Biomedical Engineering Reference
In-Depth Information
The hydrogen molecular ion is interesting in that it does not have an electron pair yet it
is stable. The obvious question is, can we give a hand-waving explanation for this stability,
one that does not rely on the presence of electron pairs?
The obvious place to look is the electron density. The electron density plots by themselves
are not particularly informative; the ground-state wavefunction shows a region of appar-
ently enhanced electron density between the nuclei, whilst the excited-state wavefunction
has less electron density in this region. It ismore interesting to look at the difference between
the electron density in the molecule and two ground-state hydrogen atoms, each of which
contains half an electron. Such plots are called
density differences
. Positive contours corres-
pond to a gain of electron density (in units of
e
) compared to two overlapped half hydrogens;
negative contours correspond to a loss. The molecule lies along the horizontal axis.
The density difference plot for the ground state (Figure 15.4) illustrates an electron
density enhancement in the region between the nuclei and so a stable molecule. Electron
density has to be redistributed, it cannot appear from nowhere and we see that the bond
region gains at the expense of regions beyond the nuclei. The excited-state wavefunction
shows the opposite effect: there is a depletion of electron density in the bond region, and
a gain of electron density beyond the nuclei (Figure 15.5). This explains the instability of
this state, so we can give a simple electrostatic explanation for the stability of the molecule
in its ground state, and its instability in the excited state.
It is a general principle that, once we have calculated an electron density from the laws
of quantum mechanics, we can analyse it by the laws of classical electrostatics.
40
0.002
0.006
30
-0.003
-0.008
-0.008
-0.003
-0.02
0.002
-0.021
-0.003
-0.026
-0.026
-0.017
-0.012
-0.008
-0.017
0.011
-0.003
-0.012
0.002-0.008
20
0.006
-0.003
0.002
10
0
0
10
20
30
40
Figure 15.4
Density difference for ground state
