5.6 Anharmonicity
121
λ =
E
(0)
i + E
(0)
j
2
±
δ
2
2
+ F 2
(5.52)
with
δ = E
(0)
i − E
(0)
j
(5.53)
The eigenfunctions may be written
ψ i = cψ
(0)
i + sψ
(0)
j
ψ j = −sψ
(0)
i + cψ
(0)
j
(5.54)
with
c =
√
4F 2 + δ 2 + δ
2
√
4F 2 + δ 2
and s =
1 − c 2
(5.55)
When two vibrational states i and j are in Fermi resonance, the rotational energy
may be written
E R (A
i , B
i , C
i ) = ψ i |H R |ψ i
= c
2
ψ
(0)
i
HR
ψ
(0)
i
+ s
2
ψ
(0)
j
HR
ψ
(0)
j
= E R (A i , B i , C i ) + s
2
E R (A j , B j , C j ) − E R (A i , B i , C i )
(5.56)
where the prime (
) indicates the rotational constants affected by the Fermi resonance.
If the resonance is not too strong and assuming that the rotational energy is a
linear function of the rotational constants (it is only a first-order approximation), the
rotational constants may be written
B
ξ
i = B
ξ
i + s
2
B
ξ
j − B
ξ
i
(5.57)
with ξ = a, b, c.
A typical example is given by the Fermi resonance between the states υ 1 = 1
and υ 2 = 2 of F 2 O for which the two states are separated by about 6.8 cm
−1 , see
Table 5.5. It is possible to estimate the rotational constants of the υ 2 = 2 state from
those of the ground state and the υ 2 = 1 state
B
ξ
(υ 2 = 2) = B
ξ
0 − 2α
ξ
2
(5.58)
Using
B
ξ
(υ 2 = 2) = B
ξ
(υ 2 = 2) + b
2
(2α
ξ
2 − α
ξ
1 )
(5.59)