Chemistry Reference
In-Depth Information
Different one-center density models used for X-ray data fitting can be considered
as extensions of the promolecule to account for local deformations due to chemical
bonding.
3 Multipole Expansion of the Electron Density
Since the atomic Hamiltonian commutes with the square of the total angular
momentum operator, the physical density of an atom in a well-defined angular
momentum state (
L
,
M
) has a well-defined RSH
ð
y
lm
ð#; fÞ¼
y
lm
ðOÞÞ
content [
26
]:
X
L
rð
r
Þ¼
r
2
l;
0
ð
r
Þ
y
2
l;
0
ðOÞ
(5)
l
Due to the RSHs forming a complete orthogonal basis, the product of any two
can be written as a linear combination of a finite number of RSHs. Thus, the density
of an atom obtained within the orbital approximation also has a finite multipole
expansion, as best seen for a single-determinant wave function composed of
one-electron spin-functions. Within the LCAO-MO approximation, the molecular
density can be decomposed into one- and two-center orbital product [
27
]. The
multipole content of each one-center density, just like for an atom, is uniquely
determined by the orbital basis.
The many-centered finite multipole expansion of the molecular or crystalline ED
was introduced by Stewart [
5
]. In his formalism, the total density
ðrð
r
ÞÞ
is decom-
posed into
pseudoatoms
:
X
X
L
r
lm
ð
rð
r
Þ¼
r
A
Þ
y
lm
ðO
A
Þ
(6)
A
l;m
whose RDFs (
r
lm
) can in principle be derived by minimization of the mean-square
residual (MSR:
w
2
) between the target and model densities:
!
2
ð
1
X
X
L
w
2
r
lm
ð
w
0
ð
r
2
d
r
¼
rð
r
Þ
r
A
Þ
y
lm
ðO
A
Þ
dr
¼
r
Þ
(7)
A
l
;
m
0
w
0
is a positive definite radial MSR obtained by integrating
w
2
over the angular
variables. Its minimization with respect to each RDF leads to a set of inhomoge-
neous linear equations:
D
y
lm
E
X
X
r
A
Þ¼ r
y
lm
L
O
¼ r
lm
ð
r
lm
ð
y
lm
r
lm
ð
r
A
Þþ
r
B
Þ
(8)
O
B
6¼
A
lm
