Chemistry Reference
In-Depth Information
Mn
2
(CO)
10
itself. One expects such a different behavior to be the rule, rather than the
exception, for bcps characterized by very low electron density values. That is to say,
the lower is the density at the bcp and the higher is the delocalization of sources, the
lower is expected to be the otherwise impressive robustness generally observed for the
S% descriptor.
5 Double Integrating the Local Source: An Unambiguous
Position-Space “Population Analysis”
In Sect.
2.2
, a population analysis based on the SF approach was introduced.
Preliminary results for the second-row H-
X
diatomics and numerical difficulties
met with this position-space electron population scheme are concisely summar-
ized in Sect.
5.1
below. A standard SF analysis for the H-
X
series was presented
earlier in Sect.
3.2.3
, where the relationship between the shape of the Laplacian
distributions along the series and the values of the atomic source contributions at
bcp densities was highlighted. By double integrating the LS functions defined
through these same Laplacian distributions, an electron population analysis
matrix
M
,having2
2 dimensions for diatomics, is obtained. In general, each
element of
M
is evaluated by letting the r coordinate to span one atomic basin
and the rp coordinate r
0
to span either this same basin (diagonal elements of
M
)
or a different atomic basin (out-of-diagonal elements of
M
). Operating this way
on H-
X
diatomics, one obtains the elements
M
(A,A) (A
Hor
X
) representing
the self-contributions to the atomic electron population of A, N(A), and the
elements
M
(A,B) (A,B
¼
B) defining the contributions to the
electron population of H from
X
or vice versa (13). In practice, the evaluation of
the full
M
matrix is simpler than that. It just requires a standard 3D integration
over each atomic basin as usually performed to get the various N(
¼
Hor
X
with A
6¼
)values,
provided one stores in advance the whole set of integration grid points for all
atoms in the molecule. By integrating the LS(r, r
0
)within
O
O
with r running over
the grid points of
and r
0
running over the whole set of integration grid points
of all atoms, one immediately gets and separately stores the elements of a full
row of the
M
matrix,
M
(
O
,
O
0
;
O
0
¼
1,
N
with
N
being the total number of atoms
in the molecule). Summing up the elements over the row equals N(
O
); hence
with just one standard 3D atomic integration, one obtains both the atomic
population of
O
and its decomposition in a self-contribution and in external
contribution terms to this population from the remaining atoms in the system.
However, this apparently simple procedure becomes often very challenging from
a numerical point of view. A serious problem arises as for the “correct” assign-
ment of atomic sources for those elements of volume integration which are
characterized by very low average electron density values and electron popula-
tions (see
infra
).
O
