Letter pubs.acs.org/NanoLett
Reply to “Comment on ‘Breakdown of Interference Rules in Azulene, a Nonalternant Hydrocarbon’” Mikkel Strange,† Gemma C. Solomon,*,† Latha Venkataraman,*,‡,§ and Luis M. Campos*,§ †
Nano-Science Center and Department of Chemistry, University of Copenhagen, Copenhagen 2100, Denmark Department of Applied Physics and §Department of Chemistry, Columbia University, New York, New York 10027, United States
‡
Nano Lett., 2014, 14 (5), 2941−2945. DOI: 10.1021/nl5010702. Nano Lett., 2015, 15. DOI: 10.1021/acs.nanolett.5b03468.
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n his comment,1 Stadler contends that it is incorrect to suggest that the graphical rules he and his coauthors presented break down for azulene. Further, he suggests that the apparent breakdown arose from an incorrect application of the rules. While we concede that the application of the rules in our paper2 was arguably incomplete, his conclusion, that “... the graphical scheme is completely general for any molecular topology in conjugated π-systems regardless of whether they are alternant or non-alternant hydrocarbons as long as the basic assumptions of its derivation are fulfilled” is nonetheless not correct as we highlight here. His comment raises a number of interesting points that we would like to clarify in this reply. The Mathematics Behind the Rules versus the Rules. Part of Stadler’s defense is rooted in the correctness of the mathematics that underlies his graphical rules. We believe distinction must be drawn between the mathematics (calculating a determinant) and the rules, a set of simple statements that allow interference effects to be predicted “without any computation and with only the simplest back-of-the-envelope calculations”. We have no reservations about the mathematics but based on our understanding3 the rules presented by Markussen and co-workers4 are incomplete and can potentially lead to erroneous conclusions. Do the Rules Break down for Azulene? Stadler correctly contends that the conclusions we drew for 1,3-Azulene were based on an incomplete application of the rules and an omission of closed-loop diagrams. We had recently also come to the conclusion that closed-loop diagrams were the missing term and prior to receiving his comment had submitted a related manuscript exploring the consequences of these diagrams,3 which can both introduce and remove destructive interference features in the transmission. We hope that the discussion that we undertake between his comment and our reply clarifies this issue for future readers. Our interpretation of the rules proposed by Markussen and co-workers4 was partly based on the examples presented in that work, where systems that had closed-loop structures were analyzed by pairing neighboring atoms. Having read through the rules again, in light of Stadler’s comment, we would like to highlight that the case of 1,3Azulene is not the only potential problem for the rules. Other substitution patterns will also “break the rules”, even as carefully recapitulated in Stadler’s comment. Namely, 1,4-Azulene, 1,6Azulene, and 1,8-Azulene. These were not systems considered in our previous experimental work and therefore did not enter the discussion. Figure 1 shows that all these systems have a © XXXX American Chemical Society
Figure 1. 1,4- 1,6-, and 1,8-azulene also present problems for the rules as presented by Stadler and co-workers (a). The azulene molecule, illustrating the numbering employed and the transmission as a function of energy from a tight-binding calculation, with clear destructive interference at the Fermi energy for the 1,4, 1,6, and 1,8 substituted variants. (b) The diagrams for each system illustrating that there is no evidence from the rules that these systems should exhibit destructive interference (in this case arising due to a cancellation of diagrams).
clear signature of destructive interference at the Fermi energy, while all the diagrams suggest that there is no destructive interference in these systems. This leaves us with further cases of apparent failure. Received: October 13, 2015
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DOI: 10.1021/acs.nanolett.5b04154 Nano Lett. XXXX, XXX, XXX−XXX
Letter
Nano Letters If the Mathematics Is Correct How Can the Rules Fail? The inherent challenge in distilling complex mathematics into simple rules is in preserving the full complexity of the problem, while maximizing the simplicity of the final presentation. We contend that there are additional details, or missing rules, that need to be included for a complete description, as described in Pedersen et al.3 Should We Use the Rules? While azulene has undoubtedly presented an interesting challenge for the rules due to its nonalternant bicyclic structure, it has also been a means by which to clarify their operation. Diagrams have a long history in chemistry5−8 and their introduction into a transport framework provided a great tool for rapidly predicting interference effects; however, these diagrams need to be refined and clearly presented to be generally applicable. As we deepen our understanding of destructive interference and the range of systems in which it can manifest, so too do subtleties emerge in how we must use these graphical tools. We firmly believe these graphical rules have an important role to play in our understanding of transport and look forward to their future development.
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AUTHOR INFORMATION
Corresponding Authors
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[email protected] Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the National Science Foundation, under Grant DMR-1206202. G.C.S. and M.S. received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC Grant Agreement No. 258806.
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REFERENCES
(1) Stadler, R. Nano Lett. 2015, DOI: 10.1021/acs.nanolett.5b03468. (2) Xia, J.; et al. Nano Lett. 2014, 14, 2941−5. (3) Pedersen, K. G. L.; Borges, A.; Hedegård, P.; Solomon, G. C.; Strange, M. Preprint, http://arxiv.org/abs/1510.06015, 2015. (4) Markussen, T.; Stadler, R.; Thygesen, K. S. Nano Lett. 2010, 10, 4260−4265. (5) Woodward, R. B.; Hoffmann, R. Angew. Chem., Int. Ed. Engl. 1969, 8, 781−853. (6) Fleming, I. Pericyclic Reactions; Oxford University Press: New York, 1999. (7) Graovac, A.; Gutman, I.; Trinajstić, N.; Ž ivković, T. Theoretica Chimica Acta 1972, 26, 67−78. (8) Hosoya, H. Theoretica Chimica Acta 1972, 25, 215−222.
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DOI: 10.1021/acs.nanolett.5b04154 Nano Lett. XXXX, XXX, XXX−XXX
