Chemistry Reference
In-Depth Information
2 Static Electron Density from X-Ray Structure Factors
Within the kinematic theory, single-crystal X-ray diffraction is treated as a coherent
elastic scattering of X-ray photons on electrons with periodic distribution (
rð
).
The key property relevant to this process is the Bragg structure factor (
F
(H))
which is the Fourier transform of the mean electron density (
hi
r
Þ
) in the unit cell
of volume
V
c
.
ð
X
f
A
ð
e
i
2
p
Hr
dr
e
i
2
p
HR
A
F
ð
H
Þ¼
h
rð
r
Þ
i
¼
H
Þ
t
A
ð
H
Þ
(1)
A
V
c
H is the scattering vector, with integral reciprocal-axis components (Miller
indices
h
,
k
and
l
), satisfying the Laue conditions (
*
*
*
Ha
¼
h
,
Hb
¼
k
,and
Hc
¼
l
) dictated by the translation symmetry of the crystal density. Assuming that the
system remains in the ground electronic state during the scattering process, the
averaging is taken only over the vibrational states. The right-hand side of (1) is
obtained within the harmonic convolution approximation to thermal smearing [
4
],
which requires the static crystalline density to be partitioned into “atomic”
densities (
r
A
), each of which rigidly follows the motion of a particular nucleus
at R
A
:
X
A
r
A
ð
rð
r
Þ¼
r
A
Þ;
r
A
¼
r
R
A
(2)
The atomic scattering factor (
f
A
) and the temperature factor (
t
A
) are, respectively,
the Fourier transform of
)
of harmonic nuclear displacements with respect to the equilibrium configuration
(u
A
¼
r
A
and the Gaussian probability density function (
P
ð
u
A
Þ
Q
A
R
A
):
2
pÞ
3
=
2
U
A
2
u
A
0
U
A
u
A
j
1
=
2
e
P
ð
u
A
Þ¼ð
j
(3)
The
components of
the mean-square displacement
amplitude matrix
) are referred to as the atomic (anisotropic) displacement parameters
u
A
u
0
A
(U
A
¼
(ADP).
Atomicity is a driving concept of X-ray diffraction analysis. It is invoked at
different stages of the structure elucidation, starting from initial phase assignment,
through structure refinement, to accurate modeling of solid-state charge distribu-
tions. The conventional structure refinement is based on the promolecule density
(
r
o
), which is the superposition of spherical atomic densities (
r
A
) derived from
quantum-chemical calculations on isolated atoms:
ð
r
Þ
X
r
o
r
A
ð
ð
r
Þ¼
r
A
Þ
(4)
A
