Is the atomic quadrupole moment of a carbon atom in graphene zero?: The case for a rational definition of the properties of atoms in a molecule

Abstract

It is generally assumed that the carbon atoms of graphitic samples and their finite analogs have sizable quadrupole moments, with the out-of-plane component ($Q^{\rm C}_{20}$ in traceless spherical coordinates) being the dominate contribution. However, there is no consensus on what the quantity is for such carbon-based systems and values reported in the literature range from $Q^{\rm C}_{20} \sim -1.14$ to $+0.79$ a.u. In this work we propose a theoretical framework in which well-defined statements can be made about properties of atoms-in-a-molecule (AIMs) even when these properties are not experimentally observable. Using this framework and the distributed multipole method basis-space iterated Stockholder atoms (BS-ISA), we show that the atomic quadrupole moment of a carbon atom in graphene is essentially zero within the limits of precision of the numerical method used. We explain how the experimentally measured atomic quadrupole moment of a graphite sample determined by Whitehouse \& Buckingham likely originated almost entirely from edge dipoles, and we propose a more realistic electrostatic model for finite graphene nanoflakes.

Figures (4)

• Figure 1: Three different physical models for the electrostatic moments of finite graphene flakes. In the GDMA model the flake possesses a large contribution from the $\mathrm{Q}_{20}^\text{C}$ moment on the carbon atoms and inward pointing edge dipoles, while in the BS-ISA model there are near-zero $\mathrm{Q}_{20}^\text{C}$ moments and outward pointing edge dipoles. The W&B model is intermediate with no edge dipoles and only $\mathrm{Q}_{20}^\text{C}$ moments on the carbon atoms. Intensity of coloration and size of text both reflect the magnitude of the quantity it represents. Red coloration indicates a positive and blue a negative contribution to the total quadrupole ($\mathrm{Q}_{20}$).
• Figure 2: Convergence of multipolar electrostatic interaction energy when a negative point charge is scanned along the principle rotation axis of dicircumcoronene. Results are reported in (a) and (b) for multipole expansions of different orders as well as for the reference results from the PBE0(AC) density. The maximum order of the expansion is given by $\mathrm{\ell_{max}}$. The insets show the results at short range on an expanded scale. Differences between the interaction energies from the various multipole expansions and the PBE0(AC) results are reported in (c) and (d).
• Figure 3: Convergence of charge penetration energy in multipolar electrostatic models when a negative point charge is scanned along the principle rotation axis of dicircumcoronene. The maximum order of the expansion is given by $\mathrm{\ell_{max}}$. The figure shows the natural log of the magnitude of energy differences reported in Figures \ref{['figure:c96h24_scan']} (c) and (d). The left image we include the charge penetration energy if one applies a W&B-like approach by simply allocating the molecular quadrupole moment over all carbon atoms equally as $\mathrm{Q}_{20}^\text{C}$.
• Figure 4: The electrostatic interaction of a negative point charge (q) with increasingly large carbon nanoflakes at a distance of $\mathrm{z}=3.4$ Å above the plane of the nanoflake. Results shown for only atomic quadrupole moments ($Q_{20}^C$) on the carbon atoms as well as those with dipoles ($Q^{\mathrm{CH}}_{1m}$) on the edge carbon atoms. The axis, $r$, is an average distance of the edge carbons to the center of mass in the carbon nanoflake. A schematic of the arrangement is included as an inset for clarity.