33 votes
Accepted

Understanding group theory easily and quickly

The figures below show you how to navigate your way round point group tables. The irreducible representations (irreps) are shown as the row of characters. A reducible representation is a collection of ...
porphyrin's user avatar
  • 30.3k
29 votes
Accepted

What is the mathematical basis behind the Jahn-Teller effect?

1. Perturbations As already mentioned, the Jahn–Teller effect has its roots in group theory. The essence of the argument is that the energy of the compound is stabilised upon distortion to a lower-...
orthocresol's user avatar
  • 70.9k
27 votes
Accepted

Why is the letter J omitted in the spdf... sequence?

Omitting j when alphabetically enumerating things has a long tradition. First of all, the alphabet did not always exist in the form we know it today. Quoting Wikipedia: After [...] the 1st century ...
mhchem's user avatar
  • 3,316
23 votes

Why is the letter J omitted in the spdf... sequence?

For the azimuthal quantum number (l) of an atom, there is no "j" because some languages do not distinguish between the letters "i" and "j". L is the total orbital quantum number in spectroscopic ...
MaxW's user avatar
  • 22.3k
21 votes

Why do we need the identity operator, E

It is not that we need the identity operator. It is just that things (character tables, irreducible representations, etc.) work the way they do. As to why they do so, I refer you to any decent ...
Ivan Neretin's user avatar
  • 31.1k
20 votes
Accepted

Why do we need the identity operator, E

What you are asking arises as a fundamental consequence of the definition of vector spaces and the operations defined for them and, specifically, symmetry groups. The identity operator does not do "...
Todd Minehardt's user avatar
18 votes
Accepted

Group theoretical condition for an integral to be zero

By Maschke's Theorem, every direct product of representations is decomposable into a direct sum of representations, that is, the function you are integrating can be rewritten as a sum of functions ...
levineds's user avatar
  • 3,070
16 votes

How do I determine the molecular vibrations of linear molecules?

As Tyberius noted, the projection formula does not work for infinite order groups (this is because the Hermitian form on characters is defined to be G-invariant by averaging over all elements in a ...
levineds's user avatar
  • 3,070
15 votes
Accepted

What are point groups and the benefits of using them?

Point groups are a very valuable tool for analysing molecules without knowing much about them. They rely on analysing a molecule’s symmetry and deducing both physical parameters and the shape of ...
Jan's user avatar
  • 67.8k
15 votes
Accepted

Using group theory to make molecular orbital diagrams

Group theory really just formalises the process that you're going through when you construct molecular orbital diagrams by inspection (though as the molecules get bigger it gets significantly harder ...
NotEvans.'s user avatar
  • 17.1k
15 votes
Accepted

What is it that is "symmetry-adapted" about symmetry-adapted perturbation theory?

The "symmetry" in symmetry-adapted perturbation theory refers to the anti-symmetry of the wave function with respect to electron exchange: $$\Psi(\mathbf{r}_1,\mathbf{r}_2) = -\Psi(\mathbf{r}_2,\...
Jan Jensen's user avatar
  • 4,809
14 votes
Accepted

How do I show that a transition is magnetic dipole allowed with group theory/symmetry?

For a state-to-state transition coupled by an operator to be allowed, the direct product of the irreducible representations (irreps) of all three components must contain the completely symmetric irrep ...
pentavalentcarbon's user avatar
13 votes
Accepted

Molecular orbital diagram and irreducible representations for dinitrogen

Because this is a diatomic molecule, there are no group orbitals. Put another way, the group orbitals are the molecular orbitals. Knowing the nitrogen atomic orbitals (AOs) and their irreducible ...
pentavalentcarbon's user avatar
12 votes
Accepted

Why are there only two irreducible representations in the C3 group?

You got everything right. It is just that two of these representations would use complex numbers ($\varepsilon=e^{2\pi i\over3}$ and $\varepsilon^2$), and we don't like that, so we stick them together ...
Ivan Neretin's user avatar
  • 31.1k
9 votes

Why do we need the identity operator, E

Symmetry operator $E$ is the lowest symmetry a molecule may possess; and indeed all objects possess this operator. Yet, among molecules revealing $E$ as the only the operation are the ones exhibiting ...
Buttonwood's user avatar
  • 29.1k
9 votes

Why do we need the identity operator, E

Well, according to this Math StackExchange post, you need the identity element because it is impossible to define an inverse element a if you don't know about the identity element e. The definition of ...
JSCoder says Reinstate Monica's user avatar
9 votes
Accepted

Why not just use the x,y,z as basis functions instead of linear/quadratic functions of these unit vectors?

In a single word, the answer is d-orbitals, as you guessed! It's very frequently useful to determine how specific orbitals on an atom transform, because this in turn determines which orbitals it can ...
orthocresol's user avatar
  • 70.9k
8 votes
Accepted

Identifying the C3, C4, S4, and S6 symmetry operations in the Oh point group

At first I thought the character table was counting both positive and negative rotations, but then the order for the $C_2'$ and $C_2$ rotations should be doubled if that were the case ($12C_2'$ and $...
Jan's user avatar
  • 67.8k
8 votes
Accepted

Notation for excited states

As has already been mentioned in comments, those letters and numbers for each state correspond to their symmetry, specifically the symmetry of the (purely electronic) wavefunction. For more ...
pentavalentcarbon's user avatar
8 votes
Accepted

Example of an achiral molecule without a plane of symmetry or inversion center?

Part 1 Are there any molecules lacking plane of symmetry/center of inversion but that are achiral due to presence of an axis of improper rotation? Yes, although such examples are very rare. In ...
orthocresol's user avatar
  • 70.9k
7 votes

How do I determine the molecular vibrations of linear molecules?

Instead of trying to use symmetry tables it is possible to calculate the vibrational normal modes directly using the equations of motion, rather as would be done for a double pendulum for example. It ...
porphyrin's user avatar
  • 30.3k
7 votes

Why is the letter J omitted in the spdf... sequence?

Distinguishing between i and j is, as others have mentioned, a rather recent phenomenon, not unlike the distinction between u and v. Especially in German when typesetting in blackletter, the glyphs ...
Jan's user avatar
  • 67.8k
7 votes

How can the ground state electron configuration for a d3 complex correspond to an F term symbol?

Free ion The ground-state term symbol is only $\mathrm{^4F}$ in the case of a free ion. If you take a closer look at the Tanabe-Sugano diagram, the $\mathrm{^4F}$ term only appears at the far left-...
orthocresol's user avatar
  • 70.9k
7 votes

Group theoretical condition for an integral to be zero

EDIT: Made this more rigorous. (I think. Feel free to critique). First, let's decompose $\psi$ into components that correspond to how its representation reduces. That is, if $$\Gamma_{\psi} = \...
Zhe's user avatar
  • 17.4k
7 votes

Group theoretical condition for an integral to be zero

I consulted my coworker who has a Ph.D. in algebraic topology. The guidance I received was pretty helpful, so here's the new way I understand the problem. There are a few steps that I might not be ...
Zhe's user avatar
  • 17.4k
7 votes
Accepted

Is there a possible distortion of XeF6 from Oh point group to reduced symmetry?

Now, there are a few things that must be stated very clearly. Firstly, it pays to be very careful when you are talking about point groups. It's careless (and confusing to others) to write $\mathrm{T_{...
orthocresol's user avatar
  • 70.9k
7 votes

Irreducible representations and system states connection

The symmetry of a state is the MO (molecular orbital) product of its occupied molecular orbitals. Let's take the example of water molecule. We can see the symmetry label of each of the MO in the ...
tobiuchiha's user avatar
7 votes

Grouping of symmetry classes in chemistry character tables

The $\sigma_\mathrm v$ mirror planes in the $C_\mathrm{3v}$ point group are themselves related by symmetry: note that they can be interchanged via rotation by 120 degrees about the preexisting $C_3$ ...
orthocresol's user avatar
  • 70.9k
6 votes
Accepted

Molecular term symbols for excited state oxygen

What are the possible term symbols for excited state oxygen with configuration $(1\pi_u)^3(1\pi^*_g)^3$ (all other occupied orbitals are closed shell)? Based on the possible values of $S$ and $\lambda$...
Ondrej Gutten's user avatar
6 votes

How do I determine the molecular vibrations of linear molecules?

I actually didn't know how to do this either, but I've found how to work with these infinite symmetry point groups. You have noticed that the reduction formula would not work properly for these groups....
Tyberius's user avatar
  • 11.7k

Only top scored, non community-wiki answers of a minimum length are eligible