## Introduction to Symmetry and Point Groups

Symmetry is especially important in carrying out molecular orbital calculations.

• To be a valid representation of the electronic structure of a molecule, the square of the wavefunction, Y2, must have the same symmetry properties as the molecule itself.

• In order for this to be true, the individual wave functions, Y, represented as linear combinations of atomic orbitals, must be either symmetric or antisymmetric with respect to the symmetry elements possess by the molecule.

Although SPARTAN and GAUSSIAN generally take care of the detailed symmetry manipulations for us, it is important that we

• Input a structure that has an appropriate symmetry

• Recognize what elements constitute that symmetry

• Know how to alter a structure to achieve or break a particular symmetry.

Useful references are:

• "Chemical Applications of Group Theory", 3rd ed., F. A. Cotton, Wiley, 1990

• "Structure and Symmetry (Readable Group Theory for Chemists)", 2nd ed., by S. F. A. Kettle, John Wiley and Sons, 1995 ( which approaches group theory from molecular structure rather than the other way around)

• "An Introduction to Molecular Orbitals", by Y. Jean and F. Volatron (translated by J. Burdett), Oxford University Press, 1993, describes the qualitative construction of molecular orbitals in accord with symmetry principles. Here is an example of such a construction.

Symmetry elements are geometric entities that are used to minipulate molecules so as to transform them from one spatial orientation into another, indistinguishable, orientation. For the analysis of the large majority of organic molecules, only two symmetry elements are needed:

• the mirror or symmetry, plane, s

• the axis of rotation (proper axis), Cn

These elements can be defined as follows:

• A mirror plane divides a molecule into mirror image halves

• More formally, a plane passing through a molecule is a symmetry plane if changing the signs of the atomic coordinates in a direction perpendicular to the plane produces an indistinguishable orientation

For example, chloroform has three symmetry planes, defined by the C-H bond and each C-Cl bond, and bisecting the opposite Cl-C-Cl angle.

Mirror planes are further classified as vertical (subscript 'v') or horizontal (subscript 'h') according to whether they contain the principal rotational axis of the molecule or are perpendicular to it.

• An axis is a proper axis of order n if a rotation of 360/n degrees produces an indistinguishable orientation of the molecule.

As shown in the Figure, the C-H bond of chloroform is colinear with a C3 axis: each rotation of 360/3 = 120 degrees leaves the molecule in an identical arrangement.

• All of the s planes contain this axis, hence they are all sv planes.

The H-H molecule has a Cn of order infinity congruent with the H-H bond, and

• an infinite number of sv containing that axis.

• One additional s perpendicular to the Cn, midway between the nuclei.

• It is a sh.

Molecules may contain two other symmetry elements, which rarely need be specified.

• A center of inversion, i, such that if the origin of the Cartesian system is placed at this point, and the signs of all Cartesians are then changed, an indistinguishable orientation is produced.

• The i typically lies at the intersection of all of the Cn and s, and hence is redundant in most organic molecules. This may not be so in some inorganic complexes.

• An improper axis, or reflection-rotation axis, Sn, likewise is redundant in any molecule that possesses other rotational symmetry

The point group or symmetry group is the name given to the collection of symmetry elements possessed by a molecule. Each common collection of elements is represented by a simple symbol, called the Schonflies notation, indicating the type of reflection symmetry and the order of the principal rotational axis.

The following decision tree, based on a chart on p. 56 of Cotton, facilitates assignment of molecules to their point groups.

To use this chart, one simply answers "yes" or "no" to the questions posed in the boxes and follows the appropriate arrows to the point group designation. Let's make use of some examples.

For example (a), the answer to the first question, "Cn only?" is yes; the molecule has no reflection symmetry. The answer to the question, "Is n > 1?" is no. The molecule has only the trivial C1 axis possessed by all molecules, and hence belongs to point group C1.

• The answer to "Sigma only?" is also "no"; the red dotted line marks a C2 axis.

• Can two C2 axes be found perpendicular to this one? No.

• Does a symmetry plane exist perpendicular to the C2 axis? No.

• Are two sv planes present? Yes - the plane of the paper and the perpendicular plane containing the C2 axis.

• Point group C2v.

Allene, example (c), yields "no" answers to the first two questions. The red dotted line marks a C2 and the plane of the virtual paper is a s.

• Are two C2 axes present perpendicular to the first one? Yes. Indicated by the blue dotted lines, they lie halfway in between the planes of the two end carbons.

• Is a perpendicular s present? No.

• A total of two sv? Yes. The previously identified plane of the paper and the perpendicular plane containing the red C2 axis.

• Point group Dnd.

Example (d), benzene, likewise yields "no" answers to the first two questions. The C6 axis marked in red is evident, and the plane of the molecule, at least, is a symmetry plane.

• Can six C2 axes be found? Yes.

• These divide into two sets. One example is shown of each: passing through opposite corners of the ring (blue) and passing through the middle of opposite sides (green).

• Is a s present perpendicular to the C6? Yes, the molecular plane.
• Point group D6h.

• The six vertical s, each containing one of the C2 axes, need not be located to specify the point group.

Example (e) illustrates a molecule having only reflection symmetry; answering "no" to the first question and "yes" to the second places it in point group Cs. (The mirror plane bisects the molecule perpendicular to the ring plane, and contains the Cl-C-H plane.)

The planar anti conformation of hydrogen peroxide in (f) has a C2 perpendicular to the virtual paper (red dot) and the plane of the paper is a s. One answers "no", "no", "no", and "yes" to arrive at C2h for the point group.

• The intersection of two reflection planes must be a symmetry axis. If the angle between the planes is p/n, the axis is n-fold.

• If a reflection plane contains an n-fold axis, there must be n - 1 other planes at angles of p/n

• Two two-fold axes separated by an angle p/n require a perpendicular n-fold axis

• A two-fold axis and an n-fold axis perpendicular to it require n - 1 additional two-fold axes separated by angles of p/n

• An even-fold axis, a reflection plane perpendicular to it, and an inversion center are interdependent; any two of these imply the existence of the third

Please refer to either of the references cited at the beginning of this page for a more complete analysis of symmetry properties and their relationship to constructing a valid set of molecular orbitals. (Orbital, rather than molecular, symmetries are usually represented with a different set of symbols, devised by Mulliken.)

Note that the Cs symmetry of the cyclopropane derivative above is the symmetry of Blake's tiger:

"Tiger, tiger, burning bright
In the forests of the night,
What immortal hand or eye
Shaped thy fearful symmetry?"

..... William Blake

It also is the symmetry (approximately) possesed by most of us critters who go about on the surface of the earth. In his book, "Climbing Mount Improbable", Richard Dawkins speculates on how this particular symmetry came to evolve. He argues that differentiating front from back follows from separating ingestion and excretory functions, and gravity provides the force to make top different from bottom. But no comparable mechanisms exist to differentiate left from right; hence, bilateral symmetry. Indeed, he suggests, a mutation that caused such a distinction would be harmful, since it might well cause the afflicted critter to move in circles!