Abstract
If simple guidelines could be established for understanding how quantum interference (QI) can be exploited to control the flow of electricity through single molecules, then new functional molecules, which exploit roomtemperature QI could be rapidly identified and subsequently screened. Recently it was demonstrated that conductance ratios of molecules with aromatic cores, with different connectivities to electrodes, can be predicted using a simple and easytouse “magic number theory.” In contrast with counting rules and “curlyarrow” descriptions of destructive QI, magic number theory captures the many forms of constructive QI, which can occur in molecular cores. Here we address the question of how conductance ratios are affected by electronelectron interactions. We find that due to cancellations of opposing trends, when Coulomb interactions and screening due to electrodes are switched on, conductance ratios are rather resilient. Consequently, qualitative trends in conductance ratios of molecules with extended pi systems can be predicted using simple ‘noninteracting’ magic number tables, without the need for largescale computations. On the other hand, for certain connectivities, deviations from noninteracting conductance ratios can be significant and therefore such connectivities are of interest for probing the interplay between Coulomb interactions, connectivity and QI in singlemolecule electron transport.
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Introduction
Understanding and exploiting roomtemperature quantum interference (QI) in single molecules is the key to creating new highperformance singlemolecule devices and thinfilm materials formed from selfassembled molecular layers. During the past decade, experimental and theoretical studies of single molecules attached to metallic electrodes have demonstrated that roomtemperature electron transport is controlled by QI within the core of the molecule^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}. Many of these demonstrations have been achieved by noting that in contrast with artificial quantum dots, where atomicscale details of the coupling of a dot to external electrodes are not known, the connectivity to the core of a single molecule may be controlled to atomic accuracy. Figure 1 shows two examples of molecules with a common anthanthrene core, connected via triple bonds and pyridyl anchor groups to gold electrodes. The anthanthrene core (represented by a lattice of 6 hexagons) of molecule 1 and the anthanthrene core of molecule 2 are connected differently to the triple bonds. Therefore it is natural to ask how the electrical conductance and interference properties of such molecules are affected by connectivity.
In a typical experiment using mechanically controlled break junctions or STM break junctions^{13,14,15,16,17,18}, fluctuations and uncertainties in the coupling to electrodes are dealt with by measuring the conductance of such molecules many thousands of times and reporting the statisticallymostprobable electrical conductance, just before the junction breaks. If \({\sigma }_{ij}\) is the statisticallymostprobable conductance of a molecule such as 1, with connectivity ij and \({\sigma }_{lm}\) is the corresponding conductance of a molecule such as 2, with connectivity lm, then it was recently predicted theoretically and demonstrated experimentally^{21,22,23} that for polyaromatic hydrocarbons (PAHs) such as anthanthrene, the statisticallymostprobable conductance ratio \({\sigma }_{ij}/{\sigma }_{lm}\) is independent of the coupling to the electrodes and could be obtained from tables of “magic numbers,” which for bipartite PAHs in the absence of electronelectron interactions, are simply tables of integers. If \({M}_{ij}\) (\({M}_{lm}\)) is the magic number corresponding to connectivity ij (lm), then this “magic ratio theory” predicts
From a conceptual viewpoint, magic ratio theory views the shaded regions in Fig. 1 as “compound electrodes”, comprising both the anchor groups and gold electrodes, and focuses attention on the contribution from the core alone. The validity of Eq. (1) rests on the following key foundational concepts^{1,2,21,22,23}:

1.
weak coupling

2.
locality

3.
connectivity

4.
midgap transport

5.
phase coherence

6.
connectivityindependent statistics
When these conditions apply, the complex and often uncontrolled contributions from electrodes and electrodemolecule coupling cancel in conductance ratios and therefore a theory of conductance ratios can be developed by focussing on the contribution from molecular cores alone.
The term “weak coupling” means that the central aromatic subunit such as anthanthrene should be weakly coupled to the anchor groups via spacers such as acetylene. ‘Locality’ means that when a current flows through an aromatic subunit, the points of entry and exit are localised in space. For example in molecule 1, the current enters at a particular atom i and exits at a particular atom j. The concept of ‘connectivity’ recognises that through chemical design, spacers can be attached to different parts of a central subunit with atomic accuracy and therefore it is of interest to examine how the flow of electricity depends on the choice of connectivity to the central subunit. The concept of ‘midgap transport’ is recognition of the fact that unless a molecular junction is externally gated by an electrochemical environment or an electrostatic gate, charge transfer between the electrodes and molecule ensures that the energy levels adjust such that the Fermi energy E_{F} of the electrodes is usually located in the vicinity of the centre of the HOMOLUMO gap and therefore transport takes place in the cotunnelling regime. In other words, transport is usually ‘offresonance’. The concept of ‘phase coherence’ recognises that in this cotunnelling regime, the phase of electrons is usually preserved as they pass through a molecule and therefore transport is controlled by QI. The condition of “connectivityindependent statistics” means that the statistics of the coupling between the anchor groups and electrodes should be independent of the connectivity to the aromatic core. When each of these conditions applies, it can be shown^{1,2,21,22} that the most probable electrical conductance corresponding to connectivity i,j is proportional to \({{G}_{ij}({E}_{{\rm{F}}})}^{2}\) where \({G}_{ij}({E}_{{\rm{F}}})\) is the Green’s function of the core alone, evaluated at the Fermi energy of the electrodes. In the absence of timereversal symmetry breaking, \({G}_{ij}({E}_{{\rm{F}}})\) is a real number. Since only conductance ratios are of interest, we define magic numbers by
where A is an arbitrary constant of proportionality, chosen to simplify magic number tables and which cancels in Eq. (1). Magic ratio theory applies to any singlemolecule junction, provided conditions 1–6 are satisfied. It represents an important step forward, because apart from the Fermi energy \({E}_{{\rm{F}}}\), no information about the electrodes is required. The question we address below is what are the precise values of the numbers \({M}_{ij}\) and how are they affected by electronelectron interactions?
In the literature, several papers discuss the conditions for destructive QI, for which \({M}_{ij}\approx 0\)^{6,9,10,11,12,13,14,15,16,17,18,24,25,26,27,28,29}. On the other hand, magic ratio theory aims to describe constructive QI, for which \({M}_{ij}\) may take a variety of nonzero values. If \(H\) is the noninteracting Hamiltonian of the core, then since the matrix \(G({E}_{{\rm{F}}})={({E}_{{\rm{F}}}H)}^{1}\), the magic number table is obtained from a matrix inversion, whose size and complexity reflects the level of detail contained in \(H\). The quantities \({M}_{ij}\) were termed “magic”^{21,22,23}, because even a simple theory based on connectivity alone yielded values, which were found to be in remarkable agreement with experiment. For example for molecule 1, the prediction was \({M}_{ij}=\,\,1\), whereas for molecule 2, \({M}_{lm}=\,\,9\) and therefore the electrical conductance of 2 was predicted to be 81 times higher than that of 1, which is close to the measured value of 79. This large ratio is a clear manifestation of quantum interference (QI), since such a change in connectivity to a classical resistive network would yield only a small change in conductance. To obtain the above values for \({M}_{ij}\) and \({M}_{lm}\), the Hamiltonian \(H\) was chosen to be
where the connectivity matrix \(C\) of anthanthrene is shown in Fig. 2b. In other words, each element \({H}_{ij}\) was chosen to be −1 if \(i,\,j\) are nearest neighbours or zero otherwise and since anthanthrene is represented by the bipartite lattice in which odd numbered sites are connected to even numbered sites only, \(H\) is block off diagonal. The corresponding core Green’s function evaluated at the gap centre \({E}_{{\rm{F}}}=0\,\,\)is therefore obtained from a simple matrix inversion \(G(0)=\,\,{H}^{1}\). Since \(H\) and therefore \({H}^{1}\) are block offdiagonal, this yields the following structure for the magic number table of the PAH core \(M=(\begin{array}{cc}0 & {\bar{M}}^{t}\\ \bar{M} & 0\end{array})\). The offdiagonal block of the magic number table \(\bar{M}\) for anthanthrene is shown in Fig. 2c. As noted above, for molecule 1, with connectivity 9–22, \({M}_{9,22}=\,\,1\), whereas for molecule 2, with connectivity 3–12, \({M}_{3,12}=\,\,9\).
Magic number tables such as Fig. 2c are extremely useful, since they facilitate the identification of molecules with desirable conductances for future synthesis. Conceptually, tables obtained from Hamiltonians such as Eq. (3) are also of interest, since they capture the contribution from intracore connectivity alone (via the matrix \(C\), comprising −1’s or zeros), while avoiding the complexities of chemistry. Although magic number tables obtained from such connectivity matrices were shown to agree qualitatively with break junction measurements of several different molecules carried out by different experimental groups^{22}, the errors in the experimental estimates of conductance ratios are rather large and the number of molecules tested is small. Therefore it is of interest to seek to improve the accuracy of magic number tables by utilising more accurate core Hamiltonians. An essential ingredient missing from the Hamiltonian of Eq. (3) is electronelectron interactions and therefore in what follows we aim to obtain improved estimates of magic numbers by including the effect of Coulomb interactions and screening. Results will be presented for a variety of graphenelike molecules, including benzene, naphthalene, anthracene, pyrene and anthanthrene.
The main outcome of this study is exemplified by the interacting magic number table \({M}^{{\rm{int}}}\) for anthanthrene, whose lower offdiagonal block \({\bar{M}}^{{\rm{int}}}\) is presented in Fig. 2d. Note that magic numbers are only defined up to a constant of proportionality, which does not affect the predicted conductance ratios. Therefore to facilitate comparison between interacting and noninteracting values, in the table of Fig. 2d, the constant is chosen to minimise the mean square deviation between the noninteracting and interacting Mtables. The latter shows for example, that in the presence of Coulomb interactions, the magic number for molecule 1, changes from −1 to \({M}_{9,22}^{{\rm{int}}}=\,\,0.44\), whereas for molecule 2, the magic number changes from −9 to \({M}_{3,12}^{{\rm{int}}}=\,\,5.44\). Hence interacting magic number theory predicts that the conductance of 2 is (−5.44/−0.44)^{2}=152 times higher than that of 1 (or more precisely 148 if magic numbers to 3 decimal places are used, as presented in the Supplementary Information (SI)). This demonstrates that the conductance ratio of 81, predicted by noninteracting magic numbers is qualitatively correct (ie to within a factor of 2). Furthermore comparison between tables c and d in Fig. 2 shows that the noninteracting magic number table captures the qualitative trends of the interacting magic number table. This qualitative agreement is remarkable, since the former can be obtained from a few lines of e.g. MATLAB code, while the latter is the result of a substantial manybody calculation. Results for both noninteracting and interacting magic number tables of a range of PAHs are presented in the SI. Our main conclusion is that noninteracting magic numbers are a useful qualitative guide for predicting conductance ratios, even in the presence of Coulomb interactions and screening.
Results
In the following numerical simulations, the transmission coefficient T_{ij}(E) describing the probability that electrons of energy E can pass from one electrode to another via sites i, j. Systems with the chiral symmetry have a symmetric energy spectrum which means that for halffilled systems the Fermi energy is at the gap centre. Therefore, conductance ratios are obtained from T_{ij}(0). To include the effects of the Coulomb interaction, we first generalise the Hamiltonian of Eq. (3) to the interacting ParrPariserPople (PPP) model^{30,31,32}. We base our treatment of the Coulomb interaction on a scheme proposed by Ohno^{33}, which obtains intersite interaction integrals by smoothly interpolating between the Hubbard integral U for zero separation between sites and an unscreened Coulomb interaction for large separations between sites. This is an established model for the aromatic molecules and yet its simplicity enables us to study the effect of interaction. Recently it was shown experimentally^{34} that molecular levels shift as a result of Coulomb interaction with image charges in the metal leads, resulting in a HOMOLUMO gap renormalization. Therefore, we also take into account additional electric potential screening, which is induced by the conducting electrodes. We model the latter as infinite parallel plates located at a distance d from each of the connection sites.
Calculations for smaller molecules (benzene, naphthalene, and anthracene) are performed using both the Lanczos exact diagonalization method^{35} and using the restricted HartreeFock (HF) approximation (for technical details see Methods and the SI). We use the latter since we consider effects of the Coulomb interaction in the simplest scheme possible (for superior approximate methods as for example GW method see^{36}). For the larger molecules (pyrene and anthanthrene) the Lanczos “calculation is not feasible. For the smaller molecules, where it is possible to compare the Lanczos method with the HF approximation, agreement was found for the HOMOLUMO gaps (within approx. 1%) and conductances (within approx. 10%) for different connectivities. This gives us confidence that use of the HF approximation for the larger molecules is valid.
As a first example, we present results for the conductance ratio of molecules with naphthalene cores (see Fig. 3a), with two different connectivities, denoted 6–9 and 3–8, whose noninteracting magic ratio is 4 [see table in Fig. 3c of the SI]. To elucidate the effects of varying the strength of interactions, we multiply all the interaction integrals by a scale factor λ and examine the effect of varying λ. The upper table in Fig. 3b shows a comparison between results obtained using HF and direct Lanczos diagonalization for different values of the scaling parameter λ, ranging from λ = 0 (noninteracting) to λ = 1 (interacting) and to the greater, unphysical value of λ = 2. For λ = 1, the lower table in Fig. 3b shows the effect of screening by electrodes at different distances d from molecule, ranging from d = d_{0}, where d_{0} is the carboncarbon bond length, to d = ∞ (no screening). Figure 3c shows that the HF approximation reproduces the exact Lanczos HOMOLUMO gap correctly for naphthalene and while there is a small discrepancy in the transmission coefficient (Fig. 3d) at the Fermi level E = 0, the HF conductance ratio is qualitatively correct, deviating appreciably from the exact value only when λ becomes much larger than the physicallyrelevant one. Note that the conductance ratio at λ = 0 is not exactly equal to the noninteracting ratio of 4 due to the presence of a small but finite coupling of the molecule to the electrodes.
The lower table in Fig. 3b shows that screening by the electrodes does not change the ratio appreciably even though the renormalization of the HOMOLUMO gap is different for different connectivities. The difference in gap renormalisation occurs, because screening is more effective when the distance between electrodes is small. The gap is thus reduced more by screening for the 6–9 connectivity, where the long axis of the molecule is parallel to electrode surfaces, than for the 3–8 connectivity, where it is perpendicular to them. If the QI between different paths through the molecule did not change, one would expect the conductance ratio to be proportional to the ratio of inverse gaps squared and therefore the conductance ratio at d/d_{0} = 1 should have increased by 31% compared to the conductance ratio in absence of screening. Here this effect is almost exactly compensated by the screening induced change of QI between different paths through the molecule.
The results in Fig. 3b show that for naphthalene, the HF and Lanczos predictions for the conductance ratio are rather close to each other and to that of (noninteracting) magic number theory. As a second example, Fig. 4b shows HF results for the conductance ratio of molecules 1 and 2 with anthanthrene cores and Fig. 4c shows their corresponding transmission functions \(T(E)\). As for naphthalene, the conductance ratio increases from the noninteracting value when interactions are present (upper table in Fig. 4b), but here the deviations from the noninteracting magic ratio of 81 are more pronounced. In contrast with naphthalene, the conductance ratio is also affected by screening: when the electrodes become closer to the molecule, the ratio drops back towards the noninteracting value. In contrast with naphthalene, the rescaling of the HOMOLUMO gaps of both connectivities would lead to an increase of the conductance ratio (by 37% for d/d_{0} = 1), so the drop of the conductance ratio can be attributed to screeninginduced change in the QI of different paths through the molecule.
To highlight the correlation (and differences) between noninteracting and interacting conductance ratios, the blue dots in Fig. 4d are plots of HF conductance ratios versus those predicted by noninteracting magic numbers for all possible pairs of connectivities and shows that there is a significant degree of correlation between the two. The main conclusion from these results and for corresponding results for other molecules (see SI) is that although Coulomb interactions and screening cause the conductance ratios to vary, in many cases the noninteracting magic ratios provide the correct qualitative trend. In the case of anthanthrene (Fig. 4), the noninteracting ratio of 81 is surprisingly close to the mostphysical conductance ratio of 79.3, which occurs at λ = 1 and a screening distance of d/d_{0} = 1.
The above results are obtained from the PPP model, which coincides with the noninteracting Hamiltonian of Eq. (3) when \(U=0\). This model preserves chiral symmetry and guarantees that the centre of the HOMOLUMO gap lies in the middle of the energy spectrum (\(E=0\)). The model captures the effect of connectivity and Coulomb interactions, without introducing complexities associated with the chemical nature of the molecules. To include the latter, we used density functional theory to compute the transmission coefficient \(T(E)\) of molecules with different connectivities attached to gold electrodes. Figure 5a shows plots of \(\mathrm{log}\,T(E)\) versus E for the 3–8 and 6–9 connectivities of naphthalene and Fig. 5b shows \(\mathrm{log}\,T(E)\) versus E for the 3–12 and 9–22 connectivities of anthanthrene. To highlight the further role of chemistry, the bottom right inset of Fig. 5b shows corresponding results when the anthanthrene core is directly coupled to gold electrodes, as shown in the bottom left inset. For energies in the shaded regions of these plots, the ratio of geometric averages of transmission coefficients approximately coincides with the noninteracting magic ratio rule (see Table 1, column 7).
PPP and DFT results for conductance ratios of benzene, anthracene and pyrene with two connectivities are presented in Figs 7–12 of the SI. Except for benzene, the conductance ratios for those connectivities were measured experimentally. Table 1 shows a comparison between these results, the noninteracting magic ratios and experiment.
The effect of interaction on conductance ratios can be roughly estimated by considering the PPP model in the infiniterange interaction limit (where the interaction integrals take the same value \(\tilde{U}\) for all pairs of sites in a molecule), which can be solved exactly for an isolated molecule. In this limit, the core Green’s function takes the form \(\tilde{G}(0)=\,{(H+\frac{1}{2}\tilde{U}{\rm{sgn}}H)}^{1}\) and can be easily evaluated from the connectivity matrix C and \(\tilde{U}\) alone, with \(H=(\begin{array}{cc}0 & C\\ {C}^{t} & 0\end{array})\) (see Supplementary Information). For \(\tilde{U}\) we take the average value of the PPP interaction integrals in a given molecule. This is a useful limit, because as shown by Table 1, for all molecules except anthanthrene the conductance ratios calculated from \(\,\tilde{G}\) correctly predict the direction in which the PPP ratio will deviate from the noninteracting magic ratio. Furthermore, the infiniterange interaction prediction is quantitatively correct within approx. 20%. Unfortunately, for anthanthrene with 3–12 and 9–22 connectivities, the infiniterange interaction limit conductance ratio is not a good approximation to the PPP ratio. We traced the latter failure to the fact that the Green’s function element corresponding to the 9–22 connectivity crosses zero as a function of the interaction strength in the vicinity of the actual value of interaction. Therefore, the conductance ratio is very sensitive to the actual form and strength of interaction for this connectivity.
The orange dots in Fig. 4d are plots of HF conductance ratios versus those predicted by infinite range interaction model for all possible pairs of connectivities and show that there is a significant improvement compared with the noninteracting magic ratios. Moreover, as shown in Fig. 16 in the SI, typically the infiniterange interaction model correctly predicts in which direction the PPP conductance ratio will deviate from the noninteracting value.
The main result contained in Figs 3,4, and Table 1 is that the noninteracting conductance ratios are typically similar to those obtained in the presence of Coulomb interactions and therefore despite their simplicity, are a useful guide for predicting conductance ratios and identifying connectivities with high or low conductance. Furthermore for small molecules, where Lanczos results for the PPP model are available, the Lanczos ratios agree with those obtained using HF.
On the other hand, there are cases where interactions cause a strong deviation from noninteracting conductance ratios. We identified several pairs of connectivities for different molecules, where this is the case. For example, for anthanthrene we predict the conductance ratio for 6–7 and 1–10 connectivities to be about 275, which is much larger than the noninteracting ratio of 16 for this pair of connectivities. Additional examples are presented in Tables 3 and 4 of the SI. These connectivities are interesting, because experimental measurement of their conductance ratios would establish that at least for certain connectivities, Coulomb interactions are needed to describe transport through such molecules.
Discussion
We have used exact (Lanczos) diagonalization, HartreeFock theory and density functional theory to examine conductance ratios of polyaromatic hydrocarbons with different connectivities to electrodes, which can be predicted using a simple and easytouse “magic number tables,” such as those shown in Fig. 2c,d (and in Figs 2–6 of the SI). We find that when Coulomb interactions and screening due to electrodes are switched on, conductance ratios are rather resilient, even though the conductances themselves vary. Consequently, although the precise numbers depend on the strength of the interaction and on screening, qualitative trends in conductance ratios can be predicted using noninteracting magic number tables. Overall the differences between HF, Lanczos and DFT predictions and variations due to screening are found to be comparable with deviations from experimental values. Therefore at the current level of experimental measurement, noninteracting magic numbers provide a useful tool for identifying molecules for subsequent experimental screening, without the need for largescale computations involving electronelectron interactions. On the other hand, we have also identified examples where conductance ratios are sensitive to interactions. These molecules would be interesting targets for future synthesis, since their conductance ratios would demonstrate that in general both QI and interactions play an important role in controlling the flow of electricity through single molecules.
Methods
When analysing the PPP model, calculations for smaller molecules (benzene, naphthalene, and anthracene) are performed using both the Lanczos exact diagonalization method^{35} and for larger molecules, where exact diagonalization is not feasible, we use the restricted HartreeFock (HF) approximation (for technical details see Methods and the SI). In both cases, the wide band approximation was used, in which the self energy due to the contacts is modelled by a single number. When including chemical details at an atomistic level, we use the SIESTA implementation of DFT combined with nonequilibrium Green’s functions, in which the full selfenergy matrix is computed.
This dual approach to modelling is needed, because correlated ab initio calculations with chemical specificity are not feasible. A similar combination of methods was utilised in^{36}, where in addition, the GW method was used. Within the PPP model, interactions are present within the molecule only, whereas interactions within the DFT meanfield treatment are present in both the molecules and electrodes.
DFTNEGF
The optimized geometry and ground state Hamiltonian and overlap matrix elements of each structure was selfconsistently obtained using the SIESTA implementation of density functional theory (DFT). SIESTA employs normconserving pseudopotentials to account for the core electrons and linear combinations of atomic orbitals to construct the valence states. The generalized gradient approximation (GGA) of the exchange and correlation functional is used with the PerdewBurkeErnzerhof parameterization (PBE) a doubleζ polarized (DZP) basis set, a realspace grid defined with an equivalent energy cutoff of 250 Ry^{35,36}. The geometry optimization for each structure is performed to the forces smaller than 40 meV/Å. The meanfield Hamiltonian obtained from the converged DFT calculation or a simple tightbinding Hamiltonian was combined with Gollum quantum transport code^{37} to calculate the phasecoherent, elastic scattering properties of the system consisting of left (source) and right (drain) leads and the scattering region. The transmission coefficient T(E) for electrons of energy E (passing from the source to the drain) is calculated via the relation \(T(E)=Tr\{{{\rm{\Gamma }}}_{R}(E){\mathscr{G}}(E){{\rm{\Gamma }}}_{L}(E){{\mathscr{G}}}^{\dagger }(E)\}\,\). In this expression,\(\,{{\rm{\Gamma }}}_{L,R}(E)=i({{\rm{\Sigma }}}_{L,R}(E){{\rm{\Sigma }}}_{L,R}^{\dagger }(E))\) describe the level broadening due to the coupling between left (L) and right (R) electrodes (which are modelled with atomic precision as shown in Fig. 5) and the central scattering region, \({{\rm{\Sigma }}}_{L,R}(E)\,\,\)are the retarded selfenergies associated with this coupling and \({\mathscr{G}}={(ESH{{\rm{\Sigma }}}_{L}{{\rm{\Sigma }}}_{R})}^{1}\) is the retarded Green’s function, where H is the Hamiltonian and S is overlap matrix. Using obtained transmission coefficient \(T(E)\), the conductance could be calculated by Landauer formula (\(\sigma ={\sigma }_{0}{\int }^{}dE\,T(E)(\,\,\partial f/\partial E)\)) where \({\sigma }_{0}=2{e}^{2}/h\) is the conductance quantum, \(f(E)={(1+\exp ((E{E}_{F})/{k}_{B}T))}^{1}\) is the FermiDirac distribution function, T is the temperature and \({k}_{B}\) is Boltzmann’s constant.
HartreeFock
The PPP Hamiltonian, \({H}^{{\rm{int}}}=\sum _{ijs}{H}_{ij}{c}_{is}^{\dagger }{c}_{js}+\frac{1}{2}\sum _{ij}{U}_{ij}({n}_{i}1)({n}_{j}1)\), contains matrix elements H_{ij} of the noninteracting Hamiltonian (for the nearestneighbour hopping integral we take γ = 2.4 eV) and interaction integrals U_{ij}, which in the absence of screening by electrodes we calculate using the Ohno interpolation^{33}: U_{ij} = U/(1 + (U/(e^{2}/4πε_{0}d_{ij}))^{2})^{−1/2} where U = 11.13 eV is the Hubbard parameter and d_{ij} is the distance between sites i and j. The interatomic distance is d_{0} = 1.4 Å. We take the image charge effects into account by analytically solving^{38} the Poisson’s equation for the electrostatic Greens function in a simplified geometry, namely we assume the electrodes are two infinite parallel plates located at a distance d away from each of the connectivity sites. We decouple the interaction terms within the restricted HF approximation, yielding renormalized hopping matrix elements \({H}_{ij}^{{\rm{H}}{\rm{F}}}={H}_{ij}{{\rm{U}}}_{{\rm{i}}{\rm{j}}}\langle {c}_{{\rm{j}}{\rm{s}}}^{\dagger }{{\rm{c}}}_{{\rm{i}}{\rm{s}}}\rangle \). The expectation value is calculated from the Slater determinant built from the occupied scattering states of a molecule attached to electrodes. We model electrodes as tightbinding chains with nearestneighbour hopping integral of 10γ. The hopping integral between the connectivity site on the molecule and the nearest electrode site is γ, leading to coupling Γ_{L,R} = 0.2γ (in the wide band limit we can neglect the energy dependence of Γ). The procedure is iterated until a selfconsistent solution is obtained. Due to the chiral symmetry possessed by the PPP Hamiltonian of our molecules, the HF Hamiltonian has the same structure as the noninteracting one, i.e. the hopping integrals between atoms on the same sublattice as well as onsite energies remain zero. Once the convergence is achieved, the conductance is calculated with the LandauerBüttiker formula^{39,40} with the transmission function T(E) read from the scattering state at energy E. We also performed unrestricted HF calculations where we allowed each sublattice to develop a magnetization. We found that the antiferromagnetic solution becomes the ground state only for interaction strengths that exceed the physically relevant ones by more than approx. 50%. For details, see the Supplementary information.
Systems with the chiral symmetry have a symmetric energy spectrum which means that for halffilled systems the Fermi energy is at the gap centre^{36}. The chiral symmetry is defined and its consequences are explained in Supplementary Note 3. There it is shown that the PPP model as well as the corresponding HartreeFock Hamiltonian have this symmetry. Chiral symmetry ensures that the energy spectrum of the molecule is symmetric with respect to the centre of the HOMOLUMO gap.
Clearly HF is an effective noninteracting theory, which creates new effective hoppings between nonneighbouring sites, which are absent from the noninteracting model. The inclusion of arbitrary longrange hoppings could significantly change the magic ratios, whereas those generated by the HF approximation using physicallyrelevant parameters do not.
Lanczos
The size of the Hilbert space grows exponentially with the size of the molecule and the exact full diagonalization of the PPP Hamiltonian is in our case limited to smallest system of benzene molecule. We therefore apply the Lanczos method^{41}, which allows for the treatment of larger systems as well as the calculation of ground state properties exactly. Within the Lanczos method one obtains the ground state ψ_{0}⟩ of an isolated molecule by starting from a random manybody state and then iteratively applying the Hamiltonian for generation of new basis states, within which the effective Hamiltonian is tridiagonal and easy to diagonalize. On the other hand, the core Green’s function G is obtained by starting the iterative procedure from \({c}_{is}\)ψ_{0}⟩ or \({c}_{is}^{\dagger }\)ψ_{0}⟩, and by calculating matrix elements between two series of Lanczos eigenstates for the Lehmann representation. The results converge within 80 iterative steps. The Green’s function \({\mathscr{G}}\) of a molecule attached to electrodes is then calculated within the elastic cotunneling approximation^{42,43}, i.e., the presence of the electrodes is taken into account with the Dyson’s equation \({{\mathscr{G}}}^{1}={G}^{1}{{\rm{\Sigma }}}_{{\rm{L}}}{{\rm{\Sigma }}}_{{\rm{R}}}\). The selfenergies Σ_{L,R} due to coupling to electrodes correspond to the same electrodemolecule couplings as in the HF calculation. The approximation is valid far from transmission resonances and above the Kondo temperature of the system. In our case both conditions are satisfied, because the Fermi level is at the centre of the HOMOLUMO gap and there is no unpaired electron in the molecule. In the elastic cotunneling approximation the conductance can again be calculated with the LandauerButtiker formula, with the transmission function obtained from \({\mathscr{G}}\) as \(T(E)={\mathrm{Tr}\{{\rm{\Gamma }}}_{{\rm{R}}}(E){\mathscr{G}}(E){{\rm{\Gamma }}}_{{\rm{L}}}(E){{\mathscr{G}}}^{\dagger }(E)\}\)^{44}.
Data Availability
All data generated or analysed during this study are included in this published article (and its Supplementary Information files).
References
Lambert, C. J. Basic Concepts of Quantum Interference and Electron Transport in SingleMolecule Electronics. Chem. Soc. Rev. 44, 875–888 (2015).
Lambert, C. J. & Liu, S.X. A Magic Ratio Rule for Beginners: A Chemist’s Guide to Quantum Interference in Molecules. ChemistryA European Journal 24, 4193–4201 (2017).
Sadeghi, H. et al. Conductance Enlargement in Picoscale Electroburnt Graphene Nanojunctions. Proc. Natl. Acad. Sci. 112(9), 2658–2663 (2015).
Sedghi, G. et al. Bennett, LongRange Electron Tunnelling in OligoPorphyrin Molecular Wires. N. Nat. Nanotechnol. 6, 517 (2011).
Zhao, X. et al. Oligo(aryleneethynylene)s with Terminal Pyrydyl Groups: Synthesis and Length Dependence of the TunnellingtoHopping Transition of SingleMolecule Conductances. Chem. Mat. 25(21), 4340–4347 (2013).
Markussen, T., Stadler, R. & Thygesen, K. S. The Relation between Structure and Quantum Interference in SingleMolecule Junctions. Nano Lett. 10(10), 4260–4265 (2010).
Papadopoulos, T., Grace, I. & Lambert, C. Control of Electron Transport Through Fano Resonances in Molecular Wires. Phys. Rev. B. 74, 193306 (2006).
Bergfield, J. P., Solis, M. A. & Stafford, C. A. Giant Thermoelectric Effect from Transmission Supernodes. ACS Nano. 4(9), 5314–5320 (2010).
Ricks, A. B. et al. Controlling Electron Transfer in DonorBridgeAcceptor Molecules Using CrossConjugated Bridges. J. Am. Chem. Soc. 132(43), 15427–15434 (2010).
Markussen, T., Schiötz, J. & Thygesen, K. S. Electrochemical Control of Quantum Interference in AnthraquinoneBased Molecular switches. J. Chem. Phys. 132, 224104 (2010).
Solomon, G. C., Bergfield, J. P., Stafford, C. A. & Ratner, M. A. When “small” Terms Matter: Coupled Interference Features in the Transport Properties of CrossConjugated Molecules. Beilstein J. Nanotechnol. 2, 862–871 (2011).
Vazquez, H. et al. Probing the Conductance Superposition Law in SingleMolecule Circuits with Parallel Paths. Nat. Nanotechnol. 7, 663–667 (2012).
Ballmann, S. et al. Experimental Evidence for Quantum Interference and Vibrationally Induced Decoherence in SingleMolecule Junctions. Phys. Rev. Lett. 109, 056801 (2012).
Aradhya, S. V. et al. Dissecting Contact Mechanics from Quantum Interference in SingleMolecule Junctions of Stilbene Derivatives. Nano Lett. 12(3), 1643–1647 (2012).
Kaliginedi, V. et al. Correlations between Molecular Structure and SingleJunction Conductance: A Case with Oligo(phenyleneethynylene)Type Wires. J. Am. Chem. Soc. 134(11), 5262–5275 (2012).
Aradhya, S. V. & Venkataraman, L. SingleMolecule Junctions Beyond Electronic Transport. Nat. Nanotechnol. 8, 399–410 (2013).
Arroyo, C. R. et al. Signatures of Quantum Interference Effects on Charge Transport Through a Single Benzene Ring. Angew. Chem. Int. Ed. 52(11), 3152–3155 (2013).
Guédon, C. M. et al. Observation of Quantum Interference in Molecular Charge Transport. Nat. Nanotechnol. 7(5), 305–309 (2012).
Manrique, D. Z., AlGaliby, Q., Hong, W. & Lambert, C. J. A New Approach to Materials Discovery for Electronic and Thermoelectric Properties of SingleMolecule Junctions. Nano Letters. 16, 1308–1316 (2016).
Manrique, D. Z. et al. A Quantum Circuit Rule for Interference Effects in SingleMolecule Electrical Junctions. Nature Communications 6, 6389 (2015).
Sangtarash, S. et al. Searching the Hearts of Graphenelike Molecules for Simplicity, Sensitivity and Logic. J. Am. Chem. Soc. 137(35), 11425–11431 (2015).
Geng, Y. et al. Magic Ratios for ConnectivityDriven Electrical Conductance of Graphenelike Molecules. J. Am. Chem. Soc. 137(13), 4469–4476 (2015).
Sangtarash, S., Sadeghi, H. & Lambert, C. J. Exploring Quantum Interference in Heteroatomsubstituted Graphenelike Molecules. Nanoscale 8, 13199–13205 (2016).
Nozaki, D. & Toher, C. Reply to Comment on “Is the Antiresonance in MetaContacted Benzene Due to the Destructive Superposition of Waves Traveling Two Different Routes around the Benzene Ring?”. J. Phys. Chem. 121, 11739–11746 (2017).
Nozaki, D., Lucke, A. & Schmidt, W. G. Molecular Orbital Rule for Quantum Interference in Weakly Coupled Dimers, LowEnergy Giant Conductivity Switching Induced by Orbital Level Crossing. J. Phys. Chem. Lett. 8, 727–732 (2017).
Zhao, X., Geskin, V. & Stadler, R. Destructive Quantum Interference in Electron transport: A Reconciliation of the Molecular Orbital and the Atomic Orbital Perspective. J. Chem. Phys. 146, 092308 (2017).
Reuter, M. G. & Hansen, T. Communication: Finding Destructive Interference Features in Molecular Transport Junctions. J. Chem. Phys. 141, 181103 (2014).
Garner, M. H., Solomon, G. C. & Strange, M. Tuning Conductance in Aromatic Moleules: Constructive and Counteractive Substituent Effects. J. Phys. Chem. C 120, 9097–9103 (2016).
Borges, A., Fung, E.D., Ng, F., Venkataraman, L. & Solomon, G. C. Probing the Conductance of the sigmaSystem of Bipyridine Using Destructive Interference. J. Phys. Chem. Lett. 7, 4825–4829 (2016).
Pariser, R. & Parr, R. G. A SemiEmpirical Theory of the Electronic Spectra and Electronic Structure of Complex Unsaturated Molecules. I. J. Chem. Phys. 21, 466 (1953).
Pople, J. A. Electron Interaction in Unsaturated hydrocarbons. Trans. Faraday Soc. A 49, 1375 (1953).
Reich, S., Maultzsch, J., Thomsen, C. & Ordejón, P. TightBinding Description of Graphene. Phys. Rev. B 66, 035412 (2002).
Ohno, K. Parameters in SemiEmpirical Theory. Theor. Chim. Acta 2, 291 (1964).
Perrin, M. L. et al. Large Tunable ImageCharge Effects in SingleMolecule Junctions. Nature nanotechnol. 8, 282 (2013).
Soler, J. M. et al. The SIESTA Method for ab initio OrderN Materials Simulation. J. Phys.: Condens. Matter 14, 2745 (2002).
Pedersen, K. G. L. et al. Quantum Interference in offresonant Transport Through Single Molecules. Phys. Rev. B 90, 125413 (2014).
Ferrer, J. et al. GOLLUM: A NextGeneration Simulation Tool for Electron, Thermal and Spin Transport. New J. Phys. 16, 093029 (2014).
Kaasbjerg, K. & Flensberg, K. Image Charge Effects in SingleMolecule Junctions: Breaking of Symmetries and NegativeDifferential Resistance in a Benzene SingleElectron Transistor. Phys. Rev. B 84, 115457 (2011).
Landauer, R. Electrical Resistance of Disordered OneDimensional Lattices. Philos. Mag. 21, 863 (1970).
Büttiker, M. FourTerminal PhaseCoherent Conductance. Phys. Rev. Lett. 14, 1761 (1986).
Jaklič, J. & Prelovšek, P. FiniteTemperature Properties of Doped Antiferromagnets. Adv. Phys. 49, 1–92 (200).
Averin, D. V. & Nazarov, Y. V. Virtual Electron Diffusion During Quantum Tunneling of the Electric Charge. Phys. Rev. Lett. 65, 2446 (1990).
Groshev, A., Ivanov, T. & Valtchinov, V. Charging Effects of a Single Quantum level in a Box. Phys. Rev. Lett. 66, 1082 (1991).
Bergfield, J. P. & Stafford, C. A. ManyBody Theory of Electronic Transport in SingleMolecule Heterostructures. Phys. Rev. B 79, 245125 (2009).
Acknowledgements
L.U., T.R., J.K., and A.R. acknowledge the support of the Slovenian Research Agency under Contract No. P10044. This work is supported by FET Open project 767187 – QuIET, the EU project BACTOFUEL and the UK EPSRC grants EP/N017188/1, EP/N03337X/1 and EP/P027156/1.H.S. and S.S. acknowledge the Leverhulme Trust (Leverhulme Early Career Fellowships no. ECF2017186 and ECF2018375) for funding.
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C.J.L., A.R., T.R. and J.H.J. conceived and conducted the project. L.U. and T.R. carried out the HartreeFock formalism, J.K. contributed results with Lanczos technique and S.S. and H.S. carried out the DFT calculations. C.J.L., L.U. and T.R. wrote the manuscript. All authors took part in the discussions and reviewed the manuscript.
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Ulčakar, L., Rejec, T., Kokalj, J. et al. On the resilience of magic number theory for conductance ratios of aromatic molecules. Sci Rep 9, 3478 (2019). https://doi.org/10.1038/s41598019399371
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DOI: https://doi.org/10.1038/s41598019399371
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