# A Simple Model of Ballistic Conduction in Multi-Lead Molecular Devices
## 6 Results
### Complete Devices of Types CSD and CBSD  
Figure 7 shows calculated curves \(T(E)\) for molecular devices of CSD and CBSD types based on cycles and linear polyenes. The first three panels show the transmission curves for CSD based on 6-, 8-, and 10-membered rings (Figure 7a-c). They show qualitatively similar behaviour, with broad maxima, shoulders, and minima associated with the eigenvalues of the respective graphs. The pattern of a central minimum for \((4N+2)\)-rings and a central maximum for \(4N\)-rings persists to larger ring sizes. Traces of the constituent devices are apparent in the lead contributions, e.g., compare ortho, meta, and para benzene devices (Figure 4a-c) with the CSD in Figure 7a. As noted in Section 4.1, transmission is non-zero across the whole band window. However, the summed current can include some very low sink-lead contributions. In the case of the 10-ring, for example, the lead transmissions at the Fermi level for leads 2, 3, 4, 5, and 6 are 0.1581, 0.0445, 0.0853, 0.0042, and 0.0674, respectively. Figure 6: Transmission curves for devices based on pentalene: Hückel theory results for (**a**) 1,3,5,7,; (**b**) 1,3,5-; (**c**) 1,3,5-; (**d**) 1,3,7-; (**e**) 1,3-; (**f**) 1,5-; and (**g**) 1,7-devices. Lead 1, at vertex 3 in all cases, is the source with transmission equal to the total envelope, shown in black. Transmissions for the second, third, and fourth leads (in the numbering scheme of Figure 3) are shown in red, green, and magenta, respectively. In the case of repetition of identical curves, later colours overwrite earlier ones. Red vertical lines show positions of ionisation poles, whilst blue lines mark attachment pole positions. The lead and connection parameters are \(\beta_{\text{L}}=2.0\) and \(\beta_{\text{L}}=1.0\). The next three panels (Figure 7d-f) show the transmission curves for CBSD based on 8-, 10-, and 12-membered rings (Figure 7a-c). Again, a pattern of alternating behaviour at the Fermi level is apparent and is confirmed by extension of the calculations to larger rings: CBSD based on \(4N\)-rings show a central maximum in total transmission, consistent with superposition of sink-lead curves with central maxima, whereas \((4N+2)\)-rings have an insulating minimum at the Fermi level, indicating vanishing of all lead contributions. As noted in Section 4.3, this is part of a general pattern based on the number of non-bonding orbitals (NBOs), or in other words, the nullity of the graph. In the \(4N\)-ring, the nullity is 2 and all centres carry non-zero density arising from any occupation of the non-bonding shell (i.e., a \(4N\)-ring is a core graph [82]). Cancellation of the \(E^{2}\) factor between numerator and denominator in the expression for lead currents therefore yields conduction at the Fermi level for these rings. The \((4N+2)\)-rings have no such cancellation. This convex/concave pattern for antiaromatic/aromatic rings is one example of a nullity-based selection rule for a family of devices. There will be many more to be found. Figure 7: Complete symmetric and complete bipartite symmetric devices. Complete symmetric devices based on (**a**) the C\({}_{6}\) ring, (**b**) the C\({}_{8}\) ring, and (**c**) the C\({}_{10}\) ring. Complete bipartite symmetric devices based on (**d**) the C\({}_{8}\) ring, (**e**) the C\({}_{10}\) ring, and (**f**) the C\({}_{12}\) ring. Complete bipartite symmetric devices based on heptatriene: (**g**) the (1,3,5,7)-device, (**h**) the (2,4,6)-device, and (i) the (4,2,6)-device. (In each case, Lead 1 is the source, with transmission shown in black. The full palette of colours is red, green, magenta, blue, and orange for sink leads, working clockwise from the source. In the case of repetition of identical curves, later colours overwrite earlier ones. For all plots, \(\beta_{\rm L}=2.0\) and \(\beta_{\rm LM}=1\). Finally, the third row (Figure 7g-i) deals with complete bipartite devices based on the heptratriene linear polyene. As shown in Figure 3, the molecular graph has two partite sets of vertices.