The model seems to spread free electrons in a box up to the Fermi level (the fuzzy limit of the Fermi-Dirac distribution changes very little). The pi and so on come then from a computation of the density of states versus a small eenrgy increment around an arbitrary energy, where states are quantized by the wave number, taking for instance an integer number of half-waves in each box dimension.

From E=p^{2}/2m, you take a thin spherical volume element in the wavenumber space k_{x} k_{y} k_{z}, compute the state density in that space (it's uniform), the area of the sphere hence the volume between |k| and |k|+d|k| which is a number of states, convert |k| in E and get a density of states per energy unit around an arbitrary energy.

Then you sum from zero to the Fermi energy (or take the Fermi-Dirac distribution if you wish...) and compare with the number of metallic electrons to deduce the Fermi level.

I suppose the model then claims that the bulk modulus results from compressing the gas of metallic electrons in a smaller volume, which increases the Fermi level. This would be of course a brutal model, because

- The lower electron sheaths bring stiffness as well

- The nuclei can rearrange under pressure

- Metallic electrons are not free in a flat potential

- Metallic electrons result from orbitals, which can adapt or even re-orient a bit under pressure

- Electrons can have magnetic interactions! Said to make Invar's abnormal modulus

Similar computations are made for semiconductors (then with a modified mass, often called "effective" for reasons obscure to me, as if there were some ineffective mass) and for white dwarf stars.

You could compare with alkaline elements to check the model. One metallic electron per atom fits Hall effect measurements rather well for them. You get their (relatively small) bulk modulus from the sound speed and the density.