# Why aren't atomic radii calculated using Schrödinger's equation?

The atomic radii are estimated using a variety of methods: most of these involve dividing their bond length by 2. But that is a very crude way of measuring atomic radii. I mean, atoms overlap each other when bonded, surely the actual atomic radius is much more than the one obtained by this method.

Why don't they simply use the Schrodinger's wave equation to calculate atomic radii. If it can predict atomic orbitals to such accuracy, surely it can predict something as simple as atomic radius?

• What is your definition of atomic radii? – user26143 Feb 8 '14 at 8:14
• I would say that the radius equals the farthest orbital in the boundary surface diagrams of the atomic orbitals. – Gerard Feb 8 '14 at 9:52
• How do you obtain the boundary surface? Take the ground state wavefunction of the hydrogen atom for instance, $e^{-r}$, where is the boundary? – user26143 Feb 8 '14 at 10:02
• I like the question, unfortunately there is nothing simple for any atom with more than one electron. I apologize if you felt offended by the "poem". – Klaus-Dieter Warzecha Feb 8 '14 at 10:38
• @KlausWarzecha: There was nothing "offending" about the poem. It was quite amusing. But it wasn't relevant. – Gerard Feb 8 '14 at 15:49

## 2 Answers

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

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

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.

• It would be helpful if you would include some explanation. – Gerard Feb 8 '14 at 9:53
• Saw this poem in my g09 output the other day. :) – LordStryker Nov 11 '14 at 22:45

I saw this in a "related questions" link.

The answer is that yes, you can use quantum mechanics to define atomic radii (e.g., van der Waals radii). Multiple people have done so with different schemes. I believe the most recent is:

"Consistent van der Waals Radii for the Whole Main Group" J. Phys. Chem. A 2009, 113, 5806–5812

Similar efforts have been made for covalent bonding radii using quantum mechanics.