Intramolecular Vibrational Energy Redistribution and Torsional Isomerization: A Model Classical and Quantum Study

Harold W. Schranz and Michael A. Collins,

Research School of Chemistry, Australian National University,

Canberra, ACT 0200, Australia.

Contents

Abstract

An initial study [1], considered the nonlinear resonant interaction resulting from kinematic coupling between the torsion mode and other modes in sequentially bonded ABBA type tetra-atomic molecules. It was found that the nonlinear resonant interactions were most likely to involve the symmetric bending mode.

In order to facilitate a quantum study [2] of the nonlinear resonance between the symmetric bend and torsion modes a reduced dimensional model was employed. The low dimensionality of the system also makes it amenable to the methods used commonly in the study of ergodic properties of nonlinear classical dynamical systems [3] e.g. surfaces of section, phase space plots. The rate of torsional isomerization is compared to the prediction of Transition State Theory, and related to the observed intramolecular vibrational energy redistribution (IVR).

The dependence of the nonlinear resonance on the relevant kinematic terms in the Hamiltonian is clearly demonstrated in both the quantum and classical studies. Whereas the mechanisms for the nonlinear resonance is essentially the same, the exact frequency matching required, and strength and timescales of the resulting energy transfer can be significantly different. The extent to which classical studies of IVR can be used to make quantitative predictions will be discussed.

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Introduction

How energy moves amongst the vibrational modes of a molecule, a process known as intramolecular vibrational energy redistribution (IVR), is crucial in determining the rates and mode specificity of chemical reactions. It is of considerable current interest and practical relevance to determine whether IVR is sufficiently globally rapid that a chemical reaction can be accurately described by statistical theories or whether a dynamical treatment is necessary.

In the present study we considered the nonlinear resonant interaction of a torsion mode with higher frequency modes. We began with a full-dimensional classical study of the vibrational dynamics of sequentially bonded tetraatomic ABBA type molecules.

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Full Dimensional Classical ABBA Study


A number of models of HOOH and CH3OOCH3 (with united methyl groups) were investigated. The results of this investigation motivated further classical and quantum studies involving reduced dimensional ABBA models.

Internal Coordinates

Internal coordinates (3 bonds, 2 bends, 1 torsion) used to describe an ABBA type molecule. The torsion angle is defined such that it equals zero at the cis-configuration and equals 180 degrees at the trans-configuration:

Potential Energy Surface

As the focus was on the role that generic coupling terms (e.g. kinematic) played in facilitating IVR, the potential energy surface was taken to be a simple sum of valence terms: three Morse bond terms, two harmonic angle bends and a torsion term. The torsion mode was either taken to be harmonic (the "harmonic ABBA model") or was based on a more realistic anharmonic cosine series expansion (the "anharmonic ABBA model").

Anharmonic torsion potential for HOOH and CH3OOCH3 models:

Kinematic Coupling

The kinematic coupling is determined by the G-matrix elements expressed as a function of the internal coordinates. The G-matrix elements involved with the torsion mode have a particularly rich structure, depending on all of the internal coordinates. Thus, the torsion mode may be coupled to other modes by nonlinear resonant interactions if the required order of coupling terms is present and if a reasonable frequency matching is satisfied. For an n:1 resonance between torsion and another coordinate, the Hamiltonian must contain terms nth order in the torsion coordinate and first order in another coordinate and the other coordinate must have a frequency n times the torsion frequency. Thus, on the basis of a Taylor series expansion of the diagonal and off-diagonal torsional G-matrix elements, it is likely that low order (e.g. 2:1) resonances will lead to more rapid and extensive IVR than higher order (e.g. 4:1) resonances.

For an anharmonic torsion mode the torsion frequency is energy dependent. However, the classical frequency and the analogous quantum energy gaps can exhibit a broad range over which the effective frequency is slowly varying. In this energy range, it is possible that sufficiently strong nonlinear resonant interactions could play a role in determining the rate and extent of IVR.

Effective torsion frequency for HOOH and CH3OOCH3 models. The solid line is the classical torsion frequency, the solid circles are quantum energy gaps and the solid triangles are averages of successive energy level gaps above the cis barrier:

Because of the symmetry of the ABBA molecule only certain modes may interact with the torsion. In the present models, the torsion mode is the lowest frequency vibration and the strongest resonant interactions resulting in significant IVR involve the symmetric bending mode.

IVR Dynamics

Ensembles of 100-1000 classical trajectories were run with initial conditions involving various levels of torsional excitation and zero total angular momentum. The following examples of IVR are given for models in which the torsion is modelled by a harmonic mode. Models involving an anharmonic torsion behave similarly for the case of the strong 2:1 resonance [1].

Torsion energy vs time for a 2:1 harmonic HOOH model following 0, 1, 2, 5, 10 or 15 quanta of torsional excitation:

Valence mode and normal mode energies vs time for a 2:1 harmonic CH3OOCH3 model following single quantum excitation of the symmetric bend:

Torsion energy vs time for a 2:1 harmonic CH3OOCH3 model following 0, 1, 2, 5, 10, 20, 30 or 40 quanta of torsional excitation or 0, 1, 2, 5, 10 or 20 quanta of symmetric bend excitation:

Torsion energy vs time for a 4:1 harmonic CH3OOCH3 model following 1, 5, 10, 20, 30 or 40 quanta of torsional excitation:

The principal conclusions of this classical investigation into the vibrational dynamics of ABBA molecules were that:

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Reduced Dimensional Quantum and Classical ABBA Study


The full-dimension classical ABBA study indicated that a reduced dimensional model involving just the symmetric bend and the torsion could be a good approximation. This reduced dimensionality facilitates a detailed quantum and classical treatment. An important question is to what extent will the corresponding classical and quantum dynamics show agreement.

Quantum and Classical IVR

For the 2:1 harmonic model of CH3OOCH3 we find that the classical and quantum dynamics are quantitatively in agreement at short times (less than 2 ps) and qualitatively in agreement at longer times.

Short and long time evolution of the torsion potential energy for a 2:1 harmonic CH3OOCH3 model:

For the 4:1 harmonic model of CH3OOCH3 we find that the classical and quantum dynamics are only qualitatively in agreement at short times and agreement is much poorer at longer times.

Short and long time evolution of the torsion potential energy for a 4:1 harmonic CH3OOCH3 model:

Computer Animations of Quantum and Classical IVR

A selection of MPEG computer animations is available to demonstrate the corresponding quantum wavepacket dynamics and classical trajectory dynamics for 2:1 and 4:1 harmonic models of CH3OOCH3. The wavepacket dynamics occur over a period of 1 ps and the classical dynamics occur over a period of 3 ps. The initial state involves pure torsional excitation of about 3000 cm-1. Animations for anharmonic models of CH3OOCH3 show similar behaviour.

Typical initial (pure torsion) state of quantum wavepacket, zero-point in symmetric bend, excitation in torsion:

2:1 harmonic symmetric bend : harmonic torsion model of CH3OOCH3

4:1 harmonic symmetric bend : harmonic torsion model of CH3OOCH3

The classical mechanics of the reduced dimensional ABBA system agrees quantitatively with the corresponding quantum mechanical study across a wide range of symmetric bend : torsion frequency ratios. The level of agreement achieved between the full dimensional classical and the reduced dimensional (classical and quantum) studies suggests that a reduced dimensional approach can be a useful and accurate approximation for strong (e.g. 2:1) resonances. In the case of weaker resonances (e.g. 4:1), which lead to IVR on a slower picosecond timescale, other processes may contribute in real systems.

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Nonlinear Resonance and Torsional Isomerization

The low dimensionality of the classical two degree of freedom ABBA model makes it amenable to an analysis of the dynamics in terms of configuration space plots and surfaces of section. If the torsion mode is modelled by a realistic double-well potential, motion between the equivalent wells constitutes a simple conformational isomerization reaction. An important question is how the rate and extent of IVR for different symmetric bend : torsion frequency ratios affects the validity of statistical predictions of the isomerization rate.

Phase Space and Surfaces of Section

When the symmetric bend : torsion frequency ratio = 5.89 the system is far from nonlinear resonance and the surface of section has a large empty central island region. As the system is moved closer to 2:1 resonance, trajectories begin to traverse more of the surface of section in an increasingly global and chaotic manner.

Surfaces of Section for anharmonic CH3OOCH3 model with an initial torsional excitation of 5 harmonic quanta: Symmetric bend : torsion frequency ratios of 1.99, 3.93, 4.42, 5.89:

The behaviour seen above for the surfaces of section is reflected in the corresponding trajectory plots in configuration space.

Configuration Space Plots for anharmonic CH3OOCH3 model with an initial torsional excitation of 10 harmonic quanta. Symmetric bend : torsion frequency ratios of 1.99, 3.93, 4.42, 5.89:

Trapping in Wells

When the symmetric bend : torsion frequency ratio is near 2, the model exhibits facile and strong IVR on a subpicosecond timescale and the rate of isomerization is that predicted by statistical theory. At the same time, it is quite common to see trajectories trapped for long periods of time, 0.5 to several ps, in either well, even though the total energy is in excess of 4000 cm-1 and the trans barrier is only about 88 cm-1.

Trapping of a typical high energy trajectory for a 2:1 anharmonic CH3OOCH3 model:

Rates of Torsional Isomerization

Statistical predictions of the average isomerization rate constant (for both directions) based on microcanonical transition state theory (TST) are compared with the dynamical rates. It is found that near at 2:1 resonance, good agreement is found regardless of the initial form of excitation (local or microcanonical). Away from strong resonances, the microcanonical excitation yields dynamical rates that agree with statistical predictions. However, the local torsional excitation leads to dynamical rates in excess of the statistical rate and closer to the rate expected for the barrier crossing of a 1D torsion.

The total average isomerization rate constant for a 2:1 anharmonic CH3OOCH3 model and a 5.89:1 anharmonic CH3OOCH3 model:

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Concluding Remarks

We have demonstrated that there is a close correspondence of the full dimensional classical ABBA system with the classical and quantum reduced dimensional systems for the case of a 2:1 resonance. In both cases, the 2:1 resonant interaction produces IVR between the torsion and symmetric bend on a very fast subpicosecond time scale, even with relatively poor frequency matching. Weaker IVR can occur by 4:1 processes on a much longer timescale.

The facile 2:1 energy transfer pathway involving the torsion and symmetric bend and torsion has been shown to directly influence the dynamics of the isomerization reaction. Clearly, the results of this study are relevant to a number of isomerization reactions, where the reaction coordinate is essentially a torsion. For example, the much studied photoisomerizations of cis- and trans-stilbene and their substituted analogs involve an ethylenic torsion as a primary component of the reaction coordinate. The interaction of this coordinate with other relatively low-frequency coordinates, such as the symmetric bend, symmetric phenyl twist and the out-of-plane phenyl bend, could determine the nature of the reaction dynamics.

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Acknowledgements

The authors gratefully acknowledge an allocation of time on the Fujitsu VP-2200 of the Australian National University Supercomputer Facility on which the sequences of data for the quantum wavepacket and classical trajectory computer animations were generated. The computer animations were created from the raw data with the able assistance of Roger Edberg of the ANUSF, using AVS version 5 on an SGI Onyx and a DEC 5000/240 platform.

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References

[1] H. W. Schranz and M. A. Collins, J. Chem. Phys. 98 (1993) 1132.

[2] M. A. Collins and H. W. Schranz, J. Chem. Phys. 100 (1994) 2089.

[3] H. W. Schranz and M. A. Collins, J. Chem. Phys. 101 (1994) 307.

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Last revised Thursday 26 February 1998 EST - Harold W. Schranz Email: Harold.Schranz@anu.edu.au