Point Groups in Chemistry

Introduction

The concept of symmetry has important implications in chemistry. The symmetry of a molecule can have consequences on the appearance of the molecule's spectra, the relative chemical reactivity of groups, physical properties such as dipole moments, and many other aspects of chemistry including the way we make use of molecular orbitals and their interactions.

The five basic symmetry operators applicable to molecules are: the identity operator (E), rotation axes (C_n), reflection planes (σ), improper axis of rotation (rotation-reflection axes)(S_n), and a center of inversion (i). (Review?)

Definition of a Group

A group, G = (..., g_i, ...), consists of a number of elements related by an operation called a group multiplication, and has the following properties:

1. The product of any two elements (operators) is in the set (i.e. the group is closed under group multiplication).
2. The associative law holds, i.e. g_i (g_k*g_l) = (g_i *g_l)g_k (i.e. the order in which the operators is applied does not matter).
3. There is a unit element (the identity operator) such that: Eg_i = g_iE = g_i.
4. There is an inverse to each element g_i^-1 such that g_i * g_i^-1 = g_i^-1 * g_i = E. (i.e. for a C₃ axis the inverse of a 120° rotation is a rotation by -240°).

Molecular Point Groups

A molecular point group is a set of symmetry operators, which when carried out on a molecule leaves it unchanged.

Schoenflies Notation:

E = identity operator.
C_n = rotation about an axis by 360°/n. The principal axis is the axis of highest n.
σ_h = reflection in a horizontal plane that lies perpendicular to the principal axis.
σ_v = reflection in a vertical plane that contains the principal axis.
σ_d = reflection is a diagonal plane that bisects the angle between two C₂ axes.
S_n = improper rotation, i.e. C_n followed by a σ_h.
i = inversion through the center of mass.

Interrelation of Symmetry Operators

There are a number of important relationships between symmetry operators. These are:

1. The intersection of two mirror planes must be a symmetry axis.
2. If the reflection plane contains an n-fold axis, there must be n-1 other reflection planes at angles of 180°/n.
3. Two C₂ axes separated by 90° require a perpendicular n-fold axis.
4. A C₂ axis and an n-fold axis require n-1 additional C₂ axes separated by 90°.
5. An even-fold axis, a reflection plane perpendicular to it, and an inversion center are interdependent. Any two implies the existence of the third.
6. An inversion center, if it exists, must occur at the point where all other symmetry operators meet.

Determining Molecular Point Groups

The following image map shows how to determine the molecular point group.

There are a number of ways to subdivide the groups, one is:

1. Type 1: No rotation axis; point groups C₁, C_s, C_i.
2. Type 2: Only one axis of rotation; point groups C_n, S_n, C_nv, C_nh.
3. Type 3: One n-fold axis and n 2-fold axes; point groups D_n, D_nh, D_nd
4. Type 4: More than one axis higher than C2; points groups T_d, O_h, I_h, K_h.

Stereotopic Relationships

When comparing different molecules three possible relationships exist:

1. The molecules are identical. They can not be distinguished under any conditions, chiral or achiral.
2. Enantiomers, which are pairs of molecules which are related as non-superimpossible mirror images. They can only be distinguished under chiral conditions. Only molecules whose point groups are C_n (n > 1), D_n (n > 1), T, O or I are chiral and can exist in enantiomeric forms.
3. Constitutional isomers and diastereomers differ in all scalar properties and are distinguishable under any conditions, chiral or achiral. Geometric isomers, which are related by the position of groups about a double bond, are a special case of diastereomers.

Molecules are chiral if their point groups do not contain S_n (n > 1) operators, otherwise they are achiral.

Finally there are a number of ways to describe the relationship between atoms (or groups) within a molecule. This is extremely important in NMR spectroscopy. Identical atoms produce the same NMR signal, and do not split one another.

1. Homotopic groups: are not distinguishable under any conditions, chirl or achiral. In order to have homotopic groups the molecule must have a finite axis of rotation. The only molecules which can not have homotopic groups are those whose point groups are C₁, C_s, C_i, C_v. In general homotopic groups are related by the rotation axis.
2. Enantiotopic groups: same in all scalar properties and only distinguishable under chiral conditions. Replacement of these atoms (or groups) will produce enantiomers. In general enantiotopic groups are related only by mirror planes.
3. Constitutionally heterotopic and diastereotopic groups: distinguishable under all conditions as they differ in all properties.

These relationships also have serious consequences in terms of chemical synthesis. Homotopic groups will react the same in all chemical reactions and produce the same product. Enantiotopic groups will react at the same rate with non-chiral reagents, but different rates with chiral reagents. They will produce pairs of enantiomers. Finally, constitutionally heterotopic and diastereotopic groups will react differently under any conditions.