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SOLUTIONSMANUAL TO ACCOMPANY ATKINS' PHYSICAL CHEMISTRY 385
�is mode is in fact not Raman active , something which is best deduced using
the symmetry arguments discussed in Topic 11E.�e rule of mutual exclusion
does not apply.
E11D.6(b) �e benzenemolecule has a centre of symmetry, so the rule ofmutual exclusion
applies. �e molecule has no permanent dipole moment but if it is distorted
into a boat-like shape the expectation is that there will be an induced dipole
and that therefore the mode will be infrared active .
According to the rule of mutual exclusion, such a mode is not Raman active .
E11D.7(b) �e exclusion rule applies only to molecules with a centre of symmetry. C2H4
does possess such symmetry, and so the exclusion rule does apply .
Solutions to problems
P11D.2 NO2+ is isoelectronic with CO2, so the molecule is linear. Around the central
N there are four electron pairs, arranged as two double bonds.
�e normal modes of NO2+ are analogous to those of CO2 shown in Fig. 11D.2
on page 452.�e oneRaman active normalmode at 1400 cm−1 is the symmetric
stretch (ν1).�e lowest frequencymode at 540 cm−1 is the (doubly degenerate)
bend (ν3): typically, bends are a lower frequencies than stretches.�e infrared
active mode at 2360 cm−1 is the asymmetric stretch (ν2).
�e weak absorption at 3735 cm−1 is likely to be either an overtone or a com-
bination band. �e most likely candidate is for one quantum to be excited
into each of the modes ν1 and ν2; such a transition would have wavenumber
1400 cm−1 + 2360 cm−1 = 3760 cm−1. �e numbers do not agree exactly due
to the e�ects of anharmonicity.
11E Symmetry analysis of vibrational spectroscopy
Answer to discussion question
Solutions to exercises
E11E.1(b) Note that there is an error in the Exercise: the displacements span A1g +2A1u +
2E1u + E1g.
�e displacements include translations and rotations. For the point groupD∞h,
(x , y) and z transform as E1u and A1u, respectively. �e rotations (Rx , Ry)
transform as E1g; for a linear molecule there is no rotation about the inter-
nuclear axis and so the symmetry species for Rz is not taken away. Taking
these symmetry species away leaves just the symmetry species of the normal
modes as A1g +A1u + E1u . �eses correspond to 4 normal modes, which is
the number expected for a linear triatomic (the E1u is doubly degenerate).