Hamiltonian Zwanzig

As shown by Zwanzig the GLE, Eq. 1, may be derived Ifom a Hamiltonian in which the reaction coordinate q is coupled bilinearly to a harmonic bath ... 

I. Generalized Langevin equation. Zwanzig s Hamiltonian, n. Evaluation of quantum rates for multi-dimensional systems, ni. Beyond the Langevin equation/quantum Kramers paradigm ... 

Another critical result, which provided a more microseopic view of the Langevin equation, was the proof by Zwanzig that when the dynamics of a system obeying the classical Hamiltonian... 

It is important to notice that the solution of the GLE depends only on y(t) and not on the particular set of parameters Ck, mk, cok that generate it through Eq. (8). In order to make this result more intelligible we should emphasize that the modes k in the Zwanzig Hamiltonian Eq. (7) do not (except in the crystalline case) refer to actual modes of the system rather, they represent a hypothetical environment " that generates the correct dynamical friction y(t) through Eq. (8), such that when entered in the GLE Eq. (2) it provides an accmate description of the dynamics. 

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 quantum version of the Hamiltonian Eq. (7) has been studied for decades in both Physics and Chemistry in the 2-level limit. If the potential energy surface (PES) is represented as a quartic double well, then the energy eigenvalues are doublets separated by, roughly, the well frequency. When the mass of the transferred particle is small (e g. electron), or the barrier is very high, or the temperature is low, then only the lowest doublet is occupied this is the 2-level limit of the Zwanzig Hamiltonian. 

The goal of studying the quantum Zwanzig Hamiltonian is to generalize these results to the case when excitations to higher doublets are possible. This detail changes the problem completely since there is no small parameter for a perturbative approach. 

This energy is the Marcus activation energy needed for symmetrizing the potential energy smface. Unlike the Marcus theory result Eq. (10), this activation energy Ea is not equal to Er/4 but smaller the reason is that in the Zwanzig Hamiltonian the transfer distance along the symmetrized PES Eq. (25) is shorter than the transfer distance for the imcoupled potential V(s). 

A lot of progress has been made in solving the quantum Zwanzig Hamiltonian and imderstanding its physical behavior in different regimes of the parameter space. Undoubtedly there are many open questions, but in the rest of this paper we will address a different question is the quantum Zwanzig Hamiltonian the appropriate model for realistic proton systems ... 

We have generalised these results to the case when the reduction of the Zwanzig Hamiltonian to a 2-level system is not appropriate. We started with the Hamiltorrian... 

In this work we shall follow the Langevin equation approach and in the spirit of Zwanzig s work we shall start from the following Hamiltonian ... 

Equation (55) shows that for this strategy to work, one needs the two ensembles to overlap in the sense [somewhat less restrictive than in the case of the Zwanzig formula, Eq. (54)] that the two single-ensemble PDFs are measurable at some common value of Jt, the most obvious candidate being the Ji 0 region intermediate between the values typical of the two ensembles. Equation (56) shows that this requirement is effectively equivalent to the condition that the probabilities of acceptance of a Hamiltonian switch can be measured, in both directions. 

At the other extreme, if the switching time xs is short, the work done is just the energy cost of an instantaneous and complete Hamiltonian switch, and one recovers the Zwanzig formula [cf. Eq. (54)]... 

Section III is devoted to illustrating the first theoretical tool under discussion in this review, the GME derived from the Liouville equation, classical or quantum, through the contraction over the irrelevant degrees of freedom. In Section III.A we illustrate Zwanzig s projection method. Then, in Section III.B, we show how to use this method to derive a GME from Anderson s tight binding Hamiltonian The second-order approximation yields the Pauli master equation. This proves that the adoption of GME derived from a Hamiltonian picture requires, in principle, an infinite-order treatment. The case of a vanishing diffusion coefficient must be considered as a case of anomalous diffusion, and the second-order treatment is compatible only with the condition of ordinary... 

In the special case where the site energies are random fluctuations, this is the Anderson model [20,21]. It is well known that Anderson used this model to prove that randomness makes a crystal become an insulating material. Anderson localization is subtly related to subdiffusion, and consequently this important phenomenon can be interpreted as a form of anomalous diffusion, in conflict with the Markov master equation that is frequently adopted as the generator of ordinary diffusion. It is therefore surprising that this is essentially the same Hamiltonian as that adopted by Zwanzig for his celebrated derivation of the van Hove and, hence, of the Pauli master equation. 

In the special case of a nondemolition interaction Hamiltonian, the master equation of the total system reduces to uncoupled systems of first order differential equations, whose dimensions are the same as the dimension Na of the ancilla Hilbert space. After having traced over the ancilla state, the master equation of the dynamic system can be expressed either as N,fh order differential equations in time or, equivalently, as Zwanzig equations with an explicit memory over the system evolution. 

The beautiful point of this formulation is that the Marcus-Levich-Dogonadze result Equation (5) is the solution of the Hamiltonian Equation (7) in the deep tunneling limit. In addition, the solution of the Hamiltonian Equation (7) in the classical limit reproduces the TST result, corrected for recrossings of the barrier and for memory effects.12 These results mean that the Zwanzig Hamiltonian provides a unified description of proton transfer reactions in all the three parameter regions defined earlier in this section. 

The Hamiltonian of Eq. (9.2) couples the reaction coordinate to the environmental oscillator degrees of freedom by terms linear in both reaction coordinate and bath degree of freedom. This is derived in Zwanzig s original approach by an expansion of the full potential in bath coordinates to second order. This innocuous approximation in fact conceals a fair amount of missing physics. We have shown [16a] that this collection of bilinearly coupled oscillators is in fact a microscopic version... 

---