This results in $$\frac{1}{{\cal R}_T(r_1)}\frac{d {\cal R}_T(r_1)}{dr_1}=-Z,$$ and $${\cal R}_T(r_1)\propto e^{-Zr_1}.$$ A similar condition applies to electron 2 as well. For orbital momenta $l > 0$ we have $$\frac{1}{{\cal R}_T(r)}\frac{d {\cal R}_T(r)}{dr}=-\frac{Z}{l+1}.$$ Similarly, studying the case $r_{12}\rightarrow 0$ we can write a possible trial wave function as $$\psi_T(\boldsymbol{R})=e^{-\alpha(r_1+r_2)}e^{\beta r_{12}}. \tag{5}$$ The last equation can be generalized to $$\psi_T(\boldsymbol{R})=\phi(\boldsymbol{r}_1)\phi(\boldsymbol{r}_2)\dots\phi(\boldsymbol{r}_N) \prod_{i < j}f(r_{ij}),$$ for a system with $N$ electrons or particles.