# Electron density definition of single, double, triple bonds?

I was reading this question and the answer by user EricBrown to it, and this got me thinking about covalent chemical bonds.

The way I was always taught is that a single bond contains 2 paired electrons, a double bond 4 etc (The Lewis dots drawings always show this nicely). In his answer Eric mentions that the whole definition of single, double etc bonds is ambiguous because there haven't been measurements of the electron density. This notion strikes me as odd, because a simple googling gives me articles like this: How similar is a molecule to another? An electron density measure of similarity between two molecular structures Int. J. Quantum Chem. 1980, 17 (6), 1185–1189. But it did make me think about the way the different types of bonds are defined.

I read up a bit on the Theory of atoms in molecules and indeed this theory only specifies 1 bond type: the line with maximum electron density between two nuclei. According to the theory this line is unique for a given pair of nuclei in a given molecule. I could imagine that a molecule which has a triple bond (6 bonding electrons in the 'classical' theoretical sense) would have a relatively high electron density along this maximum line as compared to e.g. a single bond.

My question is: if we would be able to measure the electron density in a molecule (are we?) couldn't we use this to define a relation between the value of the maximum electron density and the notion of a single, double or triple bond? Or perhaps use the spread in electron density between the nuclei for this?

• Actually, the electron density of the "second" and "third" bonds has a node at the center line. Pi and delta bonds always do. So a triple bond has no more electron density along the center line that a single bond. It will have multiple local maximae of lesser density, and probably one global maxima; the center line. Nice question though, +1. I'll have to think about this one; though doubt I can answer it beyond "quantum mechanics said so". – ManishEarth May 4 '13 at 12:29

Below are plots of the electron density for staggered ethane, ethene, and ethyne, computed from electronic structure theory. For ethene, which is planar, imagine that the hydogens are jutting forward and backward into the paper, so that the contour plots should contain any putative $\pi$ bonds. It is the orthogonal plane that still contains the two carbon atoms.

A couple of points that I wish to make:

1. You are correct (in this case for sure) that there is some association between stronger bonds and higher electron density at the C-C bond critical point $(\star)$. This was noted by Bader and co-workers.
2. There is no evidence (to my eyes) that there are any side-wise $p$ orbital interactions forming $\pi$ bonds. These look like single $\sigma$ bonds!

My question is, where do we draw the dividing line in the correlation between electron density at the bond critical point, and what constitutes single, double, and triple bonds?

Responding to @michaelm: I might have expected this, if for example, ethene were to have a "double bond" between the carbons:

• what program are u using to generate this plot using QTAIM? – DurgaDatta May 5 '13 at 3:55
• Awesome! I'm also wondering about the QTAIM code. I would love to play around with it. I agree that from this picture you could define numbers for single, double and triple bonds, but these would probably be wrong if you start looking at e.g. propanol, propenol and propynol – Michiel May 5 '13 at 6:57
• I wrote this from scratch in Mathematica. It's a bit of an "expert system" at this point, but it's the best way (for me) to get precise control over the computation and presentation layers. – Eric Brown May 5 '13 at 11:18
• It looks beautiful. What I was wondering, to put the nail in the coffin of my proposal, can you come up with an example of a molecule that has either (i) lower electron density in its single bonded than in its double bonded version or (ii) lower electron density at the double bond than Ethane has in its single bond? – Michiel May 5 '13 at 15:01
• @michielm I think what I am trying to say is, if you look at the electron density, there are no particularly distinguishing characteristics between what we call single, double, and triple! They all look just about the same! – Eric Brown May 6 '13 at 15:21

Double bonds can be detected on electron density using electron location function (ELF) methods.

This is a plot for the f-localization domain in the ethylene molecule. Surfaces are colored according to the basin they belong to: magenta for core basins, light blue for the C-H bonds, and green for the C-C banana bonds.