The Two-Pathway Viewpoint to Interpret Quantum Interference in

Article Cite This: J. Phys. Chem. C 2019, 123, 15977−15984

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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*,†,‡

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†

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 S Supporting Information *

ABSTRACT: In the study of the quantum interference effect in single molecular junctions where the molecules contain five-membered heterocycles, an implicit twopathway viewpoint can be found that assumes that the carbon backbone play the dominant role and regard the heteroatoms as a tunable factor. Until now, this viewpoint has not been systematically evaluated. In this work, we concretely divide the zeroth Green 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 antiresonances 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.

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INTRODUCTION The quantum interference (QI) effect in molecular devices has attracted great attention as an essential element in the device function, both theoretically1−7 and experimentally.8−14 In the phase-coherent regime, because of the interference of the electron wave functions 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 antiresonance between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO).7,8 Molecular switches or sensors,2−4,13 transistors,5,14 logic devices,15 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 molecules,20−22 the four-level model for π−π stacking intermolecular junctions,23 and the three-level model for anthraquinones.9,24 More generally, several analyzing methods © 2019 American Chemical Society

based on the tight-binding (TB) model serve as effective tools 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 diagram27−29 proposes a graphic method to obtain the numerator of the molecular Green function and find out the zero-conductance conditions. The magic ratio rule15,30 eliminates the singularity of the Green 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 and diradical existence.33 The antiresonance, 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 Received: April 5, 2019 Revised: June 1, 2019 Published: June 5, 2019 15977

DOI: 10.1021/acs.jpcc.9b03177 J. Phys. Chem. C 2019, 123, 15977−15984

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The Journal of Physical Chemistry C molecules was studied by implementing the orbital rule35 and magic ratio rule.36,37 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 the aromaticity of the ring or electronegativity of the 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 The influences of the heteroatom and connection in molecules whose structures are similar to dibenzothiophene (DBT) were discussed in different ways.40,43−45 The negative relationship between conductance and aromaticity was proven 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 fivemembered 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 that 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; that is, 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 examples. Analytical derivation is done under the TB approach, while the nonequilibrium Green function formalism combined with the 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 that a value h/2k2 depending on TB parameters of the heteroatom can be taken as a criterion of the domination of the C pathway.

intrinsic QI properties of a molecule can be explored by observing the zeroth Green function. Especially, the value of | 2 Gmol ij (E)| 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 wave functions of valence electrons and improved Troullier− Martins pseudopotentials to describe the atomic cores.61,62 The single-ζ (SZ) basis set is adopted for the parametrization of the TB model, while the double-ζ plus polarization basis set is used in the calculations of metal−molecule−metal junction models. The exchange and correlation functional is described by the generalized gradient approximation in the Perdew− Burke−Ernzerhof form.63 The 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 matrices obtained under SIESTA to perform the nonequilibrium Green function (NEGF) calculations. Here, the basis set describing Au changes to single-ζ plus polarization. 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 a 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.

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RESULTS AND DISCUSSION 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 function can be written as ijGC + GCHCSZHSCGC GCHCSZ yz zz Gmol = jjjj zz SC C ZH G Z k {

(1)

C −1

Here, G = (EI − H ) corresponds to the zeroth Green function matrix of the C backbone and Z = (EI − HS − H SCGC HCS) −1 corresponds to interaction between the heteroatoms in the molecule. The energy dependence is omitted for clarity. 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, that is, 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 an 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. A number of calculations are carried out under the TB approach for different connections in different thienoacenes, the results of which obey the following rule. When the C backbone exhibits CQI, |Gmol(0)|2 of the whole molecule is C

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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 function of the extended molecule and ΓL/R(E), which is called the broadening function, describes the coupling between the left/right electrode and the (extended) molecule. When the molecule is weakly coupled to electrodes, the zeroth Green function, that is, the Green function of the isolated molecule, can be adopted to predict the conductance.26,27,30 Assuming that each electrode only couples 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 2 27 T(E) = ΓLii (E)ΓRjj (E)|Gmol That means that the transij (E)| . mission probability from i to j at a certain energy is proportional to the modulus square of Gmol ij (E). As a result, 15978

DOI: 10.1021/acs.jpcc.9b03177 J. Phys. Chem. C 2019, 123, 15977−15984

The Journal of Physical Chemistry C

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Article

(3)

Figure 1. (a) Illustration of the C pathway and S′ pathway in a thionoacene. Modulus squares of Green functions of the whole molecule and the C backbone for connections indicated by arrows in (b) BDT-1 and (c) BTBT-1. Figure 2. (a) Illustration of dividing the C backbone of BDT-1 into two C−C pairs (segments 1 and 3) and a benzene segment (segment 2). (b) Segmented expression of the Green function of the C backbone at E = 0 for an intersegmental connection. (c) Shape of the Green functions between possible sites in the C−C pair and benzene. Nonzero values at E = 0 are noted as green text. Horizontal axes range from −1.2β to 1.2β, while vertical axes range from −3β−1 to 3β−1.

comparable to |G (0)| . When the C backbone exhibits DQI instead, |Gmol(0)|2 becomes much lower than the 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 the top of Figure 1b, 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 case of CQI shown on the upper panels, between HOMO and LUMO resonances, both of the 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. Additional evidence is shown in Figures S3a−c and S4a−f. On the other hand, in the case of DQI, typical antiresonances 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 antiresonances may also shift left, split, vanish, or stay near the zero energy as shown in Figures S3d−f and S4g−l. No matter how the antiresonances 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.) The 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’. Segmented Analysis. The 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 C

GijS′ = LiT(k2Z)Lj

2

Here, we adopt β as the unit of energy and Hamiltonian; thus, the unit of Green function becomes β−1. Obviously, only Z is affected by the heteroatom-dependent parameters, especially by h/2k2 at E = 0, which approximately equals 1 for S. Some certain elements of GC also play important roles in GS’. As shown in Figure 2a, the C backbone of a thienoacene can be ″cut″ into NS + 1 segments by the axis of each fivemembered ring. Resulting segments are either C−C pairs or acenes, the zeroth Green functions of which have been widely explored.26,30,64 It can be proven that the intrasegmental GCij (0) equals the Green function of the segment itself at zero energy (indicated by gij), while intersegmental GCij (0) is the product of the segmental Green functions with an extra negative sign if the number of segments is even. Taking the intersegmental connection in BDT-1 as an example, the segmented expression of GC(0) is illustrated in Figure 2b. Similarly, GSij′(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 functions proven 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 function is odd for starred−unstarred connection and even for starred−starred and unstarred− unstarred connections; (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 segmental functions, which are always zero at E = 0, there are six possible forms of Gmol ij (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 intersegmental and whether the C backbone exhibits CQI or DQI:

(2)

where 15979

DOI: 10.1021/acs.jpcc.9b03177 J. Phys. Chem. C 2019, 123, 15977−15984

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The Journal of Physical Chemistry C Table 1. Expression of the Green Function at E = 0 for Thienoacenes Containing Only One Five-Membered Ringa GC(0)

GS’(0) − gnqgip gjm

case

conditions

1

i* ∈ (1), j ∈ (1)

gij

2

i ∈ (1), j ∈ (1)

0

3

i* ∈ (1), j* ∈ (1)

0

4

i* ∈ (1), j ∈ (2)

−gipgjq

5

i ∈ (1), j* ∈ (2)

0

2(h/2k 2 + gmpgnq)

6

i ∈ (1), j ∈ (2)

0

2(h/2k 2 + gmpgnq)

2(h/2k 2 + gmpgnq)

gimgjm 2(h/2k 2 + gmpgnq) gnqgnqgipgjp 2(h/2k 2 + gmpgnq) gnqgmpgipgjq 2(h/2k 2 + gmpgnq)

gimgjn − gmpgimgjq

a The C atoms in the five-membered ring are numbered by m, p, q, and 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, an asterisk is added after the index if it belongs to starred atoms.

Figure 3. Intrasegmental 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) Left: modulus squares of Green 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: corresponding equilibrium transmission spectra calculated under the NEGF+DFT approach.

(1) Intrasegmental, 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) Intrasegmental, DQI (cases 2 and 3): The S′ pathway tends to shift the antiresonance 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) Intersegmental, CQI (case 4): The S′ pathway will always suppress the conductance of the C pathway, while the degree of suppression is small compared with the initial value. (4) Intersegmental, DQI (cases 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 antiresonance depends on the specific structure.

connections are a little higher. For meta connections in Figure 3d, the antiresonances shift to 0.23β for meta-1 in BT and to 0.16β for meta-2, which obey rule 2. 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 in 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 as that of the ortho connection in benzene instead of lower, which does not affect the conclusion. Antiresonances of meta-1 and meta-2 connections in BT are 0.22 and 0.16 eV higher than that of the meta connection in benzene, respectively, indicating that the location of the antiresonance predicted under the TB approach is only qualitatively reliable. On the other hand, intrasegmental connections are investigated by several existing thienoacenes containing more than one five-membered ring. Some symmetric connections are concerned as they are experimentally easy to achieve. 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 of 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,

Tracing back to Figure 1b,c, the four connections that satisfy the four conditions also obey these rules, although BDT-1 and BTBT-1 contain two five-membered rings. In fact, much evidence shown in Figures S3 and S4 has confirmed the generality of the rules deduced from the simplest twosegmental case. Details of the derivation in this section can be found in the Supporting Information. Comparison with NEGF+DFT Results. Below, we will perform some purposive calculations to verify these four rules. Results of intrasegmental 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, the 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 function around E = 0, while |Gmol|2 of all the other 15980

DOI: 10.1021/acs.jpcc.9b03177 J. Phys. Chem. C 2019, 123, 15977−15984

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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 panel (b). Lengths of the arrows indicate the magnitude of the bond currents.

side. Three connections in DBTDT-1 and DBTDT-2 are chosen, and the calculated transmission spectra are drawn in Figure 5b using the same color map as in 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. 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, the 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 the SZ basis set is taken as the reference of the molecular conductance. Results are shown

Figure 4. Intersegmental 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 functions of the whole molecule (solid lines) and C backbone that exhibits CQI (dashed lines) under the TB approach for connections indicated in the inset. Right: corresponding equilibrium transmission spectra calculated under the NEGF+DFT approach.

conductances of which are evidently lower than the α−α connection in BDT-1 with similar length and topology. Among these three connections, typical antiresonances exist in the α−α and β−β connections of BDT-2 where GC exhibits odd parity so that passes through the origin. (Gmol and GC curves without modulus squares 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 in Figure 4c and DBTDT-1 and DBTDT-2 as in 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 than |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. 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 intrasegmental angles do not change. We note that the antiresonances 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 by the sensitivity of an antiresonance due to a Green function sweeping over the lateral axis instead of passing through it, similar to 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 in 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

Figure 6. (a) Illustration of connections in DBT-like and thiophenelike molecules. (b) Modulus squares of the Green functions at the midgap |G(Emidgap)|2 for connections shown in panel (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. 15981

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in Figure 6b in the order of h/2k2. Specifically, we define h/2k2 = ∞ when there is only a C backbone so that GS’ becomes zero. Generally, the calculated relative conductances 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. 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 backbones. Next, S and O heteroatoms in which h/2k2 is close to 1 also lead to low ratios and so is N. In contrast, the B heteroatom with negative h/2k2 results in a 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.

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (H.D.). *E-mail: [email protected]; [email protected] (W.H.). ORCID

Xi Yu: 0000-0001-5750-7003 Huanli Dong: 0000-0002-5698-5369 Wenping Hu: 0000-0001-5686-2740 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS 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, 21473222), 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.

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CONCLUSIONS In summary, the common two-pathway viewpoint in studying molecules containing five-membered heterocycles, which assumes that the C backbone dominates the molecular conductance and the heteroatoms serve as a tunable factor, has been explicitly confirmed. The zeroth Green 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 intrasegmental connections and always suppresses the conductance for intersegmental connections. When the C backbone exhibits DQI, conductance is always relatively low. The antiresonance shifts not far away for intrasegmental connections and is highly dependent on the specific molecular structure for intersegmental connections. A value h/2k2 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 than C.

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S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.9b03177. Test of the tight-binding parameters, Green functions of more connections in more thienoacenes, properties of the segmental Green functions, and detailed derivation of the segmented analysis (PDF) 15982

DOI: 10.1021/acs.jpcc.9b03177 J. Phys. Chem. C 2019, 123, 15977−15984

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