Carbon bonding in graphite

[laser1]

Homework Statement

Describe the C-C bonding in graphite

Relevant Equations

N/A

My lecture notes says it is $\sigma + \frac{\pi}{2}$. Why is it not $\sigma + \frac{\pi}{3}$? As each electron from every carbon is shared between 3 C-C bonds

 

[mjc123]

I agree with you. Benzene has 6 pi electrons and 6 C-C bonds. Graphite has 6 pi electrons and 9 C-C bonds per 6 C atoms. You can draw 3 resonance structures; each bond is double in 1 structure and single in 2.

 

[laser1]

I agree with you. Benzene has 6 pi electrons and 6 C-C bonds. Graphite has 6 pi electrons and 9 C-C bonds per 6 C atoms. You can draw 3 resonance structures; each bond is double in 1 structure and single in 2.

Yeah that was my reasoning too. Probably just a typo. Thanks for confirming!

 

[DrDu]

This is a difficult question! The p_z orbitals do not only have bonding interactions with neighbouring atoms but also interact with the neighbouring bonds. If one assumes a perfect pairing quinoid structure, the pi system would be non-bonding.

 

[laser1]

This is a difficult question! The p_z orbitals do not only have bonding interactions with neighbouring atoms but also interact with the neighbouring bonds. If one assumes a perfect pairing quinoid structure, the pi system would be non-bonding.

Oh. I assume that my notation sigma + pi/2 etc. stuff is just a very simplified representation of it?

 

[DrDu]

During the holidays, I did some reading and was finally pointed to Coulson's book "Valence":
Using Hückel theory, he finds a pi bond order of 0.53, hence a total bond order of 1.53 between any two C-atoms. More details are available in the following article:
https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1939.0006

 

[DrDu]

As the article by Wheland (1940), cited by Coulson, does not contain any derivation of the binding energy of graphite, just stating the resonance energy of 0.58 per atom, I tried to calculate this value myself.
There are numerous texts available, where the Hückel or tight binding model is solved for graphene.
The elementary cell of graphene contains two C atoms, so the Hückel matrix for a given point in reciprocal space is 2x2 with zeros on the diagonal and $-\beta (1+\exp(i\phi_1)+\exp(i\phi_2))$ in the 12 position (and the complex conjugated value in position 21). Here $\phi_{1,2}$ are proportional to the elements of the k vector. This element can be interpreted as the sum of the matrix elements between atom 1 and 2 in the same cell, and between atom 1 in the cell considered and 2 in adjacent cells. The precise relation between $\phi$ and k is not important, as the band is full and we will have to average $\phi_1$ and $\phi_2$ over the range 0 to $2\pi$.
The orbital energies are
$$-\beta | 1+\exp(i\phi_1)+\exp(i\phi_2)| =-\beta (3+2\cos(\phi_1)+2\cos(\phi_2)+2\cos(\phi_1-\phi_2))$$.
Wolfram Alpha returns a value of 62.1626 for the double integral over $\phi_1$ and $\phi_2$ from 0 to $2\pi$ respectively. I don't think this integral has an analytic solution. Division by $(2\pi)^2$ yields finally $-\beta \times 1.57460$. This has to be multiplied by two as there are two electrons in each cell, but also divided by two to get the value per atom. This practically coincides with the value of 1.58 $\beta$ reported by Coulson.