Molecular systems model Hamiltonian

In order to demonstrate the NDCPA a model of a system of excitons strongly coupled to phonons in a crystal with one molecule per unit cell is chosen. This model is called here the molecular crystal model. The Hamiltonian of... 

Muller and Stock [227] used the vibronic coupling model Hamiltonian, Section III.D, to compare surface hopping and Ehrenfest dynamics with exact calculations for a number of model cases. The results again show that the semiclassical methods are able to provide a qualitative, if not quantitative, description of the dynamics. A large-scale comparison of mixed method and quantum dynamics has been made in a study of the pyrazine absorption spectrum, including all 24 degrees of freedom [228]. Here a method related to Ehrenfest dynamics was used with reasonable success, showing that these methods are indeed able to reproduce the main features of the dynamics of non-adiabatic molecular systems. 

Then, there are model Hamiltonians. Effectively a model Hamiltonian includes only some effects, in order to focus on those effects. It is generally simpler than the true full Coulomb Hamiltonian, but is made that way to focus on a particular aspect, be it magnetization, Coulomb interaction, diffusion, phase transitions, etc. A good example is the set of model Hamiltonians used to describe the IETS experiment and (more generally) vibronic and vibrational effects in transport junctions. Special models are also used to deal with chirality in molecular transport junctions [42, 43], as well as optical excitation, Raman excitation [44], spin dynamics, and other aspects that go well beyond the simple transport phenomena associated with these systems. 

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. 

Let us focus on molecular systems for which we know molecular Hamiltonian models, H(q,Q). Electronic and nuclear configuration coordinates are designated with the vectors q and Q, respectively x = (q,Q) = (qi,..., qn, Qi,---, Qm- For an n-electron system, q has dimension 3n Q has dimension 3m for an m-nuclei system. The wave function is the projection in configuration space of a particular abstract quantum state, namely P(x) P(q,Q), and base state func-... 

Regarding the parametrization of the Hamiltonian Eq. (7), the present approach relies on the parameters of the underlying lattice model Eq. (5). However, one could envisage an alternative approach, similar to the one described in Refs. [66-69] for small molecular systems, where a systematic diabatiza-tion is carried out based on supermolecular electronic structure calculations as described in Sec. 2.2. 

In Refs. [55, 79], the truncation at the level of Heg has been tested for several molecular systems exhibiting an ultrafast dynamics at Coin s, and it was found that this approximation can give remarkably good results in reproducing the short-time dynamics. This is especially the case if a system-bath perspective is appropriate, and the effective-mode transformation is only applied to a set of weakly coupled bath modes [55,72]. In that case, the system Hamiltonian can take a more complicated form than given by the LVC model. 

We start out with a section on the energy functionals and Hamiltonians that are relevant for molecular systems interacting with a structured environment. We continue with a section that briefly describes the correlated electron structure method, the multiconfigurational self-consistent field (MCSCF) electronic structure method. In the following section we cover the procedure for obtaining the correlated MCSCF response equations for the two different models describing molecules in structured environments. The final sections provide a brief overview of the results obtained using the two methods and a conclusion. 

The pseudopotential concept was advanced a long ago [1] and is based on the natural energetic and spatial separation of core and valence electrons. The concept allows a significant reduction in computational efforts without missing the essential physics of phenomena provided the interaction of core and valence electrons is well described by some effective (model) Hamiltonian. Traditionally, pseudopotentials are widely used in the band structure calculations [2], because they allow convenient expansions of the wavefunctions in terms of plane waves suited to describing periodical systems. For molecular and/or nonperiodical systems, the main advantage of pseudopotentials is a... 

In Section VII we conclude our results and discuss several issues arising from our proposals. We revisit our original motivation—that is, to find a simple model, in the sense of dynamical systems, that captures several common aspects of slow dynamics in liquid water, or more generally supercooled liquids or glasses. Our attempt is to make clear the relation and compatibility between the potential energy landscape picture and phase space theories in the Hamiltonian dynamics. Importance of heterogeneity of the system is discussed in several respects. Unclarified and unsolved points that still remain but should be considered as crucial issues in slow dynamics in molecular systems are listed. 

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