Extended scaled subsystem

The previous sections focused on the case of isolated atoms or molecules, where coherence is fully maintained on relevant time scales, corresponding to molecular beam experiments. Here we proceed to extend the discussion to dense environments, where both population decay and pure dephasing [77] arise from interaction of a subsystem with a dissipative environment. Our interest is in the information content of the channel phase. It is relevant to note, however, that whereas the controllability of isolated molecules is both remarkable [24, 25, 27] and well understood [26], much less is known about the controllability of systems where dissipation is significant [78]. Although this question is not the thrust of the present chapter, this section bears implications to the problem of coherent control in the presence of dissipation, inasmuch as the channel phase serves as a sensitive measure of the extent of decoherence. 

The LVC model further allows one to introduce coordinate transformations by which a set of relevant effective, or collective modes are extracted that act as generalized reaction coordinates for the dynamics. As shown in Refs. [54, 55,72], neg = nei(nei + l)/2 such coordinates can be defined for an electronic nei-state system, in such a way that the short time dynamics is completely described in terms of these effective coordinates. Thus, three effective modes are introduced for an electronic two-level system, six effective modes for a three-level system etc., for an arbitrary number of phonon modes that couple to the electronic subsystem according to the LVC Hamiltonian Eq. (7). In order to capture the dynamics on longer time scales, chains of such effective modes can be introduced [50,51,73]. These transformations, which are briefly summarized below, will be shown to yield a unique perspective on the excited-state dynamics of the extended systems under study. 

The QM/MM method, and the polarizable continuum method as well, are usually considered as prototypical examples of the so-called multi-scale approaches. They combine two different description levels for the chemical system in both cases, a quantum part interacts with a classical part. Indeed, the QM/MM method can easily be extended to multi-scale schemes that include more than two description levels. Examples of three level schemes are the QM/MM/Continuum [47] and QM/QM7 MM approaches [48, 49]. In the later case, the system is divided into two QM parts, which may be described with the same or different methods, and a classical MM part. Dielectric continuum models for liquid interfaces are already available [43,50, 51] and a QM/MM/Continuum partition could be imagined in this case too, for instance to describe a solute-solvent cluster interacting with a polarizable dielectric medium. Here, however, we will focus on the QM/QM /MM partition. There is not a general scheme for this kind of approach and different algorithms can be employed to describe the interaction between subsystems. The main issue is the calculation of the interaction between two quantum subsystems that are described at QM (possibly different) theoretical levels. 

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