In quantum chemistry application, s i stands for the four possibilities of the projection of the spin quantum number of the two electrons that can occupy a single orbital, thus s i = | 00 ⟩ ⟩ , | 10 ⟩ ⟩ , | 01 ⟩ ⟩ , | 11 ⟩ ⟩ , where the first (second) entry of these kets corresponds to the spin-up(down) electron. In quantum chemistry, A s 1 (for a given s i ) and A s N (for a given s N ) are traditionally chosen to be row and column matrices, respectively. This way, the result of A s 1 … … A s N is a scalar value and the trace operation is unnecessary. N is the number of sites (the orbitals basically) used in the simulation.