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The role of through-space interactions in modulating constructive and destructive interference effects in benzene Anders Borges, Jianlong Xia, Sheng-Hua Liu, Latha Venkataraman, and Gemma C. Solomon Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b01592 • Publication Date (Web): 26 Jun 2017 Downloaded from http://pubs.acs.org on June 28, 2017

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The role of through-space interactions in modulating constructive and destructive interference e↵ects in benzene Anders Borges,†,‡ Jianlong Xia,¶ Sheng Hua Liu,§ Latha Venkataraman,⇤,k and Gemma C. Solomon⇤,‡ †Department of Applied Physics, Columbia University, New York ‡Nano-Science Center and Department of Chemistry, University of Copenhagen, Copenhagen ¶School of Chemistry, Chemical Engineering and Life Science, Wuhan University of Technology, Wuhan 430070, China §Key Laboratory of Pesticide and Chemical Biology, Ministry of Education, College of Chemistry, Central China Normal University, Wuhan, 430079, China kDepartment of Applied Physics and Department of Chemistry, Columbia University, New York E-mail: [email protected]; [email protected]

Abstract Quantum interference e↵ects, be they constructive or destructive, are key to predicting and understanding the electrical conductance of single molecules. Here, through theory and experiment, we investigate a family of benzene-like molecules that exhibit both constructive and destructive interference e↵ects arising due to more than one contact between the molecule and each electrode. In particular, we demonstrate that the

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⇡-system of meta-coupled benzene can exhibit constructive interference and its paracoupled analog can exhibit destructive interference - and vice versa - depending on the specific through-space interactions. As a peculiarity, this allows a meta-coupled benzene molecule to exhibit higher conductance than a para-coupled benzene. Our results provide design principles for molecular electronic components with high sensitivity to through-space interactions and demonstrate that increasing the number of contacts between the molecule and electrodes can both increase and decrease the conductance.

Keywords molecular electronics, destructive interference, Scanning Tunneling Break Junction technique, quantum circuit laws Quantum e↵ects in the electrical conductance of single molecules may some day find a place in nanoscale electronic devices of practical use. A significant challenge, however lies in determining a robust set of circuit rules that reliably capture both the destructive and constructive interference e↵ects that are crucial to understanding the conductance of these systems. As an example, consider two parallel molecular wires between two electrodes. The conductance of this system is predicted to have up to four times the conductance of a single molecular wire due to constructive interference, 1–3 but one experiment measured only an imperfect ratio of 2.8. 4 Similar experiments where molecules have more than one possibility of anchoring to the same lead have also been investigated theoretically 5–9 and experimentally 10,11 but no simple links between theory and experiments have been demonstrated. As another example, consider the ⇡ system of six-membered aromatic rings where the electrodes are attached meta- with respect to each other. The conductance of such systems is expected to be low due to destructive interference, while para-coupled analogs are predicted to be better conductors. 12–16 Experiments have revealed that meta-coupled molecules in some cases do show lower conductance than the corresponding para-coupled molecules 17–19 but there have also been reports of deviation from this rule-of-thumb. 11,20–23 With such a contrast 2


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between simple model systems calculations and experimental observations, better understanding of interference e↵ects in single molecules is necessary to design molecular electronic components relying on interference. Here we probe the sources of imperfect interference in parallel molecular wires and meta-coupled benzene through single molecule conductance measurements and DFT calculations on a series of benzene derivatives. The series contains both single meta- and para-substituted benzenes and their analogs containing two benzene molecules in parallel. We show that specific through-space interactions have important and surprising consequences for electron transport across the molecules. We investigated the four molecules shown in Figure 1(a). The molecules all contain a central benzene ring functionalised with thiomethyl groups that can bind to Au electrodes through a weak and selective donor-acceptor bond. 24 The sulfurs are separated from the rings via methylene groups. This ensures weak coupling to the electrodes and also allows two benzene rings to be connected via the linker groups. This allows us to investigate the conductance of two benzene rings in parallel. The molecules 1 and 3 are connected to the binding groups in the para-position and are not expected to show destructive interference, while the molecules 2 and 4 are expected to show destructive interference due to the binding groups sitting meta- to each other. This reasoning is based on simple H¨ uckel models of the ⇡-system like the one shown in Figure 1(b). This shows six pz -like orbitals, each centered on a C and electronically coupled via hopping elements in a Hamiltonian to their closest neighbors. It also shows hopping elements between a single Au atom and two pz -orbitals on each side of the ring. The single Au atom is then electronically coupled to the rest of the Au electrodes as indicated with overlapping yellow spheres. We indicate stronger interactions with a full line and a weaker interaction with a dashed line. We can quantify the magnitude of the through-space interaction as the ratio of the two interactions

v2 . v1

Through-out this paper we refer to this as the relative

coupling. The binding groups in 1 and 2 can rotate freely around the methylene groups but in 3 and 4 they are more constrained because the binding groups connect two benzene rings. As

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Figure 1: a) Investigated molecules. b) Sketch of model system. Two pz orbitals are electronically coupled the same electrode with relative coupling vv21 . a result, we can expect the relative coupling of 1 and 2 to be di↵erent from that of 3 and 4 - and therefore also the transport properties (vide infra). We measured the conductance of the molecules using the Scanning Tunneling BreakJunction technique (STM-BJ). 25 Nano-junctions were repeatedly formed between a Au wire and a Au coated substrate under ambient conditions under a small bias in a solution of propylene carbonate (PC) containing the molecules using a modified STM setup with sub-˚ A piezo-electric control. In order to suppress the conductance of the solvent we applied a thin layer of Apiezon wax to the Au tip as described elsewhere. 26,27 We measured conductance as a function of the electrode displacement to generate conductance versus displacement traces and analyze these data using one- and two-dimensional histograms. 24 For this work, molecules 1, 3 and 4 were obtained from commercial sources and used without further purification. 2 was synthesized following known procedures as detailed in the supporting information (SI). To understand the experimental observations we simulated a large number of probable junction geometries using DFT and calculated the zero-bias conductance using a Green’s function approach and the Landauer formula. We used the ASE 28 /GPAW 29,30 package 4


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with the PBE functional 31 and a custom double-⇣ plus polarization function basis set 32 on all atoms. Using a recently reported approach 33 we partitioned the transport properties into di↵erent contributions by choosing the size of the central region and di↵erent basis set rotations. More information can be found in the SI. Briefly, the central region is taken as that of the benzene ring (binding groups are included in the electrodes) and we rotate the basis set in this region into one that maintains strict

or ⇡ symmetry for the equivalent benzene

molecule in vacuum. We also evaluate the junction Hamiltonian and self energies in the basis of pz -orbitals obtained using partly occupied maximally localized Wannier functions as reported elsewhere 34 to obtain model parameters. Conductances calculated using DFT are generally higher than those measured in experiment because DFT tends to underestimate the molecular band gap. 35,36 We present calculations without correcting for this because it allows us to estimate upper limits to the conductance of specific junction geometries. Figure 2(a) shows the experimentally obtained 2D histogram for the molecules 1-4 under a bias of -100 mV (applied to the tip). These represent the distribution of measured conductances recorded as the tip is retracted after breaking of the Au-Au contact which has a characteristic conductance of 1 G0 . We see that 1 exhibits a rather flat 2D histogram while there is a pronounced slope for 2. Figure 2(b) shows the average line profile between the vertical lines in the 2D histograms. From this we see that on average 1 is observed to be a better conductor than 2 for large displacements, corresponding to fully elongated molecular junctions. However, at small relative displacements (under 0.1 nm), the conductance of 2 is comparable or even larger than that of 1. From the line profiles we also see that both molecules exhibit two characteristic peaks with both molecules showing a peak around 10

3.5

G0 . Clearly, the simple picture of destructive interference versus constructive interference in meta- versus para-coupled benzene is insufficient to explain the transport properties of 1 and 2. In molecules 3 and 4, two benzene molecules are linked to each other through the CH2 SCH2

groups attached to the para- and meta- positions respectively. This has the

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e↵ect that the binding groups are more rigid and we should expect more well-defined features in the 2D histograms. The 2D conductance histograms of both 3 and 4 in Figure 2(a) indeed shows flatter and more well-defined features than those observed for 1 and 2. For 4 we see at least two well-defined conductance peaks. The 1D conductance profiles in Figure 2(b) shows the unexpected result that for many traces 3 is observed to be a worse conductor than 4. The results therefore show, that for a single benzene ring weakly coupled to the electrodes the meta-coupled molecule most often is a worst conductor than the para-coupled molecule, while the opposite is the case when two benzenes are coupled in parallel. To explain this we first discuss a few possible reasons why the conductance of meta-coupled benzene molecules can be high.

Figure 2: a) 2D conductance histograms of molecules 1-4 in Figure 1 based on 3000 traces at V=-100 mV. Same color scale is used in all four histograms. b) Line profiles averaged over the displacements between the two vertical lines in the corresponding 2D histogram.

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Figure 3a shows a H¨ uckel model of benzene consisting of six identical pz -orbitals in a ring electronically coupled to two electrodes through the orbitals sitting meta- to each other. It is straight forward to show that the zero-bias conductance of this system vanishes if the Fermi energy of the electrodes is equal to the on-site energy of the identical pz orbitals. In the following we use this model to explain deviation from the idealized result of the H¨ uckel model - namely that the zero bias conductance vanishes in the middle of the gap between the highest occupied and lowest unoccupied molecular orbitals (HOMO and LUMO). We shall refer to such deviation as masking. One hypothesis in the literature is that attaching binding groups to the benzene ring distorts the frontier orbitals of benzene and removes the destructive interference feature. 21,22 To the extent that binding groups only modulate the magnitude of the electronic communication between the pz -orbitals and the electrode, they cannot change the prediction of interference. Bulky binding groups, or tilted binding geometries, can also allow coupling between an electrode and more than one pz -orbital in the molecule, which can also mask the destructive interference. This mechanism is sketched in Figure 3(b) and can be thought of as a quantum mechanical short-circuit mechanism to alleviate destructive interference. For a classical system, the conductance will always increase if the number of connections to the electrode is increased. But as we are about to show, an increasing number of connections between the electrode and the molecule can both increase and decrease the conductance. We note that the through-space interactions as shown in the figure can be quantified as the ratio of the two interactions If the

v2 v1

- the relative coupling.

system of the benzene is coupled more strongly to the electrodes than the ⇡

system, transport through the orthogonal

system can dominate the current and thereby

e↵ectively mask destructive interference in the ⇡ system. This e↵ect is sketched in Figure 3(c). This was recently shown to be the case for 3,3’-bipyridine. 33 The partitioning into

and ⇡ system is trivial for benzene due to the mirror plane of the

ring, but attaching binding groups and electrodes to the molecule will break this symmetry.

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Figure 3: Representation of di↵erent sources of masking of destructive interference in metacoupled benzene. a) Ideal case where only one site is bound to each electrode. b) Allowing more sites to interact with each electrode can lead to through-space masking. c) The underlying system can carry current leading to masking. d) Realistic electrodes break perfect symmetry and the and ⇡ systems are coupled indirectly through the electrodes leading to /⇡ masking. This means that the

and ⇡ systems will be indirectly coupled via hopping elements in the

Hamiltonian via the electrodes. When this mixing is significant, the conductance, cannot be understood in terms of separate classical contributions as is the case for classical resistors in parallel. 20 Instead, both constructive and destructive interference between the system can occur. This situation is sketched in Figure 3(d) where lines indicate that

and ⇡ and

⇡ systems are coupled to each other via the electrodes. It has been proposed that many-body e↵ects can play an important role in perturbing the electronic structure of molecular junctions. 37 The most important many-body e↵ect is that of explicit electron-electron interactions which leads to screening e↵ects that decreases the molecular band gap and can shift the anti-resonances of molecules displaying destructive interference. 16 Interactions between electrons and molecular vibrations have also been shown to alleviate destructive interference 38 but calculations for benzene-type molecules indicate that the conductance of a meta-coupled benzene should still be much lower than the corresponding para-coupled benzene. 39 Vibrationally induced e↵ects are therefore not an important source of masking. We do not consider such e↵ects here and focus instead solely on sources of masking that can be captured by the Landauer formula. With these sources of masking in mind, we are now at a position to discuss the theoretical results. 8


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Table 1 shows the experimental and average calculated conductance of 168 junction geometries containing 1 and 2. The experimentally determined high- and low conductance peak of 1 and 2 were estimated by fitting Lorentzians to the peaks in Figure 2(b). The average conductance calculated by DFT overestimates the total conductance as expected. We also see the degree to which the average conductance can be understood in terms of and ⇡ and mixed /⇡ contributions as outlined in Figure 3. Table 1: Conductance of molecules 1 and 2 in units of 10 Molecule Experimental G (high/low) Avg. calculated G . contr. ⇡ contr. /⇡ contr. Six site model

1 7.1/3.8 21.6 0.1 (0.6 %) 22.0 (102.2 %) -0.6 (-2.6 %) 21.6 (100.0 %)

4

G0

2 4.1/2.0 7.3 1.2 (17.0 %) 7.1 (96.8 %) -1.0 (-13.8 %) 6.7 (91.1 %)

We see that the calculated contribution to the conductance from the

system is smaller

for 1 than for 2 as expected due to a longer tunneling path in 1. Importantly however, the conductance of 2 lies below the experimental low-conductance peak of 2. Since our calculated conductance is an upper limit to the experimental conductance this indicates a lower length limit that allows the observation of destructive interference. For a shorter molecule, destructive interference would be masked by here, the

conductance, but for the molecules studied

system cannot be the dominant contributor to transport. We see also that the

conductance through the ⇡ system is unexpectedly high for 2. This could indicate that the six site model in Figure 3 is an inadequate approximation for the ⇡ system. However, the six site model derived from DFT reproduces almost the same conductance as the full ⇡ system. This shows that

and /⇡-masking can not be the primary source masking in the calcula-

tions. A recent experimental result showed that inserting alkyl groups of di↵erent lengths between the binding group and the benzene ring influences the relative conductance of metaand para-connected benzene. 22 When the binding group was inserted directly on benzene, the meta-connected molecule was a better conductor than the para-connected benzene, but 9


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when one or two aliphatic carbons were inserted in between the binding group and the ring, the para-connected molecule was observed to the better conductor. From this analysis we can now understand that it is both the through-space interactions and the tunneling length that determines the lower threshold of conductance for these types of molecules. On the basis of our transport calculations we can conclude that the primary source of masking of destructive interference in our calculations is due to through-space masking. The conductance of the simple model system introduced in Figure 1b can be calculated analytically in the limit of weak coupling between the Au atom and the ring. This is demonstrated in the supporting information (SI). The resulting conductance as function of the relative coupling to the electrodes is shown in Figure 4. Surprisingly the line-shape is independent of the molecular band-gap but also of the specific magnitude of the coupling to Au. This allows the calculated conductance to be normalized to units of a single paracoupled benzene ring. The dashed lines separate four di↵erent regimes depending on the through-space interactions and each regime displays strikingly di↵erent conductance. In the regime labeled meta-2, the conductance is zero through-out. This means that the destructive interference is robust with respect to the through-space masking for the particular interactions shown on the principal axis. This robust destructive interference feature relies on perfect coherence across the interface between Au/binding group and the pz orbitals. When this is the case, the perfect destructive interference feature is robust. Without it, the conductance would increase with increasing coupling strength to the electrodes as is the case for classical electrodes (detailed in the SI). When the through-space interactions correspond to those in the regime labeled meta-1 we see a large variation of the conductance where the conductance ranges from zero to four times the conductance of a single para-coupled benzene ring. This striking result indicates that the conductance of a meta-coupled benzene can be higher than that of a purely paracoupled benzene ring due to through-space interactions. In the limit where v1 tends towards v2 the notion of meta- versus para-coupled benzene is meaningless. Still when the relative

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coupling is 12 , the conductance of the meta-coupled molecule is the same as that of a purely para-coupled molecule. This shows that the destructive interference is highly sensitive to through-space interactions in this regime. In the regime labeled para-1, the conductance of the para-coupled benzene ranges from one to four times the conductance of a purely para-coupled ring and is therefore also highly sensitive to the relative coupling. In the regime labeled para-2 we see that through-space interactions actually reduce the conductance of the para-coupled benzene quite e↵ectively. This striking result shows that a primarily para-coupled benzene actually can exhibit destructive interference. This destructive interference feature goes against the classical result that increasing the coupling to the electrodes can only increase the conductance - a result we also show in the SI. We note that without the interference features just described, the para-coupled benzene never exhibits a lower conductance than a meta-coupled benzene with a similar relative coupling.

Figure 4: Solid lines: Conductance as function of relative coupling for meta- and paracoupled molecules for ideal model system. Dots: Calculated conductance from DFT plotted against the calculated relative coupling. The figures below the first axis indicate which interaction with the electrodes are large (bold line) and which ones are weak (dotted line). In the figure we also show the calculated conductance normalized to the conductance of 11


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a purely para-coupled benzene for 168 fully relaxed junction geometries using DFT. Most of the points lie close to the analytical result shown as the solid line. We find that the largest deviations from the analytical result (as those for meta-2 ) occur for cramped junctions where the coupling strength to the left and right electrodes is asymmetric and there are more than two through-space interactions to the same electrode. Our calculations indicate that for fully extended junctions, only two pz -orbitals couple appreciably to the binding groups/electrodes and the analytical result is such a good approximation to the full DFT calculation that the four regimes are likely to be observed experimentally. We therefore propose that the two conductance peaks of 1 and 2 reflect the two regimes for each molecule. The low characteristic conductance of 2 would then correspond to meta2 or the first half of meta-1 where the destructive interference is robust with respect to through-space interactions. The high conductance peak then corresponds to meta-1 where through-space masking gives considerable contributions to the conductance thereby masking the destructive interference. The high conductance peak of 1 could then correspond to para1 where through-space interactions increase the conductance. Finally the low-conductance peak of 1 could correspond to para-2 where through-space coupling gives rise to destructive interference which results in a conductance comparable to the high conductance of 2. We expect the constructive and destructive interference e↵ects mapped out in Figure 4 to be sensitive to coupling between two parallel rings and also the extent to which the rings are coupled indirectly through the binding groups and electrodes. Both could a↵ect constructive interference associated with parallel benzene wires. The mechanisms for imperfect constructive interference are less intuitive than those of destructive interference but can be explained using the same types of sketches as in Figure 3. 5,6 Before we present the results of our calculations for 3 and 4 we first go through a few mechanisms that can mask the expected constructive interference of two molecular wires in parallel. The case of perfect constructive interference between two benzene molecules in parallel occurs when the two benzene molecules are coupled weakly to the same site in the electrode

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Figure 5: Sketch of sources of masking of constructive interference. Thin lines corresponding to small hopping terms in the Hamiltonian. a) Rings are weakly connected only via electrodes, a situation leading to perfect constructive interference. b) Rings are not connected via hopping elements to the electrodes (i.e. injection is not coherent) and we see incoherence masking. Coupling between the two rings either through the electrodes (c) or directly (d) leads to indirect or direct coupling masking respectively. Hamiltonian. We demonstrate this in the SI. In this case, the conductance of the two combined benzene rings is equal to four times that of a single benzene ring due to constructive interference. This is sketched in Figure 5(a). Here, the thin lines represent small hopping elements in the Hamiltonian and the thick lines represent large hopping elements in the Hamiltonian. Note that the hopping elements in the ring systems are much larger than those connecting the molecules with the electrodes. Constructive interference will be masked if the two rings are not connected weakly via the electrodes. The conductance of the two benzene rings in this case is just the classical sum of the separate conductances. This is sketched in Figure 5(b). If the molecules are coupled strongly to each other through the electrodes, for instance via the binding groups, this may also mask the constructive interference. This is sketched in Figure 5(c). Constructive interference can also be masked if the molecules are coupled directly. For two benzene molecules in parallel this may occur if the two molecules interact directly via ⇡-⇡ stacking. This situation is similar to short-circuiting two parallel wires and is sketched in Figure 5(d). Having discussed possible reasons for masking constructive interference we are ready to analyze the computational results. Table 2 shows the calculated conductance for the isomers of 3 and 4 in the regimes 13


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Table 2: Transport calculations of 3 and 4. Conductance in units of 10 4 G0 . Experimental numbers indicated with a star (*) are obtained by fitting a single Lorentzian to the broad feature of 3 in Figure 2(b). Regime

G(Expt.) G(DFT) ⇡ /⇡ L / R (eV) xL /xR GDD /GSD GDD (cut)/GSD

3 (high G) para-1

3 (low G) para-2

4 (high G) meta-2

4 (low G) meta-1

12.6* 40.6 1.1 (2.6%) 37.9 (93.5 %) 1.6 (3.9 %) 0.14/0.14 0.63/0.63 2.6 3.5

12.6* 14.5 0.1 (0.7 %) 9.9 (68.3 %) 4.5 (31.0 %) 0.16/0.16 0.56/0.58 2.5 3.9

26.9 27.6 48.6 (175.9 %) 4.6 (16.8 %) -25.6 (-92.7 %) 0.16/0.22 0.70/ 0.48 3.2 3.3

13.8 24.6 1.5 (6.3 %) 30.1 (122.1 %) -7.0 (-28.4 %) 0.21/0.21 0.45/0.45 3.3 3.9

discussed for 1 and 2 (Figure 4). More isomers are possible for 4 and we discuss this issue in the SI. We see that the calculated conductance of 3 corresponding to para-2 is lower than for the one in para-1 as one would expect due to destructive interference in the latter. We also see that the conductance of 4 for meta-1 is higher than 3 for para-2 as might be expected due to constructive interference for meta-1 and destructive interference for para-2. However, the calculated conductance for meta-2 is higher than that for meta-1. This does not agree with our finding that meta-2 should exhibit robust destructive interference with respect to through-space interactions. However, table 2 also shows the conductance partitioned into symmetry components. This shows that the conductance of meta-2 cannot be understood in terms of symmetry components. This was not observed in the other calculations despite similar binding geometries and strengths. As we show in the SI, this is due to a gateway state that dominates the conductance for this particular isomer due to its particular geometry and almost 0.5 ˚ A shorter tunneling length. As consequence, the conductance of the molecule can not be understood in terms of its ⇡ system. Something similar may occur for 2 in cramped

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junctions but this was not observed for extended molecules, probably because of the flexible nature of the linkers that allow extension of the molecule. From our transport calculations we can evaluate the ratio of conductance for the double decker molecule and the equivalent monomer (the details are provided in the SI). In the table we denote this ratio as GDD /GSD . As seen in the table, the ratio ranges from 2.6 to 3.3 which is far from the theoretical result of 4.0, which means that perfect constructive interference between the rings is not upheld. The e↵ect of indirect (Figure 5c) and direct coupling (Figure 5d) cannot be investigated separately but the e↵ect of both can be removed at the same time in our calculations and the conductance ratio can then be reevaluated (full details in SI). The result is shown as GDD (cut)/GSD in Table 2. From this we see that the conductance ratio with interactions cut between the wires almost reproduces the perfect superposition law as in the model system except for meta-2. This eliminates coherence masking as the source of imperfect constructive interference for these molecules. We can therefore associate direct and indirect coupling as the primary source of imperfect constructive interference between the two parallel molecular wires. This is consistent with calculations reported by Vazquez et al. 4 who showed that stretching the molecular junctions from their equilibrium positions increased the conductance ratio. We can therefore associate constructive/destructive interference in the meta-/para-connected molecules as the primary source of di↵erence in conductance. Importantly, the calculations show that the low conductance of the para-coupled molecules is due to destructive interference while the high conductance of the meta-coupled molecules is due to constructive interference. With these observations we are able to rationalize the experimental line profiles shown in Figure 2b. This showed a larger spread in the measured conductance for 3 and well-defined peaks for 4. The high conductance peak can be attributed to meta-2 which is due to a highconducting gateway state which does not reflect the robust destructive interference in the ⇡ system. This was not observed for 2 and we associate it with the reduced flexibility of the binding group in 3 and 4 when compared to 1 and 2(detailed in SI). The low-conductance

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peak could correspond to meta-1 where constructive interference allows the molecule to conduct better than para-1. The broad conductance peak of 3 can then be attributed to lowconductance contributions from molecules displaying destructive interference corresponding to para-2 and high-conductance molecules displaying constructive interference corresponding to para-1. To reiterate, by analyzing DFT transport calculations we have shown that the ”imperfect” interference of the molecules 1-4 can be understood in terms of di↵erent types of masking of which through-space masking is the most important. As a result of through-space masking the ⇡-system of meta-coupled benzene can exhibit constructive interference and its para-coupled analog can exhibit destructive interference - and vice versa - as through-space coupling is modulated. This allows the ⇡ system of meta-benzene to be a better conductor than that of para-benzene. We have explained this in terms of regimes that involve di↵erent through-space couplings to the electrodes and full quantum coherence across the molecular junction. Since the line-shape in Figure 4 does not depend on the band-gap of benzene our model is likely to be robust to screening e↵ects captured by higher level methods such as GW, since these primarily change the molecular band-gap. 40 The intrinsically quantum circuit rules described here are in direct conflict with both classical circuit rules where increasing interactions with the electrodes always increases the measured conductance, and also the rule-of-thumb that a meta-coupled benzene is a worse conductor than that of a para-coupled one as evidenced by our STM break-junction experiments. Our results mean that any molecular junction with more than one connection to the same electrode can exhibit interference phenomena similar to that of the ⇡-system of benzene. This knowledge should be taken into account when designing potential binding groups and highlights that more is not necessarily better: increasing the coupling to the electrodes can both increase and also decrease conductance. It also demonstrates that robust destructive interference can be turned into efficient constructive interference and vice-versa through subtle changes in through-space interactions between molecule and electrode. While it has

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been shown that destructive interference can be highly sensitive to weak through-space interactions, 41,42 we have shown that certain interactions actively protect the destructive interference inherent in a molecule or can introduce destructive interference where it was thought there was none. We imagine that the idea of robust destructive interference of meta-2 can be combined with the high sensitivity to through-space interactions of meta-1 to make sensors that are highly specific to distinct through-space interactions and rely on coherence across the electrode/molecule interface.

Acknowledgement A.B. and G.C.S. acknowledge financial support from the Danish Council for Independent Research, Natural Sciences and the Carlsberg Foundation. L.V. acknowledges support from the Packard Foundation. J.X. acknowledges financial support from National Natural Science Foundation of China (21502147) and the Fundamental Research Funds for the Central Universities(WUT: 2017II42GX ).

Supporting Information Available Derivation of analytic results, evaluation of conductance ratios, computational details, symmetry partitioning and six-site model, synthetic details of 2, discussion of isomers of 4. This material is available free of charge via the Internet at http://pubs.acs.org/.

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