Frontier Orbital Perspective for Quantum Interference in Alternant and

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Frontier Orbital Perspective for Quantum Interference in Alternant and Non-Alternant Hydrocarbons Yuta Tsuji, and Kazunari Yoshizawa J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b02274 • Publication Date (Web): 11 Apr 2017 Downloaded from http://pubs.acs.org on April 15, 2017

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The Journal of Physical Chemistry

Frontier Orbital Perspective for Quantum Interference in Alternant and Non-Alternant Hydrocarbons

Yuta Tsuji,1 and Kazunari Yoshizawa2* 1)

Education Center for Global Leaders in Molecular Systems for Devices, Kyushu University, Nishi-ku,

Fukuoka 819-0395, Japan 2)

Institute for Materials Chemistry and Engineering and International Research Center for Molecular

System, Kyushu University, Nishi-ku, Fukuoka 819-0395, Japan

*To whom correspondence should be addressed. E-mail: [email protected]

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Abstract: The wave-particle duality of electrons gives rise to quantum interference (QI) in single molecular devices. A significant challenge to be addressed in molecular electronics is to further develop chemical intuition to understand and predict QI features. In this study, an orbital rule is markedly ameliorated so that it can capture the manifestation of QI not only in alternant hydrocarbons but also in non-alternant ones. The orbital-based prediction about the occurrence of QI in a non-alternant hydrocarbon shows good agreement with experimental results. A simple perturbation theoretic line of reasoning suggests that frontier orbital phase and splitting play a pivotal role in QI phenomena.

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Introduction Exploring charge transport properties through single molecules is an active theoretical and experimental area not only in molecular electronics1,2 but also in molecular sensing.3 One of the most striking quantum effects in molecular conductance is destructive quantum interference (QI), 4,5 ,6 where conductivity is significantly suppressed. Recently, growing attention has been devoted to QI because it has the potential to open doors to many new applications, such as thermoelectric devices,7 molecular switches,8 and spin filters.9 Predicting QI in both alternant and non-alternant hydrocarbons is a goal of theoretical work in molecular electronics.10,11,12,13 As one can speculate from the governing role of molecular orbitals (MOs) for chemical structures and reactivity,14,15 QI features can also be regarded as an outcome of the frontier MO (FMO) theory. So far, we have developed a selection rule to predict QI based on the MO representation of the zeroth Green’s function.16,17,18,19 The predictability of this rule has been validated experimentally for an alternant system.20 Stadler and co-workers have pointed out the limited applicability of the orbital rule in alternant hydrocarbons.21 Yet, its applicability to non-alternant systems is effectively moot. Hence, in this study, we refine the rule so that it can predict QI not only in alternant hydrocarbons but also in non-alternant ones.

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Results and discussion Let us begin with alternant hydrocarbons, where all the atoms belong either to the starred set or to the unstarred set in such a way that no atom of one set is adjacent to another atom of the same set.22 Such a hydrocarbon is exemplified by butadiene, as shown in Figure 1. In some early studies on electron transfer via an alternant-hydrocarbon linkage, it was found that the electronic coupling between two starred atoms or two unstarred atoms is significantly suppressed. 23 , 24 This should also be true for molecular conductance, and nowadays such an electronic coupling is termed alike coupling.25 QI is expected to occur in the electron transport between a pair of atoms leading to alike coupling. On the other hand, the electronic coupling between a starred atom and an unstarred atom is expected to be large, called disjoint coupling.

Figure 1. Top: Structure of butadiene with the numbering and starring of atoms. The solid arrow indicates a conductive path, while the dashed arrow indicates a transmission path where QI occurs. Bottom: Electron transmission probability in butadiene as a function of energy calculated with the Hückel MO (HMO) method combined with the non-equilibrium Green’s 4


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function formalism.

For pedagogical clarity, let us illustrate our original orbital rule that rehearses a theoretical result already in the literature before we show how we modify it. When the source and drain electrodes are attached to the rth and sth atoms in a molecule, the molecular conductance g is approximately proportional to the absolute square of the value of the zeroth order Green’s function at the Fermi level (EF) as follows:16 g =

2 2e 2 γ L γ R G rs(0 ) (E F ) , h

(1)

where 2e2/h is the quantum of conductance and γL(R) is the coupling between the molecule and the left (right) electrode. This equation holds true for short molecules, in which coherent tunneling is dominant. 26 In longer molecules, on the other hand, QI features might be obscured due to dephasing effects. To include the effects of the electrode attachment, one needs to use the non-equilibrium Green’s function (NEGF) instead of the zeroth order Green’s function. The details of the NEGF method are described in the section 1 of the Supporting Information (SI) to this paper. The NEGF can be well approximated by the zeroth Green’s function in molecular junctions where the interaction between the molecule and the electrode is regarded to be small. The details of the relation between the NEGF and zeroth Green’s function are described in the section 2 of SI. We can calculate the zeroth Green’s function from the MO energy levels and MO coefficients as 5



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Grs(0 ) (EF ) = ∑ k

Crk Csk , EF − ε k

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(2)

where Crk is the MO coefficient on the rth atom in the kth MO and εk is the kth MO energy. If the Fermi level is assumed to be located between the energy of the highest occupied MO (εHOMO) and that of the lowest unoccupied MO (εLUMO), the Green’s function can be dominated by the contributions from the HOMO and LUMO,16,17,18,19 Cr ,HOMO C s , HOMO EF − ε HOMO

+

Cr ,LUMOC s , LUMO EF − ε LUMO

(3)

.

The validity of the location of the Fermi level is further discussed in the section 3 of SI. Note that the sign of the denominator of the HOMO term is positive, while that of the LUMO term is negative. So, if the sign of Cr,HOMOCs,HOMO is the same as that of Cr,LUMOCs,LUMO, these two cancel each other out, leading to QI. This condition for the occurrence of QI can be summarized as, (Cr,HOMOCs,HOMO)×(Cr,LUMOCs,LUMO) > 0.

(4)

When one employs this selection rule in conjunction with the Coulson’s pairing theorem,27 it will be clear that eq. 4 is always true for the case where both rth and sth atoms belong to either starred set or unstarred set. The pairing theorem states that the MO coefficients of paired MOs are the same for the starred atoms, while they have the opposite sign for the unstarred atoms. We can use this theorem to provide a prescription for the occurrence of QI based on the starring procedure. If the two atoms which connect to the electrodes belong to the starred set, Cr,HOMOCr,LUMO > 0 and Cs,HOMOCs,LUMO > 0. This leads to 6


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the condition shown by eq. 4. If the two atoms belong to the unstarred set, Cr,HOMOCr,LUMO < 0 and Cs,HOMOCs,LUMO < 0. This also leads to the condition shown by eq. 4. So these two situations correspond to a case where QI occurs. By definition these two cases correspond to the alike coupling. On the other hand, if the rth and sth atoms belong to the starred set and unstarred set, respectively, Cr,HOMOCr,LUMO > 0 and Cs,HOMOCs,LUMO < 0. This does not satisfy eq. 4, indicating that QI is not expected to occur. This pair of atoms leads to the disjoint coupling. Figure 1 shows electron transmission spectra for butadiene calculated at the Hückel MO (HMO) level of theory.28 Since there is an alike coupling between the C1 and C3 atoms, a transmission dip is observed at EF. This clearly indicates the occurrence of QI. In contrast, the coupling between the C1 and C4 atoms is disjoint, leading to a finite transmission probability there. A disjoint coupling is also expected for the connection between the C2 and C3 atoms; nevertheless, we can see a QI feature, which is what one can refer to as hard zero. The hard zero is defined as the transmission dip stemming from the disjoint coupling, while that coming from the alike coupling is defined as easy zero.29 This is not a merely nominal distinction because a dip due to the hard zero QI is likely to split into two under the influence of the though-space coupling and many body charge-charge correlations.30,31 The easy zero QIs can be easily predicted by the original orbital rule or the starring procedure, whereas it is not that easy to predict the occurrence of hard zero QIs. This is what we address in this study. 7



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For starters, we need to scrutinize why the orbital rule does not work in the connection between the C2 and C3 atoms. Looking at the coefficients of only the HOMO and LUMO of butadiene, we cannot expect any cancellation between them. If we do not neglect the contributions from the HOMO-1 and LUMO+1, in turn we note that the cancellation between the HOMO and HOMO-1 occurs, and ditto for the LUMO and LUMO+1 (see Figure 2). One might argue that the HOMO-1 and LUMO+1 would be further from EF than the HOMO and LUMO. As can be seen in Figure 2, however, one could see a relatively large orbital localization on the connecting sites in these orbitals. This leads to a large absolute value of the numerator in the Green’s function, allowing them to have the same weight as the HOMO and LUMO. It should be noted that Stadler and co-workers has performed an analysis of QI based on Larsson’s formula and pointed out that the contribution of all MOs needs to be taken into account to identify QI reliably from a MO perspective.21

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Figure 2. Frontier orbitals of butadiene (left) and schematic representation of the contributions from their orbitals to the Green’s function for the 1-4 (non-QI case), 2-3 (hard-zero QI case), and 1-3 (easy-zero QI case) connections (right). White MO lobes indicate a positive coefficient, i.e., (+), while red ones indicate a negative coefficient, i.e., (-). A paired Green’s function elements indicated by a red arrow cancel out.

The next thing we need to do is to amend our orbital rule so that it can capture the hard zero QI as well. To this end, we take a liking group to the electrode into account.32 There are some reasons why we do so. First, in practical molecular conduction experiments, people cannot directly connect the electrodes to a molecule without any linkers. One of the most widely used likers for gold electrodes is sulfur due to the formation of a strong Au-S covalent bond.33 Second, a significant localization of orbitals on the linker units near EF can be expected because the energy level of the π orbitals on the linkers (usually heteroatoms) is prone to be close to EF. In addition to such a significant localization, the next HOMOs and LUMOs would be repelled from the resultant HOMO and LUMO as a result of orbital 9



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interaction, missing out on a chance to contribute to the Green’s function with a comparable weight. Looking at the orbital interaction between a linker unit and the π-conjugated skeleton will result in an important insight into the strength of electronic coupling which can significantly affect the occurrence of QI.34,35 Figure 3a and 3b show a perturbation MO (PMO) analysis36,37 of 1,4-sulfur-terminated butadiene (1,4-STB) and 1,3-STB in terms of the union of butadiene with two S atoms, each bearing a p-π orbital. Details of the PMO method are given in the section 4 of SI. Owing to the relatively large distance between the two S atoms in the S2 fragment, the interaction between them is so small that the bonding πu and anti-bonding πg orbitals of the 3pz atomic orbitals (AOs) of S can be almost degenerate. For clarity, however, they are arranged vertically. The HMO distributions are depicted by using the HuLiS software.38

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Figure 3. Orbital interaction diagram for the PMO analysis of (a) 1,4- and (b) 1,3-STB in terms of the union of butadiene with two sulfur anchoring atoms, each bearing a p-π orbital.

A major difference between these two orbital interaction diagrams is that the non-QI system, i.e., 1,4-STB, has a HOMO-LUMO splitting; whereas the QI system, i.e., 1,3-STB has two degenerate non-bonding MOs (NBMOs). We will correlate the manifestation of QI with the splitting of the FMOs. As shown in Figure 3a, the phase of the bg HOMO on the C1 atom in butadiene is different from that on the C4 atom. Therefore, it can interact with the πg orbital of the S2 fragment, leading to the bg LUMO of 1,4-STB. On the other hand, since the phase of the au LUMO on the C1 atom in butadiene is the same as that on the C4 atom, it can interact with the πu orbital, leading to the au HOMO of 1,4-STB. In the resultant au HOMO and bg LUMO, a significant orbital localization on the S atoms can be seen. This is because the orbital levels of the S2 fragment are much closer to the resultant FMOs than those of the parent butadiene 11



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system. This is a direct consequence of the perturbation theory.39 Let us move on to scrutiny of the orbital interaction in a system where easy-zero QI occurs. As shown in Figure 3b, the orbital phase on the C1 atom in butadiene is different from that on the C3 atom in both bg HOMO and au LUMO. So they can interact with the πg orbital of the S2 fragment, resulting in the NBMO1. The πu orbital can mix with both au HOMO-1 and bg LUMO+1 of butadiene, yet their contributions are relatively small, so the lines between them are not shown. This orbital interaction gives rise to the NBMO2. If we assume each S atom bears only one π electron,40 the molecule obtained by the interaction between the main π-conjugated skeleton and the linker units can be viewed as a diradical. This is an orbital-based way of rationalizing the mathematically proved correlation between the emergence of QI and diradical existence.41 One would anticipate that the orbital interaction diagram for 2,3-STB might be similar to that for 1,4-STB because the phase of the bg HOMO on the C2 atom in butadiene is different from that on the C3 atom, while they have the same phase in the au LUMO. But it turns out that things are not so simple; as shown in Figure 4, the πg orbital of S2 is destabilized by the out-of-phase interaction with the bg HOMO of butadiene; whereas it is stabilized by the in-phase interaction with the bg LUMO+1 at the same time because it has a relatively large orbital amplitude on the interaction sites, yielding an bg orbital. The same is true for the relation of the πu orbital to the au LUMO and au HOMO-1 of butadiene. The resultant bg and 12


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au orbitals have the same energy at the HMO level, forming a degenerate pair of NBMOs.

Figure 4. Orbital interaction diagram for the PMO analysis of 2,3-STB in terms of the union of butadiene with two sulfur anchoring atoms, each bearing a p-π orbital.

Let us see how the PMO analysis works in the prediction of QI in STBs. When there is no orbital degeneracy, we can use eq. 3 as it is. However, when there are two degenerate NBMOs, the HOMO and LUMO terms need to be replaced with the NBMOs as shown below.

Cr ,NBMO1C s ,NBMO1 Cr ,NBMO2C s ,NBMO2 + . EF − ε NBMO1 EF − ε NBMO2

(5)

In this case, the opposite rule applies: QI is expected if the sign of Cr,NBMO1Cs,NBMO1 is different from that of Cr,NBMO2Cs,NBMO2 because both denominators have the same value. This prescription for the occurrence of QI can be summarized as, (Cr,NBMO1Cs,NBMO1)×(Cr,NBMO2Cs,NBMO2) < 0.

(6)

When this modified rule is applied to 1,3- and 2,3-STBs, one can find out that the rule 13



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adequately predicts the occurrence of both hard-zero and easy-zero QIs. In addition, we have also checked how the modified orbital rule works in a cyclic system (see section 5 of SI) and large π-conjugated systems (see section 6 of SI). A difficulty here arises from orbital degeneracy. Since any set of orthonormal linear combinations is an acceptable set of NBMOs, the choice of the set of NBMOs is arbitrary.37 In Figure 3b and 4, we show symmetry-adapted NBMOs, which one may call delocalized orbitals.42,43 However, one can move by a unitary transformation from delocalized MOs to localized MOs (LMOs).44,45 We have confirmed that the modified orbital rule is applicable to LMOs as well (see section 7 of SI). Finally, we turn to the predictability of QI in non-alternant systems, holding azulene up as an example because its conductive properties have attracted much attention recently.11,12,13,46,47 For example, current rectification46 and voltage-induced conductance switching and hysteresis47 are found. So it is essential to provide an insight into the transport properties of azulene in light of our modified orbital rule. Venkataraman and co-workers have carried out an experimental measurement for four azulene derivatives shown in Scheme 1.11 They found that the 5-7 connection shows the lowest conductance of the four derivatives, which can be assigned to the occurrence of QI. Let us see how one can validate this observation by means of the FMO analysis of azulene derivatives.

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Scheme 1. Structures of the four azulene derivatives measured in ref. 11. L indicates the thiochroman linker unit used for the electrode attachment (see the rightmost part).

Figure 5 shows the FMOs of four sulfur-terminated azulene (STA) derivatives: 1,3-STA, 2,6-STA, 4,7-STA, and 5,7-STA. They are a model system for the actually measured compounds. In all the four derivatives, the sign of the product of the MO coefficients on the S atoms in one FMO is different from that in the other. But there is a finite HOMO-LUMO gap in the 1,3-, 2,6-, and 4,7-derivaties, while the 5,7-derivative has two degenerate NBMOs. Therefore, QI is expected only for the 5,7-derivative, which is fully consistent with the above-mentioned experimental measurement. However, there is no discrimination between the hard-zero and easy-zero QIs in non-alternant systems due to the loss of alternancy. In SI, one can see the modified orbital rule works adequately in other non-alternant systems (see section 8 of SI). Also, additional analyses on the QI feature in azulene are shown in section 9 of SI.

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Figure 5. FMOs of the four sulfur-terminated azulene derivatives: 1,3-STA, 2,6-STA, 4,7-STA, and 5,7-STA.

Summary and Conclusions In summary, our orbital rule has been polished up so that it can predict a class of QI to which the original one cannot be applied. The amelioration has also made it possible to predict QI in non-alternant systems. The splitting between the FMOs resulting from the interaction between the π-conjugated skeleton and the linker units provides an essential insight into the conductive properties of a molecule. Now that one can quickly and easily calculate the Hückel molecular orbital on line, the cost required for the prediction of conductivity based on our scheme should be as little as other useful methods, such as drawing diradical structures41 and drawing a diagram.48 Essential symmetric natures of MOs remain 16


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almost unchanged even if one goes beyond the Hückel method, such as extended Hückel, Hartree-Fock, and DFT calculations.49 Therefore, the qualitative discussion on molecular conductance from the HMO perspective are helpful in understanding and predicting the electron transport properties of molecular wires.

Notes: The authors declare no competing financial interests.

Acknowledgements This work was supported by KAKENHI Grant numbers JP24109014 and JP15K13710 from Japan Society for the Promotion of Science (JSPS) and the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT), the MEXT Projects of “Integrated Research Consortium on Chemical Sciences”, “Cooperative Research Program of Network Joint Research Center for Materials and Devices”, “Elements Strategy Initiative to Form Core Research Center”, and JST-CREST “Innovative Catalysts”. The computation was mainly carried out using the computer facilities at Research Institute for Information Technology, Kyushu University.

Supporting Information:

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Electron transport calculations at the Hückel level, zeroth order Green’s function, the location of the Fermi level, PMO analyses, application of the modified orbital rule to p-quinodimethane, zethrene, and diphenalenyl compounds, discussion on the delocalized NBMO vs. localized NBMO, electron transport through some non-alternant hydrocarbons, additional analyses on the QI features in azulene, effects of heteroatomic linkers on the existence/absence of QI, electron transport calculations at the DFT level, and Cartesian coordinates for the structures used for the DFT calculations. This material is available free of charge via the Internet at http://pubs.acs.org.

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(14) Woodward, R. B.; Hoffmann, R. The Conservation of Orbital Symmetry; Academic Press: New York, 1970. (15) Fukui, K. Theory of Orientation and Stereoselection; Springer-Verlag: Berlin, 1975. (16) Tada, T.; Yoshizawa, K. Quantum Transport Effects in Nanosized Graphite Sheets. ChemPhysChem 2002, 3, 1035-1037. (17) Yoshizawa, K.; Tada, T.; Staykov, A. Orbital Views of the Electron Transport in Molecular Devices. J. Am. Chem. Soc. 2008, 130, 9406-9413. (18) Yoshizawa, K. An Orbital Rule for Electron Transport in Molecules. Acc. Chem. Res. 2012, 45, 1612-1621. (19) Tada, T.; Yoshizawa, K. Molecular Design of Electron Transport with Orbital Rule: toward Conductance-Decay Free Molecular Junctions. Phys. Chem. Chem. Phys. 2015, 17, 32099-32110. (20) Taniguchi, M.; Tsutsui, M.; Mogi, R.; Sugawara, T.; Tsuji, Y.; Yoshizawa, K.; Kawai, T. Dependence of Single-Molecule Conductance on Molecule Junction Symmetry. J. Am. Chem. Soc. 2011, 133, 11426-11429. (21) Zhao, X.; Geskin, V.; Stadler, R. J. Phys. Chem. 2017, 146, 092308. (22) Dewar, M. J. S. The Molecular Orbital Theory of Organic Chemistry; McGraw-Hill: New York, 1969. (23) Halpern, J.; Orgel, L. E. The Theory of Electron Transfer between Metal Ions in Bridged Systems. Discuss. Faraday Soc. 1960, 29, 32-41. (24) Richardson, D. E.; Taube, H. Electronic Interactions in Mixed-Valence Molecules as Mediated by Organic Bridging Groups. J. Am. Chem. Soc. 1983, 105, 40-51. (25) Solomon, G. C.; Andrews, D. Q.; Goldsmith, R. H.; Hansen, T.; Wasielewski, M. R.; Van Duyne, R. P.; Ratner, M. A. Quantum Interference in Acyclic Systems: Conductance of Cross-Conjugated Molecules. J. Am. Chem. Soc. 2008, 130, 17301-17308. (26) Nozaki, D.; Girard, Y.; Yoshizawa, K. Theoretical Study of Long-Range Electron Transport in Molecular Junctions. J. Phys. Chem. C 2008, 112, 17408-17415. (27) Coulson, C.A.; Rushbrooke, G. S. Note on the Method of Molecular Orbitals. Math. Proc. Cambridge Philos. Soc. 1940, 36, 193-200. (28) In SI we show transmission spectra calculated with density functional theory (DFT), all 20


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of which are consistent with the HMO results (see section 11 of SI). (29) Tsuji, Y.; Hoffmann, R.; Movassagh, R.; Datta, S. Quantum Interference in Polyenes. J. Chem. Phys. 2014, 141, 224311. (30) Solomon, G. C.; Bergfield, J. P.; Stafford, C. A.; Ratner, M. A. When “Small” Terms Matter: Coupled Interference Features in the Transport Properties of Cross-Conjugated Molecules. Beilstein J. Nanotechnol. 2011, 2, 862. (31) Pedersen, K. G. L.; Strange, M.; Leijnse, M.; Hedegård, P.; Solomon, G. C.; Paaske, J. Quantum Interference in Off-Resonant Transport through Single Molecules. Phys. Rev. B 2014, 90, 125413. (32) One might be worried that the liking group would change the QI feature. However, the qualitative trend whether QI occurs or not remains unchanged even if a linking group is added (see section 10 of SI). (33) Leary, E.; La Rosa, A.; González, M. T.; Rubio-Bollinger, G.; Agraït, N.; Martín, N. Incorporating Single Molecules into Electrical Circuits. The Role of the Chemical Anchoring Group. Chem. Soc. Rev. 2015, 44, 920-942. (34) Tsuji, Y.; Staykov, A.; Yoshizawa, K. Orbital Views of Molecular Conductance Perturbed by Anchor Units. J. Am. Chem. Soc. 2011, 133, 5955-5965. (35) Solomon, G. C. Cross-Conjugation and Quantum Interference. In Cross Conjugation: Modern Dendralene, Radialene and Fulvene Chemistry; Hopf, H., Sherburn, M. S., Eds.; Wiley-VCH: Weinheim, 2016; pp 397-412. ( 36 ) Heilbronner, E.; Bock, H. The HMO-Model and Its Applications: Basis and Manipulation; Verlag Chemie: Berlin, 1976. ( 37 ) Borden, W. T.; Davidson, E. R. Effects of Electron Repulsion in Conjugated Hydrocarbon Diradicals. J. Am. Chem. Soc. 1977, 99, 4587-4594. (38) Goudard, N.; Carissan, Y.; Hagebaum-Reignier, D.; Humbel, S. HuLiS 3.3.3 c, 2007, JDK 1.8.0_101: http://www.hulis.free.fr/ (39) Albright, T. A.; Burdett, J. K.; Whangbo, M.-H. Orbital Interactions in Chemistry, 2nd ed.; Wiley: Hoboken, 2013. (40) The accepted mechanism of Au-thiol bond formation involves the generation of a thiyl radical (RS·) to form a strong Au-S covalent bond (see ref. 33). (41) Tsuji, Y.; Hoffmann, R.; Strange, M.; Solomon, G. C. Close Relation between Quantum Interference in Molecular Conductance and Diradical Existence. Proc. Natl. Acad. Sci. U. S. A. 2016, 113, E413-E419. (42) Young, D. Computational Chemistry, A Practical Guide for Applying Techniques to Real World Problems; Wiley Interscience: New York, 2001. 21



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(43) Csizmadia, I. G. Theory and Practice of MO Calculations on Organic Molecules; Elsevier: Amsterdam, 1976. (44) Lineberger, W. C.; Borden, W. T. The Synergy between Qualitative Theory, Quantitative Calculations, and Direct Experiments in Understanding, Calculating, and Measuring the Energy Differences between the Lowest Singlet and Triplet States of Organic Diradicals. Phys. Chem. Chem. Phys. 2011, 13, 11792-11813. (45) Lahti, P. M. Magnetic Properties of Organic Materials; Marcel Dekker: New York, 1999. (46) El-Nahas, A. M.; Staykov, A.; Yoshizawa, K. First-Principles Calculations of Electron Transport through Azulene. J. Phys. Chem. C 2016, 120, 9043-9052. (47) Schwarz, F.; Koch, M.; Kastlunger, G.; Berke, H.; Stadler, R.; Venkatesan, K.; Lörtscher, E. Charge Transport and Conductance Switching of Redox-Active Azulene Derivatives. Angew. Chem. Int. Ed. 2016, 55, 11781-11786. (48) Pedersen, K. G. L.; Borges, A.; Hedegård, P.; Solomon, G. C.; Strange, M. Illusory Connection between Cross-Conjugation and Quantum Interference. J. Phys. Chem. C 2015, 119, 26919-26924. (49) Stowasser, R.; Hoffmann, R. What Do the Kohn−Sham Orbitals and Eigenvalues Mean? J. Am. Chem. Soc. 1999, 121, 3414-3420.

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Frontier Orbital Perspective for Quantum Interference in Alternant and

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