six broad principal categories: (i) Structural indices of aromaticity such as
Krygowski’s HOMA index [40–43]; (ii) magnetic properties such as Schleyer’s
NICS and its variants [44–46], NMR spectra [47, 48], or ring currents [49–51];
(iii) energetic criteria such as resonance energies and aromatic stabilization energies
[52–55]; (iv) chemical graph theory criteria as described in detail in Chap. 11 of this
book by Professor Ivan Gutman and Dr. Slavko Radenković and references therein;
(v) quantum chemical calculated measures of electron delocalization of which the
PDI and the FLU are well-known examples among several (see for example Refs.
[56–63]); and (vi) aromaticity measures derived from the ring critical point properties, the electrostatic potential, or other properties derived from the electron
density (see for example Refs. [49, 64–70]).
These aromaticity measures are generally, but not always, highly correlated and
lead at least, but again not always, to a similar ranking of local aromaticity of
various rings within molecules [71]. The discrepancies between various indices of
aromaticity is not surprising since each of these indices bring to the fore primarily a
single aspect of an essentially mutli-dimensional multi-facetted phenomenon.
The LDM codes for more than one aspect of the electron distribution in the
molecule (one-electron density and pair density) as described in Sect. 3.1 above.
Equations (3.5–3.10) show that an LDM contains information on atomic populations, atomic charges, the total number of electrons in the molecule (and their
localized and delocalized subpopulations), and also two-electron information
derived from the pair density, that is, the full atom-atom delocalization matrix of the
system. It is thus expected that the LDM codes strongly for aromaticity by virtue of
the first Hohenberg-Kohn theorem. The core question is how to get from LDMs to a
description of aromaticity?
It is proposed here to approach the problem of quantifying aromatic character
from a different angle using LDMs. Instead of attempting to measure aromaticity
directly, the similarity of a given 6-membered carbon ring in a condensed aromatic
system to benzene is taken, itself, as a measure of aromaticity. In a second
approach, the eigenvalues of the LDM of a “ring-in-a-molecule (RIM)” are taken
as predictors of the local aromatic character of that ring [72, 73].
All rings dealt with in this study are exclusively 6-membered carbon rings. Since
these rings occur mainly in polycyclic benzenoid hydrocarbons (PBHs), the number
of hydrogen atoms that are attached to a given ring varies depending on the
neighborhood of the ring in the molecule. To avoid this inconsistency due to
the variability of the atoms bonded to the carbon atoms forming the ring, we focus
the study exclusively on the carbon skeleton made of the six carbons within a given
ring.
When a carbon atom or more is shared between two rings this atom is taken
twice, once in evaluating the LDM of one ring and a second time in evaluating the
LDM of the second ring. For each molecule, we thus have a number m of LDMs
which equals the number of different 6-membered rings in the molecule.
For example, phenanthrene is broken down to three separate LDMs each representing one of its three rings as shown in Scheme 3.1.
3 Localization-Delocalization Matrices and Electron Density …
71
Krygowski’s HOMA index [40–43]; (ii) magnetic properties such as Schleyer’s
NICS and its variants [44–46], NMR spectra [47, 48], or ring currents [49–51];
(iii) energetic criteria such as resonance energies and aromatic stabilization energies
[52–55]; (iv) chemical graph theory criteria as described in detail in Chap. 11 of this
book by Professor Ivan Gutman and Dr. Slavko Radenković and references therein;
(v) quantum chemical calculated measures of electron delocalization of which the
PDI and the FLU are well-known examples among several (see for example Refs.
[56–63]); and (vi) aromaticity measures derived from the ring critical point properties, the electrostatic potential, or other properties derived from the electron
density (see for example Refs. [49, 64–70]).
These aromaticity measures are generally, but not always, highly correlated and
lead at least, but again not always, to a similar ranking of local aromaticity of
various rings within molecules [71]. The discrepancies between various indices of
aromaticity is not surprising since each of these indices bring to the fore primarily a
single aspect of an essentially mutli-dimensional multi-facetted phenomenon.
The LDM codes for more than one aspect of the electron distribution in the
molecule (one-electron density and pair density) as described in Sect. 3.1 above.
Equations (3.5–3.10) show that an LDM contains information on atomic populations, atomic charges, the total number of electrons in the molecule (and their
localized and delocalized subpopulations), and also two-electron information
derived from the pair density, that is, the full atom-atom delocalization matrix of the
system. It is thus expected that the LDM codes strongly for aromaticity by virtue of
the first Hohenberg-Kohn theorem. The core question is how to get from LDMs to a
description of aromaticity?
It is proposed here to approach the problem of quantifying aromatic character
from a different angle using LDMs. Instead of attempting to measure aromaticity
directly, the similarity of a given 6-membered carbon ring in a condensed aromatic
system to benzene is taken, itself, as a measure of aromaticity. In a second
approach, the eigenvalues of the LDM of a “ring-in-a-molecule (RIM)” are taken
as predictors of the local aromatic character of that ring [72, 73].
All rings dealt with in this study are exclusively 6-membered carbon rings. Since
these rings occur mainly in polycyclic benzenoid hydrocarbons (PBHs), the number
of hydrogen atoms that are attached to a given ring varies depending on the
neighborhood of the ring in the molecule. To avoid this inconsistency due to
the variability of the atoms bonded to the carbon atoms forming the ring, we focus
the study exclusively on the carbon skeleton made of the six carbons within a given
ring.
When a carbon atom or more is shared between two rings this atom is taken
twice, once in evaluating the LDM of one ring and a second time in evaluating the
LDM of the second ring. For each molecule, we thus have a number m of LDMs
which equals the number of different 6-membered rings in the molecule.
For example, phenanthrene is broken down to three separate LDMs each representing one of its three rings as shown in Scheme 3.1.
3 Localization-Delocalization Matrices and Electron Density …
71