In new methods 3 and 4, we use the same method as the original MC-DCFT to convert 𝜌𝜌(𝐫𝐫) and 𝐷𝐷(𝐫𝐫) into 𝜌𝜌 � 𝑎𝑎 (𝐫𝐫) and 𝜌𝜌 � 𝑏𝑏 (𝐫𝐫), but we use alternative definitions of 𝐷𝐷(𝐫𝐫) by taking advantage of the fact that eq 6 for the effective number of unpaired electrons is not unique. The effective number of unpaired electrons can be defined in various ways; by changing the definition of 𝑓𝑓(𝑛𝑛), we are able to construct alternative definitions of 𝐷𝐷(𝐫𝐫).