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358 11MOLECULAR SPECTROSCOPY
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.5
1.0
1.5
2.0
ν̃p
1
+
co
s(
2π
ν̃p
)
δ = 0.11
δ = 0.23
δ = 0.50
δ = 0.73
Figure 11.2
D11B.8 1H2molecules can exist in two forms: the para form has antisymmetric nuclear
spin wavefunctions and the ortho form has symmetric nuclear spin wavefunc-
tions. As discussed in Section 11B.4 on page 439, it follows that because of
these arrangements of the nuclear spins the ortho form must have rotational
wavefunctions restricted to odd J values. Ortho hydrogen therefore cannot
exist in the J = 0 state and hence it always has some rotational energy: that
is, it has rotational zero-point energy.
Solutions to exercises
E11B.1(b) Expressions for the moments of inertia are given in Table 11B.1 on page 431;
PH3 is a symmetric rotor and the second entry under symmetric rotors is the
required one.�e moment or inertia about the threefold axis is I∣∣
I∣∣ = 2mH(1 − cos θ)R2
= 2[(1.0078) × (1.6605 × 10−27 kg)] × (1 − cos 93.5○) × (142 × 10−12 m)2
= 7.16... × 10−47 kgm2 = 7.16 × 10−47 kgm2
�e corresponding rotational constant is given by [11B.13b–433]
à = ħ
4πcI∣∣
= 1.0546 × 10−34 J s
4π × (2.9979 × 1010 cms−1) × (7.16... × 10−47 kgm2)
= 3.91 cm−1
E11B.2(b) �ere is an error in the Exercise: the molecule is AB4, not AB3. �e required
expressions are the �rst listed under symmetric rotors in Table 11B.1 on page