Z-matrix (chemistry)

For the mathematical meaning of this term see Z-matrix (mathematics).

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 structural information content is identical, the position and orientation in space, however, is not. 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 like 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.

Example

The methane molecule can be described by the following Cartesian coordinates (in Ångströms):

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.

External links

• Parsons, Jerod; Holmes, J. Bradley; Rojas, J. Maurice; Tsai, Jerry; Strauss, Charlie E. M. (2005). "Practical conversion from torsion space to Cartesian space forin silico protein synthesis". Journal of Computational Chemistry. 26 (10): 1063. doi:10.1002/jcc.20237. PMID 15898109.
• Java implementation of the NERF conversion algorithm
• Z-Matrix to Cartesian Coordinate Conversion Page
• Chemistry::InternalCoords::Builder — Perl module to build a Z-matrix from Cartesian coordinates.