Why aren't atomic radii calculated using Schrödinger's equation?
Erwin with his psi can do
Calculations quite a few.
But one thing has not been seen:
Just what does psi really mean?


(Erich Hückel, english translation by Felix Bloch)

It is not easy to perform the calculation that you called simple. The Hamilton operator $\hat{H}$ for a multi-body problem of $K$ cores and $N$ electrons

\[
\hat{H} = - \frac{1}{2}\sum_a^K\frac{1}{M_a}\Delta_a - \frac{1}{2}\sum_i^N\frac{1}{M_i}\Delta_i - \sum_i^N\sum_a^K \frac{Z_a}{r_{ai}} + \sum_{i &lt; j}^N\frac{1}{r_{ij}}+ \sum_{a &lt; b}^K\frac{Z_aZ_b}{R_{ab}}
\]

is still a beast for $K=1$ with all the core-electron and electron-electron interactions.

So, while the question is legitimate, the calculation simply cannot be done without a lot of approximations, such as the Hartree-Fock method.

By the way, for the determination of the atomic radii of solid metals, there's x-ray diffraction, which I wouldn't call crude.