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

We wish now to minimize the above functional. We introduce again a set of Lagrange multipliers, noting that since $$\langle i | j \rangle = \delta_{i,j}$$ and $$\langle \alpha | \beta \rangle = \delta_{\alpha,\beta}$$, the coefficients $$C_{i\gamma}$$ obey the relation $$\langle i | j \rangle=\delta_{i,j}=\sum_{\alpha\beta} C^*_{i\alpha}C_{i\beta}\langle \alpha | \beta \rangle= \sum_{\alpha} C^*_{i\alpha}C_{i\alpha},$$ which allows us to define a functional to be minimized that reads $$$$F[\Phi^{HF}]=E[\Phi^{HF}] - \sum_{i=1}^A\epsilon_i\sum_{\alpha} C^*_{i\alpha}C_{i\alpha}. \tag{16}$$$$