Chemistry Reference
In-Depth Information
cylindrical polar system is the position along the axis
z
, which can take any value from
−∞
. Figure A10.4a also shows that the infinitesimal volume for this coordinate
system is
u
d
to
+∞
d
u
d
z
.
The
s
-functions for the two atomic centres still follow the form given in Equa-
tion (A10.3), but we must express the distance from each nuclear centre in terms of the
cylindrical polar coordinates. Figure 10.4c shows a general point (open circle) and the vec-
tors required to find the distance from each H atom nucleus. It is straightforward to show
that
φ
R
12
2
+
u
2
1
/
2
R
12
2
−
u
2
1
/
2
z
2
z
2
r
1
=
+
and
r
2
=
+
(A10.20)
where
r
i
is the distance of the general point from nucleus
i
and
R
12
is the internuclear
separation. These expressions assume we have placed the molecule with the bond centre
at the origin.
The angular integral in Equation (A10.19) simply gives 2
π
, and so
∞
∞
σ
g
+
|
σ
g
+
=
2
N
1g
2
1
1
π
(
s
1
+
s
2
)(
s
1
+
s
2
)
u
d
u
d
z
(A10.21)
−∞
0
These integrals are carried out in the
Mathematica
sheet for this appendix available
from the Website, As a check that we obtain unity for the normalized molecular
orbital in the new coordinate system. The integrals over
and
u
involve taking all
contributions from planes perpendicular to the molecular axis at a particular
z
value,
such as the plane illustrated in Figure 10.4b. By carrying out these integrations first
we can plot the total for each plane against
z
and so obtain a picture of the con-
tributions to the integral along the molecular axis. For this example, Figure 10.3b
and d show a comparison of the calculated total density on each plane as a func-
tion of
z
for the bonding and antibonding MOs. Each plot includes a comparison
with the same calculation for an isolated H atom at the position of the leftmost
nucleus (H
1
).
The bonding orbital shows a build-up of density compared with the isolated atom, while
the antibonding orbital gives rise to a lower total density in the internuclear region, going to
zero at the bond centre. This interpretation was also drawn from the plots of the wavefunc-
tions along the molecular axis (Figure 10.3a and c). However, as we are now integrating
over an entire slice of density at each
z
-coordinate, the curves in Figure A10.3b and d
are smoother and do not show the cusps (sudden change in gradient as we pass through
the nuclei) that are seen in the plots along the molecular axis. Notice also that the dip in
the density between the nuclei in the bonding orbital is much less than for the axial plot
(Figure A10.3b compared with Figure A10.3a), showing that the total density integrated
over planes between the H nuclei in the bonding orbital is almost constant.
In the remainder of this appendix we will use this type of plot to illustrate the contri-
butions to the calculated expectation values from planes perpendicular to the molecular
axis.
φ
