Reverse Bond-Length Alternation in Cumulenes: Candidates for

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Reverse Bond-length Alternation in Cumulenes: Candidates for Increasing Electronic Transmission with Length Marc H. Garner, William Bro-Jørgensen, Pernille D. Pedersen, and Gemma C. Solomon J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b05661 • Publication Date (Web): 01 Aug 2018 Downloaded from http://pubs.acs.org on August 3, 2018

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The Journal of Physical Chemistry

Reverse Bond-length Alternation in Cumulenes: Candidates for Increasing Electronic Transmission with Length Marc H. Garner, William Bro-Jørgensen, Pernille D. Pedersen, Gemma C. Solomon* Department of Chemistry and Nano-Science Center, University of Copenhagen, Universitetsparken 5, DK-2100, Copenhagen Ø, Denmark.

1 ACS Paragon Plus Environment

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Abstract. Single-molecule conductance generally decays exponentially with the length of the molecule when the transport mechanism is a coherent tunneling process. However, it was recently found that this length dependence can be reversed in linear conjugated molecules if the bond-length alternation is reversed. In this work we show that even-carbon cumulenes show this behavior as the bond-lengths are reversed for the dominant π system compared to the equivalent polyenes and polyynes. We explore the electronic origins of the reversed bond-length alternation in cumulenes and its relation to the length dependence of the electronic transmission. Through density functional theory and non-equilibrium Greens function calculations we predict that cumulenic wires have reverse decay of transmission with length, i.e., the decay constant β is found to be negative. As a direct consequence of the reversed bond-length alternation, the electronic transmission increases with length as the HOMO-LUMO gap rapidly narrows. Based on recent progress in cumulene synthesis, we discuss substituent strategies that may increase reverse bond-length alternation. Cumulenes stand out as promising candidates for a series of molecules that may show reverse decay of single-molecule conductance with increasing length.


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Introduction The tunneling probability of a quantum mechanical particle decays exponentially with the length of the potential energy barrier, a trend that is seen in electron transfer and transport experiments for a range of molecular systems1-7. In single-molecule junction experiments the low-bias conductance between two metal electrodes is measured through a bridging molecule8-10, essentially constituting the tunneling barrier for the system. The attenuation of the single-molecule conductance, G, with length is generally described by = · exp (− · )

(1)

where g0 is a pre-factor that depends, in part, on the interface between the molecule and the electrodes, L is length in arbitrary length units or number of atoms, and β is the exponential attenuation factor11-16. When the tunneling transmission between two electrodes through a linear πconjugated molecule is modelled at the Hückel level of theory, β can be related to the bond-length alternation = −2 ln

(2)

where β is the exponential attenuation factor, and ts and td are the resonance integrals of the formal single and double bonds, respectively.17 However, Tsuji et al. noted this relationship has a peculiar implication: If the bond-length alternation of the molecule is reversed, β will be negative, and the tunneling transmission will increase with increasing molecular length.17 In linear conjugated molecules the ground-state geometry has formal single bonds that are longer than the double bonds, and consequently ts < td. In a geometry without bond-length alternation (ts = td), the molecule is metallic and there is no decay with length. Beyond the metallic limit, a regime of reverse tunneling decay may exist for molecules where ts > td, and thus the tunneling



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transmission increases with the length of the molecule. Since the first suggestion of a reverse transmission decay system by Tada and Yoshizawa18, different reverse and decay-free systems have been suggested, including armchair graphene nanoribbons19, ethyndiyl-linked and fused oligomers of porphyrin20-24, and quinoidal molecules25-27. Still, reverse decay of single-molecule conductance with length has yet to be verified experimentally and the search for stable molecules with reverse bond-length alternation continues.28-29 Here we suggest cumulenes as a potential candidate for a series of molecules with reverse bondlength alternation in their stable ground-state geometries. While it has been duly noted in a number of studies that cumulenes show weak bond-length alternation30-41, the fact that the alternation is reversed and may correspond well to the situation described by Tsuji et al.17 has not yet been explored. In this work, we examine the possibility of reverse decay of tunneling transmission in evencarbon cumulenes due to reverse bond-length alternation. We study the molecules through density functional theory (DFT) and Hückel model calculations. All DFT calculations presented in the manuscript are done in the Atomic Simulation Environment42 (ASE) with the PBE exchangecorrelation functional43 and double-ζ plus polarization basis set for all atoms as implemented in GPAW.44-45 A benchmark study for this method is included in Supporting Information part A. The paper proceeds as follows. First, follows an introduction to the electronic structure of cumulenes. Second, we explore reverse bond-length alternation and conductance decay with simple Hückel models. Next, we present DFT calculations of the single-molecule transmission for alkene, cumulene, and alkyne series. This is followed by a brief exploration and discussion of chemical strategies for increasing the reverse bond-length alternation effect in cumulenes, and finally we summarize the findings.


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Electronic structure of cumulenes Cumulenes are linearly π-conjugated molecules with all-double bond Kekulé structures. Odd n [n]cumulenes (n = 3, 5 …), where n is the number of cumulated double bonds so that the number of carbon atoms is n+1, have been synthesized and characterized up to [9]cumulene.46-52 We will denote these simply as odd [n]cumulenes. We only treat the odd [n]cumulenes here, i.e., the evencarbon series; the odd-carbon cumulenes will be treated in separate studies due to their helical orbital structure.53-55 Scheme 1: Two Orthogonal π systems of [3]cumulene

The basis of p orbitals in cumulenes has two important features. First, as shown in Scheme 1, the two π systems are orthogonal. Different from the even [n]cumulenes, for which these can mix and form helical π orbitals as we have described in detail elsewhere55, the π systems of odd [n]cumulenes remain separated by symmetry also when the molecule is substituted. The second important feature is that the two π systems are displaced by one carbon atom relative to each other.32 The long π-system (blue in Scheme 1) spans n+1 atoms and lies out of the hydrogen plane of the molecule; the short π-system (red in Scheme 1) spans n–1 atoms and lies in the plane of the



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hydrogens. In this sense, the central double bond in [3]cumulene lies in the plane of the hydrogens, while the two double-bonds at the edges are equivalent to those we know from butadiene. In electron transport experiments the electrodes will typically primarily couple into one of the two π-systems as they are separated by symmetry. Though dependent on the specific molecular design and experimental method, it is most likely that the transport properties will be dominated by the out-of-plane π-system as this extends fully onto the terminal carbon atoms. As we shall see, it is in this π-system the bond-lengths are “reversed” in the sense that unlike the equivalent alkene and alkyne the shortest bonds are no longer at the edges of the molecule. Bond-length Alternation in Cumulenes. In Figure 1 we show the calculated structures for alltrans alkene, odd [n]cumulene, and alkyne molecules. Note that we use non-standard nomenclature for the alkenes and alkynes. For clarity, we use the same n-index for all three series, corresponding to the number of cumulated double bonds in the cumulene; the total number of carbon atoms is n+1 in all cases. In Figure 1b, the optimized structures and bond lengths are plotted for [3]alkene (butadiene), [3]cumulene (butatriene), and [3]alkyne (butadiyne). The difference between these three linear carbon molecules is just two hydrogen atoms between each. While the alkene and alkyne have the shortest bond at the edges as expected from their Kekulé structures, the cumulene has the longest bond at the edges and the shortest bond in the middle.


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Figure 1. DFT-calculated bond lengths. (a) Chemical structures of [n]alkene, [n]cumulene, and [n]alkyne, where x is a positive integer and n = 2x+1. (b) Optimized structures and bond lengths in Ångström for [3]alkene, [3]cumulene, and [3]alkyne. (c) Bond lengths of [11]alkene, [11]cumulene, and [11]alkyne. In Figure 1c, we show the calculated bond lengths for n = 11 for the three series of molecules. Again, the first and last bonds are the shortest for [11]alkene and [11]alkyne in agreement with the Kekulé structures. For [11]cumulene the bond length alternation is very small in the center of the molecule, and the molecule approaches the true cumulenic structure with equal bond lengths. The weak alternation continues for the longer members of the cumulene series and is consistently reverse compared with alkenes and alkynes31, 33, 50. Worth noting, though all bonds in the Kekulé structures of cumulenes are denoted as double-bonds, the carbon atoms are sp-hybridized and consequently the bond lengths are very similar to those of the alkynes.



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The calculated (reverse) bond-length alternation in the cumulene series is somewhat underestimated compared with calculations at higher levels of theory as presented in Supporting Information part A and crystallographic data.50 The results presented herein should therefore be considered a lower-boundary of the reverse conductance decay effect. The reverse bond-length alternation trend is consistent throughout the cumulene series studied here (up to n=19). Alkynes and cumulenes are both molecular forms of the “infinite” 1D material carbyne56-59. These have received much attention recently and their mechanical and electronic properties have been studied extensively60-65. In relation to carbyne, we note that reverse bond-length alternation is an edge effect; in a periodic structure there is either bond-length alternation or no bond-length alternation.66-68 In this sense, reverse bond-length alternation is purely a chemical concept, because chemists expect alternation between single and double bonds as drawn in Kekulé structures. Consequently, chemists expect linear conjugated molecules to start and end with double or triple bonds; if not, the molecule is a (di)radical. Frontier Molecular Orbitals. What is the origin of the reverse bond-length alternation in cumulenes? In Figure 2a we examine the frontier molecular orbitals of [5]cumulene, which consists of two orthogonal π-systems. Again, the long π-system is n+1 atoms long; the short π-system in n– 1 atoms long, now with some hyperconjugation onto the hydrogen atoms. These two π-systems are separated by symmetry and their nodal structure is displaced in space.55


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Figure 2. Electronic structure calculated with DFT. (a) Molecular structure and frontier molecular orbitals of [5]cumulene animated with iso-value = 0.02. (b) HOMO-LUMO gap as function of length for [n]alkene, [n]cumulene, and [n]alkyne. In alkenes and alkynes, bond-length alternation stabilizes the highest occupied molecular orbital (HOMO) and destabilizes the lowest unoccupied molecular orbital (LUMO), in equivalence to Peierls distortion in infinite atomic chains.69 Normal bond length alternation opens the HOMOLUMO gap and consequently the gaps are larger for alkynes than for alkenes as plotted in Figure 2b. Reverse bond length alternation has the opposite effect and therefore the HOMO-LUMO gap for cumulenes is smaller than that of alkenes. Though alkynes and cumulenes have almost the same stucture, the calculated HOMO-LUMO gap is over half an eV smaller for the cumulenes.



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In cumulenes a distortion of the bond lengths that stabilizes the HOMO will destabilize the neardegenerate HOMO–1 because its nodal structure is displaced by one atom.32 Furthermore, the HOMO–1 has significant hyperconjugation to the hydrogen atoms, thus making the normal bond length distortion more expensive. The antibonding hyperconjugative interaction means that the inplane and out-of-plane MOs switch place for the longer members of the series. The extra cost of the hyperconjugative interaction appears to be the reason the in-plane π system wins, and the bond length alternation is reversed as seen in Figure 1. Consequently, the long π system which we expect to dominate the electron transport has its bond lengths reversed, and correspond to the situation described by Tsuji et al.17

Model Calculations We now turn to model calculations to understand the relation between electronic structure and transmission decay with the Hückel method (tight binding model with only nearest neighbor interactions). We set the coulomb integral (on-site energy) to 0 eV, and for a double-bond we use resonance integral t = –3 eV. We use different variations of t based on the DFT-calculated bond lengths for the different molecules. We make three models. The first two are shown diagrammatically in Scheme 2, and are based on the bond lengths of the alkene series as described in Supporting Information part B. Here, for the formal double bonds we use td = t, and for the formal single-bonds we use ts = 0.935·t; we refer to this model as Normal-BLA because t d > t s. Scheme 2: Normal-BLA and Reverse-BLA models


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To understand the opposite limit, we reverse the model so td = 0.935·t and t s = t. This model we refer to as Reverse-BLA because ts > td, that is, the formal single-bonds are “shorter” than the formal double-bonds. However, from the data presented in Figure 1 we note that the cumulene bond lengths are not completely reversed compared with the alkene and alkyne. It is primarily the first bond that is long, after that there is only very weak bond-length alternation. Shown diagrammatically in Scheme 3, we make a model which corresponds to the bond lengths of the cumulene series, which we refer to as Cumulene-BLA. Here, there is only weak bond-length alternation in the central bonds. We use t1 = t and t2 = 0.990· t, while for the longer edge bonds we use ted = 0.968·t. This model corresponds only to the cumulene π-system that is out of hydrogen plane, i.e., the long π-system which we expect to dominate the transport properties. Scheme 3: Cumulene-BLA model.

We calculate the Landauer transmission with the non-equilibrium Green’s functions (NEGF) approach with wide-band electrodes attached to the terminal sites in our models with non-zero matrix elements of ΓL/R set to γ = 0.5 eV. In Figure 3 we examine the transmission decay. The transmission decays exponentially with length for the Normal-BLA model, and increases exponentially with length for the Reverse-BLA model as previously predicted17; the cumulene model is in-between as one would expect from equation 2. Comparing Figure 3a and 3b, there are two differences worth noting. First, the smaller HOMO-LUMO gap in Reverse-BLA moves the resonances closer to the Fermi energy, which is arbitrarily set to zero in the model. And more



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subtly, the resonance shape is wider (less decaying) for Reverse-BLA. The resonance position and broadness are the two effects that control the off-resonant transmission.

Figure 3. Transmission calculated with Hückel method for different model lengths (values of n). (a) Transmission as function of energy for the normal-BLA model. (b) Transmission as function of energy for the reverse-BLA model. (c) Transmission at the Fermi energy for reverse-BLA (orange), cumulene-BLA(red), and normal-BLA models (purple). Zeroth-order Greens Function. For Hückel models the transmission at the Fermi energy, T(EF), can be approximated using the zeroth-order Green’s function as described in the work of Yoshizawa and coworkers 18, 70-71. This analysis yields the following proportionality.


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( ) ∝

, ∙ ," $ # −

%

(3)

Here ci,L/R is the orbital coefficient of the i'th molecular orbital on the site of contact of the left (L) or right (R) electrode, and εi is the eigenvalue of the i'th molecular orbital. The numerator and denominator are not independent of each other (the eigenvectors and eigenvalues of the Hamiltonian). The decay of transmission is essentially a competition between the orbital coefficients at the contact points becoming smaller, and the HOMO-LUMO gap becoming smaller (making the inverse of the molecular orbital eigenvalues larger). There is an important link to the diradical character of the molecules here, which in the MO picture loosely corresponds to frontier orbitals with large orbital coefficients on the terminal atoms and small HOMO-LUMO gap.72-75 The connection between diradical character and transmission has been discussed in detail elsewhere,29, 72-73, 76-79

as well as its relation to the coupling between the associated magnetic centers73, 80-84.

Following this type of analysis, Tada and Yoshizawa found that reverse transmission decay in modified graphene appears due to the frontier orbital localizing on the edges.18-19, 85-86 To explore this suggestion for a cumulenic system, we decompose the zeroth order Green’s function (eq. 3) for the HOMO of the three Hückel models. In Figure 4a the square of the term for the HOMO in equation 3 is plotted. It is, to a very good approximation, proportional to the transmission at the Fermi energy (Figure 3c), as expected by considering the Coulson-Rushbrooke pairing theorem.87-88 We now look at the two factors: The square of the inverse HOMO eigenvalue (panel b, mind that EF is mid-gap) and the square of the product of the orbital coefficients (panel c). While the inverse of the HOMO eigenvalue results in a large difference between the three models, there is only a small difference between the orbital coefficients of the three. In all three cases the terminal orbital coefficients still decay significantly with length, and reverse bond-alternation only dampens this effect.



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Figure 4. Decomposition of the contribution from the HOMO to the transmission by analysis of the zeroth-order Green’s function at the Fermi energy as function of length. (a) The term of the HOMO in the zeroth-order Green’s function has the same trend as the transmission in Figure 3. (b) Square of the inverse of the HOMO eigenvalue, i.e., the denominator in the zeroth-order Green’s function. (c) Square of the product of the HOMO orbital coefficients at the left and right contact sites in, i.e., the numerator in the zeroth-order Green’s function. By direct comparison of Figure 4b and 4c, it is clear that the main contributor to the reverse decay of transmission is the smaller HOMO-LUMO gap.27 Algethami et al. also attributed the increasing transmission with length in fused porphyrin oligomers to a rapid decrease in the HOMO-LUMO gap as the oligomer length is increased.24 In this context a vanishing HOMO-LUMO gap generally


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corresponds to increasing diradical character, which may provide some chemical intuition for reverse conductance decay. It is clear that as the HOMO-LUMO gap vanishes the conductance will saturate at a value close to G0, the quantum of conductance, and the transport will become resonant; thus forming the upper limit for reverse conductance decay. In cumulenes the increasing diradical character with length can be observed in the reduced energetic cost of axial torsion as the groundstate and diradical transition-state geometries approach each other energetically.30,

89-92

The

importance of the terminal orbital coefficients and the HOMO-LUMO gap was also discussed in detail by Stuyver et al.29 on the topic of conductance decay, and Proppe and Herrmann73 who looked at electronic communication through molecular bridges in a broader context. It is clear that the increasing diradical character and vanishing HOMO-LUMO gap are intimately linked to the edge-localization of frontier orbitals described for functionalized graphene.18-19,

85-86

Further

exploration of this connection between the one- and two-dimensional cases may lead to additional insight into the physical mechanism of reverse transmission decay and to more candidate molecular systems with this intriguing property.

DFT Transmission Let us now examine the electron transport with DFT combined with the NEGF approach93. Single-molecule junctions are built by placing the molecule between two four-atom Au pyramids (tetrahedrons) on 4x4 Au(111) surfaces with periodic boundary conditions in the plane of the surfaces. The molecule in the junction is relaxed to a force threshold of 0.05 eV/Å with all Au atoms kept fixed during the optimization, and the transmission is calculated with a k-point sampling over a 4x4x1 Monkhorst-Pack mesh in the first Brilloun zone. Many studies have focused on the transport properties of atomic chains of carbon, but none of them correspond to the molecular geometries typically probed in STM break-junction experiments.41, 84, 94-98 Prasongkit et al.94 found the transmission of even-carbon cumulenic wires to



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be almost independent of length, and similar trends were seen for the magnetic coupling through such systems by Sarbadhikary et al.84 However, the type of electrode-molecule contact considerably influences the transport properties of π-conjugated molecules.99 Here we construct Au-molecule-Au junctions where the molecule is weakly bound through a donor-acceptor bond between amine groups and Au electrodes, where the contact geometries are generally well-defined and the transport is known to be off-resonant.100 In Figure 5a, optimized junction structures are shown for the eightcarbon species.

Figure 5. DFT transmission of amine-linked single-molecule junctions. (a) Optimized junction structures of amine-linked [7]alkyne, [7]cumulene, and [7]alkene. (b) Transmission plotted semilogarithmically against energy for odd [n]cumulenes, n = 3-9. c) Transmission plotted semilogarithmically against energy for [n]alkynes, n = 3-9. (d) Transmission at the Fermi energy as function of length (n) for [n]alkene (purple), [n]cumulene (red), and [n]alkyne (blue), for n = 3-19. All transmission plots are included in SI part C. The transmission of the four shortest members of the alkyne and cumulene series are plotted in Figure 5b and 5c. As expected from the model calculations the transmission decays exponentially in the middle of the HOMO-LUMO gap (around EF) for the alkyne series, while the transmission does not show clear decay for the cumulene series as the HOMO-LUMO rapidly becomes narrow with


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increasing length. Plotting the transmission at the Fermi energy against length (n) for the three series in Figure 5d, we see the trend we expect from the bond-length alternation (Figure 1); the alkyne a has larger decay constant, β, than the alkene series, while the cumulene series has a slightly negative β, i.e., reverse decay of transmission with length. It is important to note that β is subject to the energy level alignment relative to the Fermi energy which will depend on the anchor group used, and for calculations it will be method dependent. We note that there is reasonable agreement between the alkyne calculations and reported experimental values of β.101-107 Given that we underestimate the reverse bond-length alternation with the method used, in particular for the longer members of the cumulene series (see Fig. S1 and S2), we expect the result to persist at higher-levels of theory.

Increasing Reverse Bond-length Alternation in Cumulenes The bond-length alternation and conductance decay are reversed for simple functionalized cumulenes; however, these effects are fairly weak. Therefore, it is imperative to explore ways to increase these effects. The electronic structure can be manipulated chemically by using electron withdrawing and donating substituents. If donors and acceptors are placed at opposite ends of the molecule a push-pull system is formed and the charge distribution is manipulated. Consequently, resonance structures with zwitterionic character will become increasingly significant. This strategy was recently used to manipulate the cross-conjugated character of oligo(phenyleneethynylene) wires.108 The push-pull strategy has recently also successfully been applied to [3]cumulenes resulting in increasingly reversed bond-length alternation as the zwitterionic resonance structure becomes more pronounced than the cumulenic one; the two contributors shown in the top panel of Scheme 4 represent a simple (and perhaps less realistic) example of such a molecule.109-113 These



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resonance structures provide an instructive picture as to how the redistribution of charge in the molecule directly affects the bond-length alternation, as a reverse single-triple bond alternation becomes predominant in the push-pull substituted species. Scheme 4: Cumulenic and Zwitterionic Resonance Contributors of Substituted Cumulenes.

There are several substituent strategies that achieve this effect. Replacing the entire end-groups by different odd-carbon rings, forming a pentaheptafulvalene-expanded cumulene will do exactly this114-115; such species were previously found to have a substantial dipole moment as the electron structure adapts towards 6 electrons in each ring, as shown in Scheme 4 (middle).116-117 A number of substituted cumulenes show increased reverse bond-length alternation as the substituents may help stabilize different resonance contributors, such as the pentafulvalene-expanded cumulene shown at the bottom of Scheme 4.37, 50, 118-121 For this purpose a captodative strategy was explored theoretically for stabilizing quinoidal resonance structures which give rise to reverse conductance decay in recent work by Stuyver et al.122


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Figure 6. DFT-calculated bond lengths of substituted [n]cumulenes. (a) Chemical structures of [n]cumulene,

(CN)2[n]cumulene(NH2)2,

[n]cumulene-extended

pentaheptafulvalene,

and

[n]cumulene-extended pentafulvalene, where is x is a positive integer and n = 2x+1. (b) Optimized structures and bond lengths in Ångström for the four [3]cumulene species. Note the fulvalene structures are scaled smaller due to their size. (c) Bond lengths of the four [11]cumulene species. The calculated structures for the substituted cumulenes shown in scheme 4 and Figure 6 all show increased reverse bond-length alternation. In all three substituted [3]cumulenes the edge bonds are longer and the central bond is shorter compared to unsubstituted [3]cumulene. A trend that persists for the longer members of each series. The pentafulvalene expanded cumulene shows slightly weaker bond-length alternation than the other systems, but still considerably stronger than the unsubstituted cumulene. These values are in good agreement with crystallographic data of similar systems.109-110, 113, 120, 123



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Figure 7. DFT transmission of amine-linked single-molecule junctions. (a) Optimized junction structures of amine-linked [5]pentaheptafulvalene, [5]pentafulvalene, and [7]cumulene. (b) Transmission plotted semilogarithmically against energy for odd [n]pentaheptafulvalene, n = 3-9. c) Transmission plotted semilogarithmically against energy for odd [n]pentafulvalene, n = 3-9. (d) Transmission at the Fermi energy as function of length (n) for [n]cumulene (red), [n]pentaheptafulvalene (blue), and [n]pentafulvalene (purple), for n = 3-15. By increasing the reverse bond length alternation, the reverse conductance decay trend is increased. In Figure 7, the transmission of the penteheptafulvalene- and pentafulvalene-expanded cumulenes are calculated with DFT as in the previous section. The (CN)2–(NH2)2 substituted system is not included as we cannot make junctions with comparable anchor groups. The transmission trend of the expanded cumulenes is dependent on the energy level alignment relative to EF. Still, the trend we expect from the model calculations is clear. Stronger reverse bondlength alternation results in stronger reverse decay of conductance with length. We see this in the more negative β for the pentaheptafulvalene and pentafulvalene series in Figure 7d. The conductance decay in push-pull systems is likely to be bias and solvent dependent as the molecules are likely to show rectification.124-126 The electric field that the bias induces will, pending the orientation of the molecule, further enhance the push-pull effect on the bond-length alternation.


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However, it must be noted that the effect of the electric field on the bond-length alternation is predicted to be modest at low bias.127-129

Prospective Systems and Outlook It is clear that stability of the longer members of a series will be one of the main challenges for realizing reverse decay on conductance. For cumulenes this in an inherent challenge as the longer members are less stable and cannot be protected by normal strategies such as protection through sterically crowding the molecule. Ingenious strategies for protecting long linear carbons molecules are being undertaken; polyynes and cumulenes protected with rotaxane rings have been reported in recent years.49, 130-133 Multipodal platform approaches holds promise for systematic control of the contact geometry of linear one-dimensional wires, which may help avoiding dimerization by spatial separation of the wires.134-137 The general stability challenge for any molecular series showing reverse conductance decay is the increased diradical character that seem to be its prerequisite.29 Push-pull and captodative substituent strategies are promising ways to accommodate (or reduce the) diradical character.122 It remains the central challenge to accommodate molecules with increasingly small HOMO-LUMO gaps which is a common denominator for highly conducting molecules, particularly those with reverse bondlength alternation. This is in analogy to the suggested negative correlation between conductance and aromaticity in small cyclic systems.138-142 In the aromatic limit the molecules are stable with a big HOMO-LUMO gap, in the anti-aromatic limit the molecules are unstable diradicals with vanishing HOMO-LUMO gaps. Scheme 5: Quinodal and Aromatic Resonance Contributors of the Quinodimethane Unit.



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Despite the afore-mentioned stability challenges, recent synthetic efforts have successfully seen quinodimethane units fused into cumulenic chains. Oligo-quinodimethanes themselves show strong reverse decay of transmission as the aromatic sextet of each phenyl ring is an important resonance contributor in addition to the quinoidal structure as shown in Scheme 5 (top).29,

143

As length

increases the quinodal structure loses out and diradical character increases, therefore only few oligo-quinodimethane derivates have been successfully isolated.144-145 The aromatic sextets promote reverse bond length alternation when incorporated into cumulenes as shown in Scheme 5. Recent synthetic reports of quinoid-expanded [3]cumulenes all display strong reverse bond alternation in the crystal structures; in all cases terminal substituent groups that can accommodate diradical character were used.146-149 It is clear that it will be a challenge to make long oligomers of such systems, still these are prospective systems for reverse conductance decay. Transmission calculations on such systems, included in Supporting Information part D, show clear reverse decay of transmission with length. Finally, we should mention a class of cumulenes we have not yet touched upon in this paper. Research in metallocumulenes (cumulenylidene complexes) has seen great progress and may hold promise for making longer cumulenic oligomers.150-152 From the perspective of the cumulene these show reverse bond-length alternation;153-158 some approach fully reversed alkyne bond lengths.159 As seen in polyynediyl complexes, metal centers may allow for stabilization of longer chains.160 Stability can further be enhanced by using large extended ligands, which can span the full length of the polyyne wire.161-162 The extra protection has already allowed the electron transport properties of long metallopolyynes to be probed experimentally.163-165 The role of one or more metal centers in


Page 23 of 47 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


the wire adds an extra level of complexity and the transport properties of metallocumulenes will be examined in future studies.166-167

Conclusions We have reexamined the peculiar result of Tsuji et al.17 that linear conjugated molecules with reverse bond length alternation show increasing tunneling transmission as the length of the molecule is increased. We have examined how cumulenes are a class of molecule that show this behavior. Despite their all-double bond structure, cumulenic bonds are not of equal length and the bond lengths are reversed for the dominant π system that span the full length of the carbon backbone. Consequently, we predict a negative decay constant, β, for several series of cumulenes. The transmission increases as the HOMO-LUMO gap rapidly becomes narrower with length; a direct consequence of reverse bond-length alternation. Reverse bond-length alternation in cumulenes further increases if suitable substituent strategies are applied. Based on recent synthetic progress in extended cumulenes, we find there is a great number of potential cumulene series that may show reverse decay of conductance with length. Moving forward, the central challenge in realizing reverse conductance decay with length will be stabilizing the longer members of the series. Corresponding Author *E-mail: [email protected] Supporting Information The Supporting Information is available free of charge on the ACS Publications website. Benchmark study of bond-length alternation; Hückel models, Transmission Plots; Cumuleneextended quinodimethane.

ACKNOWLEDGMENTS



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TOC Figure



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Author Biographies Marc Hamilton Garner received his MSc in 2016 and is currently a PhD student in Gemma Solomon’s group at the University of Copenhagen. His research interests revolve around theoretical chemistry and electron transport with specific research interests including molecular orbital theory, substituent effects, sigma-conjugation in silanes, and quantum interference effects.

William Bro-Jørgensen is currently a BSc student in Gemma Solomon’s group at the University of Copenhagen. His research interests include molecular orbital theory and its deployment in organometallic systems.

Pernille Dalsgaard Pedersen received her BSc in 2017 and is currently a MSc student at the University of Copenhagen. Her research interests are mainly in the field of computational chemistry including its applications in atmospheric chemistry and chemical kinetics.

Gemma C. Solomon majored in Chemical Physics at the University of Western Australia before moving to the University of Sydney where she completed her BSc(Hons) in 2003, for which she was awarded the University Medal. She received her PhD in Chemistry in 2007 from the University of Sydney where she worked with Prof. Jeffrey Reimers and Prof. Noel Hush. She moved to Northwestern University, USA where she was a postdoctoral fellow with Prof. Mark Ratner until moving to the University of Copenhagen in 2010 to start her independent career as an assistant professor in Chemistry. She is currently an Associate Professor in the Nano-Science Center and Department of Chemistry at the University of Copenhagen where her research focus is


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charge and heat transport through molecules. Since 2014 she has been a Senior Editor for the Journal of Physical Chemistry A/B/C.


Reverse Bond-Length Alternation in Cumulenes: Candidates for

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