Chemistry Reference
In-Depth Information
6 Absolute Error of the Pseudoatom Density
The methods outlined above allows for the evaluation of the absolute error in the
pseudoatom ED corresponding to all three models (U-, R-, and 1STO-SPA). In
what it follows, we show how the total density of
a-glycine
can be reconstructed
with different RDFs. The MO density was calculated at the experimental geometry
[
52
] with the ADF program package [
53
] using BLYP level of theory [
54
,
55
] and a
quadruple-zeta polarized basis set (QZ4P) [
56
]. The projection of
m
-dependent
RDFs (
R
lm
) onto
m
-independent ones (
R
l
) was also done numerically to avoid
accuracy loss due to fitting.
Figure
2
displays residual EDs calculated in the plane of the carboxylate group
with reference to the target density at different level of expansions (
L
¼
0-4) using
the three sets of RDFs;
R
lm
(Fig.
2a
),
R
l
(Fig.
2b
), and
S
l
(Fig.
2c
).
It is clear that the U-SPA model terminated at the hexadecapolar level (
L
4)
can reconstruct the target density within an overall accuracy of 0.05 e/
˚
3
.The
maximum/minimum residual density of 0.043/
¼
0.047 e/
˚
3
is located in the close
vicinity of the oxygen nuclei and rapidly diminishes as the level of expansion
increases. All bonding residual contours are less than 0.02 e/
˚
3
,whichcanbe
further reduced if the multipole expansion is extended to
L
6. As expected, the
ED reconstructed by the R-SPAs has a lower accuracy; the residual density
amounts to about 0.05 e/
˚
3
in the C-O bonds, but it is much more pronounced
near to the oxygen nuclei. The most important finding is that these systematic
features cannot be accounted for even with the inclusion of higher-order multi-
poles, since the convergence stops already at the octupolar level (
L
¼
¼
3). The
1STO-SPA model appears to be a relatively modest representation of the true ED,
especially in bonds of
character. We find residual ED peaks of 0.15 e/
˚
3
in the
C-O bonds and maximum/minimum residual EDs of 0.163/
p
0.235 e/
˚
3
near to
the oxygen nuclei.
Table
1
lists percentage errors in the Laplacian at the BCP of different bonds
obtained from the reconstructed U- and R-SPA densities, as well as from the
HC-PA model fitted to the structure factors corresponding to
r
C
.Thevaluesare
2
r
c
. While the U-SPA model (using
R
lm
up to
calculated with reference to
r
L
4) well accounts for the target density, the Laplacian is rather poorly
reproduced, especially in polar bonds. A detailed analysis reveals that this is
primarily due to the limit of the level of expansion; one must extend the
U-SPA
model up to
L
¼
2
r
c
) at the C-O bond
with four significant figures. Interestingly, the R-SPAs with
m
-independent
RDFs (
L
¼
18 to obtain the “exact” Laplacian (
r
4) can occasionally perform better in predicting the bond-parallel
curvature (
l
3
) at the BCP than the U-SPAs. The error in the principal curvatures
due to the use of restricted RDFs and the error arising from the limited level of
expansion appear to cancel each other. The Laplacian values obtained from the
data-fitted HC density are, however, markedly less accurate, exhibit no correla-
tion with those from the R-SPAs, and strongly depend on the details of the
refinement.
¼
