Energy surfaces, model equations

On the basis of the structure correlation principle Gilli postulated that the isomerization reaction path, which can be inferred from the Xn vs t correlation, proceeds along the valley of the energy surface. The valley connects the reagents (here cis isomers) and the products (here trans isomers). In order to construct the energy surface model Gilli used the equation... 

Finally, in brief, we demonstrate the influence of the upper adiabatic electronic state(s) on the ground state due to the presence of a Cl between two or more than two adiabatic potential energy surfaces. Considering the HLH phase, we present the extended BO equations for a quasi-JT model and for an A -1- B2 type reactive system, that is, the geometric phase (GP) effect has been inhoduced either by including a vector potential in the system Hamiltonian or... 

Nevertheless, very-long-lived quasi-stationary-state solutions of Schrodinger s equation can be found for each of the chemical structures shown in (5.6a)-(5.6d). These are virtually stationary on the time scale of chemical experiments, and are therefore in better correspondence with laboratory samples than are the true stationary eigenstates of H.21 Each quasi-stationary solution corresponds (to an excellent approximation) to a distinct minimum on the Born-Oppenheimer potential-energy surface. In turn, each quasi-stationary solution can be used to construct an alternative model unperturbed Hamiltonian //(0) and perturbative interaction L("U),... 

The classical Morse curve model of intramolecular dissociative electron transfer, leading to equations (3.23) to (3.27), involves the following free energy surfaces for the reactant (Grx-) and product (Gr +x ) systems, respectively ... 

The MM approach contrasts with the quantum mechanical (QM) part of the modeling which is restricted to the immediate area of the active site and considers either all electrons (or sometimes just outer shell electrons) explicitly in arriving at a first principle evaluation by solving the Schrodinger equation to deduce the local potential energy surface for the active site. 

The modification of theoretical gas-phase reaction techniques to study gas-surface reactions continues to hold promise. In particular, the LEPS formalism appears to capture a sufficient amount of realistic bonding characteristics that it will continue to be used to model gas-surface reactions. One computational drawback of the LEPS-style potentials is the need to diagonalize a matrix at each timestep in the numerical integration of the classical equations of motion. The size of the matrix increases dramatically as the number of atoms increases. Many reactions of more direct practical interest, such as the decomposition of hydrocarbons on metal surfaces, are still too complicated to be realistically modeled at the present time. This situation will certainly change in the near future as advances in both dynamics techniques and potential energy surfaces continue. 

The unimolecular reactions of CH3CH2CH2O2 were studied in detail (Fig. 6) complete potential energy surfaces were generated using both DFT [B3LYP/ 6-31+G(d,p) and mPWlK/6-31+G(d,p)] and CBS-QB3 methods. As expected, 1,5-H transfer [Equation (34)] occurs with the lowest barrier, followed by simultaneous 1,4-H transfer and HO2 expulsion [Equation (31)]. The overall decompositions of each H-atom transfer product (i.e., each QOOH radical) were modeled. It... 

Equation to a general computational scheme. A theoretical models needs to be unique and well defined and, to the maximum extent possible, be unbiased by preconceived ideas. It should lead to Potential Energy Surfaces which are continuous. It is also desirable (but not required) that a theoretical model be Size Consistent and... 

Because of the simplification that results from 7 = 0 for low-energy surfaces, they are often chosen as model systems in fundamental research. Even when neglecting 7re is of questionable validity, ysv and ySc are often used interchangeably for lack of suitable data. We see in Chapter 9 —Equation (9.7), for example —how ire may be determined from experimental adsorption data. Otherwise, we generally assume 7re = 0. 

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