Intramolecular vibrational energy redistribution as state space diffusion: Classical-quantum correspondence

Abstract

We study the intramolecular vibrational energy redistribution (IVR) dynamics of an effective spectroscopic Hamiltonian describing the four coupled high frequency modes of CDBrClF. The IVR dynamics ensuing from nearly isoenergetic zeroth-order states, an edge (overtone) and an interior (combination) state, is studied from a state space diffusion perspective. A wavelet based timefrequency analysis reveals an inhomogeneous phase space due to the trapping of classical trajectories. Consequently the interior state has a smaller effective IVR dimension as compared to the edge state.Investigating the dynamics of an initially localized vibrational excitation of a molecule in terms of timescales, final destinations and competing pathways has been of considerable interest to chemical physicists for a number of decades [1,2,3,4,5,6,7]. Due to the sustained theoretical [1,2,3,7] and experimental efforts [4,5,6,7] it is only now that a fairly detailed picture of the intramolecular vibrational energy flow is beginning to emerge. Recent studies [8,9,10,11] suggest that IVR can be described as a diffusion in the zeroth-order quantum number space (also known as the state space) mediated by the anharmonic resonances coupling the zeroth-order states. The state space approach[6] makes several predictions on the observables associated with IVR. Foremost among them is that an initial zeroth-order bright state |b diffuses anisotropically on a manifold of dimension D much smaller than N (or N − 1 with energy constraint). N is the number of vibrational modes in the molecule. As a result the survival probability P b (t) exhibits power law behaviour on intermediate time scaleswith σ b = α | b|α | 4 being the dilution factor of the zeroth-order state |b and |α denoting the eigenstates of the system. Wong and Gruebele[12] explained the power law behaviour from the state space perspective by providing a perturbative estimate for D as:with n = |n i − n b | being the distance, in state space, from the state |b to other states |i and the sum is over all states |i such that |n i − n b | ≤ n. The zeroth-order quantum numbers n k are associated with the state |k and the symbol ∆ indicates a finite difference evaluation of the dimension due to the discrete nature of the state space. In practice one chooses two different distances n in the state space and evaluates eq. 2 and thus D ≈ D(n). The quantityis a measure of the number of states locally coupled to |b . The difference in the zeroth-order energies is denoted by ∆E 0 ib = E 0 i − E 0 b and V ib = i|V |b . Notice that in the strong coupling limit, ∆E 0 ib ≪ V ib , L ib ≈ 1 whereas in the opposite limit L ib ≈ 0. Thus D v (n) can range between the full state space dimension and zero [12]. For further discussions on the origin and approximations inherent to eq. 2 we refer the reader to the original reference [12]. In the context of the present study it is sufficient to note that D ∝ N loc which has been confirmed in the earlier work [12].Clearly Eq. 2 explicitly includes the various anharmonic re...

Driven coupled Morse oscillators: visualizing the phase space and characterizing the transport

“…In turn, such studies have led to researchers extending the concept of a transition state for driven systems [51] with obvious relevance to quantum control. On the other hand, attempts are also being made [52][53][54] to connect the nature of global phase space transport with the structure of the nonlinear resonance networks, also known as the Arnold web. In particular, the Arnold web is central for a classical description of the IVR dynamics.…”

Dynamical tunnelling in molecules: quantum routes to energy flow

“…Significant efforts are needed to characterize the resonance web dynamically in order to have a clear understanding of how the specific features of the web influence the IVR. Recent works [60,61,62,63,67,68,220] are beginning to focus on extracting such details in systems with three or more degrees of freedom which promises to provide valuable insights into the classical and quantum mechanisms of IVR in polyatomic molecules. Perhaps such an understanding would allow one to locally perturb specific regions of the resonance web and hence ultimately lead to control over the classical and quantum dynamics.…”

“…Following the initial studies, which were perhaps ahead of their time, far fewer efforts were made for almost a decade but there has been a renewal of interest in the problem over the last few years with the NHIMs playing a crucial role. Armed with the understanding that the notion of partial barriers, chaos, and resonances are very different in N ≥ 3 fresh insights on transition state and RRKM theories, and hence IVR, are beginning to emerge [58,59,60,61,62,63,64,65,66,67,68,69].…”

Models for Intrinsic Non-RRKM Dynamics. Decomposition of the SN2 Intermediate Cl––CH3Br

“…In the work presented here the intrinsic non-RRKM N(t) is represented by the above multi-exponential function, but it should be noted that both power law [12][13][14][15][16][17] and stretched exponential [18] expressions have also been used to describe non-exponential unimolecular decomposition. RRKM theory requires chaotic intramolecular dynamics, with ergodic behavior on the time-scale of the unimolecular reaction, and there is considerable interest in characterizing the type(s) of classical motions and resulting phase space structure(s) which give rise to intrinsic non-RRKM behavior [19][20][21][22][23][24][25][26][27][28].…”