1D: Applying symmetry and dummy atoms

drawing One way of simplifying the cyclopropane z-matrix is by making use of the symmetry of the molecule. Therefore we need the symmetry axis, perpendicular to the carbon plane, which is defined by placing two dummy atoms, one in the center of the ring, and one perpendicular above it, at an arbitrary distance of 1 A. Subsequently the carbons and hydrogens are defined with respect to these dummy atoms.

We start with the upper dummy, then the central one. These 'atoms' get XX as a label (in MOPAC nomenclature).

XX   0.00    0.00    0.00   0  0  0
XX   1.00    0.00    0.00   1  0  0
 C   0.90   90.00    0.00   2  1  0
 C   0.90   90.00  120.00   2  1  3
 C   0.90   90.00  240.00   2  1  3
 H   1.10  125.00    0.00   3  2  1
 H   1.10  125.00  180.00   3  2  1
 H   1.10  125.00    0.00   4  2  1
 H   1.10  125.00  180.00   4  2  1
 H   1.10  125.00    0.00   5  2  1
 H   1.10  125.00  180.00   5  2  1
*************************************
Questions
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The three angles of 90 degrees are not variables, but constants.
The same holds for all the dihedral angles. They follow from the definition of the dummy atoms.
drawing
These drawings illustrate the way the dihedral angles are defined.
drawing
In MOPAC, variables and constants are distinguished by a label. Values to be optimized are followed by a '1', constants get a '0'.
XX   0.00 0   0.00 0   0.00 0  0  0  0
XX   1.00 0   0.00 0   0.00 0  1  0  0
 C   0.90 1  90.00 0   0.00 0  2  1  0
 C   0.90 1  90.00 0 120.00 0  2  1  3
 C   0.90 1  90.00 0 240.00 0  2  1  3
 H   1.10 1 125.00 1   0.00 0  3  2  1
 H   1.10 1 125.00 1 180.00 0  3  2  1
 H   1.10 1 125.00 1   0.00 0  4  2  1
 H   1.10 1 125.00 1 180.00 0  4  2  1
 H   1.10 1 125.00 1   0.00 0  5  2  1
 H   1.10 1 125.00 1 180.00 0  5  2  1
The number of variables is now 15. Symmetry considerations allow us to further reduce this number.
All three C-XX2 distances will be equal because of the symmetry in the molecule. In MOPAC this is indicated at the bottom of the z-matrix (which is closed by a zero line) and SYMMETRY as a keyword on the first line:
XX   0.00 0   0.00 0   0.00 0  0  0  0
XX   1.00 0   0.00 0   0.00 0  1  0  0
 C   0.90 1  90.00 0   0.00 0  2  1  0
 C   0.90 0  90.00 0 120.00 0  2  1  3
 C   0.90 0  90.00 0 240.00 0  2  1  3
 H   1.10 1 125.00 1   0.00 0  3  2  1
 H   1.10 1 125.00 1 180.00 0  3  2  1
 H   1.10 1 125.00 1   0.00 0  4  2  1
 H   1.10 1 125.00 1 180.00 0  4  2  1
 H   1.10 1 125.00 1   0.00 0  5  2  1
 H   1.10 1 125.00 1 180.00 0  5  2  1
 0   0.00
 3,  1,  4,  5          which means that for atom 3, variable 1 
                        (the distance)
is transferred to the atoms 4 and 5 as well. 
The corresponding flags have to be changed from 1 to 0.
This symmetry could not have been implemented using the definition path suggested in the question above.

So the format is: reference atom, variable function, related atom(s).
See the MOPAC manual for the complete list of functions.
The most important ones are:
1 the distance
2 the angle
3 the dihedral angle
9 dihedral angle varies as 180 degrees plus reference dihedral
14 dihedral angle varies as negative of reference dihedral
17 bond angle varies as 180 degrees minus reference bond angle

For the hydrogens 7 to 11, both the distances and the angles can be taken from the values for H6:

6,  1,  7,  8,  9,  10,  11
6,  2,  7,  8,  9,  10,  11
So the complete input file, with z-matrix, looks like:
 AM1 T=600 SYMMETRY ....          	(keyword line)
 Cyclopropane input file	  	(comment line)
 with symmetry			  	(comment line)
XX   0.00 0   0.00 0   0.00 0  0  0  0
XX   1.00 0   0.00 0   0.00 0  1  0  0
 C   0.90 1  90.00 0   0.00 0  2  1  0
 C   0.90 0  90.00 0 120.00 0  2  1  3
 C   0.90 0  90.00 0 240.00 0  2  1  3
 H   1.10 1 125.00 1   0.00 0  3  2  1
 H   1.10 0 125.00 0 180.00 0  3  2  1
 H   1.10 0 125.00 0   0.00 0  4  2  1
 H   1.10 0 125.00 0 180.00 0  4  2  1
 H   1.10 0 125.00 0   0.00 0  5  2  1
 H   1.10 0 125.00 0 180.00 0  5  2  1
 0   0.00				(blank or zero line)
 3,  1,  4,  5 
 6,  1,  7,  8,  9,  10,  11
 6,  2,  7,  8,  9,  10,  11
Note that there are now only three 1's left in the z-matrix.

Exercise: construct a z-matrix for the chair conformation of cyclohexane with the minimum number of variables.
Check your answer.

This chapter is continued in:
1E: Creating MOPAC input files


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