Symmetry and irreducible representation

[sineontheline]

1. could anyone give sort of a qualititative explanation of how symmetry and irreducible representation are related in the context of molecular spectroscopy? like why is it so useful to count how many symmetries a molecule has and what does it have to do with irreducible represenations and spectroscopy?

2. could anyone point me to a book/website with a lucid discussion of it -- preferably an explanation fit for a physicist (like one rife with examples and explanations not like thm/remark/proof style).
im reading sternberg right now -- its good, but its still confusing.

 

[SpectraCat]

1. Well, the symmetry of a molecule is a fundamental property that determines how it's wavefunction transforms under certain physical operations. An irreducible representation defines a particular set of transformational properties under the various operations of a given symmetry group. So, from a certain point of view, the irreducible representation *is* the symmetry of a wavefunction, for most intents and purposes anyway. Spectroscopists certainly conflate the two routinely during discussions of spectroscopic measurements.

For example, the common "transition dipole selection rule" states that in order for two quantum states to be coupled by a single photon of EM radiation, the product of the initial and final wavefunctions with the dipole operator must transform as the completely symmetric representation of the point group of the system. This is because if this condition is *not* met, then the associated integral defining the "transition dipole moment" will be identically zero:
$$\left\langle\psi_{f}\right|\vec{\mu}\left|\psi_{i}\right\rangle=0$$
The group theoretical treatment of this can be particularly useful, since it often allows one to replace integrals over wavefunctions with simple algebraic expressions.

2) Pretty much any modern textbook presenting quantum mechanics or spectroscopy should have a section that explains this at some level. "Symmetry and Spectroscopy", by Harris and Bertolucci is a classic text, available cheaply from Dover, that explains this really well (IMO) from an experimentalists point of view, with lots of worked problems and examples. "Fundamentals of Molecular Symmetry" is a more recent text by Bunker and Jensen that covers a lot of the same material from a less applied point of view.

 

[Entropia]

I can provide a response to your questions about symmetry and irreducible representation in the context of molecular spectroscopy.

Symmetry is an important concept in molecular spectroscopy because it allows us to simplify the analysis of a molecule's properties. In simple terms, symmetry refers to the ability of an object to be rotated, reflected, or translated without changing its overall appearance. In the case of molecules, symmetries can be found in their shape, electronic structure, and vibrational modes.

The study of symmetry in molecules is closely related to the concept of irreducible representations. An irreducible representation is a mathematical representation of a group of symmetries that cannot be further broken down into smaller components. In other words, it is the most basic representation of a group of symmetries. In molecular spectroscopy, we use irreducible representations to describe the vibrational and electronic states of a molecule.

Counting the number of symmetries a molecule has is useful because it allows us to predict the number of possible electronic and vibrational states that the molecule can have. This information is essential for understanding the spectroscopic properties of a molecule, such as its absorption and emission spectra. Additionally, the symmetries of a molecule can also provide insight into its chemical and physical properties, such as its reactivity and stability.

As for resources, I would recommend the book "Symmetry and Spectroscopy: An Introduction to Vibrational and Electronic Spectroscopy" by Daniel C. Harris. It provides a clear and comprehensive explanation of symmetry and its application in molecular spectroscopy, with plenty of examples and explanations. Other helpful resources include "Molecular Symmetry and Group Theory" by Alan Vincent and the website "Symmetry and Spectroscopy" by Dr. Richard F. W. Bader. I hope this helps in your understanding of this topic.