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.