Chemistry Reference
In-Depth Information
At the end of a calculation in atomic units, as we always shall do, the
actual SI values can be obtained by taking into account the SI equiva-
lents (1.65).
The Hamiltonian of the hydrogenic system in atomic units will then
take the following simplified form:
1
2
r
Z
r
H
¼
2
ð
1
:
66
Þ
Fromnowon, we shall consistently use atomic units everywhere, unless
explicitly stated.
(c) The atomic two-electron system
Two electrons are attracted by a nucleus of charge
þ
Z.The
Hamiltonian will be
1
2
r
1
2
r
Z
r
1
Z
r
2
þ
1
1
r
12
H
¼
r
12
¼
h
1
þ
h
2
þ
2
2
1
2
ð
1
:
67
Þ
where
1
2
r
Z
r
h
¼
2
ð
1
:
68
Þ
is the one-electron Hamiltonian (which has the same functional
form for both electrons) and the last term is the Coulomb repulsion
between the electrons (a two-electron operator). Z
¼
2givestheHe
atom.
(d) The hydrogen molecule-ion H
2
This is a diatomic one-electron molecular system, where the electron is
simultaneously attracted by the two protons at A and B. The Born-
Oppenheimer Hamiltonian (see Chapter 9) will be
1
R
¼
1
2
r
1
r
A
1
r
B
þ
1
H
¼
h
þ
R
¼
h
A
þ
V
2
ð
1
:
69
Þ
where h
A
is the one-electron Hamiltonian (1.68) for atom A (with
Z
¼
1) and
1
r
B
þ
1
R
V
¼
ð
1
:
70
Þ
is the interatomic potential between the hydrogen atom A and the
proton B.
