To compute the helium ground state (singlet) in Hartree–Fock theory we have to solve this equation:
$$ \left[ - \frac{\Delta}{2} - \frac{Z}{r} + \frac{1}{2} \int d^3 r' \frac{\vert \Psi(\boldsymbol{r'}) \vert^2}{ \Vert \boldsymbol{r} - \boldsymbol{r'} \Vert } - \epsilon \right] \Psi(\boldsymbol{r}) = 0 \tag{1}$$
If we use the optimal purely radial function $\Psi(r)$, with self-consistent field, we obtain no less than $2 \epsilon \sim - \pu{2,86 Ha}$ (instead of the real ground state $\sim - \pu{2,90 Ha}).$
Question
What if we use a more complicated function with like $\Psi(r,\theta,\phi)$, for example a decomposition in whatever angular function (spherical harmonics, sine, cosine and so on)?
Hint
I tried minimization with the following function:
$$ \Psi \propto\mathrm e^{- a_1 r} (1 + a_2 r^2 \cos^2 \theta) \tag{2} $$
But it does not work, I achieved only $\pu{-2,855 Ha}.$
