I might have gone overboard with this one:

Unfortunately for me, I don't know any undergrad-level science. The only science I know is at the research level, and the NIST database shows hundreds of experimentally observed states. If one of you know what the undergrad chemistry textbooks say about what spin states are allowed, please add the answer for this poor soul who thought they would just get an answer to their homework question.

@NikeDattani If a random person asked me this, I would probably only discuss the spin states that correspond to the lowest-energy electron configurations, e.g.: Cu(I) is ... 3d10 which is always a singlet state (S = 0), and Cu(II) is ... 3d9 which is always a doublet state (S = 1/2). However, I don't think the question is very well-defined, so I can't say that my answer is any "better" than yours.

@orthocresol Are you sure about that? The question asks "How many spin states do these ions have" ... your answer is "one"

"one, no matter what the system is, unless there's some very special case where the triplet and singlet have the same energy"

I wouldn't say they have to have exactly the same energy, but they should at least within some (admittedly arbitrary) threshold where it makes chemical sense to discuss it, e.g. in O2.

I don't think there's any case where a triplet and singlet have the same energy, is there?

@orthocresol O2 famously has a triplet ground state

Not a singlet

So for O2 you would say "two spin states"?

Well, there's also the exact definition of what a spin state is, which I'm not sure about. A triplet state has three levels, so does that count as 1 or 3? In a similar way, oxygen has low-lying 1Σ and 1Δ excited states, so do they count as one or more?

Yea when I posted this question here, it was mainly because I needed help with what the "exact definition of a spin state" is. I have no undergrad level science knowledge, only research level, which would be the NIST reference table that shows several states at various energies.

@orthocresol Actually I wouldn't say "a triplet state has 3 levels", I'd say it's composed of 3 states with the same energy level

Seems also to be the language that Wikipedia uses, you can scroll down to the part where it says "There are three states with total spin angular momentum 1:"

Yes, that's better terminology.

Please forgive me; my own work mainly consists of big magnetic fields where they don't have the same energy. :-)

I just think the pour soul thought they'd get an answer to a homework question and now I confused them a lot more !!!

The homework question's answer is probably: 1 for Cu+ and 2 for Cu2+ (the 2 micro-states of the ground state doublet).

Perhaps, but we may also never really find out what the question setter meant. Anyway, that is not a bad thing. It has happened on Chem.SE so often, this is a prime example: chemistry.stackexchange.com/questions/33645/…

See the original form of the question (before all the edits) too. :-)

So for Cu+, the Singlet had 10 d-electrons:

Here I've changed just the first d-spin-orbital, but why that one? Aren't they all equivalent when there's no ligands?

Okay I think I get it, even in the Wikipedia definition of triplet, they ignore the degeneracy between |01+10> vs |10+01>, so in that sense, the triplet as actually composed of FOUR states not 3 !!

But in Cu+, the Triplet is composed of 7 states >??