Reactivity eigenfunctions

The results of the derivation (which is reproduced in Appendix A) are summarized in Figure 7. This figure applies to both reactive and resonance stabilized (such as benzene) systems. The compounds A and B are the reactant and product in a pericyclic reaction, or the two equivalent Kekule structures in an aromatic system. The parameter t, is the reaction coordinate in a pericyclic reaction or the coordinate interchanging two Kekule structures in aromatic (and antiaromatic) systems. The avoided crossing model [26-28] predicts that the two eigenfunctions of the two-state system may be fomred by in-phase and out-of-phase combinations of the noninteracting basic states A) and B). State A) differs from B) by the spin-pairing scheme. 

Quantum chemistry is a most useful tool to discuss the structures, physical properties, and reactivities of 7r-electron systems (61MI2 69MI1 81MI1). In quantum mechanics, an electronic state of a molecule is described using a wave function, which is an eigenfunction of a Hamilton operator. Some assumptions must be made to calculate the wave function in the case of a complicated molecule. 

This is essentially the method proposed by Cashion and Hersch bach191 and yields results of comparable accuracy to those of Porter and Karplus.207 Several more empirical methods based on the London equation (41) have been proposed and these are discussed in the next section. It has been shown that the Cashion-Herschbach energy expression may be represented as a sum of pair potentials which are spin operators, the correct energy expression being obtained after the doublet spin eigenfunction has been projected out. This spin dependent type of potential has been used to discuss reactive scattering within the framework of the Faddeev formalism.211... 

Figure 4 shows an initial wavcpackct for a reactive scattering calculation superimposed on a potential energy surface for the Li + HF LiF + H reaction [15, 108]. The initial wavcpackct is placed in the asymptotic region of the reactant channel, where there is no force between the reactant molecules. It is constructed by first calculating the desired initial vibrational-rotational eigenfunctions of the reactant... 

The GPT for eigenvalue problems developed so far pertains to integral parameters that are functionals of the static eigenfunctions. One of these parameters is the static reactivity. GPT can be extended to take into account perturbation in functionals of eigenfunctions other than the static eigenfunctions. We show here a GPT formulation for the prompt-mode reactivity [see Section II, Eq. (28)], as well as for the prompt-mode decay constant a. Applying the procedure used to derive Eq. (134) to Eq. (281. we find 49),... 

Spiegelmann and Malrieu have proposed improved versions of this procedure,where the model space is spanned by several multi-configurational zeroth-order descriptions of the various diabatic eigenfunctions. This proposal, which fits very well the architecture of the CIPSI algorithm, has received applications on Arf excited states, NaCl curve crossing, HeNe and the Cs (7p) -i- HjfX Sg ) CsH(X Zg ) + H reactive collision. ... 

R is the translational coordinate and q denotes the coordinates of all internal degrees of freedom with the channel eigenfunctions 0n(Q) 1, 2 stand for a set of asymptotic quantum numbers. This is the multichannel formulation for q) which can be used for elastic (without q and (f)ni)i inelastic, and reactive processes (with introduction of the indices 1,2 for the arrangements A+BC and AB+C in the collinear case). uin R)] l = 2,. .,7V is a square integrable basis set. UQn R) smd U n R) have the properties of incoming and outgoing waves, which can be free functions (the special form depends on the dimension of space to be included) or distorted functions. In order to regularise won and u n u n = bey are multiplied by a cut-off function f R). In case of collinear reaction one chooses... 

In terms of the definitions in Section 2, and noting in particular that the vibrational eigenfunctions av(ra) in equation (16) can be chosen to be purely real (so it is immaterial whether or not they are complex conjugated in bras), the reactive scattering version of the variational functional in equation (34) for a general scattering matrix element Sa /,aviE) becomes ... 

The reactivity measurement which is made is shown to be highly sensitive to any harmonic effects present in the theoretical model chosen. The harmonic effects arise in the expression for the spacial flux distribution in accordance with eigenfunctions of the buckling equation (wave equation). The measured reactivity, that reactivity described by the physical model, is shown to be that of the fundamental mode. 

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