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

Minimizing with respect to $$C^*_{i\alpha}$$, remembering that the equations for $$C^*_{i\alpha}$$ and $$C_{i\alpha}$$ can be written as two independent equations, we obtain $$\frac{d}{dC^*_{i\alpha}}\left[ E[\Phi^{HF}] - \sum_{j}\epsilon_j\sum_{\alpha} C^*_{j\alpha}C_{j\alpha}\right]=0,$$ which yields for every single-particle state $$i$$ and index $$\alpha$$ (recalling that the coefficients $$C_{i\alpha}$$ are matrix elements of a unitary (or orthogonal for a real symmetric matrix) matrix) the following Hartree-Fock equations $$\sum_{\beta} C_{i\beta}\langle \alpha | h | \beta \rangle+ \sum_{j=1}^A\sum_{\beta\gamma\delta} C^*_{j\beta}C_{j\delta}C_{i\gamma}\langle \alpha\beta|\hat{v}|\gamma\delta\rangle_{AS}=\epsilon_i^{HF}C_{i\alpha}.$$