Essential degrees of freedom

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

The essential degrees of freedom are found by a principal component analysis of the position correlation matrix Cy of the cartesian coordinate displacements Xi with respect to their averages xi), as gathered during a long MD run ... 

This section is used to introduce the momentum-enhanced hybrid Monte Carlo (MEHMC) method that in principle converges to the canonical distribution. This ad hoc method uses averaged momenta to bias the initial choice of momenta at each step in a hybrid Monte Carlo (HMC) procedure. Because these average momenta are associated with essential degrees of freedom, conformation space is sampled effectively. The relationship of the method to other enhanced sampling algorithms is discussed. 

We are now going to consider the case of a given timeseries Ot)t=ti,...,tN with constant sampling time t = tj+i — tj. Here, the Of do not necessarily denote the state of the molecule at time t but rather some low-dimensional observable, for example, some or all torsion angles or the set of essential degrees of freedom (if this should be available). We assume that there is an unknown metastable decomposition, say, into m sets Di,..., Dm- Using the notation introduced above, we furthermore assume t -C mins, Tjk but do not specify the relation... 

MTD is based on the assumption that the relevant processes can be described by a small set of coarse-grained variables, called collective variables (CVs), describing the essential degrees of freedom of the system. The CVs f ( r ) themselves are... 

In special cases (as in colloidal solutions) some particles can be considered as essential and other particles as irrelevant , but in most cases the essential space will itself consist of collective degrees of freedom. A reaction coordinate for a chemical reaction is an example where not a particle, but some function of the distance between atoms is considered. In a simulation of the permeability of a lipid bilayer membrane for water [132] the reaction coordinate was taken as the distance, in the direction perpendicular to the bilayer, between the center of mass of a water molecule and the center of mass of the rest of the system. In proteins (see below) a few collective degrees of freedom involving all atoms of the molecule, describe almost all the... 

However, in many applications the essential space cannot be reduced to only one degree of freedom, and the statistics of the force fluctuation or of the spatial distribution may appear to be too poor to allow for an accurate determination of a multidimensional potential of mean force. An example is the potential of mean force between two ions in aqueous solution the momentaneous forces are two orders of magnitude larger than their average which means that an error of 1% in the average requires a simulation length of 10 times the correlation time of the fluctuating force. This is in practice prohibitive. The errors do not result from incorrect force fields, but they are of a statistical nature even an exact force field would not suffice. 

As a consequence of this observation, the essential dynamics of the molecular process could as well be modelled by probabilities describing mean durations of stay within different conformations of the system. This idea is not new, cf. [10]. Even the phrase essential dynamics has already been coined in [2] it has been chosen for the reformulation of molecular motion in terms of its almost invariant degrees of freedom. But unlike the former approaches, which aim in the same direction, we herein advocate a different line of method we suggest to directly attack the computation of the conformations and their stability time spans, which means some global approach clearly differing from any kind of statistical analysis based on long term trajectories. 

As stated earlier, the main motivation for using either PCA or PCA is to construct a low-dimensional representation of the original high-dimensional data. The notion behind this approach is that the effective (or essential, as some call it [33]) dimensionality of a molecular conformational space is significantly smaller than its full dimensionality (3N-6 degrees of freedom for an A-atom molecule). Following the PCA procedure, each new... 

Thus far we have discussed the direct mechanism of dissipation, when the reaction coordinate is coupled directly to the continuous spectrum of the bath degrees of freedom. For chemical reactions this situation is rather rare, since low-frequency acoustic phonon modes have much larger wavelengths than the size of the reaction complex, and so they cannot cause a considerable relative displacement of the reactants. The direct mechanism may play an essential role in long-distance electron transfer in dielectric media, when the reorganization energy is created by displacement of equilibrium positions of low-frequency polarization phonons. Another cause of friction may be anharmonicity of solids which leads to multiphonon processes. In particular, the Raman processes may provide small energy losses. 

The computer simulation of models for condensed matter systems has become an important investigative tool in both fundamental and engineering research [149-153] for reviews on MC studies of surface phenomena see Refs. 154, 155. For the reahstic modeling of real materials at low temperatures it is essential to take quantum degrees of freedom into account. Although much progress has been achieved on this topic [156-166], computer simulation of quantum systems still lags behind the development in the field of classical systems. This holds particularly for the determination of dynamical information, which was not possible until recently [167-176]. 

We briefly repeat now the essential parts of the maximum entropy method for details we refer to the literature [167-169]. We seek to obtain information on the dynamics of the internal degree of freedom of the model from PIMC simulations. The solution of this problem is not... 

For long (infinite) /am.v-polyacclylene chains, the treatment of quantum lattice fluctuations is very complicated, because many lattice degrees of freedom couple in a non-linear way to the lowest electronic transitions. We have recently shown that for chains of up to 70 CH units, the amount of relevant lattice degrees of freedom reduces to only one or two, which makes it possible to calculate the low-energy part of the absorption spectrum in an essentially exact way [681. It remains a challenge to study models in which both disorder and the lattice quantum dynamics are considered. 

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