Three-coupled angular momenta, quantum

A term scheme is a representation of an energy level in an isolated many-electron atom, derived via the Russell-Saunders coupling scheme. In general, a term scheme is written as a collection of many-electron quantum numbers S and L. The value of S is not used directly but is replaced by the spin multiplicity, 2S +1. Similarly, the total angular momentum quantum number, L, is replaced by a letter symbol similar to that used for the single-electron quantum number 1. The term scheme is written States with a multiplicity of 1 are called singlet states, states with a multiplicity of 2 are called doublet states, those with a multiplicity of three are called triplets, those with a multiplicity 4 are called quartets and so on. Hence, S is called singlet S, and is called triplet P. 

What are the values of the total angular momentum quantum number / for a problem with three coupled sources of angular momentum, = 2, ]2 = 1, and... 

The multiplicity associated with a given angular momentum is the number of different possible projections on the z-axis. This is always equal to 1 greater than twice the angular momentum quantum number that is, multiplicity (/) = 2/ + 1. Spin multiplicity is equal to 2S + 1, and the names singlet, doublet, triplet, quartet, and so on, are attached to states with spin multiplicities of 1, 2,3, and 4, respectively. From the spin coupling just carried out, we can see that two inequivalent electrons can spin-couple to produce either a singlet state (i.e., S = 0) or a triplet state (i.e., S = 1). Three inequivalent electrons can be coupled to produce a quartet state (S = 3/2) or two different doublet states (S = 1/2). 

For a diatom-diatom complex, there are three sources of angular momentum the rotation of each of the monomers, characterised by quantum numbers ji and j2, and the end-over-end rotation of the complex as a whole. There are at least two ways of formulating the diatom-diatom problem in body-fixed coordinates. In the one which displays most clearly the similarity to atom-diatom systems, ji and j 2 are first coupled together to form a resultant h... 

---