Chemistry Reference
In-Depth Information
The HC formalism was developed with the sole purpose of extracting the static
ED of crystalline materials from high-resolution X-ray data. The experimental
effort involved was technically demanding and much more time consuming than
nowadays. Since the accessible data are limited in resolution, the analysis requires
an economic model which performs reasonably well even with a small number of
fitting parameters (at least an order of magnitude smaller than the number of data
points to be fitted). The HC-PA effectively compromises between flexibility and
locality - an obvious reason for its success. The formalism has survived a period of
over 30 years during which it has been challenged by data of increased quality and
quantity. The first point to be emphasized is that the HC-PA is structure indepen-
dent, that is, no parameters related to the local connectivity or bonding of the atom
are included in its analytic expression (9). One of its obvious limitations is the
frozen-core approximation, that is, the core polarization, which plays an important
role in balancing electrostatic forces [
31
], is not accounted for directly. This effect
is expected to be pronounced for heavier elements, since the scattering power of
their atoms is dominated by the core electrons. The radial signals due to core
polarization are extremely sharp, localized in the vicinity of the nuclei, and thus
manifest themselves mainly in the high-order reflections, for which, however, the
experimental signal-to-noise ratio is low. The extent to which the frozen-core
approximation biases the valence density parameters is thus not readily, if at all,
accessible by X-ray diffraction.
The most important sources of bias are the restrictions imposed on the valence
RDFs. The expansion/contraction of
r
V
upon charge gain/loss is accounted for by a
single scaling parameter (
k
). Simple chemical arguments, however, suggest that the
sub-shells are not uniformly deformed upon bond formations. It is also evident that
single Slater functions are not flexible enough to describe radial deformations in
molecules. Moreover, their direct construction from the ground-state minimal-basis
atomic orbitals is ambiguous. For example, no odd-order and hexadecapolar RDFs
can be assigned to a quadrupolar atom (
s,p
-valence shell) without mixing
s
and
p
orbitals according to some hybridization scheme. For atoms with outermost
d
sub-
shell, the
s
and
p
contributions from the same shell can either be included (giving
rise to all RDFs up to
l
ΒΌ
4, but with mixed orbital contributions) or kept frozen
(leading to
S
2
and
S
4
, but with pure
d
-orbital contribution). The RDF set deduced in
such a way is unbalanced, since
r
V
is derived from the atomic wave functions of
near-HF-limit, while the deformation RDFs are constructed from single Slater
orbitals. Furthermore, these functions are
m
-independent, unlike those in the gen-
eral expansion (6). This means that poles with the same angular (
l
) but different
magnetic quantum number (
m
) share the same
S
l
. This restriction prohibits distin-
guishing between
p
-orbitals involved in
p
and
s
bonding, which was recognized as
an important requirement for an adequate basis set (split-valence) already at the
early stage of developments of routine molecular orbital (MO) calculations. Last
but not least, the same set of RDFs is assigned to centers with the same atomic
number, irrespective of their local bonding situation.
The adequacy of the HC formalism and the physical content/reliability of the ED
projected onto the model from a finite set of structure factors are frequently raised
