Quantum classical Liouville equation

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. 

The quantum-classical Liouville equation can be derived by formally expanding the operator on the right side of Eq. (6) to O(h0). One may justify [4] such an expansion for systems where the masses of particles in the environment are much greater than those of the subsystem, M > tn. In this case the small parameter in the theory is p = (m/Mj1/2. This factor emerges in the equation of motion quite naturally through a scaling of the variables motivated... 

When the quantum-classical Liouville equation is expressed in the adiabatic basis, the most difficult terms to simulate come from the off-diagonal force matrix elements, which give rise to the nonadiabatic coupling matrix elements. As described above, contributions coming from this term were computed using the momentum-jump approximation in the context of a surface-hopping scheme. 

One way to simplify this term in the evolution equation is to make use of a basis that diagonalizes the force contribution [42] i.e., we represent the quantum-classical Liouville equation in a basis i R) such that... 

The quantum-classical Liouville equation in the force basis has been solved for low-dimensional systems using the multithreads algorithm [42,43]. Assuming that the density matrix is localized within a small volume of the classical phase space, it is written as linear combination of matrices located at L discrete phase space points as... 

The evolution equations for the quantities entering the right side of this equation are obtained by substitution into the quantum-classical Liouville equation. For a variety of one- and two-dimensional systems for which exact results are known, excellent agreement was found. 

The quantum-classical Liouville equation was expressed in the subsystem basis in Sec. 3.1. Based on this representation, it is possible to recast the equations of motion in a form where the discrete quantum degrees of freedom are described by continuous position and momentum variables [44-49]. In the mapping basis the eigenfunctions of the n-state subsystem can be replaced with eigenfunctions of n fictitious harmonic oscillators with occupation numbers limited to 0 or 1 A) —> toa) = 0i, , 1a, -0 ). This mapping basis representation then makes use of the fact that the matrix element of an operator Bw(X) in the subsystem basis, B y (X), can be written in mapping form as B(( (X) = (AIBy X A ) = m Bm(X) mx>), where... 

Given this correspondence between the matrix elements of a partially Wigner transformed operator in the subsystem and mapping bases, we can express the quantum-classical Liouville equation in the continuous mapping coordinates [53]. The first step in this calculation is to introduce an n-dimensional coordinate space representation of the mapping basis,... 

Carrying out the this change of representation on the quantum-classical Liouville equation and using the product rule formula for the Wigner transform... 

We now show how the mean field equations can be derived as an approximation to the quantum-classical Liouville equation (8) [9]. The Hamiltonian... 

If we substitute the above expression for pw X,t) into the quantum-classical Liouville equation we find... 

To take the quantum-classical limit of this general expression for the transport coefficient we partition the system into a subsystem and bath and use the notation 1Z = (r, R), V = (p, P) and X = (r, R,p, P) where the lower case symbols refer to the subsystem and the upper case symbols refer to the bath. To make connection with surface-hopping representations of the quantum-classical Liouville equation [4], we first observe that AW(X 1) can be written as... 

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. 

Q. Shi and E. Geva. A derivation of the mixed quantum-classical Liouville equation from the influence functional formalism. J. Chem. Phys., 121(8) 3393-3404, 2004. 

I. Horenko, B. Schmidt, and C. Schutte. A theoretical model for molecules interacting with intense laser pulses The floquet-based quantum-classical Liouville equation. The Journal of Chemical Physics, 115(13) 5733-5743, 2001. 

The quantum-classical Liouville equation is obtained from this equation by introducing scaled variables such that the characteristic momenta of the light and heavy degrees of freedom are comparable. This scaling introduces a small parameter p = (m/M)1/2 in the equations of motion. Expansion of the quantum Liouville operator to 0 p) yields the quantum-classical Liouville equation [2,4,12-20],... 

To obtain the second approximate equality we expanded the right hand side to first order in the small parameter p = mlM) ". Returning to unsealed units we have the quantum-classical Liouville equation,... 

The quantum-classical Liouville equation may be expressed in any convenient basis. In particular, the adiabatic basis vectors, a R), are given by... 

We may easily carry out a linear response theory derivation of transport properties based on the quantum-classical Liouville equation that parallels the... 

Various schemes have been proposed for the solution of the quantum-classical Liouville equation [13,21-24]. Here we describe the sequential short-time algorithm that represents the solution in an ensemble of surface-hopping trajectories [25,26]. 

I. Horenko, M. Weiser, B. Schmidt, and C. Schiitte (2004) Fully adaptive propagation of the quantum-classical Liouville equation. J. Chem. Phys. 120, pp. 8913-8923... 

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]. 

In order to propagate the Wigner density numerically, we need to choose a set of initial grid points, R ) in phase space. Having chosen a grid, the quantum-classical Liouville equation with an effective potential (EP-QCLE) becomes a set of uncoupled matrix equations, one for each trajectory ... 

As explained in the Introduction, one needs to distinguish the following kinds of surface hopping (SH) methods (i) Semiclassical theories based on a connection ansatz of the WKB wave function, " (ii) stochastic implementations of a given deterministic multistate differential equation, e.g. the quantum-classical Liouville equation, and (iii) quasiclassical models such as the well-known SH schemes of Tully and others. " In this chapter, we focus on the latter type of SH method, which has turned out to be the most popular approach to describe nonadiabatic dynamics at conical intersections. 

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. 

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