In general we end up computing the expectation value of the energy in terms of some parameters \( \alpha_0,\alpha_1,\dots,\alpha_n \) and we search for a minimum in this multi-variable parameter space. This leads to an energy minimization problem where we need the derivative of the energy as a function of the variational parameters.

In the above example this was easy and we were able to find the expression for the derivative by simple derivations. However, in our actual calculations the energy is represented by a multi-dimensional integral with several variational parameters. How can we can then obtain the derivatives of the energy with respect to the variational parameters without having to resort to expensive numerical derivations?