The state space model of vibrational energy flow: An experimental test using SEP spectra of jet-cooled thiophosgene

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Abstract

Nearly a 1000 XB transitions of jet-cooled SCCl2 have been measured by stimulated emission pumping. The observed rovibrational band contours cover the 210–330 THz region. A statistical autocorrelation analysis of the spectra answers a key question about the state space model of vibrational energy flow: is the dilution factor σ, which describes how many eigenstates participate in an optically ‘bright’ state due to anharmonic mixing, distributed according to a bimodal probability distribution P(σ) or not? Our SEP data supports a bimodal distribution predicted by state space models.

Introduction

Quantum mechanical models of vibrational energy flow (IVR) have continuously improved for 40 years [1], [2], [3], [4]. The earliest global statistical model required only the total energy E as a parameter [5]. The tier picture added the quantum number difference Δn between vibrational states [6], positing ‘preferred’ states to which energy can flow. Tiers are very useful for qualitative interpretation of data, and as a filtering method for computations [7], [8].

To obtain a predictive analytical and numerical model [3], a third parameter must be added: the direction of energy flow in state space guided by local resonances [9]. State space is the quantized version of the classical action space [1] (Fig. 1), having as coordinates vibrational quantum numbers [3], [10]. In state space, vibrational wave packets diffuse anisotropically in far fewer than 3N-6 dimensions [11], [12]. Wave packet flow requires a sufficiently high local number of coupled states Nloc[3]; the total density of states is not a relevant parameter [13].

Here, we examine jet-cooled stimulated emission pumping (SEP) spectra of SCCl2 to test a key prediction of the state space model, the shape of the probability distribution P(σ) of the dilution factor σ[14]. The dilution factor characterizes the long-time limit of energy flow [15]. For an initial ‘bright’ state ∣0 〉, σ is given by [3]σ=0t2=iIi2+2i,j>iIiIjcosωijt=iIi2.Here, ∣t〉 is the time-evolving vibrational wavepacket, composed of a superposition of eigenstates ∣i〉 that share intensities Ii = ∣〈 0∣i 〉∣2 because of an harmonic coupling to ∣0〉. σ−1 counts the effective number of eigenstates that borrow intensity from ∣0〉.

At low energy, bright states have σ  1 (no mixing), and at high energies, σ  0. But how does the distribution shift between these limits as energy is increased? Our null hypothesis will be the global statistical model. As discussed later, it predicts that P(σ) smoothly shifts from peaking near 1 to peaking near 0: the onset of energy flow is gradual. An analytical analysis of the state space model instead predicts a bimodal distribution peaked near 0 and 1 [14]:P(σ)σ-1/2(1-σ)-3/2expπγ(T)21-σ,where γ(T) rapidly increases as the IVR threshold parameter T approaches 1. A similar bimodal distribution is recovered by the Bose Statistics Triangle Rule (BSTR) model, a local random matrix model of IVR [14]. In both models energy flow tends to be ‘all or nothing’ in the threshold region: states ‘protected’ from IVR and states undergoing extensive IVR coexist.

Eq. (2) was originally compared with C–H stretching fundamental decays from several organic molecules [14], [15], and more recently with dilution factors from an experiment-derived effective Hamiltonian [16]. We now test it more directly with SEP spectra of SCCl2, a prototype molecule for studying vibronic and vibrational couplings [17], [18], with an IVR threshold region between 200 and 300 THz, and with long spectral sequences where all modes show some activity [16], [18], [19].

To compare the experimental spectra with state space and global models, we first show that the two models predict different spectral autocorrelation functions (SAFs) for overlapping sequences of bright states. We then compare the model SAFs with the experimental SAFs. The bi-exponential SAF predicted by the state space model agrees better with experiment. The agreement is not fully quantitative: there are spectral correlations among bright states beyond the energy range assumed by our model.

Section snippets

Sample

SCCl2 (Sigma–Aldrich) was used without further purification, and seeded at 1% into a He molecular beam (backing pressure 0.1 MPa, final pressure 0.3–2 × 10−2 Pa through a 0.5 mm orifice piezoelectric valve).

Pump-dump

In SEP, vibrational states on the ground electronic surface are accessed by pumping to an excited electronic state vibrational energy level, then dumping with a second laser pulse to the desired ground state level [20]. As the dumping frequency is scanned, states are detected by monitoring

Assignments

The SEP spectra dumped from the 102, 101 and 101402 states are shown in Fig. 2A. Out of nearly 1000 transitions observed, 92 bright states were assigned based on the dispersed fluorescence spectra [18] (Fig. 2A, Table 1). The assigned states range from vibrational eigenstates to significantly fragmented features.

Vibrational band contours

Contours consist of two ≈30 GHz wide lobes (blue side: P/Q branches relative to the B state; red: R branch) [19]. The characteristic contours were fitted using rotational constants from

Acknowledgment

This work was supported by a creativity extension to grant NSF CHE 99-86670.

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