These slides will tell you the importance of group theory in chemistry. Writing successful Z-matrix by hand essentially requires a deeper understanding of group theory. There is a strong correlation between the point group symmetry of the molecule and the Z-matrix associated with it. If you want to write Z-matrices without using graphical interfaces (such as molden, Jmol, etc.,) at least for small molecules then obviously you need to understand the correlation between the two so that you will be successful and won't be wasting too much of time in front of the computer.

1. Practical Implications of Group
Theory
Dr. Venkatesan S. Thimmakondu,
Department of Chemistry
BITS-Pilani, K K Birla Goa Campus

2. Introduction
• I studied group theory for the last one and
half months. Where am I going to use it?
• How a computer understands molecular
geometry?
• Obviously, we need an input to do any sort of
calculation.

3. Coordinates
• Cartesian Coordinates
• Internal Coordinates
• Or via a graphical interface (molden, molekel,
Avagadro, Jmol, Gaussview, etc.,)

4. Water Molecule in Cartesian
Coordinates
---------------------------------------------------------------------------
Z-matrix Atomic Coordinates (in bohr)
Symbol Number X Y Z
---------------------------------------------------------------------------
O 8 0.00000000 0.00000000 0.12708029
H 1 0.00000000 -1.48440166 -1.00842821
H 1 0.00000000 1.48440166 -1.00842821
---------------------------------------------------------------------------

5. Can I write the geometry of the below
molecule in Cartesian?
When we study the molecule with x-ray crystallography, Cartesian
coordinates are often the natural choice.

6. Internal Coordinates (Z-Matrix)
• To specify 2 points in space what we need?
• To specify 3 points in space what we need?
• To specify 4 points in space what we need?
• Specifying each atom of a molecule in terms
of a distance (bond length), angle (bond
angle) and torsional (dihedral) angle to other
atoms is what we call it as Z-Matrix.

7. Z-Matrix of Water
• There are more than one way of writing Z-
matrix even for small molecules.
O
H 1 R1
H 1 R1 2 A1
R1 = 0.988984834251219
A1 = 105.170884348412642

8. One can also write
• For the same water molecule:
H
O 1 R1
H 2 R1 1 A1
R1 = 0.988984834251219
A1 = 105.170884348412642
• Because, it has nothing to do with the actual
bonding.

9. Connection???
• Where is group theory here?
• Seriously, I am missing something here.
• Let’s assume that the point group symmetry
of water is not C2v but Cs.
• If so, what kind of changes I need to do in the
Z-matrix.
• How can it be Cs?

10. Z-Matrix of Water (in Cs symmetry)
O
H 1 R1
H 1 R2 2 A1
R1 = 0.988984834251219
R2 = 0.988884834251219
A1 = 105.170884348412642
• Even if there is a difference on the 4th decimal
place, it matters!

11. H2O in Cs symmetry
• 3 entries (two bond lengths and one bond
angle) found in Z-matrix
• There are 3 unique (two bond lengths and
one bond angle) internal coordinates.
• Of these, 3 will be optimized.

12. H2O in C2v Symmetry
• 3 entries (two bond lengths and one bond
angle) found in Z-matrix
• There are 2 unique (one bond length and one
bond angle) internal coordinates.
• Of these, 2 will be optimized.

13. Symmetry is essential in electronic
structure calculations
• The reasons are obviously pragmatic.
• A calculation run on a molecule whose input
structure has the exact symmetry that the
molecule should have, will tend to be faster
and will yield a “better” geometry than one
run on an approximate structure, however
close this may be to the exact one.
• You lose the symmetry, you deal with more
variables.

14. Point group?
C:CCCC
pentatetraenylidene
H
H
C
C
C:
CC
ethynylcyclopropenylidene
H
H
:C
C
C
C
C
ethynylpropadienylidene
H
H
C
H
H
C C:
3-(didehydrovinylidene)cyclopropene
H
H H H
H
H
ortho-tetradehydrobenzene meta-tetradehydrobenzene para-tetradebydrobenzene

15. Can you write a Z-Matrix for H2CO
C
O 1 R1
H 1 R2 2 A1
H 1 R2 2 A1 3 D180
R1 = 1.20
R2 = 1.10
A1 = 120
D180 = 180.0
• Why dihedral angle as 180°? Why not 120°?

16. What difference does it make?

17. Why the angle should be between 0 to
180?
• 0 degree angle? That means we are
superimposing one atom over the other.
• Why 180 degree angle is bad?
• Because, if you define 180 degree angle in
your Z-matrix, then defining dihedral angles
will be a problem.

18. Dummy Atom
• Dummy atom (X) is just a point in space and
has no significance in bonding and hence no
significance in the actual calculation. However,
we need dummy atom in the Z-matrix for the
following reasons.
• Case 1: The function of dummy atom is to
break up the problematic 180° angle into two
90° angles.
• Case 2: If there are no real atoms on a
rotational axis or mirror plane, dummy atoms
can be useful for defining the symmetry
element.

19. Think about constructing a Z-matrix for
benzene without Dummy atoms
• I am pretty sure you will realize the importance
of dummy atoms.
• Judicious use of dummy atoms and realizing the
importance of symmetry are very essential in
solving the molecular problems in a computer.
• By the way, it is possible to get the D6h
symmetry without dummy atoms for benzene.
However, during optimization it would fail.
Think about it why it happens?

20. X
C 1 RCC*
C 1 RCC* 2 A60
C 1 RCC* 3 A60 2 D180
C 1 RCC* 4 A60 3 D180
C 1 RCC* 5 A60 4 D180
C 1 RCC* 6 A60 5 D180
H 1 RXH* 2 A60 7 D180
H 1 RXH* 3 A60 2 D180
H 1 RXH* 4 A60 3 D180
H 1 RXH* 5 A60 4 D180
H 1 RXH* 6 A60 5 D180
H 1 RXH* 7 A60 6 D180
RCC = 1.3886
A60 = 60.0000
D180 = 180.0000
RXH = 2.4708
Note: An asterisk symbol (RCC*, RXH*) means that it is a variable and not a
constant. Choosing variables and constants in the right way is the key to
success in getting the desired symmetry.