Dynamical potential approach to DCO highly excited vibration

doi:10.1016/j.cplett.2007.03.062

Chemical Physics Letters

Volume 439, Issues 1–3, 4 May 2007, Pages 231-235

https://doi.org/10.1016/j.cplett.2007.03.062 Get rights and content

Abstract

The conserved polyad number of DCO vibration is employed to classify its highly excited vibrational states. The dynamical potentials are shown to be anti-Morse and harmonic along the D–C and C–O coordinates, respectively. These, together with the Lyapunov and phase space analyses, are demonstrated to interpret the state stability that in the lower energy realm according to a polyad, D–C is prone to be dissociative or chaotic while the states in the very high energy realm, are stable and regular. A model of a particle moving in an anti-Morse potential and coupled to a simple harmonic oscillator is established for DCO vibrational dynamics.

Graphical abstract

The 28 levels with P = 6 (a), the percentages of nondissociation cases for these levels (b), the dynamical potential corresponding to the bending coordinate (c) and the average Lyapunov exponents for the levels (d) are shown.

Introduction

With the advent of laser technology, the dynamics of highly excited vibration is being under exploration. DCO (deuterated formyl radical) is such a system that has been studied heavily both experimentally and theoretically [1], [2], [3], [4], [5]. Refs. [1], [2] treated the bound and resonance states based on the potential energy surface which is calculated by ab initio algorithm and their analysis was confirmed by SEP spectra. Notably, they found the destruction of the conserved quantities such as the actions on the individual bond coordinates. This led to the ambiguity in the state assignment. However, they also confirmed the 1:2:1 (D–C stretch-bend-C–O-stretch) resonance. Based on the levels (states) reported in Ref. [2], Troellsch et al. [3] employed a spectroscopic (algebraic) Hamiltonian to study its kinetics of the intramolecular vibrational energy redistribution (IVR). They claimed the coupling of the CO stretching states with the DC stretching states was found to be strong up to v = 3. However, the coupling of higher DC stretching states (v > 4) remained weak. They could confirm the decaying rates of the states just above the dissociation limit. However, for those of higher energy, their prediction was not accurate. More works on DCO vibrational dynamics were done via analysis based on periodic orbits [4] and density and phase plots [5]. These works mainly involved the mode assignment.

In this work, we will treat this issue from a global viewpoint, instead. That is, we will explore the dynamics of the whole levels sharing a common polyad number according to the 1:2:1 resonance in a dynamical phase (coset) space in which classical nonlinear dynamical ideas such as dynamical potential and Lyapunov exponent can be employed for the dynamical analysis. Furthermore, a dynamical model of an anti-Morse oscillator coupled with a simple harmonic oscillator is proposed for the DCO vibrational dynamics. Finally, a concluding remark concerning this algorithm will be discussed.

Section snippets

The dynamical phase space, polyad number and Lyapunov exponent

The algebraic Hamiltonian we used is that by Troellsch et al. [3]. We transformed it to the SU(3)/U(2) coset coordinates [6] which is a classical dynamical space for the quantum levels [7]. A point in this space corresponding to an eigenstate represents a vibrational configuration. The transformation leads to: $n_{2} = 2 (P - n_{1} - n_{3}), n_{1} = (q_{1}^{2} + p_{1}^{2}) / 2, n_{3} = (q_{3}^{2} + p_{3}^{2}) / 2$ and the coset Hamiltonian in (q₁, p₁, q₃, p₃) is: $H (q_{1}, p_{1}, q_{3}, p_{3}) = ω_{1} (\frac{p_{1}^{2} + q_{1}^{2}}{2} + \frac{1}{2}) + X_{11} {(\frac{p_{1}^{2} + q_{1}^{2}}{2} + \frac{1}{2})}^{2} + ω_{2} (2 P - p_{1}^{2} - q_{1}^{2} - p_{3}^{2} - q_{3}^{2} + \frac{1}{2}) + X_{22} {(2 P - p_{1}^{2} - q_{1}^{2} - p_{3}^{2} - q_{3}^{2} + \frac{1}{2})}^{2} + ω_{3}$

Dynamical potential

For a better exploration of the DCO vibrational dynamics, we need to calculate the dynamical potentials [10], [11], which are the effective potentials D–C, the bending and C–O experience, dynamically. This is done by varying (p₁, p₃) to obtain extreme energies E₊ (maximal) and E₋ (minimal) for each (q₁, q₃) under the constraint $p_{1}^{2} + q_{1}^{2} + p_{3}^{2} + q_{3}^{2} ⩽ 2 P$ to ensure positive n₂. E₊(q₁, q₃) and E₋(q₁, q₃) thus define the dynamical potential in which the levels sharing a common P reside. Alternatively, we can

The state dynamics under a polyad number

The dynamics of the levels under P will be discussed first. For each level, we will simulate the stability of D–C bond by choosing randomly 200 initial points in the phase space and following their trajectories for a duration of 10 ps. D–C bond is considered as stable (nondissociative), if n_D-C < 4 during this time interval. The dissociation is considered as that, once a trajectory has crossed the dividing surface in the direction of products, it will never return to the reactant region of phase

Contrast of the dynamical potentials of D–C and C–O stretches

It is interesting to note the contrast of the dynamical potentials of D–C and C–O stretches. We show them in Fig. 3a and b, respectively. The transformation from q_i to cartesian Δr_i is via $Δ r_{i} = a_{i}^{- 1} \ln {[1 - (1 - λ_{i}^{2})^{1 / 2} sgn] / λ_{i}^{2}}$ , with q_i < 0, sgn = 1; q_i > 0, sgn = −1. $λ_{i} = 1 + X_{ii} (q_{i}^{2} + 1) / ω_{i}$ and $a_{i}^{- 1}$ is the characteristic length of a Morse potential [15]. The dynamical potential of D–C stretch is an anti-Morse (upside-down) potential while that of C–O is, more or less, like an harmonic one. We, therefore,

Concluding remarks

The dynamics of scattering and resonance states is expected to be quantum mechanical. However, this does not rule out the semi-classical approach from certain aspects. In this article, we demonstrated that the coset space dynamics, Lyapunov analysis and dynamical potential are useful for the understanding of state dynamics. It is also demonstrated that the dynamical behavior of the levels as classified by the polyad number shows parallelity and similarity. This clarifies the superficial

Acknowledgements

This is supported by the National Natural Science Foundation Of China (Grant No. 20373030), the Key grant Project of Chinese Ministry of Education (No. 306020) and the Specialized Research Fund for the Doctoral Program of Higher Education. We thank Prof. H.S. Taylor for his stimulating discussion and preprint to be published in J. Phys. Chem. during his visit to Tsinghua in October, 2006.

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