Magnetization in Hartree-Fock free-electron gas

From these definitions, and from the fact that AT = A t + A i, we find 

Notice that in the simple approximation we are considering, the free-electron gas, there are no other terms involved in the total energy since we neglect the Coulomb interactions and there is no exchange interaction between spin-up and spin-down electrons. 

We recall the defiiution of the average volume per particle through the variable rs, the radius of the sphere that encloses this average volume  

While the above discussion is instructive on how exchange interactions alone can lead to non-zero magnetization, this simple model is not adequate to describe reah-stic systems. There are several reasons for this, having to do with the limitations of both the Hartree-Fock approximation and the uniform electron gas approximation. First, the Hartree-Fock approximation is not realistic because of screening and 

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