This preview shows pages 1–2. Sign up to view the full content.
Chem 120A
BornOppenheimer Approximation, part II
04/20/07
Spring 2007
Lecture 35
1
Review of the BornOppenheimer Approximation
Here we will outline the basic steps in using the BornOppenheimer approximation.
1. We begin by separating out an electronic Hamiltonian,
H
e
(
q
,
Q
)
from the total Hamiltonian,
H
(
q
,
Q
)
.
H
(
q
,
Q
) =
T
N
+
T
e
+
V
ee
+
V
NN
+
V
eN
(1)
=
T
N
+
H
e
(
q
,
Q
)
(2)
where
T
N
is the nuclear kinetic energy,
T
e
is the electron kinetic energy,
V
ee
are the electronelectron
interactions,
V
NN
are the nucleusnucleus interactions, and
V
eN
are the electronnucleus interactions.
Here
q
describes the electronic cooridinates in the molecule and
Q
describes the nuclear coordinates.
The nuclear coordinates act as parameters in the electronic Hamiltonian.
2. Next for each nuclear conﬁguration (parametrized by Q) we can solve the electronic Schr¨odinger
equation to obtain the electronic energies
H
e
(
q
,
Q
)
ψ
n
(
q
,
Q
) =
ε
n
(
Q
)
ψ
n
(
q
,
Q
)
(3)
3. For each nuclear conﬁguration we can write a different nuclear Hamiltonian which incorporates the
electronic energy eigenvalue,
ε
n
(
Q
)
, as a potential energy term. Thus each electronic energy be
comes a different potential energy surface for the nuclei (which is sometimes referred to as a Born
Oppenheimer surface).
H
n
(
Q
) =
T
N
+
ε
n
(
Q
)
(4)
4. Finally, we solve the nuclear Schr¨odinger equations to obtain the total energy
E
n
ν
and the nuclear
wavefunction
φ
n
ν
(The total wavefunction of the molecule is the product of the electronic and nuclear
wavefunctions
Ψ
n
ν
=
ψ
n
φ
n
ν
).
(
T
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '07
 Whaley
 Physical chemistry, Electron, pH

Click to edit the document details