The hamiltonian becomes then $$\hat{H}=-\frac{\hbar^2\nabla_1^2}{2m}-\frac{\hbar^2\nabla_2^2}{2m} -\frac{2ke^2}{r_1}-\frac{2ke^2}{r_2}+ \frac{ke^2}{r_{12}},$$ and Schroedingers equation reads $$\hat{H}\psi=E\psi.$$ All observables are evaluated with respect to the probability distribution $$P(\boldsymbol{R})= \frac{\left|\psi_T(\boldsymbol{R})\right|^2}{\int \left|\psi_T(\boldsymbol{R})\right|^2d\boldsymbol{R}}.$$ generated by the trial wave function. The trial wave function must approximate an exact eigenstate in order that accurate results are to be obtained.