Quantum-classical Wigner-Liouville equation

In the next section we describe how the QCL equation may be expressed in any basis that spans the subsystem Hilbert space. Here we observe that the subsystem may also be Wigner transformed to obtain a phase-space-like representation of the subsystem variables as well as those of the environment. Taking the Wigner transform of Eq. (8) over the subsystem, we obtain the quantum-classical Wigner-Liouville equation [24],... 

The classical free streaming Liouville operators are iCf1 1 = Jy and = for the light ( ) subsystem particles and (h) heavy environmental particles, respectively. The quantum-classical Wigner-Liouville equation (9) can be written in a more compact form,... 

R. Kapral and A. Sergi. Quantum-classical Wigner-Liouville equation. Ukr. Math. J., 57 749-756, 2005. 

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... 

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 ... 

The quantum-classical Liouville equation (QCLE) provides an approximate but accurate description of a quantum subsystem coupled in an arbitrary manner to a bath that can be described by classical dynamics in the absence of coupling to the quantum subsystem. The QCLE describes the time evolution of the partially Wigner transformed density matrix of the system p R,P,t) discussed above, and is given by ... 

The Wigner function has the valuable property that the time evolution equation for the quantum dynamics in the Wigner representation resembles that for the classical Liouville dynamics. Specifically, the Schrodinger equation can be transformed to [70]... 

The conceptual framework for the - semiclassical simulation of ultrafast spectroscopic observables is provided by the Wigner representation of quantum mechanics [2, 3]. Specifically, for the ultrafast pump-probe spectroscopy using classical trajectories, methods based on the semiclassical limit of the Liouville-von Neumann equation for the time evolution of the vibronic density matrix have been developed [4-8]. Our approach [4,6-8] is related to the Liouville space theory of nonlinear spectroscopy developed by Mukamel et al. [9]. It is characterized by the ability to approximately describe quantum phenomena such as optical transitions by averaging over the ensemble of classical trajectories. Moreover, quantum corrections for the nuclear dynamics can be introduced in a systematic manner, e.g. in the framework of the entangled trajectory method [10,11]. Alternatively, these effects can be also accounted for in the framework of the multiple spawning method [12]. In general, trajectory-based methods require drastically less computational effort than full quantum mechanical calculations and provide physical insight in ultrafast processes. Additionally, they can be combined directly with quantum chemistry methods for the electronic structure calculations. 

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