chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...

, the Z-matrix is a way to represent a system built of

atoms Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, an ...

. A Z-matrix is also known as an internal coordinate representation. It provides a description of each atom in a molecule in terms of its

atomic number The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of an atomic nucleus. For ordinary nuclei, this is equal to the proton number (''n''p) or the number of protons found in the nucleus of every ...

bond Bond or bonds may refer to: Common meanings * Bond (finance), a type of debt security * Bail bond, a commercial third-party guarantor of surety bonds in the United States * Chemical bond, the attraction of atoms, ions or molecules to form chemica ...

length,

bond angle Bond or bonds may refer to: Common meanings * Bond (finance), a type of debt security * Bail bond, a commercial third-party guarantor of surety bonds in the United States * Chemical bond, the attraction of atoms, ions or molecules to form chemical ...

, and

dihedral angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the un ...

, 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. 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. The name arises because the Z-matrix assigns the second atom along the Z axis from the first atom, which is at the origin. 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. While the transform is conceptually straightforward, algorithms of doing the conversion vary significantly in speed, numerical precision and parallelism. 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. Back-conversion from Cartesian to torsion angles is simple trigonometry and has no risk of cumulative errors. They are used for creating input geometries for molecular systems in many

molecular modelling Molecular modelling encompasses all methods, theoretical and computational, used to model or mimic the behaviour of molecules. The methods are used in the fields of computational chemistry, drug design, computational biology and materials sci ...

and

computational chemistry Computational chemistry is a branch of chemistry that uses computer simulation to assist in solving chemical problems. It uses methods of theoretical chemistry, incorporated into computer programs, to calculate the structures and properties of m ...

programs. A skillful choice of internal coordinates can make the interpretation of results straightforward. Also, since Z-matrices can contain molecular connectivity information (but do not always contain this information), quantum chemical calculations such as geometry

optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...

may be performed faster, because an educated guess is available for an initial

Hessian matrix In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...

, and more natural internal coordinates are used rather than Cartesian coordinates. The Z-matrix representation is often preferred, because this allows symmetry to be enforced upon the molecule (or parts thereof) by setting certain angles as constant. The Z-matrix simply is a representation for placing atomic positions in a relative way with the obvious convenience that the vectors it uses easily correspond to bonds. A conceptual pitfall is to assume all bonds appear as a line in the Z-matrix which is not true. For example: in ringed molecules like

benzene Benzene is an organic chemical compound with the molecular formula C6H6. The benzene molecule is composed of six carbon atoms joined in a planar ring with one hydrogen atom attached to each. Because it contains only carbon and hydrogen atoms, ...

, a z-matrix will not include all six bonds in the ring, because all of the atoms are uniquely positioned after just 5 bonds making the 6th redundant.

Example

The

methane Methane ( , ) is a chemical compound with the chemical formula (one carbon atom bonded to four hydrogen atoms). It is a group-14 hydride, the simplest alkane, and the main constituent of natural gas. The relative abundance of methane on Eart ...

molecule can be described by the following Cartesian coordinates (in

Ångström The angstromEntry "angstrom" in the Oxford online dictionary. Retrieved on 2019-03-02 from https://en.oxforddictionaries.com/definition/angstrom.Entry "angstrom" in the Merriam-Webster online dictionary. Retrieved on 2019-03-02 from https://www.m ...

s): C 0.000000 0.000000 0.000000 H 0.000000 0.000000 1.089000 H 1.026719 0.000000 -0.363000 H -0.513360 -0.889165 -0.363000 H -0.513360 0.889165 -0.363000 Re-orientating the molecule leads to Cartesian coordinates that make the symmetry more obvious. This removes the bond length of 1.089 from the explicit parameters. C 0.000000 0.000000 0.000000 H 0.628736 0.628736 0.628736 H -0.628736 -0.628736 0.628736 H -0.628736 0.628736 -0.628736 H 0.628736 -0.628736 -0.628736 The corresponding Z-matrix, which starts from the carbon atom, could look like this: C H 1 1.089000 H 1 1.089000 2 109.4710 H 1 1.089000 2 109.4710 3 120.0000 H 1 1.089000 2 109.4710 3 -120.0000 Only the 1.089000 value is not fixed by

tetrahedral symmetry 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection ...

References

External links

*
Java implementation of the NERF conversion algorithmC++ implementation of the NERF conversion algorithmChemistry::InternalCoords::Builder
—

Perl Perl is a family of two high-level, general-purpose, interpreted, dynamic programming languages. "Perl" refers to Perl 5, but from 2000 to 2019 it also referred to its redesigned "sister language", Perl 6, before the latter's name was offici ...

module to build a Z-matrix from Cartesian coordinates. {{Matrix classes Molecular modelling Computational chemistry