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Notes: The classical variational theory of chemical reaction rates gives the rate as the equilibrium flux of systems through a trial surface in the phase space of the reaction system. The surface divides the phase space into reactant and product regions and is varied to obtain a least upper bound for the rate of product formation. For bimolecular reactions of the type A+BC→AB+C and the high pressure limit of unimolecular reactions of the type ABC→products, we have derived expressions which give the canonical and microcanonical rate coefficients for the most general dividing surface defined by internal configuration space coordinates. For bimolecular reactions, we have also derived an expression for the energy dependent mean reaction cross section for the most general dividing surface. Expressions for the rate coefficients and mean reaction cross section for any of the more restricted formulations of the dividing surface used in earlier work, and more flexible ones, can be obtained by the substitution of two terms in the appropriate general equation; namely, the partial derivatives of the function that defines the surface with respect to the internal coordinates that define the surface. For example, the flexibility of the surface can be improved systematically by introducing terms in a power series expansion of the surface. The application of a simplex algorithm to determine the coefficients (variational parameters) in an expansion of the surface that includes terms up to the second and third order may give a surface which is close to the best one possible. The minimization procedure corresponds to a search of the potential-energy function for a reaction coordinate that satisfies the variational condition. The variationally determined dividing surface identifies regions of configuration space in which the potential energy must be accurate in order to obtain accurate classical rate coefficients. Thecalculus of variations was applied to the general equations for the rate coefficients to obtain differential equations which give the best dividing surface defined by internal coordinates for the canonical and microcanonical formulations of transition state theory. The corresponding rate coefficients or mean reaction cross section are as close as possible to convergence with the results of classical trajectory calculations. The flux can thus be minimized completely by obtaining a numerical solution for the differential equations which define the best surface or nearly minimized for a series approximation to the best surface. Computational studies are required to determine the more tractable method. By using an appropriate generating function, the procedure described in this work can be extended to reactions involving more than three bodies. The variationally determined dividing surface, reaction coordinate, and reaction surface Hamiltonian could provide a basis for a semiclassical theory of reaction rates.

Notes: The classical variational theory of chemical reaction rates gives the rate as the equilibrium flux of systems through a trial surface in the phase space of the reaction sysem. The surface divides the phase space into reactant and product regions and is varied to obtain a least upper bound for the rate of product formation. For atom–diatom reactions of the type A+BC→AB+C, we derived expressions which give the canonical rate coefficient and the microcanonical mean reaction cross section for the most general dividing surface defined by internal-configuration-space coordinates [J. Chem. Phys. 87, 5746 (1987)]. The dividing surface can be expressed as a power series in two of the internal coordinates and its flexibility can be systematically improved by introducing additional terms. We apply this variational formulation to the H+H2 and H+I2 reactions. Canonical rate coefficients are calculated using the downhill simplex algorithm to find the best values of three, six, and ten variational parameters in the first-, second-, and third-order expansions of the dividing surface. For the H+H2 reaction, canonical variational rate coefficients at 300 and 900 K show the expected improving trend for the first through third-order expansions of the dividing surface. The variational rate coefficient for the H+H2 reaction converges to the classical trajectory value at 300 K and exceeds the trajectory value at 900 K by a factor of 1.18±0.10. A reactivity map is devised to show the statistical importance of configurations on the dividing surface. For the quadratic dividing surface at 300 K, the most statistically important configuration on the dividing surface is nearly symmetric in terms of internuclear distances measured from the central H atom and has a "bond angle'' for the arrangement H–H–H of 166 deg. The power series dividing surface for both the canonical and microcanonical formulations converges to a position which is close to the symmetric dividing surface of conventional transition state theory. Canonical variational rate coefficientsfor the H+I2 reaction also show the expected improving trend with the expansion order of the dividing surface. However, the best variational rate coefficient for the H+I2 reaction exceeds the trajectory value by a factor of 1.767. The effective convergence of variational values of this ratio for the third-order expansion of the dividing surface shows that at this order, the dividing surface is nearly as good as it can be when its formulation is limited to configuration-space variables. For the quadratic dividing surface, the most statistically important configuration at 600 K has I–I and I–H internuclear separations of 5.10 and 4.65 a.u., respectively, and a bond angle for the arrangement I–I–H of 109 deg. The microcanonical formalism is applied to the H+I2 reaction and quadratic variational dividing surfaces are determined for seven values of the internal energy. The dividing surfaces show a weak dependence on the energy. The improvement obtained when the microcanonical results are used to evaluate the canonical rate coefficient at 600 K amounts to only 0.265%.

Notes: A general classical variational theory of reaction rates [J. Chem. Phys. 87, 5746 (1987)] is applied to the F+H2→FH+H reaction for a series of potential-energy functions (PEFs). The variational theory gives the rate as the equilibrium flux of phase points through a trial surface which divides reactants from products and is varied to obtain a least upper bound for the rate. This dividing surface (DS) is defined by a power-series expansion of the H–H internuclear separation (r) in internal coordinates R and θ where R is the distance between atom F and the center-of-mass of H2 and θ is angle which the H2 internuclear axis makes with a line from the center-of-mass of H2 to atom F. The angle-dependent terms in the DS make it possible to describe the dynamical stereochemistry of atom–diatom reactions in a new and useful manner. The profile of the angle-dependent minimum potential energy for reaction versus orientation angle is varied systematically in the PEF series to define a trend toward a "flatter" angle-dependent barrier. Portraits of the dynamical stereochemistry are obtained for each PEF by plotting contours of the density of variational flux on the DS. These reactivity relief maps show how the accuracy of the variational method depends on the expansion order of the DS and how the field of reactivity which surrounds the diatomic reactant expands with increasing temperature and energy. The accuracy of the variational theory was determined by comparing energy-dependent mean reaction cross sections and incremental (angle-dependent) mean reaction cross sections with results obtained by calculating classical mechanical trajectories. The DS was used to show how the accuracy of the no-recrossing assumption of transition state theory depends on orientation angle. Variational and trajectory results were used to calculate energy-dependent transmission and product coefficients. © 1999 American Institute of Physics.