Phase space mixed

For larger systems, various approximate schemes have been developed, called mixed methods as they treat parts of the system using different levels of theory. Of interest to us here are quantuin-seiniclassical methods, which use full quantum mechanics to treat the electrons, but use approximations based on trajectories in a classical phase space to describe the nuclear motion. The prefix quantum may be dropped, and we will talk of seiniclassical methods. There are a number of different approaches, but here we shall concentrate on the few that are suitable for direct dynamics molecular simulations. An overview of other methods is given in the introduction of [21]. 

Fio. 5. Region in ,-Y2 phase space with non-zero chemical source term and the mixing line. 

Second, the mapping approach to nonadiabatic quantum dynamics is reviewed in Sections VI-VII. Based on an exact quantum-mechanical formulation, this approach allows us in several aspects to go beyond the scope of standard mixed quantum-classical methods. In particular, we study the classical phase space of a nonadiabatic system (including the discussion of vibronic periodic orbits) and the semiclassical description of nonadiabatic quantum mechanics via initial-value representations of the semiclassical propagator. The semiclassical spin-coherent state method and its close relation to the mapping approach is discussed in Section IX. Section X summarizes our results and concludes with some general remarks. 

In a mixed quantum-classical calculation the trace operation in the Heisenberg representation is replaced by a quantum-mechanical trace (tTq) over the quantum degrees of freedom and a classical trace (i.e., a phase-space integral over the initial positions xq and momenta Po) over the classical degrees of freedom. This yields... 

Because the mapping approach treats electronic and nuclear dynamics on the same dynamical footing, its classical limit can be employed to study the phase-space properties of a nonadiabatic system. With this end in mind, we adopt a onemode two-state spin-boson system (Model IVa), which is mapped on a classical system with two degrees of freedom (DoF). Studying various Poincare surfaces of section, a detailed phase-space analysis of the problem is given, showing that the model exhibits mixed classical dynamics [123]. Furthermore, a number of periodic orbits (i.e., solutions of the classical equation of motion that return to their initial conditions) of the nonadiabatic system are identified and discussed [125]. It is shown that these vibronic periodic orbits can be used to analyze the nonadiabatic quantum dynamics [126]. Finally, a three-mode model of nonadiabatic photoisomerization (Model III) is employed to demonstrate the applicability of the concept of vibronic periodic orbits to multidimensional dynamics [127]. 

When there are many (quasi)bound states, healing can occur in an off-diagonal manner in that the (quasibound) state that they come back to need not be the same as the (quasibound) state that they dissociated from. Healing thus provides for an enhanced mixing of the bound phase space. 

This form (even though it is mixed with a momentum part and an energy part for the electron) clearly goes to zero at pe = 0 and also at Tc = Q and thus has a maximum in between. The shape of this function is shown in Figure 8.1. This function is often called the statistical or phase space factor for the decay. 

The state-of-the-art catalyst system is a Mo-V-Nb-Te mixed oxide [52], This catalyst is quite sensitive to its synthesis and process parameters and the automated catalyst synthesis tools described above were capable of synthesizing these and other challenging mixed metal oxides successfully. The workflow was validated by synthesizing a region of the known Mo-V-Nb-Te catalyst system phase space in the primary scale and secondary scale (Fig. 3.19a and b). Very good agreement between primary, secondary, and literature optima were obtained. One of the pri-... 

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,... 

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]. 

Burghardt, I. Dynamics of coupled Bohmian and phase-space variables a moment approach to mixed quantum-classical dynamics. J. Chem. Phys. 122 94103 (2005). 

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