Chemistry Reference
In-Depth Information
Z
lδφ
=
r
sin(
θ
)
δφ
δφ
r
δθ
l
θ
δr
δθ
r
φ
Y
X
δτ
=
rlδrδθδφ
=
r
2
sin(
θ
)
δrδθδφ
Figure A9.6
Geometry required to define a small volume element
δτ
in spherical polar coor-
dinates according to the small changes
. The sides of the small element shown
can be found using the definition of radians as the ratio of the arc length subtended by the
angle and the radius.
δ
r,
δθ
and
δφ
In effect, this proceedure sums the probability around a thin spherical shell of radius
r
and
thickness
r
, as illustrated in Figure 7.12. The limits of the integrals in this equation can
be understood with reference to Figure A9.1. The
δ
θ
derivative runs from 0 to
π
(or 180
◦
)
φ
sweeping out a half circle with its diameter on the
Z
-axis. Then, the
integral covers a
full circle from
−
π
to
, ensuring that the half circle defined by
θ
is swept through a full
π
sphere.
Our atomic orbital functions are products of radial and angular terms, and so this type
of multiple integral can be treated as a product of integrals over each of the coordinates.
For the 1s orbital, for example:
⎛
⎝
⎞
⎠
δ
π
π
χ
100
χ
100
r
2
sin(
P
100
(
r
)
δ
r
=
θ
)d
θ
d
φ
r
(A9.56)
−
π
0
Taking the angular and radial functions from Tables A9.1 and A9.2:
Z
eff
a
0
3
/
2
π
π
1
√
π
A
2
r
2
exp(
P
100
(
r
)
δ
r
=
−
ρ
r
)
δ
r
d
φ
sin
(θ)
d
θ
with
A
=
(A9.57)
−
π
0
