Minimum-energy path coordinate

Techniques have been developed within the CASSCF method to characterize the critical points on the excited-state PES. Analytic first and second derivatives mean that minima and saddle points can be located using traditional energy optimization procedures. More importantly, intersections can also be located using constrained minimization [42,43]. Of particular interest for the mechanism of a reaction is the minimum energy path (MEP), defined as the line followed by a classical particle with zero kinetic energy [44-46]. Such paths can be calculated using intrinsic reaction coordinate (IRC) techniques... 

In order to define how the nuclei move as a reaction progresses from reactants to transition structure to products, one must choose a definition of how a reaction occurs. There are two such definitions in common use. One definition is the minimum energy path (MEP), which defines a reaction coordinate in which the absolute minimum amount of energy is necessary to reach each point on the coordinate. A second definition is a dynamical description of how molecules undergo intramolecular vibrational redistribution until the vibrational motion occurs in a direction that leads to a reaction. The MEP definition is an intuitive description of the reaction steps. The dynamical description more closely describes the true behavior molecules as seen with femtosecond spectroscopy. 

MEP (IRC, intrinsic reaction coordinate, minimum-energy path) the lowest-energy route from reactants to products in a chemical process MIM (molecules-in-molecules) a semiempirical method used for representing potential energy surfaces... 

The surface in Fig. 12 demonstrates that there is little coupling between the C—F translation coordinate and the bending coordinate of the complex. Stated another way, the time scale for intramolecular vibrational redistribution between these coordinates is slow compared to the time scale for breaking the C—F bond. These conclusions are not obvious upon examination of the minimum energy path shown in Fig. 11, and indeed such diagrams, while generally instructive, can lead to improper conclusions because they hide the multidimensional nature of the true PFS. A central assumption of statistical product distribution theories... 

The SQ method extracts resonance states for the J = 25 dynamics by using the centrifugally-shifted Hamiltonian. In Fig. 20, the SQ wavefunc-tion for a trapped state at Ec = 1.2 eV is shown. The wavefunction has been sliced perpendicular to the minimum energy path and is plotted in the symmetric stretch and bend normal mode coordinates. As anticipated, the wavefunction shows a combination of one quanta of symmetric stretch excitation and two quanta of bend excitation. The extracted state is barrier state (or quantum bottleneck state) and not a Feshbach resonance. 

In Fig. 18 the transition coordinates (Section 6.6.) of the three calculated transition states are shown for illustration of the above discussion. These eigenvectors give a quantitative picture of the atomic motions (towards the minima linked by the transition states) when crossing the respective barriers along the minimum energy path. As expected the transition coordinates of the Cs- and C2 -conformations are symmetric with respect to the mirror plane and the twofold axis, respectively, indicating the conservation of these symmetry elements during the associated transitions. (The transition coordinate of the Cj-form... 

The top of the profile is maximum (saddle point) and is referred as the transition state in the conventional transition state theory. It is called a saddle point because it is maximum along the orthogonal direction (MEP) while it is minimum along diagonal direction of Fig. 9.12. The minimum energy path can be located by starting at the saddle point and mapping out the path of the deepest descent towards the reactants and products. This is called the reaction path or intrinsic reaction coordinate. 

Figure 6.3 DFT calculated energies for Ag moving along the minimum energy path between two fourfold sites on Cu(100). Energies are relative to the energy of Ag in the fourfold hollow site. The reaction coordinate is a straight line in the x y plane connecting two adjacent energy minima.

Figure 1. Lower panel Minimum energy path of the four lower adiabatic states, correlating to Li( 5 )-tHF( E+) and Li( P)+HF( S+). Also, the ionic diabatic state has been qualitatively shown. Upper panel non-adiabatic couplings between the ground and first excited electronic states along the minimum energy path, as a function the internal Jacobi coordinates describing the Li+HF entrance channel.

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