106
5 The Vibrations of Polyatomic Molecules
and, for nonlinear molecules, three coordinates to describe the rotation of the
molecule about its center of mass. There are 3N − 6 degrees of freedom left for
the vibrations. If the molecule is linear, molecular rotations can only occur about the
two axes that are perpendicular to the molecular axis. In that case, there are only 3N
− 5 degrees of freedom.
5.2 Classical Kinetic Energy of the Rigid Rotor
Let x α , y α , z α be the instantaneous coordinates of the αth atom and a α , b α , c α the values
of these coordinates at equilibrium in the molecule fixed axis system. Displacement
from equilibrium will be defined by δx α = x α −a α , δy α = y α −b α , and δz α = z α −c α .
The kinetic energy T may be written (in the following, the vectors and the matrices
will be noted in bold)
2T =
N
α=1
m α v
2
α = ˙
ξ
+ M ˙
ξ = ˙
q
+
˙
q =
3N
i=1
q
2
i
(5.2)
where + means transpose, v α is the velocity of atom α, and M is the 3N × 3N
diagonal matrix of the masses,
ξ
+
= (δx 1 , δy 1 , δz 1 , . . . , δx n , δy n , δz n )
(5.3)
and
q = M
1/2
ξ
(5.4)
It is possible to use a series expansion for the potential energy (assuming that the
displacements q i remain small)
2V = 2V 0 + 2
3N
i=1
∂ V
∂q i
0
q i +
3N
i, j=1
∂
2 V
∂q i ∂q j
0
q i q j + · · ·
(5.5)
Neglecting the higher-order terms gives the harmonic approximation that is
satisfactory as a first approximation and that we will assume in this section.
Assuming that the energy of the equilibrium configuration is zero implies V 0 =
0 as well as for the first derivative
∂ V
∂q i
0
= 0
(5.6)
so that
5 The Vibrations of Polyatomic Molecules
and, for nonlinear molecules, three coordinates to describe the rotation of the
molecule about its center of mass. There are 3N − 6 degrees of freedom left for
the vibrations. If the molecule is linear, molecular rotations can only occur about the
two axes that are perpendicular to the molecular axis. In that case, there are only 3N
− 5 degrees of freedom.
5.2 Classical Kinetic Energy of the Rigid Rotor
Let x α , y α , z α be the instantaneous coordinates of the αth atom and a α , b α , c α the values
of these coordinates at equilibrium in the molecule fixed axis system. Displacement
from equilibrium will be defined by δx α = x α −a α , δy α = y α −b α , and δz α = z α −c α .
The kinetic energy T may be written (in the following, the vectors and the matrices
will be noted in bold)
2T =
N
α=1
m α v
2
α = ˙
ξ
+ M ˙
ξ = ˙
q
+
˙
q =
3N
i=1
q
2
i
(5.2)
where + means transpose, v α is the velocity of atom α, and M is the 3N × 3N
diagonal matrix of the masses,
ξ
+
= (δx 1 , δy 1 , δz 1 , . . . , δx n , δy n , δz n )
(5.3)
and
q = M
1/2
ξ
(5.4)
It is possible to use a series expansion for the potential energy (assuming that the
displacements q i remain small)
2V = 2V 0 + 2
3N
i=1
∂ V
∂q i
0
q i +
3N
i, j=1
∂
2 V
∂q i ∂q j
0
q i q j + · · ·
(5.5)
Neglecting the higher-order terms gives the harmonic approximation that is
satisfactory as a first approximation and that we will assume in this section.
Assuming that the energy of the equilibrium configuration is zero implies V 0 =
0 as well as for the first derivative
∂ V
∂q i
0
= 0
(5.6)
so that