As these terms coincide with the chemical concepts of bond lengths, bond angles and torsion angles, the internal coordinates are well suited to describe (changes in) organic structures. A complete set of internal coordinates is called a z-matrix.
Programs like MOPAC and Gamess accept both cartesian coordinates and z-matrices as input of structures.
In a z-matrix the first atom is the origin. The second atom is defined by the distance to atom1 (d21), the third atom by a distance (to atom1 or atom2) and an angle (a321). | |
Starting with the fourth atom the dihedral angle
(t4321) is introduced. From here every atom is described by a distance, an angle and a dihedral angle, with respect to already defined atoms. | |
Example z-matrix, with numerical values added:At1 0 0 0 0 0 0 At2 1.51 0 0 1 0 0 At3 1.50 127.7 0 2 1 0 At4 1.49 110.4 61.7 3 2 1 At5 1.50 109.3 125.7 3 2 1 At6 .. ... etc. |
The last three columns contain the atom numbers in the definition path. Atom 5 is defined by a distance to atom 3, the angle 5-3-2 and the dihedral angle 5-3-2-1. |
Bond angles of 180 degrees must be avoided in a definition path, as these make the dihedral angles undetermined. (In the drawing At4 will 'disappear' behind the atoms 2 and 3 in case a432 is 180 degrees). For this purpose dummy atoms can be introduced: geometrical points that help to define atoms, but without chemical meaning.
A few points are explained in more detail in the following worked examples:
Example:
Constructing a simple z-matrix.
Exercise Interpretating a z-matrix.
This chapter is continued in:
1D: Applying symmetry and dummy atoms in
MOPAC input files
shows how dummy atoms and symmetry
rules can greatly simplify the construction of z-matrices.