Holonomic distance constraints

The first step in the DG calculations is the generation of the holonomic distance matrix for aU pairwise atom distances of a molecule [121]. Holonomic constraints are expressed in terms of equations which restrict the atom coordinates of a molecule. For example, hydrogen atoms bound to neighboring carbon atoms have a maximum distance of 3.1 A. As a result, parts of the coordinates become interdependent and the degrees of freedom of the molecular system are confined. The acquisition of these distance restraints is based on the topology of a model structure with an arbitrary, but energetically optimized conformation. 

Totally rigid models A special case of a system of particles with holonomic constraints is the rigid body. A rigid body can be thought of as a system of particles with a distance constraint between every pair of particles. A (nonlinear) rigid model has six degrees of freedom three translational and three rotational. 

The first step of the structure refinement is the appHcation of distance geometry (DG) calculations which do not use an energy function but only experimentally derived distances and restraints which follow directly from the constitution, the so-caUed holonomic constraints. Those constraints are, for example, distances between geminal protons, which normally are in the range between 1.7 and 1.8 A, or the distance between vicinal protons, which can not exceed 3.1 A when protons are in anti-periplanar orientation. 

Deng et al. treated solvated electrons and bipolarons in ammonia with SDFT-LSDA [56,57]. They used the Car-Parrinello procedure with holonomic constraints described in Section 1. Deng et al. found that the electrons of the bipolaron complex in ammonia were spatially separated by a small distance, making that complex peanut-shaped in contrast to the more spherical bipolaron observed in molten KCl [49-51]. In accordance with the KCl studies, increasing the concentration of solvated electrons induced a transition to metallic behavior. 

One of the first approaches introduced and used in condensed phases, also in the sense of the historical development, was a biasing technique known as the Blue Moon [55]. This approach was better refined over the years [56] to become more user-friendly and easily implementable in a computer code. For all the details, we refer the reader to the cited original publications. Just to summarize the essential points, let us recall that in the case of first principles dynamical simulation, the method relies on the identification of a reaction coordinate, or order parameter, = (R/) of a given subset of atomic coordinates (Lagrangean variables) R/ able to track the activated process or chemical reaction on which one wants to focus. The simplest example is represented by the distance = R/—Ry between two atoms that are expected to form or break a chemical bond. This analytical function is added to, e.g., a Car-Parrinello Lagrangean as a holonomic constraint. 

Minimization and vibrational analysis are useful for the determination of force field parameters, system preparation, and the study of many problems of biological interest. A new optimizer based on a truncated Newton method (TNPACK) that is effective for large molecules has been added to CHARMM. All minimizers, excluding TNPACK, support the use of holonomic constraints on selected bonds and angles (SHAKE). Vibrational analysis has been extended via addition of the MOLVIB module (K. Kuczera and J. Wiorkiewicz-Kuczera, unpublished), which allows for the determination of potential energy distributions and the analysis of lattice modes in combination with the CRYSTAL facility. Minimizations may also be performed in the presence of a variety of structural constraints. This allows for atomic positions, internal coordinates, interatomic distances, etc. to be fixed or constrained to specified values. Such constraint methods may be used in molecular dynamics simulations. 

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