+ and ŜG -are the spin ladder operators of the corresponding fragments that change the m F s or m G s quantum numbers of the corresponding site states by ±1 with the corresponding prefactors, e.g., s F (s F + 1) -m F s (m F s + 1)ℏ. The result of acting with Ŝ2 onto |Φ⟩ can then be obtained from standard angular-momentum algebra and reads