The Hartee-Fock equations for the Jellium model [equations 5.4 to 5.6] have plane wave solutions. These correspond to quasi-particles and have the energy dispersion relation 5.1.
The exchange hole refers to the fact that electrons of like spin avoid one another due to the Pauli exclusion principle. This means the energy is less than the Coulomb repulsion energy of an electron moving in a uniform charge density.
The total energy of all the occupied Hartree-Fock states gives an "exchange energy" which only depends on the electron density (5.14). [This expression is sometimes used in the X_alpha method of Slater and the Local Density Approximation (LDA) of Density Functional Theory (DFT)].
The HF excitation spectrum is qualitatively incorrect. It predicts that the low-temperature specific heat of metals should go like T/ln(T) whereas it is always observed to go like T. This problem can be resolved by including screening, which cuts off the long-ranged Coulomb potential (see Chapter 8).
HF energy can be used to calculate the cohesive energy of simple metals. The result is worse than the non-interacting electron model! This difficulty arises because the direct and exchange Coulomb energies almost cancel each other.
The quasi-particles have lower energy than free particles because of the exchange hole.