The Localization–Delocalization Matrix and the Electron-Density-Weighted Connectivity Matrix of a Finite Graphene Nanoribbon Reconstructed from Kernel Fragments

Bader’s quantum theory of
atoms in molecules (QTAIM) and
chemical graph theory, merged in the localization–delocalization
matrices (LDMs) and the electron-density-weighted connectivity matrices
(EDWCM), are shown to benefit in computational speed from the kernel
energy method (KEM). The LDM and EDWCM quantum chemical graph matrices
of a 66-atom C46H20 hydrogen-terminated armchair
graphene nanoribbon, in 14 (2×7) rings of C2v symmetry, are accurately reconstructed
from kernel fragments. (This includes the full sets of electron densities
at 84 bond critical points and 19 ring critical points, and the full
sets of 66 localization and 4290 delocalization indices (LIs and DIs).)
The average absolute deviations between KEM and directly calculated
atomic electron populations, obtained from the sum of the LIs and
half of the DIs of an atom, are 0.0012 ± 0.0018 e– (∼0.02 ± 0.03%) for carbon atoms
and 0.0007 ± 0.0003 e– (∼0.01
± 0.01%) for hydrogen atoms. The integration errors in the total
electron population (296 electrons) are +0.0003 e– for the direct calculation (+0.0001%) and +0.0022 e– for KEM (+0.0007%). The accuracy of
the KEM matrix elements is, thus, probably of the order of magnitude
of the combined precision of the electronic structure calculation
and the atomic integrations. KEM appears capable of delivering not
only the total energies with chemical accuracy (which is well documented)
but also local and nonlocal properties accurately, including the DIs between the fragments (crossing fragmentation lines). Matrices
of the intact ribbon, the kernels, the KEM-reconstructed ribbon, and
errors are available as Supporting Information.