Fast degrees of freedom

Some people prefer to use the multiple time step approach to handle fast degrees of freedom, while others prefer to use constraints, and there are situations in which both techniques are applicable. Constraints also find an application in the study of rare events, where a system may be studied at the top of a free energy barrier (see later), or for convenience when it is desired to fix a thennodynamic order parameter or ordering direction... 

In many cases the dynamical system consists of fast degrees of freedom, labeled x, and slow degrees of freedom, labeled y. An example is that of a fluid containing polyatomic molecules. The internal vibrations of the molecules are often very fast compared to their translational and orientational motions. Although this and other systems, like proteins, have already been treated using RESPA,[17, 34, 22, 23, 24, 25, 26] another example, and the one we focus on here, is that of a system of very light particles (of mass m) dissolved in a bath of very heavy particles (mass M).[14] The positions of the heavy particles are denoted y and the positions of the light particles rire denoted by X. In this case the total Liouvillian of the system is ... 

Eq. (122) represents a set of algebraic constraints for the vector of species concentrations expressing the fact that the fast reactions are in equilibrium. The introduction of constraints reduces the number of degrees of freedom of the problem, which now exclusively lie in the subspace of slow reactions. In such a way the fast degrees of freedom have been eliminated, and the problem is now much better suited for numerical solution methods. It has been shown that, depending on the specific problem to be solved, the use of simplified kinetic models allows one to reduce the computational time by two to three orders of magnitude [161],... 

At the antipodes of the latter description, there is a continuous need for better low-resolution models that involve, for instance, coarse graining of molecules, or implicit solvation. This need is motivated by the expectation that the free energy of a large system can be calculated with sufficient accuracy without requiring that all its components be described at the atomic level. In many cases, this is equivalent to the assumption that a mean-field approximation works, or that many fast degrees of freedom can be removed from the system, yet without any appreciable loss of... 

The separation of the time scale and the decoupling of different modes often occur in molecular systems. Eliminating the fast degrees of freedom is therefore a commonly used procedure in constructing the model of molecular systems. If... 

We seek a formulation in which we will not need to identify (to begin with) what are the slow and the fast degrees of freedom. Hence we seek an automated stabilizing algorithm. 

According to the previous section, we shall start by considering X and P as fast degrees of freedom, relaxing on a much more rapid timescale than the orientational coordinates and momenta of the solute and the solvent cage. Many different projection schemes are available to handle stochastic partial differential operators. Here we choose to adopt a slightly modified total time ordered cumulant (TTOC) expansion procedure, directly related to the well known resolvent approach. In order to make this chapter self-contained, we summarize the method in the Appendices and its application to the cases considered here and in the next section. 

All NMR interactions are represented by tensors of rank 2, and enter the static NMR Hamiltonian after averaging over the fast degree of freedoms. The lineposition and lineshape are determined by this mean tensor, which should reflect the local symmetry. How the NMR interaction tensors are transformed by the symmetry operations is consequently highly relevant to structural phase transition that are characterized by a loss of some symmetry operations. 

The objective of the present section is to provide a general ansatz to solve the Schrodinger equation for problems with several degrees of freedom. The ansatz is particularly efficient if there are slow and fast degrees of freedom. To be specific, we consider molecules where the nuclei and electrons are the slow and fast degrees of freedom, respectively, but shall keep the generality of the approach in mind. 

Arguments based on time scales were also important in suggesting the decomposition of TCFs into nested expressions, where one first calculates the correlation of fast degrees of freedom for fixed slow ones, and then one constructs the total TCFs by suitable weighted averages. This was the procedure followed to describe vibrational-rotational transitions in both impulsive and short-wavelength calculations. 

Several theoretical approaches to the description of dynamical solvent effects have been proposed within the framework of PCM of other continuum models [41]. The simplest, and most commonly used, treatment involves the definition of two limit time regimes equilibrium (EQ) and nonequilibrium (NEQ). In the former all the solvent degrees of freedom are in equilibrium with the electron density of the excited-state density, and the solvent reaction field depends on the static dielectric constant of the embedding medium. In the latter, only solvent electronic polarization (fast degrees of freedom) is in equilibrium with the excited-state electron density of the solute, while the slow solvent degrees of freedom remain equilibrated with the groimd-state electron density. In the NEQ time regime the fast solvent reaction field is ruled by the dielectric constant at optical frequency (Copt, usually related to the square of the solvent refractive index). 

In complex systems the set of fast degrees of freedom arises both from vibrations of stiff bonds and bond angles and from... 

In many cases the dynamical system consists of fast degrees of freedom, labeled x, and slow degrees of freedom, labeled y. An example is that of a fluid containing polyatomic molecules. 

In complex molecular systems, there are motions with disparate characteristic time scales fast degrees of freedom, such as bond and angle vibrations, and slow degrees of freedom, such as molecular translation and rotation. The integration time step size is determined by fast degrees... 

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