Can we have molecular transitions that change multiplicity?

Hello! Can we have molecular transitions (not restricted to electric dipole) that change the multiplicity of the electronic level i.e. ##2S+1##. For electric dipole that is strictly forbidden. For magnetic dipole, we have a term in the operator of the form ##S\cdot B## and assuming the B is along the z-axis of the lab frame and after a bit of math we get a matrix element for this of the form:

$$<S' \Sigma'|T_q^1(S)|S \Sigma>$$
where I assumed Hund case a, and ##T_q^1(S)## is the spherical tensor representation of the operator S in the molecule frame. However this operator seem to act only on ##\Sigma##, while leaving S unchanged, which would imply that we can't change S by a dipole transition either. Is this the case? How can we change S? Thank you!

Even if the operator "seems to act" only on the projection quantum numbers ##\Sigma##, you still have the all-powerful Wigner-Eckart theorem at your disposal to study matrix elements of the form that you've written. For each spherical ##q##-component of the spin tensor ##\text{T}^{(1)}_{q}(\mathbf{S})## you have
$$ \langle S' \Sigma'| \text{T}^{(1)}_{q}(\mathbf{S})|S \Sigma \rangle = (-1)^{S'-\Sigma'} \begin{pmatrix} S' & 1 & S\\ -\Sigma' & q & \Sigma \end{pmatrix} \langle S'|| \text{T}^{(1)}(\mathbf{S})||S \rangle \rm{,}$$
where the quantity on the rightmost hand-side is the reduced matrix element of the tensor, and due to the symmetry conditions imposed on the Wigner ##3j## symbol (namely, that the quantum numbers in its upper row satisfy the triangle rule for addition of angular momenta) you obtain that, in this case,
$$ \Delta S \equiv S - S' = \pm 1 \rm{,}$$
which shows you that your tensor can connect multiplicity manifolds whose spins differ by one.

As for your first question regarding the possibility of transitions between different electronic terms of different multiplicities - of course such transitions are possible (one example is the oxygen ##A##-band - look it up if you want), but the mechanisms which "allow" for the change in multiplicity in the general case depend on the symmetries of the wave functions of the relevant molecular states between which such transitions can occur.