I am trying to understand molecular term symbols of oxygen molecule. Everywhere I looked, it is claimed that the molecule has 6 microstates, three of them are singlet, and three of them are triplet. Now I can draw the diagrams for the singlet states with no problem, but I am struggling to find the tree triplet configurations. According to my microstate matrix there should be a triplet state with a total spin multiplicity of zero - how is this possible?
In a $(\pi2p)^2$ configuration the orbital $\lambda$ and spin s quantum numbers have to be tabulated and combinations removed which break the Pauli principle. The total angular momentum quantum number is $\Lambda$ and spin $\Sigma$, for example the combination
$\lambda_1 = 1, \lambda_2 =1 , s_1=1/2, s_2 =-1/2$
generates total angular momentum $\Lambda = 2, \Sigma =0$ and this is part of the $^1\Delta$ state.
There are only 6 valid combinations leading to $^1\Delta, ^3\Sigma^-,^1\Sigma^+ $ states of which $^3\Sigma^-$ is the ground state.
A vector spin diagram shows how a triplet can have spin multiplicity of zero. There is a picture of vector spin for singlets and triplets in my answer to this question; How does spin flipping of triplet carbenes occur?