Z-matrix

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

Mathematics

In the field of mathematics the class of Z-matrices are matrices which have the following property, all off-diagonal entries are less than zero. L-matrices have the additional property that all diagonal entries are greater than zero. It should be noted that these two properties are also common to another class of matrices, M- matrices, which are a subclass of Z-matrices. M-Matrices have the additional two properties: they are nonsingular and their inverses are nonnegative.

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. 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 skillfull choice of internal coordinates can make the interpretation of results straightforward. Also, since Z-matrices contain molecular connectivity 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. , Modiefied SOR-type iterative method fo 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. Philidelphia, PA. 2nd edition. page 28.