Reaction path Hamiltonians

Fehrensen B, Luckhaus D and Quack M 1999 Inversion tunneling in aniline from high resolution infrared spectroscopy and an adiabatic reaction path Hamiltonian approach Z. Phys. Chem., NF 209 1-19... 

Miller W H, Handy N C and Adams J E 1980 Reaction path Hamiltonian for polyatomic molecules J. Chem. Phys. 72 99... 

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. 

S. Lee and J. T. Hynes, Solution reaction path Hamiltonian for reactions in polar... 

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. 

Miller and co-workers [64, 67] have shown that a canonical transformation of the reaction path Hamiltonian yields the form... 

A quantum dynamical study of the Cl- + CH3 Br 5k2 reaction has been made.78 The calculations are described in detail and the resulting value of the rate constant is in much better agreement with experiment than is that derived from statistical theory, hi related work on the same reaction, a reaction path Hamiltonian analysis of the dynamics is presented.79 The same research group has used statistical theory to calculate the rate constant for the 5n2 reaction... 

Of course, there is more to a chemical reaction than its rate constant the reaction path or mechanism is also of central interest. Once again, nonequilibrium solvation is crucial in describing this path. In an equilibrium solvation picture, the solvent polarization would remain equilibrated throughout the reaction course, but this assumption is rarely satisfied for an actual reaction path, because of the same considerations noted above for the rate constant. Indeed these nonequilibrium solvation effects can qualitatively change the character of the reaction path as compared with an equilibrium solvation image. Dielectric continuum dynamic descriptions thus have an important role to play here as well. Indeed, we will employ in this contribution the reaction path Hamiltonian formulation previously developed [48,49], which can be used to generate a reaction path which is the analog in solution of the well-known Fukui reaction path in the gas phase [50], The reaction path will be discussed for both reaction topics in this contribution. 

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... 

The separation of the PES into a part determined by the reaction coordinate and a part described by a quadratic approximation in a subspace of the remaining coordinates has recently often been used, typically with the WKB approximation (236,237) Yamashita and Miller (238) utilized the reaction-path Hamiltonian method combined with the path-integral method to calculate the rate constant of the reaction of H + H2. 

In a study of the rate of isomerization of HCN to CNH, Rice and co-workers [19] suggested exploiting a reaction path Hamiltonian as a device to permit extension of classical statistical reaction rate theory from few-dimensional to many-dimensional systems. In that approach the dynamics of the reacting molecule is reduced to that of a system with a complicated but one-dimensional reactive DOF coupled with other effective DOFs. Although their calculations based on this approach yield an accurate description of the isomerization rate as... 

Further reduction of the constrained reaction path model is possible. Here we adopt a system-bath model in which the reaction path coordinate defines the system and all other coordinates constitute the bath. The use of this representation permits the elimination of the bath coordinates, which then increases the efficiency of calculation of the motion along the reaction coordinate. In particular. Miller showed that a canonical transformation of the reaction path Hamiltonian T + V) yields [38]... 

A standard reaction path Hamiltonian is derived from a quadratic expansion of the potential in the modes orthogonal to the reaction coordinate [55-59]. For zero total angular momentum this Hamiltonian is expressed as... 

Flexible RRKM theory and the reaction path Hamiltonian approach take two quite different perspectives in their evaluation of the transition state partition functions. In flexible RRKM theory the reaction coordinate is implicitly assumed to be that which is appropriate at infinite separation and one effectively considers perturbations from the energies of the separated fragments. In contrast, the reaction path Hamiltonian approach considers a perspective that is appropriate for the molecular complex. Furthermore, the reaction path Hamiltonian approach with normal mode vibrations emphasizes the local area of the potential along the minimum energy path, whereas flexible RRKM theory requires a global potential for the transitional modes. One might well imagine that each of these perspectives is more or less appropriate under various conditions. 

First, we would like to mention the reaction path Hamiltonian approach (RPH) proposed by Miller et al. (1980). In this approach the information enclosed in the RP is supplemented by some information about the shape of the PES on the 3N - 7 coordinates which are perpendicular to each point of 5, and are described in the harmonic approximation. The PES thus assumes the form ... 

As seen in the reaction path Hamiltonian, Eq. (80), the energies A and B are measured from zero. On the other hand, the linear surprisal Eq. (72) does not care about the origin in the energy coordinate. For the sake of simplicity,... 

Nonadiabatic Feshbach calculations. Using the reaction-path Hamiltonian and invoking an adiabatic separation of the reaction coordinate from all other coordinates, resonance energies and adiabatic partial widths are obtained by neglecting all off-diagonal terms of the Hamiltonian. The most important... 

Reaction-path Hamiltonian model, see text for description, reference 133, reference 135, reference 57,. .. indicates calculations not performed, reference 56, 8 references 145 146... 

V.A. Benderskii, E.V. Vetoshkin H.P. Trommsdorff (2001). Chem. Phys., 271, 165-182. Tunnelling splittings in vibrational spectra of non-rigid molecules, X. Reaction path Hamiltonian as zero-order approximation. 

However, the accuracy of the detailed calculations is still not sufficient for chemical accuracy, 1 kcal/mole for instance, and the computational expense still prohibits calculation of enough points to map out the full multidimensional PES. The use of a reaction path Hamiltonian formalism (Miller et al. 1980) would reduce the number of required points but may not be particularly appropriate since the depth of the physisorption well can often reach 0.2-0.4eV. This accelerates the molecule and thus prevents following the reaction path even at extremely low initial kinetic energies. For example, on the contour plot in Fig. 17, the well M of nearly 0.4 eV will cause acceleration along Z and not along r, even though the latter is close to the reaction coordinate near point D. 

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