Relevance and Determination of the reflection symmetry +/- in Spectroscopy?

If there are two electrons located in a shell (which contains two subshells) the molecule can either have a singlet state (if both electron are in the same subshell; because of the Pauli principle) or a triplett (if the two electrons are in the two different subshells).


The sign + or - means symmetric or antisymmetric with respect to reflection at a mirror plane that contains the molecular axis (for $\Sigma$ states only) of a diatomics.


Let's look how to determine if a state is of + or - symmetry:



> 
> $\mathbf{For~oxygen~in~the~ground~state:}$
> 
> 
> $\Psi\_{\mathrm{a}}^{\mathrm{R}}(q\_1,1\_2) = \sqrt{2}i\chi (z\_1,\rho\_1)\chi(z\_2,\rho\_2)\sin(\varphi\_1-\varphi\_2) \qquad \mathrm{for~the}~^3\sum~\mathrm{term}$ 
> 
> 
> and
> 
> 
> $\Psi\_{\mathrm{s}}^{\mathrm{R}}(q\_1,1\_2) = \sqrt{2}\chi (z\_1,\rho\_1)\chi(z\_2,\rho\_2)\cos(\varphi\_1-\varphi\_2) \qquad \mathrm{for~the}~^1\sum~\mathrm{term}$ 
> 
> 
> 


([image](https://i.stack.imgur.com/pwhCu.png))


It's easy to determine if a state is of +/- symmetry if we know the mathematical description of the radial wave function $\Psi^R (q\_1 ,q\_2)$ (with the two coordinates $q\_i$ of the two electrons), because exchanging $\varphi\_2$ with $\varphi\_1$ in $cos(\varphi\_1-\varphi\_2)$ part stays the same and therefore + and for the $sin(\varphi\_1-\varphi\_2)$ (function the sign changes and therefore we get - symmetry.


**What is my question now?** 


I've some troubles to determine the +/- symmetry of a state (of a diatomics for light atoms and therefore only for s and p orbitals) if I have no mathematical description or tabular data given.


I guess it should be possible to directly derive the +/- symmetry by knowing the electronic configuration.


1. **A specific example of oxygen**.


I read that in the two electrons in a subshell (that means the same molecular orbital shell and the same spatial orbital) of either $\lambda$ = +1 or -1 and therefore a singlet gives a state with + symmetry.


I'm wondering now: Has a singlet state always + symmetry and a triplet state always - symmetry? I'mn interested to understand why this is the case. I guess this has something to do with spatial and spin part.


I don't know if I'm on the right track, but I think for the ground state two electrons in the same spatial orbital (=same subshell) must have opposite spins (because of Pauli principle) and I guess this means having antisymmetric spin-part. Because the product of spin and spatial part must be antisymmetric I expect the spatial part to be symmetric, what is labeled with g(from german gerade). But I guess this has nothing to do with +/-.


For the first excited state of oxygen [electronic configuration ...$(1\Pi\_u)^3(1\Pi^\*\_g)^3$] there exists six states: $^1{\Sigma} ^{-}\_u$,$^1{\Sigma} ^{+}\_u$,$^3{\Sigma} ^{-}\_u$,$^3{\Sigma} ^{+}\_u$,$^1{\Delta} \_u$,$^3{\Delta} \_u$.


This is also one case where my rule singlet gives + symmetry not work. I'm now thinking in which case we have $^1{\Sigma} ^{-}\_u$ and in which case we have $^1{\Sigma} ^{+}\_u$. The two electrons have opposite spin. One electron has $\lambda\_1=+1$ and the other has $\lambda\_2=-1$. I guess that the sign +/- depends if the electron with $\lambda\_1=+1$ is in a $(1\Pi\_u)$ or in a $(1\Pi^\*\_g)$ orbital.


2. **Symmetry properties from a table.**


Instead of determining the + or - symmetry we can use tables which shows the possible terms for a specific electronic configuration:



> 
> $\pi\_g \otimes \pi\_g = \sigma\_g^- \oplus \sigma\_g^+ \oplus \delta\_g$
> 
> 
> 


([image](https://i.stack.imgur.com/Kq94P.png))


This relation shows that two electrons both in $\pi$ orbitals results in these three different states. Sometimes this terms are written in upper case letters, sometimes they are written in lower case letters. This seems to be arbitrarily choosen.


3. **Practical relevance of the +/- symmetry in Spectroscopy.**


In Spectroscopy there are two important selection rules: The Parity selection rule and the Angular momentum selection rule.


I read that a transition between two states must undergo a change of the parity from + to - and vise versa. This is the Parity selection rule. The +/- is the basis for the parity selection rule.


Therefore I would expect that the g/u symmetry (symmetric/antisymmetric with respect to an inversion on the centre-of-mass) is the basis of the Angular Momentum selection rule. 


Is this true?


