The Two-Pathway Viewpoint to Interpret Quantum Interference in

Subscriber access provided by Nottingham Trent University

C: Energy Conversion and Storage; Energy and Charge Transport

The Two-Pathway Viewpoint to Interpret Quantum Interference in Molecules Containing Five-Membered Heterocycles: Thienoacenes as Examples Yang Li, Xi Yu, Yonggang Zhen, Huanli Dong, and Wenping Hu J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b03177 • Publication Date (Web): 05 Jun 2019 Downloaded from http://pubs.acs.org on June 5, 2019

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

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

The Journal of Physical Chemistry

The Two-Pathway Viewpoint to Interpret Quantum Interference in Molecules Containing Five-Membered Heterocycles: Thienoacenes as Examples Yang Li†, Xi Yu‡, Yonggang Zhen†, Huanli Dong†*, Wenping Hu†‡* †Beijing National Laboratory for Molecular Science, Key Laboratory of Organic Solids, Institute of Chemistry, Chinese Academy of Sciences, Beijing 100190, China ‡Tianjin Key Laboratory of Molecular Optoelectronic Sciences, Department of Chemistry, School of Science, Tianjin University and Collaborative Innovation Center of Chemical Science and Engineering, Tianjin 300072, China

1 ACS Paragon Plus Environment

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

Page 2 of 34

ABSTRACT In the study of quantum interference effect in single molecular junctions where the molecules contain five-membered heterocycles, an implicit two-pathway viewpoint can be found which assume the carbon backbone play the dominant role and regard the heteroatoms as a tunable factor. Till now this viewpoint has not been systematically evaluated. In this work, we concretely divide the zeroth Green's functions of thienoacenes covalently bonded to metal electrodes into two pathways under the tight-binding approach, and discuss the relationship between them. The fact that heteroatoms do not change the grounding color of the interference set by the carbon backbone is theoretically demonstrated. By investigating the carbon backbone segmentally, the impact of the heteroatoms on the molecular conductance and the feature of the anti-resonances can be further specified. Moreover, a value determined only by common tight-binding parameters of the heteroatom is proposed as a criterion of the reliability of the two-pathway viewpoint. First principles calculations combined with existing experimental reports further corroborate our conclusions.


Page 3 of 34


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



Page 4 of 34

INTRODUCTION Quantum interference (QI) effect in molecular devices has attracted great attentions as an essential element in device function, both theoretically1-7 and experimentally8-14. In the phase-coherent regime, because of the interference of the electron wavefunctions flowing through the molecule, constructive or destructive QI (in short CQI or DQI) affects the molecular conductance. A well-known example of DQI is the ultralow conductance of meta-coupled benzene due to the anti-resonance between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO).7-8 Molecular switches or sensors2-4,

13,

transistors5,

14,

logic devices15 and thermoelectric

devices16-17 have already been presented based on this phenomenon, taking advantage of either the ultralow conductance or the deep slope of the logarithmic transmission coefficient. Recently, it has also been applied to isomer recognition.18 Moreover, molecular segments exhibiting DQI can act as the bridge of a donor-bridge-acceptor system to tune the rate of photoinitiated charge separation and recombination.19 To find a simple and intuitive model is always a research topic in studying molecular junctions. Some few-level models are sufficient to explain the QI effect in specific types of molecules, such as the two-level model for T-shaped molecules20-22, the four-level model for π-π stacking inter-molecular junctions23 and the three-level model for anthraquinones9,

24.

More generally, several

analyzing methods based on the tight-binding (TB) model serve as effective tools 4 ACS Paragon Plus Environment

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


to interpret the topology dependent charge transport mechanism. For example, the orbital rule25-26 correlates the conductance with the molecular orbital expansion coefficients, so that provides a way to examine the interaction between different molecular orbitals. The Markussen-Stadler-Thygesen (MST) diagram27-29 proposes a graphic method to obtain the numerator of the molecular Green's function and find out the zero-conductance conditions. The magic ratio rule15, 30 eliminates the singularity of the Green's function so that the conductance can be investigated analytically. In alternant hydrocarbons, connections exhibiting DQI are easy to be discriminated. Because of the nearly identical atomic sites, nearly identical hopping integrals and the pairing theorem of molecular orbitals, the aforementioned general methods can be simplified to provide convenient selection rules, while there are other identifying methods based on the Kekulé structure31-32, diradical existence33, etc. The anti-resonance, if exists, is always at the midgap under the simplest approach. In contrast, regular rules in heterocyclic compounds are more complicated. Recently, much research effort has been spent on these systems as they bring in more tunable factors.34 For example, QI in some nitrogen substituted aromatic molecules were studied implementing orbital rule35 and magic ratio rule36-37 respectively. Conductance of molecules containing five-membered heterocycles, as another research interest, is also related to QI. In thiophene-like molecules with different heteroatoms, the conductance has been correlated with either aromaticity of the ring or electronegativity of the 5 ACS Paragon Plus Environment


Page 6 of 34

heteroatom depending on the connection.38-40 There are also attractive works focusing on the complicated charge transport in fused-ring molecules containing five-membered heterocycles.41-42 Influences of heteroatom and connection in molecules whose structures are similar with dibenzothiophene (DBT) were discussed in different ways.40, 43-45 The negative relationship between conductance and aromaticity was proved to be broken in expanded porphyrins.46-47 Furthermore, rules of QI in several benzodithiophene (BDT) derivatives with alike topology48 and the connection dependent conductance in a series of molecules with a dithiophene core49 have been studied recently. From the studies where the molecules contain five-membered heterocycles, we can find a common two-pathway viewpoint. It treats the C backbone as a determining factor of conductance, and considers the heteroatoms as an influencing factor. For example, chemists naturally think the conductance is lower when a linear conjugated path through C-C covalent bonds cannot be drawn,43, 48-49

and the charge transport in DBT-like molecules has already been qualitatively

divided into a pathway through the C backbone and the other pathway through the heteroatom.44-45 This viewpoint holds true in all the works we mentioned above, i.e., conductance of a molecule containing five-membered heterocycles is always relatively lower when the C backbone exhibits DQI, and vice versa. However, it has not been explicitly demonstrated yet. The aim of this work is to elaborate this two-pathway viewpoint and discuss the impact of each pathway on the molecular conductance. The widely studied thienoacenes48-52 are taken as 6 ACS Paragon Plus Environment

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


examples. Analytical derivation is done under the TB approach, while non-equilibrium Green’s function formalism combined with density functional theory (in short the NEGF+DFT approach)53-59 is used to verify the results. We confirm that the heteroatoms do not change the grounding color of the interference set by C, and suggest that the interactive pattern between the two pathways can be analyzed more specifically. At last, we prove a value h/2k2 depending on TB parameters of the heteroatom can be taken as a criterion of the domination of the C pathway. METHODS In a metal-molecular-metal junction, the electronic transmission coefficient can be obtained by the Landauer formula60 T(E) = Tr[ΓL(E)Gr(E)ΓR(E)Gr†(E)]. Here

Gr(E) is the retarded Green’s function of the extended molecule, and ΓL/R(E), which is called broadening function, describes the coupling between left/right electrode and the (extended) molecule. When the molecule is weakly coupled to electrodes, the zeroth Green’s function, i.e., Green’s function of the isolated molecule, can be adopted to predict the conductance.26-27, 30 Assuming that each electrode only couple to one atomic orbital of the atom covalently bonded to it, for example, the left electrode with site i and the right electrode with site j, the transmission coefficient reduces to T(E) = ΓLii(E)ΓRjj(E)|Gmolij(E).2, 27 That means the transmission probability from i to j at a certain energy is proportional to the modulus square of Gmolij(E). As a result, intrinsic QI properties of a molecule can be explored by observing the zeroth Green’s function. Especially, the value of 7 ACS Paragon Plus Environment


Page 8 of 34

|Gmolij(E)|2 at the Fermi energy can be chosen as a reference of the low-bias conductance of the molecule. In this work, ground state electronic properties of the interested systems are obtained by the SIESTA package which uses finite-range numerical atomic basis sets to construct the wavefunctions of valence electrons and improved Troullier– Martins pseudopotentials to describe the atomic cores.61-62 Single-ζ (SZ) basis set is adopted for the parameterization of the TB model, while double-ζ plus polarization (DZP) basis set is used in the calculations of metal-molecule-metal junction models. Exchange and correlation functional is described by the generalized gradient approximation (GGA) in the Perdew–Burke–Ernzerhof (PBE) form.63 Real space grid cutoff is set to 200.0 Ry. In geometry optimization, conjugate gradient relaxation is performed until the atomic forces are less than 0.03 eV Å-1. TranSiesta56 implemented in SIESTA is employed to obtain the electronic transmission properties. It uses the Hamiltonian and overlap matrix obtained under SIESTA to perform the nonequilibrium Green’s function (NEGF) calculations. Here the basis set describing Au changes to single-ζ plus polarization (SZP). When one-dimensional Au electrodes are used, we construct nonperiodic junction models with two buffer layers. When three-dimensional Au electrodes are used, periodic junction models are constructed where Au(111) layers with a (4×4) transverse supercell are used as electrodes. The anchor groups connect with


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


an Au atom on the hollow site above the Au(111) surface. K-point sampling is generated as a 4×4 grid in the transverse directions. RESULTS AND DISCUSSION The two pathways. First of all, we will decompose the electronic transmission of a thienoacene into two pathways under the TB approach. Dividing the Hamiltonian matrix into the C block HC, the S block HS and the interaction blocks HCS/SC, the zeroth Green’s function can be written as 𝐆mol =

(

𝐆C + 𝐆C𝐇CS𝐙𝐇SC𝐆C 𝐆C𝐇CS𝐙 𝐙𝐇SC𝐆C 𝐙

)

(1)

Here GC = (EI - HC)-1 corresponds to the zeroth Green’s function matrix of the C backbone, and Z = (EI - HS - HSCGCHCS)-1 corresponds to interaction between the heteroatoms in the molecule. The energy dependence is omitted for short. In covalently bonded thienoacenes, anchor groups only connect with C atoms, so that we concentrate on the top-left block of Gmol. It is the sum of two terms, i.e., the C backbone term GC and the other term contributed by both the heteroatoms and the C backbone. We can define the latter term as GS’ which indicates a S dependent pathway. These two pathways are illustrated in Figure 1a. We take the hopping integral of all the C-C covalent bonds as -β and the C-S bonds as -kβ. The on-site energy of C is set to zero and that of S is -hβ. Practically, we use k = 0.60, h = 0.72 and β = 2.53 eV. These parameters are testified by DFT calculations in the Supporting Information.



Page 10 of 34

Figure 1. (a) Illustration of the C pathway and S’ pathway in a thionoacene. (b-c) Modulus squares of Green's functions of the whole molecule and the C backbone for connections indicated by arrows in BDT-1 (b) and BTBT-1 (c). A number of calculations are carried out under the TB approach for different connections in different thienoacenes, results of which obey the following rule. When the C backbone exhibits CQI, |Gmol(0)|2 of the whole molecule is comparable to |GC(0)|2. When the C backbone exhibits DQI instead, |Gmol(0)|2 become much lower than other connections in the same molecule. Because the practical energy alignment between the molecular levels and the Fermi energy is uncertain, the value around the zero energy can serve as a reference of conductance. For example, we choose two connections in the molecule BDT-1 as illustrated on top of Figure 1b, respectively meeting the two conditions. Corresponding |Gmol|2 and |GC|2 curves are drawn below. Similarly, curves of another two connections in BTBT-1 are drawn in Figure 1c. In the cases of CQI shown on the upper panels, between HOMO and LUMO resonances both of the 10 ACS Paragon Plus Environment

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


curves are relatively high. |Gmol(0)|2 is a little higher than |GC(0)|2 for BDT-1 and a little lower for BTBT-1, but on the same order. More evidences are shown in Figure S3a-c and S4a-f. On the other hand, in the cases of DQI, typical anti-resonances also exist in between HOMO and LUMO resonances of the |Gmol(E)|2 curves, although they deviate from the zero energy to about 0.28β for BDT-1 and 0.14β for BTBT-1. In more situations, the anti-resonances may also shift left, split, vanish or stay near the zero energy as shown in Figure S3d-f and Figure S4g-l. No matter how the anti-resonances perform, the values of |Gmol(0)|2 are always relatively low. These results have confirmed the regular rules summed out from existing investigations, corroborating the two-pathway viewpoint. Between the two terms of Gmol, the character of GC is easy to obtain by existing methods because of the alternant feature of the C backbone. (Alternant means the C sites can be divided into starred and unstarred ones so that covalent bonds only connect sites of different types.) Character of GS' is more complicated. In order to prove the empirical rule, it is necessary to discuss the explicit relationship between GC and

GS'. The segmented analysis. Expression of GS' of a certain connection can be further expanded. If we number the two C atoms adjacent to Su by mu and nu (and the other two C atoms in the five-membered ring are pu and qu as noted in Figure 2a), the (i,j) element of GS’ is 𝐺S𝑖𝑗′ = 𝐋T𝑖 (k2𝐙)𝐋j (2) 11 ACS Paragon Plus Environment


Page 12 of 34

where

{

𝐙=

1

[(

𝐸

h

]

)

―1

+ 𝐈N ― 𝐘 2k2 2k2 2k2 S 𝑌𝑢𝑣 = 𝐺C𝑚𝑢,𝑚𝑣 + 𝐺C𝑛𝑢,𝑛𝑣 + 𝐺C𝑚𝑢,𝑛𝑣 + 𝐺C𝑛𝑢,𝑚𝑣 𝐺C𝑚1,𝑖 + 𝐺C𝑛1,𝑖 𝐋𝑖 = 𝐺C𝑚2,𝑖 + 𝐺C𝑛2,𝑖 …

(

)

Here we adopt β as the unit of energy and Hamiltonian, thus the unit of Green's function becomes β-1. Obviously, only Z is affected by the heteroatom-dependent parameters, especially by h/2k2 at E = 0 which approximately equals to 1 for S. Some certain elements of GC also play important roles in GS'.

Figure 2. (a) Illustration of dividing the C backbone of BDT-1 into two C-C pairs (segment 1 and 3) and a benzene segment (segment 2). (b) Segmented expression of the Green's function of the C backbone at E = 0 for an inter-segmental connection. (c) Shape of the Green’s functions between possible sites in C-C pair and benzene. Non-zero values at E = 0 are noted as green texts. Horizontal axes range from -1.2β to 1.2β while vertical axes range from -3β-1 to 3β-1. As shown in Figure 2a, the C backbone of a thienoacene can be "cut" into

NS+1 segments by the axis of each five-membered ring. Resulting segments are 12 ACS Paragon Plus Environment

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


either C-C pairs or acenes, zeroth Green’s functions of which have been widely explored.26,

30, 64

It can be proved that the intra-segmental GCij(0) equals to the

Green's function of the segment itself at zero energy (indicated by gij), while inter-segmental GCij(0) is the product of the segmental Green’s functions with an extra negative sign if the number of segments is even. Taking a inter-segmental connection in BDT-1 as an example, the segmented expression of GC(0) is illustrated in Figure 2b. Similarly, GS'ij(0) can also be written segmentally according to eq 2. In a simplest two-segmental molecule containing only one five-membered ring, the segmental form of GS' is simple because Li and Z reduce to scalars. Herein we point out four laws of the segmental Green's functions proved in the Supporting Information: Starred site m and unstarred site p are adjacent sites on the "edge" of the C-C pair or acene which can be shared with a five-membered ring, then: (A) The parity of the Green's function is odd for starred-unstarred connection, and even for starred-starred and unstarred-unstarred connection; (B) 0 < gmp ≤ 1; (C) 0 ≤ |gip| ≤ 1, 0 ≤ |gjm| ≤ 1; (D) |gipgjm| ≤ |gij| if i is starred and j is unstarred. After eliminating the odd functions which are always zero at E = 0, there are six possible forms of Gmolij(0) listed in Table 1 without loss of generality. Under laws B-D, four interactive patterns between GC and GS' can be summed out according to whether the connection is intra- or inter- segmental, and whether the C backbone exhibits CQI or DQI:



Page 14 of 34

(1) Intra-segmental, CQI (Case 1): The S’ pathway may either enhance or suppress the conductance of the C pathway, but will not change the order of the conductance because GS’(0) is always much smaller than GC(0). (2) Intra-segmental, DQI (Case 2 and 3): The S’ pathway tend to shift the anti-resonance of the C pathway not far away from E = 0 because GCij(E) is based on the odd function gij(E) not very flat near E = 0, and GS'(0) is always a small value. (3) Inter-segmental, CQI (Case 4): The S’ pathway will always suppress the conductance of the C pathway while the degree of the suppression is small compared with the initial value. (4) Inter-segmental, DQI (Case 5 and 6): The conductance is relatively low because GS'(0) is small compared with typical GC(0) in the same molecule, and the performance of the anti-resonance depends on the specific structure. Table 1. Expression of the Green’s function at E = 0 for thienoacenes containing only one five-membered ring. case

conditions

GC(0)

i* ∈ (1) gij

1

j ∈ (1) i ∈ (1) 2

0

j ∈ (1)

GS'(0)

―𝑔 𝑛𝑞 𝑔 𝑖 𝑝𝑔 𝑗 𝑚 2(h/2k2 + 𝑔 𝑚𝑝𝑔 𝑛𝑞 )

𝑔 𝑖 𝑚𝑔 𝑗 𝑚 2(h/2k2 + 𝑔 𝑚𝑝𝑔 𝑛𝑞 )


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


i* ∈ (1) 3

0

j* ∈ (1) i* ∈ (1) -gipgjq

4

j ∈ (2) i ∈ (1) 5

0

j* ∈ (2) i ∈ (1) 6

0

j ∈ (2)

𝑔 𝑛𝑞 𝑔 𝑛𝑞 𝑔 𝑖 𝑝𝑔 𝑗 𝑝 2(h/2k2 + 𝑔 𝑚𝑝𝑔 𝑛𝑞 )

𝑔 𝑛𝑞 𝑔 𝑚𝑝𝑔 𝑖 𝑝𝑔 𝑗 𝑞 2(h/2k2 + 𝑔 𝑚𝑝𝑔 𝑛𝑞 )

𝑔 𝑖 𝑚𝑔 𝑗 𝑛 2(h/2k2 + 𝑔 𝑚𝑝𝑔 𝑛𝑞 )

―𝑔 𝑚𝑝𝑔 𝑖 𝑚𝑔 𝑗 𝑞 2(h/2k2 + 𝑔 𝑚𝑝𝑔 𝑛𝑞 )

The C atoms in the five-membered ring are numbered by m, p, q, n in serial. Here

m and p belong to segment (1) while m and q are classified as starred atoms in the alternant backbone. In the second column, a market * is added after the index if it belongs to starred atoms. Tracing back to Figure 1b-c, the four connections which satisfy the four conditions also obey these rules, although BDT-1 and BTBT-1 contain two five-membered rings. In fact, more evidences shown in Figure S3 and S4 have confirmed the generality of the rules deduced from the simplest two-segmental case. Details of the derivation in this section can be found in the Supporting Information.



Page 16 of 34

Comparison with NEGF+DFT results. Below we will perform some purposive calculations to verify these four rules. Results of intra-segmental connections are shown in Figure 3. In some molecules containing benzene segments, the |Gmol|2 curves of ortho, para and meta connections inside the segments are compared with those in benzene (black dashed line). Obviously, ortho and para connections shown on the left of Figure 3b-c obey rule (1). That is, |Gmol|2 of the ortho-1 connection in BT is a little lower than the segmental Green's function around E = 0, while |Gmol|2 of all the other connections are a little higher. For meta connections in Figure 3d, the anti-resonances shift to 0.23β for meta-1 in BT, and to 0.16β for meta-2, which obey rule (2).

Figure 3. The intra-segmental case. (a) Illustration of a BT coupled to 1D Au electrodes through an alkynyl line and a thiol anchor group on each side. (b-d) 16 ACS Paragon Plus Environment

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


Left: Modulus squares of Green's functions of the whole molecule (colored solid lines) and the benzene segment (black dashed lines) under the TB approach for connections indicated in the inset. Right: The corresponding equilibrium transmission spectra calculated under the NEGF+DFT approach. Then we construct junction models for NEGF+DFT calculations by coupling the molecule to one-dimensional (1D) Au electrodes through an alkynyl line and a thiol anchor group on each side like Figure 3a. The calculated equilibrium transmission spectra are shown on the right of Figure 3b-d. Because the energy alignment between molecular levels and the Fermi energy calculated by DFT is known to be unreliable for thiol anchors,30, 36 we compare the T(E) curves at about 1 eV above the Fermi energy with the |Gmol|2 curves at E = 0. Generally, results obtained under the TB approach are in good accordance with the transmission spectra. There are also some small differences. For example, the value of T(E) of the ortho-1 connection in BT at 1 eV is nearly the same with that of the ortho connection in benzene instead of lower, which do not affect the conclusion. Anti-resonances of meta-1 and meta-2 connections in BT are respectively 0.22 eV and 0.16 eV higher than that of the meta connection in benzene, indicating that the location of the anti-resonance predicted under the TB approach is only qualitatively reliable. On the other hand, intra-segmental connections are investigated by several existing thienoacenes containing more than one five-membered rings. Some symmetric connections are concerned as they are experimentally easy to achieve. 17 ACS Paragon Plus Environment


Page 18 of 34

First, four connections in BDT-1 and BDT-2 shown in Figure 4b are chosen. Under the TB approach, the calculated ratio of |Gmol(0)|2 between the α-α connection in BDT-2 and BDT-1 is about 0.063, in good agreement with the experimental conductance ratio 0.065.48 Between the HOMO and LUMO resonances, the |Gmol|2 curve of α-α connection in BDT-1 (green line) is slightly lower than the corresponding |GC|2 curve (dark green dashed line), just as rule (3) states. All the other three connections in Figure 4b fit the condition of rule (4), conductance of which are evidently lower than the α-α connection in BDT-1 with similar

length

and

topology.

Among

these

three

connections,

typical

anti-resonances exist in the α-α and β-β connections of BDT-2 where GC exhibit odd parity so that pass through the origin. (Gmol and GC curves without modulus square can be found in Figure S12.) However, this is not a general conclusion because the S’ pathway may totally change the shape of GC, especially when GC is ultralow in a wide range near the origin. More calculations are performed adopting BTBT-1 and BTBT-2 as Figure 4c, and DBTDT-1 and DBTDT-2 as Figure 4d. Here para and meta are determined by the connections of the outmost benzene segments. According to the slightly lower values of |Gmol(0)|2 compared with |GC(0)|2 when the C backbones exhibit CQI, and the evidently lower conductance of connections when the C backbones exhibit DQI, rules (3) and (4) are further confirmed.


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


Figure 4. The inter-segmental case. (a) Illustration of a BDT-1 coupled to 1D Au electrodes through an alkynyl line and a thiol anchor group on each side (b-d) Left: Modulus squares of Green's functions of the whole molecule (solid lines) and C backbone which exhibits CQI (dashed lines) under the TB approach for connections indicated in the inset. Right: The corresponding equilibrium transmission spectra calculated under the NEGF+DFT approach. Transmission spectra using 1D Au electrodes calculated under the NEGF+DFT approach are shown on the right. To construct the structural model for the C backbone, we delete the S atoms, passivate the dangling bonds by H and do not perform further geometry optimization so that the intra-segmental angles do not change. We note that the anti-resonances in the T(E) curves of meta connections in BTBT-1 and DBTDT-1 are not predicted by the TB model. It can be explained 19 ACS Paragon Plus Environment


Page 20 of 34

by the sensitivity of an anti-resonance due to a Green’s function sweeping over the lateral axis instead of passing through it, similar with the case of hard zero in alternant hydrocarbons.64 Except for this, the TB and DFT results agree with each other. Junction models using three-dimensional (3D) Au electrodes are also constructed as Figure 5a. In these junctions, molecular cores couple with electrodes through an alkynyl line and a 2,3-dihydrobenzo[b]thiophene anchor group on each side. Three connections in DBTDT-1 and DBTDT-2 are chosen, the calculated transmission spectra drawn in Figure 5b using the same colormap with Figure 4d. Except for the locational change of the Fermi energy due to the different anchor groups, results using 3D electrodes do not lead to further differences. Moreover, equilibrium currents from the left electrode to the right one are decomposed to bond currents,7 which are shown in Figure 5c. It is clear that the bond currents through the S atoms are evidently lower than those through the C backbone when the C backbone exhibits CQI, which is the case of the para connection in DBTDT-1. In contrast, when the C backbone exhibits DQI, the bond currents to some extent display the figures of the complicated S’ pathway.


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


Figure 5. (a) Illustration of a DBTDT-1 coupled to 3D Au electrodes through an alkynyl line and a 2,3-dihydrobenzo[b]thiophene group on each side. (b) Equilibrium transmission spectra of three connections in DBTDT-1 and DBTDT-2. (c) Bond currents at the Fermi level from the left electrode to the right one for connections in (b). Lengths of the arrows indicate the magnitude of the bond currents. Reliability of the two-pathway viewpoint for different heteroatoms. At last we will extend our results to more kinds of heteroatoms. According to eq 2, impact of the species of the heteroatom is largely dependent on the parameter h/2k2 as an important member of the denominator of GS’. We choose a series of connections in DBT-like and thiophene-like molecules as illustrated in Figure 6a, which have been widely investigated.38-40, 43-45 B, N, O and S are chosen as the heteroatoms in comparison with a pure C backbone, while the value of |Gmol|2 in the middle of the HOMO and LUMO obtained under DFT using SZ basis set is taken as the reference of the molecular conductance. Results are shown in Figure 6b in the order of h/2k2. Specifically, we define h/2k2 = ∞ when there is only a C backbone 21 ACS Paragon Plus Environment


Page 22 of 34

so that GS' becomes zero. Generally the calculated relative conductance of similar connections with different heteroatoms agree with theoretical and experimental results, while the different trend of 1-3 connection in thiophene-like molecules39 can be ascribed to the rapid change of the transmission spectrum and the roughness of the midgap assumption.

Figure 6. (a) Illustration of connections in DBT-like and thiophene-like molecules. (b) Modulus squares of the Green's functions at the midgap |G(Emidgap)|2 for connections shown in (a) in the order of the TB parameter h/2k2 of the heteroatom. (c) Ratios of |G(Emidgap)|2 between connections in the same molecules where the C backbones exhibit DQI and CQI. Reliability of the two-pathway viewpoint is evaluated by the ratio of |Gmol(Emidgap)|2 between connections where the C backbones exhibit DQI and CQI in the same molecule. This is because the former one mainly depends on the S’ pathway while the latter one mainly depends on the C pathway. The C pathway should not be seen as a dominant factor of the interference feature if in a molecule some of the ratios are high, and consequently the two-pathway viewpoint is no longer reliable enough. According to Figure 6c, the ratios are always low for the C 22 ACS Paragon Plus Environment

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


backbones. Next, S and O heteroatoms in which h/2k2 are close to 1 also lead to low ratios, and then is N. In contrast, B heteroatom with negative h/2k2 results in high ratio between the 2-3 and 1-4 connections in borole, and also between the meta and para connections in dibenzoborole, exceeding 0.5. As a result, the two-pathway viewpoint should be used with caution for negative h/2k2. CONCLUSION In summary, the common two-pathway viewpoint in studying molecules containing five-membered heterocycles, which assume the C backbone dominate the molecular conductance and the heteroatoms serve as a tunable factor, have been explicitly confirmed. Zeroth Green’s function of the molecule, the modulus square of which is proportional to the molecular conductance, is divided into a C backbone term GC and a heteroatom dependent term GS’. Connections are further classified according to the type of QI of the C backbone and the segmental ascription of the connecting atoms, where segments are separated by the axis of each five-membered ring. Four rules of the interaction between the two pathways are summed out. When the C backbone exhibits CQI, conductance of the whole molecule is always relatively high. The S’ pathway may either enhance or suppress the conductance of the C pathway for intra-segmental connections, and always suppresses the conductance for inter-segmental connections. When the C backbone exhibits DQI, conductance is always relatively low. The anti-resonance shifts not far away for intra-segmental connections, and is highly dependent on the specific molecular structure for inter-segmental connections. A value h/2k2 23 ACS Paragon Plus Environment


Page 24 of 34

depending on common TB parameters can be taken as a criterion of the domination of the C pathway. The two-pathway viewpoint is reliable for high h/2k2, and should be used carefully for low h/2k2, especially when it is negative indicating higher on-site energy of the heteroatom compared with C.

ASSOCIATED CONTENT Supporting Information. The following file is available free of charge. Test of the tight-binding parameters, Green's functions of more connections in more thienoacenes, properties of the segmental Green's functions and detailed derivation of the segmented analysis. AUTHOR INFORMATION Corresponding Author * Email: [email protected]. * Email: [email protected], [email protected]. Notes The authors declare no competing financial interest. ACKNOWLEDGMENT


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


The authors acknowledge financial support from the Ministry of Science and Technology of China (2017YFA0204503, 2016YFB0401100), the National Natural Science Foundation of China (51725304, 51633006, 21875259, 51733004, 61890943), the Strategic Priority Research Program (XDB12000000) of the Chinese Academy of Sciences, the Youth Innovation Promotion Association of the Chinese Academy of Sciences and the National Program for Support of Top-notch Young Professionals. REFERENCES 1.

Emberly , E. G.; Kirczenow, G., Antiresonances in molecular wires. J. Phys.:

Condens. Matter. 1999, 11, 6911-6926. 2.

Baer, R.; Neuhauser, D., Phase Coherent Electronics: A Molecular Switch

Based on Quantum Interference. J. Am. Chem. Soc. 2002, 124, 4200-4201. 3.

Cardamone, D. M.; Stafford, C. A.; Mazumdar, S., Controlling Quantum

Transport through a Single Molecule. Nano Lett. 2006, 6, 2422-2426. 4.

Andrews, D. Q.; Solomon, G. C.; Goldsmith, R. H.; Hansen, T.; Wasielewski,

M. R.; Duyne, R. P.; Ratner, M. A., Quantum Interference: The Structural Dependence

of

Electron

Transmission

through

Model

Systems

and

Cross-Conjugated Molecules. J. Phys. Chem. C 2008, 112, 16991-16998. 5.

Ke, S. H.; Yang, W., Quantum-Interference-Controlled Molecular Electronics.

Nano Lett. 2008, 8, 3257-3261. 6.

Solomon, G. C.; Andrews, D. Q.; Goldsmith, R. H.; Hansen, T.; Wasielewski,

M. R.; Duyne, R. P.; Ratner, M. A., Quantum Interference in Acyclic Systems: 25 ACS Paragon Plus Environment


Page 26 of 34

Conductance of Cross-Conjugated Molecules. J. Am. Chem. Soc. 2008, 130, 17301-17308. 7.

Solomon, G. C.; Herrmann, C.; Hansen, T.; Mujica, V.; Ratner, M. A.,

Exploring local currents in molecular junctions. Nat. Chem. 2010, 2 (3), 223-228. 8.

Aradhya, S. V.; Meisner, J. S.; Krikorian, M.; Ahn, S.; Parameswaran, R.;

Steigerwald, M. L.; Nuckolls, C.; Venkataraman, L., Dissecting contact mechanics from quantum interference in single-molecule junctions of stilbene derivatives.

Nano Lett. 2012, 12 (3), 1643-1647. 9.

Guedon, C. M.; Valkenier, H.; Markussen, T.; Thygesen, K. S.; Hummelen, J.

C.; van der Molen, S. J., Observation of quantum interference in molecular charge transport. Nat. Nanotechnol. 2012, 7 (5), 305-309. 10. Arroyo, C. R.; Tarkuc, S.; Frisenda, R.; Seldenthuis, J. S.; Woerde, C. H.; Eelkema, R.; Grozema, F. C.; van der Zant, H. S. J., Signatures of quantum interference effects on charge transport through a single benzene ring. Angew.

Chem. Int. Ed. 2013, 52 (11), 3152-3155. 11. Garner, M. H.; Li, H.; Chen, Y.; Su, T. A.; Shangguan, Z.; Paley, D. W.; Liu, T.; Ng, F.; Li, H.; Xiao, S.; Nuckolls, C.; Venkataraman, L.; Solomon, G. C., Comprehensive suppression of single-molecule conductance using destructive sigma-interference. Nature 2018, 558 (7710), 415-419. 12. Liu, J.; Huang, X.; Wang, F.; Hong, W., Quantum Interference Effects in Charge Transport through Single-Molecule Junctions: Detection, Manipulation, and Application. Acc. Chem. Res. 2019, 52 (1), 151-160. 26 ACS Paragon Plus Environment

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


13. Huang, B.; Liu, X.; Yuan, Y.; Hong, Z. W.; Zheng, J. F.; Pei, L. Q.; Shao, Y.; Li, J. F.; Zhou, X. S.; Chen, J.; Jin, S.; Mao, B. W., Controlling and Observing of Sharp-Valleyed Quantum Interference Effect in Single Molecular Junctions. J. Am.

Chem. Soc. 2018, 140, 17685-17690. 14. Bai, J.; Daaoub, A.; Sangtarash, S.; Li, X.; Tang, Y.; Zou, Q.; Sadeghi, H.; Liu, S.; Huang, X.; Tan, Z.; Liu, J.; Yang, Y.; Shi, J.; Meszaros, G.; Chen, W.; Lambert, C.; Hong, W., Anti-resonance features of destructive quantum interference in single-molecule thiophene junctions achieved by electrochemical gating. Nat.

Mater. 2019, 18, 364-369. 15. Sangtarash, S.; Huang, C.; Sadeghi, H.; Sorohhov, G.; Hauser, J.; Wandlowski, T.; Hong, W.; Decurtins, S.; Liu, S. X.; Lambert, C. J., Searching the Hearts of Graphene-like Molecules for Simplicity, Sensitivity, and Logic. J. Am. Chem. Soc. 2015, 137 (35), 11425-11431. 16. Bergfield, J. P.; Solis, M. A.; Stafford, C. A., Giant Thermoelectric Effect from Transmission Supernodes. ACS Nano 2010, 4, 5314-5320. 17. Strange, M.; Seldenthuis, J. S.; Verzijl, C. J. O.; Thijssen, J. M.; Solomon, G. C., Interference enhanced thermoelectricity in quinoid type structures. J. Chem. Phys. 2015, 142, 084703. 18. Zhang, Y. P.; Chen, L. C.; Zhang, Z. Q.; Cao, J. J.; Tang, C.; Liu, J.; Duan, L. L.; Huo, Y.; Shao, X.; Hong, W.; Zhang, H. L., Distinguishing Diketopyrrolopyrrole Isomers in Single-Molecule Junctions via Reversible Stimuli-Responsive Quantum Interference. J. Am. Chem. Soc. 2018, 140 (21), 6531-6535. 27 ACS Paragon Plus Environment


Page 28 of 34

19. Ricks, A. B.; Solomon, G. C.; Colvin, M. T.; Scott, A. M.; Chen, K.; Ratner, M. A.; Wasielewski, M. R., Controlling Electron Transfer in Donor-Bridge-Acceptor Molecules Using Cross-Conjugated Bridges. J. Am. Chem. Soc. 2010, 132, 15427– 15434. 20. Papadopoulos, T. A.; Grace, I. M.; Lambert, C. J., Control of electron transport through Fano resonances in molecular wires. Phys. Rev. B 2006, 74 (19), 193306. 21. Stadler, R.; Markussen, T., Controlling the transmission line shape of molecular t-stubs and potential thermoelectric applications. J. Chem. Phys. 2011,

135 (15), 154109. 22. Nozaki, D.; Sevincli, H.; Avdoshenko, S. M.; Gutierrez, R.; Cuniberti, G., A parabolic model to control quantum interference in T-shaped molecular junctions.

Phys. Chem. Chem. Phys. 2013, 15 (33), 13951-13958. 23. Frisenda, R.; Janssen, V. A.; Grozema, F. C.; van der Zant, H. S.; Renaud, N., Mechanically controlled quantum interference in individual pi-stacked dimers.

Nat. Chem. 2016, 8 (12), 1099-1104. 24. Markussen, T.; Schiotz, J.; Thygesen, K. S., Electrochemical control of quantum interference in anthraquinone-based molecular switches. J. Chem. Phys. 2010, 132 (22), 224104. 25. Yoshizawa, K.; Tada, T.; Staykov, A., Orbital Views of the Electron Transport in Molecular Devices. J. Am. Chem. Soc. 2008, 130, 9406-9413. 26. Yoshizawa, K., An Orbital Rule for Electron Transport in Molecules. Acc.

Chem. Res. 2012, 45, 1612-1621. 28 ACS Paragon Plus Environment

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


27. Markussen, T.; Stadler, R.; Thygesen, K. S., The relation between structure and quantum interference in single molecule junctions. Nano Lett. 2010, 10 (10), 4260-4265. 28. Markussen, T.; Stadlerb, R.; Thygesen, K. S., Graphical prediction of quantum interference-induced transmission nodes in functionalized organic molecules.

Phys. Chem. Chem. Phys. 2011, 13, 14311-14317. 29. Pedersen, K. G. L.; Borges, A.; Hedegård, P.; Solomon, G. C.; Strange, M., Illusory Connection between Cross-Conjugation and Quantum Interference. J.

Phys. Chem. C 2015, 119 (48), 26919-26924. 30. Geng, Y.; Sangtarash, S.; Huang, C.; Sadeghi, H.; Fu, Y.; Hong, W.; Wandlowski, T.; Decurtins, S.; Lambert, C. J.; Liu, S. X., Magic ratios for connectivity-driven electrical conductance of graphene-like molecules. J. Am.

Chem. Soc. 2015, 137 (13), 4469-4476. 31. Fowler, P. W.; Pickup, B. T.; Todorova, T. Z.; Myrvold, W., Conduction in graphenes. J. Chem. Phys. 2009, 131 (24), 244110. 32. Stuyver, T.; Fias, S.; De Proft, F.; Geerlings, P., Back of the Envelope Selection Rule for Molecular Transmission: A Curly Arrow Approach. J. Phys. Chem. C 2015, 119 (47), 26390-26400. 33. Tsuji, Y.; Hoffmann, R.; Strange, M.; Solomon, G. C., Close relation between quantum interference in molecular conductance and diradical existence. Proc.

Natl. Acad. Sci. U S A 2016, 113 (4), E413-E419.



Page 30 of 34

34. Low, J. Z.; Sanders, S. N.; Campos, L. M., Correlating Structure and Function in Organic Electronics: From Single Molecule Transport to Singlet Fission. Chem.

Mater. 2015, 27 (16), 5453-5463. 35. Li, X.; Staykov, A.; Yoshizawa, K., Orbital views of the electron transport through heterocyclic aromatic hydrocarbons. Theo. Chem. Accounts 2011, 130 (4-6), 765-774. 36. Sangtarash, S.; Sadeghi, H.; Lambert, C. J., Exploring quantum interference in heteroatom-substituted graphene-like molecules. Nanoscale 2016, 8 (27), 13199-13205. 37. Liu, X.; Sangtarash, S.; Reber, D.; Zhang, D.; Sadeghi, H.; Shi, J.; Xiao, Z. Y.; Hong, W.; Lambert, C. J.; Liu, S. X., Gating of Quantum Interference in Molecular Junctions by HeteroatomSubstitution. Angew. Chem. Int. Ed. 2017, 56, 173-176. 38. Chen, W.; Li, H.; Widawsky, J. R.; Appayee, C.; Venkataraman, L.; Breslow, R., Aromaticity decreases single-molecule junction conductance. J. Am. Chem. Soc. 2014, 136 (3), 918-920. 39. Yang, Y.; Gantenbein, M.; Alqorashi, A.; Wei, J.; Sangtarash, S.; Hu, D.; Sadeghi, H.; Zhang, R.; Pi, J.; Chen, L.; Huang, X.; Li, R.; Liu, J.; Shi, J.; Hong, W.; Lambert, C. J.; Bryce, M. R., Heteroatom-Induced Molecular Asymmetry Tunes Quantum Interference in Charge Transport through Single-Molecule Junctions. J.

Phys. Chem. C 2018, 122 (26), 14965-14970.


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


40. Borges, A.; Solomon, G. C., Effects of Aromaticity and Connectivity on the Conductance of Five-Membered Rings. J. Phys. Chem. C 2017, 121 (15), 8272-8279. 41. Cai, Z.; Awais, M. A.; Zhang, N.; Yu, L., Exploration of Syntheses and Functions of Higher Ladder-type p-Conjugated Heteroacenes. Chem 2018, 4, 2538-2570. 42. Cai, Z.; Zhang, N.; Awais, M. A.; Filatov, A. S.;Yu, L., Synthesis of Alternating Donor–Acceptor Ladder-Type Molecules and Investigation of Their Multiple Charge Transfer Pathways. Angew. Chem. Int. Ed. 2018, 57, 6442-6448. 43. Liu, H.; Ni, W.; Zhao, J.; Wang, N.; Guo, Y.; Taketsugu, T.; Kiguchi, M.; Murakoshi, K., Nonequilibrium Green's function study on the electronic structure and transportation behavior of the conjugated molecular junction: terminal connections and intramolecular connections. J. Chem. Phys. 2009, 130 (24), 244501. 44. Klausen, R. S.; Widawsky, J. R.; Su, T. A.; Li, H.; Chen, Q.; Steigerwald, M. L.; Venkataraman, L.; Nuckolls, C., Evaluating atomic components in fluorene wires.

Chem. Sci. 2014, 5 (4), 1561-1564. 45. Gantenbein, M.; Wang, L.; Al-Jobory, A. A.; Ismael, A. K.; Lambert, C. J.; Hong, W.; Bryce, M. R., Quantum interference and heteroaromaticity of para- and meta-linked

bridged

biphenyl

units

in

single

molecular

conductance

measurements. Sci. Rep. 2017, 7 (1), 1794.



Page 32 of 34

46. Stuyver, T.; Perrin, M.; Geerlings, P.; De Proft, F.; Alonso, M., Conductance Switching in Expanded Porphyrins through Aromaticity and Topology Changes. J.

Am. Chem. Soc. 2018, 140 (4), 1313-1326. 47. Stuyver, T.; Fias, S.; Geerlings, P.; De Proft, F.; Alonso, M., Qualitative Insights into the Transport Properties of Hückel/Möbius (Anti)Aromatic Compounds: Application to Expanded Porphyrins. J. Phys. Chem. C 2018, 122 (34), 19842-19856. 48. Zhang, Y.; Ye, G.; Soni, S.; Qiu, X.; Krijger, T. L.; Jonkman, H. T.; Carlotti, M.; Sauter, E.; Zharnikov, M.; Chiechi, R. C., Controlling destructive quantum interference in tunneling junctions comprising self-assembled monolayers via bond topology and functional groups. Chem. Sci. 2018, 9 (19), 4414-4423. 49. Zhang, J.; Sun, W.; Liu, H.; He, Y.; Zhao, J., Effects of terminal connection and molecular length on electron transport in linear conjugated molecular wires.

Comp. Mater. Sci. 2014, 87, 100-106. 50. Tagami, K.; Tsukada, M.; Wada, Y.; Iwasaki, T.; Nishide, H., Electronic transport of benzothiophene-based chiral molecular solenoids studied by theoretical simulations. J. Chem. Phys. 2003, 119 (14), 7491-7497. 51. Cai, Z.; Lo, W. Y.; Zheng, T.; Li, L.; Zhang, N.; Hu, Y.; Yu, L., Exceptional Single-Molecule Transport Properties of Ladder-Type Heteroacene Molecular Wires. J. Am. Chem. Soc. 2016, 138 (33), 10630-10635. 52. Hedström, S.; Matula, A. J.; Batista, V. S., Charge Transport and Rectification in Donor–Acceptor Dyads. J. Phys. Chem. C 2017, 121 (35), 19053-19062. 32 ACS Paragon Plus Environment

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


53. Hohenberg, P.; Kohn, W., Inhomogeneous Electron Gas. Phys. Rev. 1964, 136 (3B), B864-B871. 54. Kohn, W.; Sham, L. J., Self-Consistent Equations Including Exchange and Correlation Effects. Phys. Rev. 1965, 140 (4A), A1133-A1138. 55. Meir, Y.; Wingreen, N. S., Landauer formula for the current through an interacting electron region. Phys. Rev. Lett. 1992, 68 (16), 2512-2515. 56. Brandbyge, M.; Mozos, J.-L.; Ordejón, P.; Taylor, J.; Stokbro, K., Density-functional method for nonequilibrium electron transport. Phys. Rev. B 2002, 65 (16), 165401. 57. Xue, Y.; Dattab, S.; Ratnera, M. A., First-principles based matrix Green’s function approach to molecular electronic devices: general formalism. Chem. Phys. 2002, 281, 151-170. 58. Rocha, A. R.; Garcia-Suarez, V. M.; Bailey, S. W.; Lambert, C. J.; Ferrer, J.; Sanvito, S., Towards molecular spintronics. Nat. Mater. 2005, 4 (4), 335-339. 59. Li, R.; Zhang, J.; Hou, S.; Qian, Z.; Shen, Z.; Zhao, X.; Xue, Z., A corrected NEGF+DFT approach for calculating electronic transport through molecular devices: Filling bound states and patching the non-equilibrium integration. Chem.

Phys. 2007, 336 (2-3), 127-135. 60. Imry, Y.; Landauer, R., Conductance viewed as transmission. Rev. Mod. Phys. 1999, 71, S306-S312. 61. Troullier, N.; Martins, J. L., Efficient pseudopotentials for plane-wave calculations. Phys. Rev. B 1991, 43 (3), 1993-2006. 33 ACS Paragon Plus Environment


Page 34 of 34

62. Soler, J. M.; Artacho, E.; Gale, J. D.; García, A.; Junquera, J.; Ordejón, P.; Sánchez-Portal, D., The SIESTA method for ab initio order-N materials simulation.

J. Phys.: Condens. Matter. 2002, 14, 2745-2779. 63. Perdew, J. P.; Burke, K.; Ernzerhof, M., Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865-3868. 64. Tsuji, Y.; Hoffmann, R.; Movassagh, R.; Datta, S., Quantum interference in polyenes. J. Chem. Phys. 2014, 141 (22), 224311.


The Two-Pathway Viewpoint to Interpret Quantum Interference in

Recommend Documents