[spirochete]

Is there a qualitative way to determine relative orbital coefficients in conjugated systems that only contain carbon, such as allyl radical and butadiene? I get the idea that it may have to do with drawing waves that go through the nodes, but when I try to do it myself by hand it doesn't always reach the correct answer. I attached some examples of what I mean.

I am asking if it is possible or simple to do so without thinking about any of the mathematical equations, or just memorizing what the pictures look like. If so, could somebody describe the logic that I should use.

[Corribus]

Orbital coefficients are calculated by considering the resonance/exchange, coulomb and overlap integrals of all atomic orbitals in the system. They represent the fraction of the MOs composed of the AOs located on each respective nucleus. To a first approximation, coefficients can be (and usually are) calculated via a simple Huckel MO treatment of the out of plane p-orbitals, which assigns simple values of the various component integrals based on some crude approximations. This treatment can be done by hand, although for large molecules it is not a trivial exercise. Some familiarity with matrix math and group theory can be helpful at simplifying the calculations. Butadiene is pretty simple; anthracene more difficult. More accurate calculations of orbital coefficients require more refined approximations of the integrals - an approach that usually requires a computer and some software designed for this purpose.

https://en.wikipedia.org/wiki/H%C3%BCckel_method

Are there specific molecules you need to calculate orbital coefficients for?

[spirochete]

Thank you for the detailed response! But I was asking if there is a qualitative way to do it. Imagine that you have never studied physical chemistry, and that you don't even know how to do simple algebra, just to make it extra fun. No math. Is there any logic that derives the relative sizes (just very roughly, like "big/small/small/big") of the coefficients for simple systems such as allyl, butadiene, allyl conjugated with an extra alkene, etc.

Also, the symmetry parts are obvious to me. So that's not the issue.

[Corribus]

Picture this in your head: the electron wavefunction is sinusoidal function spread over the entire molecule. In the LCAO-MO approach, the molecular orbital wavefunction is approximated as an array of atomic orbitals - each originating on one of the nuclear positions - that blend together. The orbital coefficient reflects the magnitude of contribution of each atomic orbital and can be visualized as the intensity of the wavefunction at the point in space where the nucleus is. So if you can picture in your head what the wavefunction is and how it is spread over the nuclei, you could, in principle draw or specify the relative magnitude of the orbital coefficient.

This is easiest for linear molecules in which the molecular orbital wavefunction is a single dimension function. The upper figure you posted basically tells you how to do this. If you draw the appropriate sin function (the wavefunctions of a 1D particle in a box are basically just sin functions – and make sure they are normalized) on a piece of paper, with the nuclei evenly spread nuclei underneath it, you can use that as a guide to draw how big the atomic orbital should be. The wavefunction extends beyond the terminal positions, keep that in mind. No math required, assuming you know how to draw a sin function well. (You can even quantify it using the equation provided there – note that you cannot just use vanilla sin functions to do this; there is a coefficient out front that arises due to the requirement that wavefunctions – and orbital coefficients – be normalized.)
This becomes quite a bit more difficult with nonlinear systems, in which the molecular wavefunctions are two dimensional and more complicated in form – although in principle the same idea holds.

However, this does all kind of miss the point that there is no one right set of orbital coefficients. The orbital coefficients are tied to the energy eigenvalues and the approximations one makes to the integrals mentioned in the earlier post. If you change those, the orbital coefficients can change quite a bit in magnitude.

(Apologies for the deleting an earlier post. I wasn’t satisfied with my original response.)

[spirochete]

Thank you very much again. It would seem that I am extremely bad at drawing these sin curves by hand, and that is my issue. In fact, previous to making this post I had even written out a hypothetical method that I attempted, but failed to be reproducible for me:

"You draw evenly spaced dots for each atom, but also imagine 2 additional dots, one on each side of the end atoms. Then draw the waves. The wave changes sign each time it reaches a node. Then, the magnitude (vertical height) of the wave above the X-axis predicts predicts how large each orbital is."

[Corribus]

Well yes this is more or less what you do, but it's not as easy to draw an accurate sin function by hand as you might think. If your curvature is off, it'll affect your results.