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

We can rewrite this equation as (changing dummy variables) $$\sum_{\beta} \left\{\langle \alpha | h | \beta \rangle+ \sum_{j}^A\sum_{\gamma\delta} C^*_{j\gamma}C_{j\delta}\langle \alpha\gamma|\hat{v}|\beta\delta\rangle_{AS}\right\}C_{i\beta}=\epsilon_i^{HF}C_{i\alpha}.$$ Note that the sums over greek indices run over the number of basis set functions (in principle an infinite number).