Mixed quantum-classical methods large systems

The main goal in the development of mixed quantum classical methods has as its focus the treatment of large, complex, many-body quantum systems. While applications to models with many realistic elements have been carried out [10,11], here we test the methods and algorithms on the spin-boson model, which is the standard test case in this field. In particular, we focus on the asymmetric spin-boson model and the calculation of off-diagonal density matrix elements, which present difficulties for some simulation schemes. We show that both of the methods discussed here are able to accurately and efficiently simulate this model. 

Linear and nonlinear infrared spectroscopy are powerful tools for probing the structure and vibrational dynamics of molecular systems." In order to take full advantage of them, however, accurate models and methods are required for simulating and interpreting spectra. A common approach for modeling spectra is based on computing optical response functions (ORFs)." Unfortunately, exact calculations of quantum-mechanical ORFs are not feasible for most systems of practical interest due to the large number of DOF. Instead, mixed quantum-classical methods ean provide suitable alternatives." " " ... 

The main practical problem in the implementation of the mixed quantum-classical dynamics method described in Section 4.2.4 is the nonlocal nature of the force in the equation of motion for the stationary-phase trajectories (Equation 4.29). Surface hopping methods provide an approximate, intuitive, stochastic alternative approach that uses the average dynamics of swarm of trajectories over the coupled surfaces to approximate the behavior of the nonlocal stationary-phase trajectory. The siu--face hopping method of Tully and Preston and Tully describes nonadiabatic dynamics even for systems with many particles. Commonly, the nuclei are treated classically, but it is important to consider a large niunber of trajectories in order to sample the quantum probability distribution in the phase space and, if necessary, a statistical distribution over states. In each of the many independent trajectories, the system evolves from the initial configuration for the time necessary for the description of the event of interest. The integration of a trajec-... 

In this context, one of the most efficient approaches is based on mixed quantum-classical dynamics in which the nonadiabatic effects are simulated using Tully s surface hopping (TSH) method [13, 14]. It is applicable to a large variety of systems ranging from isolated molecules and clusters to complex nanostructures... 

In the harmonic method and its extensions it is always assumed that the amplitudes of the molecular vibrations about their equilibrium positions and orientations remain small. It will be illustrated by several examples in Sect. 2.3 that this is often not realistic for molecular crystals. In the plastic phases there is not even a well-defined equilibrium orientation of the molecules, but also in the ordered phases the librational amplitudes may become substantial. The motions in such systems have been studied by classical methods, in particular the Molecular Dynamics method and the Monte Carlo method [73]. The advantage of these methods is that they can also be applied to study liquids, and the melting of solids, and other systems (glasses, solutions, mixed crystals) which have lost translational periodicity. Large amplitude motions in molecular crystals can also be studied quantum mechanically, however, by the methods described below. 

The proper representation of solvents in quantum chemical (QC) calculations is of crucial importance for the future success of QC because the vast majority of technical and biological chemistry takes place in fluid systems, while QC has been developed for isolated molecules for 40 years. Because of the extremely large number of molecules necessary for a realistic description of a solvent environment and the exponential increase of the costs of QC calculations with increasing size of the system, a direct extension of QC to such systems appears to be impossible in general, although first steps towards that goal have been made by the Car-Parrinello method (see Combined Quantum Mechanical and Molecular Mechanical Potentials and Combined Quantum Mechanics and Molecular Mechanics Approaches to Chemical and Biochemical Reactivity). Mixed classical quantum methods could... 

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