Chemistry Reference
In-Depth Information
A10.9 Optimal Radial Decay of Molecular Orbitals
In Problem A10.1 we proposed a trial form for the 1s orbital in which the decay with
distance from the nucleus is controlled by a parameter
ζ
:
s
tr
=
ζ
3
/
2
|
exp(
−
ζ
r
)
(A10.31)
π
It was also shown that the decay constant that was consistent with the virial theorem
gave the lowest total energy for the atomic 1s orbital. We can now apply this to the
molecular case by recalculating the kinetic, potential and total energies for the electron
in the
orbital in H
2
+
fixed at the experimental bond length but for a range of decay
constants. The result is shown in Figure A10.11.
|
1
σ
g
+
ζ
= 1.7
ζ
= 0.5
Energy/Ha
〈
T
(H
2
+
)〉
1
σ
g
0.5
-4
-2
0
2
4
-4
-2
0
2
4
0.6
0.8
1.0
1.2
1.4
1.6
ζ
/ bohr
−1
-0.5
〈
T
(H
2
+
)〉
1
σ
g
+ 〈
U
(H
2
+
)〉
1
σ
g
ζ
= 1.238
-1.0
-1.5
〈
U
(H
2
+
)
〉
1
σ
g
-4
-2
0
2
4
U
(H
2
+
)
Figure A10.11
Plot of expectation values for the potential energy
1
σ
g
, kinetic energy
T
(H
2
+
)
σ
g
+
MO of
H
2
+
at a nuclear separation
1
σ
g
and total energy of an electron in the
|
1
of R
12
=
2
bohr against the basis function decay factor
ζ
. The diagrams inset to the side of
the plot show the MO density at the
ζ
values indicated compared with the
ζ
=
1
distribution
(dashed lines).
orbital becomes more spread out; this has the effect of
reducing the kinetic energy, as the MO has less curvature than when we use the AO decay
constant in the basis. The more diffuse orbital also increases the potential energy, as the
electron spends more time away from the nuclei. This wins out and the energy increases
compared with the
For
ζ<
1 bohr
−
1
,the
|
1
σ
g
+
ζ
=
1 reference.
orbital becomes more compact and the potential energy
goes down. Of course, this also confines the electron more closely and increases the cur-
vature of the wavefunction, and thus the kinetic energy. Near to unity the net effect is a
lowering of the total energy, and a minimum is found for
For
ζ>
1 bohr
−
1
,the
|
1
σ
g
+
1.238 bohr
−
1
. At higher
values of the decay constant the kinetic energy increases more rapidly than the potential
falls.
ζ
=
