The configurations of single and double bonds in polycyclic hydrocarbons are abstracted as Kekul\'e states of graphs. Sending a so-called soliton over an open channel between ports (external nodes) of the graph changes the Kekul\'e state and therewith the set of open channels in the graph. This switching behaviour is proposed as a basis for molecular computation. The proposal is highly speculative but may have tremendous impact. Kekul\'e states with the same boundary behaviour (port assignment) can be regarded as equivalent. This gives rise to the abstraction of Kekul\'e cells. The basic theory of Kekul\'e states and Kekul\'e cells is developed here, up to the classification of Kekul\'e cells with $\leq 4$ ports. To put the theory in context, we generalize Kekul\'e states to semi-Kekul\'e states, which form the solutions of a linear system of equations over the field of the bits 0 and 1. We briefly study so-called omniconjugated graphs, in which every port assignment of the right signature has a Kekul\'e state. Omniconjugated graphs may be useful as connectors between computational elements. We finally investigate some examples with potentially useful switching behaviour.
Deep Dive into Kekule Cells for Molecular Computation.
The configurations of single and double bonds in polycyclic hydrocarbons are abstracted as Kekul'e states of graphs. Sending a so-called soliton over an open channel between ports (external nodes) of the graph changes the Kekul'e state and therewith the set of open channels in the graph. This switching behaviour is proposed as a basis for molecular computation. The proposal is highly speculative but may have tremendous impact. Kekul'e states with the same boundary behaviour (port assignment) can be regarded as equivalent. This gives rise to the abstraction of Kekul'e cells. The basic theory of Kekul'e states and Kekul'e cells is developed here, up to the classification of Kekul'e cells with $\leq 4$ ports. To put the theory in context, we generalize Kekul'e states to semi-Kekul'e states, which form the solutions of a linear system of equations over the field of the bits 0 and 1. We briefly study so-called omniconjugated graphs, in which every port assignment of the right sig
In the quest for smaller and smaller computational elements, we may hope to arrive one day at the level of molecules. Controllable electrical conductance within molecules is the realm of chemistry. One proposal in this direction is to use socalled π-conjugation in polycyclic hydrocarbons, as studied in Marleen van der Veen's PhD thesis "π-Logic" [17].
The development of this field of π-conjugation and π-logic needs several abstractions that belong to branches of mathematics like graph theory and linear algebra. In mathematics, the symbol π is primarily associated with the circumference and the area of the circle. The term “conjugation” has also several connotations in mathematics. In this paper, conjugation means the constructive interaction between a pair of neighbouring (carbon-carbon) π-orbitals, leading to delocalisation of the electrons.
The basic physical idea is that the configuration of single and double bonds in certain polycyclic ‘aromatic’ [note: these are polyunsaturated hydrocarbons, usually referred to as polycyclic aromatic hydrocarbons (PAHs), but they need not be aromatic in the strict chemical sense] hydrocarbons influences the electrical conductivity between points of the molecule, and can be influenced by electrical signals over channels in the molecule. In other words, the molecule can serve as a switch. Since it was Kekulé who proposed, in 1865, the alternating single and double bonds in the benzene ring (one of the simplest cyclic hydrocarbons), we prefer to associate the basic ideas to be exposed here with the name Kekulé.
The polycyclic hydrocarbons we are considering have boundary atoms that can serve as ports to probe and modify the electronic properties of the molecules. The electrical resistance between two ports is low when there is a path of alternating single and double bonds between them [20]. By sending a so-called soliton over the alternating path, the single and double bonds along the path are toggled [6]. Such toggling of an alternating path may open or close other alternating paths in the molecule. This is the switching behaviour alluded to. The toggling can also be done by chemical means (for example a redox reaction [1,19]).
In our abstraction of the molecule, the graph of the atoms and bonds is kept fixed, while it is allowed to change the multiplicities (single or double) of the bonds. A configuration of bonds such that every internal node has precisely one double bond is called a Kekulé state. Nodes with precisely one edge to the remainder of the graph are called ports. The port assignment of a Kekulé state describes the multiplicities of the bonds at the ports. A pair of ports is called a channel. A channel is called open (low resistance) in a Kekulé state if there is an alternating path between its ports.
Another reason for naming the configurations Kekulé states is that they are to represent closed shell molecules (‘Kekulé structures’), i.e., molecules in which all electrons are paired, as opposed to non-Kekulé states.
The term ‘Kekulé state’ has its chemical equivalent in ‘resonance structure’ (or ‘resonance contributor’), with the restriction of being a system that is closedshell and without charges. The number of different Kekulé states of the graph is a measure of the stability of the molecule. In this sense, Kekulé states represent all structures that are regarded in the Valence Bond Theory [8,7,2,15] of molecules (as opposed to Molecular Orbital Theory, that also includes charged and open shell configurations as parts for the total quantum mechanical description of the electronic structure of a molecule).
The Kekulé state is a debatable abstraction. The actual quantum-mechanical state is a weighted superposition of many states, in which the Kekulé states have high weights; e.g., the two Kekulé states of the benzene hexagon are just two components of a single quantum-mechanical state. This does not matter for the switching behaviour described, however, because it turns out that all Kekulé states with the same port assignment have the same open (closed) channels between ports, see Theorem 2 in section 2.2. It follows that the effect of sending a soliton over an open channel only effects the port assignment, and consists of toggling the port assignment only at the ports of the channel.
We introduce Kekulé cells to capture this behaviour. More precisely, we introduce a mathematical concept cell that captures the behaviour, and Kekulé cells are those that can be obtained from graphs with single and double bonds in them.
A serious physical objection is that the Kekulé states form a qualitative characteristic of the state, whereas the precise energy levels of the various eigenstates are quantitative. This objection must be dealt with when the qualitative investigations are leading to actual technical proposals.
Van der Veen et al. [18] have first proposed certain π-conjugated systems that can act as ‘soldering points’ for molecular wires in the sense th
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