To find the derivatives of the local energy expectation value as function of the variational parameters, we can use the chain rule and the hermiticity of the Hamiltonian.

Let us define $$ \bar{E}_{\alpha_i}=\frac{d\langle E_L\rangle}{d\alpha_i}. $$ as the derivative of the energy with respect to the variational parameter \( \alpha_i \) We define also the derivative of the trial function (skipping the subindex \( T \)) as $$ \bar{\Psi}_{i}=\frac{d\Psi}{d\alpha_i}. $$