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

$$ \bar{E}_{\alpha}=\frac{d\langle E_L[\alpha]\rangle}{d\alpha}. $$

as the derivative of the energy with respect to the variational parameter \( \alpha \) (we limit ourselves to one parameter only). In the above example this was easy and we obtain a simple expression for the derivative. We define also the derivative of the trial function (skipping the subindex \( T \)) as

$$ \bar{\psi}_{\alpha}=\frac{d\psi[\alpha]\rangle}{d\alpha}. $$