# How to determine bond length and angle in a conjugated pi-system (polymethine)?

I have a polymethine molecule with let's say 9 $$\ce{C}$$ atoms.

$$\ce{H-CH=CH-CH=CH-CH=CH-CH=CH-CH3}$$

How can I calculate the length of the whole molecule?

I need this number to compute the wavelengths the molecule will absorb (model of linear potential well).

So what are the average bond lengths and bond angles in this molecule and how are they calculated?

EDIT: Another example for polymethine (just the red part of it):

So what are the average bond lengths and bond angles in this molecule and how are they calculated?

We can find the bond lengths in the literature. The carbon-carbon double bond length is 1.338 Å, typical for a double bond (reference). The carbon-carbon single bond length is 1.454 Å, shorter than expected due to resonance. All of the carbons are $\ce{sp^2}$ hybridized, which means that all of the bond angles should be ~120°.

Using this information and vector addition you can determine the length of any polymethine.

• 1.338Â are 0.1338nm, right? Then every double bond is 0.1338nm * sin(60°) = 0.1159nm and every single bond is 0.1454nm * sin(60°) = 0.1259nm projected to the axis through the molecule chain. That would lead me to the conclusion that my molecule above (4 C=C and 4 C-C bonds) is about 0.9672nm long. Or did I misunderstand you? – Byte Commander Apr 20 '15 at 16:41
• Right on the nm. Couldn't you just add the full C=C bond length plus 0.1454 * cos(60°) ? – ron Apr 20 '15 at 16:54
• I don't think so. I need the length of the molecule, which is the distance between the C atom at each end, assuming the molecule has alternating bonding angle directions (looks like the example image added to the question). – Byte Commander Apr 20 '15 at 17:07
• Your way makes sense and I get the same final number as you. – ron Apr 20 '15 at 17:54
• @InternetGuy No, resonance structures involving charge separation do not count nearly as much as neutral resonance structures in terms of describing the molecule. – ron Dec 13 '17 at 19:02

Think in a chain of triangles.

Given that the distance between C1 and C2 (= $a$), the distance between C2 and C3 (= $b$), and the bond angle $\gamma$ are known, the distance between C1 and C3 is

$c = \sqrt{a^2 + b^2 -2ab\cos\gamma}$

(cosine rule)

If you have a site license (or a torrent) to use ChemDraw you can generate a 3D model of such a structure and optimize it to find the most stable configuration. It will give you the bond angles on each to a high degree of accuracy. You can use the MM2 function to optimize and find bond lengths as well. ChemDraw will give you a simple list. It can also generate bond rotations, but you can bet that the conjugated pi system will remain planar, as the energy barrier to rotation about the sp2 carbon bonds is very high.

If you utilize a simple one-dimensional particle-in-a-box model, the length L of the "box" would be the length of the conjugated system. This would be the path along which electrons are conjugated. It's not exactly the jagged line between all the sp2 carbons, but it's pretty close, so you can say L=(number of bonds in conjugated system)x(average length of these bonds). Note that this is ABSOLUTELY NOT the distance between the C1-C3-C5-etc. carbons the other guy mentioned--I don't think he understands exactly what you're trying to calculate.

Using this model, you can calculate the wavelength of maximum absorption from the quantum numbers of HOMO-LUMO transitions. Look at the equation:

In this equation nf and ni are the electronic quantum numbers of the final and initial states of a transitioning electron, h is Planck's constant, m is the mass of an electron, and L is as described earlier. The quantity in parentheses simplifies to N+1, as ni=N/2 and nf=N/2+1, where N is the number of pi electrons in the conjugated system. To understand this conceptually, the ground state of a molecule will be populated such that the N/2 lowest energy levels will be filled (since electrons fill them in pairs), and all higher energy levels will be empty. When it absorbs light, one of its electrons jumps from the highest filled energy level (HOMO, with ni=N/2) to the lowest unfilled level (LUMO, with nf=N/2+1). It's important to understand that if an electron is promoted, it can't simply skip an energy level, so if you know the number of pi electrons you also know what the HOMO-LUMO transition will be. If you can count the number of pi electrons in the conjugated system (ex. 1,6-diphenyl-1,3,5 hexatriene has 3 double bonds in its box, which means 6 pi electrons), then you can use this equation to find the desired wavelength of maximum absorption. Surely you've seen the classical equation:

where c is the speed of light. If you substitute this into the first equation you should be able to solve for the wavelength of maximum absorption. Mind your units!