Research School of Chemistry **Machine** VP

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

The focus of the project is on unimolecular and related reactions in the gas phase. The aim is the efficient calculation of accurate energy and angular momentum resolved rate constants k(E,J) on global molecular potential energy surfaces.

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?

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

For comparison with the classical trajectory calculations, a new and general statistical method is proposed which takes advantage of our recently derived procedure for efficiently sampling the phase space (the space of configuration and momentum coordinates) of a reactant molecule at a well defined energy E and angular momentum J (an EJ ensemble). The phase space integrals involved in evaluating k(E,J) are difficult to calculate because:

1) They are of dimensionality 6N where N is the number of atoms in the molecule, e.g. for a molecule with 10 atoms the integrals are 60 dimensional.

2) The presence of singularities in the integrand in terms of the two delta functions specifying the energy E and angular momentum J.

Our proposed method obviates these problems by:

1) Integrating separately over the molecular momenta thereby *halving the
dimensionality of the integral* and yielding a *nonsingular integrand*
which satisfies *energy and angular momentum conservation*.

2) Utilizing a Markov walk procedure to integrate over the configuration space (Fig. 2). Especially for larger molecular systems, it is necessary to enhance the efficiency of the sampling by use of importance and umbrella sampling techniques.

3) Significant performance advantages would accrue by *vectorisation of the
time intensive Markov sampling* in the method. In its basic form, a single
very long Markov walk (>10^{7} steps) is undertaken to sample the
configuration space (Fig. 2a). On the VP2200, a better way is to startup many
(~1000) shorter walks (Fig. 2b) which should yield a *performance enhancement
of around 2 orders of magnitude*. This would enlarge the accessible range
of molecules that can be treated.

4) In addition, the *variational minimization of the rate constant* k(E,J)
with respect to the location of the critical surface qC separating products
from reactant *can be performed in a single calculation*. This is shown
in Fig. 2 where the Markov walk(s) can pass through as many (typically, about
10-20) critical surfaces as desired.

5) The treatment of the *reaction coordinate* will be *generalised*
to account for reactions such as torsional isomerisations and reactions that
involve the simultaneous variation of a number of internal coordinates.

Figure 2.

a) Scalar implementation

of a statistical method

b) Vector implementation

of statistical method

Statistical and dynamical calculations of the rate of isomerisation of disulphane (HSSH) have been performed but the analysis of the results is not yet complete. Extension of the above comparisons to other larger systems, e.g. the isomerisation of methyl isocyanide, is underway.

**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 numerically intensive, the former requiring large amounts of CPU time.