Quantum classical coupling

Quantum-classical coupling through Bohmian particles... 

Shiang J J et al 1998 Cooperative phenomena in artificial solids made from silver quantum dots the importance of classical coupling J. Phys. Chem. 102 3425... 

In molecular dynamics applications there is a growing interest in mixed quantum-classical models various kinds of which have been proposed in the current literature. We will concentrate on two of these models the adiabatic or time-dependent Born-Oppenheimer (BO) model, [8, 13], and the so-called QCMD model. Both models describe most atoms of the molecular system by the means of classical mechanics but an important, small portion of the system by the means of a wavefunction. In the BO model this wavefunction is adiabatically coupled to the classical motion while the QCMD model consists of a singularly perturbed Schrddinger equation nonlinearly coupled to classical Newtonian equations, 2.2. 

Various kinds of mixed quantum-classical models have been introduced in the literature. We will concentrate on the so-called quantum-classical molecular dynamics (QCMD) model, which consists of a Schrodinger equation coupled to classical Newtonian equations (cf. Sec. 2). 

In this paper, we consider the symplectic integration of the so-called Quantum-Classical Molecular Dynamics (QCMD) model. In the QCMD model (see [11, 9, 2, 3, 6] and references therein), most atoms are described by classical mechanics, but an important small portion of the system by quantum mechanics. This leads to a coupled system of Newtonian and Schrddinger equations. 

Mixed Quantum/Classical Approach Coupled Chromophores and the Time Averaging Approximation... 

H2O, where N/2 molecules have N OH stretch chromophores. In this case we need to label the transition dipoles and frequencies by an index i that runs from 1 to N. In addition, in general these chromophores interact, with couplings (in frequency units) coy. In this case the above mixed quantum/classical formula can be generalized to [95 98]... 

Within the mixed quantum/classical approach, at each time step in a classical molecular dynamics simulation (that is, for each configuration of the bath coordinates), for each chromophore one needs the transition frequency and the transition dipole or polarizability, and if there are multiple chromophores, one needs the coupling frequencies between each pair. For water a number of different possible approaches have been used to obtain these quantities in this section we begin with brief discussions of each approach to determine transition frequencies. For definiteness we consider the case of a single OH stretch chromophore on an HOD molecule in liquid D2O. 

Theoretical calculations for ultrafast neat water spectroscopy are difficult to perform and difficult to interpret (because of the near-resonant OH stretch coupling). One classical calculation of the 2DIR spectrum even preceded the experiments [163] Torii has calculated the anisotropy decay [97], finding reasonable agreement with the experimental time scale. Mixed quantum/ classical calculations of nonlinear spectroscopy for many coupled chromo-phores is a daunting task. We developed the TAA for linear spectroscopy, and Jansen has very recently extended it to nonlinear spectroscopy [164]. We hope that this will allow for mixed quantum/classical calculations of the 2DIR spectrum for neat water and that this will provide the context for a molecular-level interpretation of these complex but fascinating experiments. 

The Golden Rule approach has been used for many years by Levich, Dogonadze, and co-workers (39, 40), who have stressed the difference between the roles of "quantum" (high-frequency) and "classical" coupling modes in discussing the theory of electron transfer, and by a number of subsequent workers (16, 41). 

As a last example of a molecular system exhibiting nonadiabatic dynamics caused by a conical intersection, we consider a model that recently has been proposed by Seidner and Domcke to describe ultrafast cis-trans isomerization processes in unsaturated hydrocarbons [172]. Photochemical reactions of this type are known to involve large-amplitode motion on coupled potential-energy surfaces [169], thus representing another stringent test for a mixed quantum-classical description that is complementary to Models 1 and II. A number of theoretical investigations, including quantum wave-packet studies [163, 164, 172], time-resolved pump-probe spectra [164, 181], and various mixed... 

A mixed quantum classical description of EET does not represent a unique approach. On the one hand side, as already indicated, one may solve the time-dependent Schrodinger equation responsible for the electronic states of the system and couple it to the classical nuclear dynamics. Alternatively, one may also start from the full quantum theory and derive rate equations where, in a second step, the transfer rates are transformed in a mixed description (this is the standard procedure when considering linear or nonlinear optical response functions). Such alternative ways have been already studied in discussing the linear absorbance of a CC in [9] and the computation of the Forster-rate in [10]. 

The mixed quantum classical description of EET can be achieved in using Eq. (49) together with the electronic ground-state classical path version of Eq. (50). As already indicated this approach is valid for any ratio between the excitonic coupling and the exciton vibrational interaction. If an ensemble average has been taken appropriately we may also expect the manifestation of electronic excitation energy dissipation and coherence decay, however, always in the limit of an infinite temperature approach. 

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. 

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]. 

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