Hamiltonian mixed quantum-classical

In this section, we introduce the model Hamiltonian pertaining to the molecular systems under consideration. As is well known, a curve-crossing problem can be formulated in the adiabatic as well as in a diabatic electronic representation. Depending on the system under consideration and on the specific method used, both representations have been employed in mixed quantum-classical approaches. While the diabatic representation is advantageous to model potential-energy surfaces in the vicinity of an intersection and has been used in mean-field type approaches, other mixed quantum-classical approaches such as the surfacehopping method usually employ the adiabatic representation. 

In the last few years we have witnessed the successful development of several methods for the numerical solution of multi-dimensional quantum Hamiltonians Monte Carlo methods centroid methods,mixed quantum-classical methods, and recently a revival of semiclassical methods. We have developed another approach to this problem, the exponential resummation of the evolution operator. - The rest of this Section will explain briefly this method. 

Be aware of the fact that we have to consider the non-Markovian version of the quantum master equation to stay at a level of description where the emission rate, Eq. (39), can be deduced. Moreover, to be ready for a translation to a mixed quantum classical description a variant has been presented where the time evolution operators might be defined by an explicitly time-dependent CC Hamiltonian, i.e. exp(—iHcc[t — / M) has been replaced by the more general expression Ucc(t,F). 

The last line defines the mixed quantum-classical Liouville operator C. The W subscripts denote a partial Wigner transform of an operator or density matrix. The phase space variables of the bath are (R,P) and the partial Wigner transform of the total hamiltonian is given by,... 

In order to obtain a more compact formulation of the mixed quantum-classical equations we use a Hamilton-Jacobi-like formalism for the propagation of the quantum degree of freedom as in earlier studies [23], A similar approach has been introduced by Nettesheim, Schiitte and coworkers [54, 55, 56], TTie formalism presented here is based on recent investigations of the present authors [23], This formalism can be summarized as follows. Starting from the Hamiltonian Eqn. (2.2) and averaging over the x- and y-mode, respectively, gives... 

A delicate point in such mixed quantum-classical schemes is the self-consistency between the classical and quantum motions. The time-dependent variation of the electronic Hamiltonian arising from the nuclear dynamics results in electronic transitions. These quantum transitions, on the other hand, influence the forces on the nuclei, often with dramatic dynamical consequences. 

The validity of this mixed quantum-classical scheme is far from obvious, and important questions regarding its applicability may be raised. For example, does this coupled system of quantum and classical degrees of freedom conserve the total energy as it should (the answer is a qualified yes it may be shown that the sum of the energy of the classical system and the instantaneous expectation value of the Hamiltonian of the quantum system is a constant of motion of this dynamics). Experience shows that in many cases this scheme provides a good approximation, at least for the short-time dynamics. 

Consider an atom A colliding with a diatomic molecule BC. The mixed quantum classical Hamiltonian for the system is... 

A mixed quantum-classical approach can now be obtained by treating the rotational motion of the three vectors r, classically and the three variables p, q1 and % quantally. In order to further simplify the Hamiltonian in this case, we introduce the transformed wave function... 

Fig. 1. A Schematic illustration of the partitioning of a many-body system in mixed quantum/-classical calculations. The partitioning shows the division of the system into three regions the quantum motif, the classical region and the boundary region. This partitioning describes the division of the effective Hamiltonian given in Eq. (12)...

The system paritioning and Hamiltonian specifications as described above present a mathematical/computational problem which must be addressed to carry out mixed quantum/classical dynamics simulations. This involves the quantum mechanical method used to compute the expectation value of the energy and forces in the quantum motif as well as the treatment of the quantum motif/classical region interfacial atoms. In this section we provide a review of some of the techniques used in doing such calculations. 

Proteins, however, must function at biological temperatures, and to be useful, the Davydov soliton must survive at these temperatures. The first difficulty faced by the Davydov/Scott model was the question of the thermal stability of the Davydov soliton. The Davydov/Scott Hamiltonian includes two systems one, the amide I vibration, is treated as a quantum mechanical entity and the second, the vibrations of the peptide groups as a whole (or the changes in the hydrogen bond lengths) are very often treated classically, an approximation that shall be designated here as the mixed quantum-classical approximation. The first simulations of the Davydov/Scott model at finite temperature were performed within the mixed quantum/classical model and coupled the classical part of the system to a classical bath. The result was that the localized excitation dispersed in a few picoseconds at biological temperatures. However, this result clashed with another obtained in Monte... 

Mixed quantum-classical mixed representation of the Hamiltonian and wavefunctions... 

Presumably the most straightforward approach to chemical dynamics in intense laser fields is to use the time-independent or time-dependent adiabatic states [352], which are the eigenstates of field-free or field-dependent Hamiltonian at given time points respectively, and solve the Schrodinger equation in a stepwise manner. However, when the laser field is approximately periodic, one can also use a set of field-dressed periodic states as an expansion basis. The set of quasi-static states in a periodic Hamiltonian is derived by a Floquet type analysis and is often referred to as the Floquet states [370]. Provided that the laser field is approximately periodic, advantages of using the latter basis set include (1) analysis and interpretation of the electron dynamics is clearer since the Floquet state population often vary slowly with the timescale of the pulse envelope and each Floquet state is characterized as a field-dressed quasi-stationary state, (2) under some moderate conditions, the nuclear dynamics can be approximated by mixed quantum-classical (MQC) nonadiabatic dynamics on the field-dressed PES. The latter point not only provides a powerful clue for interpretation of nuclear dynamics but also implies possible MQC formulation of intense field molecular dynamics. 

Mixed quantum-classical Hamiltonian in an optical field... 

Modem first principles computational methodologies, such as those based on Density Functional Theory (DFT) and its Time Dependent extension (TDDFT), provide the theoretical/computational framework to describe most of the desired properties of the individual dye/semiconductor/electrolyte systems and of their relevant interfaces. The information extracted from these calculations constitutes the basis for the explicit simulation of photo-induced electron transfer by means of quantum or non-adiabatic dynamics. The dynamics introduces a further degree of complexity in the simulation, due to the simultaneous description of the coupled nuclear/electronic problem. Various combinations of electronic stmcture/ excited states and nuclear dynamics descriptions have been applied to dye-sensitized interfaces [54—57]. In most cases these approaches rely either on semi-empirical Hamiltonians [58, 59] or on the time-dependent propagation of single particle DFT orbitals [60, 61], with the nuclear dynamics being described within mixed quantum-classical [54, 55, 59, 60] or fuUy quantum mechanical approaches [61]. Real time propagation of the TDDFT excited states [62] has... 

Electron-Molecule Scattering Mixed Quantum-Classical Methods Path Integral Methods Photodissociation Dynamics Rates of Chemical Reactions Reaction Path Hamiltonian and its Use for Investigating Reaction Mechanisms Reactive Scattering of Polyatomic Molecules Time-dependent Multiconfigurational Hartree Method. 

Consider a mixed quantum q) - classical (Q) system, where the quantum Hamiltonian H q Q) depends parametrically on the positions of classical particles... 

We begin by assuming that the quantum mechanical system under study may be well-approximated by a mixed quantum lassical system driven by a classical radiation field, in which the photo-active DOF (i.e. chromophore) are treated quantum mechanically and the photo-inactive DOF (i.e. environment) are treated classically. The time-dependent Hamiltonian for this system is given by ... 

In the mixed Quantum Mechanical/Molecular Mechanical (QM/MM) hese methods, the solvent is treated as a set of classical particles, whose interactions are described by means of classical equations. The coupling between solute and solvent is accomplished by using a modified Hamiltonian [62-73] (see equation 19). 

In order to illustrate the approximations involved when trying to mix quantum and classical mechanics we consider a simple system with just two degrees of freedom r and R. The R coordinate is the candidate for a classical treatment - it is, e.g., the translational motion of an atom relative to the center of mass of a diatomic or polyatomic molecule. This motion is slow compared to the vibrational motion of the diatom - here described by the r coordinate. Thus if we treat the latter quantum mechanically we could introduce a wavefunction t) and expand this in eigenstates of the molecule, i.e., eigenstates to the hamiltonian operator Hq for the isolated molecule. Thus we have... 

The basic strategy for the QM/MM method lies in the hybrid potential in which a classical MM potential is combined with a QM one (Field et al. 1990). The energy of the system, , is calculated by solving the Schrbdinger equation with an effective Hamiltonian, for the mixed quantum mechanical and classical mechanical system... 

The effective Hamiltonian for the mixed quantum and classical system is divided into three terms (Lyne et al. 1999)... 

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