Effects of Aromaticity and Connectivity on the ... - ACS Publications

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Effects of Aromaticity and Connectivity on the Conductance of FiveMembered Rings Anders Borges and Gemma C. Solomon* Nano-Science Center and Department of Chemistry, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark S Supporting Information *

ABSTRACT: Even though five-membered rings, for example, thiophene, are ubiquitous in organic and molecular electronics, as a class of molecules, they resist a simple interpretation. Generally containing four sp2-hybridized carbon atoms, the fifth position can be filled by any number of substituents. This flexibility leads to a diverse range of electronic properties, but also presents a challenge for deriving a model description. Starting from a noninteracting Hamiltonian obtained from Kohn−Sham density functional theory calculations, we derive an effective four-site model that provides a unified description of these systems. We rationalize the zero-bias conductance of these molecules in terms of aromaticity and connectivity. The conductance was found to be highly sensitive to connectivity, as for benzene, but we also found the conductance to be sensitive to aromaticity. The model predicts the same relative conductance as reported in prior experiments in almost all cases and provides a link between chemical intuition and single-molecule conductance. The method used to develop the model is general and could be applied to other types of molecules.

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INTRODUCTION When presented with a molecule, a chemist would like to have some degree of chemical intuition that allows him/her to make predictions about its properties. As an example, consider benzene. As part of a general chemistry education, one learns that the six-electron π system is stabilized compared with hexatriene through aromaticity. Most chemists also remember that substituents influence meta positions differently from para and ortho positions. As this knowledge pertains to electronic communication through the system, unsurprisingly, some of this intuition can be of value for the design of molecular electronic components. However, a direct relationship between concepts such as aromaticity and substitution patterns and the molecular conductance is not trivial. To properly justify any such connection, it is necessary to employ a theoretical framework that directly relates the experimental observable with the hypothesis being investigated. This has been successfully done for the case of the conductance of a single benzene ring linked to two electrodes through binding groups attached in the meta and para positions. This can be achieved with a Hückel model of the π system for benzene, and predictions that the conductance of the meta-coupled molecule is much lower than that of the corresponding para-coupled molecule because of destructive interference1−5 have also been borne out by experiment.6−8 Furthermore, this prediction is robust toward increasing levels of sophistication in the model.9 As soon as one tries to extend these ideas more generally, however, problems arise. Consider the case of the (relatively) ubiquitous five-membered ring. Unlike benzene, it is not trivial to write down a simple Hückel model for five-membered conjugated molecules that can be related to experiment and is robust toward increasing levels of sophistication in the model. © XXXX American Chemical Society

This is due, in part, to the mixing of the pz orbitals on the four C atoms in the ring with those on a heteroatom (as in furane and thiophene) or those in chemical bonds (as in dimethylcyclopentadiene). In the latter case, the electrons, in turn, also communicate with the σ system of the molecule, which complicates the construction of a model. One way to proceed is to construct effective Hamiltonians by the use of Green’s function methods.10 These methods are particularly useful for these molecules because it is the pz orbitals involved in the double bonds that interact with the binding groups and, ultimately, the metallic leads. If a relationship between the pz orbitals and the conductance could be found, there would be a theoretical framework that could be compared with experiments. In this work, we derive this relationship for fivemembered rings resembling those depicted in Scheme 1. This scheme shows four pz orbitals electronically coupled to each Scheme 1. (Left) Sketch of the Molecular Junction, Including the Au Leads, σ System of the Molecule, Heteroatom (X), pz Orbitals, and Definition of the Atom Numbering and (Right) Model System Where the Effects of the Environment on the pz Orbitals Have Been Projected Out and the Energy-Dependent Parameters of H̃ are Defined

Received: January 10, 2017 Revised: March 29, 2017 Published: March 29, 2017 A

DOI: 10.1021/acs.jpcc.7b00283 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

membered ring contribute significantly to the transport. In this case

other, a set of metallic leads, depicted as two Au tips, and the rest of the molecule, which includes a heteroatom, X, and the underlying σ system depicted as green spheres. A similar description was derived by Hybertsen et al.11 In that case, a relationship between the splitting of the highest occupied molecular orbital (HOMO) and HOMO − 1 and conductance was derived for a molecule containing amine linkers bound to a few Au atoms.11 It was assumed that the conductance of the molecule was well represented by a two-site model, which is not always the case.12 A similar method was used by Herrmann and Elmisz to evaluate the splitting of molecular orbitals and relate the splitting to the conductance.13 In this work, we investigate the correlations between connectivity, aromaticity, and the electrical conductance of five-membered rings. We first explain how to construct effective Hamiltonians in an intuitive basis from which the zero-bias conductance can be calculated. We then demonstrate the method on a series of molecules displaying different aromaticities and connectivities. Finally, we compare our results with the experimental results available in the literature.

r R a T (E) ≈ Γ LppGpq Γ qq Gqp

where the indices p and q denote the pz orbitals centered on two corners of the ring. We order the basis functions such that the four pz orbitals are the first four basis functions in the central region. The effects of the non-pz orbitals on the pz orbitals can be included by invoking an energy-dependent molecular self-energy, Σm. This will reduce the size of the central region to that of the four pz orbitals at the cost of extra complexity in the energy dependence of Σm. The matrix element Grpq can then be re-expressed as r Gpq (E) = [ES4 − H4 − Σ4L(E) − ΣR4 (E) − Σm(E)]−pq1

(3)

where S4 is the overlap matrix between the four pz orbitals and equivalently for H4, ΣL4 (E), and ΣR4 (E) and Σm(E) = [ES4M − H4M − ΣL4M(E) − ΣR4M(E)]g rm(E)

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[ESM4 − HM4 − ΣLM4(E) − ΣRM4(E)]

THEORY For the connection between pz orbitals and conductance, we turn to the Landauer formalism, with which one can treat semiinfinite systems such as a molecule attached to electrodes. We consider an infinite system described by a noninteracting electronic Hamiltonian such as that obtained from Kohn− Sham density functional theory. We work in a nonorthogonal local basis that can be partitioned into the left lead, the central region, and the right lead, where the leads are periodic far from the central region. We assume that the central region is electronically coupled to both leads through off-diagonal elements in the Hamiltonian and overlap matrix but that the electrodes are not coupled to each other in this way. We furthermore assume that we can transform the Hamiltonian and overlap matrix to a local normalized basis set that conforms to our intuition. By this, we mean that we can construct a linear combination of the basis functions that lives up to our expectation of a pz orbital centered on four corners of our molecule containing a five-membered ring. At equilibrium, we are free to choose the size of our central region as we wish. As the central region, we choose all basis functions centered on the five-membered ring including our pz orbitals. All basis functions centered outside the ring are taken to belong to the left and right leads, as appropriate. This includes potential binding groups. The zero-bias conductance of this electrode−molecule− electrode system, G, can be calculated as G = G0T(EF), where G0 is the quantum of conductance and T(EF) is the transmission evaluated at the Fermi energy of the infinite system as T (E ) =

(4)

is the molecular self-energy that projects out the effects of all other basis functions on the four pz orbitals. Here, the subindex 4M refers to the off-diagonal submatrix between the four pz orbitals and the remaining basis functions in the five-membered ring. The subindex M4 refers to its conjugate transpose. grm is the Green’s function of the non-pz basis functions in the fivemembered ring and is calculated as grm(E) = [ESm−Hm − ΣLm − ΣRm]−1. Here, Sm is the overlap matrix of the non-pz basis functions in the five-membered ring and similarly for Hm, ΣLm, and ΣRm. From Grpq(E), the zero-bias conductance can be evaluated according to eq 2. This requires the inversion of [ES4 − H4 − ΣL4 (E) − ΣR4 (E) − Σm(E)]. At equilibrium, there exists a 4 × 4 model system with the same zero-bias conductance described by the effective Hamiltonian H̃ = H4 + Re[ΣL4 (EF)] + Re[ΣR4 (EF)] + Re[Σm 4 (EF)], with the purely imaginary lead self-energy Σ̃ = Im[ΣL4 (EF)] + Im[ΣR4 (EF)] + Im[Σm 4 (EF)]. The procedure is illustrated in Scheme 1. The left-hand side represents the complicated system with the four pz orbitals coupled to the heteroatom (X), the σ system, and the Au leads. It also defines the numbering of the pz orbitals. The right-hand side depicts the effective four-site Hamiltonian, which, due to symmetry, is characterized by only two effective on-site elements, E1(EF) and E2(EF), and the coupling elements connecting the sites, H̃ 12 = H̃ 34 = −a(EF), H̃ 23 = −b(EF), and H̃ 14 = c(EF). We are interested in how the connectivity of this effective four-site model is reflected in the conductance. If we assume an orthonormal basis and use the Fermi energy as the zero-point energy, we can evaluate Grpq in the limit of weak coupling to the leads and far from molecular resonances as Grpq(EF) = [−H̃ − ̃ −1 Σ̃ ]−1 pq ≈ [−H]pq . Using Cramer’s formula, this expression can be evaluated as

∑ Γ ijLGjkr Γ klRGlia ijkl

(2)

(1)

Here, Gr(E) = [ES − H − ΣL(E) − ΣR(E)]−1 is the retarded Green’s function; S and H are the overlap matrix and Hamiltonian, respectively, of the central region; and ΣL/R represents the left/right lead self-energies, for which ΓL/R = −2Im(ΣL/R). The sum in eq 1 runs over all basis function in the central region. Consider the case where only terms that include elements between two pz orbitals centered on the five-

r Gpq (E F ) ≈

( −1)b + a detqp( −H̃ ) det( −H̃ )

(5)

Here, detqp(−H̃ ) is the determinant of −H̃ with the qth row and pth column removed. Because the zero-bias conductance is proportional to absolute square of Grpq(EF) and the connectivity indices appear only in the numerator, the relative conductance B


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The Journal of Physical Chemistry C of identical five-membered rings connected differently to the leads is proportional to |detqp(−H̃ )|2. Because of the simplicity of the four-site model, this equation can be evaluated analytically or in the spirit of Stadler and co-workers.14−16 detqp(−H̃ ) can be evaluated graphically using a method that relates a sum of diagrams by a set of rules to detqp(−H̃ ). Because we do not make explicit use of the diagrams, we refer the reader to the literature for the rules used to construct them. Instead, we work with the analytic expression of detqp(−H̃ ). We have now outlined how to evaluate the contribution to the conductance from the pz orbitals that we assume are responsible for the conductance through the molecule. This is a sensible assumption for molecules that are short enough to be dominated by off-resonant tunneling transport and long enough to be dominated by current through the π system in accordance with eq 2. In the Supporting Information (SI), we demonstrate that this approximation is sensible for a realistic molecule bound to two Au electrodes.

electrode22 and the binding groups.23,24 In the SI, we show that using Au electrodes and conjugated amine linkers lead to a Fermi energy shifted by roughly −0.5 eV with respect to the center of the band gap. The validity of our approach relies on the assumptions that we have made along the way. Using the Landauer formula and the single-particle picture on top of DFT does not give the correct conductance, even if the exact functional is known.25,26 Furthermore, the result is sensitive to the functional being used. Generally, inaccuracy in the functional leads to underestimation of the molecular band gap.27,28 As a result, we likely overestimate the predicted conductance. Despite these issues, DFT usually predicts the correct trends in conductance. Because we focus on qualitative predictions and chemical trends, we believe that our results hold meaning even in comparisons with experiments. Furthermore, we evaluate the contribution to transport assuming that only the pz orbitals are electronically coupled to the leads. The partly occupied maximally localized Wannier functions used as pz orbitals obey a stationary criterion and are therefore well-defined for the gas-phase molecule and unique for every molecule. Attaching binding groups might shift the alignment with frontier orbitals but also their shape. We address these issues in the SI and confirm that our assumptions are sensible. For short conjugated binding groups, we imagine that our model realistically captures the qualitative features of the chemistry.

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PRACTICAL ASPECTS We used density functional theory (DFT) as implemented in the ASE17/GPAW18,19 packages to obtain a realistic electronic structure of a number of molecules containing five-membered rings. The geometries of the molecules were relaxed to within 0.02 eV/Å using the Perdew−Burke−Ernzerhof (PBE) functional and a grid spacing of 0.18 Å with 6 Å of vacuum, periodic boundary conditions, Γ-point sampling, and a double-ζ plus polarization functions (dzp) basis set.20 A self-consistent DFT calculation for the optimized molecule was executed to obtain the Kohn−Sham operator and overlap matrix in the local basis. The converged calculation was fed to the Wannier module of ASE to obtain the pz orbitals as the maximally localized partly occupied Wannier functions.21 This calculation requires that the number of unoccupied states to include in the localization algorithm be set. Inspired by Thygesen et al.,21 we treat this number as a parameter and find the number that maximizes the localization per state, which we found is often the one that yields the sought-after pz orbitals in terms of the atom-centered basis. We then express the Kohn−Sham operator and overlap matrix in the basis of these four pz orbitals. Although constructed to be orthonormal, the maximum off-diagonal overlaps are up to 10−5. We therefore Löwdin orthogonalize the pz orbitals to within numerical accuracy to obtain truly orthogonal pz orbitals. This allows for the construction of a basis with the same span as the full dzp basis set through the Gram−Schmidt orthogonalization of the remaining N − 4 basis functions to the pz orbitals, where N is the number of atomcentered basis functions in the calculation. Next, we assume that only the pz orbitals can couple to the leads through an empirical energy-independent wide-band selfenergy ΣLpp = ΣRqq = iγ, where γ = 0.1 eV. We show in the SI that this is accurate. This corresponds to weak coupling to the electrodes. It allows for the construction of grm without knowledge of the coupling to the leads from the non-pz orbitals. From grm, one can construct Σm(E), which is entirely real. Because the calculation is performed on gas-phase molecules, it is necessary to specify a Fermi energy. This value depends on the electrode being used but lies somewhere between the highest occupied and lowest unoccupied Kohn−Sham orbitals of the gas-phase molecule. For lack of a better choice, we set the Fermi energy to be the midpoint of these two energies. In more realistic systems, it depends on the nature of the

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RESULTS We begin by investigating the energy dependence of the effective Hamiltonian for three sample molecules and relate it to chemical intuition prevalent in the chemical literature. Next, we investigate the correlation between aromaticity and the effective Hamiltonian. We then proceed to an investigation of the conductance as a function of connectivity, and finally, we relate our findings to experimental results in the literature. Energy Dependence of the Effective Hamiltonian. Figure 1a shows the energy dependence of the effective Hamiltonian element H̃ 14 for thiophene (red), 5,5-dimethylcyclopentadiene (green), and the hypothetical compound borole (blue). Thiophene is considered to be aromatic because the S atom is sp2-hybridized and contributes two electrons to the π system such that the ring fulfills Hückels 4n + 2 rule.29 Borole is considered to be antiaromatic because the pz orbital centered on B is unoccupied and therefore does not ”mix” with the occupied pz orbitals centered on the C atoms.30 It can therefore be thought of as fulfilling the 4n rule associated with antiaromatic systems, which makes it highly reactive.29,31,32 5,5-Dimethylcyclopentadiene is considered to be nonaromatic29,33 because the quaternary carbon has all of its electrons involved in σ bonds and, therefore, is effectively noncyclic. It is evident that the effective coupling H̃ 14 is close to zero across the energy range for 5,5-dimethylcyclopentadiene, whereas there is a large energy dependence for thiophene and borole. We find that this effective coupling originates almost exclusively from the molecular self-energy. For thiophene, the effective coupling is positive in the band gap and changes sign at about −2.5 eV. This indicates the existence of a local energy level below the energy levels of the pz orbitals that interacts with the sites labeled 1 and 4. This picture is consistent with the idea that a pz orbital on S mediates coupling across the ring. For borole, the sign of the effective coupling is negative in the band gap, and there is a sign change at about +2.9 eV. This indicates the existence of an unoccupied local C


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Figure 2. Investigated molecules and resulting effective four-site Hamiltonian, H̃ . Colors of spheres indicate on-site energies, whereas colors and widths of arrows indicate off-diagonal elements in H̃ . The color code is shown at the bottom, and all values are given in eV.

from the color bar. Also shown is the value of the effective coupling between the sites represented as arrows with a color and width corresponding to the value of the effective coupling. The molecules are sorted according to the parameter c, which is the effective coupling between sites 1 and 4 in the middle of the band gap. This happens to also be the order in which the molecules are sorted according to the aromaticity as predicted by the nucleus-independent chemical shifts (NICSs), as we show in the SI. The NICS value describes the tendency of a ring to display paratropic or diatropic ring currents in the presence of a magnetic field.34 Hence, the aromaticity correlates with the effective coupling, H̃ 14 = c. The molecules range from from antiaromatic (cyclobutadiene) to nonaromatic (5,5dimethylcyclopentadiene) to aromatic (thiophene). As can be seen in the figure, the magnitude of the effective coupling between the sites is larger than the effective on-site energies, but the difference in magnitude decreases with increasing aromaticity and antiaromaticity. Cyclobutadiene is special, however, because it displays equivalent on-site energies similarly to the nonaromatic species despite being antiaromatic. We shall see later that this has important consequences for its zero-bias conductance. One can also see that c ranges from −2.8 eV for cyclobutadiene to +2.1 eV for X = N-Ph. Arrows between the four sites are drawn only if the magnitude of the corresponding effective Hamiltonian element is larger than 0.1 eV. This makes it evident that there is practically no effective electronic coupling between non-neighboring sites in the fourmembered ring, which indicates that our constructed pz orbitals are very local. It is tempting to think of the effective Hamiltonian as a Hückel model from which one can calculate a band gap and molecular orbitals. It is important, however, to remember that the effective Hamiltonian is energy-dependent and, therefore, does not afford the same interpretation as a Hückel model. Still, the effective Hamiltonian at the Fermi energy corresponds to a Hückel model coupled to a set of wide-band leads that produces the same zero-bias conductance as the full Hamiltonian. This affords a way to think about the effective

Figure 1. (a) Effective coupling between the pz orbitals closest to the heteroatom. (b) Effective on-site energy of the pz orbital closest to the heteroatom.

energy level that interacts with sites 1 and 4 and is consistent with an unoccupied pz orbital centered on B. Figure 1b shows the effective on-site elements for site 1 for the three molecules. This shows that the effective on-site energy at the Fermi energy for site 1 is positive for the aromatic thiophene molecule, smaller for the nonaromatic 5,5-dimethylcyclopentadiene, and even smaller for the antiaromatic borole. Having investigated the energy dependence of the effective Hamiltonian, we note that, according to eqs 2 and 5, it is only the value of the effective Hamiltonian at the Fermi energy that determines the zero-bias conductance. Relationship between Aromaticity and the Effective Hamiltonian. To investigate the correlation between aromaticity and the effective Hamiltonian, we calculated the effective Hamiltonian at the center of the band gap for a family of molecules spanning the full spectrum from antiaromatic through nonaromatic to aromatic. The resulting effective Hamiltonians are visualized in Figure 2. Here, the effective on-site energies of the pz orbitals are shown as spheres with a color representing the value of the on-site energy as indicated D


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The Journal of Physical Chemistry C Hamiltonians from a molecular orbital perspective. For instance, the effective coupling between on-site elements in the effective Hamiltonian of 5,5-dimethylcyclopentadiene resembles the Hückel model of butadiene because of bond alternation and a lack of interaction between sites 1 and 4. That of borole resembles a Hückel model of cyclobutadiene with strong bond alternation and the same sign for all electronic coupling elements. The latter causes it to have a small band gap. The aromatic compounds are characterized by a positive effective 1,4 coupling while the remaining couplings are negative. This corresponds to a Möbius aromatic Hückel model where the 1,4 interaction increases the band gap.35 This is where the limitations of considering the model Hamiltonian as a Hückel model for the system are most evident. Molecules such as thiophene and pyrrole are five-center, six-electron aromatic molecules, not four-site Möbius aromatic systems. By treating them as four-site systems, we force the molecular selfenergy to renormalize the sites closest to the heteroatom by changing their on-site energies and coupling them to reproduce the correct matrix element Grpq. Luckily, this renormalization is small in the middle of the band gap when compared to the hopping parameters H12 and H23. This affords the luxury of thinking of the aromaticity parameter c as a perturbation to nonaromaticity. Still, we stress that caution should be exercised in interpreting the effective Hamiltonian as a Hückel model. Relationship between the Effective Hamiltonian and the Zero-Bias Conductance. To determine the conductance, we must consider how the molecules are connected to the leads. Figure 3a shows the four different ways in which electrodes can be connected to five-membered rings. It is rewarding to first consider the case of weak effective coupling between sites 1 and 4, as is the case for nonaromatic molecules. In this limit, attaching the leads in sites 1,4 and 1,2 corresponds to linearly conjugated species, which are typically considered relatively good conductors. Connecting the leads to sites 2,3 and 2,4 corresponds to cross-conjugated species, which are known to display destructive interference.36−38 This means that, in the limit of weak effective coupling between sites 1 and 4 (c), we expect low conductance for nonaromatic molecules when the leads are connected in the 1,3 and 2,3 positions. To investigate the effects of the 1,4 effective coupling, we show also in Figure 3 the Markussen−Stadler−Thygesen (MST) diagrams and the value of detqp(−H̃ ) at the Fermi energy under the assumption that only neighboring sites in the four-membered rings are electronically coupled. For the 1,4connected molecule, there are only two terms/diagrams, which evaluate to det41( −H̃ ) = −a 2b + cE2 2

Figure 3. (a) Sketches indicating connectivities, resulting MST diagrams, and detqp(−H̃ ). (b) Calculated zero-bias conductance as a function of c for the molecules shown in the inset.

In this case, the first term, a3, is larger than the corresponding first term in the 1,4-connected molecule when there is bond alternation. This means that, for the nonaromatic molecules, where E1 ≈ E2 ≈ c = 0, the conductance of the 1,2-connected molecule should be slightly larger than that of the 1,4connected molecule. Figure 3b shows the calculated zero-bias conductance, which is roughly proportional to |detqp(−H̃ )|2. Here, one can see that, indeed, the 1,2-connected molecule shows slightly higher conductance than the 1,4-connected nonaromatic molecules but also for all other species. We turn next to the 1,3-connected molecule, for which the terms/diagrams evaluate to det31( −H̃ ) = −acE2 − abE1

(6)

One can see that both terms contain an on-site energy, and the conductance should therefore vanish due to destructive interference when E1 = E2 = 0, as is roughly the case for the nonaromatic molecules and cyclobutadiene. This is not the case for the antiaromatic molecules (except cyclobutadiene) and aromatic molecules, where the on-site-energies are different from the assumed Fermi energy. In Figure 3b, one can see that, indeed, the conductance is much lower than for the 1,4- and 1,2-connected molecules. As expected, the conductance increases as the on-site energies differ from zero. For cyclobutadiene, the conductance is very low because the onsite energies are all roughly zero. Finally, we turn to the 2,3-connected molecules, for which

Here, a and b describe the effective coupling between sites 1 and 2 (the formal double bond) and sites 2 and 3 (the formal single bond). For the antiaromatic systems, where c is positive and the on-site energy E2 is positive, the two terms have opposite sign (the exception being cyclobutadiene). The first term is therefore much larger than the second term for our molecules. The same is the case for aromatic systems, where c is negative and E2 is negative. For the 1,2-connected molecule, there are three terms/ diagrams, which evaluate to det 21( −H̃ ) = a3 + aE1E2 + abc

(8)

(7) E


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The Journal of Physical Chemistry C det32( −H̃ ) = bc 2 − bE2 2 + a 2c

(9)

Here, one can see that there are two diagrams/terms that do not contain on-site loops and that these terms depend on c. This means that 2,3-connected molecules should be expected to be very sensitive to the parameter c and less so to the on-site energy. This is also reflected in Figure 3, where the nonaromatic molecules exhibit very low conductance and high dependence on c. Finally, we can comment on the zero-bias conductance as a function of aromaticity. For the 1,4- and 1,2-connected molecules, there is a tendency for a general decrease in conductance as c increases. As shown in the SI, c also correlates with the aromaticity. We can therefore correlate an increasing aromaticity with a decreasing conductance for these types of molecules. Relation to Experiments. The idea that increasing aromaticity decreases molecular conductance was presented by Breslow and Foss in 200839 and investigated experimentally in a series of studies. In one, it was reported that the conductance of derivatives of 5,5-dimethylcyclopentadiene, furan, and thiophene showed decreasing conductance as a function of aromaticity. This is in good agreement with our predictions for 1,4-connected molecules as outlined in the previous section. A more recent work demonstrated that a heavily substituted analogue of the ketone (X = CO), shown in Figure 2 with c = −1.2 eV, actually showed lower conductance than an aromatic thiophene analogue and a nonaromatic analogue of 5,5-dimethylcyclopentadiene. This is inconsistent with our model, which predicts that antiaromatic molecules should conduct well. A full DFT calculation including linker groups and Au also predicts a high conductance for this molecule, as we show in the SI. This indicates that the discrepancy between experiment and calculation might be due to the failure of DFT to predict the correct qualitative conductance. It should be noted, however, that the ketone exhibits destructive interference just outside the band gap and that this destructive-interference feature is highly sensitive to the on-site energy of the pz orbital centered on oxygen. Because DFT is known to underestimate the band gap, it is likely that the discrepancy arises as a result of a failure of DFT to correctly predict interference features for this molecule. We stress that this error is not related to our method but rather is inherent in DFT. In any case, the ketone is a special case because it features two sp2-hybridized atoms connected to the four pz orbitals. Further work explored the conductance of biphenylene.40 This molecule can be interpreted as an antiaromatic cyclobutadiene molecule that is chemically protected by having two of the four bonds embedded in benzene rings. The molecule is shown in Figure 4a to the far left. Thiomethyl groups were used as binding groups and attached on the opposite side of the ring, as indicated by dashed lines. The conductance of this molecule was compared with that of the fluorene, also shown to its immediate right in Figure 4a. The central five-membered ring in this molecule was considered to be nonaromatic, as shown in Figure 2, where c is roughly zero for the molecule without benzene rings. It was found that the conductance values of the biphenylene and fluorene were approximately the same. This went against the authors’ hypothesis that antiaromaticity should lead to high conductance, and it was speculated that the benzene rings decreased the antiaromaticity to an extent that made the system nonaromatic. We interpret the result using a different picture. Consider the sketch of the molecules shown

Figure 4. Conductance of fluorene type wires. (a) Experimentally measured molecules and their corresponding effective Hamiltonian. (b) Model Hamiltonian which can be reduced to the effective Hamiltonian on the right. All on-site elements are assumed to be zero. Thin lines indicate a hopping element − t and a wiggly line indicate a hopping element of c. Thick lines indicate a hopping element of −2t. Connectivity is indicated with a dashed line corresponding to a wideband self-energy of −iγ on the indicated site. (c) Conductance as a function of c as indicated in a).

in Figure 4b. This sketch represents a Hückel model of the molecules connected to two leads through wide-band leads with strength iγ through the sites indicated. The lines correspond to hopping elements in the Hamiltonian between sites with identical on-site energies set to zero. Using the method outlined in the Theory section, it is easily shown that the conductance of this model is equivalent to the conductance of the four-site model indicated to the right of the sketch. This shows that the effects of projecting out the eight pz orbitals in the molecule corresponds to doubling the hopping element that makes up one side of the benzene molecule (the formal double bond in the five-membered ring) while all other hopping elements remain unchanged. It is also seen that, whereas the magnitude of the coupling to the leads is unchanged, the connectivity of the four-site model is such that the site sitting meta to the leads is the only site that is coupled to the leads in the four-site model. This shows that the molecule probed in the experiment actually corresponds to a F


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concept of antiaromaticity, nonaromaticity, and aromaticity with positive, zero, and negative signs of the parameter c. As for benzene, we found that the investigated five-membered rings can show robust signs of destructive interference in the conductance when the leads are connected in the 1,3 and 2,4 positions. When connected in the 1,4 and 1,2 positions, the conductance is dominated by the molecular band gap and, hence, the parameter c. Antiaromatic systems are therefore predicted to be better conductors than nonaromatic systems, which, in turn, are predicted to be better conductors than aromatic systems. We compared our results to those available in the literature and found qualitative agreement except for the antiaromatic cyclopentadienone (X = CO), for which the predicted conductance was higher than expected. We showed that the apparent failure is not due to our method but rather inherent in DFT. Our results reveal that the conductances of single molecules are intricately connected to their topologies and that simple effective Hückel models can be used to make sense of the large differences in conductance observed in experiments. Our model provides a minimal basis on which a discussion of the chemistry and physics can be based in terms of the parameters a, b, c, E1, and E2. Furthermore, the method provides a direct link between the DFT calculations on a conjugated gas-phase molecule, a simple Hückel model, and the zero-bias conductance. The method of projecting out the effects of non-pz orbitals while maintaining the span of the full basis set can be used to develop simple models for other advanced molecular topologies without making unjustified assumptions.

1,4-connected four-site model exhibiting heavy bond alternation. Figure 4a also shows the equivalent four-site model predicted by projecting out the effects of all other basis functions in the DFT calculation of the molecules without binding groups attached. One can see that the parameter a is roughly doubled, as expected, and all other parameters are similar to those shown in Figure 2. This is exactly the result predicted for the model system. In Figure 4c, we show the calculated conductances of the molecules in Figure 4a. This shows a conductance that is largely insensitive to the aromaticity parameter c for the 1,4connected effective Hamiltonian. This explains why a similar conductance was observed in experiment. Attaching the leads at the 2,3 or 1,3 positions is predicted to have a much larger dependence on the aromaticity parameter c according to eqs 9 and 8. Had the experimentalists compared these molecules instead, our simple model predicts a much larger difference in conductance due to the difference in the aromaticity parameter c. The figure also shows the calculated conductances for the other connectivities. This figure indeed shows a difference in conductance of the two molecules of up to 2 orders of magnitude. Another work41 investigated the effects of fluorene derivatives similar to the three molecules shown on the right in Figure 4a. Here, thiomethyls were used as binding groups in the places indicated by the dashed lines. X = N-Ph was used in the experiments, but because of complications with band entanglement, we simplified that system to NH. The simple model shown in Figure 4b indicates that the molecules can be interpreted as 2,3-connected five-membered rings exhibiting strong bond alternation. Experimentally, it was found that the N-Ph molecule was a much better conductor than the aromatic furan analogue and the nonaromatic Si−H2 analogue. From Figure 2, one can see that c = 2.1 for X = N-Ph, whereas the value is 1.2 for X = O and −0.6 for X = Si. According to eq 9, the conductances of these molecules should depend highly on the aromaticity parameter c, through the terms bc2 and a2c. This dependence is reflected in the calculated conductances in Figure 4c, which shows the same relative conductances as observed in experiment. In the experimental article, the results were explained in terms of splitting of the HOMO and HOMO − 1, which, when a two-site model is assumed for the conductance, is proportional to the square root of the conductance.11 Our results here demonstrate that the lower (but similar) conductances of the molecules with X = Si−H2 and X = O can be understood as a manifestation of imperfect destructive interference masked by the aromaticity parameter c. This masking is highly sensitive to c and is therefore less pronounced for the molecule with X = NH or X = N-PH for which the magnitude of c is larger. Figure 4 shows that, if the equivalent molecule with X = C−H2 had been measured, the observed conductance could have been much lower than for the other molecules.

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ASSOCIATED CONTENT

S Supporting Information *

This material is available free of charge via the Internet at http://pubs.acs.org/. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.7b00283. Validity of assumptions, correlation of c with aromaticity, and realistic calculation for ketone (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Gemma C. Solomon: 0000-0002-2018-1529 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Roald Hoffmann for helpful suggestions. We acknowledge financial support from the Danish Council for Independent Research, Natural Sciences, and the Carlsberg Foundation.

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CONCLUSIONS We have investigated the zero-bias conductances of fivemembered rings in terms of aromaticity and connectivity. We derived a model for the conductance of five-membered rings assuming that the Hamiltonian was noninteracting and that four pz orbitals were responsible for coupling to the leads. Using DFT, we obtained the pz orbitals as maximally localized partly occupied Wannier functions and extracted parameters for our model. On the basis of the parameters, we correlated the

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REFERENCES

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