Graph-theoretical exploration of the relation between conductivity and connectivity in heteroatom-containing single-molecule junctions

Single-molecule junctions exhibit unique electron-transport properties that are not observed in extended systems, such as solids and polymers.^1–10 One of the unique transport properties is the phenomenon of interference between transporting electrons. This is known as quantum interference (QI) and occurs due to the quantum phase of the electrons caused by the wave-particle duality.^8–11 There are two types of QI: constructive QI, which occurs when the electron phases are identical,^9,12 and destructive QI, which occurs when the electron phases are opposite.^10,13–15 In this Communication, we will refer to destructive QI simply as QI.

A graph-theoretic approach has been proposed as one of the methods of predicting whether QI will occur in single-molecule junctions.¹⁶ Graph theory is a field of mathematics that utilizes a graph, a mathematical tool for modeling pairwise relationships between objects.¹⁷ A graph is a set of vertices and edges in which an edge may connect two vertices or commence from a vertex and return to it. Graph theory can be applied to chemistry using the adjacency matrix A,^16,18–23 which is a matrix containing entries of +1 if a pair of atoms is connected and 0 if it is not. By treating atoms as vertices and bonds as edges [see Fig. 1(a)], the topological properties of molecules can be revealed.¹⁸ 

QI can be explained using the concept of a walk on the molecular graph.^16,17,24,25 In the graph represented by A, the number of walks of length k commencing at node i and terminating at j is equal to the (i, j)-th element of the kth power of the adjacency matrix, i.e., (A^k)_ij.²⁴ Using molecular graphs for alternant hydrocarbons, which do not contain any odd-membered rings and heteroatoms, their single-molecule conductivity can be related to the presence of odd-length walks on their graphs [see Fig. 1(a) for an example].^16,26 In benzene, there are odd-length walks between the two carbon atoms in the para-position [Fig. 1(b), left], whereas even-length walks exist between the two carbon atoms in the meta-position [Fig. 1(b), right]. Therefore, the molecular junction becomes conductive and insulative when the electrodes are connected to two carbon atoms in para- and meta-positions with respect to each other, respectively.^27–30 

This is a highly intuitive interpretation of QI. However, this method cannot be directly applied to molecules containing heteroatoms because the electron–hole symmetry is frequently broken in such molecules.^31,32 In this study, we used a graph-theoretical method named Sachs graphs to overcome this problem and attempt to establish a novel interpretation of QI by relating the conduction behavior of single-molecule junctions containing a heteroatom to walks on the graph.

A brief review of the molecular conductance theory is provided here for reference. The conductance of a single molecule, g, can be calculated from Landauer’s formula^33,34 combined with the nonequilibrium Green’s function (NEGF).^3,35,36 Landauer’s formula is given by

where the (i, j)-th entry corresponds to the ith and jth sites in the molecule connected to the two electrodes, 2e²/h is the quantum conductance, and T_ij (E_F) is the transmission probability at the Fermi energy. The transmission probability is proportional to the (i, j)-th entry of the zeroth-order Green’s function (ZOGF), G_ij, as follows:³⁷ 

Considering that the Fermi energy is located at E = 0, the ZOGF simplifies to the following form:²⁶ 

In this study, we employ the Hückel Hamiltonian, H, to describe the electronic structure of π systems. If the Coulomb integrals of all the sp² carbon atoms are identical and their values can be set to zero without loss of generality, all the diagonal elements of H become zero. H can be written as

where β is the resonance integral between adjacent carbon 2pπ orbitals and A is the adjacency matrix for the molecular graph. If β is used as the unit of energy, the Hamiltonian can be simplified as H = A. Consequently, the ZOGF can be rewritten as

The conductive properties of molecular junctions can be predicted by examining the inverse of the adjacency matrix. Therefore, the topology of the π-conjugated network can be linked to the molecular conductivity.

The topological properties of a graph can be revealed by examining the characteristic polynomial, p(λ), which is defined as follows:³⁸ 

where I is the identity matrix, N is the total number of sp² carbon atoms in the π-conjugated network, and a_n is the coefficient of the (N–n)-th power term of λ.

Considering the molecular junctions of closed-shell alternant hydrocarbons, N is necessarily an even number. Applying the Cayley–Hamilton theorem³⁹ to Eq. (6) yields

Combining Eqs. (5) and (7), the ZOGF is obtained in the form of a series expansion in terms of A.

The coefficients in the characteristic polynomial can be determined from the topology of the molecular graph according to Sachs formulation.^20,40 This process involves the use of Sachs graphs that have two components: a complete graph K₂ consisting of two vertices and a cycle C_m consisting of m vertices.⁴⁰ 

For alternant hydrocarbons lacking heteroatoms, the coefficients of the terms in the characteristic polynomial can be calculated using the Sachs formula^20,40,41 as follows:

where s is a Sachs graph, S_n is the set of all Sachs graphs having n vertices, c(s) is the number of components in s (i.e., the total number of K₂’s and C_m’s included), and r(s) is the number of cycles having three or more nodes in s (i.e., the total number of C_m’s for m ≥ 3). By definition, a₀ = 1.⁴⁰ 

If S_n is devoid of Sachs graphs containing odd-membered rings (that is, if the molecule represented by the chemical graph corresponds to an alternant hydrocarbon), it consists solely of Sachs graphs containing even numbers of vertices. In other words, S_n is an empty set for odd values of n, resulting in a_n = 0. Since Sachs graphs may be unfamiliar to beginners, an example using benzene is provided in Fig. 2(a). The reader may discover therefrom how the method functions. A good guide to how to draw Sachs graphs to obtain the coefficients of the terms in the characteristic polynomial can be found in the literature.⁴² 

Due to alternancy, a_n is zero for odd values of n. For even values of n, a_n contributes to the characteristic polynomial and thereby influences the ZOGF. Therefore, the right-hand side of Eq. (7) for alternant hydrocarbons consists solely of odd powers of A. Furthermore, the exponent of A is N − n − 1, which is odd solely when n is even. Consequently, the expression for the ZOGF, shown in Fig. 1(a), contains solely odd powers of A.

The elements of the nth power of A enumerate the number of walks commencing at a particular node of the graph and terminating at the point of commencement or at a different node in n steps. In case only walks of even length exist between nodes i and j, the (i, j)-th entry of the ZOGF for alternant hydrocarbons is zero because the ZOGF lacks even powers of A.¹⁶ In such cases, the molecular conductance is predicted to be small. This is the graph-theoretical explanation of QI.

After establishing these fundamentals, we considered the effect of introducing a heteroatom on the mathematics developed up to this point. To calculate the characteristic polynomial for a molecular graph with a heteroatom introduced, the adjacency matrix for the graph needs to be defined. As discussed below, the diagonal component corresponding to the heteroatom in the adjacency matrix for the molecular graph has a non-zero value. For this reason, hetero-conjugated molecules can be represented by edge-weighted graphs in which one or more edges are weighted to distinguish them from the remaining edges. The weighted edge commences at the node corresponding to the heteroatom and terminates at its point of origination. The weight is identified by a parameter α that is specific to the type of the heteroatom. The weighted edge for a heteroatom in a molecular graph is visually identified by a loop (a cycle of a single node, also known as a self-loop).^18,21,38 An edge-weighted graph is shown in Scheme 1 for illustrative purposes.

The adjacency matrix for the molecule shown in Scheme 1 can be written as

The diagonal entries of the adjacency matrix can be traced back to the Coulomb integrals for the corresponding atoms. Considering that the heteroatom X is described by the Coulomb integral α_X, which is different from that for the carbon atom, α_C, the parameter α is related to the Coulomb integrals as follows:^21,43

Since the bond strength between the heteroatom and the neighboring carbon atom is usually different from that of the carbon–carbon bond, the off-diagonal elements corresponding to the heteroatom–carbon atom bond in the adjacency matrix may have a value different from 1. However, for simplicity, this will not be taken into account in the following. As examined in the supplementary material, the effect of the off-diagonal elements on the conduction properties can be said to be small in a qualitative sense.

The introduction of a heteroatom causes electronegativity perturbation, which breaks the electron–hole symmetry characteristic of alternant hydrocarbons.⁴⁴ The mid-point of the HOMO–LUMO gap, at which we assume that the Fermi energy is located, does not necessarily coincide with E = 0. However, since we are considering the introduction of a single heteroatom, the change in Fermi energy, if any, will not be significant. Therefore, it would be reasonable to set the Fermi energy of the hetero-conjugated system at E = 0. Similar assumptions have been made in previous studies.^31,45,46 In this work, we will attempt to understand molecular conductance using Eqs. (5)–(7). We are currently conducting a graph-theoretical study of QI at energies different from E = 0, the results whereof will be reported elsewhere.

The coefficients of the terms in the characteristic polynomial, calculated using Eq. (8), need to be modified when a heteroatom is introduced. For this purpose, a modified Sachs graph has been proposed in the literature as follows:²¹ 

where l(s) is the number of self-loops in s. The number of self-loops is not accounted for in r(s). In this study, we initially investigated molecules lacking odd-membered rings. Therefore, the Sachs graphs having odd numbers of nodes arise solely due to the self-loop corresponding to the heteroatom [see Fig. 2(b) for an example].

For a molecule containing a single heteroatom, l(s) can have a value of 0 or 1. l(s) = 1 when n is odd and l(s) = 0 when n is even. Therefore, l(s) may be expressed as $1−−1n2$⁠. From the preceding discussion, it is clear that the ZOGF for an alternant hydrocarbon having one carbon atom replaced by a heteroatom can be expressed as⁴⁷ 

In this equation, there are even powers of A, which are absent from the ZOGF expression for alternant hydrocarbons lacking heteroatoms. Furthermore, the even powers of A in this equation are always multiplied by α.

The odd powers of A in Eq. (12) are not multiplied by α, and their coefficients can be traced back to Eq. (8). Therefore, the coefficients of the odd-powered terms of A in Eq. (12) are identical to those calculated for the molecule that can be obtained by reconverting the introduced heteroatom into a carbon atom. In other words, the introduction of the heteroatom does not change the coefficients of the odd powers of A in the ZOGF.

The effect of the heteroatom on the ZOGF appears primarily through the coefficients of the even powers of A. However, it is not only the even powers of A that are involved in the process of incorporating the heteroatom effect into the ZOGF. With the introduction of the heteroatom, a self-loop is added to the molecular graph. Therefore, walks traversing the self-loop must also be considered. A single step is required to traverse the self-loop. Therefore, each time one passes through the self-loop, the length of the walk increases by one.

If solely even-length walks exist between two nodes on a bipartite graph, conduction between them is forbidden because of QI. When a heteroatom is introduced, even powers of A appear in the ZOGF formula, indicating that conduction between the two nodes can occur. In addition, if there is a self-loop arising from the heteroatom in between the even-length walk, it can change into an odd-length walk after traversing it once, contributing to the odd powers of A in the ZOGF formula. This is another reason due to which conduction between the two nodes becomes possible.

In recent years, researchers have focused on the conductance of heterocyclic compounds and particularly on their QI properties.^31,48–52 Eq. (12) aids in understanding the conduction properties of even-membered heterocyclic compounds containing a single heteroatom. Conductance calculations³¹ and measurements⁴⁸ have been performed previously for heterocyclic analogs of benzene. Hereinafter, we consider whether Eq. (12) rationally explains the findings of the aforementioned studies.

We consider that the benzene-like heterocycle is numbered as shown in Scheme 1. When electrodes are connected to the second and fourth sites or the second and sixth sites, meta-connected molecular junctions are formed. However, it has been experimentally confirmed that the conductance of the aforementioned molecular junctions is higher than that of a meta-connected molecular junction of benzene.⁴⁸ This experimental result is interpreted as a loss of QI properties due to the introduction of the heteroatom. Furthermore, when electrodes are connected to the third and fifth sites, the measured conductance is nearly identical to that of the meta-connected benzene molecular junction. Sangtarash et al.⁵³ conducted a theoretical study, wherein they reduced the problem of QI in molecular junctions containing a heteroatom to a problem of parity of connection sites, by means of an elegant formulation using the Dyson equation. Their results were also consistent with those of the experiments.⁴⁸ 

Hereinafter, we evaluate whether the formulation developed up to this point can provide insights into the experimental results. From Eq. (12), a polynomial can be obtained that can express the ZOGF for the heterocyclic analog of benzene as follows:

As a consequence of the form of Eq. (12), Eq. (13) contains even powers of A that are multiplied by the heteroparameter α.

The matrix representation of the ZOGF for the molecule considered can be written as follows:

This equation can be obtained by inverting the adjacency matrix shown in Eq. (9) or by substituting it into the A’s in Eq. (13). The absolute value of the matrix element $Gij$ provides an approximate guide for predicting whether the conductance will be high or low when the electrodes are connected to the ith and jth sites of the molecule.

Considering the case in which the electrodes are connected at the meta-position with respect to each other, that is, electron transport occurs between sites separated solely by even-length walks along the ring aside from the walk through the self-loop, we can perform the following analysis. The values of the elements of G corresponding to the connections 2–4, 2–6, and 4–6 are α/4 or −α/4. In these connection patterns, the conductance increases with an increase in the absolute value of α. This implies that the conductance depends on the nature of the heteroatom. The value of the elements of G corresponding to the connections 1–3, 1–5, and 3–5 is 0. This implies that in these connection patterns, the QI properties were maintained despite the introduction of the heteroatom.

To complete this line of investigation, we consider the case in which the electrodes are connected in the para-position with respect to each other, that is, electron transport occurs between sites separated solely by odd-length walks along the ring aside from the walk through the self-loop. The value of the corresponding element of G is always 1/2. The value of the ZOGF did not change with the value of the heteroatom parameter.

The above is merely a reproduction of experimentally verified facts using the Hückel method. What we really want to do is not just to invert A to obtain G but to provide insight into why the matrix elements of G should take the values they do. To this end, we will hereinafter analyze the difference between conductive and insulative connection modes using the path-counting scheme developed in the preceding parts of this Communication.

The meta-connections of electrodes in the heterocyclic analog of benzene can be of three types (see Fig. 3): hereinafter, the 3–5 insulative connection is referred to as meta1, and the 2–4 and 2–6 conductive connections are referred to as meta2 and meta3, respectively.

When the electrodes are attached to two carbon atoms in the meta-position, there are only even-length walks between the connected sites. However, because a self-loop exists in the heterocycle, odd-length walks occur while traversing between the two sites.

Initially, we consider walks traversing the 3–5 connection (meta1), as shown in Fig. 3(a). There is one walk of length 2, resulting in (A²)_3,5 = 1; five walks of length 4, resulting in (A⁴)_3,5 = 5; and one walk of length 5 that includes one passage through the self-loop, resulting in (A⁵)_3,5 = α. The cancellation between the contributions from the 2-step, 4-step, and 5-step walks is complete, as shown at the bottom of Fig. 3(a), and the corresponding matrix element of the ZOGF becomes zero.

Thereafter, we consider walks traversing the 2–4 connection (meta2), as shown in Fig. 3(b). There is one walk of length 2, resulting in (A²)_2,4 = 1; five walks of length 4, resulting in (A⁴)_2,4 = 5; and two walks of length 5 that include one passage through the self-loop for each, resulting in (A⁵)_2,4 = 2α. The cancellation between the contributions from the 2-step, 4-step, and 5-step walks is incomplete, as shown at the bottom of Fig. 3(b), and the corresponding matrix element of the ZOGF becomes −α/4. The difference in conduction behavior between meta1 and meta2 can be attributed to the difference in the number of walks of length 5, which includes one passage through the self-loop corresponding to the heteroatom. Therefore, the resulting finite conductance in meta2 depends on the parameter of the heteroatom.

Finally, we consider walks traversing the 2–6 connection (meta3), as shown in Fig. 3(c). There is one walk of length 2, resulting in (A²)_2,6 = 1; one walk of length 3 that includes one passage through the self-loop, resulting in (A³)_2,6 = α; six walks of length 4, including one walk with two passes of the self-loop, resulting in (A⁴)_2,6 = α² + 5; and seven walks of length 5 that consist of one walk with three passes of the self-loop and six walks with one pass of the self-loop, resulting in (A⁵)_2,6 = α³ + 6α. The cancellation between the contributions from the 2-step, 3-step, 4-step, and 5-step walks is incomplete, as shown at the bottom of Fig. 3(c), and the corresponding matrix element of the ZOGF becomes α/4. Meta3 is different from the other connections in that there is a 3-step walk with one pass through the self-loop. Furthermore, there are walks that traverse the self-loop not once but twice or thrice. These walks result in terms α² and α³. However, they are not included in the final ZOGF formula because they cancel each other out. A similar analysis has been performed for the para connection as well, and it is shown in the supplementary material for this Communication.

In summary, we have investigated the conductance through a heteroatom-containing single-molecule junction by utilizing the concept of the Sachs graph and counting walks on a molecular graph. To incorporate the effect of the heteroatom on the conductance, a weighted self-loop was introduced into the molecular graph. We have shown that the ZOGF in such a situation can be expressed as the sum of the ZOGF for the parental graph lacking the heteroatom and the polynomial of even powers of the adjacency matrix multiplied by the heteroatom parameter. Because an experimental report on the measurement of conductance in several symmetrically distinct meta-type connection modes in a heterocyclic analog of benzene has been published previously, we attempted to determine the cause of the difference in conduction behavior by counting the walks on the molecular graph corresponding to the heterocycle on the basis of the derived ZOGF formula. In meta-connected molecular junctions, the contributions of walks of different lengths to the ZOGF formula may or may not cancel each other out. In the former case, the ZOGF becomes zero and QI occurs, thereby reducing the conductance, whereas in the latter case, the conductance is higher. With the introduction of the heteroatom, even-length walks are able to contribute to the ZOGF; additionally, an odd number of passes through the self-loop corresponding to the heteroatom changes the parity of the length of walks on the graph corresponding to the parental structure without the heteroatom. This is the origin of the difference in conductance between molecular junctions consisting of meta-connected benzene and heterocyclic molecules.

See the supplementary material for the ZOGF calculation for the heterocyclic analog of benzene with weighted off-diagonal components and application of the graph theoretic path-counting approach to the para-connection of molecular junctions consisting either of benzene or its heterocyclic analog.

This work was supported by KAKENHI (Grant Nos. JP17K14440, JP17H03117, and JP21K04996) from the Japan Society for the Promotion of Science (JSPS) and the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT) through the MEXT projects of Integrated Research Consortium on Chemical Sciences, the Cooperative Research Program of Network Joint Research Center for Materials and Devices, the Elements Strategy Initiative to Form Core Research Center and by JST-CREST JPMJCR15P5 and JST-Mirai JPMJMI18A2. The computations in this work were primarily performed using the computer facilities at the Research Institute for Information Technology, Kyushu University. Y.T. acknowledges a JSPS Grant-in-Aid for Scientific Research on Innovative Areas [Discrete Geometric Analysis for Materials Design (Grant No. JP20H04643) and Mixed Anion (Grant No. JP19H04700)].