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 \( d\mathbf{\tau}=dx_1dr_2\dots dr_A \).