Spin Density
 
For open-shell situations or for some closed-shell UHF situations non zero spin densities exist and can be calculated.
 
The definition of a closed-shell situation has subtleties, but a simplistic definition of it might be that the probability of finding a spin-up electron is  
identical to the probability of finding a spin-down electron, everywhere in space. Thus, in contrast to the total electron density defined as the sum of alpha and beta densities above, it is convenient to define the spin density as the difference between the alpha spin density and the beta spin density.
 
The spin density defines the excess probability of finding spin-up over spin-down electrons at a point in space and is zero everywhere for closed-shell RHF situations.