Chemistry Reference
In-Depth Information
),
contributions, which in the case of (11) may be then both conveniently expressed in
terms of the elements of an electron “population” matrix
M
.
N(
O
) turns out to be decomposable in an inner, N
i
(
O
), and an outer, N
o
(
O
X
ðO; O
0
Þ:
N
ðOÞ¼
N
i
ðOÞþ
N
o
ðOÞ¼
M
ðO; OÞþ
M
(13)
O
0
6¼O
M
(
O
,
O
) represents the contribution of
O
to determining the electron population
O
0
) is the contribution to this same population from
of its own basin, while
M
(
O
,
O
0
.
M
(
O
0
,
basin
O
), instead, represents the contribution of
O
in determining the
O
0
and, usually,
M
(
O
0
)
O
0
,
population of
). At variance with standard
population analyses, the matrix
M
is thus generally not symmetric, although it
will be clearly so in specific cases (e.g., the homonuclear diatomics). In the case of
(12), the outer contribution would consist of a sum over the interatomic surfaces of
O
O
,
6¼
M
(
O
(Scheme
1
).
The matrix
M
(13) defines a full “population analysis,” based only on the
observable
of the global contributions from each chemical residue R linked to
O
2
and without resorting, as it is generally the case, to any discrete
representation of the one electron density matrix on atomic bases and to population
analyses derived thereof (e.g., Mulliken Population analysis, MPA) [
17
]. As antici-
pated earlier, the quotes enclosing “population analysis” signify the profound
difference between an atomic population given in terms of a sum of contributions
to the density from the atomic basis set functions (e.g., MPA) and the present one
given in terms of contributions from well-defined portion of spaces
determining
that
population. For the sake of simplicity, we will, however, omit the quotes from now
on when referring to either type of populations. The matrix
M
, analogously to the
local source and to the source function, can be obtained from experimental electron
densities. It thus represents a unique chance to compare theoretical and experimen-
tal electron populations on an unbiased basis, regardless of the different descriptions
- atomic basis sets, pseudoatom densities [
10
], etc. - adopted to derive them. It is
worth noting the formal analogy between the proposed population analysis and that
one would obtain by defining the boundaries of atoms according to the Hirshfeld
surfaces (HS) introduced by Spackman and Byrom [
18
], and by then using the
so-called Hirshfeld's stockolder partitioning [
19
] to apportion, among the various
atoms, the electron density within the HS bounded basins. This procedure would
automatically lead to a nonsymmetric
M
matrix formally analogous to that obtained
by the double integration of the local source and would equally be applicable, on the
same grounds, to both theoretical and experimental densities. An interesting aspect
r
r
R
1
R
2
R
4
W
Scheme 1 Atomic group
O
and its linked chemical
residues R
R
3
