Comment on “Is the Antiresonance in Meta-Contacted Benzene Due to

Comment pubs.acs.org/JPCC

Comment on “Is the Antiresonance in Meta-Contacted Benzene Due to the Destructive Superposition of Waves Traveling Two Different Routes around the Benzene Ring” Rudolf Sýkora and Tomás ̌ Novotný* Department of Condensed Matter Physics, Faculty of Mathematics and Physics, Charles University, Ke Karlovu 5, CZ-121 16 Prague 2, Czech Republic

■

J. Phys. Chem. C2017, 121 (21), 11739−11746. DOI: 10.1021/acs.jpcc.6b11951 CRITIQUE here we only state them at energy εα: G13A =

The authors of ref 1 question and argue against the “interference-in-real-space” explanation of zero transmission of a meta-contacted benzene ring. They offer (and prefer) an alternative “interference-in-energy-space” picture. While the latter viewpoint is a legitimate one, their arguments against the former are not in accord with their own method and are hence unfounded. The authors’ argument, expressed in the excerpt: “the relative phase shift between path A and path B is not π at E = εα, as shown in Figure 4d,g”, appearing toward the end of the paragraph below their eq 3, is incorrect, together with the figures themselves. Following the authors’ method and definitions, we claim their Figure 4d should look more like the full blue line in Figure 1. Most importantly, the phase difference is π around E = εα,

i

G13B = − 2γ ; notice the π phase difference.

We mention that while it is useful to compare the phases of G in the β23 = β and E = εα casesince in that case the amplitudes of the Green functions are the sameit is not that much so if either of these conditions is not met. Particularly, when β23 ≠ β, although the phases are still mutually shifted by π at E = εα, GA13 + GB13 ≠ 0. On the other hand, we know that the exact G13 = 0. It is in this sense that we agree with the authors saying that the “simple” interference picture (i.e., that the propagation happens exclusively through one or the other branch) fails, and excursions to both branches must be accounted for, especially for small γ, as the original article also comments on. Finally, we note that the graphs of Green-function components f m in the article’s Figure 3 for β23 = β and β23 = 0 are swapped. (And graphs’ x-axes were probably desired to generally mean (E − εα)/|β|.) A,B

■

ADDITIONAL COMMENT

The authors study a benzene ring with just one perturbed bond, β23, and find that the transmission at energy εα remains zero in the meta configuration, independently of the bond value. In fact, one may perturb arbitrarily all the bonds and still observe the same result, as long as the on-site energies are kept equal. One way to see this follows from Figure 2, and the formal expansion

Figure 1. Phase difference between paths A and B in the limit of an isolated molecule, γ → 0 (full lines), and for four values (colors) of “scissor-cutting” β23/β. For completeness we also show a graph for a finite γ = β and β23 = β (dashed blue line).

contrary to the authors’ published result of zero phase difference. (Were it not for the associated discussion in the text, we would just believe that the y-axes were erroneously shifted by π.) The existence of phase-π destructive interference in real space at energy εα in the meta-contacted (unperturbed) benzene is discussed in ref 2see eqs 9 and 10 and the surrounding paragraphs therewhich although cited by the authors (ref 58) did not raise enough alert to reconsider their result. Actually, analytic formulas can be derived for GA, GB; © XXXX American Chemical Society

i ββ23 , γ β 2 + β232

Figure 2. Perturbed benzene ring as a sum of an unperturbed ring A and a “perturbation” B. Received: July 19, 2017 Published: August 10, 2017 A

DOI: 10.1021/acs.jpcc.7b07120 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Comment

The Journal of Physical Chemistry C G13(E = εα) = (εα − H )−1 = (εα − A − B)−1 1 1 1 1 1 1 = + B + B B εα − A εα − A εα − A εα − A εα − A εα − A 0 + ··· = G13 + G10iBij Gj03 + G10iBij Gjk0 Bkl Gl03 + ··· = 0

Gij0 ≡

( ) 1 εα − A

is known to only connect indices of opposite

ij

parity; Bij connects nearest neighbors and thus changes parity, too; and the total number of G0’s and B’s in each term of the expansion is odd. A more rigorous way is to realize that this all is, actually, just a manifestation of the Coulson−Rushbrook theorem: ref 3, chapter 6, especially the last-but-one paragraph on page 100.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]ff.cuni.cz. ORCID

Rudolf Sýkora: 0000-0003-1123-389X Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS This work was supported by the Czech Science Foundation by Grant No. 16-19640S. REFERENCES

(1) Nozaki, D.; Toher, C. Is the Antiresonance in Meta-Contacted Benzene Due to the Destructive Superposition of Waves Traveling Two Different Routes around the Benzene Ring. J. Phys. Chem. C 2017, 121, 11739−11746. (2) Lambert, C. J. Basic concepts of quantum interference and electron transport in single-molecule electronics. Chem. Soc. Rev. 2015, 44, 875−888. (3) Coulson, C. A.; Mallion, R. B. Huckel Theory for Organic Chemists; Academic Press, 1978; p 182.

B

DOI: 10.1021/acs.jpcc.7b07120 J. Phys. Chem. C XXXX, XXX, XXX−XXX