Dynamical equations of quantum-classical systems

Nonadiabatic dynamics in open quantum-classical systems: Forward-backward trajectory solution

“…We assume that the dynamics of such a system is described by the quantum-classical Liouville equation 39,42,46,[55][56][57][58][59] . For a quantum operatorB W (X), which depends on the classical phase space variables X = (R, P ) = (R 1 , R 2 , ..., R N b , P 1 , P 2 , ..., P N b ) of the bath, this evolution equation takes the form,…”

Communication: Quantum dynamics in classical spin baths

“…When such coordinates are canonically conjugate momenta and positions, one approach to treat these situations is provided by the quantum-classical Liouville equation. [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] Nevertheless, there are also many interesting cases when the bath is described in terms of classical spins, e.g., in order to model complex molecules where magnetic effects are important. 29,30 Such models can be studied by means of Monte Carlo methods or by molecular dynamics simulations.…”

“…29,30 Such models can be studied by means of Monte Carlo methods or by molecular dynamics simulations. 31,32 The scope of this Communication is to generalize the quantum-classical Liouville equation [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] in order to study the dynamics of quantum systems embedded in baths of classical spins. This is naturally achieved within a quantumclassical theory formulated by means of generalized antisymmetric brackets.…”

“…In practice, fast bath decoherence alleviates this problem and, indeed, the canonical version of the quantum-classical bracket, or of the quantum-classical Liouville equation, is used for many applications in chemistry and physics. [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] Moreover, it is worth reminding that the non-Lie (or, as they are also called, non-Hamiltonian) brackets, with their lack of time translation invariance, are also used to impose thermodynamical (such as constant temperature and/or pressure) [36][37][38] and holonomic constraints 39,40 in classical molecular dynamics simulations. 31,32 In order to represent the abstract equation (6) the quantum-classical Hamiltonian of Eq.…”

Nonadiabatic Dynamics of Condensed Phase Rate Processes

“…The retention of quantum coherence is a limitation of such algorithms, and methods have been developed to incorporate decoherence effects. 12,20,21 The nonadiabatic approach that we describe in this Account is based on the quantum-classical Liouville equation, [22][23][24][25][26][27][28][29][30][31] which specifies the evolution of the density matrix or an observable for a quantum mechanical system embedded in an environment that can be described classically. With such an evolution equation in hand, we then show how the rates of nonadiabatic chemical processes can be computed and discuss the nature of nonadiabatic quantum-classical dynamics.…”