Chemistry Reference
In-Depth Information
In this case the combinations are
Y
11c
=
N
c
(
Y
1,
−
1
−
Y
1,1
)
Y
11s
=
i
N
s
(
Y
1,
−
1
+
Y
1,1
)
and
(A9.78)
=
AN
c
2sin(
θ
) cos(
φ
)
=
AN
s
2sin(
θ
)sin(
φ
)
where the second construction depends on the fact that i
2
1 and we have made further
use of Equation (A9.17). The subscripts refer to
l
, the magnitude of
m
l
and the type of
linear combination used: 'c' signifies that cos(
=−
) is present.
The imaginary functions were normalized so that the integration over all space of the
corresponding probability gives 1, but this will not be the case for the linear combinations
in Equation (A9.78). So we have included normalization constants
N
c
and
N
s
which must
be determined. This can be done quite neatly by making use of the orthogonality of the
spherical harmonic functions. We require
φ
) survives and 's' that sin(
φ
Y
11c
|
Y
11c
=
1
(A9.79)
The left-hand side can be written in terms of the imaginary spherical harmonics:
Y
11c
|
Y
11c
=
N
c
(
Y
1
−
1
−
Y
11
)
|
(
Y
1
−
1
−
Y
11
)
(A9.80)
Y
11c
|
Y
11c
=
N
c
(
Y
1
−
1
|
Y
1
−
1
+
Y
11
|
Y
11
−
Y
11
|
Y
1
−
1
−
Y
1
−
1
|
Y
11
)
But we know that the original functions are normalized and orthogonal, so the overlap of
like functions is one and overlap between functions is zero, so we have
1
√
2
2
N
c
=
1
and so
N
c
=
(A9.81)
The same arguement can be used to show that
N
s
takes the same value. These normalization
constants have been included in Table A9.1.
A9.10 Cartesian Forms of the Real Angular Functions
Now, functions in the spherical polar coordinate system can be referenced back to the
Cartesian coordinates via the relationships
x
=
r
sin(
θ
) cos(
φ
)
y
=
r
sin(
θ
)sin(
φ
)
(A9.82)
z
=
r
cos(
θ
)
=
+
+
r
2
x
2
y
2
z
2
which can be shown from the geometry of Figure A9.1. Looking at the linear combinations
of Equation (A9.78), conversion to Cartesian coordinates gives
2
N
c
3
2
1
/
2
2
N
s
3
2
1
/
2
x
y
x
2
x
2
Y
11c
=
and
Y
11s
=
π
π
+
y
2
+
z
2
+
y
2
+
z
2
(A9.83)
