Constrained systems

H. Van Damme, P. Levitz, and L. Gatineau, in Chemical Reactions in Organic and Inorganic Constrained Systems, R. Setton, ed., Kluwer Academic, Hingham, MA, 1986. 

It is possible to devise extended-system mediods [79, 82] and constrained-system methods [88] to simulate the constant-A/ r ensemble using MD. The general methodology is similar to that employed for constant-... 

Even further complications are to be expected for general systems of the type (3). These are related to the approximation of the slowly varying solution components and other related quantities of (3) for k —> oo by the corresponding solution of the constrained system DAE... 

In general, the solution components of the DAE (4) are the correct limits (as K —> oo) of the corresponding slowly varying solution components of the free dynamics only if an additional (conservative) force term is introduced in the constrained system [14, 5]. It turns out [3] that the midpoint method may falsely approximate this correcting force term to zero unless k — 0 e), which leads to a step-size restriction of the same order of magnitude as explicit... 

The main disadvantage of this approach arises when the limit constrained system is different from (4), as mentioned in the introduction and demonstrated in 5 for our second model problem. 

The standard numerical integrators for the constrained system (12) are the SHAKE scheme [23], which extends the Verlet method (2),... 

The most commonly used method for applying constraints, particularly in molecula dynamics, is the SHAKE procedure of Ryckaert, Ciccotti and Berendsen [Ryckaert et a 1977]. In constraint dynamics the equations of motion are solved while simultaneous satisfying the imposed constraints. Constrained systems have been much studied in classics mechanics we shall illustrate the general principles using a simple system comprising a bo sliding down a frictionless slope in two dimensions (Figure 7.8). The box is constrained t remain on the slope and so the box s x and y coordinates must always satisfy the equatio of the slope (which we shall write as y = + c). If the slope were not present then the bo... 

CE uses holonomic constraints. In a constrained system the coordinates of the particles 5t independent and the equations of motion in each of the coordinate directions are cted. A second difficulty is that the magnitude of the constraint forces is unknown, in the case of the box on the slope, the gravitational force acting on the box is in the ction whereas the motion is down the slope. The motion is thus not in the same direc-s the gravitational force. As such, the total force on the box can be considered to arise wo sources one due to gravity and the other a constraint force that is perpendicular to otion of the box (Figure 7.8). As there is no motion perpendicular to the surface of the the constraint force does no work. 

The force and moment ia a constrained system can be estimated by the cantilever formula. Leg MB is a cantilever subject to a displacement of and leg CB subject to a displacement Av. Taking leg CB, for example, the task has become the problem of a cantilever beam with length E and displacement of Av. This problem caimot be readily solved, because the end condition at is an unknown quantity. However, it can be conservatively solved by assuming there is no rotation at poiat B. This is equivalent to putting a guide at poiat B, and results ia higher estimate ia force, moment, and stress. The approach is called guided-cantilever method. 

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].)... 

The theorem under discussion is a particular case of a very general principle, which was stated by Maxwell (1871) in the form that a force producing alteration of the state of a constrained system is always greater than a similar force producing the same alteration in an unconstrained system. ... 

Tolbert LM, Solntsev KM (2002) Excited-state proton transfer from constrained systems to super photoacids to superfast proton transfer. Acc Chem Res 35 19-27... 

Kupfer, R., Poliks, M.D. and Brinker, U.H. (1994). Carbenes in constrained systems. 2. First carbene reactions within zeolites-solid state photolysis of adamantane-2-spiro-3 -diazirine. J. Am. Chem. Soc. 116, 7393-7398... 

Our initial studies focused on the transition metal-catalyzed [4+4] cycloaddition reactions of bis-dienes. These reactions are thermally forbidden, but occur photochemically in some specific, constrained systems. While the transition metal-catalyzed intermole-cular [4+4] cycloaddition of simple dienes is industrially important [7], this process generally does not work well with more complex substituted dienes and had not been explored intramolecularly. In the first studies on the intramolecular metal-catalyzed [4+4] cycloaddition, the reaction was found to proceed with high regio-, stereo-, and facial selectivity. The synthesis of (+)-asteriscanoHde (12) (Scheme 13.4a) [8] is illustrative of the utihty and step economy of this reaction. Recognition of the broader utiHty of adding dienes across rc-systems (not just across other dienes) led to further studies on the use of transition metal catalysts to facilitate otherwise difficult Diels-Alder reactions [9]. For example, the attempted thermal cycloaddition of diene-yne 15 leads only... 

The theory of Brownian motion for a constrained system is more subtle than that for an unconstrained system of pointlike particles, and has given rise to a substantial, and sometimes confusing, literamre. Some aspects of the problem, involving equilibrium statistical mechanics and the diffusion equation, have been understood for decades [1-8]. Other aspects, particularly those involving the relationships among various possible interpretations of the corresponding stochastic differential equations [9-13], remain less thoroughly understood. This chapter attempts to provide a self-contained account of the entire theory. 

In this section, we use simple phenomenological arguments to construct a diffusion equation for constrained systems, in a notation that is common to both... 

Brownian motion of a constrained system of N point particles may also be described by an equivalent Markov process of the Cartesian bead positions R (f),..., R (f). The constrained diffusion of the Cartesian coordinates may be characterized by a Cartesian drift velocity vector and diffusivity tensor... 

The behavior of a constrained system may thus be correctly described by a 3A -dimensional model with a mobility AT P, an initial distribution that is confined to within an infinitesimal region around the constraint surface, and an equilibrium distribution Peq(6) whose value at each point the constraint surface is proportional to the desired value of giving... 

The connection between a diffusion equation and a corresponding Markov diffusion process may be established through expressions for drift velocities and diffusitivies. The drift velocity for both unconstrained and constrained systems may be expressed in an arbitrary system of coordinates in the generic form... 

To describe a constrained system of N particles, we must instead take be the constrained mobility and take eq cx within the constraint... 

A generalized set of reciprocal vectors for a constrained system is defined here to be any set off contravariant basis vectors b, ..., b- and K covariant basis... 

The present analysis builds directly on three previous analyses of SDEs for constrained systems by Fixman [9], Hinch [10], and Ottinger [11]. Fixman and Hinch both considered an interpretation of the inertialess Langevin equation as a limit of an ordinary differential equation with a finite, continuous random force. Both authors found that, to obtain the correct drift velocity and equilibrium distribution, it was necessary to supplement forces arising from derivatives of C/eff = U — kT n by an additional corrective pseudoforce, but obtained inconsistent results for the form of the required correction force. Ottinger [11] based his analysis on an Ito interpretation of SDEs for both generalized and Cartesian coordinates, and thereby obtained results that... 

The inertialess Langevin equation for the Cartesian bead positions of a constrained system may be formulated as a limit of a set of ODEs containing rapidly varying constraint forces, which must be chosen so as keep the instantaneous bead velocities always tangent to the constraint surface. We consider a Langevin equation... 

The preceding definition of a kinetic SDE reduces to that given by Hiitter and Ottinger [34] in the case of an invertible mobility matrix X P, for which Eq. (2.268) reduces to the requirement that Zap = K. In the case of a singular mobility, the present definition requires that the projection of Z p onto the nonnull subspace of K (corresponding to the soft subspace of a constrained system) equal the inverse of within this subspace, while leaving the components of Z p outside this subspace unspecified. 

In both traditional and kinetic interpretations of the Cartesian Langevin equation for a constrained system, one retains some freedom to specify the hard and mixed components of the force variance tensor Several forms for Z v have been considered in previous work, corresponding to different types of random force, which generally require the use of different corrective pseudo forces ... 

Here, is the mobility tensor in the chosen system of coordinates, which is a constrained mobility for a constrained system and an unconstrained mobility for an unconstrained system. As discussed in Section VII, in the case of a constrained system, Eq. (2.344) may be applied either to the drift velocities for the / soft coordinates, for which is a nonsingular / x / matrix, or to the drift velocities for a set of 3N unconstrained generalized or Cartesian coordinates, for a probability distribution (X) that is dynamically constrained to the constraint surface, for which is a singular 3N x 3N matrix. The equilibrium distribution is. (X) oc for unconstrained systems and... 

---