1C: Internal coordinates

Apart from cartesian (and normal) coordinates, internal coordinates are being used. They describe the positions of the atoms in terms of distances, angles and dihedral angles, with respect to an origin atom.

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.
Note that 3N-6 variables are used: there are six zero's in the upper right corner of the matrix. The orientation of the structure in space is not specified. The six 'missing' variables correspond to the three translations and three rotations of the whole structure (with respect to three axes) which do not change the energy of system and can therefore be omitted.
A visualization program displaying a z-matrix contains instructions as to where to place the second (e.g. 'along pos. x-axis') and third atom (e.g. 'in x-y plane').

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.


This is the first chapter: Types of coordinates, z-matrices, input files
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