5.5 Coriolis Interaction
117
by a direct diagonalization of the Hamiltonian. When there is a Coriolis interaction
between two vibrational states υ k and υ l , there is a non-diagonal term in υ
υ k , υ l |H|υ k + 1, υ l − 1 = 2i B
ξ
e ζ
ξ
kl
ω l
ω k
+
ω k
ω l
(υ k + 1)υ l
4
P ξ + · · ·
(5.36)
ζ
ξ
kl is the Coriolis-zeta constants which couples the vibrations υ k and υ l trough the ξaxis (a, b, c). The spectra are strongly perturbed and difficult to assign. Furthermore,
the least-squares system of equations becomes strongly nonlinear and it is difficult
to obtain accurate parameters. When the Coriolis interaction is not too strong, it is
possible to take it into account by a second-order perturbation calculation. For the
α-constants (see (6.2) of Chap. 6), it gives the following term
α
ξ
k (Cor.) = −2
B
ξ
e
2
ω k
l
3ω
2
k + ω
2
l
ζ
ξ
kl
2
ω
2
k − ω
2
l
(5.37)
When the α-constants are summed to determine the equilibrium rotational
constants (see Chap. 6 and (6.23)), the resonance contribution disappears
1
2
k
α
ξ
k =
l>k
B
ξ
e ζ
ξ
kl
2 (ω k − ω l )
2
ω k ω l (ω k + ω l )
(5.38)
In theory, even if the Coriolis interaction is handled only approximately, the equilibrium rotational constants should not be affected. However, (5.38) is only a firstorder approximation and it fails in case of a strong Coriolis interaction. A typical
example is found in trans formic acid, HCOOH of symmetry C s . Table 5.3 shows the
difficulty to determine reliable values for the individual α-constants for the states υ 7
= 1 (OCO scissor mode at 626.17 cm
−1 ) and υ 9 = 1 (COH torsion at 640.73 cm
–1 )
whereas the sum α 7 + α 9 is correctly determined. These two states are coupled
through strong A- and B-type Coriolis resonances. Table 5.4 gives the values of
α
ξ
6 + α
ξ
8 for states υ 6 = 1 (in-plane C-O stretch at 1104.85 cm
–1 ) and υ 8 = 1 (outof-plane C-H wag at 1033.47 cm
–1 ) that are also coupled by strong A- and B-type
Coriolis resonances and demonstrates the failure of (5.38).
Although the Coriolis interaction is a harmonic phenomenon, its analysis can be
quite difficult and it is often easier to calculate the rovibrational correction from an
ab initio force field using (5.38).
117
by a direct diagonalization of the Hamiltonian. When there is a Coriolis interaction
between two vibrational states υ k and υ l , there is a non-diagonal term in υ
υ k , υ l |H|υ k + 1, υ l − 1 = 2i B
ξ
e ζ
ξ
kl
ω l
ω k
+
ω k
ω l
(υ k + 1)υ l
4
P ξ + · · ·
(5.36)
ζ
ξ
kl is the Coriolis-zeta constants which couples the vibrations υ k and υ l trough the ξaxis (a, b, c). The spectra are strongly perturbed and difficult to assign. Furthermore,
the least-squares system of equations becomes strongly nonlinear and it is difficult
to obtain accurate parameters. When the Coriolis interaction is not too strong, it is
possible to take it into account by a second-order perturbation calculation. For the
α-constants (see (6.2) of Chap. 6), it gives the following term
α
ξ
k (Cor.) = −2
B
ξ
e
2
ω k
l
3ω
2
k + ω
2
l
ζ
ξ
kl
2
ω
2
k − ω
2
l
(5.37)
When the α-constants are summed to determine the equilibrium rotational
constants (see Chap. 6 and (6.23)), the resonance contribution disappears
1
2
k
α
ξ
k =
l>k
B
ξ
e ζ
ξ
kl
2 (ω k − ω l )
2
ω k ω l (ω k + ω l )
(5.38)
In theory, even if the Coriolis interaction is handled only approximately, the equilibrium rotational constants should not be affected. However, (5.38) is only a firstorder approximation and it fails in case of a strong Coriolis interaction. A typical
example is found in trans formic acid, HCOOH of symmetry C s . Table 5.3 shows the
difficulty to determine reliable values for the individual α-constants for the states υ 7
= 1 (OCO scissor mode at 626.17 cm
−1 ) and υ 9 = 1 (COH torsion at 640.73 cm
–1 )
whereas the sum α 7 + α 9 is correctly determined. These two states are coupled
through strong A- and B-type Coriolis resonances. Table 5.4 gives the values of
α
ξ
6 + α
ξ
8 for states υ 6 = 1 (in-plane C-O stretch at 1104.85 cm
–1 ) and υ 8 = 1 (outof-plane C-H wag at 1033.47 cm
–1 ) that are also coupled by strong A- and B-type
Coriolis resonances and demonstrates the failure of (5.38).
Although the Coriolis interaction is a harmonic phenomenon, its analysis can be
quite difficult and it is often easier to calculate the rovibrational correction from an
ab initio force field using (5.38).