Quantum Monte Carlo: the helium atom

The helium atom consists of two electrons and a nucleus with charge \( Z=2 \). The contribution to the potential energy due to the attraction from the nucleus is $$ -\frac{2ke^2}{r_1}-\frac{2ke^2}{r_2}, $$ and if we add the repulsion arising from the two interacting electrons, we obtain the potential energy $$ V(r_1, r_2)=-\frac{2ke^2}{r_1}-\frac{2ke^2}{r_2}+ \frac{ke^2}{r_{12}}, $$ with the electrons separated at a distance \( r_{12}=|\boldsymbol{r}_1-\boldsymbol{r}_2| \).