We can then take the derivatives of $$ \sigma_E = \langle E^2\rangle - \langle E\rangle^2, $$ with respect to the variational parameters. The derivatives of the variance can then be used to defined the so-called Hessian matrix, which in turn allows us to use minimization methods like Newton's method or standard gradient methods.

This leads to however a more complicated expression, with obvious errors when evaluating integrals by Monte Carlo integration. Less used, see however Filippi and Umrigar. The expression becomes complicated $$ \bar{E}_{ij} = 2\left[ \langle (\frac{\bar{\Psi}_{ij}}{\Psi}+\frac{\bar{\Psi}_{j}}{\Psi}\frac{\bar{\Psi}_{i}}{\Psi})(E_L-\langle E\rangle)\rangle -\langle \frac{\bar{\Psi}_{i}}{\Psi}\rangle\bar{E}_j-\langle \frac{\bar{\Psi}_{j}}{\Psi}\rangle\bar{E}_i\right] +\langle \frac{\bar{\Psi}_{i}}{\Psi}E_L{_j}\rangle +\langle \frac{\bar{\Psi}_{j}}{\Psi}E_L{_i}\rangle -\langle \frac{\bar{\Psi}_{i}}{\Psi}\rangle\langle E_L{_j}\rangle \langle \frac{\bar{\Psi}_{j}}{\Psi}\rangle\langle E_L{_i}\rangle. $$

Evaluating the cost function means having to evaluate the above second derivative of the energy.