Reaction surface Hamiltonian

Carrington and Miller (235) developed a method called the reaction-surface Hamiltonian for reactions with large amplitudes perpendicular to the reaction path and for some types of reactions with bifurcation of the reaction path. In contrast to the reaction-path Hamiltonian method, in the reaction-surface Hamiltonian method two coordinates are extracted from the complete coordinate set. One coordinate describes motion along the reaction path and the second one describes the large-amplitude motion. Potential energy in space of the remaining 3JV — 8 coordinates perpendicular to the two-dimensional reaction surface is approximated by quadratic functions. It... 

As an alternative that solves the kinetic coupling problem. Miller and co-work-ers suggested an all-Cartesian reaction surface Hamiltonian [27, 28]. Originally this approach partitioned the DOF into atomic coordinates of the reactive particle, such as the H-atom, and orthogonal anharmonic modes of what was called the substrate. If there are N atoms and we have selected reactive coordinates there will he Nyi = 3N - G - N-g harmonic oscillator coordinates and the reaction surface Hamiltonian reads... 

Another obvious defect of both the RLM and BCRLM models is that they assume a collinearly dominated reaction intermediate. While the potential energy surfaces for many collision systems do favor collinear geometries, there are of course many reactions which do not. Extensions of the BCRLM model are therefore needed to treat noncollinear systems, perhaps along the lines defined by the Carrington and Miller reaction surface Hamiltonian theory. 

The BCRLM is by its very nature constrained to treating collinearly dominated reaction processes. One could extend the method to non-colllnear systems by Including effective potential terms and more complicated kinetic energy operators to represent the motion of the reacting system along its (bent) minimum energy path from reactants to products. This is indeed an example of the Carrington and Miller reaction surface Hamiltonian theory, which at present is probably the most fruitful approach for noncollinear systems. 

Thus, the apparently most accurate theoretical estimate of the barrier to proton transfer in a malonaldehyde molecule, determined as a difference between the energies of the structures XIa and XIc, is so far 4.3-5.0 kcal/mol. This value explains well fast (k > 10 s" ) tautomerization XIa F XIb observed in solution by the NMR method. Note, however, that calculations by means of a reaction surface Hamiltonian constructed for malonaldehyde [63] gave the barrier of 6.6 0.5 kcal/mol. 

LAM = large amplitude motion RP = reaction path RPH = reaction path Hamiltonian RSH = reaction surface Hamiltonian SAM = small amplitude motion SRP = specific reaction parameter SRPH = solution reaction path Hamiltonian. 

Although intrinsic reaction coordinates like minima, maxima, and saddle points comprise geometrical or mathematical features of energy surfaces, considerable care must be exercised not to attribute chemical or physical significance to them. Real molecules have more than infinitesimal kinetic energy, and will not follow the intrinsic reaction path. Nevertheless, the intrinsic reaction coordinate provides a convenient description of the progress of a reaction, and also plays a central role in the calculation of reaction rates by variational state theory and reaction path Hamiltonians. 

A very perceptive treatment of chemical reaction dynamics, called the reaction path Hamiltonian analysis, states that the reactive trajectory is determined as the minimum energy path, and small displacements from that path, on the potential-energy surface [64-71]. The usual analysis keeps the full dimensionality of the reacting system, albeit with a focus on motion along and orthogonal to the minimum energy path. It is also possible to define a reaction path in a reduced dimensionality representation. 

This paper draws a parallel between the (full) six-dimensional H + H2O —> H2 -I- OH and the (reduced) seven-dimensional H -l- CH4 —> H2 + CH3 abstraction reactions. In Sec. 2, we briefly present the initial state TD quantum wave packet approach for the A -I- BCD and X + YCZ3 reactions. The Hamiltonians, body-fixed (BE) parity-adapted rotational basis functions, initial state construction and wave packet propagation, and extraction of reaction probabilities, reaction cross sections, and thermal rate coefficients from the propagated wave packet to compare with experiments are discussed. In Sec. 3 we briefly outline the potential energy surfaces used in the calculations. Some... 

In a pivotal development. Miller, Handy and Adams [12] derived the classical Hamiltonian for a simple potential based on the MEP. The idea of the reaction path Hamiltonian is, conceptually, to consider the potential as a trough or as a stream bed along with 3N-7 harmonic walls that are free to close in or widen out as one proceeds along the trough. The potential energy surface is approximated as the potential energy of the MEP Vo(s) plus a quadratic approximation to the energy in directions perpendicular to the MEP,... 

For large systems (systems with more than four atoms) it is necessary to use methods which reduce the coordinate space for which we have to know the potential energy surface. Since chemical reactions are at most three or four-center reactions, the obvious partitioning is to treat the motion of the three or four atoms defining the reaction center by some of the methods described and the remaining motion using a small-amplitude description identical to the one used in the reaction path Hamiltonian method. In the... 

Chapter 2, Michael L. McKee and Michael Page address an important issue for bench chemists how to go from reactant to product. They describe how to compute reaction pathways. The chapter begins with an introduction of how to locate stationary points on a potential energy surface. Then they describe methods of computing minimum energy reactions pathways and explain the reaction path Hamiltonian and variational transition state theory. 

A problem that sometimes occurs in reaction-path Hamiltonians, especially for bend potentia1s, is the bifurcation of the reaction path. This occurs when a harmonic frequency becomes imaginary, and for the Raff surface this occurs for bends on both sides of the saddle point. initio calculations can be helpful in determining if the bifurcation is an artifact of the form of the analytic potential function or if it is present in the actual system. When the MEP bifurcates it is probably best to base the RPH on a reference path centered on the ridge between two equivalent MEP s. l This requires extra effort when computing vibrational energy levels since the vibrational potential becomes a double-minimum one, but it probably reduces mode-mode coupling, which (see Sect. 2) is hard to treat accurately. 

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