A skillful choice of internal coordinates can make the interpretation of results straightforward. They are used for creating input geometries for molecular systems in many molecular modelling and computational chemistry programs. Back-conversion from Cartesian to torsion angles is simple trigonometry and has no risk of cumulative errors. These matter because macromolecular chains, such as polymers, proteins, and DNA, can have thousands of connected atoms and atoms consecutively distant along the chain that may be close in Cartesian space (and thus small round-off errors can accumulate to large force-field errors.) The optimally fastest and most numerically accurate algorithm for conversion from torsion-space to cartesian-space is the Natural Extension Reference Frame method. While the transform is conceptually straightforward, algorithms of doing the conversion vary significantly in speed, numerical precision and parallelism. Z-matrices can be converted to Cartesian coordinates and back, as the structural information content is identical, the position and orientation in space, however is not meaning the Cartesian coordinates recovered will be accurate in terms of relative positions of atoms, but will not necessarily be the same as an original set of Cartesian coordinates if you convert Cartesian coordinates to a Z matrix and back again. The name arises because the Z-matrix assigns the second atom along the Z axis from the first atom, which is at the origin. However, it is convenient to write a Z-matrix in terms of bond lengths, angles, and dihedrals since this will preserve the actual bonding characteristics. It provides a description of each atom in a molecule in terms of its atomic number, bond length, bond angle, and dihedral angle, the so-called internal coordinates, although it is not always the case that a Z-matrix will give information regarding bonding since the matrix itself is based on a series of vectors describing atomic orientations in space. A Z-matrix is also known as an internal coordinate representation. In chemistry, the Z-matrix is a way to represent a system built of atoms. For the mathematical meaning of this term see Z-matrix (mathematics).
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