How do we determine the values for S?

The correct answer describes the process of determining the total spin quantum number (S) for a system of electrons in an atom or ion. In quantum mechanics, especially in the context of electron configuration, S represents the total spin of the system, which is derived from the individual spins of the electrons.

Each electron has a spin quantum number of either +1/2 or -1/2, and when you have multiple electrons, the total spin quantum number S reflects all their contributions. The values of S can be obtained by considering the combinations of these spins. For a two-electron system, for example, if both electrons have spins of +1/2, the total spin S can be 1 (triplet state). If one electron has spin +1/2 and the other has spin -1/2, the total spin S becomes 0 (singlet state). Thus, the possible values for S indeed take the form of s1+s2, s1+s2-1, and so on, reflecting all possible combinations of the spins from the individual electrons.

This interpretation of S emphasizes the combinatorial nature of electron spins and is foundational for understanding more complex systems in quantum mechanics and their magnetic properties. The accurate calculation is crucial