Chemistry Reference
In-Depth Information
(i)
(a)
0.4
(b)
0.0
-0.4
(ii)
0.4
0.0
-0.4
-20
-10
0
10
20
Position (Å)
-20
-10
0
10
20
Figure 2.3
The wavefunctions of the (a) symmetric and (b) anti-symmetric confined
state in the coupled shallow wells of fig. 2.2(a), calculated for central barrier
widths of (i)
b
=
5 Å and (ii)
b
=
2 Å. The solid lines show the exact wave-
functions, while the dashed lines are a variational estimate assuming each
wavefunction to be a linear combination of isolated well wavefunctions.
lines) with the energy levels calculated using eq. (1.37) and the variational
wavefunctions of eq. (2.12) (dotted lines): agreement between the two
remains good to small well separation
b
. This ability to use isolated atomic
wavefunctions as basis states in variational calculations explains in large
part why atomic properties play such a major role in determining trends
in the observed chemical and physical properties of molecules and solids.
We note that the variational method does break down here as
b
0,
particularly for the excited states in the deeper well (fig. 2.4(b)). This is not
surprising as at
b
→
=
0, eqs (2.9) and (2.10) reduce to
k
tan
(
ka
)
=
κ
and
k
cot
(
ka
)
=−
κ
(2.13)
respectively, which are just the equations determining the even and odd
energy levels in a well of width 2
a
.
In summary, through the example of the double quantumwell, we have
shown
1
how molecular energy levels evolve continuously from those of
isolated atoms, as the atoms are brought closer together;
2
how repulsion between energy levels on neighbouring atoms can lead
to the formation of bonding and anti-bonding states;
