electron density, a representative set of gradient vector field lines traced by the
gradient of the electron density, the intersections of interatomic surfaces (IASs)
with the plane of the figure, the set of bond paths that are coplanar with the plane of
the figure, and the bond critical points each of which lies simultaneously on the IAS
and the associated bond path. For atoms exposed on the molecular surface (and
hence that extend to infinity), the atomic basins are usually delimited by the
intersections of their IASs with the outer isodensity contour of ρ vdW = 0.001 atomic
unit (a.u.), the van der Waals envelope (1 a.u. of electron density = 1 electron per
cubic bohr).
As explained above, numerical integration (using readily available robust software such as Keith’s AIMAll [18]) yields atomic quantum mechanical averages of
properties such as atomic electron populations (N(Ω)), number of electrons localized within the basin (Λ(Ω)), and number of electrons delocalized (shared) between
one atomic basin and every other basin in the molecule (δ(Ω,Ω’)).
The number of electrons delocalization (shared) between atomic basins Ω i and
Ω j can be measured by the delocalization index (DI), δ(Ω i ,Ω j ). For a closed-shell
molecule, the DI at the Hartree-Fock level of theory is defined [19]:
dðX i ; X j Þ ¼ 2 F
a
ðX i ; X j Þ
þ 2 F
b
ðX i ; X j Þ
;
ð3:1Þ
where
F
r
ðX i ; X j Þ ¼ À
X occ
k
X occ
l
Z
X i
Z
X j
u
Ã
k ðr 1 Þu l ðr 1 Þu
Ã
l ðr 2 Þu k ðr 2 Þdr 1 dr 2
ð3:2Þ
¼ À
X occ
k
X occ
l
S kl ðX i Þ S lk ðX j Þ
ð 3:3Þ
is the Fermi correlation, and where S kl (Ω i ) = S lk (Ω i ) is the overlap integral of two
spin orbitals φ k and φ l within Ω i , and where σ refers to spin (α or β). For single
determinantal methods, the first order density matrix—printed in standard electronic
structure software—is sufficient to determine all properties since it fixes the second
order density matrix. For post-Hartree-Fock methods, the Müller approximation is
used by AIMAll, the software used to obtain the LIs and DIs, to obtain an
approximate second-order density matrix from the first order density matrix.
If i = j in Eqs. 3.2 and 3.3, (S kl (Ω i ) S lk (Ω j ) → [S kl (Ω i )]
2 ), then both integrals are
over the same atomic basin giving the total Fermi correlation for the electrons
contained within that basin. At the limit of total localization this double integral
approaches–N
σ (Ω i ), the negative of the σ-spin population of Ω i . This limit is
reached only when atoms are infinitely separated since in any molecule electrons in
a given atomic basin always exchange with electrons in every other atomic basin
3 Localization-Delocalization Matrices and Electron Density …
57
gradient of the electron density, the intersections of interatomic surfaces (IASs)
with the plane of the figure, the set of bond paths that are coplanar with the plane of
the figure, and the bond critical points each of which lies simultaneously on the IAS
and the associated bond path. For atoms exposed on the molecular surface (and
hence that extend to infinity), the atomic basins are usually delimited by the
intersections of their IASs with the outer isodensity contour of ρ vdW = 0.001 atomic
unit (a.u.), the van der Waals envelope (1 a.u. of electron density = 1 electron per
cubic bohr).
As explained above, numerical integration (using readily available robust software such as Keith’s AIMAll [18]) yields atomic quantum mechanical averages of
properties such as atomic electron populations (N(Ω)), number of electrons localized within the basin (Λ(Ω)), and number of electrons delocalized (shared) between
one atomic basin and every other basin in the molecule (δ(Ω,Ω’)).
The number of electrons delocalization (shared) between atomic basins Ω i and
Ω j can be measured by the delocalization index (DI), δ(Ω i ,Ω j ). For a closed-shell
molecule, the DI at the Hartree-Fock level of theory is defined [19]:
dðX i ; X j Þ ¼ 2 F
a
ðX i ; X j Þ
þ 2 F
b
ðX i ; X j Þ
;
ð3:1Þ
where
F
r
ðX i ; X j Þ ¼ À
X occ
k
X occ
l
Z
X i
Z
X j
u
Ã
k ðr 1 Þu l ðr 1 Þu
Ã
l ðr 2 Þu k ðr 2 Þdr 1 dr 2
ð3:2Þ
¼ À
X occ
k
X occ
l
S kl ðX i Þ S lk ðX j Þ
ð 3:3Þ
is the Fermi correlation, and where S kl (Ω i ) = S lk (Ω i ) is the overlap integral of two
spin orbitals φ k and φ l within Ω i , and where σ refers to spin (α or β). For single
determinantal methods, the first order density matrix—printed in standard electronic
structure software—is sufficient to determine all properties since it fixes the second
order density matrix. For post-Hartree-Fock methods, the Müller approximation is
used by AIMAll, the software used to obtain the LIs and DIs, to obtain an
approximate second-order density matrix from the first order density matrix.
If i = j in Eqs. 3.2 and 3.3, (S kl (Ω i ) S lk (Ω j ) → [S kl (Ω i )]
2 ), then both integrals are
over the same atomic basin giving the total Fermi correlation for the electrons
contained within that basin. At the limit of total localization this double integral
approaches–N
σ (Ω i ), the negative of the σ-spin population of Ω i . This limit is
reached only when atoms are infinitely separated since in any molecule electrons in
a given atomic basin always exchange with electrons in every other atomic basin
3 Localization-Delocalization Matrices and Electron Density …
57