It must be emphasized that this D2 procedure actually provides a correction to
the DFT approximations in order to recover the correct interaction energies for
reference van der Waals systems, and not a direct evaluation of the dispersion
energy. Indeed, some DFT approximations can partly (and often fortuitously)
include dispersion effects [92]. Nonetheless, in case of functionals that are known to
fully miss dispersion (typically B3LYP), it is reasonable to presume that the correction will be close to the real dispersion energy. In consequence, the dispersion
interaction energy between any atom pair will be evaluated by the B3LYP-D2
correction.
16.2.2 Basics of QTAIM: Critical Points
QTAIM also includes a complementary local approach to atomic interactions, by
focusing on the so-called critical points (CPs), defined as points in real space where
the electron density gradient vanishes. CPs can be classified by their rank r and their
signature s, this last number being the sum of the signs of the density Hessian
eigenvalues. For instance, in this (r, s) nomenclature, (3, −3) CPs correspond to
attractors (in general nuclei). The Poincaré-Hopf (PH) relationship imposes that the
numbers of the different BCP types are linked according to (for a non-periodic
system):
n ð3;À3Þ À n ð3;À1Þ þ n ð3; þ 1Þ þ n ð3;À3Þ ¼ 1:
ð16:18Þ
Of particular importance are (3, −1) CPs. They correspond to a minimum of
density along a direction and to a maximum in the two perpendicular directions. At
a molecular equilibrium geometry, they are called “bond critical points” (BCPs) and
then enable to determine which atoms are bonded in the orthodox QTAIM sense:
two atoms are bonded if and only if a BCP is present between them, the union of the
two GPs linking the BCP to each nucleus being named the “bond path”. Note that
such a characterization has raised some controversies in the last decades (see Ref.
[93] for a recent discussion). However, the existence of a bond path is a fundamental feature that has been explicitly mentioned in the IUPAC definition [1] of
halogen bonds.
Noteworthy, BCPs properties usually provide valuable information about the
bond itself. For instance, covalent bonds are characterized by high electron density
values (ρ c ) at BCPs, while ρ c is typically very low for non-covalent bonds like
hydrogen, agostic, halogen, and van der Waals ones [94–97]. Another important
descriptor is the BCP density laplacian value that quantifies the electron accumulation (negative values) or depletion (positive values) at the BCP. Thus, very
negative values are observed for covalent bonds (“shared interactions” in Bader’s
vocabulary), while ionic (“closed-shell” interactions) ones feature positive values.
Many relationships between interaction energies and (q c ; r
2
q c ) have been reported
in the literature.
16 Following Halogen Bonds Formation …
441
the DFT approximations in order to recover the correct interaction energies for
reference van der Waals systems, and not a direct evaluation of the dispersion
energy. Indeed, some DFT approximations can partly (and often fortuitously)
include dispersion effects [92]. Nonetheless, in case of functionals that are known to
fully miss dispersion (typically B3LYP), it is reasonable to presume that the correction will be close to the real dispersion energy. In consequence, the dispersion
interaction energy between any atom pair will be evaluated by the B3LYP-D2
correction.
16.2.2 Basics of QTAIM: Critical Points
QTAIM also includes a complementary local approach to atomic interactions, by
focusing on the so-called critical points (CPs), defined as points in real space where
the electron density gradient vanishes. CPs can be classified by their rank r and their
signature s, this last number being the sum of the signs of the density Hessian
eigenvalues. For instance, in this (r, s) nomenclature, (3, −3) CPs correspond to
attractors (in general nuclei). The Poincaré-Hopf (PH) relationship imposes that the
numbers of the different BCP types are linked according to (for a non-periodic
system):
n ð3;À3Þ À n ð3;À1Þ þ n ð3; þ 1Þ þ n ð3;À3Þ ¼ 1:
ð16:18Þ
Of particular importance are (3, −1) CPs. They correspond to a minimum of
density along a direction and to a maximum in the two perpendicular directions. At
a molecular equilibrium geometry, they are called “bond critical points” (BCPs) and
then enable to determine which atoms are bonded in the orthodox QTAIM sense:
two atoms are bonded if and only if a BCP is present between them, the union of the
two GPs linking the BCP to each nucleus being named the “bond path”. Note that
such a characterization has raised some controversies in the last decades (see Ref.
[93] for a recent discussion). However, the existence of a bond path is a fundamental feature that has been explicitly mentioned in the IUPAC definition [1] of
halogen bonds.
Noteworthy, BCPs properties usually provide valuable information about the
bond itself. For instance, covalent bonds are characterized by high electron density
values (ρ c ) at BCPs, while ρ c is typically very low for non-covalent bonds like
hydrogen, agostic, halogen, and van der Waals ones [94–97]. Another important
descriptor is the BCP density laplacian value that quantifies the electron accumulation (negative values) or depletion (positive values) at the BCP. Thus, very
negative values are observed for covalent bonds (“shared interactions” in Bader’s
vocabulary), while ionic (“closed-shell” interactions) ones feature positive values.
Many relationships between interaction energies and (q c ; r
2
q c ) have been reported
in the literature.
16 Following Halogen Bonds Formation …
441