Bond stretching constraints

Third, the use of coupled holonomic constraints having more complex forms than bond-stretch constraints highlights the limitations of the methods, which are not obvious when only simple bond-stretch constraints are employed. 

With the additional EEM restriction that [Pg.91]

The most general form of holonomic constraint is nonlinear in the particle positions. Even the simple bond-stretch constraint is nonlinear. Consequently, Eq. [39] is in general a system of / coupled nonlinear equations, to be solved for the / unknowns (7). This nonlinear system of equations must be contrasted with the linear system of equations Eqs. [10] and [11] (which is also in general part of the method of undetermined parameters) used in the analytical method to solve for the Lagrangian multipliers and their derivatives. A solution of Eq. [39] can be achieved in two steps ... 

Linearization and iteration The nonlinear system of equations, Eq. 57, is linearized and solved for a first estimate solution of [7], as discussed in connection with Eq. [39]. The solution is then inserted in the retained quadratic terms, and the linear system is solved for an improved estimate of the I7). This iterative procedure is repeated until the 7 converge within a desired tolerance. For the bond-stretch constraint, there is just one nonlinear (quadratic) term in its Taylor expansion (see later, Eq. [95]), and the linearization and iteration procedure is a fairly good approximation, justified even for relatively large corrections. For the bond-angle and torsional constraints, with infinite series Taylor representations, tighter limits are imposed on the allowable constraint... 

The simultaneous matrix solution Eq. [62] of the matrix method is replaced by iterations over the sequence of constraints. The SHAKE algorithm consists of an iterative loop inside which the constraints are considered individually and successively. That is, the constraints are decoupled. SHAKE was initially described for the case of bond-stretch constraints and later generalized to handle general forms of holonomic constraint. The algorithm is discussed here for the case of general holonomic constraints Beginning with the starting point of the matrix method, Eq. [56], a solution can be achieved in three steps. 

Equation [67] is generally nonlinear in even in the common case of a bond-stretch constraint. 

This section begins by applying the method of undetermined parameters, with the basic Verlet algorithm, to the treatment of bond-stretch constraints. Since they are the most common type of constraint in MD simulations and were... 

Consider a system of N particles and / general holonomic constraints, and assume that bond-stretch constraints (/ < /) are present. For bond-stretch constraints, the general holonomic constraint, Eq. [1], takes the special form... 

Now we use the treatment in the preceding section to give a physical picture of the SHAKE process of resetting the coordinates to satisfy bond-stretch constraints. If the bond-stretch constraint Eq. [95] had been formulated, instead, in terms of the bond-stretch internal coordinated (i.e., magnitude of the distance between the two atoms), the following bond-stretch SHAKE displacements would have been found from Eq. [77] ... 

From Eq. [101] it is seen that 8r. and 8r,. are again in the directions of the line connecting the two atoms at the preceding time step tg and pointing inward. Hence the current displacements are in the direction of the Wilson unit vectors, with = 27 " ]r.(fp) - r (tQ). A later section details the correspondence between the formulations of the bond-stretch constraint in terms of the bond-stretch internal coordinate and in terms of the square of the interatomic distance. 

The SHAKE tolerance for the bond-stretch constraints is the same in both procedures [(s/ ) = 10 ], and the bond-angle tolerance in the angle-constraint approach is obtained from Eq. [119] [(af) 10 ]. The accuracies of the two approaches are comparable for the adopted values of tolerances. SPC/E designates a modified simple point charge model. 

If the form of the bond-stretch constraint is chosen to be in terms of the magnitude of the bond stretch rather than its square (i.e., as in Eq. [95]), the convergence for the triangulation bond stretch is reached when (dB) (sf) X B, where (sf) is the bond-stretch magnitude tolerance factor or relative bond-stretch magnitude tolerance. A relation between (sf) and (sf) is easily obtained. The tolerance on the bond stretch... 

The solution for the 7 and It)] using SHAKE was termed RATTLE by Andersen. The material here is new in the sense that the original formulation of RATTLE was for bond-stretch constraints only. The treatment is general-... 

As with SHAKE, RATTLE is illustrated in this chapter for the internal coordinate constraints of bond stretch, angle bend, and torsion. For bond-stretch constraints and the first stage of RATTLE, inserting the constraint a ( rj) of Eq. [95] into Eq. [131] yields... 

The obtained anomaly (Figs. 11.19 and 11.20a) results from the increase of connectivity arising from the tetrahedral to octahedral silicon coordination change [125] under pressure (Fig. 11.20b). This increase in connectivity leads in fact to additional stress induced by an increase of the number bond-stretching constraints equal to nf (Si)s=rsi/2 which is found to increase from 2 at ambient pressure to 2.05 at... 

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