Local Pathways in Coherent Electron Transport through Iron Porphyrin

Oct 28, 2010 - ... Northwestern UniVersity, 2145 Sheridan Road, EVanston, Illinois 60208-3113, United States ... Electron transport through transition...
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J. Phys. Chem. C 2010, 114, 20813–20820

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Local Pathways in Coherent Electron Transport through Iron Porphyrin Complexes: A Challenge for First-Principles Transport Calculations† Carmen Herrmann,* Gemma C. Solomon, and Mark A. Ratner Department of Chemistry, Northwestern UniVersity, 2145 Sheridan Road, EVanston, Illinois 60208-3113, United States ReceiVed: July 30, 2010; ReVised Manuscript ReceiVed: October 11, 2010

We investigate the coherent electron transport properties of a selection of iron porphyrin complexes in their low-spin and high-spin states, binding the system to metallic electrodes with three different substitution patterns. We use a study of the local transmission through the complexes and their molecular orbitals to show the role of the various components of the molecular structure in mediating electron transport. While there are energies where the metal center and the axial ligands participate in transport, in the off-resonant energy range, these components simply form a scaffold, and the transport is dominated by transmission through the porphyrin macrocyle alone. This is still true when going from the low-spin to the high-spin state, except that now, an additional iron-centered MO contributes to transport in the formerly off-resonant region. It is found that while the choice of the exchange-correlation functional can strongly influence the quantitative results, our qualitative conclusions hold irrespective of the functional employed. 1. Introduction Electron transport through transition-metal complexes is a significant topic in molecular electronics from both an experimental1-10 and a theoretical perspective.11-19 These systems open up new areas of chemistry with different molecular topologies20-23 and multiple stable charge and spin states24,25 of a molecule in a junction. The role of the metal center in the transport properties is unclear, including the extent to which it takes an active role in mediating transport as opposed to acting as a scaffolding element supporting a structure of surrounding ligands that carry the current. Experimental observation of the Kondo effect in transition-metal complexes suggests that, at least in some cases, the metal center plays a critical role in transport.1,2,6 However, this need not always be the case. A first step in answering this question is to map the path that the tunneling current takes through a metal complex. This information can be helpful in deciding which parts of the chemical structure should be modified, either directly or indirectly, to achieve the desired changes in the transport behavior. Beyond their biological and technological significance, ligands such as porphyrins, corrins, or phthalocyanines are of special interest because they build stable complexes with different metal centers, frequently with several accessible spin states. Further, a vast array of substitution patterns is possible, and the spatial relation between two attachment sites to the electrodes can, in principle, be controlled chemically.22,23 The synthetic flexibility offered by these systems means that the electronic coupling can be tuned over a large range, providing suitable candidates for many types of electronic devices.26 On the basis of density functional theory (DFT) calculations, porphyrin systems have been proposed as chemical switches27 and molecular current routers.28 Their electron transport properties have been investigated in numerous experiments, including single-molecule studies on bare porphyrins connected to electrodes at meso positions.29 Metal complexes †

Part of the “Mark A. Ratner Festschrift”. * To whom correspondence should be addressed. E-mail: c-herrmann@ northwestern.edu. Fax: +1-847-467-4996. Tel: +1-847-467-4984.

with cyclic conjugated systems as ligands are also the subject of numerous scanning tunneling microscopy (STM) studies, where the rings are usually adsorbed flat on the surface.24,30,31 Metal-free substituted porphyrins have been investigated in different orientations of the π-system with respect to the surface ranging from planar to perpendicular.32 Spatially resolved conductance and tunneling current maps of phthalocyanine molecules adsorbed on surfaces have been measured with STM techniques.33,34 Here, we study the local contributions to transmission in iron(II) porphyrin complexes connected to electrodes in a singlemolecule junction setup and how these contributions depend on the electron’s energy and the connection to the electrodes. We assume that electron transport occurs in the off-resonant coherent tunneling regime; it should be kept in mind that the conditions under which this is a good assumption for a given system are sometimes unclear. Due to the complexity of the problem under study, in particular, its nonequilibrium nature, we make several further approximations (see section 2 and the Appendix). This implies that our study is meant as a qualitative guide rather than providing quantitative accuracy. Close to the molecular orbital (MO) energies (resonances), it may be expected that the transmission is dominated by the atoms on which the MO is mainly located. It is an open question though what happens in the intermediate (off-resonant) energy regions. To answer this question, a decomposition of the transmission into atomic contribution provides valuable information. Comparing results obtained with two different density functionals, one hybrid and one pure, illustrates how difficult it may be to extract quantitative information on electron transport properties from state-of-the-art first-principles calculations. Qualitative conclusions, on the other hand, are found to be less sensitive to the parameters of the electronic structure calculations. After giving an outline of the theory employed in section 2, the systems under study are introduced in section 3. Further details on the theoretical approach and the methodology may be found in the Appendix A and in ref 35. Transmission, central subsystem MOs (CMOs), and local transmission are discussed for the iron porphyrin molecules first in the low-spin and then

10.1021/jp107147x  2010 American Chemical Society Published on Web 10/28/2010

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Figure 1. (a) Side view of the same-side beta-connected structure in two different spin states. The low-spin state has an extra CO ligand compared to the high-spin state. Isolated molecules optimized as dithiols with BP86/TZVP. C atoms are in brown, N in blue, O in red, H in white, and Au (electrodes) and Fe (center) in yellow. (b) Detailed view of the adsorption site (for the sake of clarity, only one S atom of the molecule is shown).

Figure 2. Connection schemes under study, illustrated for the low-spin species. Isolated molecules are optimized as dithiols with BP86/TZVP. The color scheme for the molecular structures is as that in Figure 1.

in the high-spin state in section 4, and a summary and outlook are provided in section 5. 2. Theory We will assume that the Landauer-Imry scheme36,37 holds, so that the current Is(V) for electrons of spin s ∈ {R,β} can be calculated from the integral of a transmission function Ts(E), where the energy (E) range to be integrated over is determined by the Fermi level EF of the electrodes in equilibrium and the bias voltage V (assuming a symmetric voltage drop over the junction and zero temperature)

Is(V) )

e h

+eV/2 dE Ts(E) ∫EE-eV/2 F

F

(1)

where A and B are atoms located to the left or right, respectively, of the surface through which transport is calculated. (Within the implementation of local transmission employed in this work, the local transmission contributions are forced to originate and end at atoms.) We model the gold junctions by attaching Au9 clusters to each side of the molecule, mimicking hollow-site adsorption. We then calculate the electronic structures for these systems using spin-restricted and spin-unrestricted Kohn-Sham DFT42,43 for the low-spin and high-spin complexes, respectively, and postprocess44 the converged Fock and overlap matrices to obtain the transmission from a Green’s function approach,45 where the central subsystem is treated explicitly and the environment is described by self energies46-48 (see Appendix A.2 and ref 35 for further details). 3. Systems under Study

e being the unit charge and h Planck’s constant (see Supporting Information for further details). This relationship holds in the coherent tunneling regime, that is, for low temperatures and short molecular bridges with a large separation between the molecular one-particle energy levels and the Fermi energies of the electrodes (“off-resonant” conditions).38 The number of electrons on the molecule is assumed to be constant in time. Broadly, the transmission measures the propensity for an electron to tunnel through the junction. In principle, Ts depends not only on the energy but also, generally to a lesser extent, on the bias voltage. We neglect the voltage dependence here and use the zero bias transmission throughout. The zero voltage differential conductance may be estimated from the transmission at EF. We decompose the transmission into local (atomic) contributions39-41

Ts(E) )

∑

A∈L,B∈R

Ts,AB(E)

(2)

The porphyrin systems have either imidazole and carbon monooxide (CO) or only imidazole as the axial ligands (see Figure 1), mimicking the coordination environment of COmyoglobin and myoglobin. The axial ligands contribute to the transmission only occasionally and over a small energy range, so that our results may be expected to be valid for other axial ligands as well. We took the models to be in the spin ground state of the corresponding myoglobin system, which is low spin (S ) 0) for the CO-containing species and high spin (S ) 2) for the CO-free system.49 The molecules are connected to gold electrodes via thiolate groups through either meso or beta positions (Figure 2). In the latter case, there are two possibilities; the sulfur atoms may be located either on the same side or on opposite sides of one of the porphyrin’s symmetry planes. The synthetically most accessible substitution scheme is the meso position, whereas selective substitution in the beta position may be a challenge.50,51 Our focus is therefore on the meso-connected porphyrin. The beta-connected species are included here to illustrate to what extent our conclusions depend on the linker

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Figure 3. Isodensity plots of CMOs of the meso-connected structure in the low- and high-spin states. B3LYP/LANL2DZ. The color scheme for the molecular structures is as that in Figure 1, except that C atoms are in black here.

scheme. Additional data on those structures may be found in the Supporting Information. Although we are focusing on the molecular contributions to transport, we included three gold atoms on each side in the central subsystem. This is because when including the iron porphyrin only, significant ghost transmission35 occurs even with a moderately sized double-ζ basis set. Ghost transmission manifests as an increased baseline for the transmission across a large energy range and can be very sensitive to such small changes in the computational model.35 4. Results and Discussion First, we will study the local contributions to transmission for the three low-spin systems employing the hybrid density functional B3LYP,52,53 which features a 20% exact exchange admixture, and compare the results with the pure functional BP86.54,55 Going from a hybrid functional to a pure one usually implies that the energetic gap between the highest occupied (HOMO) and the lowest unoccupied MO (LUMO) decreases.

Since peaks in the transmission are often located close to CMO energies, it may be anticipated that changing the functional will result in quantitative changes in the transmission. (CMOs were obtained by solving the secular equation for the central subsystem blocks of the Fock and overlap matrices calculated for the full gold cluster-molecule-gold cluster system.) Thus, it is important to ask whether there are qualitative conclusions, such as the nature of the local transmission at energies in offresonant energy ranges, which can be drawn irrespective of the functional employed. Eventually, we will discuss whether these conclusions also hold when going to a system with a high-spin ground state. In most calculations, if not mentioned otherwise, a basis set of double-ζ quality (LANL2DZ) is employed. This is first because of convergence difficulties when using basis sets of triple-ζ quality. Second, the LANL2DZ basis set may be expected to provide a sufficiently good quality for the purpose of this study. Furthermore, using a smaller basis set gives a clearer picture when plotting local transmission because the basis functions are more localized on the atoms on which they are

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Figure 4. Transmission calculated for the three systems under study in the different spin states with two different exchange-correlation functionals. The BP86 data for the meso-connected structure in the high-spin state are missing due to convergence problems in the self-consistent field algorithm.

centered and therefore are more suitable for the definition of atoms in molecules. Representative data showing that employing a larger basis set does not affect our qualitative results are provided in the Supporting Information. 4.1. Low-Spin Iron Porphyrin Systems. In the low-spin state, the B3LYP/LANL2DZ energies of CMOs which are located on the iron porphyrin system are either below about -5 eV or above about -2 eV (see left column in Figure 3 for a representative example), corresponding to the HOMO and LUMO, respectively, in the isolated iron porphyrin. This is reflected in the transmission (upper left panel of Figure 4), which has peaks at approximately these energies, with lower transmission in between. The interval between those peaks will be referred to as the off-resonant region. The transmission curves through the two beta-connected systems are very similar to each other, with a number of peaks at energies lower than -5 eV and dipping as low as 10-2 in the off-resonant region. For the meso-connected form, the overall energy dependence of the transmission is comparable to the beta-connected cases from a qualitative point of view, but the off-resonant transmission is around an order of magnitude higher. Also, the peaks at around -5 eV are much sharper in the beta-connected cases, which may be attributed to the fact that the CMOs responsible for those peaks (located at -5.05, -5.04, and -5.11 eV, for the opposite- and same-side beta-connected and the mesoconnected structures, respectively), have larger coefficients on the sulfur atoms for the meso-connected structure and are thus more strongly coupled to the electrodes (see Figure 3 and Figure 1 in the Supporting Information). The diagonal shape of the CMO at -5.84 eV in the meso structure suggests that a beta connection scheme would lead to stronger coupling of this CMO to the electrodes (compare the MO at -6.20 eV in Figure 1 in the Supporting Information), but due to the large number of transmission peaks and features in that energy region, this broadening cannot be seen very clearly. Following a discussion of the MOs contributing to the features in the transmission, complementary information can be obtained from a local decomposition of the transmission. Whereas a MO can be localized on any part of the molecule, local transmission has to fulfill the requirement that the sums of contributions over pairs of atoms to the left and right of any surface have to add up to the total transmission (compare eq 2), which puts constraints on the degree of localization that is possible. In the upper panels of Figure 5, the through-metal and through-ligand contributions to the transmission are plotted for

the three connection schemes under study in the low-spin state, using the B3LYP functional. The local contributions are calculated by summing up all local transmission values TAB, for which A is located to the left and B to the right of three particular interfaces for the three molecular systems. We assign the components to particular sets depending on whether A ) Fe (“Fe”) or A * Fe (“other”) and then take the absolute value (for plots of the signed contributions, see Figure 13 in the Supporting Information). The red dotted curves indicate the sum of all local transmission contributions across the interface (“sum”), for which A and B are located within the central subsystem and which deviate slightly from the total transmission because the electrode atoms outside of the central region are not taken into account. For all three cases, and in particular for the meso-connection scheme, the through-metal contribution is well below the through-ligand contributions in the off-resonant range, whereas in the energy region below about -5 eV, the metal plays a much more important role (see the local transmission plots in Figures 6-12 in the Supporting Information). The energies at which this occurs approximately correspond to those of the CMOs centered on the iron atom. In the lower-energy region, both total and local transmissions oscillate strongly, which may be related to several energetically close-lying CMOs. Given the uncertainties inherent in our methodology, it seems not advisible to discuss those changes, and in particular the ringcurrent-like features, in detail. For the meso-connected case (see upper panel in Figure 6), representative local transmission plots are given to illustrate how the local transmission pattern may change considerably in the region below -5 eV, whereas it is essentially stable throughout the off-resonant energy range. We are assuming here that the energies above the off-resonant region are too high to be experimentally relevant. As far as the comparison between the transmission curves obtained with the B3LYP and BP86 functionals is concerned (upper versus lower panels in Figure 4), as expected in B3LYP, the peaks corresponding to the porphyrin LUMO are shifted to lower energies, and the ones corresponding to the porphyrin HOMO are shifted to higher energies. This can also be seen in the CMO plots, with energies of -4.82 (HOMO) and -2.78 eV (LUMO) for the same-side beta-connected case (Figure 2 in the Supporting Information). The overall shape of the transmission curves is comparable for the two functionals, except for the opposite-side beta-connected structure, which features a more pronounced dip at the lower-energy end of the offresonant region in the BP86 transmission. This may be attributed

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Figure 5. Absolute contributions from the iron center (Fe) and from the remaining atoms (other) to the transmission T (total) across the interface indicated by the blue line in the upper panels. The color scheme for the molecular structures is as that in Figure 1.

to iron-dominated CMOs at -4.92, -4.87, and -4.58 eV, two of which are energetically located in the porphyrin HOMO-LUMO gap (see Figure 3 in the Supporting Information), whereas they are well below the HOMO and HOMO-1 when employing the B3LYP density functional. When using BP86 for the sameside beta-connected case, iron-dominated CMOs can be found at -4.87, -4.88, and -4.59 eV, and accordingly, there are small metal contributions close to those energies in the local transmission (see Figure 7 in the Supporting Information). However, these CMOs do not qualitatively alter the transmission. This discussion illustrates how sensitive the transmission can be to CMO shifts and rearrangements. Overall, despite the quantitative changes in the transmission and the qualitative changes in the orbital ordering, the conclusions drawn for local transmission contributions from the B3LYP results also hold for the BP86 calculations; whereas at energies corresponding to metal-centered MOs the local transmission has considerable through-metal contributions, at other energies and in particular in the off-resonant region, throughligand terms dominate (see lower panels in Figure 5). 4.2. High-Spin Iron Porphyrin Systems. When comparing the B3LYP/LANL2DZ CMOs for the meso-connected structure in the high-spin structure (see Figure 3) to the ones for the lowspin structure, it becomes obvious that there is one metalcentered CMO between the porphyrin HOMO and LUMO, located at -4.62 (R electrons) or -4.67 eV (β electrons). This CMO is clearly visible as an extra peak in the transmission at -4.61 and -4.68 eV (upper middle and right panel of Figure

4). Comparing the shapes of these CMOs suggests that the peak in the R transmission is broader than the one in the β transmission due to the stronger coupling between the R MO and the gold electrode as compared to that with the β MO. Accordingly, at the energy of this extra CMO, there is a large through-iron contribution in the local transmission (see lower part of Figure 6). Similar conclusions can be drawn for the betaconnected structures (see Supporting Information). From the BP86 transmission (lower panels in Figure 4) and its local contributions (Figures 7, 11, 12, and 14 in the Supporting Information), it can be seen that although there are considerable differences with respect to the detailed shape of the transmission, the same general conclusion can be drawn as that from the B3LYP calculations, namely, that going from the low-spin to the high-spin structure may introduce additional peaks in the transmission, which correspond to iron-centered MOs and thus to important iron contributions to the local transmission, but that otherwise the local transmission in the off-resonant region follows the porphyrin ligand. 5. Conclusion In summary, this theoretical study of coherent electron tunneling through iron porphyrin junctions suggests that off resonance (for energies around -2 to -5 eV or slightly lower), tunneling occurs mainly through the porphyrin ligand, whereas for lower energies, tunneling through the metal center is more significant. There is strong correlation between the energies of

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Figure 6. Representative local transmission plots at different energies for the meso-connected system in the low-spin and high-spin states. B3LYP/ LANL2DZ. The diameter of the arrows represents the value of the local transmission for a pair of atoms. Red and blue arrows indicate local transmission contributions in the same or opposite direction, respectively, as the overall current. For each structure, the largest local transmission contribution is normalized, that is, the largest arrow in each plot has the same diameter. The value of the overall transmission is given next to each plot. The color scheme for the molecular structures is as that in Figure 1, except that C atoms are in black and Fe atoms are in red.

CMOs centered on the iron atom and local transmission patterns with dominant iron contributions. Going from the low-spin to the high-spin system by removing the CO ligand introduces additional peaks in the transmission which correspond to ironcentered CMOs with energies between the porphyrin ring HOMO and LUMO. This link between CMOs and local transmission suggests the question of what can be learned from local transmission that cannot be learned from analyzing the CMOs. While close to resonances, both schemes give qualitatively similar answers to the question which path does the electron take, local transmission can cover the whole energy range, so that it is also possible to extract information on off-resonant energy ranges. For the form of a given molecular orbital to dominate across a large energy range, the molecular resonance has to be broadened significantly through strong coupling to the metallic electrodes. In the systems studied here, even when metalcentered molecular resonances were near or even in the offresonant region, they did not result in metal-dominated local transmission over an appreciable energy range. In fact, resonances with dominant contributions through the metal center were generally narrow across the entire energy range studied here. This raises a question for future work as to how to design metal complexes where the metal d orbitals are strongly coupled to the electrodes. In all cases, the meso-connected form transmission is about an order of magnitude larger (over the off-resonant energy range) than the one for the beta-connected variants. Still, the linker scheme does not affect our conclusions on local contributions to the transmission.

The comparison of B3LYP and BP86 results illustrates that one has to be careful with interpreting the results of transport calculations too quantitatively. Qualitative results may be stable with respect to changes of parameters in the electronic structure calculation as well as to chemical changes such as removing a ligand accompanied by a change in the spin ground state. It would be interesting to complement the present study by investigating other ligands such as phthalocyanines and corrins or spin crossover systems and by studying the effect of changing the metal center and the connection scheme in more detail. Also, a comparison to systems adsorbed on surfaces at various angles32 would provide very helpful information. Such studies can be a first step toward elucidating, by comparing theoretical and experimental results, the conditions under which particular synthetic approaches give structures in which electron transport by coherent tunneling can proceed along particular pathways. From the experimental side, techniques such as vibrational IETS, shot noise measurements, and the recently developed STMbased methods to study spatially resolved conductance33 may prove very valuable for this purpose. Acknowledgment. C.H. gratefully acknowledges funding through a Forschungsstipendium by the Deutsche Forschungsgemeinschaft (DFG). This material is based upon work supported as part of the ANSER, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award Number DE-SC0001059. We thank J. Subotnik for a customized version of Qchem.

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Appendix A. Methodology. A.1. Electronic Structure Calculations. The dithiol molecules were optimized with an overall charge of zero, using a triple-ζ quality basis set with one set of polarization functions (TZVP56,57 and the BP86 density functional,54,55 employing the Qchem58 quantum chemistry program package, with a maximum atomic displacement in subsequent optimization steps below 1.2 × 10-3 Å. After optimization, the thiol groups’ hydrogen atoms were stripped off, and the structures were placed between two Au9 clusters with a sulfur-gold distance of 2.48 Å.59 Au-Au distances were set to their value in extended gold crystals (2.88 Å). A tight criterion of 10-8 au for the direct inversion of the iterative subspace (DIIS) error was chosen in all Qchem calculations, whereas a criterion of 10-6 au was chosen for the largest element of the commutator of the Fock matrix and the density matrix in the ADF calculations. The density matrix of the Au9-molecule-Au9 systems was calculated from the molecular orbital (MO) coefficients in each step of the self-consistent field (SCF) algorithm, as usually done for closed systems, and not from the lesser Green’s function, which would take the open nature of the system into account.46,47,60 In other words, the nonequilibrium Green’s function (NEGF) formalism was not employed, and thus, the presence of the semiinfinite metal electrodes was not accounted for in the SCF algorithm. No periodic boundary conditions were assumed. Spinunrestricted Kohn-Sham density functional theory calculations were performed using a locally modified version of Qchem, or ADF61 when indicated. Atom-centered basis sets of either contracted Gaussian (Qchem) or Slater-type (ADF) of double-ζ quality (LANL2DZ effective core potential (ECP) with matching basis sets as implemented in Qchem;62 DZ61) or triple-ζ quality with one set of polarization functions (TZVP,56,57 TZP61) were employed. All ADF calculations were carried out within the zeroth-order regular approximation (ZORA) to describe scalar relativistic effects and used a small frozen core, comprising the 1s shell of carbon, nitrogen, and oxygen atoms and the 1s-4d shells of gold atoms. Stuttgart ECPs63 were used for the gold atoms in the TZVP calculations. For the transport calculations, either the B3LYP exchange-correlation functional,52,53 which features 20% Hartree-Fock exchange, or the pure density functional BP8654,55 was employed. A.2. Electron Transport Calculations. Transmission functions were obtained by postprocessing output from electronic structure calculations on these finite-size electrode-molecule-electrode systems using routines written in our laboratory.44 In the Green’s function approach, Ts is calculated from a trace over matrices describing the coupling of a central region64 to the left and right electrodes, ΓL/R,s, and the central system subblock of the retarded and advanced Green’s functions of the electrode-molr/a 48,65 , ecule-electrode system GC,s r a Ts(E) ) tr(ΓR,s(E)GC,s (E)ΓL,s(E)GC,s (E))

(3)

the advanced Green’s function being the complex conjugate of r are calculated from the overlap and the retarded. ΓX,s and GC,s Fock matrices of a finite cluster electrode-molecule-electrode system

ΓX,s ) -2 Im[(ESXC - HXC,s)†gX,s(ESXC - HXC,s)]

(4)

-1 1 1 r GC,s ) ESC - HC,s + i ΓR,s + i ΓL,s 2 2

(

)

(5)

The Fock and overlap matrices of the electrode-moleculeelectrode system are divided into central, left-electrode, and right-electrode regions. SXC and HXC,s denote the coupling block of electrode X and the molecule in the overlap and Fock matrix, respectively, while the molecule (or “central region”) subblocks of these matrices are indicated by the subscript C. The three gold atoms closest to the sulfur atom were included in the central region to avoid ghost transmission.35 The Green’s function matrices of the isolated electrodes X (X ∈ {L,R}) were calculated in the wide band limit (WBL) approximation

(gX)ij ) -i · π · LDOSconst · δij

(6)

that is, the local density of states (LDOS) was assumed to be independent of the energy, and furthermore, the same LDOS value of 0.036 eV-1, obtained from the s band LDOS for bulk gold from DFT calculations66 was assigned to all basis functions. Although this is a rather crude approximation, it typically works very well for electrode metals such as gold, which feature a comparatively flat LDOS distribution around the Fermi energy (when plotted as a function of energy) and whose conduction properties are dominated by the broad s band. For example, the same qualitative trends were observed for the transmission of an iron complex,27 regardless of whether the electrodes were described in the WBL approximation or with a more sophisticated tight-binding model. Energies were not shifted by the system’s Fermi energy. This is because the correct choice of the Fermi energy is not obvious for calculations of the type performed here. The Fermi function of bulk gold (-5.5 eV67) may be regarded as a rough approximation, although the fact that in practical calculations, metal clusters of finite size are used to model the bulk electrodes may shift this value. Local transmission values between atoms A and B, TAB, were calculated according to39-41

Ts,AB )

∑

Hs,µν Im(GsrΓsLGsa)νµ

(7)

µ∈A,ν∈B

where µ and ν denote the largely atom-centered basis functions which are obtained after carrying out a symmetric (Lo¨wdin) orthogonalization in the central subsystem. MOs were plotted using Molden,68 with isodensity values of 0.03. B. Resonant versus Off-Resonant Transport. The CMO plots and the transmission curves displayed in Figures 3 and 4 suggest that there may be CMOs close to the electrodes’ Fermi energy (that is, in resonance), which for bulk gold is -5.5 eV.67 If this is the case, charging and electron-phonon coupling may play an important role in controlling the transport properties, which is not accounted for in the Landauer model.38 Due to the approximations made in the description of the semi-infinite electrodes and of the molecular electronic structure employed here (see Supporting Information), it is not clear precisely where the Fermi energy to be used in eq 1 is located.69 The value for bulk gold may only serve as a rough guideline. Furthermore, the choice of the exchange-correlation functional and other parameters of the calculation may affect all one-electron energy

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