**Principal Investigator**
Harold W Schranz **Project** s10

Research School of Chemistry **Machine** VP

**Efficient Calculation of Statistical and Dynamical
Reaction Rates for Large Dimensional Molecular Systems**

Knowledge of how quickly chemical reactions occur is an essential ingredient in the rigorous modelling of combustion, industrial and atmospheric reaction systems. This project focuses on the development of new methods for the accurate prediction of rate constants for chemical reaction. Crucial to this development is a more complete understanding of how and on what timescale energy moves about a molecule.

The current dominant theories of unimolecular reaction
are statistical. A fundamental assumption is that the timescale
on which energy moves about a reactant molecule is much shorter
than the timescale for reaction. It is assumed that intramolecular
vibrational energy redistribution (IVR) is globally rapid throughout
the molecular phase space (Fig. 1). Further assumptions are made
in the application of such statistical theories. Traditionally,
reactant molecules are modelled as a set of separable harmonic
oscillators and rigid rotors. Angular momentum conservation is
often neglected or approximated.

Figure 1. Statistical evolution
of trajectories through molecular phase space for a unimolecular
reaction

It has been widely thought that the assumption of
rapid IVR referred to above is valid for sufficiently large polyatomics.
Much of the supporting evidence for this view comes from indirect
experimental studies of IVR and comparisons of statistical and
dynamical calculations. However, in recent studies, we have shown
that *even in the presence of fast IVR rates between some modes
the reaction dynamics can be extremely nonstatistical.* Secondly,
most comparisons of statistical and dynamical calculations have
made simplifying assumptions which render the comparisons ambiguous.
Frequently, the potential energy surface used for the dynamical
calculations is approximated by a normal mode analysis for the
statistical calculations. In addition, commonly used initial state
selection procedures used for the dynamical calculations often
cause artifacts such as short time transients which need to be
deconvoluted from the true dynamical behaviour.

It is apparent that in order to clearly identify the presence or absence of statistical behaviour in a chemical reaction it is necessary to compare statistical and dynamical calculations performed for exactly the same model under the same conditions.

**What are the basic questions addressed?**

How to calculate a rate constant for reaction?

When are statistical theories valid?

The role of statistical and dynamical behaviour in chemical reactions?

How to take account of non-statistical effects in a general theory?

Can we take advantage of non-statistical effects e.g. mode specific excitations of reactants causing product rate and yield enhancement?

How and on what timescale does energy move around within a molecule?

**What are the results to date and future of the
work?**

Significant progress has been made on a number of related research topics: firstly, how and how fast energy moves around an isolated molecule; secondly, how efficiently energy moves between molecules; and thirdly, how the process of energy transfer influences the subsequent dynamics of chemical reaction. A further topic involves the efficient calculation of certain statistical mechanical quantities used in current statistical rate theories.

How energy moves around a molecule

An initial classical study 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. This finding stimulated our later detailed quantum and classical
dynamical studies which were facilitated by employing a reduced
dimensional model (Fig. 2). Furthermore, the observed rate of
torsional isomerisation is compared to the predictions of Transition
State Theory. The importance of statistical or dynamical behaviour
is thus related to the observed extent of intramolecular vibrational
energy redistribution (IVR).

Figure 2. Wavepacket time evolution an initial excited (20 quanta) torsion state into an excited symmetric bend state for a quantum two-mode model. (a) t=0.0 ps; (b) t=0.05 ps; (c) t=0.1 ps; (d) t=0.3 ps.

These studies are being extended to larger molecular
systems. A potential energy surface for benzene which incorporates
the dominant potential couplings was constructed on the basis
of *ab initio* data. The surface was employed for a full
dimensional classical mechanical molecular dynamics study of IVR.
Comparisons were made with recent experimental observations regarding
the extent and timescale of IVR involving the ring modes. The
linewidths found experimentally were instrument limited at 1 cm-1
for a range of excited ring modes for excitations of between 1200
and 8200 cm-1 yielding
an upper limit on the IVR rate of 0.094 ps-1.
This result is consistent with preliminary trajectory calculations
which reveal an initially rapid decay followed by slow IVR at
longer times.

How energy moves between molecules

In gas-phase unimolecular reactions, intramolecular energy transfer in isolated molecules competes with intermolecular energy transfer between molecules. The extent to which either energy transfer process can be described by statistical theories is dependent on the extent of intra- and intermolecular coupling present. The role of such couplings in determining the nature and extent of energy transfer was examined in a model study of arrays of one dimensional monatomic chains under tensile stress. It was noted that chain arrays (where chains are only weakly coupled to each other) show greatly increased nonstatistical effects.

Intermolecular (collisional) energy transfer was reviewed in the context of unimolecular reaction rate theory. The concept of a collision is found to differ between statistical theories and reality or simulations; some care is required in comparisons. Results obtained in detailed classical trajectory simulations of small molecule collisions were surveyed and discussed with an emphasis on nonergodic effects (e.g. angular momentum conservation) and the dependence of energy transfer on the hardness and capture strength of the intermolecular potential.

How energy transfer influences the subsequent reaction dynamics

A classic prototype reaction in the study of unimolecular reactions is the isomerisation of methyl isocyanide. Most experimental studies have given rate constants consistent with statistical behaviour even though trajectory studies have predicted non-statistical and mode specific behaviour. Our more detailed theoretical study has indicated that the nature of the potential energy surface and the type of initial excitation can affect the extent of IVR and hence the subsequent reaction dynamics (Fig. 3).

Fig. 3. Statistical (TST) and
dynamical (trajectory) calculations of the rate of isomerisation
of CH3NC compared with
experimental results.

In summary, some detailed insights are being gained
into how the process of (intramolecular and intermolecular) energy
transfer depends on the nature of the molecular system and how
this behaviour can influence the (statistical versus dynamical)
nature of the chemical reaction.

** New method for calculating anharmonic densities
of states**

A key element in the application of statistical theories of chemical reaction is the calculation of statistical quantities such as the density of states for a reactant molecule. In order to faciliate the application of statistical theories, we developed a simulation method for the estimation of anharmonic densities of states of classical molecular models. The method is based on the equilibrium energy distribution established in an uncoupled dimer of the anharmonic molecule and a reference molecule whose density of states is known analytically as a function of energy. Applications to one-dimensional chain molecules and small clusters of atoms joined by Morse bonds indicate that the method is both simple and reliable.

**What computational techniques are used and why
is a supercomputer required?**

For the classical dynamical simulations, large numbers of trajectories need to be generated by integrating the classical equations of motion. In the statistical studies, long Markov walks need to be performed in order to properly sample the high dimensional phase space of the reactant molecule. Both types of calculations are numericallly intensive, the former requiring large amounts of CPU time but vectorizes well. A further type of calculation performed are quantum dynamical simulations of energy transfer. The current method does not vectorize well, due to the highly complex nature of the matrix elements, but use of a supercomputer is essential as memory requirements are larger than available on workstations at the RSC.

**Publications**

*Collisional Energy Transfer in Unimolecular Reactions.
Statistical Theory and Classical Simulation*,
S. Nordholm and H.W. Schranz, in Advances in Chemical Kinetics
and Dynamics*,* (John R. Barker ed.), JAI Press, 1995, Vol**.
2A**, pp. 245-281.

*The Fragmentation of One Dimensional Monatomic
Chains under Tension - Simulation and Statistical Theory,*
K. Bolton, S. Nordholm, and H. W. Schranz, Stuart Rice Festschrift,
*J. Phys. Chem.*, **99**, 2477-2488 (1995).

*Intramolecular Vibrational Energy Redistribution
and Torsional Isomerization: A Model Classical and Quantum Study*,
H. W. Schranz and Michael A. Collins, in Proceedings of the First
Electronic Computational Chemistry Conference, edited S. Bachrach,
CD-ROM version, (ARInternet , Landover MD, 1995).

*On the Estimation of Anharmonic Densities of States
of Molecules,* L. Ming, S. Nordholm, and
H. W. Schranz, Chem. Phys. Lett*.* submitted