Angle-bend constraints

The physical effects of introducing constraints into a molecular model have been discussed by several authors. > This chapter is concerned mainly with the methods of constraint dynamics. In addition to descriptions of bond-stretch and angle-bend constraints, dihedral (or torsional) constraints are explicitly considered. Torsional modes generally have frequencies comparable to those of other modes, and the weak coupling condition is not satisfied in this case. Hence the constraint approximation is not justified for torsion. This fact is particularly important because torsional motions play a major role in conformation interconversion in small molecules as well as polymers, and constraining them can seriously alter the dynamics of the original, unconstrained system. 

Figure 1 SHAKE displacements for angle-bend constraint. The displacements are shown relative to the molecular configuration at the preceding time step t and have the (negative) directions of the Wilson vectors for an angle-bend internal coordinate. The displacements of the end atoms are perpendicular to the bond directions at and the displacement of the apex atom points in the direction of the sum of their Wilson vectors.

The tolerances for bond-stretch and angle-bend constraints are usually expressed as multiples of the constraint bond-stretch and bond-angle values, respectively (st) = (sf) X B, and (at) = (af) X , where (sf) is the stretch factor or relative bond-stretch tolerance and (af) is the angle-bend factor or relative angle-bend tolerance. Substitution of the tolerance expressions into Eq. [118] yields an analogous relation between the tolerance factors ... 

For angle-bend constraints and the first stage of RATTLE, inserting the constraint ffj ([r]) of Eq. [102] into Eq. [131] gives... 

Again, the method of undetermined parameters deserves a detailed discussion in its most general form because most researchers are interested in its implementation with different integration algorithms (e.g., basic Verlet,- i velocity Verlet ), with holonomic constraints of various types (e.g., bond-stretch, angle-bend, and torsional constraints), and with particular techniques of solution (e.g., the ma-... 

Typically, holonomic constraints in MD simulations are formulated in terms of internal coordinates, such as bond-stretch, angle-bend, and torsion. The general constraint equation, Eq, [1], can then be written as follows ... 

The expressions for the s. are available from molecular vibration theory, for the common internal coordinates of bond-stretch, angle-bend, and torsion. As described earlier, s. points in the direction in which a displacement of particle i produces the greatest change in Equation [77] provides a physical picture of the SHAKE displacements for internal coordinate constraints. In the language of Wilson vectors, the singularity condition for internal coordinate constraints takes the form... 

Point 1 clearly holds for both the matrix method and SHAKE. Because the matrix method is also an iterative approach (albeit a coupled one), the essence of the remarks of point 2 also holds for the matrix method. As stated in point 1, constraining an angle bend with an angle constraint is still applicable when total rigidity is unacceptable, and triangulation is then not a feasible... 

Enforcement of bond length constraints typically allows the time step to be increased by a factor of 2 or 3. Angles may also be frozen by adding a distance constraint on atoms that are 1,3 relative to each other. Angle bending, however, affects calculated properties more than bond stretching and fixing them may often introduce... 

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