New method 3: In this method, the effective number of unpaired electrons is defined by 𝑓𝑓(𝑛𝑛) = min(𝑛𝑛, 2 -𝑛𝑛) (12) and 𝜌𝜌 � 𝑎𝑎 (𝐫𝐫), 𝜌𝜌 � 𝑏𝑏 (𝐫𝐫), and 𝐷𝐷(𝐫𝐫) are defined according to eqs 2, 3, and 5 using the newly defined 𝑓𝑓(𝑛𝑛) in eq 12. The definition of 𝐷𝐷(𝐫𝐫) according to eqs 5 and 12 is equivalent to using the unpaired electron density matrix 𝐔𝐔 of Head-Gordon. Equation 12 maintains the desirable property that a natural orbital occupancy of one indicates there is exactly one unpaired electron in the orbital.