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Characterizing Destructive Quantum Interference in Electron Transport

Destructive quantum interference in electron transport through molecules provides an unconventional route for suppressing electric current. In this work we introduce"interference vectors"for each interference and use them to characterize the interference. An interference vector may be an orbital of the bare molecule, in which case the interference is very sensitive to perturbation. In contrast, an interference vector may be a combination of multiple molecular orbitals, leading to more robust interference that is likelier to be experimentally observable. Our characterization scheme quantifies these two possibilities through the degree of rotation and also assigns an order to each interference that describes the shape of the Landauer-B\"uttiker transmission function around the interference. Several examples are then presented, showcasing the generality of our theory and characterization scheme, which is not limited to specific classes of molecules or particular molecule-electrode coupling patterns.


I. INTRODUCTION
Molecules have been suggested as components of electrical circuits in the ongoing drive for device miniaturization [1][2][3]. To this end, both experimental and theoretical studies have investigated a molecule's ability to conduct electric current when sandwiched between two electrodes to form a junction [2]. In addition to such applied interest, these molecular junctions have also proven to be fundamentally valuable for investigating the mechanical strength of chemical bonds [4] and single-molecule chemical reactions [5].
Owing to the molecules' nanometer-scale dimensions, quantum mechanical effects are inherent in transporting electrons across these junctions. Molecular orbitals are broadened into finite-lifetime resonances when the molecule is connected to the electrodes, and the alignment of these resonances relative to the junction's Fermi energy strongly correlates with the junction's conductivity. Destructive quantum interference (DQI) is one notable exception to this principle. Some molecules (such as benzene) have multiple paths [6,7] for transporting electrons across the junction, and these paths may destructively interfere with each other to suppress or even block current [8][9][10][11]. From an applications perspective, DQI may result in good molecular insulators [3].
Conjugated hydrocarbons are commonly employed, with DQI present in cyclic molecules, cross-conjugated molecules, and molecules with pendant groups (see [11] and references therein). The common theme is that DQI primarily depends on the molecule's electronic structure and where the electrodes contact the molecule [11,18]. For example, a benzene molecule produces DQI when connected to the electrodes in the meta or ortho configurations, but not para [10].
With this observation, many guidelines have been developed for predicting molecules and molecule-electrode configurations that exhibit DQI [19][20][21][22][23][24][25][26], including some graphical approaches [27][28][29]. These guidelines build physical intuition by relating DQI to either the real-space paths through the molecule (an atomic orbital-like approach) or the isolated molecule's orbitals [30]. Regardless, they tend to focus only on the existence of DQI, and are most applicable to alternant hydrocarbons [31], where Hückel or tight-binding representations for the molecule with simple molecule-electrode connections are common.
In this work, we go beyond predicting only the existence of DQI in a molecular junction and develop a broadly-applicable characterization scheme for DQI. Of primary interest is the ability to predict and classify the "robustness" of DQI; that is, the likelihood that the DQI will be experimentally observable. Essentially, DQI produces roots in the Landauer-Büttiker transmission function (vide infra), and we recently derived an eigenvalue problem for finding these roots [11]. Our main contribution here is an analysis of the corresponding eigenvectors, which we term "interference vectors". These interference vectors possess geometric properties that predict the line shape of the transmission function around DQI, thereby allowing us to characterize DQI.
We develop and showcase our analysis of interference vectors through several examples of increasing complexity. Our key findings include a relationship between bound states in a molecular junction and DQI in the same molecule if it were wired to the electrodes in a different configuration, the prediction of so-called supernodes [32] in oligomeric molecules, and the importance of "coherence" in the molecule-electrode coupling when nontrivial configurations are employed [25,33]. We also apply our analysis to DQI occurring at complex energies, which do not appear to present fundamentally new chemical insights. Our analysis is widely applicable because it builds upon a general theory of DQI [11] that is not limited to conjugated hydrocarbons or simple molecular models. We thus put DQI on similar theoretical footing as a resonance analysis for locating highly-conductive molecular junctions; both analyses now possess an eigenvalue problem with physically-meaningful eigenvectors.
We hope this will lead to new chemical and physical intuition for predicting, understanding, and exploiting DQI in electron transport processes.
The layout of this paper is as follows. Section II first overviews the pertinent details of Landauer-Büttiker theory for electron transport and then discusses DQI in benzene, the quintessential prototype for such effects in electron transport through molecules. We present our method for characterizing and analyzing DQI in Section III; this is the principal contribution of the present work. Section IV then applies our framework to numerous examples, including benzene, anthracene derivatives, and cross-conjugated molecules. Finally, we summarize and conclude in Section V.

II. BACKGROUND: LANDAUER-BÜTTIKER THEORY AND BENZENE
In this section we review DQI in a benzene molecule as described by a tight-binding model. This system has become the standard example of DQI in molecular electron transport, and a detailed analysis of it is presented in [10]. Herein we summarize the pertinent details, which will provide context and an early example for the analysis we develop in Section III. A full description of this model can be found in Section IV, the Supplemental Information, and [10]. However, before we discuss electron transport through benzene, we must first introduce the Landauer-Büttiker theory for electron transport.

A. Landauer-Büttiker Theory
Within the limit of coherent scattering, electron transport through molecules is described by the Landauer-Büttiker formalism [2,34,35]. The transmission function, T (E), is the key quantity, which essentially describes the probability that an electron with energy E successfully tunnels from one electrode to the other through the molecule. In the limit of zero applied bias, the steady-state conductance through the electrode-molecule-electrode junction is where G 0 ≡ 2e 2 /h is the quantum of conductance and E F is the Fermi energy of the junction.
From a theoretical perspective, the transmission function is obtained from the Hamiltonian of the isolated molecule, H 0 , and self-energies, Σ L/R (E), that describe how the molecule couples to the left/right electrode. The self-energies are effectively open-system boundary conditions on the molecule. Then, where is the Green function of the molecule (as modified by the electrodes) and is the spectral density for coupling the molecule to the left/right electrode. As a rough rule-of-thumb, the transmission function peaks at an energy E if E is the real part of an eigenvalue of H 0 + Σ L (E) + Σ R (E); that is, there is a molecular resonance at E. Real eigenvalues indicate the presence of bound states (i.e., molecular orbitals that do not couple to either electrode) in the molecular junction [36,37], which are inconsequential to steadystate transport and can be neglected. In what follows, we focus on the transmission function instead of conductance so that we can look at many possible behaviors with a only a few examples. The conductance can always be obtained by evaluating T (E) at the Fermi energy.
It is commonly assumed that each electrode only couples to one site of the molecule-that there is only one conduction channel through the junction-such that DQI perfectly reflects electrons that enter the junction with energy E. E is called the location of DQI. More mathematically, DQI at E means the junction yields T (E) = 0 when rank(Σ L (E)) = 1 or rank(Σ R (E)) = 1, where rank(Σ L/R (E)) can be regarded as the number of "bonds" between the molecule and the left/right electrode. Identifying DQI is thus tantamount to finding roots of the transmission function [38]. [11] discusses an approach for describing DQI when there is more than one channel through the junction.  This width of DQI in the transmission function leads to the idea of "robustness". Suppose the junction's Fermi energy is close to 0 eV in the meta-benzene system. DQI is nearby; the transmission and thus conductance will be very low such that the DQI would be observable.
This makes the DQI robust. On the other hand, if the Fermi energy were near ±2.5 eV, FIG. 1. Transmission functions for electron transport through a benzene molecule with electrodes connected in the para (red), meta (green), and ortho (blue) configurations. Meta-benzene has the most robust (widest signature in T (E)) destructive interference effect, at E = 0 eV, followed by the effects at E ≈ ±3.5 eV in ortho-benzene, and by those at E ≈ ±2.5 eV in both ortho-and meta-benzene. This figure is modified, with permission, from [11], copyright 2014, AIP Publishing LLC.
the DQI might not be experimentally observed due to its narrow energy range. Such DQI is less robust.
Hansen et al. [10] further classified these instances of DQI in benzene as either "multipath" or "resonance". Multi-path DQI stems from competing paths around the benzene molecule. The two paths essentially cancel each other through destructive interference, resulting in zero transmission. Contrasting, resonance DQI comes from the molecule's electronic structure alone, indicating substructure within H 0 . All DQI in the benzene configurations is multi-path except the instance through meta-benzene at E = 0 eV, which is resonance.
Some of the logic for distinguishing multi-path and resonance DQI in [10] is inextricably linked to the cyclic structure of benzene. Knowing that acyclic molecules can also exhibit DQI [39], one of our goals in the present discussion is to generalize this classification. We will ultimately show in Sections III and IV that resonance DQI (as generalized) is more robust than multi-path DQI. Each also has a distinct signature in the molecule's electronic structure. In this way, our characterization helps predict the experimental observability of DQI.

III. RESULTS
The benzene example in the previous section demonstrates the various types of DQI and, in the case of cyclic molecules, provides a scheme for classifying them. In this section we discuss DQI more broadly and generalize the characterization scheme. The following discussion is the primary contribution of this work.
Without making assumptions about the molecule or the molecule-electrode couplings (that is, without restricting our attention to conjugated hydrocarbons, tight-binding models, or junctions with a single conduction channel), DQI is described by a generalized eigenvalue problem [11]. To reach this result, we must distinguish the parts of the molecule that directly couple to a particular electrode from the parts that do not. The kernels of Γ L (E) and The examples in Section IV will help illustrate this idea.
Mathematically, this condition is described by the generalized eigenvalue problem where I is the identity. The notation (O) A→B denotes an operator restriction where the operator O is only applied to state vectors from A and the resulting state vectors are projected into B. In all but pathological cases, this equation can be simplified [40] to We immediately see that DQI is primarily caused by substructure within the molecular Hamiltonian; the electrodes only serve to identify the parts of the molecule that are not coupled to each electrode. Finally, owing to the asymmetry between left and right electrodes, the left eigenvectors will be generally unrelated to the right eigenvectors, and are described by Our previous work [11] focused only on the existence and locations of DQI, where it was sufficient to find eigenvalues E that satisfy Eq. (2). But Eq. (2) also associates a left FIG. 2. Graphical description of our characterization scheme for DQI. On the horizontal axis is the "degree of rotation" of DQI, θ, given by Eq. (3). When θ = 0, the left and/or right interference vector is a molecular orbital and the DQI is classified as bound-state. When θ = π/2, DQI results from an anti-resonance within the molecular Hamiltonian, generalizing the resonance-type DQI from [10]. Intermediate degrees of rotation are also possible, evincing a continuous domain with bound-state and anti-resonant DQI as opposite extremes. On the vertical axis is the "order" of DQI, which describes the shape of the transmission function around the DQI. Most DQI is firstorder, although higher orders have been theoretically predicted [32] and are called supernodes.
eigenvector |ϕ L and a right eigenvector |ϕ R with each instance of DQI. These eigenvectors, which we call "interference vectors" in the following discussion, provide the means to generalize the characterization scheme of Hansen et al. [10] from cyclic molecules to arbitrary molecules. Furthermore, each interference vector can be expanded in the molecular orbital basis, thereby revealing the participation of each molecular orbital in the DQI. Before proceeding, we note that the left and right interference vectors for the same instance of DQI are essentially unrelated to each other. This is a general property of generalized eigenvalue problems [41,42].
There are two key properties of the interference vectors and eigenvalues that lead to our generalized characterization scheme, which is graphically summarized in Figure 2. We here define these properties and state the characterization scheme. Rationale and additional details will be presented alongside examples in Section IV.
(i) The "degree of rotation" of the interference vectors, defined by From vector calculus, either arccos expression in Eq. (3) is the angle that the respective interference vector is rotated by EI − H 0 [43], with this angle defined as 0 if the interference vector is an eigenvector of H 0 for eigenvalue E. As we will see in the examples, θ relates to the robustness of DQI: θ = 0 produces a very narrow effect in the transmission function, whereas a larger θ (up to a maximum of θ = π/2) creates wider effects in T (E).
(ii) The "order" of the interference vector, which comes from the defectiveness of the DQI's eigenvalue in Eq. (2). (Recall from linear algebra that a degenerate eigenvalue may lack a complete set of linearly independent eigenvectors, in which case it is called defective. The Kronecker canonical form [41,42] helps identify these cases.) The order predicts the shape of T (E) around DQI located at E i . If the order is n = 1, 2, 3, . . ., then a Taylor Second-and higher-order DQI have been previously discussed [32]; however, they are very sensitive to perturbation such that most DQI is firstorder.
• If one or both of the interference vectors are molecular orbitals such that θ = 0, the DQI is called "bound-state".
• DQI is "anti-resonant" when θ = π/2. In this case, both interference vectors are rotated 90 • by EI − H 0 , meaning they can be regarded as molecular anti-resonances.
Resonance-type DQI from [10] fall into this category.
• Intermediate values of θ are not given special names, but would belong to the multipath class from [10].

IV. DISCUSSION
We now present several example systems that showcase our characterization scheme for DQI. First is a simple model that exhibits the development and utility of the degree of rotation. We then proceed to more chemically-relevant examples, including benzene, crossconjugated molecules, anthracene derivatives, and molecules with non-trivial couplings to the electrodes. Development of the order metric will be presented alongside the cross-conjugated molecules.
For simplicity, all of our examples employ tight-binding models, even though the theory and characterization scheme are more general. Unless noted otherwise, each "atom" in the molecule has a single orbital with an on-site energy of ε = 0 eV and couples to its nearest neighbors with β = −2.5 eV. Finally, because the magnitude of electrode-molecule coupling is not germane to DQI [see Eq. (2)], we invoke the wide-band limit [44] for the electrodes.
Matrix elements for sites where the molecule couples to an electrode are −iΓ = −0.1i eV. Full details about our models and computations can be found in the Supplemental Information.

A. Three-Site Model
Our opening example is a three-site tight-binding model, as pictured in Figure 3. Although this model might be a representation of propene, we do not place any physical or chemical significance on the results. Rather, this simple example is intended to motivate the degree of rotation and to demonstrate the different types of DQI that can occur in more physically-meaningful systems.  site. We can immediately draw several conclusions about the transport properties of this molecule when connected in some configurations. Suppose both electrodes couple to this middle site (i.e., the 2,2 configuration). There will be a bound state [36,37] in the junction because the MO |ψ 2 does not couple to either electrode and thus does not participate in transport. If we move exactly one electrode so that it couples to a different site (e.g., the 1,2 configuration), then there will be DQI at that orbital's energy. The MO is still decoupled from one electrode, and Figure 3(b) verifies a narrow instance of DQI at E = ε = 0 eV.
Furthermore, as shown in Figure 3(d), one of the interference vectors for this DQI is the MO itself. Because this interference vector is a MO that forms a bound state in a different molecule-electrode configuration, we call this "bound-state" DQI. Knowledge of the transport through one configuration can provide information on other configurations.
The nodal structure of the interference vectors is also important for characterizing DQI.
By construction in Eq. (2), each interference vector will be decoupled from at least one electrode (having a node at those sites), with corresponding left and right interference vectors . As we might expect from the above discussion, this instance of DQI is much more robust than that in the 1,2 configuration. We also see that the interference vectors are not MOs; instead, |ϕ L = |ϕ R = (|ψ 1 + |ψ 3 )/ √ 2. From our definition, this cannot be bound-state DQI.
The degree of rotation θ quantifies this nodal structure within the interference vectors by examining geometric properties of the interference vectors. Specifically, we look at the rotation of an interference vector by EI−H 0 , where E is the location of the specific instance of DQI and the angle of rotation is [43] arccos assuming ϕ L/R is neither the zero vector nor an eigenvector of EI − H 0 (in which case the angle is defined to be 0). In the case of bound-state DQI, at least one of the interference vectors is a MO-that is, it is an eigenvector of both H 0 and EI − H 0 -such that it is not rotated by EI − H 0 . Thus, |ϕ L or |ϕ R having a rotation of 0 indicates bound-state DQI.
In contrast, the interference vectors for DQI at E = ε in the 2,2 configuration are rotated π/2 by EI − H 0 ; that is, (EI − H 0 ) ϕ L/R is orthogonal to ϕ L/R .
It is straightforward to show that an interference vector that is decoupled from both electrodes will be rotated π/2 by EI − H 0 . Because of the nodes at both electrodes, it will also be rotated π/2 by EI correspond to molecular resonances in the junction, there's a notational appeal to classifying these interference vectors as anti-resonant DQI. The caveat, as can be seen in the bound-state DQI from our three-site example, is that both the left and right interference vectors must be rotated by π/2 for the instance of DQI to be considered antiresonant.
We finally arrive at the degree of rotation θ defined in Eq. Finally, before moving on to the next example, we discuss one other application for the degree of rotation. In addition to quantifying the nodal structure of the interference vectors of DQI, it also reveals some insight into the steepness of the transmission function around the DQI. Consider the Taylor series of T (E) around the location of each instance of DQI in the three-site model. For the bound-state DQI in the 1,2 configuration, The leading term of the transmission function is quadratic (indicating first-order DQI), and its coefficient does not depend on Γ, the molecule-electrode coupling strength. In a likewise fashion, the transmission function around the anti-resonant DQI in the 2,2 configuration is These leading coefficients now contain both Γ and β. When combined with other examples (see below), it appears that Γ and β are competing factors in these leading coefficients.
One of them is missing in bound-state DQI, and they are "in-phase" with each other in anti-resonant DQI. That is, they essentially appear as ratios. As we will see in the next example, intermediate DQI will have them be "out-of-phase".

B. Benzene
No characterization of DQI in molecules would be complete without showcasing benzene.
We briefly discussed this system in Section II, and Figure 1 shows the transmission functions for benzene connected in the ortho, meta, and para configurations. In the end, the analysis of DQI in benzene is very similar to that of the three-site model in the previous example. The only complicating factor is the degeneracy of benzene's highest-occupied MO and lowestunoccupied MO. As we will now see, this issue is straightforwardly handled.
Similar to our analysis of DQI in the three-site model, we begin with an examination of benzene's MOs, which are depicted in Figure 4(a). The MOs labeled |ψ 3 and |ψ 5 each have two nodes such that the para configuration will have bound states at E = ε ± β = ∓2.5 eV.
Moving exactly one electrode off of these nodes produces a system in either the ortho or meta configuration. Such a system should exhibit bound-state DQI at the energies of |ψ 3 (ε + β = −2.5 eV) and |ψ 5 (ε − β = 2.5 eV), which is verified in Figure 1. Figure 4(b) then shows the interference vectors for one of these cases. As we would expect, the interference vectors are MOs (thus having θ = 0); however, they do not exactly match |ψ 3 or |ψ 5 .
Instead, the interference vectors are linear combinations either of |ψ 2 and |ψ 3 or of |ψ 4 and |ψ 5 , which are still eigenvectors of the molecular Hamiltonian due to degeneracy of its respective eigenvalues.
In addition, we see that the DQI at E = ε = 0 eV in meta configuration is anti-resonant.
Its interference vectors are displayed in Figure 4(b), and are both decoupled from both electrodes. Finally, the DQI at E = ε ± √ 2β in the ortho configuration provides our first example of DQI that is neither bound-state nor anti-resonant. These instances of DQI have which is still quadratic and, as we expect for anti-resonant DQI, has β and Γ in-phase with each other. The bound-state DQI in ortho-or meta-benzene produce β is now missing from the coefficients, again supporting our classification as bound-state Both Γ and β are present in the coefficient, and its denominator suggests they are out-ofphase with each other.

C. Cross-Conjugated Molecules: Order of DQI
All of our examples so far have demonstrated first-order DQI. We now discuss higherorder DQI with the "comb" molecules of [45], which may be coarse-grained representations of cross-conjugated oligomers [46]. These molecules are schematically depicted in Figure   5(a). As mentioned in Section III, the order of DQI comes from the geometric degeneracy of the eigenvalue in Eq. (2) and ultimately relates to the shape of the transmission function around the DQI. Specifically, nth-order DQI at E i produces where C is a constant that depends on the molecule-electrode coupling strength (Γ) and/or molecular structure (e.g., β in our tight-binding models). This level of defectiveness is the order of DQI.
When n = 1, the eigenvalue is not defective; E = ε is a simple eigenvalue of Eq. (2) with a single (left or right) eigenvector. The DQI is thus first-order and Consistent with anti-resonant DQI, the leading coefficient in the Taylor expansion is an "in-phase" relationship between the coupling strength and the molecule structure. If we near the DQI. Higher-order DQI (resulting in supernodes) is more robust than lower-order DQI, but is also rarer than lower-order DQI (as discussed in the main text). (b) Interference vectors for DQI in these systems with n = 1, n = 2, and general n. All DQI in these comb oligomers is anti-resonant.
increase n to 2, E = ε becomes a doubly-degenerate eigenvalue, but it still has a single (left or right) interference vector. In linear algebra terms, the eigenvalue is defective. The DQI remains anti-resonant, but the shape of the transmission function around the DQI changes to This trend continues as the number of monomers ("teeth") in the comb molecule increases.
The DQI maintains a single (left or right) interference vector, but its degeneracy and order increase. In turn, the transmission function becomes flatter, and thus more robust, around the DQI: This result is evident in Figure 5(a).
Polymerization generally increases the order of DQI if the monomer exhibits DQI and the monomers are suitably connected [32,45,46]. In this sense, polymerization could be one means to designing systems with more robust DQI; others are discussed in [32]. We note, however, that high-order DQI is very sensitive to perturbation and is thus rare [32]. Any form of disorder, perhaps due to molecular vibrations, will likely reduce the DQI to firstorder. The DQI can still be relatively more robust; the transmission function will become quadratic around the DQI but with a small coefficient compared to higher-order terms.
As a mathematical aside, the order of DQI is the size of the Kronecker block for the DQI in the Kronecker canonical form [41,42]

D. Anthracene Derivatives: Complex Energies
Having developed and described our characterization scheme for DQI, we now apply it to two physical setups that are not well understood. First is DQI at complex energies [11,38,47], and then nontrivial molecule-electrode couplings [25,33] in the next subsection.
It was shown in [11,38] that DQI can occur at complex energies. The transmission function does not drop to zero along the real axis in such an event, but instead exhibits a minimum near the real part of the complex energy. The blue curve in Figure 6(c) shows an example of transmission around complex DQI energies. In general, DQI at a complex energy occurs when two instances of DQI with real energies "collide" and the locations travel off into the complex plane [38]. In this way, if a molecule exhibits DQI at two nearby energies, slight perturbations may cause the DQI to move off into the complex plane, thus increasing transmission [47].
We use tight-binding models of anthracene derivatives from [28] to demonstrate this effect.
As schematically depicted in Figures 6(a)    Most prominently, we see that changes in ε sg do not change the number of instances of DQI (note that the complex energies come in conjugate pairs). In some cases [ Figure 6(a)], DQI at complex energies emerge after two instances of DQI collide. In other cases [ Figure   6(b)], there can be a "band" of DQI at complex energies.
To gain more insight into the nature of DQI at complex energies, we examine DQI near and at the circled "collision" in Figure 6(a). The transmission functions around this DQI for ε sg = 0.84 eV, ε sg = 0.89 eV, and ε sg = 0.94 eV are displayed in Figure 6(c). Right interference vectors for DQI at these values of ε sg are also displayed in Figure 6(d In the end, an inspection of the interference vectors in Figure 6(d) shows very few changes with ε sg . There are certainly small quantitative differences (see the Supplemental Infor- In contrast, the generalized eigenvalue problem in Eq. (2) places no limitations on the molecule-electrode coupling. Therefore, our present definition of interference vectors and characterization scheme for DQI is readily applicable.
The chief subtlety with nontrivial molecule-electrode couplings is that it becomes insufficient to only specify the molecular sites that couple to each electrode. Consider the tight-binding model for butadiene in Figure 7. As depicted, one electrode couples to a single site (enumerated as site 1) but the other couples to two sites (sites 2 and 4). Similar systems are discussed in [25]. The self-energy for coupling to the second electrode might appear as where |χ j is the AO on the jth site. On the other hand, the self-energy may also be In both cases, the electrode only couples to sites 2 and 4 of the molecule, but the first self-energy has a rank of 2 and the second has a rank of 1. (Recall that the rank of the self-energy can be loosely interpreted as the number of "bonds" between the electrode and the molecule.) Hansen and Solomon [33] refer to these couplings as incoherent and coherent, respectively.
Such incoherent coupling essentially means that each molecular site independently interacts with the electrode, whereas the two sites' interactions are coordinated in the coherent case.
This seemingly-small distinction has significant effects on the transmission function and on DQI in the molecule, as displayed in Figure 7(a). Second-order, anti-resonant DQI is present at E = ε = 0 eV when the molecule is coherently coupled to the electrodes. There is no DQI, not even at complex energies, when the molecule is incoherently coupled to the electrodes.
This example highlights two points. First, nontrivial molecule-electrode couplings are more nuanced than the usual cases where each electrode couples to a single molecular site.
Coherent coupling can lead to qualitatively different transport properties from incoherent coupling. Second, our characterization of DQI readily generalizes to cases of nontrivial coupling. The generalized eigenvalue problem in Eq. (2) makes no assumptions about the style of coupling-only requiring Ker(Γ L/R )-and provides interference vectors that can be analyzed in an identical fashion. Accordingly, Figure 7(b) shows the interference vectors for the coherently-coupled butadiene molecule.

V. CONCLUSIONS
In this work we developed a characterization scheme for DQI in electron transport through molecules. DQI is generally described by a generalized eigenvalue problem [11], Eq. (2), which also associates eigenvectors with DQI. These "interference vectors" are the basis for our DQI characterization scheme. On one hand, the interference vectors can be decomposed in the MO basis, thereby revealing the participation of each MO in DQI. On the other hand, they also have geometric properties that predict the robustness of DQI. We specifically analyzed two of these properties, order and degree of rotation, which form the basis of our characterization scheme. The order describes the shape of the transmission function around the DQI and the degree of rotation quantifies the nodal structure of the interference vectors where the electrodes couple to the molecule. Increased order and degree of rotation both lead to more robust DQI.
We then explored the utility of interference vectors and this characterization scheme with several model systems. For example, we found that DQI at a complex energy appears to be of more applied interest than fundamental. As the locations of DQI move into the complex plane, the interference vectors remained essentially unchanged. In the end, this style of analysis places DQI on a similar footing with more common analyses of molecular resonances in transport. Both the peaks and valleys (those caused by DQI) of the transmission spectra-the resonances and anti-resonances, respectively-are described by eigenvalue problems involving the molecular Hamiltonian and the self-energies.
Because DQI does not depend on the magnitude of molecule-electrode coupling, but only on where the molecule couples to the electrodes [see Eq. (2)], DQI is a manifestation of substructure within the molecular Hamiltonian. The application of group theory to this substructure may reveal deeper physical and chemical insights that could lead to better molecular insulators or transistors. On a more fundamental level, such an analysis may be useful for describing chemical reactivity. It is well known in organic chemistry that meta sites in benzene tend to be less reactive than ortho or para sites. Electron transport appears to follow these trends, and it would be interesting to combine this substructure analysis with transition-state theory to better understand such chemical reactions.