Partition function Hamiltonian modes

Flexible RRKM theory and the reaction path Hamiltonian approach take two quite different perspectives in their evaluation of the transition state partition functions. In flexible RRKM theory the reaction coordinate is implicitly assumed to be that which is appropriate at infinite separation and one effectively considers perturbations from the energies of the separated fragments. In contrast, the reaction path Hamiltonian approach considers a perspective that is appropriate for the molecular complex. Furthermore, the reaction path Hamiltonian approach with normal mode vibrations emphasizes the local area of the potential along the minimum energy path, whereas flexible RRKM theory requires a global potential for the transitional modes. One might well imagine that each of these perspectives is more or less appropriate under various conditions. 

Up to now, in the frustrated nematic systems the pseudo-Casimir force has only been determined for the simplest structure with a uniform director field d < dc = Xh Xp, where Ah Ap is the extrapolation length of the honieotropic substrate and Ap the extrapolation length of the degenerate in-plane anchoring which preserves the full rotational symmetry about the substrate normal) [12]. Then, within the bare director description the correlation length in the Hamiltonian (8.28) is constant and the partition function of the fluctuation modes can be derived analytically. The derivation of the force in the bent-director and biaxial structures is more complex due to the deformed equilibrium order. 

The basic theoretical framework for understanding the rates of these processes is Fermi s golden rule. The solute-solvent Hamiltonian is partitioned into three terms one for selected vibrational modes of the solute, including the vibrational mode that is initially excited, one for all other degrees of freedom (the bath), and one for the interaction between these two sets of variables. One then calculates rate constants for transitions between eigenstates of the first term, taking the interaction term to lowest order in perturbation theory. The rate constants are related to Fourier transforms of quantum time-correlation functions of bath variables. The most common... 

In an elegant paper, by Moleslq and Moran, a fourth-order perturbative model is suggested and developed for the study of photoinduced IC. The authors stress that in case of a similar timescale for the electronic and nuclear motions, a second-order perturbation scheme, a la Fermi, will fail. Additionally, the model, as suggested here, in the case of a dominant promoting mode, can exclusively be parameterised from experimental data. The method is based on a three-way partition of a model Hamiltonian—system, bath and system-bath interaction. Subsequent use of a time correlation function approach facilitates the evaluation of rate formulas. This analysis is applied to a three-level model system containing a ground state, an optical active excited state and an optical dark state, the latter two share a CDC. In their paper the model is used to analyse the initial photoinduced process of alpha-terpinene. The primary conclusion of the study is that the most important influence on the population decay (Gaussian versus exponential) is the rate at which the wavepacket approaches the CIX of the two exeited states. 

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