Z-matrix (chemistry)
Encyclopedia
In chemistry
, the Z-matrix is a way to represent a system built of atoms. 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
, 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. 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 information content is identical. While the transform is conceptually straight forward, 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.)
They are used for creating input geometries for molecular systems in many molecular modelling
and computational chemistry
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
may be performed faster, because an educated guess is available for an initial Hessian
matrix, 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. For ringed molecules link benzene
, a z-matrix cannot include all six bonds in the ring, because they are linearly dependent, and this is a problem with Z-matrices.
molecule can be described by the following Cartesian coordinates (in Ångström
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
.
Chemistry
Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....
, the Z-matrix is a way to represent a system built of atoms. 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
Atomic number
In chemistry and physics, the atomic number is the number of protons found in the nucleus of an atom and therefore identical to the charge number of the nucleus. It is conventionally represented by the symbol Z. The atomic number uniquely identifies a chemical element...
, bond
Chemical bond
A chemical bond is an attraction between atoms that allows the formation of chemical substances that contain two or more atoms. The bond is caused by the electromagnetic force attraction between opposite charges, either between electrons and nuclei, or as the result of a dipole attraction...
length, bond angle, and dihedral angle
Dihedral angle
In geometry, a dihedral or torsion angle is the angle between two planes.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection...
, 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
Cartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...
and back, as the information content is identical. While the transform is conceptually straight forward, 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.)
They are used for creating input geometries for molecular systems in many molecular modelling
Molecular modelling
Molecular modelling encompasses all theoretical methods and computational techniques used to model or mimic the behaviour of molecules. The techniques are used in the fields of computational chemistry, computational biology and materials science for studying molecular systems ranging from small...
and computational chemistry
Computational chemistry
Computational chemistry is a branch of chemistry that uses principles of computer science to assist in solving chemical problems. It uses the results of theoretical chemistry, incorporated into efficient computer programs, to calculate the structures and properties of molecules and solids...
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
Optimization (mathematics)
In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....
may be performed faster, because an educated guess is available for an initial Hessian
Hessian matrix
In mathematics, the Hessian matrix is the square matrix of second-order partial derivatives of a function; that is, it describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named...
matrix, 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. For ringed molecules link benzene
Benzene
Benzene is an organic chemical compound. It is composed of 6 carbon atoms in a ring, with 1 hydrogen atom attached to each carbon atom, with the molecular formula C6H6....
, a z-matrix cannot include all six bonds in the ring, because they are linearly dependent, and this is a problem with Z-matrices.
Example
The methaneMethane
Methane is a chemical compound with the chemical formula . It is the simplest alkane, the principal component of natural gas, and probably the most abundant organic compound on earth. The relative abundance of methane makes it an attractive fuel...
molecule can be described by the following Cartesian coordinates (in Ångström
Ångström
The angstrom or ångström, is a unit of length equal to 1/10,000,000,000 of a meter . Its symbol is the Swedish letter Å....
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
Tetrahedral symmetry
150px|right|thumb|A regular [[tetrahedron]], an example of a solid with full tetrahedral symmetryA regular tetrahedron has 12 rotational symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.The group of all symmetries is isomorphic to the group...
.
External links
- Comparison of algorithms for conversion of internal to Cartesian coordinates
- Java implementation of the NERF conversion algorithm
- Z-Matrix to Cartesian Coordinate Conversion Page
- Chemistry::InternalCoords::Builder — PerlPerlPerl is a high-level, general-purpose, interpreted, dynamic programming language. Perl was originally developed by Larry Wall in 1987 as a general-purpose Unix scripting language to make report processing easier. Since then, it has undergone many changes and revisions and become widely popular...
module to build a Z-matrix from Cartesian coordinates.