note for a symmetric (in I ↔ J) spin-difference matrix S IJσ this is double counting . This algorithm scales as O ∼ N 2 , with N being the number of CI states (which in-turn scales exponentially). In principle, the sparsity of the CI algorithm scales suggests a more optimum approach. For an example, let's consider we have a system of 3 spin-orbitals (SO) per spin-type, and 2 ↑electrons and 1 ↓-electron system. This system has 9 possible FCI states (let's enumerate them 1 thru 9). This system has the following connectivity given in figures 3 and 4.