If the basis set is infinite, eq. 2.10 would not be an approximation. In practice, the basis set needs to be finite which results in an error introduced by this approach. For AS-ALD, either atomic orbitals are used as basis sets (see section 2.8.1 on molecular approaches) or plane-wave basis sets (see section 2.8.2). HF is historically very important and conceptually very close to KS-DFT. In practice, it is less accurate than DFT but as "costly" and thus not used anymore. However, it is the starting point for the electron correlation methods discussed next.