## Hartree-Fock by varying the coefficients of a wave function expansion

In Eq. (12), restated here $$E[\Phi] = \sum_{\mu=1}^A \langle \mu | h | \mu \rangle + \frac{1}{2}\sum_{{\mu}=1}^A\sum_{{\nu}=1}^A \langle \mu\nu|\hat{v}|\mu\nu\rangle_{AS},$$ we found the expression for the energy functional in terms of the basis function $$\phi_{\lambda}({\bf r})$$. We then varied the above energy functional with respect to the basis functions $$|\mu \rangle$$. Now we are interested in defining a new basis defined in terms of a chosen basis as defined in Eq. (13). We can then rewrite the energy functional as $$$$E[\Phi^{HF}] = \sum_{i=1}^A \langle i | h | i \rangle + \frac{1}{2}\sum_{ij=1}^A\langle ij|\hat{v}|ij\rangle_{AS}, \tag{14}$$$$ where $$\Phi^{HF}$$ is the new Slater determinant defined by the new basis of Eq. (13).