What is the maximum value for S when considering two electrons?

In the context of quantum mechanics and electron configurations, the maximum value of the total spin quantum number ( S ) for a system of two electrons can be evaluated based on their intrinsic spin, which is ( \frac{1}{2} ) for each electron.

For two electrons, the spins can combine in different ways. They can either be in a singlet state or a triplet state. The singlet state corresponds to a total spin of ( S = 0 ), where the spins are paired oppositely (one spin-up and one spin-down), resulting in cancellation of their spin contributions. In contrast, the triplet state corresponds to a total spin of ( S = 1 ), which occurs when the spins are aligned (both spins up or one spin up and one spin down but in a way that maintains a combined total spin).

However, when considering the term maximum value of ( S ) for a set of two electrons in a system, the possible values derived from their individual spins are limited by the maximum combined alignments. Specifically, the total spin quantum number ( S ) can take values from ( |S_1 + S_2| ) to ( |S_1 - S_