# Do the hydrogen atom's electron orbitals have Gaussian probability density functions?

In this article they show the following diagram: Are all the diagrams in the little boxes really just Gaussian probability density functions with mean and variance (or covariance)? If not, what kind of density functions are these? Is there a name?

• an elementary search would provide the answer so maybe you can clarify why or what properties of those figures made you suggest that a Gaussian would work here? – ZeroTheHero May 15 '20 at 0:56
• Related question: How do orbitals exist in an atom? – BioPhysicist May 15 '20 at 0:59

## 2 Answers

They are not Gaussian functions. The wavefunction is actually given in the top right corner of the diagram. You can see that it's the product of an exponential $$e^{-\rho/2}$$, with a radial function $$\rho^l$$, multiplied by two "special" functions; the $$L$$ is a generalized Laguerre polynomial, and the $$Y_{lm}$$ is a spherical harmonic. The probability density is the mod$$^2$$ of this function. The integers $$n$$, $$l$$, and $$m$$ are quantum numbers that specify each orbital, and are written in the brackets underneath each plot.

Are all the diagrams in the little boxes really just Gaussian probability density functions with mean and variance (or covariance)?

Well, you could approximate them with Gaussian-like densities, but that'd be a poor approximation. In fact, Gaussian function is the ground state of the quantum harmonic oscillator (SHO), so you'd be modelling your hydrogen atom as a system of electron and nucleus connected by a spring (similarly to Hooke's atom, but with only one electron).

Excited states of SHO would, additionally to the Gaussian function, be multiplied by Hermite polynomials and spherical harmonics.

If not, what kind of density functions are these? Is there a name?

They are composed of functions of two named classes:

1. Spherical harmonics, which describe the rotational degrees of freedom,
2. Coulomb functions for the radial degree of freedom.

Now, the functions you get directly when multiplying these two are not probability densities: rather, they are probability amplitudes. To get the probability density $$\rho$$ from a probability amplitude $$\psi$$, you should take the square modulus of the latter:

$$\rho=|\psi|^2.$$

• "Well, you could approximate them with Gaussian-like densities, but that'd be a poor approximation." Quantum chemists would like to have a word with you :P – Godzilla May 15 '20 at 18:07
• @Godzilla123 I've seen such comments before in the context of variational methods :) Yes, you can approximate a wavefunction with lots of Gaussians with different means and variances, but that's not the same as trying to approximate the ground state of hydrogen with a single Gaussian. It would be in any case a poor approximation, even for energy, since you only have one parameter here: variance. – Ruslan May 15 '20 at 18:12
• True, I was just being facetious :) In any case the shape of the wavefunction (or god forbid any derivatives) is very badly approximated no matter how many Gaussians you put in. Gaussian expansions only work well for expectation values. – Godzilla May 15 '20 at 18:17