50
2 Molecular States
ˆ
H
(k)
n ˆ
H
(k)
rot + ˆ
H
(k)
vib
(2.108)
ˆ
H
(k)
rot =
1
2
J n (I
(k)
)
−1 J n
(2.109)
where J n is the nuclear angular momentum and I
(k) is the inertia tensor for the nuclear
configuration corresponding to R
(k)
eq . The vibrational Hamiltonian ˆ
H
(k)
vib has the same
functional form as ˆ
H
(k)
n of Eq. (2.58), but is written in terms of the Cartesian coordinates in the rotating reference frame where I
(k) is diagonal (i.e., in the principal axis
system of the molecule). In this way any roto-vibrational coupling between nuclear
internal and rotational motions is neglected. Note however that these couplings are not
connecting different electronic states, so that they are not able to promote electronic
transitions. For simplicity, in the following sections we will drop the superscript
(k)
indicating the electronic state to which the vibrational and rotational Hamiltonians
are referred.
2.5.1 Rotational States
In the rotating reference frame referred above the Hamiltonian ˆ
H rot simplifies to
ˆ
H rot =
ˆ
J
2
nξ
2I ξξ
+
ˆ
J
2
nη
2I ηη
+
ˆ
J
2
nζ
2I ζ ζ
(2.110)
where ξ, η, ζ label the three components of the rotating frame. The eigenfunctions
of ˆ
H rot are the rotational functions, which depend on the Euler angles α, β, and γ
relating the fixed reference frame (x, y, z) to the rotating one (ξ, η, ζ ). As a consequence of the isotropy of space the angular momentum (in the present case the
nuclear angular momentum) is a conserved quantity. Then, the rotational Hamiltonian ˆ
H rot commutes with ˆ
J
2
n and ˆ
J nz ; its eigenstates ψ
J M
rot can be written as linear
combinations of the rigid spherical rotor wavefunctions, which represent a complete
basis and correspond to the Wigner functions D
(J )
K M
ψ
J M
rot =
J
K =−J
C
(J )
K D
(J )
K M (α, β, γ ) .
(2.111)
The Wigner functions are common eigenfunctions of the set of commuting operators
ˆ
J
2
n , ˆ
J nz and ˆ
J nζ , so that
ˆ
J
2
n D
(J )
K M =
2 J (J + 1)D
(J )
K M
ˆ
J nz D
(J )
K M = M D
(J )
K M
ˆ
J nζ D
(J )
K M = K D
(J )
K M .
(2.112)
2 Molecular States
ˆ
H
(k)
n ˆ
H
(k)
rot + ˆ
H
(k)
vib
(2.108)
ˆ
H
(k)
rot =
1
2
J n (I
(k)
)
−1 J n
(2.109)
where J n is the nuclear angular momentum and I
(k) is the inertia tensor for the nuclear
configuration corresponding to R
(k)
eq . The vibrational Hamiltonian ˆ
H
(k)
vib has the same
functional form as ˆ
H
(k)
n of Eq. (2.58), but is written in terms of the Cartesian coordinates in the rotating reference frame where I
(k) is diagonal (i.e., in the principal axis
system of the molecule). In this way any roto-vibrational coupling between nuclear
internal and rotational motions is neglected. Note however that these couplings are not
connecting different electronic states, so that they are not able to promote electronic
transitions. For simplicity, in the following sections we will drop the superscript
(k)
indicating the electronic state to which the vibrational and rotational Hamiltonians
are referred.
2.5.1 Rotational States
In the rotating reference frame referred above the Hamiltonian ˆ
H rot simplifies to
ˆ
H rot =
ˆ
J
2
nξ
2I ξξ
+
ˆ
J
2
nη
2I ηη
+
ˆ
J
2
nζ
2I ζ ζ
(2.110)
where ξ, η, ζ label the three components of the rotating frame. The eigenfunctions
of ˆ
H rot are the rotational functions, which depend on the Euler angles α, β, and γ
relating the fixed reference frame (x, y, z) to the rotating one (ξ, η, ζ ). As a consequence of the isotropy of space the angular momentum (in the present case the
nuclear angular momentum) is a conserved quantity. Then, the rotational Hamiltonian ˆ
H rot commutes with ˆ
J
2
n and ˆ
J nz ; its eigenstates ψ
J M
rot can be written as linear
combinations of the rigid spherical rotor wavefunctions, which represent a complete
basis and correspond to the Wigner functions D
(J )
K M
ψ
J M
rot =
J
K =−J
C
(J )
K D
(J )
K M (α, β, γ ) .
(2.111)
The Wigner functions are common eigenfunctions of the set of commuting operators
ˆ
J
2
n , ˆ
J nz and ˆ
J nζ , so that
ˆ
J
2
n D
(J )
K M =
2 J (J + 1)D
(J )
K M
ˆ
J nz D
(J )
K M = M D
(J )
K M
ˆ
J nζ D
(J )
K M = K D
(J )
K M .
(2.112)