Molecular Symmetry

As symmetry is fundamental for a plenty of chemical and physical properties of atoms, ions and molecules, this aspect gets more and more attention for anyone who tries to get a deeper understanding for the properties and the reactions of chemical compounds.

E.g., a prerequisite for piezoelectricity is that a crystal displays a certain symmetry. The phenomenon is fundamental for the use of quartz crystals in clocks and the synchronization of electronic devices. On the other hand, a phenomenon like optical activity is associated with the absence of certain symmetry elements.

The correspondence between symmetry and many chemical and physical properties is based on the fact that all wavefunctions - without regard of what the describe mathematically, the distribution of electrons, vibrations of a molecule, nuclear resonance - have to obey some conditions which, in turn, are based on the symmetry of the framework of nuclei.

Discussions on symmetry are of enormous importance in modern chemistry because problems in symmetry can be solved with mathematical methods. They are taken from some branch of linear algebra, the group theory. In contrast to difficult algebraic approaches necessary when establishing important theorems, the tools employed in group theory are simple but surprising fruitful with respect to the results. Considerations of symmetry and solutions based on group theory were important for the proceedings in quantum chemistry and spectroscopy and help nowadays a lot to understand the dynamics of concerted polycentric reactions.

Another useful application of molecular symmetry and the corresponding nomenclature lies in an exact description of structures. Symbols introduced deliver unambigous structural information and are more precise than long texts. For example, the symbol D_4h states that the ion [Ni (CN)₄]^2-

1. is planar
2. that any of the four groups Ni-CN are linear
3. that all angles C-Ni-C are 90°
4. all cyanogroups and all bonds between nickel and carbon are equivalent.

As the use of symmetry symbols spreads more and more in literature, it is indispensable to become familiar with the basics of symmetry, simply to comprehend what is published. We will proceed with a general consideration of this topic and continue with the rules applied in molecular symmetry.

Symmetry and Conservation Laws

•  Consequences of symmetry are the known conservation laws

(a) Homogenity of time → conservation of energy (t®t_o+t does not affect equations in dynamics)
(b) Homogenity of space → conservation of momentum
(c) Isotropy of space → conservation of angular momentum

Symmetry and properties

• Symmetry accounts for properties of matter, e.g. the electric dipol moment of homonuclear diatomic molecules:

Assumption: molecule A-A displays a dipole moment μ parallel to the positive z-axis:




Symmetry conclusion A-A must display the same dipol moment in the opposite direction:




Conclusion: The molecule displays no dipole moment in direction z, i.e. μ_z = 0!

Symmetry in chemistry

• Molecules and solids often display remarkable symmetry
• Symmetry: Objects and laws are not affected by transformation in space.

Example: Geometric symmetry of objects in a three-dimensional space.

1. H₂ molecule

The assemblage of nuclei is invariant respective reflection on plane σ:



2. C₆₀-Fulleren (GIF, VRML, PDB)

The assemblage of nuclei is invariant respective rotation, reflection, inversion and rotatory reflections.



Only four vibrational frequencies have been observed although 3N-6 = 3·60-6 = 174 degrees of freedom exist. Why?

3. Polyacetylen

The assemblage of nuclei is invariant respective translation.



Therefore, the electronic wavefunctions are so-called Bloch functions: Ψ_n = e^ikx u_n
e^ikx is a planar wave and  u_n(x) = u_n(x+t) is a function periodic in space

Synopsis

Symmetry reduces the effort in describing molecules and, in turn, simplifies mathematical treatment of processes. Therefore, group theory is indispensible in theoretical chemistry and became a celebrated tool as it provided approaches to a list of problems

• The symmetry of atomic orbitals determines the symmetry and the character of chemical bonding.
• The symmetry of molecular orbitals can be described by MULLIKEN symbols. An investigation of symmetry allows to decide whether the electronic state of a molecule changes under some kind of excitation, e.g. to predict without calculations symmetry-allowed or symmetry-forbidden transitions in UV spectroscopy.
• The symmetry of orbitals of an electronic configuration allows derivation of the term symbol for a system with a number of electrons.
• It is possible to derive the orientation of the dipol moment by considering the symmetry of a molecule.
• If a chemical reaction occurs and the symmetry elements remain unaffected while the reaction coordinate proceeds, orbitals of the educt transform to orbitals of the product of the same symmetry. (Woodward Hoffmann rule).
• In vibrational spectroscopy, normal modes can be characterized by their symmetry (irreducible representations).

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