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380 11MOLECULAR SPECTROSCOPY
P11C.12 �e expectation is that the separation between the lines in the P and R branches
will be approximately constant at 2B̃; however the separation of the �rst line of
the R branch, ν̃R(0), from the �rst line of the P branch, ν̃P(1), is approximately
4B̃. Examination of the data shows that the separation between the lines at
2906.25 cm−1 and at 2865.14 cm−1 is approximately twice that between the
other lines.�erefore 2906.25 cm−1 is assigned as the �rst line of the R branch,
the transition J = 1 ← 0, and 2865.14 cm−1 is assigned as the �rst line of the P
branch, the transition J = 0 ← 1. �e assignment of the other lines follows by
working out from these.
In the method of combination di�erences, lines sharing a common upper (or
lower) J level are identi�ed and a plot is made of the di�erence between the
wavenumbers of these pairs of lines against J. According to [11C.15b–448] a
plot of ν̃R(J) − ν̃P(J) against (J + 1
2 ) will have slope 4B̃1, and according to
[11C.15a–448] a plot of ν̃R(J − 1) − ν̃P(J + 1) against (J + 1
2 ) will have slope
4B̃0. �e data are shown in the tables and the corresponding lines are plotted
in Fig. 11.9.
J ν̃R(J)/cm−1 ν̃P(J)/cm−1 [ν̃R(J) − ν̃P(J)]/cm−1 J + 1
2
0 2 906.25 0.5
1 2 925.92 2 865.14 60.78 1.5
2 2 944.99 2 843.63 101.36 2.5
3 2 963.35 2 821.59 141.76 3.5
4 2 981.05 2 799.00 182.05 4.5
5 2 998.05
J ν̃R(J)/cm−1 ν̃P(J)/cm−1 [ν̃R(J−1) − ν̃P(J+1)]/cm−1 J + 1
2
0 2 906.25 0.5
1 2 925.92 2 865.14 62.62 1.5
2 2 944.99 2 843.63 104.33 2.5
3 2 963.35 2 821.59 145.99 3.5
4 2 981.05 2 799.00 4.5
5 2 998.05
Both plots show good straight lines. For the plot of ν̃R(J) − ν̃P(J) the slope
is 40.42 and hence B̃1 = (slope/4) cm−1 = 10.11 cm−1 . For the second plot
of ν̃R(J − 1) − ν̃P(J + 1) the slope is 41.68 and hence B̃0 = (slope/4) cm−1 =
10.42 cm−1 .
P11C.14 For a harmonic potential the virial theorem ([7E.14–278]) states that ⟨V⟩ =
⟨Ek⟩.�e total energy is the sum of the kinetic and potential energies, therefore
Etot = ⟨V⟩ + ⟨Ek⟩. It therefore follows that Etot = 2⟨V⟩.