Hamiltonian modes partitioning

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... 

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. 

As an alternative that solves the kinetic coupling problem. Miller and co-work-ers suggested an all-Cartesian reaction surface Hamiltonian [27, 28]. Originally this approach partitioned the DOF into atomic coordinates of the reactive particle, such as the H-atom, and orthogonal anharmonic modes of what was called the substrate. If there are N atoms and we have selected reactive coordinates there will he Nyi = 3N - G - N-g harmonic oscillator coordinates and the reaction surface Hamiltonian reads... 

The Hamiltonian will now be summarized for one overtone excited CH oscillator interacting with the N — 1 ring modes. Normal coordinates for the ring Q2,. . . , Q.v and for the overtone excited oscillator Q, are defined by uncoupling the overtone excited oscillator from the ring. The final form for the Hamiltonian then contains terms (both potential and kinetic) which couple the CH stretch mode to the ring modes. The derivation of the vibrational Hamiltonian was presented in Section II.C of Benzene I (103), and we will only summarize the final result here. The vibrational Hamiltonian may be partitioned into the terms... 

The integrals provide a natural definition of the term "mode" that is appropriate in the context of reaction, and they are a consequence of the (local) integrability in a neighborhood of the equilibrium point of saddle-center-----center stability type. Moreover, the expression of the normal form Hamiltonian in terms of the integrals provides us a way to partition the "energy" between the different modes. We will provide examples of how this can be done in the following. 

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. 

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. 

The separation of the full Hamiltonian can be arbitrary. If the system is a single large molecule then one may collect the most relevant modes into Hsystem, and the remaining ones form HBath- However, if our system is a small molecule embedded in an environment, the partition is obvious. 

It seems intuitively reasonable, therefore, that one should be able to partition the F degrees of freedom into two sets one containing the reaction coordinate and the few vibrational modes that are strongly coupled to it (the "system ), and the other containing the remaining (perhaps very many) modes that are only weakly affected by the reaction (the "bath ). The "system" thus consists of the reaction coordinate, mode 1, and the vibrational modes k=2,...,f, say, and the "bath" are the remaining modes k = F + l,...,F. As is customary in such developments, the Hamiltonian is divided as... 

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