Quantum interference effects in conjugated molecules have been well-explored, with benzene frequently invoked as a pedagogical example. These interference effects have been understood through a quantum interference map in which the electronic transmission is separated into interfering and non-interfering terms, with a focus on the π-orbitals for conjugated molecules. Recently, saturated molecules have also been reported to exhibit destructive quantum interference effects; however, the very different σ-orbital character in these molecules means that it is not clear how orbital contributions manifest. Herein, we demonstrate that the quantum interference effects in conjugated molecules are quite different from those observed in saturated molecules, as demonstrated by the quantum interference map. While destructive interference at the Fermi energy in the π-system of benzene arises from interference terms between paired occupied and virtual orbitals, this is not the case at the Fermi energy in saturated systems. Instead, destructive interference is evident when contributions from a larger number of non-paired orbitals cancel, leading to more subtle and varied manifestations of destructive interference in saturated systems.

Quantum interference (QI) is a phenomenon that can be harnessed to increase or decrease the electronic conductance of single molecules. Originally reported for benzene in the context of electron transfer,1 destructive QI effects in molecules were first explored theoretically2–6 before suppressed conductance was experimentally observed in mono-layers7–10 and single molecules.11–14 

Due to its utility in controlling electron transport, there has been a significant focus on developing rules to predict when QI effects will dominate, for example, the graphical rules,15 curly arrow rules,16,17 and molecular orbital rules.18–20 While these rules allow us to readily determine whether a molecule is likely to exhibit suppressed conductance due to destructive interference, they do not provide more insight into “what interferes with what” in these systems.

The archetypal example of QI is Young’s double-slit interference experiment, where light passes through slits in two barriers. The first barrier has a single slit and the second has two slits. On the wall behind the second barrier, an interference pattern is observed.21 This experiment is shown in Fig. 1(a). Wave–particle duality means that the light displays characteristics of both classically defined waves and particles. In the double-slit experiment, the interference pattern is due to the phase difference of the waves caused by the differing lengths of the optical paths as light passes through the two slits in the second barrier.

FIG. 1.

Illustration of Young’s double-slit experiment and its likeness to electronic transport through molecules. (a) Light waves passing through double slits, causing an interference pattern on the wall behind them. (b) Electrons moving as waves, passing through the molecular orbitals of a molecule and then causing interference in the electronic transmission.

FIG. 1.

Illustration of Young’s double-slit experiment and its likeness to electronic transport through molecules. (a) Light waves passing through double slits, causing an interference pattern on the wall behind them. (b) Electrons moving as waves, passing through the molecular orbitals of a molecule and then causing interference in the electronic transmission.

Close modal

Similarly, wave–particle duality also governs the behavior of electrons. When the size of the device is comparable to the electronic phase coherence length, QI effects can manifest. This QI can dramatically change the electronic transport through molecular-sized devices, as shown in Fig. 1(b). The QI can be destructive or constructive depending on whether the phase difference between interfering paths are π or 0.22 

We use the language of “paths” to draw an analog with the two-slit experiment, although no such paths in physical space are evident through the molecule. While there are mathematical arguments to support the idea that the cyclic structure of a benzene ring can represent the two paths,3,4 destructive QI is also present in acyclic molecules, challenging this interpretation. Instead, QI in molecules can be understood in terms of interfering contributions between pairs of molecular orbitals22 (MOs). While the MOs of the isolated molecule are perturbed by the coupling to the electrodes, the coupling to the electrodes is typically small. This means that the molecular conductance orbitals (MCOs), which are the eigenbasis of the molecular Hamiltonian including the self-energies of electrodes, are similar to the MOs or eigenbases of the isolated molecules. Therefore, a distinction is generally not made between MOs and MCOs in practice.22,23

When describing the MOs of organic molecules, it is useful to distinguish between the σ- and π-orbitals. Destructive QI in the π-system of conjugated molecules is frequently observed in calculations as a sharp transmission dip and has been reported in many systems, such as graphene-like molecules,24–26π-stacked molecules,27–29 and cross-conjugated molecules.5 The molecular orbital analysis by conducted Gunasekaran et al.22 provided a method to visualize the constructive and destructive QI between pairs of π-MOs. Typically, the σ-orbitals are not considered in the methods for predicting QI in conjugated molecules as σ-orbital contribution decreases extremely rapidly with length and is only significant in certain limited situations.30 

Recently, destructive QI has also been identified in the σ-systems of molecules, such as saturated silanes,31,32 the bicyclo[2.2.2]octane class of molecules,33 and cyclo-alkanes.34 The QI in σ-systems has been utilized to engineer short molecules with comprehensive suppression of the electronic transmission.31,32,34 Given the differences in the orbital character of the π-orbitals of conjugated molecules and the σ-orbitals of saturated systems, a natural question to ask is, is the orbital picture of destructive QI in σ-systems similar to that seen in π-systems or do we expect differences?

In this work, we first outline the expectations for benzene (a π-system) with different substitution patterns to illustrate how QI maps visualize QI. We then apply this method to butane whose transport properties are calculated using the ladder C model to analyze QI in σ-systems. Finally, we compare the QI between the π-system and σ-system. We see that the orbital contributions to QI in π-systems and σ-systems are quite different, with the QI in π-systems caused by phase alteration and in σ-systems caused by the change in the width and position of the individual MO transmission peaks.

To investigate how QI in σ-systems might differ from that in π-systems, we study a model for butane as shown in Fig. 2(a) as an example. The transport properties of a σ-system can be calculated using the ladder C model.35,36 While this model is parameterized for a permethylated silane chain, we use the parameters to represent a generic saturated system that we will refer to butane for simplicity. With this model, the Hamiltonian for butane is written as

ϵτprim0000τprimϵτgem0τvic00τgemϵτprim0000τprimϵτgem00τvic0τgemϵτprim0000τprimϵ.
(1)

Here, ϵ = 0 eV are the on-site energies, with the primary integrals τprim = −3.5 eV, the geminal integrals τgem = −1.1 eV, and the vicinal integrals τvic = 0.11 − 0.7 cos θ eV where θ is the dihedral angle. For θ = 180°, butane is in an all-trans configuration, and for θ = 0°, it is in a cis-configuration. Using the model Hamiltonian of Eq. (1), the transmission can be calculated as

T(E)=Tr[ΓLGΓRG],
(2)

where G/G is the retarded/advanced Green’s function of the molecule. The retarded Green’s function is given by

G=EH+i2(ΓL+ΓR)1,
(3)

where E is energy and

ΓL(R)=ρVL(R)VL(R),
(4)

where ρ is the density of state of the electrode, which is set to be 1 and VL(R) describes the coupling between the molecule and the left (right) electrode whose single non-zero coupling element is set to −0.5 eV.

FIG. 2.

(a) Butane with its connection to the electrodes denoted by dashed lines. (b) Ladder C model of butane. The red, blue, and green arrows denote the primary (τprim), geminal (τgem), and vicinal (τvic) interaction between orbitals, respectively. (c) Transmission for the butane model at selected dihedral angles θ calculated using the ladder C model, where the inset shows the structures with θ = 0° and 180°, in this case, with hydrogen substituents in the dashed positions for clarity. (d) The energetic ordering of MOs for the butane model at selected dihedral angles.

FIG. 2.

(a) Butane with its connection to the electrodes denoted by dashed lines. (b) Ladder C model of butane. The red, blue, and green arrows denote the primary (τprim), geminal (τgem), and vicinal (τvic) interaction between orbitals, respectively. (c) Transmission for the butane model at selected dihedral angles θ calculated using the ladder C model, where the inset shows the structures with θ = 0° and 180°, in this case, with hydrogen substituents in the dashed positions for clarity. (d) The energetic ordering of MOs for the butane model at selected dihedral angles.

Close modal

In the following, the method to decompose and calculate the transmission will be introduced in a way to enable easy visualization of whether there is constructive or destructive interference between any two MOs. For a more detailed description, we refer to Gunasekaran et al.22 In the MO basis, the transmission can be expressed as the contributions from all the individual MOs, which can be further split into a sum over the non-interfering parts, Ti, and a sum over the interfering parts, Tij,

T=iTi+i>jTij.
(5)

Here, both Ti and Tij are obtained from the Q matrix, defined as

Q(E)=(PΓLGP)(P1ΓRGP1)T,
(6)

where P and P−1 are two matrices constructed from the eigenvectors that diagonalize G and G, respectively, and denotes the entry-wise product. With this Q matrix, the non-interfering terms and interfering terms can be expressed as

Ti=Qii
(7)

and

Tij=Qij+Qji,
(8)

respectively.

We can complement this analysis with a calculation of the transmission phase of the individual MOs. The transmission coefficient can be expressed in terms of individual contributions of each MO,37 

T(E)=|iti(E)|2=ijtj(E)ti*(E),
(9)

where ti(E)=νil(E)giir(E)νir(E) describes the transmission coefficient through each individual MO in which νil(E)=(VLU)i and νir(E)=(U1VR)i. In this equation, U is the eigenvector of G. The giir(E) is the retarded Green’s function of system G expressed in its diagonal basis. Finally, we obtain

T(E)=ijtjti*=ijQij,
(10)

and the phase of an individual MO is given by

θi(E)=arctanIm(ti(E))Re(ti(E)).
(11)

Using the above-mentioned methods, we will show how QI can be better understood using both QI maps and the phase of individual MOs.

First, we revisit the case of benzene, following the approach of Gunasekaran et al.22 to the QI map to outline the expectations from conjugated molecules and for comparing with the following butane study. We consider two kinds of connections with the electrodes, as shown in Fig. 3(a). For benzene, we use the Hückel model with nearest neighbor coupling and the value of β is set to be −1 eV. Benzene has six π orbitals with two degenerate pairs of orbitals as the frontier orbitals. Here, we refer to the frontier orbitals as the HOMO and HOMO′ and LUMO and LUMO′ to highlight their degenerate nature and distinguish the non-degenerate HOMO−1 and LUMO+1 orbitals (that in the usual numbering would be HOMO−2 and LUMO+2). The connection to the electrodes results in a small splitting between the degenerate pairs and influences the energetic ordering of the MOs. For para-substituted benzene, HOMO and LUMO are the frontier MOs, whereas for meta-substituted benzene, HOMO′ and LUMO′ are the frontier MOs. The difference in the electrode positions results in a significant change in the transmission, as shown in Fig. 3(c). More specifically, benzene with a meta connection exhibits a sharp dip at EF, whereas that with a para connection exhibits a flat curve around EF.

FIG. 3.

Modified from Ref. 22. (a) Electrode contact sites for molecular junctions of para- and meta-substituted benzene. (b) The energetic ordering of MOs for para- and meta-substituted benzene. (c) Transmission for para- (pink) and meta-substituted benzene (blue). QI maps for (d) para- and (e) meta-substituted benzene at the Fermi level. The color scale for each QI map is normalized to Qmax = max|Qij| and is different for each QI map.

FIG. 3.

Modified from Ref. 22. (a) Electrode contact sites for molecular junctions of para- and meta-substituted benzene. (b) The energetic ordering of MOs for para- and meta-substituted benzene. (c) Transmission for para- (pink) and meta-substituted benzene (blue). QI maps for (d) para- and (e) meta-substituted benzene at the Fermi level. The color scale for each QI map is normalized to Qmax = max|Qij| and is different for each QI map.

Close modal

The decompositions of the transmission into interfering and non-interfering contributions are shown in Figs. 3(d) and 3(e), respectively. This QI map provides a diagrammatic visualization of the Q matrix calculated using Eq. (6). The squares along the main diagonal (highlighted with bold lines) represent the non-interfering transmission terms. As these terms are always positive, they will either be red or gradually change to white when decreasing close to 0. The off-diagonal squares illustrate the QI between different MOs. They may either be red, denoting constructive QI, or blue, denoting destructive QI. The shade of the color indicates the magnitude of the term. White squares indicate that the magnitude of these items is close or equal to 0. In the case of para-substituted benzene, constructive QI dominates the contribution to the transmission, while for meta-substituted benzene, the destructive QI cancels exactly with the non-interfering terms. In both cases, the maps are dominated by interactions between the two frontier orbitals, with lesser contributions from all other orbitals. This result reproduces the earlier work of Gunasekaran et al. on this system.22 

The electronic transmission for butane calculated using the ladder C model is shown in Fig. 2(c). The transmission varies significantly with the change in the dihedral angle. At a dihedral angle of 180°, there are no anti-resonances, whereas two anti-resonances are present at ±2.23 eV for a dihedral angle of 0°. When the dihedral angle is increased from 0° to 49°, the anti-resonances move closer to the Fermi energy, while the transmission decreases and vanish altogether at a dihedral angle of 50°. With the further increase in the dihedral angle, the transmission increases again and becomes a broad valley.

To understand the anti-resonances of butane, QI maps are used to analyze the electronic transmission, as shown in Fig. 4. Specifically, the QI maps for the cases with dihedral angles of 0° and 180° are presented at three different energies: EF (with E = 0 eV) and the energies of the two antiresonances for 0° (E = ±2.23 eV). In our model, the σ-system of butane has 6 σ-orbitals, and the QI map is a 6 × 6 block matrix.

FIG. 4.

QI map of butane for the case with (a) θ = 0° and E = −2.23 eV; (b) θ = 180° and E = −2.23 eV; (c) θ = 0° and E = 0 eV; (d) θ = 180° and E = 0 eV; (e) θ = 0° and E = 2.23 eV; and (f) θ = 180° and E = 2.23 eV.

FIG. 4.

QI map of butane for the case with (a) θ = 0° and E = −2.23 eV; (b) θ = 180° and E = −2.23 eV; (c) θ = 0° and E = 0 eV; (d) θ = 180° and E = 0 eV; (e) θ = 0° and E = 2.23 eV; and (f) θ = 180° and E = 2.23 eV.

Close modal

We first focus on the θ = 0° case. At E = 0 eV, the contributions from the lowest unoccupied MO +1 (LUMO+1) and highest occupied MO−1 (HOMO−1) contribute the most to the electronic transmission. These contributions are seen as four dark-red squares in the QI map: two non-interfering terms and two constructively interfering terms. Other destructively interfering terms are much smaller, primarily between the HOMO and LUMO and these two dominant orbitals. At E = −2.23 eV, the non-interfering contributions from the HOMO and the HOMO−1 increase, while the contributions from the LUMOs decrease nearly to zero. On the contrary, at E = 2.23 eV, the tendency is opposite, where the non-interfering contributions from the LUMOs are dominant. The destructive interference between HOMO and HOMO−1 for −2.23 eV and that between LUMO and LUMO+1 for 2.23 eV are very strong (see the blue blocks), reminiscent of the two orbital interference pictures observed for meta-substituted benzene. There are two differences between the interference effects observed at ±2.23 eV in butane and the interference feature at 0 eV in meta-substituted benzene. First, the non-interfering contributions from the two orbitals in butane are unequal, whereas they are equal in meta-substituted benzene. Second, the interference feature in meta-substituted benzene is midway between the occupied and virtual orbitals, that is, midway between the orbitals that interfere destructively, whereas for butane, the interference feature appears at an energy above/below the pairs of occupied/virtual orbitals that are dominant.

The QI maps of θ = 180° are both similar to and different from those of θ = 0°. The similarity is in the color structure, that is, red terms remain red in both cases and similarly blue terms remain blue, while the differences come from the relative magnitudes of the terms (here represented by the darkness of the color) and this gives a visual appearance of a significant difference. At E = 0 eV, the HOMO and LUMO constructively interfere as seen by the dark red squares in Fig. 4. There are destructive interference terms between these frontier orbitals and the HOMO−1 and LUMO+1; however, the constructive QI terms between the HOMO and LUMO and the non-interfering terms do not cancel with the destructive interference terms. Thus, the transmission at E = 0 eV does not go to zero. For the QI maps calculated at E = ±2.23 eV, the transport is dominated by a single orbital with a single non-interfering term from either the HOMO or LUMO (the energetically proximate orbital in each case). Interestingly, this is the closest approximation we see to the single-orbital/single-level tunneling model that has sometimes been invoked to explain the differences in transport properties between different molecules.

While θ = 180° and θ = 0° are clearly distinct limits for butane, it is also interesting to examine the behavior at θ = 49° and θ = 50°. As shown in Fig. 2, at these dihedral angles, a very small angular change results in qualitatively different transmission, with the two sharp dips present at θ = 49° disappearing at θ = 50° with only a single “V-” shaped dip remaining. We note here that in terms of experimental observables (e.g., current and conductance), this change is not significant as the transmission is extremely low in both cases. The significance in this qualitative change in the transmission is in how sure one can be in labeling the low transmission coming from destructive interference if we consider transmission alone, as these sharp dips are a definitive feature. We also note here that there are other situations where we are confident that low transmission results from destructive interference, despite the absence of a sharp dip, for example, if transport through the σ-system masks destructive interference in the π transport30 or if we see ring current reversal in a local current picture.38 

Fig. 5 illustrates the QI maps for θ = 49° and θ = 50° at E = ±0.38 eV (the energies of the interference features for θ = 49°) and at E = 0 eV. The most striking feature of all these maps is how similar they are. In both cases, the map at E = 0 eV is more symmetric in terms of the contributions from occupied and virtual orbitals, while the maps at E = ±0.38 eV show some asymmetry; however, this asymmetry is not very pronounced. While there are undoubtedly quantitative differences between the two dihedral angles, it is not possible to see this from the plots. In all cases, the similar color structure that was seen for θ = 180° and θ = 0° is retained, and the similarity is more readily apparent as the quantitative differences are reduced.

FIG. 5.

QI map of butane for the case with (a) θ = 49° and E = −0.38 eV; (b) θ = 50° and E = −0.38 eV; (c) θ = 49° and E = 0 eV; (d) θ = 50° and E = 0 eV; (e) θ = 49° and E = 0.38 eV; and (f) θ = 50° and E = 0.38 eV.

FIG. 5.

QI map of butane for the case with (a) θ = 49° and E = −0.38 eV; (b) θ = 50° and E = −0.38 eV; (c) θ = 49° and E = 0 eV; (d) θ = 50° and E = 0 eV; (e) θ = 49° and E = 0.38 eV; and (f) θ = 50° and E = 0.38 eV.

Close modal

Figures 4 and 5 paint a different orbital picture of destructive interference in butane from what was seen in benzene. First, in butane, more orbitals are involved (4) compared with benzene (2) despite the significantly larger HOMO–LUMO gap (over 4 eV for butane vs 2 eV for benzene). Second, the change from a system with clear destructive interference effects (θ = 0 − 49°) to a system with suppressed transmission (θ = 50 − 180°) results from a change in the magnitude of the various orbital contributions rather than any contributions changing sign. Together, these two differences mean that the QI maps are not as simple to interpret for saturated systems. This is not an indication of any failure of the QI map but simply an indication that the interference effects in saturated systems are more subtle as the system can vary continuously from being dominated by constructive to destructive interference without any dramatic change in the orbital character.

To better understand the nature of the orbital contributions, we compute the phase and transmission through individual MOs. For benzene, the phases are shown in Fig. 6 and the MO contributions to the transmission (|ti|2) are shown in Fig. 7. We can understand the differences between meta and para substitution by considering the differences in the transmission and phase of three pairs of orbitals: the HOMO and LUMO, HOMO′ and LUMO′, and HOMO−1 and LUMO+1. For each case, these pairs of orbitals are symmetry-related (from the Coulson–Rushbrooke orbital pairing theorem39). This results in equal widths for the transmission peaks from paired orbitals, as well as identical lineshapes for the phase as a function of energy (shifted in energy).

FIG. 6.

The phase of transmission through each MO of meta- and para-substituted benzene for (a) HOMO−1; (b) LUMO+1; (c) HOMO′; (d) LUMO′; (e) HOMO; and (f) LUMO. The solid line is for para-substituted benzene, while the dashed line is for meta-substituted benzene.

FIG. 6.

The phase of transmission through each MO of meta- and para-substituted benzene for (a) HOMO−1; (b) LUMO+1; (c) HOMO′; (d) LUMO′; (e) HOMO; and (f) LUMO. The solid line is for para-substituted benzene, while the dashed line is for meta-substituted benzene.

Close modal
FIG. 7.

The transmission probability from each MO of meta- and para-substituted benzene for (a) HOMO−1; (b) LUMO+1; (c) HOMO′; (d) LUMO′; (e) HOMO; and (f) LUMO. The solid line is for para-substituted benzene, while the dashed line is for meta-substituted benzene.

FIG. 7.

The transmission probability from each MO of meta- and para-substituted benzene for (a) HOMO−1; (b) LUMO+1; (c) HOMO′; (d) LUMO′; (e) HOMO; and (f) LUMO. The solid line is for para-substituted benzene, while the dashed line is for meta-substituted benzene.

Close modal

In order to understand the QI maps in Figs. 3(d) and 3(e), we consider the relative phase of the orbital pairs at E = 0 eV. In each case, the para contributions are in phase and differ by 0/2π, while the meta contributions are out of phase and differ by π. The relative transmission through each orbital differs significantly for each pair of orbitals and also for the two different substitution patterns. For para-substituted benzene, only four of the six orbitals have non-zero transmission, with the HOMO and LUMO being approximately an order of magnitude more transmissive at E = 0 eV than the HOMO−1/LUMO+1. All six orbitals contribute to the meta-substituted system, but in this case, the HOMO′/LUMO′ pair is approximately an order of magnitude more transmissive than the other pairs. Comparing with Figs. 3(d) and 3(e), we can conclude that the color intensity is related to the magnitude of the orbital transmission, while the sign/color of the interference terms is related to the phase difference between the orbital contributions at that energy (π phase difference giving destructive/blue contributions and 0/2π giving constructive/red contributions). Indeed, if we extend our analysis beyond the three pairs of orbitals, it is clear that the phase difference between two orbitals generally predicts the color of the interference terms in the QI map, for example, meta HOMO′ is π out of phase with the HOMO and HOMO−1.

Fig. 8 shows the phase of the transmission of each MO for butane for both the all-trans (θ = 180°) and the all-cis (θ = 0°) configurations. It is clear that the phase of the MOs of the σ-system only varies slightly when the dihedral angle changes. This minor change is in contrast with the π-system where the different electrode positions have a substantial impact on the resulting phase. The individual orbital contributions to the transmission for butane are shown in Fig. 9. Again, the differences are subtle for the different dihedral angles with peaks shifting in energy and changes to the width.

FIG. 8.

The phase of transmission through each MO of butane with the dihedral angle being θ = 180° and θ = 0° for (a) HOMO−2; (b) LUMO+2; (c) HOMO−1; (d) LUMO+1; (e) HOMO; and (f) LUMO. The solid line is for θ = 180°, while the dashed line is for θ = 0°.

FIG. 8.

The phase of transmission through each MO of butane with the dihedral angle being θ = 180° and θ = 0° for (a) HOMO−2; (b) LUMO+2; (c) HOMO−1; (d) LUMO+1; (e) HOMO; and (f) LUMO. The solid line is for θ = 180°, while the dashed line is for θ = 0°.

Close modal
FIG. 9.

The transmission probability from each MO of butane with the dihedral angle being θ = 180° and θ = 0° for (a) HOMO−2; (b) LUMO+2; (c) HOMO−1; (d) LUMO+1; (e) HOMO; and (f) LUMO. The solid line is for θ = 180°, while the dashed line is for θ = 0°.

FIG. 9.

The transmission probability from each MO of butane with the dihedral angle being θ = 180° and θ = 0° for (a) HOMO−2; (b) LUMO+2; (c) HOMO−1; (d) LUMO+1; (e) HOMO; and (f) LUMO. The solid line is for θ = 180°, while the dashed line is for θ = 0°.

Close modal

While the Coulson–Rushbrooke orbital pairing theorem39 is generally applied to the π-system of conjugated molecules, the mathematical form of the ladder C model we employ means that it also applies to butane. In this case, however, it is very clear that the interference is not between the paired orbitals (shown as rows in Fig. 8: HOMO−2 and LUMO+2, HOMO−1 and LUMO+1, and HOMO and LUMO) as the phase does not differ. Instead, from Figs. 4, 5, and 8, it is clear that the balance between constructive and destructive interference is only shifted by the varying magnitude of the destructive interference between the pair of HOMO−1 and LUMO+1 with primarily the HOMO and LUMO (and to a lesser extent the HOMO−2 and LUMO+2). The orbital pairing theorem means that the energetic position of paired orbitals will always be symmetrical above/below 0 eV and the widths of the transmission peaks are identical. Together, this means that when interference is between paired orbitals, the interference dip must also be at 0 eV. On the other hand, when the interference is between non-paired orbitals, the details of the balance between orbital positions and weights mean that the interference features can shift in energy and appear/disappear, as seen in Fig. 2(c).

We can highlight this difference more explicitly by writing the total transmission as a the modulus squared of the complex transmission coefficients ti from the individual orbitals37,40ti=Tieiθi, where Ti is the amplitude and θi is the phase. The total transmission probability for the σ-system or the π-system is, thus, the modulus squared of the sum of the complex transmission coefficients for all MOs,

T=|t1+t2+t3+t4+t5+t6|2.
(12)

Away from the transmission resonances, the transmission phase is 0 or ±π so ti=±Ti. As the transmission phase is unchanged when the dihedral angle changes for butane, Eq. (12) can be written in a large energy range around the Fermi level (at least in [−2.0, 2.0] eV) as

T=|T1T2+T3+T4T5+T6|2,
(13)

where the sign in front of each term is determined by the phase indicated in Fig. 8, “+” for 0 and “−” for ±π. With the change in the dihedral angle from 0° to 180°, transmission dips may appear or disappear in the energy range [−2.0, 2.0] eV but the phase of each MO is constant; consequently, the transmission dips can only be caused by the change in the width and position of individual MO contributions to the transmission. Conversely, the sign of each term in Eq. (12) for the π-system benzene will change depending on the connection to the electrodes giving a different picture of the orbital contributions to interference in this case.

By comparing the QI maps of a π-system (benzene) and a σ-system (butane), as well as the transmission phase of the orbital contributions, we have shown that the orbital contributions to constructive/destructive interference are quite different in these two cases. Namely, the transmission dip in the transmission for meta-substituted benzene arises due to phase differences between paired MOs. These contributions cancel exactly as the orbital pairing theorem guarantees that they are equidistant from 0 eV and that the individual contributions to the transmission are of equal magnitude. In contrast, for the σ-system of butane, the transmission dip is due to cancellation between non-paired orbitals as the position and width of individual orbital contributions to the transmission changes when the dihedral angle changes. Together, these results provide a richer picture of interactions and contributions to destructive interference, and an indication of how orbitals might be manipulated to exert fine control of quantum interference effects in molecules.

We gratefully acknowledge financial support from the National Natural Science Foundation of China under Grant No. 11974355 (X.Z.) and the China Scholarship Council (N.C.). This project received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation program (Grant Agreement No. 865870).

The authors have no conflicts to disclose.

Ning Cao: Conceptualization (equal); Methodology (equal); Software (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). William Bro-Jørgensen: Software (supporting); Writing – review & editing (equal). Xiaohong Zheng: Supervision (equal); Writing – review & editing (equal). Gemma C. Solomon: Conceptualization (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article.

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