During the development of our code we need to make several checks. It is also very instructive to compute a closed form expression for the local energy. Since our wave function is rather simple it is straightforward to find an analytic expressions. Consider first the case of the simple helium function $$ \Psi_T(\boldsymbol{r}_1,\boldsymbol{r}_2) = e^{-\alpha(r_1+r_2)} $$ The local energy is for this case $$ E_{L1} = \left(\alpha-Z\right)\left(\frac{1}{r_1}+\frac{1}{r_2}\right)+\frac{1}{r_{12}}-\alpha^2 $$ which gives an expectation value for the local energy given by $$ \langle E_{L1} \rangle = \alpha^2-2\alpha\left(Z-\frac{5}{16}\right) $$