Is there any metrics to quantitatively determine the amount of pi-conjugation in an organic material (polymer or small molecules) from geometric parameter, such as, bond lengths? How to compare, let's say, between two molecules which have higher or lower pi-conjugation than the other?
I am assuming you mean by "conjugation", electron delocalisation. If so, in my opinion, the answer is discrete: it is either delocalised or it is localised. It cannot be half-way. Part of the reason is delocalisation implies the wave function associated with electron motion occurs over the delocalised area. A wave has to have a uniform potential, otherwise there would be different velocities in different parts, which would lead to a refractive index difference, which in turn would lead to wave interference and within a few cycles, the reinforcement-cancellation would lead to localised over a smaller zone. Alternatively, take the original Schrödinger equation - in this ψ must have one and only one kinetic and potential energy.
Therefore, there is no "partial conjugation". I know you will see this in the textbooks, justified by computational theory, for cyclopropane. What the text books don't add that the computational programs that "proved" this also assumed it. That is just plain wrong, and the odd properties of cyclopropane are a consequence of the polarisation field arising from bending bonds.
You can use UV spectroscopy, but you have to know what you are doing. A UV excited state involves a change of electric moment (if there were no change, from Maxwell's electromagnetic theory, there would be no signal intensity.) Accordingly, anything else that interacts with that electric moment and stabilizes it will give a change in both the intensity and the wavelength of the signal and it is possible to confuse this with conjugation.
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