We want to perform a Variational Monte Carlo calculation of the ground state of two electrons in a quantum dot well with different oscillator energies, assuming total spin \( S=0 \). Our trial wave function has the following form

$$ \begin{equation} \psi_{T}(\boldsymbol{r}_1,\boldsymbol{r}_2) = C\exp{\left(-\alpha_1\omega(r_1^2+r_2^2)/2\right)} \exp{\left(\frac{r_{12}}{(1+\alpha_2 r_{12})}\right)}, \tag{25} \end{equation} $$

where the variables \( \alpha_1 \) and \( \alpha_2 \) represent our variational parameters.

Why does the trial function look like this? How did we get there? This is one of our main motivations for switching to Machine Learning.