Chemistry Reference
In-Depth Information
correct behaviour is obtained at large
R
12
. We can also see from this plot that, at relatively
large separations (above 3.5 bohr), the main attraction results from a decrease in the elec-
tron's kinetic energy which acts against the repulsion generated by the potential terms.
Only as the bond length approaches the observed H
2
+
internuclear separation does the
potential energy switch over to negative values, and the confinement of the electron gives
a net positive kinetic energy. Thus, at the optimal bond length the attractive contribution
to the bond formation energy comes from the potential energy terms.
The same procedure can be followed for the first excited state, and the energy compo-
nents and total as function of
R
12
are shown in Figure A10.12. At each point the decay
constant has been optimized to give the lowest possible energy for the electron in the anti-
bonding
orbital. The total energy is positive everywhere, tending to zero only at
large
R
12
. So, the excited state of H
2
+
is unstable with respect to dissociation. The poten-
tial energy is negative down to almost the H
2
+
bond length, where it is practically zero.
The molecule is actually destabilized by the increased kinetic energy in the antibonding
state.
The value obtained for the optimal decay constant reflects the electron response to the
potential of the nuclei. Since the electron distribution in the antibonding orbital is quite
different to that in the bonding state, we obtain different basis function decay constants
for the two orbitals. The optimal values of
|
2
σ
u
+
as a function of the internuclear separation
is plotted in Figure A10.13. We have already seen that for the ground state the basis func-
tions in the
ζ
orbital contract as the molecule is formed, leading to a lowering of the
potential energy. In the excited state, the values of
|
1
σ
g
+
ζ
that minimize the total energy in the
|
orbital for
R
12
below around 4 bohr are less than 1 bohr
−
1
. This corresponds to an
expansion of the MO compared with that obtained with the atomic basis functions, which
will tend to decrease the kinetic energy by confining the electron less closely.
2
σ
u
+
(a)
(b)
/bohr
-1
ζ
Energy/Ha
0.6
1.5
Total
0.4
1.0
〈
T
(H
2
+
)〉
2
σ
u
+
- 〈
T
(H)〉
0.2
0.5
R
12
/bohr
2
4
6
8
U
(H
2
+
)
〈
〉
2
σ
u
+
-
〈
U
(H)
〉
0
2
4
6
8
R
12
/bohr
Figure A10.13
The optimized values of the basis decay factor
ζ
as a function of internuclear
separation R
12
for the
|
1
σ
g
+
(solid line) and
|
2
σ
u
+
(dashed line) MOs.
An important point to note from this study of a relatively simple system is that the AO
functions as derived for isolated atoms are not the ideal basis for constructing MOs. In
'real' molecular systems the orbital shape adapts to the potential it experiences. For quan-
titative work we require basis sets that can reproduce this by responding to the potential
