Overtone transitions in vibrational absorption spectra for which \(\Delta
n=+2,+3, \ldots\) are forbidden for the harmonic potential \(V=(1 / 2) k x^{2}\)
because \(\mu_{x}^{m n}=0\) for \(|m-n| \neq 1\) as shown in Section \(19.4 .\)
However, overtone transitions are allowed for the more realistic anharmonic
potential. In this problem, you will explore how the selection rule is
modified by including anharmonic terms in the potential. We do so in an
indirect manner by including additional terms in the expansion of the dipole
moment \(\mu_{x}\left(x_{e}+x\right)=\mu_{0 x}+x\left(d \mu_{x} / d
x\right)_{r_{e}}+\ldots\)
but assuming that the harmonic oscillator total energy eigenfunctions are
still valid. This approximation is valid if the anharmonic correction to the
harmonic potential is small. You will show that including the next term in the
expansion of the dipole moment, which is proportional to \(x^{2},\) makes the
transitions \(\Delta n=\pm 2\) allowed.
a. Show that Equation (19.8) becomes
$$\begin{array}{l}
\mu_{x}^{m 0}=A_{m} A_{0} \mu_{0 x} \int_{-\infty}^{\infty}
H_{m}\left(\alpha^{1 / 2} x\right) H_{0}\left(\alpha^{1 / 2} x\right)
e^{-\alpha x^{2}} d x \\
\quad+A_{m} A_{0}\left(\frac{d \mu_{x}}{d x}\right)_{x=0}
\int_{-\infty}^{\infty} H_{m}\left(\alpha^{1 / 2} x\right) x
H_{0}\left(\alpha^{1 / 2} x\right) e^{-\alpha x^{2}} d x \\
\quad+\frac{A_{m} A_{0}}{2 !}\left(\frac{d^{2} \mu_{x}}{d x^{2}}\right)_{x=0}
\int_{-\infty}^{\infty} H_{m}\left(\alpha^{1 / 2} x\right) x^{2}
H_{0}\left(\alpha^{1 / 2} x\right) e^{-\alpha x^{2}} d x
\end{array}$$
b. Evaluate the effect of adding the additional term to \(\mu_{x}^{m n} .\) You
will need the recursion relationship
\\[
\alpha^{1 / 2} x H_{n}\left(\alpha^{1 / 2} x\right)=n H_{n-1}\left(\alpha^{1 /
2} x\right)+\frac{1}{2} H_{n+1}\left(\alpha^{1 / 2} x\right)
\\]
c. Show that both the transitions \(n=0 \rightarrow n=1\) and \(n=0 \rightarrow
n=2\) are allowed in this case.
