Reaction valley

In the linear approximation, since the cone is elliptic (see discussion in the preceding section) two steep sides (see Figure 14b) exist in the immediate vicinity of the apex of the cone. As one moves away from the apex along these steep directions, real reaction valleys (as in Figure 14a rather than approximate ones) develop, leading to final photoproduct minima. Thus in reality the first-order approximation will break down at larger distances, and there will be more complicated cross sections and more than two relaxation channels. Also there are symmetric cases (such as H3) in which the tip of the cone can never possibly be described by Eq. [8] because one has three equivalent relaxation channels from the very beginning of the tip of the cone. 

Kraka E, Cremer D (2010) Review of computational approaches to the potential energy surface and some new twists, the unified reaction valley approach URVA. Acc Chem Res 43 591-601... 

Figure 6.13 The Si-So conical intersection of 2i/-azirine. Si and So PESs calculated by the CASSCF method using the VDZ basis implemented in M0LPR096 as a function of Xi and X2 in units of 10 Vamu bohr and contour diagram for Sq. The arrows point toward the reaction valleys that develop on the Sq surface.

The generalized adiabatic internal modes are essential for the unified reaction valley analysis (URVA) developed by Konkoli, Kraka, and Cremer to investigate reaction mechanisms and reaction dynamics [22,52]. As an example for the application of the generalized adiabatic internal modes, the hydrogenation reaction of the methyl radical is shortly discussed here ... 

Figure 1. Schematic representation of the PES for a collinear A + BC AB + C reaction as a function of the AB and BC interatomic distances (a) viewed from above, showing the reaction valley between the repulsive walls at short bond lengths and the plateau region corresponding to fragmentation (f>) viewed from the side, showing the rise and fall of the valley floor.

Let me say at the outset that there is not just one and only one useful definition of a reaction path. Rather, the concept of a reaction valley or reaction swath [11] has real utility in those simple cases where the reaction dynamics takes place in a restricted region such as the valleys of Figs. 1 and 2. Depending on how we wish to perform dynamical calculations, for example, various definitions of a reaction path might be appropriate. 

Energy Surfaces, Advances in Molecular Electronic Structure Theory, T. H. Dunning, Jr., Ed., JAl Press, Greenwich, CT, 1990, pp. 129-173. Characterization of Molecular Porenrial Energy Surfaces Critical Points, Reaction Paths, and Reaction Valleys. 

Curvature coupling constant Bks links the motion along the reaction valley with the normal modes orthogonal to the IRC (Fig. 14.7) ... 

CALCULATION AND CHARACTERIZATION OF REACTION VALLEYS FOR CHEMICAL REACTIONS... 

ABSTRACT. The calculation and characterization of molecular potential energy surfaces for polyatomic molecules poses a daunting challenge even in the Age of Supercomputers. We have written a program, STEEP, which computes reaction paths (IRCs) for chemical reactions and characterizes the reaction valley centered on the IRC. This approach requires that only a swath of the potential surface be determined, a computationally tractable problem even for many-atom systems. We report ab initio reaction paths/valleys for two abstraction reactions the OH + H2 reaction, which is a simple, direct process and the H + HCO reaction which can proceed along two distinct pathways, a direct pathway and an addition-elimination pathway. We find that the reaction path/valley method provides many insights into the detailed dynamics of chemical reactions. 

A natural compromise is to characterize just the reaction valley leading from reactants through the transition state to products, i.e., the (3N-7)-dimensional valley centered upon the reaction path. The idea of characterizing a reacting system by a coordinate set based on such a concept has frequently been used in chemistry see, e.g., the papers listed in Ref. 5. Miller, Handy Adams (6) placed this approach on a firm theoretical foundation by deriving the nuclear Hamiltonian in a set of (3N-6) reaction path coordinates, namely, the IRC and the (3N-7) vibrational modes transverse to the reaction path. In this approach the reaction valley is... 

Together these two types of terms fully define the (harmonic) reaction valley. 

For the Li + HF reaction Dunning, Kraka Fades (7) showed that the features of the reaction valley can be readily understood in terms of the changes in the electronic structure of the system as it evolves during the reaction. For the OH + H2 reaction they showed that the terms in the reaction path Hamiltonian provide a rationale for many of the qualitative features of reaction dynamics, including such fine effects as the deposition of reactant vibrational excitation into product vibrational modes. The reaction valley approach thus provides a direct connection between the electronic structure of the system, the potential energy surface and the reaction dynamics. 

In order to determine the reaction valley the IRC first has to be calculated. Since the IRC is the steepest descent path it is the solution of the differential equation ... 

In this section we present two examples which illustrate the utility of the reaction valley model for describing the eneigetics, dynamics and mechanisms of chemical reactions. The first is a simple abstraction reaction with a single valley leading from reactants to products. Here we will focus on vibrational energy consumption and disposal in chemical reactions. The second is a not-so>simple abstraction reaction involving two radicals. For this reaction there are two valleys which lead to products one based on a direct abstraction pathway, the other on an addition elimination pathway. Here we will focus on the relative features of the two valleys. 

In summary, the reaction path does not have direct physical meaning. It is an artificial chemical instrument, a fiction of chemical thinking, but it is extremely valuable in overcoming the dimensionality dilemma. The structure of the reaction valley can be characterized by the RP itself, and by the frequencies of the transverse directions (i.e. by the steepness of the "walls of the valley"). Theories using these kinds of information represent RP concepts. This often works well because the trajectories, or wave-packets, representing the chemical system, are concentrated within the valleys. 

Miller et al. (1980) described the Hamiltonian for a reacting molecular system by using a reaction valley description of the energetics and dynamics in polyatomic systems. The whole valley floor with its inherent slope and widths of the valley is utilized to give us explanations for a variety of features of reaction dynamics. This includes specifically the dynamic coupling between the motion along the RP and the transverse vibration, i.e. the description of the energy flow from translation to vibration and vice versa. 

Investigation of the reaction valley in the harmonic approximation At each path point, the orthogonal directions to the RP are described by a quadratic (harmonic) approximation of the potential V R), which implies the calculation of the second derivatives of Vf/f) with regard to the internal coordinates. A coupling of translational and vibrational motions of the reaction complex can be described, which is the basis for a more quantitative investigation of reaction mechanism and reaction dynamics. Calculations can be done for most of the reaction systems considered by approach (2). Of course, a routine, inexpensive calculation of the matrix of second derivatives of V(R) is desirable. 

Investigation of the reaction valley by considering anharmonic corrections To calculate the correct shape of the reaction valley in directions orthogonal to the RP, third and fourth derivatives of V R) with regard to the internal coordinates R have to be calculated, which leads to a drastic increase of computational cost and limits this approach to relatively small reaction complexes. 

Investigation of reaction surfaces and reaction hypersurfaces If the translational motion of the reaction complex couples with other LAMs of the complex, the RP can be sharply curved in the regions with strong coupling. The actual trajectory of the reaction complex deviates far from the RP and a correct dynamic description can only be achieved if the reaction valley is extended to a minimum energy reaction surface or hypersurface, which embeds all LAMs. Calculations needed to describe the reaction (hyper) surface can become rather expensive and, therefore, this approach is limited to... 

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