Linear Response and Measures of Electron Delocalization ... - CRM2

Linear Response and Measures of Electron Delocalization Current Organic Chemistry, 2011, Vol. 15, No. 20 3613
F
Cl
O
O
1.3489
1.7571 1.4697 1.2269
1.1630 N
1.1738
N
P
N
O
133.25
o
O
O
129.77
o
O
O
O
O
O
O
O
S
O
1.1569
1.1601
1.7375
1.2857
1.2916
C
1.7336
C
C
C
F
o
F
Cl
108.27 113.20
o
Cl
S
S
O
O
F
Cl
O
O
1.3010 1.7453 1.4050 1.4134
B
B
S
B
F
F
Cl
Cl
O
O
O
O
Fig. (1). Schematic illustration of the family of Y-conjugated systems. The numbers are the bond lengths. After Ref. [43].
ΑAA
2.5
2.0
1.5
1.0
0.5
0.0 0.5 1.0 1.5 2.0 2.5
N 2
A c
ΑAB
0.0
0.2
0.4
0.6
0.8
1.0 0.8 0.6 0.4 0.2 0.0
N A N B c
Fig. (2). Correlation of Bader atomic population fluctuations (a) and atom-atom fluctuations (b) with the corresponding charge-flow polarizabilities for Y-
conjugated systems. On (a) green points (dashed line) are central atoms, red points (solid lines) are the oxygen ligands, and other type of ligand atoms are
marked by blue points (dotted line). On (b) only the bonds with the central atom are considered. The linear regression lines were calculated by excluding a few
outlier points, indicated by different colors. After Ref. [43].
population and it can be considered as a domain localization index,
giving the number of electrons localized in a
[2]. A similar
analysis can be performed [40] from the viewpoint of the domain
averaged Fermi hole (DAFH) concept [8, 41, 42].
The analog of the charge-conservation sum rule for the
delocalization indices is a simple consequence of the normalization
of the exchange hole: ˆN 2 a
c
=
ˆN ba a
ˆNb c
and we can
anticipate the following proportionality:
ˆN a 2 c
aa
and ˆN a
ˆNb c
ab
(a b) (18)
The proportionality factor is in principle system-dependent and
it behaves like a kind of mean excitation energy, characteristic to
the atom pair. The expected proportionality between (static) charge
flow polarizabilities and delocalization indices (and on-site chargecharge
polarizabilities) has been tested for a family of related Y-
conjugated compounds [43], shown on Fig. (1).
It is clear from Fig. (2) that both the one- and two-center
indices are in fact proportional to the corresponding charge-flow
polarizabilities. In the one-center case the atoms have been
classified in different categories: central B, C, N, S, P atoms (with
the exclusion of the carbon atoms of CS 3 and COCl 2 ), oxygen
ligand atom and halogen (F,Cl) and S ligand atoms. For the twocenter
case only formally bonded atom pairs are shown, since the
too weak bonding/polarizability does not lead to interpretable
correlations. Disregarded of the above mentioned outliers, very
good linear regressions could be found, indicating that for these
special bonds there is a unique mean excitation energy. In some
cases the delocalization indices as well as the single-center
localization indices are quite low, below 0.5. One can consider such
values as indicators for the presence of strongly ionic bonding.
The benzene molecule is an example where the Fermi-hole
extends farther than the nearest-neighbour atoms and the charge
density response remains large even in the para (1,4) position. The
-electron system of the benzene is usually considered as a