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Equiconducting molecular conductors
Chemical Physics Letters 465 (2008) 142–146
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Equiconducting molecular conductors P.W. Fowler *, B.T. Pickup, T.Z. Todorova Department of Chemistry, University of Sheffield, Sheffield S3 7HF, UK
a r t i c l e
i n f o
Article history: Received 13 August 2008 In final form 20 September 2008 Available online 25 September 2008
a b s t r a c t Non-isomorphic single-molecule conductors can have transmission functions that are identical at all electron energies, within the Hückel tight-binding approximation. These equiconducting cases follow from the graph-theoretical nature of the transmission function, and the existence of isospectral chemical graphs. Systematic constructions of equiconducting molecular graphs are presented, relying on the notions of isospectral pairs and isospectral vertices. These allow identification of non-isomorphic equiconducting connection patterns for a given molecule, and infinite families of equiconducting non-isomorphic molecules. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction The function T(E) describes the transmission of a ballistic electron of energy E through an unsaturated system treated as a single-molecule conductor [1–9]. In the Hückel tight-binding approximation, this function is essentially graph-theoretical in nature, being determined by combinations of characteristic polynomials [9]. For a model single-molecule device where wire-molecule contacts occur through single atoms, T(E) depends on four such polynomials, those of the molecular graph and three vertexdeleted subgraphs, reflecting both the molecular structure and the positions of its connections to the wires. The present note shows that within the tight-binding model it is possible to find distinct (i.e., non-isomorphic) combinations of molecules and connections that yield identical functions T(E). We will call such cases equiconducting. Generic types of equiconducting molecular conductors are identified, and explicit constructions of infinite series of examples are presented. 2. Background A model molecular device is shown in Fig. 1, where left and right wires are connected via single terminal atoms L and R to contact atoms L and R within the molecule. In the SSP tight-binding model [5,6], the transmission function is [9]
TðEÞ ¼
~~t ~sv ~Þ 4 sin qL sin qR ðu iq iq ~ ~þv ~ j2 e Rt e Lu
jeiðqL þqR Þ~s
ð1Þ
where qL and qR are the wavevectors of the electron waves in left ~; v ~ depend on and right contacts, respectively. The functions ~s; ~t; u E, and are (apart from multiplicative parameters) the characteristic
* Corresponding author. E-mail address: P.W.Fowler@sheffield.ac.uk (P.W. Fowler). 0009-2614/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2008.09.048
polynomials s; t; u; v of the molecular graphs G, G with vertex L deleted, G with vertex R deleted, and G with vertex L and R, respectively. The transmission function has an implicit parametric depen In dence on G, L; R and can be written more explicitly as TðE; G; L; RÞ. a shorthand notation the characteristic polynomials are vðG; L; RÞ For ‘degenerate’ contacts, where both sðGÞ; tðG; LÞ; uðG; RÞ; T(E) would be given wires are linked to a single contact atom, L ¼ R, by (1) with t = u and v set to zero. A noteworthy feature of (1) is that the RHS has intrinsic permutational symmetry between the contacts. If it is supposed that left and right wires are detached and re-attached, L to R and R to ¼ L, the predicted transmission is unaffected, i.e., TðE; G; L; RÞ TðE; G; R; LÞ. In general, however, a given molecular graph may have many distinct transmission functions. In the absence of molecular symmetry, a graph with n vertices has n(n 1)/2 distinct pairs of vertices and so at most n(n 1)/2 distinct functions and at most n ‘degenerate’ functions where L ¼ R. where L – R, When the molecular structure has non-trivial point group symmetry, the number of distinct pairs of vertices [10] is given by the number of copies of the totally symmetric representation C0 in the reducible representation [C(v)]2 C(v) derived from C(v), the permutation representation of the vertices. C(v) has character v(R) equal to the number of vertices unshifted under operation R, and the symmetric square [C(v)2] has character v½2 ðRÞ ¼ 1 fv2 ðRÞ þ vðR2 Þg. The number of distinct vertices is given by the 2 number of copies of C0 in C(v). It is clear that these symmetry considerations generally act to reduce the maximum number of distinct transmission functions. However, within the tight-binding approximation, it is possible to find cases where connection patterns are unrelated by symmetry but nonetheless yield identical transmission functions. Furthermore, cases can be found where different molecules give identical transmission functions. These observations lead to the idea of equiconducting molecular conductors, the topic of the present Letter.
P.W. Fowler et al. / Chemical Physics Letters 465 (2008) 142–146
αL
αL βL
_
L
β LL
_ L
αR
α
α
M
_ R
_
β RR
R
αR βR
Fig. 1. Schematic molecular electronic device represented in a tight-binding approach. bL ; bR are the resonance integrals on the wire, and bLL ; bRR the resonance integrals for the connections between terminal wire atoms (L, R) and contacting In the SSP Hamiltonian [9] model, the effects of the atoms in the molecule ð L; RÞ. semi-infinite wires are replaced by complex weights on single source and sink vertices L and R. The scaled characteristic polynomials used in Eq. (1) are ~R u; v ~L b ~L ¼ b2 =bL ; b ~R ¼ b2 =bR [9]. ~R v where b ~L t; u ~¼b ~¼b ~s ¼ s; ~t ¼ b LL RR
3. Isospectrality It will be useful to consider isospectrality of graphs, vertices and pairs of vertices [11–14]. The notation G i, G ij and G i j is used for graphs in which, respectively, vertex i, the edge between vertices i and j and the pair of vertices i and j have been deleted from G. A graph with a single distinguished vertex i is said to be rooted at i. The notation GiH denotes the graph formed by coalescence of G and H at a single common vertex i. Likewise, Gi1H1i2H2 denotes the graph formed by coalescence of i1 with the root of H1 and i2 with the root of H2, i1 and i2 being vertices of G. The characteristic polynomial of graph G will be denoted u(G); the four vðG; L; RÞ used in characteristic polynomials sðGÞ; tðG; LÞ; uðG; RÞ; (1) are then u(G), uðG LÞ; /ðG RÞ, and /ðG L RÞ, respectively. Aut(G) is the automorphism group of G, i.e., the group formed by those vertex permutations that preserve all edges of G. Two graphs G1 and G2 are said to be isospectral [15] if they have the same characteristic polynomials (and hence the same adjacency eigenvalues). Of course, this is trivially the case if the graphs are isomorphic, (i.e., identical apart possibly from labelling of the vertices). The interesting case is when G1 and G2 are nonisomorphic. Two vertices i1, i2 e G are said to be isospectral if the graphs produced by deletion of each vertex in turn are isospectral, i.e., if u(G i1) = u(G i2). This property implies that rooting of any arbitrary graph H at i1 and i2 in turn also produces isospectral graphs [12]. Again the interesting case is where i1 and i2 are non-equivalent under Aut(G), as then Gi1 and Gi2 are typically non-isomor-
143
phic. Isospectrality can also be defined for vertices i1 e G1, i2 e G2, in different, but necessarily isospectral [12], graphs G1, G2; the defining relation is then u(G1 i1) = u(G2 i2). Some authors distinguish the case where the two vertices belong to the same graph by the term endospectrality [13], and some insist on restricting isospectrality to non-equivalent vertices of a single graph ([12] but not [11]). A related concept is that of pseudosimilarity of vertices [16]: two vertices i1 and i2 that are not related by an automorphism are pseudosimilar if the graphs G i1 and G i2 are isomorphic; pseudosimilar vertices are clearly also isospectral, but isospectral vertices are not necessarily pseudosimilar. We can also build a notion of isospectrality of pairs [12]. Suppose i1 and i2 are isospectral vertices of G, as are j1 and j2. Then, u(G i1) = u(G i2) and u(G j1) = u(G j2). If, in addition to these necessary consequences of vertex isospectrality, u(G i1 j1) = u(G i2 j2), then {i1, j1} and {i2, j2} are said to be isospectral pairs. In analogy with isospectrality of vertices, it can be shown [14] that this definition implies that arbitrary perturbation in turn of corresponding vertices of the pairs {i1, j1} and {i2, j2} also produces isospectral graphs. Again, the interesting cases are those where the two pairs are not equivalent under Aut(G). The special case of what might be called ‘hinged’ isospectral pairs occurs when i1 and i2 are identical, so that u(G i1) = u(G i2) becomes trivially true, and the two conditions are then u(G j1) = u(G j2) and u(G i1 j1) = u(G i1 j2). The definition of isospectrality of pairs can be extended to the case where {i1, j1} e G1 and {i2, j2} e G2 with G1 and G2 different graphs. Fig. 2 illustrates some well known molecular graphs that contain isospectral vertices and pairs. (Many other examples are listed in [14]). G(1) is the skeleton of styrene, G(2) and G(3) are the ‘Zivkovic´ pair’ of isospectral graphs [17], G(4) and G(5) are the skeletons of isospectral pentalene derivatives [18], derived by single-vertex deletions from G(6). Details of the isospectrality properties of these graphs and some of their combinations are listed in Table 1. Many other 4. Isospectrality and equiconduction The relevance of these definitions to molecular conductors arises from the following simple idea: to devise two arrangements that have identical transmission at all energies, it is sufficient to i ði ¼ 1; 2Þ such that the four characteristic Li ; R find choices Gi ; polynomials, in the notation of [9], obey pairwise equalities
Fig. 2. Carbon skeletons of molecules discussed in this Letter. In the text, the corresponding graphs are denoted G(1)–G(6).
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Table 1 Isospectral vertices and pairs in graphs G(1)–G(6) and between selected combinations Graph
Isospectral vertices
Isospectral pairs
G(1) G(2) G(3) G(4) G(5) G(6)
2 and 6 6 and 9 None None None 2 and 5; 10 and 11
{4, 2} and {4, 6} None None None None {2, 8} and {5, 8}; {2, 10}and {5, 11}; {2, 11}and {5, 10}; {8, 10}and {8, 11}
G(2) and G(3)
G(4) and G(5)
2 in G(2) and 3 in G(3); 4 in G(2) and 4 in G(3); 6 in G(2) and 2 in G(3); 9 in G(2) and 2 in G(3); 10 in G(2) and 1 in G(3);
2 in G(4) and 1 in G(5); 5 in G(4) and 2 in G(5); 8 in G(4) and 4 in G(5); 10 in G(4) and 9 in G(5);
{2, 4} in G(2) and {3, 5} in G(3); {2, 6} in G(2) and {6, 2} in G(3); {2, 9} in G(2) and {3, 2} in G(3); {2, 10} in G(2) and {3, 1} in G(3); {4, 6} in G(2) and {4, 2} in G(3); {4, 8} in G(2) and {4, 8} in G(3); {4, 9} in G(2) and {4, 9} in G(3); {4, 10} in G(2) and {5, 1} in G(3); {6, 9} in G(2) and {2, 9} in G(3); {6, 10} in G(2) and {2, 10} in G(3); {9, 10} in G(2) and {2, 1} in G(3). {2, 5} in G(4) and {1, 6} in G(5); {2, 8} in G(4) and {1, 4} in G(5); {2, 10} in G(4) and {1, 9} in G(5); {5, 8} in G(4) and {2, 8} in G(5); {5, 10} in G(4) and {2, 10} in G(5); {8, 10} in G(4) and {4, 9} in G(5)
The vertex numbering for each graph is shown in Fig. 2. The notation i1 and i2 means that the two vertices i1 and i2 are isospectral; the notation {i1, j1} and {i2, j2} means that pairs {i1, j1} and {i2, j2} are isospectral, with i1 corresponding to j1, and i2 to j2. Vertices (pairs) that are equivalent under the automorphism group are not listed.
sðG1 Þ ¼ sðG2 Þ tðG1 ; L1 Þ ¼ tðG2 ; L2 Þ
ð3Þ
1 Þ ¼ uðG2 ; R 2Þ uðG1 ; R 1 Þ ¼ vðG2 ; L2 ; R 2 Þ vðG1 ; L1 ; R
ð5Þ
equal numbers of bonds. The converse is of course not true: pairs connected by paths of equal length are not always isospectral. An example of equiconductors of this kind is furnished by the styrene molecule, 1 (Fig. 2). In G(1), vertices 2 and 6, unrelated by symmetry, are isospectral [14], and hinged pairs {4, 2} and {4, 6}, again unrelated by symmetry, are isospectral [11]. Equiconducting devices based on these hinged pairs are illustrated in Fig. 3a. The four characteristic polynomials needed to construct for these devices are TðE; G; L; RÞ
uðGð1ÞÞ ¼ ðE2 1ÞðE2 2ÞðE4 5E2 þ 2Þ uðGð1Þ 2Þ ¼ uðGð1Þ 6Þ ¼ EðE2 1Þ2 ðE2 4Þ uðGð1Þ 2 4Þ ¼ uðGð1Þ 4 6Þ ¼ E2 ðE2 1ÞðE2 3Þ The transmission functions for symmetric devices (those in ~R ¼ c1 bÞ based ~L ¼ b which the wires have identical parameters b on the hinged pairs are
TðE; Gð1Þ; 2; 4Þ ¼ TðE; Gð1Þ; 4; 6Þ ¼ N=D where
N ¼ ð4c2 E2 ÞE2 ðE2 1Þ2 ðE2 2Þ2 and
_ L
a
_ L
_ R _ R
ð2Þ ð4Þ
_ L
b
_ R
At this point it is not specified whether molecular graphs G1 and G2 are the same or different.
_ L
4.1. Single-molecule equiconductors We consider equiconductors arising from different connection patterns for a given molecule. In this case, G1 ¼ G2 ¼ G. An easy way to generate equiconducting patterns of connection of G follows from the definition of isospectral pairs. Given two isospectral pairs {i1, j1} and {i2, j2}, equiconductors are constructed by choosing 1 ¼ j for one device and 2 ¼ j for the other. EviL1 ¼ i1 ; R L2 ¼ i2 ; R 1 2 dently, then
_ R
_ L
c
_ R
_ R
_ L
sðG1 Þ ¼ sðG2 Þ ¼ uðGÞ and
tðG1 ; L1 Þ ¼ uðG i1 Þ ¼ uðG i2 Þ ¼ tðG2 ; L2 Þ
d
_ L
_ L _
as vertices i1 and i2 are isospectral, and
R
1 Þ ¼ uðG j Þ ¼ uðG j Þ ¼ uðG2 ; R 2Þ uðG1 ; R 1 2 as j1 and j2 are isospectral, and finally
1 Þ ¼ uðG i1 j Þ ¼ uðG i2 j Þ ¼ vðG2 ; L2 ; R 2 Þ vðG1 ; L1 ; R 1 2 as the pairs {i1, j1} and {i2, j2} are isospectral. A similar argument applies for hinged pairs, where i1 and i2 are identical. Notice that the shortest path from i1 to j1 has the same length as the shortest path from i2 to j2 (See Theorem 4.2 of [14]), so isospectral pairs within a given molecule are always connected across
_
R Fig. 3. Pairs of equiconductors produced by the constructions described in the text: (a) using isospectral pairs of connections in a single molecule, (b) attaching fragments to isospectral vertices of a single molecule, (c) attaching distinct fragments to isospectral pairs in a single molecule, (d) attaching fragments to each of two pairs that are isospectral between two molecules.
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The transmission function TðE; Gð1Þ; 2; 4Þ ¼ TðE; Gð1Þ; 4; 6Þ is plotted in Fig. 4. For unsymmetrical devices, the expressions are more complicated, but the transmissions remain equal to each other. An easy construction for ‘degenerate’ molecular devices (those uses isospectrality of vertices rather than pairs. If where L ¼ R) i1 and i2 are isospectral vertices of G, then TðE; G; i1 ; i1 Þ ¼ TðE; G; i2 ; i2 Þ, and the devices formed when both wires are attached to i1 and then to i2. in turn, are equiconductors. For the example of styrene, choice of i1 = 2, i2 = 6 gives, on insertion of t ¼ u ¼ uðGð1Þ 2Þ ¼ uðGð1Þ 6Þ and v = 0 into (1),
TðE; Gð1Þ; 2; 2Þ ¼ TðE; Gð1Þ; 6; 6Þ ¼ N 0 =D0 with
N0 ¼ ð4c2 E2 ÞE2 ðE2 4Þ2 ðE2 1Þ2 and Fig. 4. Transmission curve for the two equiconductors derived from styrene, G(1), by making connections across isospectral pairs {2, 4} and {6, 4}, with parameters ~R ¼ b2 =bR ¼ c1 b; c ¼ 1:4; b ¼ 1. Positions of molecular eigenvalues ~L ¼ b2 =bL ¼ b b LL RR are shown as solid circles.
D0 ¼ ðE2 2Þ2 ðE4 5E2 þ 2Þ2 c4 2E2 ðE2 1Þ2 ðE2 3ÞðE2 4Þ ðE4 6E2 þ 4Þc2 4.2. Single-parent families of equiconductors
D ¼ u2 ðGð1ÞÞc4 E2 ð2E14 30E12 þ 177E10 532E8 þ 881E6 807E4 þ 380E2 72Þc2 þ E4 ðE2 1ÞðE2 3ÞðE8 10E6 4
In the previous example, the equiconductors were based on non-isomorphic connections of a single molecular graph. It is also
2
þ 32E 39E þ 15Þ:
a
a
b
L
a
R
a
a
A
A
A
A
A
L
R
A
B
A
B
A
L
R
B
L
b
R
L
R
b
R
L
b
L
b
B
B
B
B
R
Fig. 5. Infinite families of equiconductors: (a) Equiconductors based on attachment of fragments to isospectral vertices 5 in G(4) and 2 in G(5) (equivalently, to isospectral vertices 10 and 11 of G(6)). a and b are arbitrary, but fixed vertices of a molecular graph D. For arbitrary D, all four composite molecules are non-isomorphic but have equal transmission at all energies; (b) ‘Polymeric’ equiconductors. A and B are copies of G(4) and G(5) rooted at vertices i1 and i2 (where i1 in G(4) is isospectral with i2 in G(5)) and fused by these vertices to the backbone. The sequence is arbitrary: the transmission function of the fully substituted backbone depends on its length, but not on the ordering or partition of the side units between A and B.
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P.W. Fowler et al. / Chemical Physics Letters 465 (2008) 142–146
possible to construct examples in which equiconducting singlemolecule devices are based on molecules with non-isomorphic graphs. One construction relies on finding isospectral vertices i1, i2 of a base graph G. Take an arbitrary rooted graph H in which vertices L and R (not necessarily distinct) have been fixed, and form the graphs Gi1H and Gi2H. By the definition of vertex isospectrality, the two product graphs are isospectral, u(Gi1H0 ) = u(Gi2H0 ) for H L R, and the corresponding devices L; H R; H0 ¼ H; H are equiconducting. For example, choice of G(2) as the base graph, with H the 3 ¼ 3 and coalescence of vertex chain rooted at vertex 1 with L ¼ 2; R 1 of H with vertices 6 and 9 of G(2), in turn, gives the equiconductors shown in Fig. 3b. A second construction relies on finding isospectral pairs {i1, j1}, {i2, j2} in the base graph G. Take two arbitrary rooted graphs H1 (not necessarily distinct) freely from the and H2 and choose L; R combined set of vertices of H1 and H2. Form Gi1H1 and Gj2H2: their characteristic polynomials satisfy (2)–(5), and the devices are easily shown to be equiconductors. Formally speaking, this construction includes the case of the single-molecule equiconductors, if H1 = H2 is a single vertex. For example, the two graphs based on coalescence of 3- and 4chains to the hinged isospectral pairs of styrene are equiconductors (see Fig. 3c). 4.3. Multiple-parent families of equiconductors So far, isospectrality of vertices and pairs within a single base molecular graph has been considered as a means of finding equiconductors. In the widest definition, vertices or pairs belonging to different base molecules may be isospectral [12,13]. For example, vertices 5 in G(4) and 2 in G(5) are isospectral in this sense, as are pairs {5, 10} in G(4) and {2, 10} in G(5). Table 1 lists vertices and pairs that are isospectral between G(2) and G(3) and between G(3) and G(4). This ‘intermolecular’ isospectrality raises several new possibilities for vertices and pairs. For example, it follows from the definition that if i1 and i2 in G1 are isospectral, and i1 is isospectral with i3 in G2, then so is i2. Similarly, if {i1, j1} and {i2, j2} are isospectral pairs in G1 and one of them is isospectral with {i3, j3} in G2, then so is the other. One construction of equiconductors uses intermolecular isospectral vertices. Attachment of an arbitrary rooted graph H to isospectral vertices i1 in G1 and i2 in G2 leads to equiconducting (not necessarily disL and R devices G1i1H and G2i2H, provided that tinct) are fixed vertices of H. A special case is where H is a single which yields equiconducting degenerate devices vertex L ¼ R, based on G1 and G2. Another construction uses pairs that are isospectral in the intermolecular sense. Isospectral pairs {i1, j1} in G1 and {i2, j2} in G2 to lead to equiconducting devices G1i1H1j1H2 and G2i2H1j2H2 for arbitrary rooted graphs H1 and H2 with fixed choice of L and (not necessarily distinct) from the combined vertex sets R of H1 and H2. Fig. 3d shows such an example constructed from G(2) and G(3), with H1 and H2 containing 2 and 7 vertices, respectively.
5. Ringing the changes Given the various constructions described in the preceding section, it is clear that there are many ways to use them in combination to produce more complicated instances of equiconducting devices. Fig. 5a shows a simple example in which isospectrality between vertices of G(4) and G(5), or equivalently isospectrality of vertices within G(6), is exploited to produce an infinite family of quartets of equiconductors. There are many other possibilities: all ‘polymer’ devices produced by coalescing rooted copies of G(4) and G(5) in any order onto a polyene backbone (Fig. 5b) is equiconducting if the root vertices are isospectral between G(4) and G(5). (The proof is by induction: consider first the naked polyene chain as an attachment H that could be fused in turn to the root in either G(4) and G(5). Both side-chains give equiconductors, by the definition of isospectral vertices. Now take the polyenes plus side-chain as the attachment, and so on.) 6. Conclusion This study has shown that, within the simplified tight-binding model, there are multiple possibilities for equiconducting molecular devices. The mathematical concept of graph, vertex and pair isospectrality can be used as a design principle. More sophisticated treatments that include, e.g., many-electron interactions, r–p interaction and a more realistic treatment of the interaction of the molecule and the metal, will lift this degeneracy of transmission functions, but it is hoped that the results here give some intuitive understanding which may be useful in real cases. Acknowledgements PWF thanks the Royal Society/Wolfson Research Merit Award Scheme for financial support. TZT thanks the University of Sheffield for a Postgraduate Studentship. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
[17] [18]
C. Joachim, M.A. Ratner, Proc. Natl. Acad. Sci. 102 (2005) 8801. V. Mujica, M. Kemp, M.A. Ratner, J. Chem. Phys. 101 (1994) 6849. V. Mujica, M. Kemp, M.A. Ratner, J. Chem. Phys. 101 (1994) 6856. M. Ernzerhof, F. Goyer, M. Zhuang, P. Rocheleau, J. Chem. Theory Comput. 2 (2006) 1291. F. Goyer, M. Ernzerhof, M. Zhuang, J. Chem. Phys. 126 (2007) 144104. M. Ernzerhof, J. Chem. Phys. 127 (2007) 204709. S.K. Maiti, Phys. Lett. A 366 (2007) 114. S.K. Maiti, Chem. Phys. 331 (2007) 254. B.T. Pickup, P.W. Fowler, Chem. Phys. Lett. 459 (2008) 198. P.W. Fowler, J. Chem. Soc. Faraday Trans. 91 (1995) 2241. G. Rücker, C. Rücker, J. Chem. Inf. Comput. Sci. 31 (1991) 422. C. Rücker, G. Rücker, J. Math. Chem. 9 (1992) 207. M. Randic´, M. Barysz, J. Nowalowski, S. Nikolic´, N. Trinajstic´, J. Mol. Struct. (THEOCHEM) 185 (1989) 95. W.C. Herndon, M.L. Ellzey Jr. Tetrahedron 31 (1975) 99. D.M. Cvetkovic´, M. Doob, H. Sachs, Spectra of Graphs, Academic Press, New York, 1979. J. Lauri, R. Scapellato, Topics in Graph Automorphisms and Reconstruction, London Mathematical Society Student Texts 54, Cambridge University Press, Cambridge, 2003. T. Zivkovic´, N. Trinajstic´, M. Randic´, Mol. Phys. 30 (1975) 517–532. J.R. Dias, Molecular Orbital Calculations Using Chemical Graph Theory, Springer, Berlin, 1993 (p. 28. See also example 26 in [14]).
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Equiconducting molecular conductors P.W. Fowler *, B.T. Pickup, T.Z. Todorova Department of Chemistry, University of Sheffield, Sheffield S3 7HF, UK
a r t i c l e
i n f o
Article history: Received 13 August 2008 In final form 20 September 2008 Available online 25 September 2008
a b s t r a c t Non-isomorphic single-molecule conductors can have transmission functions that are identical at all electron energies, within the Hückel tight-binding approximation. These equiconducting cases follow from the graph-theoretical nature of the transmission function, and the existence of isospectral chemical graphs. Systematic constructions of equiconducting molecular graphs are presented, relying on the notions of isospectral pairs and isospectral vertices. These allow identification of non-isomorphic equiconducting connection patterns for a given molecule, and infinite families of equiconducting non-isomorphic molecules. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction The function T(E) describes the transmission of a ballistic electron of energy E through an unsaturated system treated as a single-molecule conductor [1–9]. In the Hückel tight-binding approximation, this function is essentially graph-theoretical in nature, being determined by combinations of characteristic polynomials [9]. For a model single-molecule device where wire-molecule contacts occur through single atoms, T(E) depends on four such polynomials, those of the molecular graph and three vertexdeleted subgraphs, reflecting both the molecular structure and the positions of its connections to the wires. The present note shows that within the tight-binding model it is possible to find distinct (i.e., non-isomorphic) combinations of molecules and connections that yield identical functions T(E). We will call such cases equiconducting. Generic types of equiconducting molecular conductors are identified, and explicit constructions of infinite series of examples are presented. 2. Background A model molecular device is shown in Fig. 1, where left and right wires are connected via single terminal atoms L and R to contact atoms L and R within the molecule. In the SSP tight-binding model [5,6], the transmission function is [9]
TðEÞ ¼
~~t ~sv ~Þ 4 sin qL sin qR ðu iq iq ~ ~þv ~ j2 e Rt e Lu
jeiðqL þqR Þ~s
ð1Þ
where qL and qR are the wavevectors of the electron waves in left ~; v ~ depend on and right contacts, respectively. The functions ~s; ~t; u E, and are (apart from multiplicative parameters) the characteristic
* Corresponding author. E-mail address: P.W.Fowler@sheffield.ac.uk (P.W. Fowler). 0009-2614/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2008.09.048
polynomials s; t; u; v of the molecular graphs G, G with vertex L deleted, G with vertex R deleted, and G with vertex L and R, respectively. The transmission function has an implicit parametric depen In dence on G, L; R and can be written more explicitly as TðE; G; L; RÞ. a shorthand notation the characteristic polynomials are vðG; L; RÞ For ‘degenerate’ contacts, where both sðGÞ; tðG; LÞ; uðG; RÞ; T(E) would be given wires are linked to a single contact atom, L ¼ R, by (1) with t = u and v set to zero. A noteworthy feature of (1) is that the RHS has intrinsic permutational symmetry between the contacts. If it is supposed that left and right wires are detached and re-attached, L to R and R to ¼ L, the predicted transmission is unaffected, i.e., TðE; G; L; RÞ TðE; G; R; LÞ. In general, however, a given molecular graph may have many distinct transmission functions. In the absence of molecular symmetry, a graph with n vertices has n(n 1)/2 distinct pairs of vertices and so at most n(n 1)/2 distinct functions and at most n ‘degenerate’ functions where L ¼ R. where L – R, When the molecular structure has non-trivial point group symmetry, the number of distinct pairs of vertices [10] is given by the number of copies of the totally symmetric representation C0 in the reducible representation [C(v)]2 C(v) derived from C(v), the permutation representation of the vertices. C(v) has character v(R) equal to the number of vertices unshifted under operation R, and the symmetric square [C(v)2] has character v½2 ðRÞ ¼ 1 fv2 ðRÞ þ vðR2 Þg. The number of distinct vertices is given by the 2 number of copies of C0 in C(v). It is clear that these symmetry considerations generally act to reduce the maximum number of distinct transmission functions. However, within the tight-binding approximation, it is possible to find cases where connection patterns are unrelated by symmetry but nonetheless yield identical transmission functions. Furthermore, cases can be found where different molecules give identical transmission functions. These observations lead to the idea of equiconducting molecular conductors, the topic of the present Letter.
P.W. Fowler et al. / Chemical Physics Letters 465 (2008) 142–146
αL
αL βL
_
L
β LL
_ L
αR
α
α
M
_ R
_
β RR
R
αR βR
Fig. 1. Schematic molecular electronic device represented in a tight-binding approach. bL ; bR are the resonance integrals on the wire, and bLL ; bRR the resonance integrals for the connections between terminal wire atoms (L, R) and contacting In the SSP Hamiltonian [9] model, the effects of the atoms in the molecule ð L; RÞ. semi-infinite wires are replaced by complex weights on single source and sink vertices L and R. The scaled characteristic polynomials used in Eq. (1) are ~R u; v ~L b ~L ¼ b2 =bL ; b ~R ¼ b2 =bR [9]. ~R v where b ~L t; u ~¼b ~¼b ~s ¼ s; ~t ¼ b LL RR
3. Isospectrality It will be useful to consider isospectrality of graphs, vertices and pairs of vertices [11–14]. The notation G i, G ij and G i j is used for graphs in which, respectively, vertex i, the edge between vertices i and j and the pair of vertices i and j have been deleted from G. A graph with a single distinguished vertex i is said to be rooted at i. The notation GiH denotes the graph formed by coalescence of G and H at a single common vertex i. Likewise, Gi1H1i2H2 denotes the graph formed by coalescence of i1 with the root of H1 and i2 with the root of H2, i1 and i2 being vertices of G. The characteristic polynomial of graph G will be denoted u(G); the four vðG; L; RÞ used in characteristic polynomials sðGÞ; tðG; LÞ; uðG; RÞ; (1) are then u(G), uðG LÞ; /ðG RÞ, and /ðG L RÞ, respectively. Aut(G) is the automorphism group of G, i.e., the group formed by those vertex permutations that preserve all edges of G. Two graphs G1 and G2 are said to be isospectral [15] if they have the same characteristic polynomials (and hence the same adjacency eigenvalues). Of course, this is trivially the case if the graphs are isomorphic, (i.e., identical apart possibly from labelling of the vertices). The interesting case is when G1 and G2 are nonisomorphic. Two vertices i1, i2 e G are said to be isospectral if the graphs produced by deletion of each vertex in turn are isospectral, i.e., if u(G i1) = u(G i2). This property implies that rooting of any arbitrary graph H at i1 and i2 in turn also produces isospectral graphs [12]. Again the interesting case is where i1 and i2 are non-equivalent under Aut(G), as then Gi1 and Gi2 are typically non-isomor-
143
phic. Isospectrality can also be defined for vertices i1 e G1, i2 e G2, in different, but necessarily isospectral [12], graphs G1, G2; the defining relation is then u(G1 i1) = u(G2 i2). Some authors distinguish the case where the two vertices belong to the same graph by the term endospectrality [13], and some insist on restricting isospectrality to non-equivalent vertices of a single graph ([12] but not [11]). A related concept is that of pseudosimilarity of vertices [16]: two vertices i1 and i2 that are not related by an automorphism are pseudosimilar if the graphs G i1 and G i2 are isomorphic; pseudosimilar vertices are clearly also isospectral, but isospectral vertices are not necessarily pseudosimilar. We can also build a notion of isospectrality of pairs [12]. Suppose i1 and i2 are isospectral vertices of G, as are j1 and j2. Then, u(G i1) = u(G i2) and u(G j1) = u(G j2). If, in addition to these necessary consequences of vertex isospectrality, u(G i1 j1) = u(G i2 j2), then {i1, j1} and {i2, j2} are said to be isospectral pairs. In analogy with isospectrality of vertices, it can be shown [14] that this definition implies that arbitrary perturbation in turn of corresponding vertices of the pairs {i1, j1} and {i2, j2} also produces isospectral graphs. Again, the interesting cases are those where the two pairs are not equivalent under Aut(G). The special case of what might be called ‘hinged’ isospectral pairs occurs when i1 and i2 are identical, so that u(G i1) = u(G i2) becomes trivially true, and the two conditions are then u(G j1) = u(G j2) and u(G i1 j1) = u(G i1 j2). The definition of isospectrality of pairs can be extended to the case where {i1, j1} e G1 and {i2, j2} e G2 with G1 and G2 different graphs. Fig. 2 illustrates some well known molecular graphs that contain isospectral vertices and pairs. (Many other examples are listed in [14]). G(1) is the skeleton of styrene, G(2) and G(3) are the ‘Zivkovic´ pair’ of isospectral graphs [17], G(4) and G(5) are the skeletons of isospectral pentalene derivatives [18], derived by single-vertex deletions from G(6). Details of the isospectrality properties of these graphs and some of their combinations are listed in Table 1. Many other 4. Isospectrality and equiconduction The relevance of these definitions to molecular conductors arises from the following simple idea: to devise two arrangements that have identical transmission at all energies, it is sufficient to i ði ¼ 1; 2Þ such that the four characteristic Li ; R find choices Gi ; polynomials, in the notation of [9], obey pairwise equalities
Fig. 2. Carbon skeletons of molecules discussed in this Letter. In the text, the corresponding graphs are denoted G(1)–G(6).
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Table 1 Isospectral vertices and pairs in graphs G(1)–G(6) and between selected combinations Graph
Isospectral vertices
Isospectral pairs
G(1) G(2) G(3) G(4) G(5) G(6)
2 and 6 6 and 9 None None None 2 and 5; 10 and 11
{4, 2} and {4, 6} None None None None {2, 8} and {5, 8}; {2, 10}and {5, 11}; {2, 11}and {5, 10}; {8, 10}and {8, 11}
G(2) and G(3)
G(4) and G(5)
2 in G(2) and 3 in G(3); 4 in G(2) and 4 in G(3); 6 in G(2) and 2 in G(3); 9 in G(2) and 2 in G(3); 10 in G(2) and 1 in G(3);
2 in G(4) and 1 in G(5); 5 in G(4) and 2 in G(5); 8 in G(4) and 4 in G(5); 10 in G(4) and 9 in G(5);
{2, 4} in G(2) and {3, 5} in G(3); {2, 6} in G(2) and {6, 2} in G(3); {2, 9} in G(2) and {3, 2} in G(3); {2, 10} in G(2) and {3, 1} in G(3); {4, 6} in G(2) and {4, 2} in G(3); {4, 8} in G(2) and {4, 8} in G(3); {4, 9} in G(2) and {4, 9} in G(3); {4, 10} in G(2) and {5, 1} in G(3); {6, 9} in G(2) and {2, 9} in G(3); {6, 10} in G(2) and {2, 10} in G(3); {9, 10} in G(2) and {2, 1} in G(3). {2, 5} in G(4) and {1, 6} in G(5); {2, 8} in G(4) and {1, 4} in G(5); {2, 10} in G(4) and {1, 9} in G(5); {5, 8} in G(4) and {2, 8} in G(5); {5, 10} in G(4) and {2, 10} in G(5); {8, 10} in G(4) and {4, 9} in G(5)
The vertex numbering for each graph is shown in Fig. 2. The notation i1 and i2 means that the two vertices i1 and i2 are isospectral; the notation {i1, j1} and {i2, j2} means that pairs {i1, j1} and {i2, j2} are isospectral, with i1 corresponding to j1, and i2 to j2. Vertices (pairs) that are equivalent under the automorphism group are not listed.
sðG1 Þ ¼ sðG2 Þ tðG1 ; L1 Þ ¼ tðG2 ; L2 Þ
ð3Þ
1 Þ ¼ uðG2 ; R 2Þ uðG1 ; R 1 Þ ¼ vðG2 ; L2 ; R 2 Þ vðG1 ; L1 ; R
ð5Þ
equal numbers of bonds. The converse is of course not true: pairs connected by paths of equal length are not always isospectral. An example of equiconductors of this kind is furnished by the styrene molecule, 1 (Fig. 2). In G(1), vertices 2 and 6, unrelated by symmetry, are isospectral [14], and hinged pairs {4, 2} and {4, 6}, again unrelated by symmetry, are isospectral [11]. Equiconducting devices based on these hinged pairs are illustrated in Fig. 3a. The four characteristic polynomials needed to construct for these devices are TðE; G; L; RÞ
uðGð1ÞÞ ¼ ðE2 1ÞðE2 2ÞðE4 5E2 þ 2Þ uðGð1Þ 2Þ ¼ uðGð1Þ 6Þ ¼ EðE2 1Þ2 ðE2 4Þ uðGð1Þ 2 4Þ ¼ uðGð1Þ 4 6Þ ¼ E2 ðE2 1ÞðE2 3Þ The transmission functions for symmetric devices (those in ~R ¼ c1 bÞ based ~L ¼ b which the wires have identical parameters b on the hinged pairs are
TðE; Gð1Þ; 2; 4Þ ¼ TðE; Gð1Þ; 4; 6Þ ¼ N=D where
N ¼ ð4c2 E2 ÞE2 ðE2 1Þ2 ðE2 2Þ2 and
_ L
a
_ L
_ R _ R
ð2Þ ð4Þ
_ L
b
_ R
At this point it is not specified whether molecular graphs G1 and G2 are the same or different.
_ L
4.1. Single-molecule equiconductors We consider equiconductors arising from different connection patterns for a given molecule. In this case, G1 ¼ G2 ¼ G. An easy way to generate equiconducting patterns of connection of G follows from the definition of isospectral pairs. Given two isospectral pairs {i1, j1} and {i2, j2}, equiconductors are constructed by choosing 1 ¼ j for one device and 2 ¼ j for the other. EviL1 ¼ i1 ; R L2 ¼ i2 ; R 1 2 dently, then
_ R
_ L
c
_ R
_ R
_ L
sðG1 Þ ¼ sðG2 Þ ¼ uðGÞ and
tðG1 ; L1 Þ ¼ uðG i1 Þ ¼ uðG i2 Þ ¼ tðG2 ; L2 Þ
d
_ L
_ L _
as vertices i1 and i2 are isospectral, and
R
1 Þ ¼ uðG j Þ ¼ uðG j Þ ¼ uðG2 ; R 2Þ uðG1 ; R 1 2 as j1 and j2 are isospectral, and finally
1 Þ ¼ uðG i1 j Þ ¼ uðG i2 j Þ ¼ vðG2 ; L2 ; R 2 Þ vðG1 ; L1 ; R 1 2 as the pairs {i1, j1} and {i2, j2} are isospectral. A similar argument applies for hinged pairs, where i1 and i2 are identical. Notice that the shortest path from i1 to j1 has the same length as the shortest path from i2 to j2 (See Theorem 4.2 of [14]), so isospectral pairs within a given molecule are always connected across
_
R Fig. 3. Pairs of equiconductors produced by the constructions described in the text: (a) using isospectral pairs of connections in a single molecule, (b) attaching fragments to isospectral vertices of a single molecule, (c) attaching distinct fragments to isospectral pairs in a single molecule, (d) attaching fragments to each of two pairs that are isospectral between two molecules.
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The transmission function TðE; Gð1Þ; 2; 4Þ ¼ TðE; Gð1Þ; 4; 6Þ is plotted in Fig. 4. For unsymmetrical devices, the expressions are more complicated, but the transmissions remain equal to each other. An easy construction for ‘degenerate’ molecular devices (those uses isospectrality of vertices rather than pairs. If where L ¼ R) i1 and i2 are isospectral vertices of G, then TðE; G; i1 ; i1 Þ ¼ TðE; G; i2 ; i2 Þ, and the devices formed when both wires are attached to i1 and then to i2. in turn, are equiconductors. For the example of styrene, choice of i1 = 2, i2 = 6 gives, on insertion of t ¼ u ¼ uðGð1Þ 2Þ ¼ uðGð1Þ 6Þ and v = 0 into (1),
TðE; Gð1Þ; 2; 2Þ ¼ TðE; Gð1Þ; 6; 6Þ ¼ N 0 =D0 with
N0 ¼ ð4c2 E2 ÞE2 ðE2 4Þ2 ðE2 1Þ2 and Fig. 4. Transmission curve for the two equiconductors derived from styrene, G(1), by making connections across isospectral pairs {2, 4} and {6, 4}, with parameters ~R ¼ b2 =bR ¼ c1 b; c ¼ 1:4; b ¼ 1. Positions of molecular eigenvalues ~L ¼ b2 =bL ¼ b b LL RR are shown as solid circles.
D0 ¼ ðE2 2Þ2 ðE4 5E2 þ 2Þ2 c4 2E2 ðE2 1Þ2 ðE2 3ÞðE2 4Þ ðE4 6E2 þ 4Þc2 4.2. Single-parent families of equiconductors
D ¼ u2 ðGð1ÞÞc4 E2 ð2E14 30E12 þ 177E10 532E8 þ 881E6 807E4 þ 380E2 72Þc2 þ E4 ðE2 1ÞðE2 3ÞðE8 10E6 4
In the previous example, the equiconductors were based on non-isomorphic connections of a single molecular graph. It is also
2
þ 32E 39E þ 15Þ:
a
a
b
L
a
R
a
a
A
A
A
A
A
L
R
A
B
A
B
A
L
R
B
L
b
R
L
R
b
R
L
b
L
b
B
B
B
B
R
Fig. 5. Infinite families of equiconductors: (a) Equiconductors based on attachment of fragments to isospectral vertices 5 in G(4) and 2 in G(5) (equivalently, to isospectral vertices 10 and 11 of G(6)). a and b are arbitrary, but fixed vertices of a molecular graph D. For arbitrary D, all four composite molecules are non-isomorphic but have equal transmission at all energies; (b) ‘Polymeric’ equiconductors. A and B are copies of G(4) and G(5) rooted at vertices i1 and i2 (where i1 in G(4) is isospectral with i2 in G(5)) and fused by these vertices to the backbone. The sequence is arbitrary: the transmission function of the fully substituted backbone depends on its length, but not on the ordering or partition of the side units between A and B.
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possible to construct examples in which equiconducting singlemolecule devices are based on molecules with non-isomorphic graphs. One construction relies on finding isospectral vertices i1, i2 of a base graph G. Take an arbitrary rooted graph H in which vertices L and R (not necessarily distinct) have been fixed, and form the graphs Gi1H and Gi2H. By the definition of vertex isospectrality, the two product graphs are isospectral, u(Gi1H0 ) = u(Gi2H0 ) for H L R, and the corresponding devices L; H R; H0 ¼ H; H are equiconducting. For example, choice of G(2) as the base graph, with H the 3 ¼ 3 and coalescence of vertex chain rooted at vertex 1 with L ¼ 2; R 1 of H with vertices 6 and 9 of G(2), in turn, gives the equiconductors shown in Fig. 3b. A second construction relies on finding isospectral pairs {i1, j1}, {i2, j2} in the base graph G. Take two arbitrary rooted graphs H1 (not necessarily distinct) freely from the and H2 and choose L; R combined set of vertices of H1 and H2. Form Gi1H1 and Gj2H2: their characteristic polynomials satisfy (2)–(5), and the devices are easily shown to be equiconductors. Formally speaking, this construction includes the case of the single-molecule equiconductors, if H1 = H2 is a single vertex. For example, the two graphs based on coalescence of 3- and 4chains to the hinged isospectral pairs of styrene are equiconductors (see Fig. 3c). 4.3. Multiple-parent families of equiconductors So far, isospectrality of vertices and pairs within a single base molecular graph has been considered as a means of finding equiconductors. In the widest definition, vertices or pairs belonging to different base molecules may be isospectral [12,13]. For example, vertices 5 in G(4) and 2 in G(5) are isospectral in this sense, as are pairs {5, 10} in G(4) and {2, 10} in G(5). Table 1 lists vertices and pairs that are isospectral between G(2) and G(3) and between G(3) and G(4). This ‘intermolecular’ isospectrality raises several new possibilities for vertices and pairs. For example, it follows from the definition that if i1 and i2 in G1 are isospectral, and i1 is isospectral with i3 in G2, then so is i2. Similarly, if {i1, j1} and {i2, j2} are isospectral pairs in G1 and one of them is isospectral with {i3, j3} in G2, then so is the other. One construction of equiconductors uses intermolecular isospectral vertices. Attachment of an arbitrary rooted graph H to isospectral vertices i1 in G1 and i2 in G2 leads to equiconducting (not necessarily disL and R devices G1i1H and G2i2H, provided that tinct) are fixed vertices of H. A special case is where H is a single which yields equiconducting degenerate devices vertex L ¼ R, based on G1 and G2. Another construction uses pairs that are isospectral in the intermolecular sense. Isospectral pairs {i1, j1} in G1 and {i2, j2} in G2 to lead to equiconducting devices G1i1H1j1H2 and G2i2H1j2H2 for arbitrary rooted graphs H1 and H2 with fixed choice of L and (not necessarily distinct) from the combined vertex sets R of H1 and H2. Fig. 3d shows such an example constructed from G(2) and G(3), with H1 and H2 containing 2 and 7 vertices, respectively.
5. Ringing the changes Given the various constructions described in the preceding section, it is clear that there are many ways to use them in combination to produce more complicated instances of equiconducting devices. Fig. 5a shows a simple example in which isospectrality between vertices of G(4) and G(5), or equivalently isospectrality of vertices within G(6), is exploited to produce an infinite family of quartets of equiconductors. There are many other possibilities: all ‘polymer’ devices produced by coalescing rooted copies of G(4) and G(5) in any order onto a polyene backbone (Fig. 5b) is equiconducting if the root vertices are isospectral between G(4) and G(5). (The proof is by induction: consider first the naked polyene chain as an attachment H that could be fused in turn to the root in either G(4) and G(5). Both side-chains give equiconductors, by the definition of isospectral vertices. Now take the polyenes plus side-chain as the attachment, and so on.) 6. Conclusion This study has shown that, within the simplified tight-binding model, there are multiple possibilities for equiconducting molecular devices. The mathematical concept of graph, vertex and pair isospectrality can be used as a design principle. More sophisticated treatments that include, e.g., many-electron interactions, r–p interaction and a more realistic treatment of the interaction of the molecule and the metal, will lift this degeneracy of transmission functions, but it is hoped that the results here give some intuitive understanding which may be useful in real cases. Acknowledgements PWF thanks the Royal Society/Wolfson Research Merit Award Scheme for financial support. TZT thanks the University of Sheffield for a Postgraduate Studentship. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
[17] [18]
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