J. Phys. Chem. 1995,99, 2732-2752
2732
The Molecular Transition State: From Regular to Chaotic Dynamics I. Burghardt and P. Gaspard” Service de Chimie Physique and Centre for Nonlinear Phenomena and Complex Systems, Universit6 Libre de Bruxelles, Campus Plaine, Code Postal 231, Boulevard du Triomphe, 1050 Bruxelles, Belgium Received: June 30, 1994; In Final Form: September 8, 1994@
The semiclassical methods of periodic-orbit quantization and equilibrium-point quantization are compared from a theoretical point of view and in the application to the transition-state resonances of a collinear dissociation model for the HgI2 molecule. The classical dynamics, semiclassical quantization, and quantum resonance states of this system are discussed in detail. The system features a transition from classically regular to chaotic dynamics, the transition being initiated by a pitchfork-type bifurcation which may be characterized in terms of a normal form. In the regular regime, the results of equilibrium-point quantization and periodicorbit quantization are shown to be in good agreement. We discuss the results of periodic-orbit quantization for the chaotic regime, which we recently reported (Burghardt, I.; Gaspard, P. J. Chem. Phys. 1994, 100, 6395.), under the aspect of the symbolic dynamics which provides a classification scheme for the periodic orbits. The symbolic dynamics is shown to be attached to the phase-space structure of a Smale horseshoe, for which we provide numerical evidence in the Hg12 system. We compare with an alternative symbolicdynamics scheme, which is based on the symmetry of the system and allows for the reduction of the dynamics to a fundamental domain. We thus obtain a comprehensive picture of the classical and quantum properties of the dissociation process.
1. Introduction The molecular transition state associated with the saddle-point region of a dissociative potential energy surface (pes) is characterized by a spectrum which features resonances, Le., metastable quantum states of finite lifetime. Such resonances are observable in the frequency domain in photodetachment spectroscopy’ or as the fine structure superimposed on the broad background of a photodissociation cross section2 and in timedomain experiments, in particular femtosecond transition-state spectroscopy; in terms of recurrences of a decaying wavepacket which was initially excited from the molecule’s ground state to the saddle-point region of the dissociative pes. In the semiclassical limit, the recurrences can be associated with the periods of classical periodic orbits (po’s). This constitutes the basis of periodic-orbit quantization, as introduced by Gutzwiller in terms of the trace f o r m ~ l a . ~The . ~ trace formula provides us with a semiclassical quantization condition for classically chaotic systems. In this paper we will consider the classical dynamics, semiclassical quantization, and quantum resonance states related to the dissociation dynamics of a collinear triatomic molecule. We consider a direct dissociation process, at energies above the single barrier associated with the saddle point of the dissociative pes, such that tunneling effects may be disregarded. We refer to a two-degree-of-freedom model of Hg12,3-637which exhibits a transition between classically regular and chaotic regimes, as we described recently.* To provide a guideline for the following discussion,Figure 1 illustrates the relation between quantum-mechanical resonances and classical unstable periodic orbits, for the simple, separable, and thus classically regular, model of a harmonic oscillator coupled to a parabolic barrier. We find complex energies E,,, = ho(n p14) - iM(m 112), which define a progression of equally spaced resonances in the complex energy plane (in the lower half of the second Riemann ~ h e e t ) .The ~ lifetime, z, can be associated with the
+
@
+
Abstract published in Advance ACS Abstracts, February 1, 1995.
longest-lived resonances (m = 0), such that z = A-I. The autocomelation function (Figure 1b) features recurrences at the classical period T = 2nI/w, which is the period of a classical unstable periodic orbit confined to the “bound’ direction. The imaginary frequency (4 determines ) the exponential decay of the recurrences’ amplitude and is directly related to the classical Lyapunov exponent I of the periodic orbit; I provides a measure of the po’s linear stability and, thereby, a measure of the rate at which neighboring trajectories escape from the po’s vicinity. Figure IC shows the unstable periodic orbit characteristic of the system’s classical dynamics; in this regular system, it is the only existing PO. From the trace formula, we may derive a quantization condition, which we give here in terms of a zeta functionlo (see section 2.3.2):
=O I
where S(E) is the classical action, A(E) is the stability eigenvalue which is here energy independent, A = exp[AU, and p is the Maslov index which indicates the discrete phase jumps that are induced in the semiclassicalquantities by the way neighboring trajectories wind around the periodic orbit.’’ For the unstable PO in the harmonic system we consider, this quantization condition reduces to the expression for the complex energy eigenvalues quoted above. Consideration of this simple case raises the question of the effect of anharmonicities, in particular the question of whether or not anharmonicities can be treated in the framework of a perturbation approach. We will find that periodic-orbit theory gives an answer to this question in the sense that it distinguishes unambiguously between regular regimes, where a finite number of po’s exist and perturbation expansions can be applied, and chaotic regimes, which are characterized by an infinite number of po’s. In our analysis of HgI2, we have carried out the periodicorbit quantization for both the regular and chaotic regimes of
0022-3654/95/2099-2732$09.Q0lQ 0 1995 American Chemical Society
Molecular Transition State
J. Phys. Chem., Vol. 99, No. 9, 1995 2733
E
1.1
Figure 1. Two-degree-of-freedom model of a harmonic oscillator coupled to a parabolic barrier. Part a schematically illustrates the saddle-point region. The spectrum, Le., the absorption cross section @E), shown on the left-hand side, features broadened peaks the center frequency of which corresponds to the harmonic-oscillator energy levels indicated on the right-hand side. (b) The autocorrelation function (@(O)l@(t)) shows recurrences at the classical period T = 2z/z/w.Note the alternating sign of the recurrences, which is a result obtained for the one-dimensional harmonic oscillator. The recurrences’ amplitude decreases exponentially with the Lyapunov exponent 1. (c) Contour plot for the potential V(q1,rJ = l/~kllq?- ‘/2klr12, where 111 is the coordinate along the unique periodic orbit sustained by the potential (the PO is indicated in the figure), and r~ is the coordinate perpendicular to the PO. The values of the force constants kll and k l correspond to the harmonic saddle-point approxi-mationwith respect to the HgI2 potential, eq 28; the range of the contour plot is Arl = A(ql ri/2) = h0.6 8, and Ar2 = A(q1 - rl/2) = lt0.6 A.
+
HgI2, with the objective of comparing the results with those of a quantum-mechanical wavepacket calculation. Our analysis* has highlighted the following features: (i) Classical dynamics. We find a regular classical regime at low energies and a chaotic regime at high energies. In the low-energy regime, the symmetric-stretch orbit is the only existing periodic orbit and is unstable. Quantization based on this PO is similar to the harmonic “saddle-point quantization” illustrated in Figure 1, augmented by anharmonic corrections. In the high-energy regime, an infinite number of unstable, isolated periodic orbits exist. They constitute a fractal repeller (implying that the invariant set of trajectories that are trapped by the dynamics is nonattracting and occupies a vanishing area in phase space), which is associated with the phase-space structure of a Smale horseshoe. The transition between the regular and chaotic regimes proceeds through a series of bifurcations over a narrow energy band. The last homoclinic tangency (see section 4)marks the onset of purely hyperbolic dynamics (no KAM tori in phase space). (ii) The resonance spectrum. The resonances resulting from semiclassical periodic-orbit quantization for the regular and chaotic regimes agree well with the spectrum obtained from wavepacket propagation, except in the vicinity of the bifurcation zone. The quantum-mechanical system appears to “smooth over” the discontinuity in the classical dynamics; this effect is indeed well-known for cases where narrow resonance zones occur in the classical dynamics.I2 The resonances’ progression qualitatively carries on into the chaotic regime, even though the spacings and lifetimes appear more irregular than in the lowenergy regime. In this paper, we are going to report further on the HgI2 system, with a focus on several aspects which are of general relevance to the approach of semiclassical quantization: (i) Quantization in the regular and chaotic regimes (section 2). We propose a theoretical framework which accommodates two different quantization schemes, Le., periodic-orbit quantization and equilibrium-point quantization (a first impression of how the two schemes converge has been given above). Starting from the time domain, the quantum-mechanicalpropagator, K(t)
= exp(-iht/h) is written in terms of Feynman’s path integral, which may be evaluated by the stationary-phase method. The stationary points associated with the propagator are (i) the (stable or unstable) equilibrium points and (ii) the periodic orbits. The semisided-Fourier transform of the propagator is the Green operator G(E), the poles of which are the, real or complex, quantum-mechanical eigenvalues. We will show that there are two different ways of deriving G(E) from the propagator. One of them uses an expansion around equilibrium points, while the other proceeds again by the stationary-phasemethod and results in a sum over periodic orbits. The first method leads to a quantization scheme which is equivalent to the Van Vleck and Birkhoff normal form^.'^-'^ This provides an extension of the simple harmonic model of Figure 1, by including anharmonicities. We may speak of an effective separability of the motions in the “bound” and “dissociative” directions, in the sense that “good” quantum numbers, namely, those pertaining to the harmonic model, are preserved. This remains true as long as the global dynamics is determined by the local features of the dynamics around the saddle equilibrium point and no bifurcations have occurred (this coincides with the fact that the transformation to Birkhoff or Van Vleck normal forms diverges at a bifurcation). The second quantization scheme is the periodic-orbit quantization first proposed by G~tzwiller,~ with h-corrections as shown by Gaspard et a1.I6 This scheme applies to classically chaotic regimes where the global dynamics no longer resembles the dynamics about an equilibrium point. The chaotic dynamics is not structureless, however, but is in correspondence with the topology of the invariant set, Le., the periodic orbits of the repeller. Indeed, the periodic orbits form a generic phase-space object called a Smale hor~eshoe,’~ which provides a symbolic dynamic^'^^'^ to classify the periodic orbits. The symbolic dynamics provides the basis for approximation schemes like the “cycle expansion”20,2’which may reduce the original sum over an infinite number of periodic orbits to a summation over few fundamental PO’Swhich constitute the building blocks for all higher-order PO’S. (ii) In section 3, we apply equilibrium-point quantization and periodic-orbit quantization to the collinear dissociation model
Burghardt and Gaspard
2734 J. Phys. Chem., Vol. 99, No. 9, 1995 of HgI2, in the classically regular regime. We compare analytical expressions for the normal form associated with equilibrium-point quantization and for the zeta-function quantization, as applied to the unique PO of the regular regime. We recall the numerical results of periodic-orbit quantization, which we reported earlier,s and show that good agreement is obtained between equilibrium-point quantization, periodic-orbit quantization, and the results of a quantum-mechanical wavepacket calculation. (iii) In section 4, we discuss the transition to the chaotic regime. We focus on the initial bifurcation and its description in terms of a resonant normal form, on the role of homoclinic and heteroclinic intersections of the stable and unstable manifolds, and on the criterion of the system having traversed the last homoclinic tangency in order to enter the fully chaotic regime. (iv) The chaotic regime; Smale horseshoe and symbolic dynamics (section 5). For the HgI2 system, we present numerical evidence for the three-branched Smale horseshoe that characterizes the chaotic dynamics. This allows us to estimate the regime of validity of the semiclassical periodic-orbit quantization, which is based on the assumption of purely hyperbolic dynamics. We discuss two different possibilities of formulating a symbolic dynamics for the system. The first is directly associated with the phase-space structure of the Smale horseshoe, while the other makes explicit use of symmetry, i.e., reflection symmetry in coordinate space, which allows for a reduction of the system to a “fundamental domain”.22 (v) The quantum-mechanical spectrum and eigenfunctions (section 6). The resonance eigenstates for the harmonic model introduced above are discussed and compared with the HgI2 eigenstates. The latter feature a nodal pattern along the symmetric-stretch PO, but do not appear to show high intensities associated with the other fundamental PO’S of the classically chaotic regime. As a conclusion and discussion, we summarize the properties of the HgI2 system and give an outlook on the application to higher-dimensional systems.
2. Equilibrium-Point Quantization vs Periodic-Orbit Quantization To obtain a unified derivation of the above-mentioned quantization schemes, Le., quantization in terms of equilibrium points and in terms of periodic orbits, it is convenient to start from the time-domain description and derive the relevant energydomain quantities, in particular the Green operator and the level density, from Fourier-transform relationships. In the following two subsections, we will show how the quantum-mechanical propagator in the path-integral representation is evaluated by the stationary-phase method, including corrections in h which are obtained by Taylor expansion around the critical points, Le., equilibrium points and periodic orbits. Subsection 2.3 addresses the derivation of the Green operator as the semisided Fourier transform of the time-domain propagator and states the expressions for tr G(E) pertaining to equilibrium-point quantization and periodic-orbit quantization. The presentation we give here is based on the more detailed account of ref 23. In the study of molecular systems, the consideration of both equilibrium points and periodic orbits is a necessity because both types of orbits can be sustained by molecular potentials. This is in contrast with several classically chaotic systems recently studied, like the hard-disk scatterers24 or the helium atom25and hydrogen negative ion.26 For these systems, the potential has no isolated equilibrium points such that equilibrium-point quantization of nonbound systems is not applicable. The method of equilibrium-point quantization has recently been
applied to interpret the photoabsorption spectrum of ozone.29 The method has also been used to calculate tunneling probabilities for systems where tunneling provides the dominant contribution to the reaction rate.27 2.1. Path Integral and Stationary-PhaseIntegration. Our starting point is the Feynman path integral which r!presents the quantum-mechanical propagator, K( r ) = exp( -iIfT/h), in coordinate space. The quantity of interest is the trace of the propagator, the semisided Fourier transform of which yields the trace of the Green operator G(E),
6(E)..----- SadTexp(j$T]k(T) E-H ih 0
(2)
if Im E > 0. The quantum-mechanical, real or complex, eigenvalues of the Hamiltonian are the poles of the Green operator and therefore the poles of its trace. In the time domain, the trace of the propagator can be expressed as a path integral by discretization of the time interval T into N time steps At = T/N N- I
with the action of the path defined in terms of the Lagrangian
Uild = ( W 4 * - V(q),
which is a discrete version of the integral W = f L dt. The semiclassical approximation resides in the evaluation of the path integral by the stationary-phase method, which presupposes that the kernel is rapidly oscillating &e., the action W is large on the scale of h), such that non-negligible contributions arise only from the neighborhood of critical points, Le., extrema of the kernel. Let us consider, as an example, a one-variable integral with an arbitrary function @(x) having one stationary point, where a&(xo) = 0. The stationary-phase integral is given by
with the notation &’ = .
-&&
Figure 2. Feynman diagrams contributing to the coefficients C1 and C2 of the h and h2corrections of eq 14.
2.2.1. Equilibrium Points. For an equilibrium point at position q,, the classical action is given by Wcl = -TV(q,) = -E,T. The Feynman path integral can be evaluated in normalmode coordinates around the equilibrium point. At the leading order, the quantum amplitude is thus simply related to the normal-mode frequencies wk, which are familiar from the perturbation techniques of molecular spectroscopy.The normal-mode frequencies may be real or imaginary; e.g., for the case of the molecular transition state, the frequency corresponding to the reaction coordinate is imaginary. In the following, we choose to simplify the notation by referring throughout to real frequencies wk, with the understanding that wk has to be replaced by f d k for an unstable direction. The higher orders can be calculated in terms of the highorder derivatives of the potential$, = PV/aQ,,...aQ,, characterizing the anharmonicities near the equilibrium point. As discussed above, we also need the classical Green matrix of the Jacobi-Hill equation associated with the equilibrium point. Thanks to the decomposition of the linear motion in normal modes, the Green matrix is diagonal with the elements
The contribution of an equilibrium point to the trace of the propagator is then given by
Let us shortly comment on an alternative formulation of the linear stability problem, namely, in phase space rather than coordinate space. Rewriting the Jacobi-Hill equation in terms of two first-order differential equations in dq and dp, we obtain where the coefficient of the first anharmonic correction is
+
for a Hamiltonian of the form H = b2/2) V(q). If a perturbation transverse to the periodic orbit is considered, it is convenient to choose a coordinate system with directions parallel and transverse to the flow. As shown in ref 30, the linear stability problem then reduces to a 2 x 2 problem which, by integration over the period Tp of the periodic orbit p , directly yields the monodromy matrix, which occurs in Gutzwiller’strace formula4, and its generalization, eq 23 below. 2.2. Time Domain: Trace of the Propagator. With these concepts in mind, let us turn to the results of the stationaryphase evaluation of the time-domain propagator. In molecular systems, the vibrational dynamics unfolds around both equilibrium points and periodic orbits, so that the trace of the propagator can be decomposed accordingly,
We first consider the contribution of the equilibrium points.
Each coefficient C, of the anharmonic corrections can be expressed in terms of Feynman diagrams as shown in Figure 2. A vertex with n legs corresponds to a derivative of order n of the potential, +,,..in. A line corresponds to a Green function gk(t,J) connecting two vertices. Integrals are thereafter carried out on all the different times associated with the different vertices. In this way, it is possible to obtain the anharmonic corrections to the energy levels. By expanding the coefficients Cn(Z‘)in Taylor series of the variables z, = exp(-iu,Z‘) and by exponentiating terms which are linear in the time T, we can transform the trace of the propagator into a multiperiodic function of time of the form
Burghardt and Gaspard
2736 J. Phys. Chem., Vol. 99, No. 9, 1995
with
f
E ~ , . .= . ~E, ,
+ AE, + Chw/(nj+
I/J
+
j= 1
f J
