The local energy as function of the variational parameters defines now our objective/cost function.

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 (with the notation \( \langle E[\boldsymbol{\alpha}]\rangle =\langle E_L\rangle \))

$$ \bar{E}_{\alpha_i}=\frac{d\langle E_L\rangle}{d\alpha_i}, $$

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

$$ \bar{\Psi}_{i}=\frac{d\Psi}{d\alpha_i}. $$