Chemistry Reference
In-Depth Information
) of the
A
-centered multipole expansion (l.h.s.) at
any radial grid point is given as a linear combination of pseudoatom RDFs (r.h.s.)
taken at the same radial grid point. The projection, driven by the angular overlap
The molecular RDF (
r
lm
ð
r
A
Þ
integral of RSHs
y
lm
j
y
lm
i
O
, yields “exact” molecular moments up to
L
, without
necessarily reproducing the total density exactly. The procedure leads to deloca-
lized RDFs, each depending on the nuclear geometry and the level of expansion (L)
at each center [
28
].
4 The Standard Pseudoatom Formalism
Since the above derivation of RDFs from a finite set of X-ray structure factors is
not feasible, a parameterized version of the expansion has been developed. In the
popular Hansen-Coppens formalism [
6
] (referred to here as the
standard
or HC-
pseudoatom model (HC-PA), the RDFs are predefined as follows:
P
lm
S
l
ðk
0
r
r
00
¼ r
C
þ
P
00
r
V
ðkr
Þ;
r
lm
¼
Þ;
l
>
0
;
l
m
þ
l
(9)
The monopole part is taken as a combination of the spherical Hartree-Fock
frozen-core (
r
C
) and normalized valence densities [
29
](
r
V
) of the isolated atom,
while those in the deformation term (
S
l
) are density normalized Slater functions:
l
n
l
þ
3
r
n
l
e
l
r
S
l
¼
(10)
ð
n
l
þ
2
Þ!
l
) are deduced from energy-optimized single-zeta
Hartree-Fock (HF) atomic wave functions [
30
]. The radial deformation of the
valence density is accounted for by the expansion-contraction variables (
k
and
k
0
).
The RSHs are density normalized and expressed in nucleus-centered local frames:
ð
The parameters (
n
l
and
ð
d
d
d
0
;m
ðOÞ
O ¼
1
and
d
l>
0
;m
ðOÞ
O ¼
2
(11)
O
O
The Fourier transform of the pseudoatom ED enters into the structure-factor
expression (1) as a complex but analytic scattering factor:
4
p
X
l
þl
i
l
J
l
ð
=k
0
Þ
f
ð
H
Þ¼
f
C
ð
H
Þþ
P
V
f
V
ð
H
=kÞþ
H
P
lm
d
lm
ðO
H
Þ
(12)
m
¼
l
where,
f
C
and
f
V
are the Fourier transform of
r
C
and
r
V
, respectively,
J
l
is the
l
-order Fourier-Bessel transform of
S
l
, and
O
H
encompasses the local polar angles
). The static ED parameters,
P
lm
,
k
and
k
0
, are optimized,
together with the nuclear positions and ADPs, in a least-squares (LS) refinement
against a set of measured structure-factor amplitudes.
of the scattering vector (
H
