# A Simple Model of Ballistic Conduction in Multi-Lead Molecular Devices
## 1 Introduction  
The study of molecular conduction and molecule-scale devices is and has been an active area at the boundary between chemistry and physics for at least half a century [1], and by now, it has accumulated a substantial number of textbooks in the literature, e.g., [2, 3, 4, 5, 6]. Over this period, different theoretical approaches have emerged. Techniques for calculation have mainly favoured Green's Function (GF) methods, both equilibrium and non-equilibrium GFs [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. Especially in chemistry, there has been a sustained development of scattering models for the ballistic conduction of electrons though molecular systems, with an emphasis on conduction through channels governed by energies and symmetry characteristics of orbitals [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. An advantage of such pictures is that they lend themselves to even simpler qualitative modelling of the effects of electron energy, relative interaction strengths, contact placement, and qualitative features of molecular electronic structures, as treated in e.g., [31, 32, 33, 34] and as illustrated by many examples in our own work, as cited below. One successful model of this type is the Source-and-Sink-potential (SSP) model proposed and developed by [PERSON] and co-workers in tandem with their DFT approach to an a priori calculation of conduction [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45]. The SSP approach replaces the doubly infinite system of molecule and leads by the finite system of a molecule dressed with a source and a sink pseudo atom equipped with complex potentials, hence in the tight-binding model replacing an \((n+\infty)\times(n+\infty)\) matrix problem by an \((n+2)\times(n+2)\) problem [40, 55]. These ideas have their roots in earlier work [56, 57] and have parallels in approaches such as SSM [58] and CAP [59]. A special feature of the SSP model in its graph theoretical incarnation [55] is that it gives a useful qualitative account of the two-lead device, giving selection rules for conduction, rationalisation of the sensitivity of current to placing of leads; models forcomposite devices; and a general classification of types of conduction behaviour, such as equiconduction, omniconduction, omni-insulation, and perfect reflection [60; 61; 62; 63; 64; 65; 66; 67; 68; 69]. Although the graph theoretical version of SSP is based on the Huckel (tight-binding) model of electronic structure, many of its most useful features persist at higher levels of theory. Conduction of the Huckel device can be analysed in terms of internal channels based on molecular orbitals of the central molecule, which are either _active_ or _inert_[63]. It can be shown [70] that, as the level of theory is raised, a similar analysis applies to Hartree-Fock MO channels and, then at the second-order (or higher) Green's Function level, to internal channels associated with Dyson orbitals. This evolution of the channel picture is consistent with the standard Meir-Wingreen [19] and Landauer-Buttiker [71; 72] expressions for elastic conduction of electrons through molecules. [PERSON]'s Function approaches have been applied to various multi-lead configurations at various levels of theory. For instance, Reference [59] quotes a dozen examples, and public domain codes such as TRANSIESTA [73] allow for calculations with many leads. Explicit high-level calculations of transmission for individual systems are undoubtedly valuable, but there is still a role for simple models that can identify broad trends for families of systems, can give a framework for interpretation, and can identify cases for more detailed analysis by a priori/ab initio methods. In line with this philosophy, our aim in the present paper is to generalise the two-lead SSP formalism to deal with multi-lead devices, retaining its many advantages. To be concrete, we call this version of SSP the SMSP (source-and-multiple-sink-potential) model.