Chemistry Reference
In-Depth Information
describing bond stereochemistry in polyatomic molecules, with particular
emphasis on the H
2
O molecule. A few applications of Pauling' semiem-
pirical theory of
electrons in conjugated and aromatic hydrocarbons
conclude the chapter.
p
9.1 THE BORN-OPPENHEIMER APPROXIMATION
This concerns the separation in molecules of the motion of the light
electrons from the slow motion of the heavy nuclei.
We want to solve the molecular wave equation
H
Y ¼
W
Y
ð
9
:
1
Þ
where H is the molecular Hamiltonian (in atomic units):
!
H
¼
X
a
X
1
2M
a
r
1
2
r
2
2
a
þ
i
þ
V
en
þ
V
ee
þ
V
nn
i
ð
9
:
2
Þ
¼
X
a
1
2M
a
r
a
þ
H
e
þ
V
nn
2
In the expression above, the first term is the kinetic energy operator
for the motion of the nuclei,
1
the term in parentheses is the electronic
Hamiltonian H
e
and the last term is the Coulombic repulsion between the
point-like nuclei in the molecule.
Since wave equation (9.1) was too difficult to solve, Born and Oppen-
heimer (1927) suggested that, in a first approximation, the molecular
wave function
Y
could be written as
Yð
x
;
q
ÞY
ð
x
;
q
ÞY
ð
q
Þ
ð
9
:
3
Þ
e
n
where
Y
e
is the electronic wavefunction, an ordinary function of the
electronic coordinates x and parametric in the nuclear coordinates q.
Y
e
is a normalized solution of the electronic wave equation
2
H
e
ð
9
:
4
Þ
Y
¼
E
e
Y
;
hY
jY
i¼
1
e
e
e
e
1
in units of the electron mass.
2
Which must be solved for any nuclear configuration specified by {q}.
M
a
is the mass of nucleus
a
