# Are the shapes of atomic orbitals direct consequence of the Schrödinger equation?

I am trying to understand whether the shapes of the orbitals are inevitable given the standard model. They would probably change if we change the fine tuning of the fundamental physical constants, right?

Do I understand correctly that the shape of atomic orbitals determines large part of the chemical properties of the element? Could we then say the chemical properties are inevitable consequence of geometry of atomic orbitals?

The shape of the orbitals are derived by the Schroedinger equation, more precisely if you look at the hydrogen atom you can separate the variables, the angular part actually determines the angular part of the orbitals (quantum numbers l and m), the radial part determines essentially the energy levels and the radial part of the orbitals (quantum number n).

Most of what you see in pitoresque drawings of orbitals is actually the angular part which is actually the spherical harmonics. Those ones are not dependent on the planck constant or the mass of the electron.

Those shapes of the spherical harmonics are essentially due to the radial simmetry of the angular part of the equation and solutions of the same types are available for other systems (e.g. oscillations of a 3D elastic ball), or the helmoltz equation, in that sense inevitable is quite a big and dramatic word.

The radial part of the equation instead is bound with the energy and therefore is directly related to the planck constant and the mass of the electrons. From this standpoint if these constants were slightly different you would have slightly different energy levels and slightly different spectras, but it would be to be proven that the full construction of chemistry falls apart.

If you extend a little further for atoms with multiple electrons the radial part is no longer decoupled from the angular part and it depends on two quantum numbers now n and l, and the angular part is no longer like pure spherical harmonics, but let's say "deformed" ones.

Being too strict and interpret literally "the shape of an atom" or electrons as clouds, or two electrons per orbital, or imagine semiclassical coherent states literally is also in general not a good idea because electrons can exchange between each other, and they have quite subtle de-localized behaviours.

From a chemistry perspective you typically look at molecular orbitals, there are many of those, with different angular distributions, and there are some approximations in which they are qualitatively similar to atomic orbitals, in some other cases these are linear combinations of orbitals or slater determinants (as mentioned in the other answers).

Therefore you can derive almost all chemical properties from the orbitals, more precisely from multiparticle delocalized orbitals, plus few other considerations about spin and statistics, and ultimately from approximated or numerical solutions of the Schroedinger equation.

Fine tuning for the standard model is a totally different animal, it is intuitively the fact that certain series terms in perturbative expansions cancel out almost perfectly each other, this is quite odd because it often happens for big terms that you would consider almost infinite.

Just a small change of parameters such as the renormalized masses or the fundamental constants makes some terms of the series not cancel out anymore and therefore diverge. Therefore it would be nicer to have a theory that produce convergence not by highly oscillating terms canceling out but with small terms smoothly converging, and again if that is possible at all is an open problem.

I am trying to understand whether the shapes of the orbitals are inevitable given the standard model.

The atomic electron orbitals of atomic physics come about because the atomic potential is spherically symmetric: $$V(r) = \frac{-Z|e|^2}{r}\;,$$ where $$Z$$ is the atomic number, e is the electronic charge, and $$r$$ is the distance from the atomic nucleus.

Because this potential is spherically symmetric, the single-particle solutions can be described in terms of: 1) a radial wavefunction $$R_{n\ell}(r)$$ that depends only on the distance from the nucleus; 2) a spherical harmonic $$Y_{\ell,m}(\theta, \phi)$$ that depends on the angle. The overall single-particle solution is the product of these two terms.

The square of the $$Y_{\ell, m}$$ is often what is plotted to show the "shape" of the orbital.

Note also that single-particle orbitals are not the "real" solution for any case other than atomic hydrogen (or other single electron ions). In all other cases, the true wavefunction is typically approximated as an antisymmetric combination of orbitals (e.g., "Slater determinant").

Could we then say the chemical properties are inevitable consequence of geometry of atomic orbitals?

You might try to say this, but you would be wrong about some chemical properties. For example, there is no way to properly understand Hund's Rules just using single-particle orbitals. You need to look at combinations of Slater determinants and take into account the interaction between electrons.

In general, there is much more to chemistry than just atomic orbitals. But often people try to futz it all into a single-particle theory involving orbitals with ad hoc modifications. See, for example, most everything in density functional theory (other than Kohn's original papers).

They would probably change if we change the fine tuning of the fundamental physical constants, right?

I mean, yeah.

The equation you are trying to solve to find "orbitals" ($$\phi$$) is: $$-\frac{\hbar^2}{2m_e}\nabla^2\phi - \frac{Z|e|^2}{r}\phi = E\phi$$

So, if you fiddle around with parameters like e, $$m_e$$, and $$\hbar$$, you will get different-sized orbitals, different single-particle energy levels, etc. For example, you can think of the inverse Rydberg as setting a length scale for atomic physics. If you mess with the parameters, you mess with this length scale.

• Any single-electron ion (or atom for H) can be described nicely by the spherical harmonics. Once you get to two electrons you can write a book as Bethe and Salpeter did... Apr 26 at 21:48

The shape of hydrogen-like atomic orbitals follows directly from the $$1/r^2$$ Coulomb force between the electron and nucleus plus the Schrodinger equation.

As long as you don’t change that you don’t change the shape of atomic orbitals. For “mild” tunings of standard model parameters the only effect you have will be to change the Bohr radius which will make orbitals bigger or smaller. For more aggressive tunings you may reach points where higher order effects (fine structure, lamb shift, nuclear structure etc.) become more dominant compared to the Coulomb force. If you enter that regime then orbital shapes might change and chemistry might start to be qualitatively different.