Constraint (computational chemistry)

In computational chemistry, a constraint algorithm is a method for satisfying the Newtonian motion of a rigid body which consists of mass points. A restraint algorithm is used to ensure that the distance between mass points is maintained. The general steps involved are: (i) choose novel unconstrained coordinates (internal coordinates), (ii) introduce explicit constraint forces, (iii) minimize constraint forces implicitly by the technique of Lagrange multipliers or projection methods. Constraint algorithms are often applied to molecular dynamics simulations. Although such simulations are sometimes performed using internal coordinates that automatically satisfy the bond-length, bond-angle and torsion-angle constraints, simulations may also be performed using explicit or implicit constraint forces for these three constraints. However, explicit constraint forces give rise to inefficiency; more computational power is required to get a trajectory of a given length. Therefore, internal coordinates and implicit-force constraint solvers are generally preferred. Constraint algorithms achieve computational efficiency by neglecting motion along some degrees of freedom. For instance, in atomistic molecular dynamics, typically the length of covalent bonds to hydrogen are constrained; however, constraint algorithms should not be used if vibrations along these degrees of freedom are important for the phenomenon being studied.

== Mathematical background == The motion of a set of N particles can be described by a set of second-order ordinary differential equations, Newton's second law, which can be written in matrix form $\mathbf {M} \cdot {\frac {d^{2}\mathbf {q} }{dt^{2}}}=\mathbf {f} =-{\frac {\partial V}{\partial \mathbf {q} }}$ If M constraints are present, the coordinates must also satisfy M time-independent algebraic equations $g_{j}(\mathbf {q} )=0$ This problem was studied in detail by Joseph Louis Lagrange, who laid out most of the methods for solving it. The simplest approach is to define new generalized coordinates that are unconstrained; this approach eliminates the algebraic equations and reduces the problem once again to solving an ordinary differential equation. Such an approach is used, for example, in describing the motion of a rigid body; the position and orientation of a rigid body can be described by six independent, unconstrained coordinates, rather than describing the positions of the particles that make it up and the constraints among them that maintain their relative distances. The drawback of this approach is that the equations may become unwieldy and complex; for example, the mass matrix M may become non-diagonal and depend on the generalized coordinates. A second approach is to introduce explicit forces that work to maintain the constraint; for example, one could introduce strong spring forces that enforce the distances among mass points within a "rigid" body. The two difficulties of this approach are that the constraints are not satisfied exactly, and the strong forces may require very short time-steps, making simulations inefficient computationally. A third approach is to use a method such as Lagrange multipliers or projection to the constraint manifold to determine the coordinate adjustments necessary to satisfy the constraints.
Finally, there are various hybrid approaches in which different sets of constraints are satisfied by different methods, e.g., internal coordinates, explicit forces and implicit-force solutions.

== Internal coordinate methods == The simplest approach to satisfying constraints in energy minimization and molecular dynamics is to represent the mechanical system in so-called internal coordinates corresponding to unconstrained independent degrees of freedom of the system. For example, the dihedral angles of a protein are an independent set of coordinates that specify the positions of all the atoms without requiring any constraints. The difficulty of such internal-coordinate approaches is twofold: the Newtonian equations of motion become much more complex and the internal coordinates may be difficult to define for cyclic systems of constraints, e.g., in ring puckering or when a protein has a disulfide bond. The original methods for efficient recursive energy minimization in internal coordinates were developed by Gō and coworkers. Efficient recursive, internal-coordinate constraint solvers were extended to molecular dynamics. Analogous methods were applied later to other systems.

== Lagrange multiplier-based methods ==

In most of molecular dynamics simulations that use constraint algorithms, constraints are enforced using the method of Lagrange multipliers. Given a set of n linear (holonomic) constraints at the time t, $d_{k}$ The forces due to these constraints are added in the equations of motion, resulting in, for each of the N particles in the system $\lambda _{k}$ From integrating both sides of the equation with respect to the time, the constrained coordinates of particles at the time, $t+\Delta t$
, are given, ${\hat {\mathbf {x} }}_{i}(t+\Delta t)$ To satisfy the constraints $\lambda _{k}$
. This system of

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Category

Computational Physics - Physics