Chemistry Reference
In-Depth Information
With this value of
ζ
at a bond length of 2 bohr we find
T
(H
2
+
)
U
(H
2
+
)
1
σ
g
+
=
0.5871 Ha
1
σ
g
+
=−
1.1726 Ha Ha
and so for the virial test we have
T
(H
2
+
)
0.5871
1.1726
=−
=−
0.5003
U
(H
2
+
)
For the optimized decay constant, the virial theorem is satisfied within the significance
of the data. The total energy is also lower than that obtained with the rigid orbital model
and we now have an estimated bond formation energy of
224 kJ mol
−
1
.
This is now remarkably close to the 255 kJ mol
−
1
for the bond energy given in Table 7.1,
considering that we have neglected the zero-point vibrational energy and the basis is still
quite simple (see Appendix 11).
Note that now the electron kinetic energy is greater than the atomic reference state
and stability is provided by the lower potential energy. The shrinking of the AOs around
the nuclei increases the electron-nuclear favourable interactions and now outweighs the
nuclear-nuclear repulsion.
A plot of the bond formation energy as a function of the nuclear separation with the
decay constant optimized is given in Figure A10.12a. The minimum for the total energy
has now shifted to 2 bohr and the potential well is deeper than for the same calculation
with the AO value of
−
0.0855 Ha or
−
. However, at large separation the potential is now incorrect, as the
decay constant is too large for the atomic state.
ζ
(a)
(b)
Energy/Ha
Energy/Ha
0.10
0.15
T
(H
2
+
)
〈
〉
1
σ
g
+
-
〈
T
(H)
〉
0.10
0.05
U
(H
2
+
)
〈
〉
1
σ
g
+
-
〈
U
(H)
〉
0.05
2
4
6
8
R
12
/bohr
2
4
6
8
-0.05
Total,
ζ
= 1
-0.05
-0.10
-0.15
R
12
/bohr
-0.10
-0.15
Total
Total
T
(H
2
+
)
〈
〉
1
σ
g
+
-
〈
T
(H)
〉
U
(H
2
+
)
〈
〉
1
σ
g
+
-
〈
U
(H)
〉
-0.20
Figure A10.12
(a) Plot of kinetic energy, potential energy contributions of and the total bond
formation energy as a function of R
12
using basis functions optimized at R
12
=
2
bohr, i.e. rigid
orbitals with
1
)is
included for comparison. (b) Plot of the energy contributions and total with the basis decay
constant obtained by minimizing the total energy at each R
12
value.
ζ
=
1.238
; the total energy plot for the rigid atomic basis functions (
ζ
=
for each internu-
clear separation so that the electron wavefunction responds to the changing environment
as the molecule is formed. This has been done in Figure A10.12b, which shows that the
A more accurate picture can be obtained by optimizing the value of
ζ
