The z-matrix, as used in the MOPAC input file.

  C    0.00000000  0    0.0000000  0    0.0000000  0    0    0    0 
  C    1.51738930  1    0.0000000  0    0.0000000  0    1    0    0
  H    1.11721396  1  107.8021607  1    0.0000000  0    2    1    0
  H    1.11205843  1  111.3722406  1 -122.3562469  1    1    2    3
  H    1.11783779  1  107.4581078  1  115.7104335  1    1    2    3
  H    1.11692070  1  111.0770073  1    0.0000000  0    1    2    3
  H    1.11712199  1  111.0150133  1    0.8782819  1    2    1    4 
  H    1.11702994  1  110.9935404  1  120.0975124  1    2    1    4

This is the z-matrix describing the eclipsed conformation of ethane in internal coordinates.

internal coordinates

We recognize one line per atom, each line starting with an atom label. The first atom is the starting point (origin). It is followed by a series of zero's. The second atom is defined by a distance to this atom; in this example the normal carbon-carbon single bond length of 1.51... A. The third atom is defined by a distance and an angle. Again we recognize the normal value for alkanes. From the fourth atom onwards, a distance, an angle and a dihedral angle are required to define the position. We speak of internal coordinates. The last three numbers on each row show the definition path: for carbon 4 the distance is to atom 1, the angle is 4-1-2, and the dihedral angle 4-1-2-3. One basic rule has to be obeyed when constructing z-matrices: all atoms have to be uniquely defined, using only atoms that have already been defined. No number higher than the current line can be used in a definition path.

dihedral angle

The dihedral angle may require an explanation. The dihedral angle H4-C1-C2-H3 is the angle between the bonds H4-C1 and C2-H3, when looking along the C1-C2 bond. A view along the central bond results in fact in a Newman projection. An anti-clockwise rotation from H4 to H3 makes the angle negative, as in this example. For H5 the rotation is more or less the same, but clockwise, and thus positive. In this case the dihedral angle is the same as what chemists call the torsion angle. (Dihedral angle is more general than torsion angle: any four points define a dihedral angle, while a torsion refers to four connected atoms.) This illustrates the advantage of the z-matrix over e.g. cartesian (x,y,z) coordinates: the description contains 'chemical' entities as bond lengths, bond angles and torsion angles.

Questions

Press LM/RM/shiftLM on structure.

How many variables are used to describe a molecule of N atoms? Is the orientation in space defined? What is the relation between the dihedral angles 4-1-2-3 and 3-2-1-4? Which structure do I get if I change the sign of all the torsion angles? Complete the drawing of the structure above, adding labels to the atoms. It is clear that H6 and H3 are eclipsed (dihedral angle close to zero), but how are the hydrogens 7 and 8 orientated?

the flags 0, 1, -1 reaction path

A last item to mention is the '0' or '1' after each parameter value. This is a flag for the MOPAC program, either to treat a value as a constant ('0'), or to optimize the value so as to minimize the energy ('1'). A third value for this flag, not present in the example above, is '-1', indicating that the value should be systematically varied. The way this is done is defined elsewhere in the input file, as we will see later. Such a path calculation allows us to simulate conformational changes and reactions, and is therefore used at several points in this tutorial.

2. The MOPAC input file.