Continuum limit representations

The continuum limit of the Hamiltonian representation is obtained as follows. One notes that if the friction function y(t) appearing in the GLE is a periodic function with period T then Eq. 4 is just the cosine Fourier expansion of the friction function. The frequencies coj are integer multiples of the fundamental frequency and the coefficients Cj are the Fourier expansion coefficients. In practice, the friction function y(t) appearing in the GLE is a decaying function. It may be used to construct the periodic function y(t T) = Y(t TiT)0(t-... 

Until the mid-eighties it was generally accepted that the STGLE was a reasonable representation of the dynamics which allowed for dynamically induced corrections to the rate predicted by the TST method. Kramers theory was considered to be complementary but essentially different from TST. It was well understood that the STGLE is a continuum limit of a Hamiltonian in which the solute interacts nonlinearly with a harmonic bath... 

Figure 7.1 Representation of the phase diagram for a pure fluid such as water. The shaded area is the continuum tlirough wliich we can continuously vary the properties of the fluid. The liigh-pressure and liigh-temperature limits shown here are arbittary. They depend only on the capabilities of the experimental apparatus and the stability of the apparatus and the fluid.

In the quantum mechanical continuum model, the solute is embedded in a cavity while the solvent, treated as a continuous medium having the same dielectric constant as the bulk liquid, is incorporated in the solute Hamiltonian as a perturbation. In this reaction field approach, which has its origin in Onsager s work, the bulk medium is polarized by the solute molecules and subsequently back-polarizes the solute, etc. The continuum approach has been criticized for its neglect of the molecular structure of the solvent. Also, the higher-order moments of the charge distribution, which in general are not included in the calculations, may have important effects on the results. Another important limitation of the early implementations of this method was the lack of a realistic representation of the cavity form and size in relation to the shape of the solute. 

Figure 6.1-3 Schematic representation of continuum resonance Raman scattering for the Br2 molecule. The incident laser frequency (o o) is in resonance with the continuous states of the repulsive 77 excited state and the repulsive part of the bound B(- 77o-i- ) state, which is above the dissociation limit at around 20 000 cm (Baierl and Kiefer, 1981).

In order to prove equation (6), it is more convenient to work from equation (3) than from the limiting relation given in equation (5) and also to introduce the Lagrangian representation [10]. For any continuum X, let the three parameters af identify the individual point particles of continuum K for... 

Since the dielectric continuum representation of the solvent has significant limitations, the molecular dynamics simulation of PCET with explicit solvent molecules is also an important direction. One approach is to utilize a multistate VB model with explicit solvent interactions [34-36] and to incorporate transitions among the adiabatic mixed electronic/proton vibrational states with the Molecular Dynamics with Quantum Transitions (MDQT) surface hopping method [39, 40]. The MDQT method has already been applied to a one-dimensional model PCET system [39]. The advantage of this approach for PCET reactions is that it is valid in the adiabatic and non-adiatic limits as well as in the intermediate regime. Furthermore, this approach is applicable to PCET in proteins as well as in solution. 

A different example is the H3 system. Here the ground state of H3 is an equilateral triangle. Recent experiments of Carrington and Kennedy indicate that has a high density of quasibound states embedded in the continuum above the H ground electronic state dissociation limit. These states or at least some of them should be representable as stable periodic and quasiperiodic orbits. 

The initial attempts to account theoretically for solvent effects were made in the 1930s [1], but the suitability of these models for understanding chemical events in condensed phases was limited by their intrinsic simplicity. Thus, it was not until the 1970s, when continuum models were implemented within the quantum mechanical framework [2], that an accurate theoretical representation of solvent effects became possible. The last decade of this century has witnessed the spectacular growth of this new area of research [3]. It is expected that in the next century continuum methods will be the most used approach for the study of solvent effects in chemical systems. 

The obvious limitations of the continuum representation of the solvent necessitated the development of microscopic models of the surroundings. Whereas for liquid phases this task is not trivial at all, for structurally well-characterized environments, like proteins [190, 207] or crystals [208] it is possible to calculate the reaction field from the polarizability distribution [209]. Assuming the existence of strongly bound solvent... 

But, is a continuum representation of solvent always sufficient May we neglect specific solute-solvent interactions such as H-bonds And what about reactions in which the solvent plays the role of a reactant Furthermore, do species in solution interact, e.g., a charged catalyst with its counterion, and how would this affect the catalyst s reactivity To address these and similar questions, we first have to extend the model systems considered to include an explicit representation of solvent and of those molecular species which may have an impact on reactivity. A first step in this direction is the use of cluster-continuum models in which a reduced number of explicit solvent molecules are introduced in the model. This approach has been successfully applied in computational studies of the organometallic reactivity [12-14], yet it suffers from some limitations [15]. How many solvent molecules should be explicitly included Is the first solvation sphere enough In order to mimic bulk conditicHis, models have to include enough solvent molecules to fully solvate the solute. These models, which are built to reproduce experimental densities, are generally treated as periodically repeating units in order to remove the explicit/ continuum (or vacuum) boundary. 

---