doany, F. E.; Zewail, A. H. J . Chem. Phys. 1986.84, 1932. Scherer, N. F.; Zewail, A. H. J . Chem. Phys. 1987, 87, 97. (49) Uzer, T.; Hynes, J. T.; Reinhart, W. P. Chem. Phys. Lett. 1985, 117, 600; J . Chem. Phys. 1986, 85, 5791. Uzer, T.; Hynes, J. T. In Stochasticity and Intramolecular Redistribution of Energy; Lefebvre, R., Mukamel, S., Eds.; NATO Series Vol. 200; Reidel: Dordrecht, 1987. (50) Sumpter, B. G.; Thompson, D. L. J . Chem. Phys. 1985, 82, 4557; 1987, 86, 2805. (51) Getino, C.; Sumpter, B. G.; Santamaria, J.; Ezra, G. S., manuscript in preparation.
Sumpter et al.
Figure 1. The rigid twister dynamical model for HOOH. The torsional motion is described by the angle 6. All other internal vibrations are frozen. The z axis is parallel to the 00 bond, while the x axis bisects the angle between the O H bonds.
the 6voH-overtone-induced dissociation of rotationless H202 molecules,49 the work of Sumpter and Thompson shows that internal rotation becomes more important at higher excitation energies.I3 (The importance of coupling of internal rotation to stretching and bending modes has recently been stressed by Spears torsion appears to be important and H ~ t c h i n s o n . ~ Moreover, ~) in the intramolecular dynamics of rotating H202;recent trajectory studies of the overtone-induced dissociation of rotating peroxide suggest that rotation-torsion interaction is very strong prior to decomp~sition.~’ Sumpter and Thompson13 have investigated the effects of molecular rotation on the intramolecular vibrational dynamics of hydrogen peroxide, using both a separable model potential surface and a more realistic surface generated from a combination of ab initio calculations and spectroscopic data. For both surfaces, addition of rotational energy equivalent to a temperature of 1500 K was found to enhance the rate of energy flow out of an initially excited O H local stretch overtone and significantly increase the rate of subsequent dissociation. Interaction between torsional vibrations and molecular rotation was found to be essential; removal of the torsional potential, leading to free internal rotation, reduced considerably the effects of rotation on overtone relaxation and diss~ciation.’~ Coupling of the torsional vibration to other modes has been suggested as a possible explanation for nonstatistical variation in the O H product state distributions for 6voH-overtone-induced di~sociation.~’Dubal and Crim53were able to describe the coarse structure of the overtone vibration predissociation spectrum of hydrogen peroxide using a vibration-torsion model in which the low-frequency torsion and the high-frequency O H stretching vibrations were separated adiabatically. Combination torsion/OH stretching excitations appear in the overtone dissociation spectrum, due to the dependence of the trans internal rotation barrier height on the level of O H excitation. Internal rotation modes have been found to play an important role in IVR. For example, close-coupling calculations of level energies and widths of predissociating van der Waals molecules indicate that internal rotation is active in facilitating dis~ociation?~ The experiments of Parmenter and c o - w o r k e r ~indicate ~~ that excitation of methyl torsional motion in p-fluorotoluene induces extensive mode scrambling. The recent studies by Spears and Hutchinson have established the importance of coupling of internal rotation to stretch and bend modes in promoting isomerization in HNNH.52.56 The present study concerns the interaction of large-amplitude torsional motion with molecular rotation. Using classical trajectories and Poincare surfaces of section, we investigate the phase space structure of a rigid twister Hamiltonian describing rotation-torsion motion in hydrogen peroxide.& The dynamical model (52) Spears, L. G.; Hutchinson, J. S . J . Chem. Phys. 1988, 88, 250. (53) Dubal, H.-R.; Crim, F. F. J . Chem. Phys. 1985, 83, 3863. (54) Hutson, J. M.; LeRoy, R. J. J . Chem. Phys. 1983, 78, 4040. (55) Longfellow, R. J.; Parmenter, C. S . J . Chem. SOC.,Faraday Trans. 2, in press. Moss, D. B.; Ewing, G. C.; Parmenter, C. S. J . Chem. Phys. 1987, 86, 51. (56) Spears, L. G.; Hutchinson, J. S. J . Chem. Phys. 1988, 88, 240.
The Journal of Physical Chemistry, Vol, 92, No. 26, 1988 7195
A Rigid Twister Model of Hydrogen Peroxide
-
00
720
1440
2160
2880
3600
@ Figure 2. Torsional potential V ( 4 ) [eq 51 versus 4 for uOH = 0. described by the rigid twister Hamiltonian is shown in Figure 1. The twisting motion to which we refer is the internal rotational mode associated with the angular coordinate 4. All other internal vibrations are frozen, while overall rotation is allowed. The phase space of the rigid twister model is found to be much richer than that for the rigid bender with several kinds of rotation-torsion motions possible. We study the classical phase space structure as a function of energy and angular momentum and investigate the effects of deuteriation and variations in torsional barrier height. The importance of particular couplings in the Hamiltonian is elucidated; it is found that there is an effective cancellation between centrifugal and Coriolis coupling terms (cf. ref 7-9). This paper is organized as follows: The “rigid twister” Hamiltonian for hydrogen peroxide is described in section 11, along with the methods used in the trajectory calculations. Results are presented in section 111, and discussion and conclusions are given in section IV. 11. Rigid Twister Hamiltonian In the ground electronic state, internal rotation of the O H groups about the 0-0bond in hydrogen peroxide is hindered by a potential of the form shown in Figure 2.46 At low energies, torsional motion is librational in character, while at higher energies free internal rotation becomes possible. Considerable effort has been devoted to determining the torsional potential energy function. Hunt et al. studied the far-infrared absorption spectrum of H 2 0 2 and determined trans and cis potential barrier heights of 386 and 2460 cm-I, respectively, and an equilibrium dihedral angle of 11 1.5’ relative to the cis c ~ n f i g u r a t i o n .The ~ ~ potential energy function was expressed in the form of a truncated Fourier cosine series in the angle 4 with coefficients chosen to give the appropriate barrier heights. A contact transformation was used to put the torsion-rotation Hamiltonian into the form of an asymmetric top plus internal rotation with dihedral angle dependent coupling, and diagonalization of the Hamiltonian matrix to second order via perturbation theory was sufficient to account for the Q-branch shapes in the far-infrared region. Subsequent ab initio calculat i o n have ~ ~ ~provided further support for the form of the potential energy function used by Hunt et al. This potential function is used in our rigid twister model Hamiltonian for hydrogen peroxide. The rigid twister dynamical model is motivated by the fact that the frequencies associated with torsional transitions are very low compared to bending (1200-1420 cm-’) and stretching (875-3600
crn-’) frequencies. In addition, G-matrix elements5*coupling the torsion to other vibrational modes in the absence of rotation are small. Adiabatic separation of the torsional mode from the other vibrational modes is therefore a useful a p p r o x i m a t i ~ n . ~The ~-~~ rigid twister model describes the interaction between large-amplitude torsional motion and overall molecular rotation of hydrogen peroxide (cf. Figure 1). The Hamiltonian we use is a classical version of that given by Hunt et al.;46details of the derivation can be found in ref 46. The torsion-rotation Hamiltonian of ref 46 has four degrees of freedom (three rotations and one vibration). To reduce the dimensionality of the problem we employ the Augustin-Miller canonical t r a n s f ~ r m a t i o n ,which ~ ~ eliminates angle variables conjugate to the conserved actions j (magnitude of the total angular momentum) and m (magnitude of the projection of j onto a space-fixed axis) and reduces the number of rotational degrees of freedom to one. It has been used previously to study the classical and semiclassical mechanics of the asymmetric top,35”7 the rigid bender,22,z3,41 and rotating-vibrating t r i a t o m i c ~ . ~In~the ? ~ Au~ gustin-Miller transformation the body-fixed components of the angular momentum vector j are written as functions of variables j , k , and x j, =
-0’2
- k2)1/2 sin x
(la)
jY =
-(j2
- k2)I/2 cos x
(1b)
j, = k
(IC)
where ( k , ~are ) canonical variables for rotation. The expressions for the body-fixed components of j in eq 1 are substituted directly into the Hamiltonian of interest. From la-c together with the torsion-rotation Hamiltonian of ref 46, the final form of the rigid twister Hamiltonian for hydrogen peroxide is H,,, = y2(j2- k 2 ) [ B ( 4 )sin2 x
+ C(4) cos2 x] + Y2A(4)kz(j2- k2)1/zD(4) k cos
Hrot..vib = -2F(4)P4(j2 - k 2 ) 1 / sin 2
(2a) (2b)
Hvib = 2G($)P@’ + v(4)
(2c)
+ Htot-vib + Hvib
(24
Htot
=
Htot
where the functional forms of the inertial parameters A(4)-G(4) are
4 4 ) = @/(@-I - S2)
(3a)
B(4) = o / ( w - 4
(3b)
r/(@r - S2) D(4) = -S/(Pr - 62)
(3c) (34
F ( 4 ) = -€/(a?) - €2)
(3e)
G(4) = a / ( w - c2)
(30
C(4) =
with60 a = A.
+ A’sin2 (4/2)
@ = (Ao+ Co) - Co sin2 (4/2) y = C,,
+ C’sin2 (4/2)
6 = Dosin (4/2) e
= Do
COS
(4/2)
~ ( 4=) A ’ - C’sin2 (4/2) (57) Helminger, P.; Bowman, N . C.; DeLuca, F. C. J. Mol. Spectrosc. 1981, 85, 120. Ewig, C. S.; Harris, D. 0. J. Chem. Phys. 1970, 52, 6268. Dunning, T. H.; Winter, N. W. J. Chem. Phys. 1975, 63, 1847. Jansen, L. J. Chem. Phys. 1985,82, 3322. Carpenter, J. E.; Weinhold, F. J. Phys. Chem. 1986,90,6405. Botschwina, P.; Meyer, W.; Senkov, A. M. Chem. Phys. 1976. 15, 25.
x
x
~~
(4a) (4b) (4c) (44 (4e) (4f)
~
(58) Wilson, E. B. Jr.; Decius, J. C.; Cross, P.C . Molecular Vibrations;
Dover: New York, 1979. (59) Hougen, J. T.; Bunker, P. R.; Johns, J. W. C. J. Mol. Spertrosc. 1970, 34, 136.
7196 The Journal of Physical Chemistry, Vol. 92, No. 26, 1988
Sumpter et al.
TABLE I: Parameters for the Rigid Twister Hamiltonian Eu 2-6 RoH 0.965 A Roo = 1.464 8, = 99.4' & = 111.5' mH = 1.007825 amu mo = 15.99492 amu V,,,,, = 386 cm-I; V , , = 2460 cm-' (u = 0) V,,,, = 850 cm-I; V,,, = 2910 cm-I ( u = 6) A. = 31.114 X kg A-2 A ' = 2.998 X kg A-2 Co = 2.820 X lo-*' kg A-2 C' = 0.178 X 10-27kg A-2 Do = -2.587 X kg A-2
g- t --
~
00
720
, 1440
, 2160
2880
I
l
3600
dded
0
z
I
b
With the choice of the z axis as the axis of quantization (see Figure l), k is approximately a conserved quantity for k j . The torsional potential is taken to be a truncated Fourier series 3
V ( 4 )=
Ea, cos ( i $ ) 1=0
(5)
SI
0
with Fourier coefficients a0
I
00
= M[(Vcis + Vtrans) X (sin2 40 + 2 cos4 40) + z(Vc1s - vtrans) cos3 $01 = M[(vcis - vtrans) X (72 - 7 2 cos2 4 0 + 2 cos4 $0) - (vcis + vtrans)
720
1440
2160
$0)
I
3600
dded 2 4
-
L" ~
COS 401 0
a2 = M[(Vc,s + VtransI(1 - 3 COS'
I
2880
- 2(J'c,s - vtrans) cos3 $01
,
00
720
1440
2160
2880
00
720
1440
2160
2880
3600
a, = M[(Vcis + !"trans) COS 40 + (vas - vtrans)('/z + '/z cos2 40)l (6) where the equilibrium dihedral angle 4o = 11 1S0, the barrier heights Vc,, = 2460 cm-I, and Vlrans= 386 cm-', and M = [4 sin4 +03-'. Values of the constants in eq 2-6 are given in Table I, and the dependence at the quantities A ( 4 ) - F(4)on the dihedral angle 4 is shown in Figure 3. The diagonal elements (I,, Iy, I,) of the inertia tensor I are also shown as functions of 4 in Figure 3. The x axis is taken to bisect the internal angle 4, and the z axis is aligned parallel to the 0-0 bond (see Figure 1). A-type motion corresponds to rotation about the axis of least inertia, which is the z axis. C-type motion corresponds to rotation about the axis of greatest inertia, which is the y axis at the equilibrium angle 4 = $o. However, both I, and Iy display large variations with +, and for $ C 108' or 4 > 252' we have I, > I,. Rotation-torsion coupling appears in the rigid twister Hamiltonian in two forms. First, as noted above, the inertial parameters are functions of the dihedral angle; this will be referred to as centrifugal coupling. Second, internal rotation generates an instantaneous angular momentum about the x axis which leads to the appearance of the cross term Hrot-nb;this will be referred to as Coriolis coupling. By removing the 4 dependence of various terms in the rigid twister Hamiltonian or by eliminating certain terms altogether it is possible to study separately the effects of the centrifugal and Coriolis couplings (see section IIIC). In addition, the trans torsional barrier height can be adjusted [see eq 61 to the values reported by Dubal and Crim for different levels of O H overtone e x c i t a t i ~ n . ~ ~ The rotational-vibrational phase space for the rigid twister model of hydrogen peroxide is four-dimensional, (x,+,k,P&, so that surfaces of section6] can be used to study the phase space structure (cf. studies of the rigid bender phase space in ref 22,23). All trajectories of the Hamiltonian of eq 2 at a fixed value of the total energy E lie on a three-dimensional manifold in the fourdimensional phase space, and there are therefore three independent (60) Equation 4b of the text is a corrected version of the equation for E ( x ) in Appendix I of ref 46a. (61) Lichtenberg, A. J.; Lieberman, M. A. Regular and Stochastic Motion; Springer: New York, 1983.
.
I
3600
dded
--.
I
i
00
720
1
I
I
1440
2160
$wed
2880
3600
l
~
A Rigid Twister Model of Hydrogen Peroxide phase space coordinates. If a slice through the three-dimensional manifold is taken by fixing the value of one of the coordinates, for example, the dynamics in the full phase space induces a symplectic (area-preserving) map of the p-q space corresponding to the other degree of freedom onto it~elf.6~ In each of the surfaces of section shown in the present paper, 100 interations of this so-called return map as determined by numerical integration of trajectories in the full phase space are plotted for a variety of initial conditions. Study of the properties of the return map via the surface of section then provides information on the phase structure a t the chosen value of the total energy. In particular, it is possible to identify the presence of invariant tori6’ corresponding to regular motion, and of stochastic or ir.regular motion.61 Trajectories that lie on invariant tori fill out smooth invariant curves in the surface of section, indicating the existence of a conserved quantity (the classical analogue of a good quantum number) in addition to the Hamiltonian itself. The tori may be nonresonant or resonant, where resonant tori appear as “islands” on the surface of section and are associated with frequency commensurabilities between the zeroth-order modes.61 On the other hand, irregular trajectories appear to fill out two-dimensional regions of the surface of section, indicating the lack of any conserved quantity other than the energy. For system with two degrees of freedom, invariant curves form absolute barriers to energy transport.6l The existence of large irregular phase space components allows extensive energy flow between zeroth-order modes, on a time scale determined by the magnitude of the flux through the most impermeable bottleneck (cantorus62 or broken separatrix”) dividing the regions of interest. [Note that trajectories in large, low-order resonance zones can exhibit significant quasi-periodic exchange of energy between zeroth-order modes; cf. Figure 7.1 Determination of the location of irregular regions and invariant curves in the surface of section therefore provides a global characterization of the intramolecular (in this case, rotation-torsion) dynamics at a given energy. The configuration space for the rigid twister Hamiltonian is a torus, periodic in x with period 2 a and in 4 with period 47r (cf. eq 6). Some trajectories on this torus are plotted below (see Figure 6). The rotational surface of section is defined by passage of the torsional angle 4 through the equilibrium value 4,, (mod 4a). (There is of course a symmetry-related minimum at 4 = a - &). At each intersection, the roots of the equation for P4 are calculated to determine whether the phase point lies on the appropriate branch of the momentum (chosen arbitrarily). The resulting surface of section is symmetric with respect to reflection about x = a but, due to the linear cross term eq 2b, it is not invariant under the transformation k -k, x -x. Similarly, the vibrational surface of section is defined by passage of x through the value x = 0 (mod 2a), again choosing a particular branch of the momentum k . The angle 4 describes internal rotation about the 0-0 bond, and both the torsional potential V(4)and the vibrational Hamiltonian Hnbare periodic in 4 with period 27r. However, the full rotation-vibration Hamiltonian is periodic in 4 with period 4n (see eq 2-4), as is the vibrational surface of section. This comes about as follows: when the angle 4 goes from 0 to 2a, say, the molecule returns to its initial internal configuration; however, it is now rotated by a about the z axis (see Figure 1). To return the molecule to its initial orientation in space, the angle must go from 0 to 4a, so that the internal configuration space is covered twice. The two halves of the surface of section (from 0 to 2 a and from 2 a to 47r, respectively) are therefore associated with the same range of internal configurations but correspond to different orientations of the molecule with respect to the (fixed) angular momentum vector j. Within each half of the vibrational surface of section, there is a double-well phase space structure, associated with motion in the vicinity of the two trans configurations separated by the low trans barrier.
- -
(62) MacKay, R. S.;Meiss, J. D.;Percival, I. C. Physica 0 (Amsterdam) 1984, 130, 5 5 . (63) Channon, S. R.;Lebowitz, J. Ann. N Y Acad. Sci. 1980, 357, 108. MacKay, R. S.; Meiss, J. D.; Percival, I. C. Physica D (Amsterdam) 1987, 270, 1.
The Journal of Physical Chemistry, Vol. 92, No. 26, 1988 7197
00
20
40
BO
Angle(rad)
Figure 4. (a) The uncoupled torsional phase space. This plot shows contours of the vibrational Hamiltonian Hnb[eq 2c] in the (p+,6 ) plane. D-, E-, F- and C-type regions are labeled. (b) The uncoupled asymmetric rigid rotor phase space. This plot shows contours in the ( k , x ) plane of the Hamiltonian H,,, [eq 2c] with = bo. C$
-
Neither of the coefficients D ( 4 ) and F ( 4 ) is invariant under 4 27r; this is consistent with the observation that the two halves of the vibrational surface of section are not identical (cf. Figures 5 and 8). This lack of symmetry is discussed further below. Since the cis barrier height is quite low (2460 cm-l), librational motion can coexist with free internal rotation (no turning point in 4) at moderate excitation energies. The behavior of the rotational variable x is librational for C-type motion and rotational for A-type motion.36
+
111. Rotation-Torsion Phase Space and Dynamics
A . Uncoupled Phase Space. We first examine the structure of the uncoupled torsion and asymmetric rotor phase space. Figure 4a is a plot of the vibrational phase space, i,e., contours of Hvib. Figure 4b shows the phase space of the rigid asymmetric rotor corresponding to H 2 0 2and is obtained by drawing contours of H,,, with 4 = &. The vibrational phase space of the uncoupled rigid twister model (Figure 4a) has the structure characteristic of systems undergoing isomerization in double well potential^.^^,^^ (We note that our rigid twister Hamiltonian has the same form as the “rigid” nbutane model discussed in ref 65b.) In the uncoupled problem there are four distinct types of vibrational trajectory, which we call D, E, F, and G type. We will discuss these trajectory types with reference to one particular half of the vibrational phase space, 0 5 $I 5 2 ~ Trajectories . classified as D or E type correspond to librational motions localized about 4 = 11 1.5’ (D type) or 4 = 248.5O (E type), respectively. F-type trajectories pass between these two degenerate potential energy wells but remain localized in the range of angle 0 I4 S 2a; Le., they are unable to surmount the cis barrier; they undergo so-called trans-torsional motion.50 (64)Gray, S. K.;Rice, S.A. J . Chem. Phys. 1987, 86, 2020. (65) (a) DeLeon, N.;Berne, B. J. J. Chem. Phys. 1981, 75, 3495. (b) Berne, B. J.; DeLeon, N.; Rosenberg, R. 0.J . Phys. Chem. 1982,86, 2166.
Sumpter et al.
7198 The Journal of Physical Chemistry, Vol. 92, No. 26, 1988 0
00
05
IO
x/zn Eyre 5. (a) Vibrational (6,p,) surface of section for the rigid twister Hamiltonian q 2 with energy E = 3292 cm-' and angular momentum j = 20. (b) Corresponding rotational (k,x ) surface of Smlim.
G-type trajectories are those for which the 9 motion is rotational in nature, able to pass freely over the cis barrier. In the uncoupled phase space there is a separatrix6' dividing localized D- and E-type trajectories from F-type motions, and another dividing F-type trajectories from isomerizing C-type motion. These separatrices are shown as dashed lines in Figure 4a. There are three types of rigid rotor The first, A type, occurs a t high rotational energies (top and bottom of Figure 4b) and camsponds to rotation of the molecule around the body-fixed I axis (Figure 1). where the trajectory has the topology of a rotation in the angle x. The second type of motion, C type, occurs at lower rotational energies for fixed j and occupies the two island regions in the rotational phase space: it corresponds to rotation of the molecule about the body-fixed y axis (at the equilibrium configuration, see above). There are two stable fixed points associated with C-type motion located a t k = 0, x = 0, and x = 1. A C-type trajectory is associated with librational motion in x. The third type of motion, 9-type, defines a separatrix dividing A- from C-type motion. There are two unstable fixed points at k = 0, x = 112, and 3 r f 2 , corresponding to rotation about the axis of intermediate inertia, the x axis. Hydrogen peroxide is very nearly a prolate symmetric top (I = -0.9944), with one rotational constant ( A = 10.03 cm-') much larger than the other two ( B = 0.8690 cm'l, C = 0.8433 cm-I). Both the rotational energy and the corresponding rotational frequency therefore increase rapidly with k at fixed j. In the absence of torsion-rotation coupling, a trajectory remains of a fixed type for all time. In the presence of coupling, the separatrices break up,6I" and it is possible for trajectories initially localized in a given region of vibrational phase space to pass to another region of phase space and so change type. Such changes in trajectory type are associated with vibration-rotation energy transfer. We now examine the fully coupled rigid twister phase space for various energies and angular momenta. 6 . Coupled Phase Space: Variation wifh E, j. 1 . E = 3292 Em-'. Surfaces of section for the fully coupled rigid twister Hamiltonian are defined as described in section 11. In Figure 5 we show vibrational (part a ) and rotational (part b) surfaces of section for a total energy of E = 3292 cm-' (1.5 X IO" hartree) and j = 20 (units of h ) . [This is enough energy to overcome both the cis and trans torsional barriers, but is only 24% of the energy required to break the G O bond in hydrogen peroxide.] To help interpret Figure 5. some representative configuration space trajectories a t the same energy and angular momentum are shown
together with their surfaces of section in Figure 6. The rotational surface of section (Figure Sb) is noticeably distorted from the uncoupled phase space, with two rotation-vibration resonance zones prominent and thin stochastic layers visible. The vibrational phase space (Figure Sa) exhibits more clearly a divided phase space6' in which both quasi-periodic and apparently chaotic motion are present. There are tori corresponding to each of the four types of torsional motion in the uncoupled system: D and E (Figure 6a), F (Figure 6b), and G (Figure 6c). The trajectory shown in Figure 6a corresponds to a trapping torus (in the terminology of DeLeon and Berne6J) with respect to isomerization between the D and E wells, while the trajectory of Figure 6b is both a crossing t o r u P with respect to the D/E isomerization and a trapping torus with respect to passage over the cis barrier. The trajectory of Figure 6c is a crossing torus for passage over the cis barrier. (Note the small number of points on the vibrational surface of section, due to the small rotational frequency a t low k.) There is a thin band of chaos in Figure 5a at the boundary between the quasiperiodic D, E, and F regions; a nonquasi-periodic trajectory which crosses the D / F separatrix is shown in Figure 6d. Two types of resonant trajectories are embedded in the stochastic layer. The first gives rise to triangular islands located at 4 = r (Figure 6e) and corresponds to w(rotation) 2w(torsion). A trajectory of this type hops between the upper and lower islands in the surface of section, eventually filling out two closed C U N ~ S . The trajectory passes between the D and E wells, and so is a resonant crossing (DIE) trajectory. This crossing resonance is seen a t a variety of energies and angular momenta (cf. Figures 8a,c,e,g, and loa, and 1 le) associated with quasi-periodic resonant regions of phase space. The s a n d remnance zone is associated with the islands located at the outer edges of the D and E regions. There are two islands, one on either side of the trans-torsional barrier (4 = r), which correspond to distinct trajectories. One of these resonant trajectories is shown in Figure 6f: it corresponds to a 1:1 resonant o(libration)) trapping torus. (w(rotation) As noted in the previous section, the two halves of the vibrational surface of section (Figure Sa) are not identical. Whereas a 1:2 crossing resonance does not appear in the right half of the section ( 2 r 5 4 C 4r), regular regions of comparable size corresponding to 1:l torsion-rotation resonances are present. Figure 5 shows the phase space structure for the rigid twister model of hydrogen peroxide a t moderate energy and angular momentum. Although I : I resonant quasi-periodic trajectories exhibit periodic flow of energy back and forth between rotation and vibration (Figure 7), the absence of global stochasticity precludes extensive rotationvibration energy transfer. At lower values of the angular momentum and energy, the rotation-torsion phase space is more regular. In what follows we study the rigid twister rotation-torsion dynamics a t higher values of j and E, corresponding to exceedingly small fractions of the molecules in thermal equilibrium at 300 K. (The fraction of HOOH molecules with j = 20 is approximately 1.6 X a t T = 300 K, and 2.6 X at T = 600 K). While angular momentum selection rules prohibit the direct excitation of low-j molecules to high-j states, it is conceivable that highly rotationally excited HOOH molecules could be formed in collisions ofO('D) atoms with H 2 0 molecules. for example. In any case, it is of interest to investigate the consequences of large j values for rotation-torsion interaction. 2. E = 8778 cm-I. We next examine the rigid twister phase space at a higher energy, E = 8778 cm-' (4 X IO" hartree). and angular momentum, j = 45, and study the effects ofdeuteriation and variation of the torsional barrier height. The fraction of HOOH molecules with j = 45 is approximately 1.8 X IO-'at T a t T = 600 K. Surfaces of section for = 300 K, and 8.6 X the rigid twister model are shown for HOOH (Figure 8a.b), DOOH (Figure 8c,d), DOOD (Figure 8e,f). and for HOOH with a trans torsional barrier corresponding to that reported by Dubal and Crims' for the fifth O H (Suo") overtone excitation (Figure 8g,h). The phase space structure of Figure 8a.b is similar to that in Figure 5a.b. The 2:I crossing resonance zone is particularly
-
-
The Journal of Physical Chemisfry, Vol. 92. No. 26. 1988 7199
A Rigid Twister Model of Hydrogen Peroxide
1
0
-I
---
Figure 6. Trajectories in (6.x ) space and skated mlational and vibrational surfaces of section for energy and angular momentum values corresponding to Figure 5 . (a) Quasi-periodic D-type (laealized below trans barrier). (b) Quasi-periodic F-type (above trans barrier, below cis barrier). (c) Quasi-periodic C-type (above cis barrier). (d) Irregular trajectory crossing D/Fseparatrir (trans barrier). (e) Quasi-periodic 1 :2 remnant D/E crossing trajectory. (0 Quasi-periodic I:Iresonant trapping trajectory.
prominent. In addition, narrow resonant islands appear in the G region. A resonant trajectory of this type is shown in Figure 9. There is an increase in the fraction of chaotic phase space; in fact, there appear to be no F-type trapping tori. Note that the narrow bands of chaos in the rotational phase space (Figure 8b)
a t high rotational energies (lkl)correspond to the chaotic D and E regions in Figure Sa. Examination of individual non-quasi-periodic trajectories initiated inside the D or E regions indicates the existence of trapping on picosecond time scales by both the D,E/F and FIG broken
7200 The Journal of Physical Chemistry, Vol. 92, No. 26, 1988
Y
a
d -
00
,
1
I
20
40
60
40
60
Time(vs)
3
8 a 00
20
nme(Ps> Figure 7. Rotational (a) and vibrational (b) energies for the 1:l resonant trajectory of Figure 6f. TABLE 11: Decay Probabilities and Lifetimes for Ensembles of Trajectories Initially Localized in the D Region of Vibrational Phase Space (SeeFigure 9)" 3' % unreacted after 5 ps lifetime/ps HOOH 91 4.5 1.4 25 HOOD 0.9 DOOD 8 HOOH~ 100 a A trajectory is considered to have reacted when it first appears on the vibrational surface of section in the angle range 2n 5 + I4n. bTorsional barrier for vOH = 6.53
s e p a r a t r i c e ~ .Localization ~~ of globally chaotic trajectories inside the broken separatrices is clearly apparent in the surface of section of Figure loa, which shows the accumulated intersections from 0 to 1.5 ps of an ensemble of 400 trajectories initially localized in the D region (Figure lob). The innermost, darkest region is filled with quasi-periodic trajectories, which are trapped for all time in the D region. The outer, lighter rings correspond to non-quasi-periodic trajectories trapped by the broken D,E/F separatrix. On a time scale of 1.5 ps, a small fraction of the ensemble has crossed the D / F separatrix, and a few trajectories have also crossed the F / G separatrix and passed over the cis barrier. However, no trajectories have found their way into the innermost chaotic region on the right-hand side of the plot, although they will do so on longer time scales. Surfaces of section for the partially (HOOD) and fully (DOOD) deuteriated species are shown in Figure 8, parts c,d and e,f, respectively. The changes in phase space structure upon deuteriation are quite subtle. A slight increase in the fraction of chaotic phase space from HOOH to HOOD to DOOD is apparent. However, the major difference occurs in the extent of trapping of globally chaotic trajectories by broken separatrices; localization of trajectories initiated in the D or E regions is much less pronounced in the deuteriated peroxides than for HOOH. This point is illustrated in Table 11, where the probability that a trajectory started on one half of the surface of section (0 I4 < 2a) will end up in the other half (2a I C#J < 4r) after 5 ps is given for all three isotopic species, together with effective lifetimes for passage across the cis barrier. Deuteriation leads to significantly reduced localization, and we conclude that the effective torsion-rotation interaction is stronger for the deuteriated analogues of HOOH, at constant total energy. This trend may be rationalized as follows. Isotopic substitution changes the effective mass for torsional motion, and increases the
Sumpter et al. TABLE 111: Largest Fourier Coefficients of Coupling Terms A (+), B ( $ J )D , ( $ J )and , F ( + ) in the Rigid Twister Hamiltonian Eq 2 (cm-I) B C D F H O O T 1.59 X 10-I -1.62X IO-' 1.78 9.88 X IO-' HOOD 2.22 X IO-' -2.19 X IO-' 1.74 9.81 X DOOD 2.72 X IO-I -2.70 X IO-' 1.74 9.87 X
moments of inertia, leading to a reduction in the rotational constants. [Vibrational and rotational (fixed k ) frequencies therefore both decrease. Since the rotational frequency decreases faster than the vibrational frequency upon deuteration, the 2:l resonance zone (for example) shifts to higher values of k, leaving the overall appearance of the rotational and vibrational surfaces of section essentially unchanged.] The most important terms responsible for rotation-torsion interaction are the centrifugal couplings inand D ( 4 ) [eq 2a], and the Coriolis term volving B ( 4 ) , e(+), involving F(4) [eq 2b], with the strength of the rotation-vibration coupling determined by the magnitude of the respective Fourier coefficients. In Table 111 we show the dominant Fourier coefficients of B, C, D,and F for the three isotopic species. Although the magnitudes of the Fourier coefficients of F and D remain almost unchanged upon deuteration, the magnitudes of the largest Fourier coefficients of B and C both increase steadily from HOOH to HOOD to DOOD. In other words, the inverses of the moments of inertia I , and Zy undergo larger variations with 4 upon deuteriation, increasing the centrifugal contribution to the rotationvibration coupling. Dubal and Crim found that the effective torsional barrier height depends upon the level of O H overtone e ~ c i t a t i o n and , ~ ~ determined the torsional barrier heights in hydrogen peroxide for vOH = 0-6 using an adiabatic separation of the torsional mode from the O H stretches. Figure 8g,h shows the phase space structure for the rigid twister model of HOOH with a trans torsional barrier of 850 cm-' and a cis torsional barrier of 2910 cm-', corresponding to the values determined by Dubal and Crim for the 6vOHstate. With the higher trans torsional barrier invariant curves corresponding to F-type crossing tori reappear, so that, as discussed in section 11, non-quasi-periodic trajectories can be localized for all time below the cis barrier. Changing the barrier heights therefore brings about a qualitative change in the rotation-torsion dynamics. Figure 11 illustrates the effects of increasing the total angular momentum at constant total energy ( E = 8778 cm-I) on the phase space structure of the rigid twister model, t o j = 60 (Figure 1la,b) a n d j = 80 (Figure 1lc,d), respectively. (We emphasize again that at 300 K a negligible fraction of the molecules will have such large angular momenta.) At j = 80, there is a transition to nearly global chaos, although regions of quasi-periodic resonant motion embedded in the stochastic sea are visible in the vibrational phase space (Figure 1IC). It is interesting to note that quasi-periodic C-type rotational tori separated by a well-defined separatrix from resonant/chaotic A-type trajectories are present in Figure 11, b and d. (The invariant curves at small k on the rotational surface of section correspond to invariant curves on the outer edges of the vibrational surface of section.) The appearance of large-scale chaos in the high k , A-type region contrasts with the rigid bender model of water.22,23In the rigid bender phase space, large-scale stochasticity first appears in the vicinity of the separatrix between A- and C-type motion and spreads outward the A-type traject o r i e ~ . For ~ ~ the , ~ ~rigid twister model of HOOH, the rotational frequencies are so low (and the moments of inertia so large) for C-type rotational motion that there is an effective time scale separation between the torsional and rotational motions, leading to quasi-periodic motion at low k . C. Coupled Phase Space: Centrifugal versus Coriolis Couplings. The importance of Coriolis and centrifugal couplings (cf. section 11) in the rigid twister model of hydrogen peroxide can be assessed by setting various terms in the Hamiltonian equal to zero or to their values at the equilibrium torsional angle &. Figure 12a,b shows vibrational (Figure 12a) and rotational (Figure 12b) surfaces of section in the absence of centrifugal coupling, at E = 8778 cm-', j = 45. That is, the coefficients A ,
The Journal of Physical Chemistry. Val. 92. No. 26. 1988 7201
A Rigid Twister Model of Hydrogen Peroxide 0
-1 - 1
I
00
00
20
i1.n
0.5
1.0
00
x/2n
,
4 20
.. .I -
1.0
1.0
05
10
X/ZX
51 ~~~
0.0
00
1.0
2.0
05
10
X/ZX
0.0
1.0
2.0
.Y/27r ..
Figure 8. (a) Vibrational (+,pa)surfaceof section for the rigid twister Hamiltonian eq 2 for HOOH with energy E = 8778 cm-' and angular momentum j = 45. (b) Corresponding rotational (k.x) surface of section. ( c ) As for (a), HOOD. (d) As for (b). HOOD. (e) As for (a). DOOD. (0 As for (b). DOOD. (8) As for (a). except that the torsional barriers V, and V,,, have been increased to 850 and 2910 cm-'. respectively. (h) As for (b). except that the torsional barriers V, and V,- have been increased to 850 and 2910 cm-', respectively.
B, C, D,and C in the rotation-torsion Hamiltonian of eq 2 are fixed a t their equilibrium values, g = go. The phase space struclure is substantially less regular than for the full Hamiltonian (Figure 8a,b). In Figure 12c.d, we show surfaces of section in the absence of Coriolis coupling, Le., the coefficient F i n eq 2b is set equal to zero. Again, the phase space is readily seen to be appreciably more irregular than that for the full Hamiltonian. The increase in irregularity in setting either ofthe two sources of rotationvibration coupling in the rigid twister equal to zero suggests that there is a cancellation of centrifugal and Coriolis perturbations in the full Hamiltonian. Such a cancellation is analogous to the rotational decoupling described by Shirtss and to the observation that potential energy coupling tends to cancel the effects of kinetic coupling in stretch-bend problem^.^.^' (66) Green. W. H.; Lawrancc. W.
D.: Mwre, C.8. J . Chem. Phys. 1987.
86.6WO. (67)Garcia-Ayllon. A.; Santamaria. J.: Ezra, G. S. J. Chem. Phys. 1988. 89. 801.
Figure 12efshows vibrational and rotational surfaces of section in the case that coefficients A, B. C, and G are set equal to their equilibrium values (g = go),while D and F retain their full @ dependence. These surfaces of section are very similar to those of the full Hamiltonian, suggesting that the 6 dependence of D is the dominant centrifugal coupling term (cf. the magnitudes of the Fourier coefficients in Table 111). The surfaces of section of Figure 12g,h. obtained by setting D = D(g=&), are very similar to those in Figure 12a-j. further supporting the conclusion that there is a balance between the centrifugal and Coriolis couplings. It would be of interest to study the classical rotation-torsion dynamics of molecules containing symmetric top rotors, such as ethane, for which the moments of inertia are independent of the dihedral angle.
IV. Summary and Conclusions In this paper we have studied the classical dynamics of interaction between large-amplitude torsional motion and molecular rotation in a rigid twister model of hydrogen peroxide. Use of
Sumpter et al.
7202 The Journal of Physical Chemistry. Vol. 92. No. 26. 1988
..
8
.. .
...
8
+/2T
P! 0
05
2.0
1.0
00
00
'
LO
x/z'2?r
0
00
10
2.0
@/ZT
Figure 9. Trajectory in (Q. x ) space and associated rotational and vi-
brational surfaces of section for a resonant trajectory appearing in the G-type (above cis barrier) region of Figure Sa. 3
00
I 0.0 ~~
~~~~~
-
0i
X/ZT
+ J L
\\
I
~
~~
I
LO
c
Ryre 11. (a) Vibrational (6, p,) surface of section for the rigid twister Hamiltonian eq 2 for HOOH with energy E = 8778 cm-' and angular momentum j = 60. (b) Corresponding rotational (k,x ) surface of section. (c) As for (a), j = 80. (d) As far (b). j = 80.
. ....
.: .,,:
$3
13-
'
,
"'. ..
.,.:
.
