How Do Term Symbols Reflect Electron Spin States and Degeneracy?

[osskall]

Hi!

I've finally made some effort to understand spectroscopic term symbols ([tex]^{2S+1}L_J[/tex]) and after having thought a few times that I'd understood but then running into trouble again, now I'm fairly convinced that I got it right. But some things bother me though.

Apparently I was locked too much in the idea that e.g. triplet states have to have unpaired electrons. I thought the number of unpaired electrons was reflected in the value of S. Hence I couldn't imagine a triplet microstate (of e.g. the configuration [tex]p^2[/tex]) with [tex]M_S=0[/tex] as one of the states having only paired electrons but had to rather think of it as something virtual/unimaginable in between the all-unpaired-spins-up and all-unpaired-spins-down microstates. But it seems like the number of unpaired electrons for a certain microstate is only related to [tex]M_S[/tex] and that some of the states unpaired electrons actually do belong to a triplet state.
And thus similarly, for e.g. the [tex]^4[/tex]S states of [tex]p^3[/tex], there are, besides the obvious states [tex]\underline{\uparrow~~}~\underline{\uparrow~~}~\underline{\uparrow~~}[/tex] and [tex]\underline{\downarrow~~}~\underline{\downarrow~~}~\underline{\downarrow~~}[/tex] two microstates where one electron has opposite spin to the other two. There are 6 such microstates with [tex]M_L=0[/tex] which all look similar, but, yet, two of them (one with [tex]M_S=1/2[/tex] and one with [tex]M_S=-1/2[/tex]) will have the same energy as the above mentioned states with all spins paired and the other four will have higher energy (belonging to either [tex]^2[/tex]D or [tex]^2[/tex]P where in any case all states only have one unpaired electron). This is somewhat hard to grasp for me. According to this, states that look completely similar are splitted and states that have either (in this case) 3 or 1 unpaired electrons and thus should have different energy are actually degenerated...
Is this correct?

When it comes to deriving the term symbols, there's nothing to it any more, but understanding it seems to be something totally different.

PS: I'm now basing my understanding of term symbols on http://www.chem.ufl.edu/~itl/4412/lectures/ATermSym.html , where the approach also is illustrated graphically for [tex]p^2[/tex] and [tex]p^3[/tex], since this approach no longer leads me to any contradictions except for the one mentioned here.

 

[Zefram]

Hi!

I can understand your confusion and it's great that you have taken the time to really understand spectroscopic term symbols. It can be a tricky concept to grasp, but once you have a solid understanding, it becomes much easier to apply it in your research.

To answer your question, yes, your understanding is correct. The number of unpaired electrons is only related to M_S and not S. As you mentioned, for a microstate with M_S=0, it is possible to have all paired electrons in a triplet state. This is because the value of M_S only refers to the total spin of the electrons, not their individual spins. Therefore, in a triplet state, the total spin is still 1 even if all the electrons are paired.

Similarly, for the ^4S states of p^3, the four microstates with M_L=0 will have the same energy because they all have the same value of M_S (1/2 or -1/2). The other two microstates with M_L=0 and M_S=0 will have higher energy because they belong to ^2D or ^2P states where there is only one unpaired electron.

I think the key to understanding this concept is to remember that the term symbols represent the total angular momentum of the system, which includes both orbital and spin angular momentum. Orbital angular momentum is related to M_L and spin angular momentum is related to M_S. So, the term symbols are not just based on the number of unpaired electrons, but also take into account the arrangement of those electrons in terms of their orbital and spin angular momentum.

I hope this helps clarify your understanding. Keep up the good work in your studies of spectroscopic term symbols!

 

[ravenprp]

Hello!

Your understanding of term symbols and their relationship to degeneracy is correct. The number of unpaired electrons is reflected in the value of M_S, not S. This means that a triplet state, for example, can have unpaired electrons with M_S=0, as you mentioned. The number of unpaired electrons does not necessarily determine the spin state of the atom or molecule. Additionally, as you pointed out, states with different numbers of unpaired electrons can have the same energy, leading to degeneracy.

It is important to note that the term symbols represent the total angular momentum of a system, which includes both orbital angular momentum (L) and spin angular momentum (S). Therefore, even if two states have the same number of unpaired electrons, they can have different term symbols if their total angular momentum is different. This is why some states with different numbers of unpaired electrons can have the same energy, as their total angular momentum is the same.

I understand that understanding term symbols and degeneracy can be challenging, but it seems like you have a good grasp on the concept. Keep practicing and referring to reliable sources, like the one you mentioned, to solidify your understanding. Good luck!