The goal of sp2graph project is to provide tools for analyzing the bond-order of sp2-hybridized carbon nanostructures. This is particular interesting for the interpretation of recent experiments performed with scanning probe microscopy on graphene-based nanostructures synthesized on metallic surfaces.
sp2graph was initiated by Pedro Brandimarte and Thomas Frederiksen at DIPC in June 2018.
API documentation is available at sp2graph.
Given the xyz geometry of a planar carbon-based structure (sp2-hybridized), sp2graph aims to provide:
- all possible Kekulé representations
- the most stable Clar structure according to Clar's π-sextet theory
- allow for user defined initial constrains, by imposing:
- single or double bonds at specified connections
- radicals at specific sites
- estimation of the most stable Kekulé structure through simple nearest-neighbor tight-binding model
Carbon sp2-bonded structures, in particular polycyclic aromatic hydrocarbons (PAHs), are often represented graphically according to the Kekulé bond formulas, where the four valence electrons of a carbon can form single or double bonds with neighboring atoms (see figures below). In this representation only the carbon-carbon bonds are drawn, being carbon-hydrogen bonds typically omitted (i.e. carbons with only three bonds or less are implicitly assumed to make extra bonds with hydrogen atoms). Structures exhibiting a radical character, such as open-shell states, are usually marked with a dot next to the carbons containing unpaired π-electrons.
The Kekulé representation of any given sp2 carbon-based system is not unique. The benzene molecule C6H6, for example, has two possible representations as shown in the left side of the figure below. Therefore, the electronic structure of a system is given by a combination of all possible Kekulé bond formulas (Wassmann et al, 2010). For benzene the two complementary Kekulé structures form an aromatic π-sextet (right side of the figure below), that represents the delocalization of the six π-electrons.
benzene: Kekulé resonant structures (left) and the corresponding Clar sextet (right).
Aromatic sextets are also known as Clar's sextets since the formulation of the Clar's π-sextet rule which states that the representation with largest number of disjoint aromatic sextets (known as Clar's formula or structure) is the most stable configuration and, therefore, the most representative for characterizing the system properties (Solà, 2013). A given system can present different equivalent Clar's formulas, i.e. different representations having the same (largest) number of aromatic sextet's. In such case, the system is described by a superposition of all equivalent Clar structures. A typical example is the anthracene molecule shown below, where a Clar's sextet can be found in one of the three rings.
anthracene: Kekulé resonant structures (left) with the corresponding Clar sextets (right).
For the phenanthrene, on the other hand, only one Clar structure can be found with two aromatic sextets (see image below). The Clar's rule can be further applied, in absence of radicals, to find the most stable structure among alternant PAHs (Yeh and Chai, 2016), i.e. isomers containing the same number of six-membered rings. For instance, by comparing the Clar structures from anthracene and phenanthrene one would expect the later to be more stable. Indeed this is in agreement with standard density functional theory calculations (DFT), where the phenanthrene is found to be more stable by 0.2 eV.
phenanthrene: Kekulé resonant structures (left) with the corresponding Clar sextets (right).
NB: There should be four Kekulé structures for the lower Clar structure with two sextets.
Interestingly, by performing structural relaxations with simple reactive empirical bond order (REBO) potentials optimized for hydrocarbons (Brenner et al, 2002) one would predict wrongly the total energy of anthracene and phenanthrene to be approximately equal, because the local bonding configurations of the carbon atoms are nearly identical. Therefore, this simple example illustrates the strength of the Clar's π-sextet rule.
Although conceptually simple, apply the Clar's sextet to larger sp2 structures can be rather complex. Furthermore, when drawing Clar structures to maximize the number of sextets, one may encounter configurations with carbon atoms with radical character. As a rule of thumb, three additional sextets need to be added to the Clar structure to compensate for the energy cost associated with creating two radicals (Das and Wu, 2015). For extended structures these rules leave many possibilities and thus call for a systematic and automated approach.
The Kekulé diagrams shown in the figures above can be described mathematically by a graph G(V,E), a fundamental combinatorial object defined by a set of vertices V and a collection of edges E connecting distinct vertex pairs. In our case, the vertices are given by the carbon atoms and the edges by the chemical bonds among them.
A simple method for representing a graph is the so-called adjacency matrix A, with dimensions VxV and where the element Aij is nonzero only if there is an edge connecting from vertex i to j. For instance, the naphthalene molecule could be represent as:
Graph representation of a sp2 structure: A naphthalene molecule with respective graph and adjacency matrix representations.
In the adjacency matrix above, a carbon-carbon bond between vertices i and j is expressed by having Aij=1, while for the absence of bond as Aij=0. The graph of any sp2 structure in general has a number of properties, to list a few:
- it is an undirected graph, meaning that an edge is defined by a pair of distinct vertices independently of their order (ij or ji), which is expressed by the symmetry of the adjacency matrix;
- it is a connected graph, i.e., for any pair of distinct vertices there is always a path between them;
- any vertex is connected to 2 or 3 other vertices (i.e., any vertex has 2 or 3 incident edges), which makes the adjacency matrix representation sparse;
- any vertex belongs to at least one cycle or, in other words, a vertex will never be hanging alone from the graph since this would mean a sp1-hybridized carbon.
Note that given the set of xyz coordinates of a sp2 structure, the set of vertex and edges can be uniquely defined. For instance, a carbon-carbon bond can be assigned to every two carbons with distance lower than 1.43 Å. However, the adjacency matrix as defined above contains no information about where the single and double bonds are. There are several ways to include such information, and a simple one would be to define Aij=2 if there is a double bond between vertices i and j. The edges containing a double bond are not uniquely defined though, for the same reason a sp2 system has different Kekulé structures. According to this definition, one possibility for the naphthalene molecule would be:
Graph representation of a Kekulé structure: A naphthalene Kekulé structure with respective graph and adjacency matrix representations.
Therefore, in addition to the properties listed before one can include:
- any vertex is connected to at least another vertex through a double bond, i.e., for any row i from the adjacency matrix there is one and only one element with Aij=2.
More generally, a Kekulé structure K could be represented by an unique adjacency matrix A plus another matrix B that contains only information about the double bonds. In this way, for any row i from the "double-bond" matrix B there is one and only one element with Bij=1. For the naphthalene Kekulé structure above this translates to:
Graph of a Kekulé structure given by adjacency plus double bonds: Naphthalene Kekulé structure represented by an adjacency matrix added to a "double bond" matrix.
As a first attempt to solve the problem of finding all possible Kekulé structures of a given carbon sp2 system, we develop a backtracking algorithm that resembles a combination of the well known depth-first search algorithm for visiting all vertex of one Kekulé structure together with a depth-first search -like algorithm for walking through the different Kekulé structures. In this procedure we keep tracking of the visited vertices stored in a list K and a FIFO (First In First Out) queue Q for the neighbors of the vertices under analysis. It starts by insetting in Q the vertex with lowest degree (in our case any vertex has degree 2 or 3) and lowest index:
- if Q is empty, then K contains a possible Kekulé structure;
- else:
- unqueue the last element l of Q and,
- if l is already in K, then go to the first step;
- else:
- append l to K;
- get a list n from all neighbors from l that are not yet in K;
- if the size of n is 1, then append to Q and go to the first step;
- else there are two possible ways to assign a double bond, so we append the neighbors to Q rolling their order and go back to the first step in each case;
- unqueue the last element l of Q and,
However, there are several problems with this approach. First that the same Kekulé structure can be found more than one time. Therefore, at the point where one possible structure is found one has to check if a similar one has not been found before. The ideal situation would be to find all possible Kekulé structures by passing through all vertices just once.
To install sp2graph the following packages are required:
- Python >=3.6
- NumPy >= 1.13
- Matplotlib >= 2.2.2
Manual installation is performed with the command:
python setup.py install --prefix=<prefix>
# or
python setup.py install --home=<my-python-home>
One may also wish to set the following environment variables:
export PYTHONPATH=<my-python-home>/lib/<python-version>/site-packages
export LD_LIBRARY_PATH=$LD_LIBRARY_PATH:<my-python-home>/lib
Contributions are highly appreciated!
If you find any bug please form a bug report/issue.
If you have a fix please consider adding a pull request.
Financial support from Spanish AEI (FIS2017-83780-P "GRANAS") is acknowledged.