Choice of trial wave function for Helium: Assume $r_1 \rightarrow 0$. $$E_L(\boldsymbol{R})=\frac{1}{\psi_T(\boldsymbol{R})}H\psi_T(\boldsymbol{R})= \frac{1}{\psi_T(\boldsymbol{R})}\left(-\frac{1}{2}\nabla^2_1 -\frac{Z}{r_1}\right)\psi_T(\boldsymbol{R}) + \mathrm{finite \hspace{0.1cm}terms}.$$ $$E_L(R)= \frac{1}{{\cal R}_T(r_1)}\left(-\frac{1}{2}\frac{d^2}{dr_1^2}- \frac{1}{r_1}\frac{d}{dr_1} -\frac{Z}{r_1}\right){\cal R}_T(r_1) + \mathrm{finite\hspace{0.1cm} terms}$$ For small values of $r_1$, the terms which dominate are $$\lim_{r_1 \rightarrow 0}E_L(R)= \frac{1}{{\cal R}_T(r_1)}\left(- \frac{1}{r_1}\frac{d}{dr_1} -\frac{Z}{r_1}\right){\cal R}_T(r_1),$$ since the second derivative does not diverge due to the finiteness of $\Psi$ at the origin.