# Comparing 3-D quantities (e.g. electron densities)

Let's assume a scenario:

We have extracted electron density F for ground electronic state S of molecule A and electron density G for a frank-condon excited state E of the same molecule A. We have extracted both F and G in a standard Gaussian cube format such that:

1. The voxels of cubes are sufficiently fine to account for any significant distortion in electron cloud.
2. The cube dimensions are sufficiently large that the electronic density lying outside the cube can be truncated to zero.

Having these conditions, are there any metrics via which we can differentiate and highlight difference between these cubes? We can project the charge density to partial charges on atoms of A or maybe use general physical features like number of nodes, phase, etc., but are there any other metrics for 3-D data that we can use to numerically quantify the difference between F and G?

(For ex.: For a 2-D distribution we can statistical dispersion to calculate skewness and variance of the dataset.)

• Mathematically, the squared norm $||F-G||_2^2$ (i.e. the integral of |F-G|^2 over the real space) obviously counts as one, but I don't know if it suits your purpose. Dec 18, 2022 at 8:33
• @wzkchem5 I am not sure how much unique that metric can be... Although it is reasonable choice.
– mykd
Dec 18, 2022 at 8:44
• Maybe you could compute multipole (dipole, quadrupole, etc.) moments to quantitatively describe the shapes of the charge density distributions. This sounds like what you’re interested in. You’ll have to take into account the atomic nuclear charges too so that the system is charge neutral. Dec 18, 2022 at 23:34
• @StephenWeitzner Thanks for your suggestion! However I was looking for a methodology where I do not have to either decompose or project the charge density. I am also trying to compare the "exact density" between different density functionals, so I was looking for a general method to compare electron densities.
– mykd
Dec 19, 2022 at 11:08
• In case things start to get long here, don't accept the prompt to create a new chat room. Just continue the discussion in an existing room and link to it here. I can handle moving over/deleting existing comments.
– Tyberius
Dec 28, 2022 at 0:06