Low-Lying Potential Energy Surfaces - American Chemical Society

Chapter 17

The Challenge of High-Resolution Dynamics: Rotationally Mediated Unimolecular Dissociation of HOCl 1

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Downloaded by STANFORD UNIV GREEN LIBR on September 14, 2012 | http://pubs.acs.org Publication Date: August 14, 2002 | doi: 10.1021/bk-2002-0828.ch017

Joel M . Bowman , Sergei Skokov , Shengli Zou , and Kirk Peterson 1

Department of Chemistry and Cherry L . Emerson Center for Scientific Computation, Emory University, Atlanta, G A 30322 Current address: Intel Corporation, 2200 Mission College Boulevard, Santa Clara, C A 95052 Department of Chemistry, Washington State University and Environmental Molecular Science Laboratory, Pacific Northwest National Laboratory, Richland, WA 99352 2

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We review methods to calculate molecular resonances (also known as quasibound states), and illustrate them for the unimolecular dissociation of H O C l --> Cl+OH, for which recent experiments have found dramatic fluctuations of the dissociation rate for HOCl(6v ) with the rotational quantum numbers J and K . The calculations do capture the large fluctuations of the dissociation rate with respect to the H O C l total angular momentum, and a simple, general model is presented to rationalize these results. Calculated rates are also presented for HOCl(7v and 8v ) and compared with very recent experiments. Limited, new calculations o f HO Cl dissociation show a dramatic isotope effect on the dissociation rate. OH

OH

OH

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© 2002 American Chemical Society

In Low-Lying Potential Energy Surfaces; Hoffmann, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2002.

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Introduction Unimolecular reactions are an important and ubiquitous process in chemistry. The textbook treatment of such reactions is based on a mierocanonieal, statistical theory such as R R K or R R K M theory (1). These theories apply to an energized molecule with energy in excess of say the dissociation energy of some fragment channel, or an isomerization process. Thus, these theories are not appropriate for a state-specific, unimolecular process, where the term state-specific refers to the energized molecule in a single quantum state. Such states are referred to as resonances. These metastable quantum states form the rigorous foundation of unimolecular rate theory, and this is the focus of this chapter. The relationship between these quantum states and statistical theories have been explored by a number of authors, and a recent, elegant approach to this relationship has been presented by Miller and co-workers (2), based on "random matrix" theory. Experimentally, it is difficult to prepare a molecule in a specific quantum state from which a unimolecular process can occur, and this has only been achieved in rare cases. One very striking recent example of this are the beautiful experiments done by the Rizzo (3-6) and Sinha (7,8) groups on the unimolecular dissociation of HOC1 (to form OH+C1) prepared in high OH-overtone states with complete rotational resolution. The most extensive experiments done by these groups determined the unimolecular dissociation lifetimes for rotating HOC1(6V H) as a function of J and K . In general, the lifetimes were determined to be of the order of hundreds of microseconds, depending on J and K , corresponding to a resonance width of the order of 10" cm" . Further the lifetime showed significant and intriguing fluctuations with J and K . These experiments stimulated theoretical work by us (9-14), independently by Schinke and co-workers (15-17), and recently both groups (18) to rigorously model this unimolecular dissociation. Ab /mft'o-based potential energy surfaces were constructed by these groups, and used in quantum dynamics calculations to obtain the real energies and widths of the HOC1 resonances for OH-overtones. The results of our calculations and their interpretation will be reviewed below. However, before describing that work, we present a short overview of the theory and calculation of unimolecular resonances. 0

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Short review of resonance theory and calculations Resonances are eigenstates of a molecule that are not stationary in time, but which are nevertheless initially highly localized in space. The eigenenergies of resonances are complex, with the real part being the physical energy of the resonance and the imaginary part related simply to the width (see below). Generally the imaginary part is much smaller in magnitude than the real part. These complex energies form a discrete spectrum, and thus resonances share



348 some characteristics in common with true bound states, and for that reason they are also referred to as quasibound states. This picture of resonances applies to non-overlapping resonances, where non-overlappping refers to resonances where the real part of the resonance energies of adjacent resonances are separated by many times the magnitude of the imaginary part. If this criterion does not apply then some of the simple characteristics of resonances summarized below may not hold, and so we will restrict the discussion to nonoverlapping resonances. In the time-dependent picture, the resonance wavefunction is given byΨ(ί) = Ψ(0)βχρ[-ΐ(Ε-ΐΓ/2)ί,where Ε - ϊΓ/2 is the complex energy eigenvalue of the resonance, mentioned above. Ψ(0) is the resonance wave function at t = 0. Under "low resolution" Ψ(0) looks like a bound state, but under "high resolution" this wavefunction actually has an oscillatory "tail" in the asymptotic region describing one or more dissociation channel. This split personality of resonances wavefunctions makes them fascinating to study, but also makes their calculations quite difficult, because they span a large region in configuration space. A n example of a molecular resonance wavefunction that displays this behavior is shown in Fig. 1. A s seen in this case, the "tail" is of much smaller magnitude than the main portion of the wavefunction which is located in the bound, strong interaction region of space. The first rigorous calculations of molecular resonances were done using coupled channel scattering methods. This approach, while completely rigorous, is difficult and laborious, because it requires a search for the scattering energies that produce abrupt changes in the phase of the scattering matrix (19). Currently, L methods that obtain resonance energies and wavefunctions directly are commonly used. These methods are like bound-state approaches, but differ from them in two essential ways. First, a basis in the dissociative degree(s) of freedom must extend to the non-interacting region. (This is not necessary for ordinary bound state calculations.) Second, a method must be introduced to accurately describe the exponential decay of the wavefunction in time. This can be done by explicitly propagating a wavepacket with a damping function that acts in the near asymptotic region, or by the analytic continuation of the Hamiltonian, H , to the complex plane. The latter approach has been used for a number of years by either the complex coordinate method (20), and more recently by the introduction of negative imaginary (absorbing) potentials in the near asymptotic region. It appears that the use of negative imaginary potentials for the explicit calculation of molecular resonances was first introduced by Jolicard and Leforestier (21). Since we use negative imaginary potentials in our calculations, we present some the details of their use next. Negative imaginary potentials are introduced to deal with the problem of reflection of the wavefunction from the edges of a finite grid or L basis (22). Thus, these potentials are only non-zero in the asymptotic region and within several deBroglie wavelengths of the end of the grid. For molecular resonances 2

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i

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F i g . l Radial part of resonance wavefunction clearly showing bound state charatcter in the strong interaction region and oscillatory, continuum character in the non-interacting region.


350 with a single fragmentation channel an often used form o f the negative imaginary potential -iU(R) is 0.

{

U(R)=\

_

R


M

R

Low-Lying Potential Energy Surfaces - American Chemical Society

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