Chemistry Reference
In-Depth Information
usually assume that the potential energy surface is harmonic, i.e. the change of energy as
the molecule vibrates is proportional to the square of the displacement from the minimum
energy point.
To picture this we can define a coordinate
q
for any vibrational state which is a sin-
gle value to indicate how far from equilibrium the oscillation has moved. An example
for the symmetric stretch in H
2
O is shown in Figure 6.2. Here, the vibrational motion
involves the H atoms moving from/toward the O atom in phase with one another while the
O atom moves only along the
C
2
axis. The idea of the normal mode coordinate
q
is that it
describes where in this collective motion the atoms are at any given time. In the harmonic
approximation the potential energy for this vibrational motion is proportional to
q
2
.
E
n
=
1
E
ph
=
hv
ph
3
E
1
=
hv
2
n
=
0
1
E
0
=
hv
2
q
Figure 6.2
The normal mode coordinate for the symmetric stretch mode (A
1
) of water.
The illustrations in Figure 6.2 show that for
q
negative, to the left of the minumum in
the potential energy curve, the H atoms are moving toward the O, while for
q
positive they
are moving away.
Each mode has a set of energy levels that form a regular ladder of states, with energies
E
n
given by
E
n
=
n
2
h
1
+
ν
(6.1)
Here,
n
is a quantum number taking values 0, 1, 2 ....etc.,thevibrational frequency,
v
,is
in s
−
1
units of the mode and
h
is the Planck constant (6.626
×
10
−
34
J s). The lowest energy
state (
n
1) differ in energy and in the amplitude of the
oscillation. In the higher energy state the atoms can move further from the minimum point
before bond strain forces cause them to return.
=
0) and the first excited state (
n
=
