## Definitions and notations

Let us denote the ground state energy by $$E_0$$. According to the variational principle we have $$E_0 \le E[\Phi] = \int \Phi^*\hat{H}\Phi d\mathbf{\tau}$$ where $$\Phi$$ is a trial function which we assume to be normalized $$\int \Phi^*\Phi d\mathbf{\tau} = 1,$$ where we have used the shorthand $$d\mathbf{\tau}=dx_1dr_2\dots dr_A$$.