Constrained dynamics

The method presented in the next section is an attempt to overcome the barrier due to the highest frequencies whatever their origin. Although it has been implemented and tested for unconstrained dynamics only, there is no fundamental reason why it cannot be applied to overcome the less restrictive time step barrier arising in constrained dynamics. 

A variety of techniques have been introduced to increase the time step in molecular dynamics simulations in an attempt to surmount the strict time step limits in MD simulations so that long time scale simulations can be routinely undertaken. One such technique is to solve the equations of motion in the internal degree of freedom, so that bond stretching and angle bending can be treated as rigid. This technique is discussed in Chapter 6 of this book. Herein, a brief overview is presented of two approaches, constrained dynamics and multiple time step dynamics. 

Although constrained dynamics is usually discussed in the context of the geometrically constrained system described above, the same techniques can have many other applications. For instance, constant-pressure and constant-temperature dynamics can be imposed by using constraint methods [33,34]. Car and Parrinello [35] describe the use of the extended Lagrangian to maintain constraints in the context of their ab initio MD method. (For more details on the Car-Parrinello method, refer to the excellent review by Gain and Pasquarrello [36].)... 

Ciccotti, G. Kapral, R. Vanden-Eijnden, E., Blue moon sampling, vectorial reaction coordinates, and unbiased constrained dynamics, Chem. Phys. Chem. 2005, 6, 1809-1814... 

Starting from the transition state it was expected the reaction would evolve either forward to the products or backward to the reactants. During the unconstrained CPMD simulations, however, the system was always found to evolve towards the reactants. Because of this it was necessary to apply constrained dynamics to the principal coordinate reaction (the distance between WAT oxygen and GTP y-phosphorus) this enabled investigation of the system evolution towards the products (Fig. 2.7). 

Barth, E., Leimkuhler, B., Reich, S. Time-reversible variable-stepsize integrator for constrained dynamics. SIAM J. Sci. Comput. 1999, 21,1027-44. 

Sundermeyer, K. (1983). Constrained Dynamics (Springer-Verlag, Berlin). 

Prior to the simulation at finite temperature, the system must be heated up to the target temperature and thermally equilibrated. The temperature should be distributed among all the normal modes in the system. Thermal equilibration usually requires running dynamics for a long period of time (of the order of picoseconds). This time may be shortened if the warm-up procedure does not displace the system far from equilibrium. Thus, the warm-up may be realized by a sequence of kinetic energy pulses, followed by a short relaxation (free dynamics). If these pulses are orthogonal, then different normal mode become excited. It should be emphasized also at this point, that prior to the constrained dynamics simulation, the warm-up and equilibration should be performed with the same constraints that will be used in the sampling simulation. 

In this section we discuss first in general the techniques used to simulate the chemical reactions with relatively high activation barriers. After that we provide a more detailed discussion of the constrained dynamics approach. Special emphasis will be put on MD along the intrinsic reaction coordinate (IRC), illustrated by a few simple examples.33... 

Constrained Dynamics, Thermodynamic Integration, and Free-energy Barriers... 

The idea of constrained dynamics performed for a set of points along such a reaction path , i.e. for a set of fixed values of the reaction coordinate, A, is not specific to MD. Similar approaches have been commonly used in computational studies based on static quantum-chemical calculations. Such approaches are known as linear transit calculations, reaction path scans, etc. A set of constrained geometry optimizations with the constraint driving the system from reactants to products is a popular way to bracket a transition state, for instance. 

Such an a posteriori approach can be implemented quite easily and naturally using the machinery of constrained dynamics. The point is in using a proper constraint that freezes the motion along the predetermined, reference reaction path. Such a constraint was defined,33 based on the fact that in order to freeze the motion in a direction given by a vector rref, the projection of the displacement vector r on must be zero, rmrre = 0. 

The choice of reaction path definition used as the reference for such a constrained dynamics is arbitrary any path may be used in practice. However, a natural choice in order to ensure that the simulation moves along the bottom of the potential energy valley connecting reactants/products with TS is the intrinsic reaction path (IRP) of Fukui.46,47 IRP by definition goes along the bottom of such a valley. IRP simply corresponds to a steepest descent path in a mass-weighted coordinates ... 

One example that demonstrates the role of this type of Lewis acid site in surface chemistry is a study of the mechanism of water dissociation over the clean a-Al203(0001) surface by Hass and coworkers [100]. They used the BLYP functional in the CPMD code to allow the free energy of dissociation to be estimated using constrained dynamics [101]. The initial adsorption mode involves the coordination... 

The background theory for estimating free energy barriers using constrained dynamics is covered in more detail in a similar study of dimethyl ether formation from methanol in zeolites Hytha, M., Stich, L, Gale, J.D., Terakura, K. 

Constraint dynamics is just what it appears to be the equations of motion of the molecules are altered so that their motions are constrained to follow trajectories modified to mclude a constraint or constraints such as constant (total) kinetic energy or constant pressure, where the pressure in a dense adsorbed phase is given by the virial theorem. In statistical mechanics where large numbers of particles are involved, constraints are added by using the method of undetermined multipliers. (This approach to constrained dynamics was presented many years ago for mechanical systems by Gauss.) Suppose one has a constraint g(R, V)=0 that depends upon all the coordinates R=rj,r2...rN and velocities V=Vi,V2,...vn of all N particles in the system. By differentiation with respect to time, this constraint can be rewritten as l dV/dt -i- s = 0 where I and s are functions of R and V only. Gauss principle states that the constrained equations of motion can be written as ... 

The dynamics arising from these equations of motion should be compared with the constrained dynamics of Eq. (35). 

The equations of motion for constrained dynamics in Cartesian coordinates are derived in complete generality by Bae and Haug [55,56]. A slightly simpler derivation specific for fixed bond lengths and bond angles can be found in [39]. What follows is a simple sketch of one particular implementation of molecular dynamics with holonomic constraints. 

Figure 7 demonstrates how damping the parameter dynamics improves the accuracy of the nuclear trajectory. The parameter trajectories under damped dynamics, shown in Figure 8, are quite smooth in comparison to the constrained dynamics (Figure 6). Fewer time steps are required to discretize the damped parameter trajectories, although small parameter masses cause numerical instability even though the trajectories are smooth.

A. J. Liu and S. R. Nagel, Jamming and Rheology Constrained Dynamics on Microscopic and Macroscopic Scales, Taylor and Francis, London, 2000. 

Palmer R G, Stein D L, Abrahams E and Anderson P W1984 Models of hierarchically constrained dynamics for glassy relaxation Phys. Rev. Lett. 53 958-61... 

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