Z-matrix

There are apparently two uses for the term Z-matrix.

Mathematics

In the field of mathematics, the class of Z-matrices are those matrices which have all off-diagonal entries less than zero. If the Jacobian of a cooperative dynamical system is $J$, then $(-J)$ is a Z-matrix by definition.

Two related classes are L-matrices and M-matrices. L-matrices have the additional property that all diagonal entries are greater than zero. M-matrices have several equivalent definitions, one of which is as follows: a Z-matrix is an M-matrix if it is nonsingular and its inverse is nonnegative. All matrices that are both Z-matrices and P-matrices are nonsingular M-matrices.

Chemistry

The Z-matrix is a way to represent a system built of atoms. 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, the information content is identical. 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.

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

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  240.0000 

References

• Huan T., Cheng G., Cheng X., Modified SOR-type iterative method for Z-matrices. Applied Mathematics and Computation, Volume 175 Issue 1, 1 April 2006, pages 258-268.
• Saad, Y. Iterative methods for sparse linear sustems. Society for Industrial and Applied Mathematics. Philadelphia, PA. 2nd edition. page 28.