![]() ![]() Valence angles: one iterates over the pairs of consecutive edges of the graph. Given a molecular graph, see package Molecular_covalent_structure, all primitive internal coordinates are generated as follows:īond lengths: one iterates over the edges of the graph. In the sequel, we focus on primitive internal coordinates (PIC) and delocalized internal coordinates (DIC). They form a complete and non redundant set of internal coordinates,Įach such coordinate is a linear combination of a number of all internal coordinates. In a nutshell, these coordinates, which stem for a linear transformation between redundant internal coordinates and Cartesian coordinates, are characterized as follows: These algorithms remain complex, though.įollowing the spirit of natural internals, yet with much simpler generation algorithms, delocalized internal coordinates were finally proposed. To reduce the coupling, both harmonic and anharmonic, between internal coordinates, natural internal coordinates were designed, and algorithms to derive them proposed. As illustrated by example exple-ic-fluoroethylene, such coordinates are usually redundant, so that deciding of a non redundant set of coordinates does not admit a unique solution. Primitive internal coordinates are encoded in the graph topology. The reader is referred to the excellent overview provided in the Q-Chem user manual ( Q-Chem and more specifically here). The choice of the representation depends on the problem tackled, and is of paramount importance when geometric optimization (minimization of the potential energy) is carried out. It is even more so in the presence of cycles. Internal coordinates: primitive, natural, delocalizedĪs illustrated by example exple-ic-fluoroethylene, the choice of a coordinate system to represent a molecule may be non trivial. Z-matrix representation of fluoroethylene from Fig. Inset: indices of the atoms for the z-matrix representation of Fig. (D) Fluoroethylene: 5 bond lengths, 6 valence angles, 4 dihedral angles. (B) A molecular graph with 3 covalent bond lengths and 3 valence angles (C) Cyclobutane: 4 bond lengths, 4 valence angles, 4 dihedral angles. (A) A molecular graph with 4 covalent bond lengths, 2 valence angles, and 1 dihedral angle. See examples exple-ic-one, exple-ic-cyclobutane and exple-ic-fluoroethylene for details. Ĭovalent structure and coordinates: Cartesian coordinates, internal coordinates, and degrees of freedom. But since, it turns out that the 15 internal coordinates have three redundancies. For the latter, observe that once one has fixed the bond, one can form 4 tuples (whence four dihedral angles) by taking the Cartesian products of the groups of atoms bonded to the two carbons. It is easily seen that there are 5 bond lengths, 6 valence angles, 4 dihedral angles. Note that an improper angle can be thought as the off planarity angle of atom with respect to the plane. The improper dihedral angle is the angle between the planes and (Fig. ![]() Pick a second atom the define a hinge, e.g. fig-ic-dh (A)).įor improper angles, consider a central atom connected to three atoms, say. fig-ic-bl-va (B)).ĭihedral angles come into two guises: proper and improper.įor proper angles, consider four consecutive atoms on a path: the dihedral angle in the angle between the two planes defined by the first three and the last three atoms (Fig. Valence angles are defined by around a particle participating in two such bonds, thereby defining an angle (Fig. īonds are defined by two points connected in the molecular covalent graph~(Fig. Internal coordinates represent the geometry of a molecule in terms of bond lengths, valence angles, and dihedral angles. We consider a molecular graph, as introduced in the package Molecular_covalent_structure. Pre-requisites: representing molecular conformations Structure generation: generating a novel structure given a known structure – aka a move set. Structure minimization: minimizing the potential energy of a system by following its negative gradient. Such coordinates are cornerstones of the following tasks, , :Įnergy calculations: computing the potential energy of a (macro-)molecular system, using a so-called force field, see the package Molecular_potential_energy. This package discusses the representations used to represent molecular conformations. ![]()
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