Two aspects of the problems of calculating steepest descent paths and locating stationary points on surfaces E(X), which are sources of some confusion in the literature, are addressed. These include writing proper expressions for the gradient and Hessian, and their transformation properties relative to coordinate transformations, based on the invariance of the surface E(X). The appropriate transformation is derived, based on a constrained energy minimization condition, to achieve what we call the Hessian eigenvalue representation. This not only allows decoupling of the variables, but also points to the minimization direction and preserves the eigenvalues of the Hessian. These results allow one to use the steepest descent path and stationary point location algorithms in any coordinate system and obtain invariant results. The validity of these considerations are also confirmed through numerical examples. The stationary condition with constrained kinematic path length is also shown to yield a Hessian eigenvalue representation for the normal modes for small vibrations. Lastly, we have constructed a mathematically consistent definition of mass-weighted Cartesians where the intrinsic reaction path of Fukui is a steepest descent path. 0
Self-consistent-field and configuration interaction calculations are performed to study the C2u interactions of Be(2s2p 3P), Be(2s2p ), Mg(3s3p 3P), and Mg(3s3p ) with H2. Bound 3B2 and 1B2 complexes are found for BeH2, while for MgH2 only the singlet complex is bound. The attractive 7B2 surface of MgH2 is shown to cross the Mg(7S) + H2 7Aj surface and to lie 18 kcal/mol below the 3At surface in the vicinity of the complex minimum. The implications of this crossing for the Mg(xP) + H2 -*• MgH(2S) + H reaction dynamics are discussed.
Dynamics of classical systems often involve finding a path of evolution of the system between chosen initial and final configurations such as reactant and product states. We develop a method for calculating the dynamics of such classical systems posed as boundary value (configurations) problems. This method is based on recasting the principle of stationary action into a computationally tractable form which can be applied to a wide variety of boundary problems. We demonstrate that a path of minimum action does not always exist except for a short enough path. However, saddle points of the action can reveal interesting dynamical pathways. We give examples from particle mechanics and applications to reaction mechanisms for the H+H2→H2+H reaction.
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