Group Theory is a branch of the mathematical field of algebra. One important application, the theory of symmetry groups, is a powerful tool for the prediction of physical properties of molecules and crystals. It is for example possible to determine whether a molecule can have a dipole moment. Many important predictions of spectroscopic experiments (optical, IR or Raman) can be made purely by group theoretical considerations.
1.1: Introduction to SymmetryYou will already be familiar with the concept of symmetry in an everyday sense. If we say something is ‘symmetrical’, we usually mean it has mirror symmetry, or ‘left-right’ symmetry, and would look the same if viewed in a mirror. Symmetry is also very important in chemistry. Some molecules are clearly ‘more symmetrical’ than others, but what consequences does this have, if any? 1.2: Symmetry Operations and Symmetry ElementsA symmetry operation is an action that leaves an object looking the same after it has been carried out. Each symmetry operation has a corresponding symmetry element, which is the axis, plane, line or point with respect to which the symmetry operation is carried out. The symmetry element consists of all the points that stay in the same place when the symmetry operation is performed. for example, in a rotation, the line of points that stay in the same place constitute a symmetry axis. 1.3: Symmetry Classification of Molecules- Point GroupsIt is only possible for certain combinations of symmetry elements to be present in a molecule (or any other object). As a result, we may group together molecules that possess the same symmetry elements and classify molecules according to their symmetry. These groups of symmetry elements are called point groups (due to the fact that there is at least one point in space that remains unchanged no matter which symmetry operation from the group is applied). 1.4: Symmetry and Physical PropertiesCarrying out a symmetry operation on a molecule must not change any of its physical properties. It turns out that this has some interesting consequences, allowing us to predict whether or not a molecule may be chiral or polar on the basis of its point group. 1.5: Combining Symmetry Operations - ‘Group Multiplication’Now we will investigate what happens when we apply two symmetry operations in sequence. As we shall soon see, the order in which the operations are applied is important. 1.6: Constructing higher groups from simpler groupsA group that contains a large number of symmetry elements may often be constructed from simpler groups. 1.7: Mathematical Definition of a GroupA mathematical group is defined as a set of elements together with a rule for forming new combinations within that group. The number of elements is called the order of the group. For our purposes, the elements are the symmetry operations of a molecule and the rule for combining them is the sequential application of symmetry operations investigated in the previous section. The elements of the group and the rule for combining them must satisfy certain criteria. 1.8: Review of MatricesAn n×m matrix is a two dimensional array of numbers with n rows and m columns. This Module addresses basic definitions and operations of matrices are are particularly relavant for symmetry aspects. 1.9: Transformation matricesMatrices can be used to map one set of coordinates or functions onto another set. Matrices used for this purpose are called transformation matrices. In group theory, we can use transformation matrices to carry out the various symmetry operations considered at the beginning of the course. As a simple example, we will investigate the matrices we would use to carry out some of these symmetry operations on a model vector. 1.10: Matrix Representations of GroupsThe symmetry operations in a group may be represented by a set of transformation matrices, one for each symmetry element. Each individual matrix is called a representative of the corresponding symmetry operation, and the complete set of matrices is called a matrix representation of the group. The matrix representatives act on some chosen basis set of functions, and the actual matrices making up a given representation will depend on the basis that has been chosen. 1.11: Properties of Matrix RepresentationsNow that we’ve learnt how to create a matrix representation of a point group within a given basis, we will move on to look at some of the properties that make these representations so powerful in the treatment of molecular symmetry. 1.12: Reduction of Representations IA block diagonal matrix can be written as the direct sum of the matrices that lie along the diagonal. Note that a direct sum is very different from ordinary matrix addition since it produces a matrix of higher dimensionality. 1.13: Irreducible representations and symmetry speciesWhen two one-dimensional irreducible representations are seen to be identical, they have the ‘same symmetry’, transforming in the same way under all of the symmetry operations of the point group and forming bases for the same matrix representation. As such, they are said to belong to the same symmetry species. There are a limited number of ways in which an arbitrary function can transform under the symmetry operations of a group, giving rise to a limited number of symmetry species. 1.14: Character TablesA character table summarizes the behavior of all of the possible irreducible representations of a group under each of the symmetry operations of the group. In many applications of group theory, we only need to know the characters of the representative matrices, rather than the matrices themselves. Luckily, when each basis function transforms as a 1D irreducible representation there is a simple shortcut to determining the characters without having to construct the entire matrix representation. 1.15: Reduction of representations IIThe formation of bonds is dependent on symmetries of the constituent atomic orbitals. To make full use of group theory in the applications we will be considering, we need to develop a little more ‘machinery’. Specifically, given a basis set we need to find out: (1) How to determine the irreducible representations spanned by the basis functions and (2) How to construct linear combinations of the original basis functions that transform as a given irreducible representation/symmetry species. 1.16: Symmetry Adapted Linear Combinations (SALCs)Once we know the irreducible representations spanned by an arbitrary basis set, we can work out the appropriate linear combinations of basis functions that transform the matrix representatives of our original representation into block diagonal form (i.e. the symmetry adapted linear combinations). Each of the SALCs transforms as one of the irreducible representations of the reduced representation. 1.17: Determining whether an Integral can be Non-zeroAs we continue with this course, we will discover that there are many times when we would like to know whether a particular integral is necessarily zero, or whether there is a chance that it may be non-zero. We can often use group theory to differentiate these two cases. 1.18: Bonding in DiatomicsYou will already be familiar with the idea of constructing molecular orbitals from linear combinations of atomic orbitals from previous courses covering bonding in diatomic molecules. It turns out that the rule that determines whether or not two atomic orbitals can bond is that they must belong to the same symmetry species within the point group of the molecule. 1.19: Bonding in Polyatomics- Constructing Molecular Orbitals from SALCsIn the previous section we showed how to use symmetry to determine whether two atomic orbitals can form a chemical bond. How do we carry out the same procedure for a polyatomic molecule, in which many atomic orbitals may combine to form a bond? Any SALCs of the same symmetry could potentially form a bond, so all we need to do to construct a molecular orbital is take a linear combination of all the SALCs of the same symmetry species. 1.20: Calculating Orbital Energies and Expansion CoefficientsCalculation of the orbital energies and expansion coefficients is based on the variation principle, which states that any approximate wavefunction must have a higher energy than the true wavefunction. This follows directly from the fairly common-sense idea that in general any system tries to minimize its energy. If an ‘approximate’ wavefunction had a lower energy than the ‘true’ wavefunction, we would expect the system to try and adopt this ‘approximate’ lower energy state. 1.21: Solving the Secular EquationsAny set of linear equations may be rewritten as a matrix equation Ax = b, which can be classified as simultaneous linear equations or homogeneous linear equations, depending on whether b is non-zero or zero. the set of equations only has a solution if the determinant of A is equal to zero. The secular equations we want to solve are homogeneous equations, and we will use this property of the determinant to determine the molecular orbital energies. 1.22: Summary of the Steps Involved in Constructing Molecular OrbitalsThe eight steps use to construct construct arbitrary molecular orbitals of polyatomic systems. 1.23: A more complicated bonding exampleGroup theory can be used to construct the molecular orbitals of molecules using a basis set consisting of all the valence orbitals. This is demonstated for water and rquuqires consideration of proper representation and characters of matrices and the extracted SALCs. 1.24: Molecular Vibrationsvibrational motions of polyatomic molecules are much more complicated than those in a diatomic. Since changing one bond length in a polyatomic will often affect the length of nearby bonds, we cannot consider the vibrational motion of each bond in isolation; instead we talk of normal modes involving the concerted motion of groups of bonds. Group theory may be used to identify the symmetries of the translational, rotational and vibrational modes of motion of a molecule. 1.25: Summary of applying group theory to molecular motionsA summary of the steps involved in applying group theory to molecular motions is given. 1.26: Group theory and Molecular Electronic StatesA molecular orbital is that it is a ‘one electron wavefunction’, i.e. a solution to the Schrödinger equation for the molecule. An electronic state is defined by the electron configuration of the system, and by the quantum numbers of each electron contributing to that configuration. The symmetry of an electronic state is determiend by taking the direct product of the irreducible representations for all of the electrons involved in that state. 1.27: Spectroscopy- Interaction of Atoms and Molecules with LightWe have already used group theory to learn about the molecular orbitals in a molecule. In this section we will show that it may also be used to predict which electronic states may be accessed by absorption of a photon. We may also use group theory to investigate how light may be used to excite the various vibrational modes of a polyatomic molecule. 1.28: SummaryHopefully this text has given you a reasonable introduction to the qualitative description of molecular symmetry, and also to the way in which it can be used quantitatively within the context of group theory to predict important molecular properties.