Quantum-classical Liouville

The goal of this chapter is twofold. First we wish to critically compare—from both a conceptional and a practical point of view—various classical and mixed quantum-classical strategies to describe non-Born-Oppenheimer dynamics. To this end. Section II introduces five multidimensional model problems, each representing a specific challenge for a classical description. Allowing for exact quantum-mechanical reference calculations, aU models have been used as benchmark problems to study approximate descriptions. In what follows, Section III describes in some detail the mean-field trajectory method and also discusses its connection to time-dependent self-consistent-field schemes. The surface-hopping method is considered in Section IV, which discusses various motivations of the ansatz as well as several variants of the implementation. Section V gives a brief account on the quantum-classical Liouville description and considers the possibility of an exact stochastic realization of its equation of motion. 

V. QUANTUM-CLASSICAL LIOUVILLE DESCRIPTION A. General Idea... 

The similar appearance of the quantum and classical Liouville equations has motivated several workers to construct a mixed quantum-classical Liouville (QCL) description [27 4]. Hereby a partial classical limit is performed for the heavy-particle dynamics, while a quantum-mechanical formulation is retained for the light particles. The quantities p(f) and H in the mixed QC formulation are then operators with respect to the electronic degrees of freedom, described by some basis states 4> ), and classical functions with respect to the nuclear degrees of freedom with coordinates x = x, and momenta p = pj — for example. 

Figure 17. Initial decay of the adiabatic population probability obtained for Model I. Compared are quantum results (thick line) and standard (thin full line) and energy-conserving (dotted line) quantum-classical Liouville results.

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... 

The last line defines the mixed quantum-classical Liouville (super)operator C. 

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... 

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