Recall: ZX=iY, ZY=−iX: Beta part = 1 4(X3X2 +Y 3Y2) +i(X 3Y2 −Y 3X2)) 4 Alpha Part (a † 1a0):From the single excitation derivation: Alpha part = 1 4(X1 −iY 1)(X0 +iY 0)

Group by spin species:Beta Part (a † 3a2): a† 3a2 = 1 4(X3 −iY 3)Z2Z1Z0 ·(X 2 +iY 2)Z1Z0 TheZ 1Z0 terms appear twice (Z 1Z0 ·Z 1Z0 =I), so they cancel out: a† 3a2 = 1 4(X3 −iY 3)Z2