Chemistry Reference
In-Depth Information
or molecule, with the lowest energy levels being filled first. If each of the
quantum wells or 'atoms' in fig. 2.2(a) has one electron in the level bound
in the quantum well, then the overall energy will be reduced by form-
ing the coupled quantum well, where the two electrons can occupy the
lowest energy level, as in the diatomic hydrogen molecule, H
2
. This level
is referred to as the bonding level. By contrast, if there are already two
electrons in the highest filled level of each isolated well or atom, as for
helium (He), the second element in the Periodic Table, it will cost energy
to form a molecule, with two of the electrons going into the lower (bond-
ing) level and the other two into the upper (anti-bonding) level. Hence He
gas is made up of isolated He atoms rather than He
2
or more complicated
molecules.
(3)
Core and valence levels
: Returning to the deep well in fig. 2.2(b), we
see for moderate
b
that the splitting between the highest energy levels is
considerably larger than is the case for the lower energy levels. This can
be understood from eqs (2.9) and (2.10) if we note that the magnitude of
κ
={
2
increases as the energy decreases, going towards
the bottom of the quantum well. In this situation, tanh
2
1
/
2
m
(
V
0
−
E
)/
}
)
are much closer to 1 for the lower energy levels than for those nearer to
the well maximum, and so the deep states are far less perturbed from their
isolated well values compared to the higher levels. The same is true in
molecules and solids where the deeper energy levels, referred to as core
levels, are largely unperturbed and do not take part in bonding. Hence,
despite having different numbers of core electrons, gaseous flourine (F),
chlorine (Cl) and iodine (I) all exist as diatomic molecules, F
2
,Cl
2
and I
2
,
as all have the same number of valence electrons.
(4)
Linear combinations of atomic orbitals
: Figure 2.3(a) and (b) show the
wavefunctions (solid lines) for the shallow-well bonding and anti-bonding
energy levels at selected values of interwell separation,
b
. It can be seen
even for small
b
that the wavefunctions are virtually indistinguishable
from symmetric and anti-symmetric combinations of the isolated quantum
well functions (indicated by the dotted lines). We may write the coupled
well wavefunction
(κ
b
)
and coth
(κ
b
ψ(
x
)
as
ψ
(
)
=
α(φ
(
)
+
φ
(
))
x
x
x
s
L
R
(2.12)
ψ
(
x
)
=
β(φ
(
x
)
−
φ
(
x
))
a
L
R
for the symmetric and anti-symmetric cases, respectively, where
φ
(
x
)
and
L
φ
are eigenstates of
a
n isolated left-hand and right-hand well respec-
tively, and
(
x
)
R
/
√
2 for large separation. We see from fig. 2.3 that this
linear combination of isolated well wavefunctions (i.e. 'atomic orbitals')
should act as a good variational guess at the molecular wavefunctions up
to relatively small well separations
b
. This is indeed confirmed by fig. 2.4(a)
and (b), which compare the exact double well energy levels of fig. 2.2 (solid
α
=
β
=
1
