Quantum-classical Liouville condensed phase

Abstract. In this chapter we discuss approaches to solving quantum dynamics in the condensed phase based on the quantum-classical Liouville method. Several representations of the quantum-classical Liouville equation (QCLE) of motion have been investigated and subsequently simulated. We discuss the benefits and limitations of these approaches. By making further approximations to the QCLE, we show that standard approaches to this problem, i.e., mean-field and surface-hopping methods, can be derived. The computation of transport coefficients, such as chemical rate constants, represent an important class of problems where the QCL method is applicable. We present a general quantum-classical expression for a time-dependent transport coefficient which incorporates the full system s initial quantum equilibrium structure. As an example of the formalism, the computation of a reaction rate coefficient for a simple reactive model is presented. These results are compared to illuminate the similarities and differences between various approaches discussed in this chapter. 

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. 

In quantum-classical Liouville (QCL) dynamics the partition of the system into bath and subsystem is motivated by the observation that for many condensed phase processes it is essential to account for the quantum mechanical character of only a few light (characteristic mass m) degrees of freedom the remaining heavy (characteristic mass M) degrees of freedom may be treated classically to a high degree of accuracy. 

We show how the quantum-classical evolution equations of motion can be obtained as an approximation to the full quantum evolution and point out some of the difficulties that arise because of the lack of a Lie algebraic structure. The computation of transport properties is discussed from two different perspectives. Transport coefficient formulas may be derived by starting from an approximate quantum-classical description of the system. Alternatively, the exact quantum transport coefficients may be taken as the starting point of the computation with quantum-classical approximations made only to the dynamics while retaining the full quantum equilibrium structure. The utility of quantum-classical Liouville methods is illustrated by considering the computation of the rate constants of quantum chemical reactions in the condensed phase. 

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]. 

Quantum-Classical Liouville Dynamics of Condensed Phase Quantum Processes... 

In the following sections we show how the quantum-classical Liouville equation and quantum-classical expressions for reaction rates can be deduced from the full quantum expressions. The formalism is then applied to the investigation of nonadiabatic proton transfer reactions in condensed phase polar solvents. A quantum-classical Liouville-based method for calculating linear and nonlinear vibrational spectra is then described, which involves nonequilibrium dynamics on multiple adiabatic potential energy surfaces. This method is then used to investigate the linear and third-order vibrational spectroscopy of a proton stretching mode in a solvated hydrogen-bonded complex. 

In these examples the dynamics is not confined to a single adiabatic potential energy surface so that the full quantum dynamics of the entire system must be followed in order to obtain the observable of interest. For large systems, typical of condensed phase applications, this is a computationally difficult, if not impossible, task. For this reason, we focus our attention on quantum-classical descriptions where such limitations are much less severe. In particular, the formulation based on the quantum-classical Liouville equation is the topic of the remainder of this chapter. 

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