4.6 Linear Molecule
87
4.6 Linear Molecule
The linear molecule is a special case of the symmetric molecule with I z = 0 and I x
= I y . Thus, the rotational energy is
E J = B J (J + 1)−D J J
2
(J + 1)
2
+ · · ·
(4.33)
as for the diatomic molecule.
4.7 Rotational Spectra of Linear and Symmetric Tops
in Excited Vibrational States
A linear molecule with N atoms has 3 N−5 modes of vibration, but the bending
modes are always doubly degenerate because the deformation of the angle may
occur in two orthogonal planes. For instance, in the case of a non-symmetric linear
molecule XYZ, there are two non-degenerate parallel modes ν 1 and ν 3 which do not
affect the linearity and correspond to the stretching vibrations of the bonds XY and
YZ, and there is a doubly degenerate bending mode ν 2 which corresponds to the
deformation of the angle XYZ. The rotational spectra of the non-degenerate states
behave in the same way as a ground state. On the other hand, as the molecule is no
more linear in the bending state, there is an angular momentum p z = along the
symmetry axis with the rotational quantum numbers = ±υ, ±(υ − 2), . . . ± 1 or
0. is analogous to the K quantum number of the symmetric top. The energy in a
degenerate state may be written
E υ = B υ J (J + 1) + (A υ −B υ )
2
+ · · ·
(4.34)
The term A υ
2 is treated as vibrational energy, and the rotational energy is
E υ = B υ [J (J + 1) −
2 ]−D υ [J (J + 1)−
2
]
2
+ · · ·
(4.35)
With the selection rule J = 1 and = 0. Finally, the rotational frequency is
ν = 2B υ (J + 1)−4D υ (J + 1)
3
+ · · ·
(4.36)
Equation (4.36) is identical to the equation used for the ground vibrational state
but with the difference that J = ||, || + 1, || + 2, . . . In other words, for an excited
bending state with || = 1, the lowest value of J is 1. States with = 0 are labeled
states, those with = ±1, ±2, ±3 are called , , states, respectively.
In a symmetric top molecule, there are also degenerate vibrations, in particular
those involving motions of the atoms which are off-axis. For a prolate top, the term
87
4.6 Linear Molecule
The linear molecule is a special case of the symmetric molecule with I z = 0 and I x
= I y . Thus, the rotational energy is
E J = B J (J + 1)−D J J
2
(J + 1)
2
+ · · ·
(4.33)
as for the diatomic molecule.
4.7 Rotational Spectra of Linear and Symmetric Tops
in Excited Vibrational States
A linear molecule with N atoms has 3 N−5 modes of vibration, but the bending
modes are always doubly degenerate because the deformation of the angle may
occur in two orthogonal planes. For instance, in the case of a non-symmetric linear
molecule XYZ, there are two non-degenerate parallel modes ν 1 and ν 3 which do not
affect the linearity and correspond to the stretching vibrations of the bonds XY and
YZ, and there is a doubly degenerate bending mode ν 2 which corresponds to the
deformation of the angle XYZ. The rotational spectra of the non-degenerate states
behave in the same way as a ground state. On the other hand, as the molecule is no
more linear in the bending state, there is an angular momentum p z = along the
symmetry axis with the rotational quantum numbers = ±υ, ±(υ − 2), . . . ± 1 or
0. is analogous to the K quantum number of the symmetric top. The energy in a
degenerate state may be written
E υ = B υ J (J + 1) + (A υ −B υ )
2
+ · · ·
(4.34)
The term A υ
2 is treated as vibrational energy, and the rotational energy is
E υ = B υ [J (J + 1) −
2 ]−D υ [J (J + 1)−
2
]
2
+ · · ·
(4.35)
With the selection rule J = 1 and = 0. Finally, the rotational frequency is
ν = 2B υ (J + 1)−4D υ (J + 1)
3
+ · · ·
(4.36)
Equation (4.36) is identical to the equation used for the ground vibrational state
but with the difference that J = ||, || + 1, || + 2, . . . In other words, for an excited
bending state with || = 1, the lowest value of J is 1. States with = 0 are labeled
states, those with = ±1, ±2, ±3 are called , , states, respectively.
In a symmetric top molecule, there are also degenerate vibrations, in particular
those involving motions of the atoms which are off-axis. For a prolate top, the term