In this article we have used the density functional tightbinding method with self-consistent charges (SCC-DFTB), which was implemented in the DFTB+ code, 1 to perform the individual calculations. 2,3 The SCC-DFTB is an approximate quantum chemical approach where the Kohn-Sham density functional is expanded to second order around a reference electron density. The reference density is obtained from selfconsistent density functional calculations of weakly confined neutral atoms within the

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... ed gradient approximation (GGA). The confinement potential is optimised to anticipate the charge density and effective potential in molecules and solids. A minimal valence basis set is used to account explicitly for the two-centre tight-binding matrix elements within the DFT level. The double counting terms in the Coulomb and exchange-correlation potential, as well as the intra-nuclear repulsion are replaced by a universal short-range repulsive potential. All structures have been fully relaxed with a conjugate gradient methodology until forces on each atom were minimized to be less than 10 −4 a.u. (i.e. ≈5 meV/Å). In all the calculations, the "PBC" set of parameters is used to describe the contributions from diatomic interactions of carbon. 4 This method has been shown to provide good and reliable results for nanographene in the past, 5-7 and data from these studies has been used in the statistical analysis presented in the main text. For convenience, the data from previous works are presented graphically here; reproduced with permission from the original publications. As pointed out in the main text this study has focussed on using a Boltzmann distribution, but samples grown using different methods and synthesis conditions may be distributed in different ways. Certainly many graphene nanoflakes are produced under kinetically driven conditions, and a variety of different distributions are possible; including the thermodynamic distribution. To give some indication of the impact using different distributions the results for the Boltzmann distribution are compared to a frequency distribution and a Gaussian (normal) distribution in the table below.