As will be utilized in the 3 and 4 electron cases, the subscript i will label spin functions sharing a fixed S, M S , and N . When all possible spin-coupling combinations are constructed among the N electrons, a linear combination of such states can recapture the exact spin-pure energy of the system. For any 2-electron system in a minimal basis, the exact singlet spin function with S z = 0 can be written in terms of geminals as: