Imagine two correlated charged particles which have opposite charges. How can we write a total correlated wave function that describes these two particles?

I know that simplest or the most intuitive way to treat such problem is product wave function which has been multiplied by a correlation factor (or Geminal), $$ \Psi= \psi_1 \psi_2 *G$$ where $\psi_1$ stands for wave function of particle 1 and $\psi_2$ for particle 2. But I mean a more creative way to including correlation other than product form.