practical implications of group theory in chemistry
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Practical Implications of Group Theory
Dr. Venkatesan S. Thimmakondu, Department of Chemistry
BITS-Pilani, K K Birla Goa Campus
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.
Coordinates
• Cartesian Coordinates• Internal Coordinates• Or via a graphical interface (molden, molekel,
Avagadro, Jmol, Gaussview, etc.,)
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 ---------------------------------------------------------------------------
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.
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.
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
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.
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?
Z-Matrix of Water (in Cs symmetry)
OH 1 R1H 1 R2 2 A1
R1 = 0.988984834251219R2 = 0.988884834251219A1 = 105.170884348412642
• Even if there is a difference on the 4th decimal place, it matters!
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.
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.
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.
Point group?
C:CCCC
pentatetraenylidene
H
H
CC
C:CC
ethynylcyclopropenylidene
H
H
:CC
CC
C
ethynylpropadienylidene
H
H
C
H
H
C C:
3-(didehydrovinylidene)cyclopropene
H
H H H
H
H
ortho-tetradehydrobenzene meta-tetradehydrobenzene para-tetradebydrobenzene
Can you write a Z-Matrix for H2CO
CO 1 R1H 1 R2 2 A1H 1 R2 2 A1 3 D180
R1 = 1.20R2 = 1.10A1 = 120D180 = 180.0
• Why dihedral angle as 180°? Why not 120°?
What difference does it make?
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.
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.
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?
XC 1 RCC*C 1 RCC* 2 A60C 1 RCC* 3 A60 2 D180C 1 RCC* 4 A60 3 D180C 1 RCC* 5 A60 4 D180C 1 RCC* 6 A60 5 D180H 1 RXH* 2 A60 7 D180H 1 RXH* 3 A60 2 D180H 1 RXH* 4 A60 3 D180H 1 RXH* 5 A60 4 D180H 1 RXH* 6 A60 5 D180H 1 RXH* 7 A60 6 D180
RCC = 1.3886A60 = 60.0000D180 = 180.0000RXH = 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.
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