Quantum-classical dynamics

U. Schmitt and J. Brinkmann. Discrete time-reversible propagation scheme for mixed quantum classical dynamics. Chem. Phys., 208 45-56, 1996. 

Thus, we see that in order to obtain the mean field equations of motion, the density matrix of the entire system is assumed to factor into a product of subsystem and environmental contributions with neglect of correlations. The quantum dynamics then evolves as a pure state wave function depending on the coordinates evolving in the mean field generated by the quantum density. As we have seen in the previous sections, these approximations are not valid and no simple representation of the quantum-classical dynamics is possible in terms of single effective trajectories. Consequently, in contrast to claims made in the literature [54], quantum-classical Liouville dynamics is not equivalent to mean field dynamics. 

The quantum mechanical forms of the correlation function expressions for transport coefficients are well known and may be derived by invoking linear response theory [64] or the Mori-Zwanzig projection operator formalism [66,67], However, we would like to evaluate transport properties for quantum-classical systems. We thus take the quantum mechanical expression for a transport coefficient as a starting point and then consider a limit where the dynamics is approximated by quantum-classical dynamics [68-70], The advantage of this approach is that the full quantum equilibrium structure can be retained. 

R. Kapral. Progress in the theory of mixed quantum-classical dynamics. Annu. Rev. Phys. Chem., 57 129-157, 2006. 

R. Kapral and G. Ciccotti. Mixed quantum-classical dynamics. J. Chem. Phys., 110 8919-8929, 1999. 

D. Mac Kernan, G. Ciccotti, and R. Kapral. Trotter-based simulation of quantum-classical dynamics. J. Phys. Chem. B, 112 424, 2008. 

The difficulty in simulating the full quantum dynamics of large many-body systems has stimulated the development of mixed quantum-classical dynamical schemes. In such approaches, the quantum system of interest is partitioned into two subsystems, which we term the quantum subsystem, and quantum bath. Approximations to the full quantum dynamics are then made such that... 

While the two methods are, at face value, quite different in the ways in which full quantum dynamics is reduced to quantum-classical dynamics, there are common elements in the manner in which they are simulated. The Trotter-based scheme for QCL dynamics makes use of the adiabatic basis and is based on surface-hopping trajectories where transitions are sampled by a Monte Carlo scheme that requires reweighting. Similarly, ILDM calculations make use of the mapping hamiltonian basis and also involve a similar Monte Carlo sampling with reweighting of trajectories in the ensemble used to obtain the expectation values of quantum operators. 

Tully JC. Mixed quantum-classical dynamics mean-field and surface-hopping. In Classical and Quantum Dynamics in Condensed Phase Simulations, ed. B.J. Berne, G. Ciccotti, D.F. Coker. Chapter 21. Singapore World Scientific, 1998. 

Abstract. A rigorous derivation of quantum-classical equations of motion is still lacking. The framework proposed so far to describe in a consistent way the dynamics of a mixed quantum-classical system using systematic approximations have failed. A recent attempt to solve the inconsistencies of quantum-classical approximated methods by introducing a group-theoretical approach is discussed in detail. The new formulation which should restore the consistency of the proposed quantum-classical dynamics and statistical mechanics will be shown to produce, instead, a purely classical description. In spite of that, the discussed approach remains interesting since it could produce non-trivial formulations. 

The question then arises if a convenient mixed quantum-classical description exists, which allows to treat as quantum objects only the (small number of) degrees of freedom whose dynamics cannot be described by classical equations of motion. Apart in the limit of adiabatic dynamics, the question is open and a coherent derivation of a consistent mixed quantum-classical dynamics is still lacking. All the methods proposed so far to derive a quantum-classical dynamics, such as the linearized path integral approach [2,6,7], the coupled Bohmian phase space variables dynamics [3,4,9] or the quantum-classical Li-ouville representation [11,17—19], are based on approximations and typically fail to satisfy some properties that are expected to hold for a consistent mechanics [5,19]. 

The property of the new (Lie) brackets (44) of being correct in the known full quantum and full classical limits may reasonably convince ourselves that the intermediate situation, in which hi —> h and h2 —> 0, generates quantum-classical dynamics. If the assumedly quantum-classical limit is performed on Ph1,h2(9i, 92), we obtain... 

The derivation of a consistent mixed quantum-classical dynamics discussed in this paper was first proposed in Ref. [15] and commented and clarified in Ref. [1], This derivation is based on a group-theoretical formulation of quantum and classical mechanics, which introduces a very elegant and formally rigorous mathematical apparatus and allows to directly obtain classical mechanics as the limit for h —> 0 of quantum mechanics, in the Heisenberg representation of quantum dynamics. 

Agostini, F., Caprara, S., Ciccotti, G. Do we have a consistent non-adiabatic quantum-classical dynamics . Europhys. Lett. 78 30001 (2007). 

Hughes, K.H., Parry, S.M., Parlant, G., Burghardt, I. A hybrid hydrodynamic-liouvillian approach to mixed quantum-classical dynamics application to tunneling in a double well. J. Phys. Chem. A. Ill 10269 (2007). 

In a second example the discrete time-reversible propagation scheme for mixed quantum-classical dynamics is applied to simulate the photoexcitation process of I2 immersed in a solid Ar matrix initiated by a femtosecond laser puls. This system serves as a prototypical model in experiment and theory for the understanding of photoinduced condensed phase chemical reactions and the accompanied phenomena like the cage effect and vibrational energy relaxation. It turns out that the energy transfer between the quantum manifolds as well as the transfer from the quantum system to the classical one (and back) can be very well described within the mixed mode frame outlined above. 

Another reduction is possible when one wishes to treat only one or a few degrees of freedom quantum-mechanically while the rest of the system can be treated still in a classical way. First pioneering studies along such lines treated the problem of electron solvation in molten salts and liquid ammonia. But, it must be noted that, when one studies the dynamics of quantum degrees of freedom coupled to a classical environment, particular care is required This mixed quantum-classical dynamics has subtle features, and is still an active area of research. 

Consequently, quantum-classical d mamics does not possess a Lie algebraic structure and this leads to pathologies in the general formulation of quantum-classical dynamics and statistical mechanics as we shall see below [5,6]. 

Furthermore, the evolution of a composite operator in quantum-classical dynamics cannot be written exactly in terms of the quantum-classical evolution of its constituent operators, but only to terms 0 h). To see this consider the action of the quantum-classical Liouville operator on the composite operator Cw = Bw l + hAl2i)Aw- A straightforward calculation shows that... 

While these features lead to some pathologies in the formulation of quantum-classical dynamics and statistical mechanics, the violations are in terms of higher order in h for the bath (or, better, the mass ratio /i), so that for systems where quantum-classical dynamics is likely to be applicable the numerical consequences are often small. We remark that almost all quantum-classical schemes suffer from these problems, although these deficiencies are often not highlighted. 

The density matrix p R,P) is not stationary under quantum-classical dynamics. Instead, the equilibrium density of a quantum-classical system has to be determined by solving the equation itpwe = 0. An explicit solution of this equation has not been found although a recursive solution, obtained by expressing the density matrix pwe in a power series in ft or the mass ratio p, can be determined. While it is difficult to find the full solution to all orders in ft, the solution is known anal3dically to 0 h). When expressed in the adiabatic basis, the result is [5]... 

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