In addition to the posts about current chemistry news that have been appearing here, the next couple of months are going to see several discussions of articles from the physical chemistry literature. This is the first of those posts.
Our first article is an historic one, “A Quantum-Mechanical Theory of Light Absorption of Organic Dyes and Similar Compounds” by Hans Kuhn, Journal of Chemical Physics (1949), Vol. 17, pp. 1198-1212. In this article Kuhn presents a theory of how to treat the π electrons in conjugated polymethine chains as a “free electron gas” or in the language of current physical chemistry texts, as “particles in a box.” This paper (and several others Kuhn wrote) is the foundation for a lab commonly performed in undergraduate physical chemistry courses on the absorption spectra of conjugated dyes.
The basic assumptions of the model are quite simple: (1) the electrons in the delocalized π electron orbitals are “free” to move along the polymethine chain (in the sense that they experience a constant potential as they move along until they get to the “end” of the chain where the potential rises steeply) and (2) the motion is one-dimensional. As long as these assumptions are correct, then modeling the energies as a one-dimensional particle in a box seems sensible.
Kuhn discusses the first assumption by comparing it to a somewhat more realistic model of a symmetrical sinusoidally varying potential in Figures 7 and 8. A comparison of the energy level diagrams shown in Figure 8(a)and 8(b) is reassuring, but I find it very irritating that he shows absolutely none of the calculations underlying 8(a) but instead refers you to another paper.
Kuhn never discusses the second assumption directly, instead justifying it indirectly through the claimed success of the model at reproducing experimental trends. As I read it, part of me wants to object that a zig-zag path is not one dimensional or that the electron distribution described by a π orbital has width and height as well as length. Would modeling the orbital as a two or three dimensional box give significantly better results?
Posted by Michael Fuson
In response to Dr. Fuson's critique, I think modeling the orbital as a three dimensional box instead of a 1D PIAB would almost certainly give significantly better results. On page 1201 Kuhn claims that his experimentally measured f value is likely low because he calculated his theoretical f from an all trans-configuration hypothetical dye molecule. In reality, the dye molecule was probably a cis-trans-isomer mixture. This problem is a direct result of Kuhn oversimplifying the shape of the orbitals in his approximation. Instead of a zig-zag misleadingly fixed at a 124 degree angle, a 2D model could account for different bond angles in a plane. A 3D boundary would more closely approximate the actual shape of the pi "electron cloud." The 3D boundary would allow for the wave/particle to exist in space around the direct C-C=C line and would more closely describe the non-linear shape of the electron orbitals. However, although it seems apparent that the use of either a 3D or 2D model would increase accuracy, the simplicity of Kuhn's model and its ability to predict experimental trends is certainly very useful. If the electrons were contained within a system that was closer to one-dimensional (as in maybe an ethyne molecule? etc.), Kuhn's 1D model would likely be an even better predictor of experimental trends because it would more closely approximate electron movement. Even so, in molecules with more complex bonding, the 1D model would be an insufficient replacement for a 3D model.
ReplyDelete--Frank May
Although I agree the modeling is severely lacking mathematical proof and the understanding of how complex the system really is, I believe this theory is great for aiding in the understanding of how particles move between bonding in a conjugated system. I agree that the molecule picked should be one slightly more simple which does not have the issue of being zig-zag rather than linear in motion. My suggestion to Kuhn would be to go a little more in depth in this paper and mention that the 1D particle in a box thought cannot work because the electron distribution described by π orbital is not 1D but rather as Dr. Fuson stated has a width and a hight as well. I believe the transition from 1 dimension to 3 dimensions does not need to include much mathematical proof given that the 1 dimensional thought can put the math in a format that can then be carried over to the 3 dimensional box. Further if the error between the experimental values and theoretical values could be justified by value lost from excluding the other 2 dimensions of the pi orbitals his argument would be a lot more believable. If more molecules were also tested (ethyne, butyne, and even benzene), I believe the justification of value difference coming from the 1D model over the 3D model would be strongly supported.
ReplyDelete~David Calhoun
I am not so sure that modeling this as a 3D-particle in a box would give much better results than a 1D-PIAB. It very well could, but I’m not positive it would be worth the added complexity. On page 1201, the example where he says Cis-trans isomer is responsible for a lower value, he was off by 80 Angstroms. Which when calculated out, is about 8 nm (unless I messed that up somewhere which is definitely possible). When we are talking about dyes, emitting wavelengths in the visible range (400-700 nm ), 8 nm isn’t that much considering the general differences between a cis and a trans bond. If I had to guess, I would have said the results would be much farther off than that before reading them with my own eyes. Also, according to figure 5, his assumptions hold up much better than bond orbital method, and molecular orbital methods. Therefore, I would say his assumptions do have some substance to them and the experimental data certainly backs that up. On the other hand, if we were talking about electromagnetic radiation of a much smaller wavelength emitted from a molecule, 8 nm could be a lot and these assumptions might not hold up so strongly. In conclusion, I think this is a great way to start thinking about electrons flowing around pi bonds.
ReplyDelete-Nate Scheidler
To add to the discussion of the 1D vs. 3D model, the use of PIAB does not seem too far-fetched. Despite the trans positioning of the bonds, the conjugated chains are all planar. Hence, the motion is still 1D. While the motion is not a straight line from one edge of the molecule("box" if you will) to the other, the motion is a combination of lines. The motion is still continued within a single plane. The 1D model does seem justified as (an admittedly simplistic) model of the system. Aren't simple models that can adequately explain the behavior of a system the basis of many theories in chemistry though? Perhaps a fuller understanding could be understood with a 3D model; however, this both complicates the math and may not make a significant difference in the calculated theoretical values in the long run.
ReplyDeleteBeyond the 1D/3D debate, as I read this paper, I found myself much more interested in the unsymmetrical dyes that Kuhn mentions than the symmetric ones he originally discusses. While the symmetric dyes are certainly an interesting place at which to start, the asymmetric dyes do not allow for us to use the assumption of a free-electron gas for quantitative approximation, as Kuhn discusses on 1205. However, as Kuhn begins to discuss the assumptions we can make, I was distressed to find that he (as Dr. Fuson mentioned) only referred us to another paper rather than discussing the approximations involved in deriving equation 9. If he spends time justifying the free-electron gas model for the symmetric dyes, some space should have been spent justifying the assumptions for the asymmetric dyes, even if this discussion ends up being a short paraphrase of the other paper. While I am not arguing about the accuracy of the model as it does simplify to the appropriate equation for symmetric dyes, as the paper is now, I find myself asking what assumptions were necessary in developing a model for the asymmetric dyes.
- Annelise
I agree with Nate that this theory is very useful and does not yield poor results, especially for the time when it was published. However, Kuhn describes the π orbitals as an electron gas cloud but describes their motion as one dimensional which sounds quite contradictory. I think that David brings up a good point that mathematically it is very concievable to carry over the 1D theory to a 3D model. In response to the end of Frank's post, I think it would be unreasonable to try the same experiment on a linear alkyne because the molecule would not be conjugated and therefore the molecule would not absorb light the same way and the π electrons would not behave the same either. Not to mention that the electrons would still exist in a three dimensional cloud. I think the best way to get improved results is to consider Dr. Fuson's suggestion to attempt a 3D model.
ReplyDelete-Adam Murray
For the argument made in this paper, a 1-D to 3-D transition seems a little overkill for the point that Kuhn seemed to be making. However, if the theory is going to be more thoroughly checked, then I think it might be appropriate to discard the assumption of 1-D motion and replace it with a more accurate 3-D model. Further, if you compare the length of the dye molecule to the "zig-zag" motion within it, the length L is significantly large compared to the variation of the electron motion, which is loosely mentioned on p. 1210 about V0 slightly decreasing as j increases. For a simple model, I think this approximation seems fair for the context it was published in.
ReplyDeleteI, too, find myself questioning where Eqn. 9 was derived from. I wonder if there were any assumptions made about the structure of the molecule when deriving this equation, and if those assumptions are significant enough to explain variations in experimental versus theoretical calculations.
I'm intrigued by how varying the different substituents on either end of the dye molecule can change its range of absorbtivity. That idea has a vast number of real world application, but that is hardly discussed in that section starting on p. 1210.
The author also mentions that the light fluoresced by the dye molecules is maximum when the incident light is parallel with the axes of the molecules (p. 1203). This makes me wonder if this can be (or is already) used as a way to polarize specific wavelengths of light, at least if I understand that section correctly.