Hamiltonians condensed phase reactions

We are now at the point where a quantum theory of condensed phase reactions may be developed. The Zwanzig Hamiltonian Eq. (7) has a natural quantum analog that consists in treating the Hamiltonian quantum-mechanically. In the rest of this paper we shall call this quantum analog the quantum Kramers problem. 

The standard language used to describe rate phenomena in condensed phases has evolved from Kramers one dimensional model of a particle moving on a one dimensional potential, feeling a random and a related friction force. In Section II, we will review the classical Generalized Langevin Equation (GEE) underlying Kramers model and its application to condensed phase systems. The GLE has an equivalent Hamiltonian representation in terms of a particle which is bilinearly coupled to a harmonic bath. The Hamiltonian representation, also reviewed in Section II is the basis for a quantum representation of rate processes in condensed phases. Eas also been very useful in obtaining solutions to the classical GLE. Variational estimates for the classical reaction rate are described in Section III. 

The two-state model (TSM) provides a very basic description of quantum transitions in condensed-phase media. It limits the manifold of the electronic states of the donor-acceptor complex to only two states participating in the transition. In this section, the TSM will be explored analytically in order to reveal several important properties of ET and CT reactions. The gas-phase Hamiltonian of the TSM reads... 

As an application of this formalism, we consider a two-level quantum system coupled to a classical bath as a simple model for a transfer reaction in a condensed phase environment. The Hamiltonian operator of this system, expressed in the diabatic basis L), P), has the matrix form [43]... 

What we have produced so far is an approximate Hamiltonian designed to study chemical reactions in complex condensed phases. We also have a mathematical method to evaluate quantum propagation using this Hamiltonian. We as yet have no practical method to compute observables such as rates. The flux correlation... 

A wide variety of problems are amenable to the Redfield methodology in addition to those discussed here. Some of the most important, in our view, are as follows. First, problems involving the interaction of strong laser fields with a condensed-phase system are often difficult to solve because the construction of a small, physically intuitive zeroth-order quantum subsystem Hamiltonian is difficult the numerical methods described above will make it possible to expand the size of the quantum subsystem and allow the problem to be attacked much more easily. A second class of problems involves relaxation of complex systems (e.g., vibronic or vibrational relaxation of a molecule in a liquid) [42,43, 72]. A third class of problems would be concerned with chemical dynamics in which the system could not be described easily by a single reaction coordinate, for example, general proton transfer reactions [98] or the isomerization of retinal in bacteriorhodopsin [120]. A low-dimensional system probably is adequate for these cases, but a nontrivial number of quantum levels will still be required. 

Despite the demands presented by such a calculation, a number of researchers have used ab initio models to treat the electronic and nuclear degrees of freedom for the quantum motif in molecular mechanics, energy minimization studies. Examples of this include the self-consistant reaction field methods developed by Tapia and coworkers [42-44], which represent only the quantum motif explicitly and use continuum models for the environmental effects (classical and boundary regions), and the methods implemented by Kollman and coworkers [45] in their studies of condensed phase (chemical and biochemical) reaction mechanisms. In both of these implementations the expectation value of the quantum motif Hamiltonian, defined in Eqs. (11) and (14) above, is treated at the Hartree Fock level with relatively small basis sets. 

In this chapter we have focused on Hamiltonian dynamics, which describe the dynamics of molecules in the gas phase. To analyze the chemical reaction in the condensed phase, the NF chemical reaction theory has been recently extended to the Langevin equation and the generalized Langevin equa-... 

Parameter A can be chosen to be any pointer of variation within the system for example, a torsional rotation to be followed in small angular steps or, more radically, the change of a solute molecule into a solvent molecule, of a hydroxyl group into a methyl group, or the actual transmutation of a reactant into a product in a chemical reaction, provided a suitable hamiltonian is available. In principle one could chose the starting state as the ideal gas, use equation 9.25 to calculate the exact free energy by statistical mechanics, and then use 9.26 or 9.27 to turn on the intermolecular potential and obtain the value of the free energy of a condensed phase. 

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