Condensed-phase system quantum

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. 

In many cases, in order to compute the dynamics of condensed phase systems, one invokes a basis representation for the quantum degrees of freedom in the system. Typically, one computes the dynamics of these systems in order to obtain quantities of interest, such as an average value, A(t) = Tr [Ap(t)], or a correlation function, as will be discussed below. Since such averages are basis independent one may project Eq. (8) onto any convenient basis. This is in principle a nice feature, and one that is often exploited to aid in calculations. However, it is important to note that the basis onto which one chooses to project the QCLE has important implications on how one goes about solving the resulting equations of motion. Ultimately the time-dependent average value of an observable is expressed as a trace over quantum subsystem... 

The QCL approach discussed thus far in this chapter provides a good approximation to the quantum dynamics of condensed phase systems. Most often other approximate quantum-classical methods, such as mean field and surface-hopping schemes, have been commonly employed to treat the same class of problems as the QCLE. These methods are attractive due to their computational simplicity however, many important quantum features, such as quantum coherence and correlations, are not properly handled in these approaches. In this section we discuss these methods and show that starting from the QCLE, an approximate theory in its own right, further approximations lead to these other approaches. 

We have presented some of the most recent developments in the computation and modeling of quantum phenomena in condensed phased systems in terms of the quantum-classical Liouville equation. In this approach we consider situations where the dynamics of the environment can be treated as if it were almost classical. This description introduces certain non-classical features into the dynamics, such as classical evolution on the mean of two adiabatic surfaces. Decoherence is naturally incorporated into the description of the dynamics. Although the theory involves several levels of approximation, QCL dynamics performs extremely well when compared to exact quantum calculations for some important benchmark tests such as the spin-boson system. Consequently, QCL dynamics is an accurate theory to explore the dynamics of many quantum condensed phase systems. 

Combinations of quantum mechanical (QM) methods for the description of the active site with a molecular mechanics (MM) treatment of the environment in the so-called QM/MM methods became the method of choice within the last decade, although already proposed in 1976 by Levitt and Warshel [2], These methods allow for a realistic description of condensed phase systems since they represent the microscopic environment with a QM treatment of the active site recent comprehensive reviews of these approaches can be found in Refs. [3,4,5,6,7],... 

For condensed phase systems. Density Functional Theory (DFT) methods constitute the optimal compromise between accuracy and efficiency of all ab initio methods available today. The key point of DFT is to show that the exact quantum mechanical total energy is a functional... 

In this approach, the system is partitioned into a part described quantum mechanically (the ion plus hydration waters) and the other treated by molecular mechanics. A detailed description of the QM/MM method as implemented for condensed phase system is provided by Field et al. [231]. Interactions inside the hydration complex are calculated using ab initio Born-Oppenheimer dynamics [228], while all the other interactions are modeled by classical pair potentials. 

In this section we present some applications of the LAND-map approach for computing time correlation functions and time dependent quantum expectation values for realistic model condensed phase systems. These representative applications demonstrate how the methodology can be implemented in general and provide challenging tests of the approach. The first test application is the spin-boson model where exact results are known from numerical path integral calculations [59-62]. The second system we study is a fully atomistic model for excess electronic transport in metal - molten salt solutions. Here the potentials are sufficiently reliable that findings from our calculations can be compared with experimental results. 

Variations on this surface hopping method that utilize Pechukas [106] formulation of mixed quantum-classical dynamics have been proposed [107,108]. Surface hopping algorithms [109-111] for non-adiabatic dynamics based on the quantum-classical Liouville equation [109,111-113] have been formulated. In these schemes the dynamics is fully prescribed by the quantum-classical Liouville operator and no additional assumptions about the nature of the classical evolution or the quantum transition probabilities are made. Quantum dynamics of condensed phase systems has also been carried out using techniques that are not based on surface hopping algorithms, in particular, centroid path integral dynamics [114] and influence functional methods [115]. 

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