The potential energy surface

In Chapter 1, it was shown that in one step in the reaction between H2 and CI2 a chlorine radical reacts with a molecule of H2. If we speculate about the structure of this three-body species, we realize that repulsions will be minimized if the structure is linear. Therefore, it is reasonable to assume that the elementary reaction step can be represented as shown in the sequence 

FIGURE 4.3 Potential energy curve for a diatomic molecule. 

For a linear triatomic transition state, it is assumed that a second potential energy curve results so that the total energy is a function of two bond distances. Therefore, a diagram can be constructed that shows energy on one axis (usually chosen to be the vertical axis), one of the bond distances on another, and the second bond distance on the third axis, which generates a three-dimensional energy surface. If we suppose the reaction 

A more fundamental problem with DFT methods is that gradient corrections to the local-density approximation are always required to give reasonable reaction barrier heights, as the LDA always overestimates the binding of the free molecule. There are, sadly, many variations on the gradient correction scheme, and which to choose is often a question of which is known to work best for the atomic species present. Estimates of barrier heights can vary considerably according to the choice of GGA [52], 

There has recently been an upsurge of interest in the classical approach as a complement to full quantum dynamics. In many cases, it is found that the agreement between classical and quantum results for the dissociation probability is acceptable if not perfect [55-60], This is illustrated in Fig. 7, which shows the quantum dissociation probability computed for the H2/Cu(l 0 0) system compared to classical and quasiclassical results [57]. In quasiclassical calculations, we use classical methods to 

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Figure Al.4.6. A cross-section of the potential energy surface of PH. The coordinate p is defined in figure Al.4.5.

For a molecule that has no observable tiumelling between minima on the potential energy surface (i.e., for a... 

In the statistical description of ununolecular kinetics, known as Rice-Ramsperger-Kassel-Marcus (RRKM) theory [4,7,8], it is assumed that complete IVR occurs on a timescale much shorter than that for the unimolecular reaction [9]. Furdiemiore, to identify states of the system as those for the reactant, a dividing surface [10], called a transition state, is placed at the potential energy barrier region of the potential energy surface. The assumption implicit m RRKM theory is described in the next section. 

State I ) m the electronic ground state. In principle, other possibilities may also be conceived for the preparation step, as discussed in section A3.13.1, section A3.13.2 and section A3.13.3. In order to detemiine superposition coefficients within a realistic experimental set-up using irradiation, the following questions need to be answered (1) Wliat are the eigenstates (2) What are the electric dipole transition matrix elements (3) What is the orientation of the molecule with respect to the laboratory fixed (Imearly or circularly) polarized electric field vector of the radiation The first question requires knowledge of the potential energy surface, or... 

Wei C M, Gross A and Scheffler M 1998 Ab initio calculation of the potential energy surface for the dissociation of H2 on the sulfur-covered Pd(IOO) surface Phys. Rev. B 57 15 572... 

The parameters used in the above expressions for the potential energy surfaces and the calculations are given in Table 1 of [60],... 

At this point, it is important to note that as the potential energy surfaces are even in the vibrational coordinate (r), the same parity, that is, even even and odd odd transitions should be allowed both for nonreactive and reactive cases but due to the conical intersection, the diabatic calculations indicate that the allowed transition for the reactive case ate odd even and even odd whereas in the case of nomeactive transitions even even and odd odd remain allowed. 

The adiabatic picture is the standard one in quantum chemistry for the reason that, not only is it mathematically well defined, but it is also that used in ab initio calculations, which solve the electronic Hamiltonian at a particular nuclear geometry. To see the effects of vibronic coupling on the potential energy surfaces one must move to what is called a diabatic representation [1,65,180, 181]. 

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