Hartree-Fock theory of free electrons

The Hartree and exchange potentials give the following potential for electron-electron interaction in the Hartree-Fock approximation  

The first term is the total Coulomb repulsion potential of electrons common for all states / i(r), while the second term is the effect of fermionic exchange, and is different for each state (, (r). 

To elucidate the physical meaning of the approximations introduced above we will consider the simplest possible case, that is one in which the ionic potential is a uniformly distributed positive background. This is referred to as the jellium model. In this case, the electronic states must also reflect this symmetry of the potential, which is uniform, so they must be plane waves  

We also define certain useful quantities related to the density of the uniform electron gas the wave-vectors have a range of values from zero up to some maximum 

It is often useful to express equations in terms of another quantity, which is defined as the radius of the sphere whose volume corresponds to the average volume per electron  

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