Chemistry Reference
In-Depth Information
a classical perspective) if the material is raised to a nonzero temperature is that
the atoms in the material will vibrate about their equilibrium positions. From a
more correct quantum mechanical perspective, the vibrations that are possible
around an atom's equilibrium position contribute to the material's energy
even at 0 K via zero-point energies. In many instances, these vibrations can
be measured experimentally using spectroscopy, so the frequencies of
vibrations are often of great interest. In this chapter, we look at how DFT cal-
culations can be used to calculate vibrational frequencies.
5.1
ISOLATED MOLECULES
We will begin with a simple example, the vibrations of an isolated CO mol-
ecule. More specifically, we consider the stretching of the chemical bond
between the two atoms in the molecule. For convenience, we assume that
the bond is oriented along the
x
direction in space. The bond length is then
defined by
b ¼ x
C
x
O
, where
x
C
and
x
O
are the positions of the two
atoms. A Taylor expansion for the energy of the molecule expanded around
the equilibrium bond length,
b
0
, gives
b b
0
þ
b b
0
þ
b
0
)
2
d
2
E
dE
db
1
2
(
b
E ¼ E
0
þ
(
b
b
0
)
(5
:
1)
db
2
The first derivative term is exactly zero because it is evaluated at the energy
minimum. So, for small displacements about the equilibrium bond length,
b
0
,
E ¼ E
0
þ a=
b
0
)
2
db
2
]
b b
0
. This approach,
which neglects the higher order terms in the Taylor expansion, is called the
harmonic approximation
.
How do the atoms move within this harmonic approximation? Treating the
C nucleus as a classical particle following Newton's law, we have
F
C
¼ ma
C
with
F
C
¼ @
where
a ¼
[
d
2
E
2(
b
=
dt
2
. Similar equations may be written for
the position of the O nucleus. A little algebra shows that the equation of motion
for the overall bond length is
x
C
and
a ¼ d
2
x
C
=
E
=@
(
b
(
t
)
d
2
b
(
t
)
dt
2
¼ a
m
C
þ
m
O
m
C
m
O
b
0
)
:
(5
:
2)
The solution of this equation is
b
(
t
)
¼ b
0
þ a
cos
vt
where
a
is an arbitrary
constant and
r
m
C
þ
m
O
m
C
m
O
v ¼
a
:
