# 7.4: Rotational and Vibrational Modes

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We can now study the effective one-body Schrödinger equation for the nuclear motion by setting $$E_j ({\bf R}) = E^g ({\bf R})$$ for the ground state. Because $$E^g (R)$$ only depends on the magnitude of $$R$$ it represents an effective central potential, so the solutions are of the form

$F^g (\underline{R}) = \frac{1}{R} \mathcal{R}_{NL} (R) Y_{LM_l} (\theta , \phi ) \nonumber$

where $$Y_{LM_l} (\theta , \phi )$$ are the spherical harmonics and the function $$\mathcal{R}_{NL}(R)$$ satisfies the radial equation

$\left[ − \frac{\hbar^2}{2\mu_{12}} \left( \frac{d^2}{dR^2} − \frac{L(L + 1)}{R^2} \right) + E^g (R) − E \right] \mathcal{R_{N L}} = 0 \nonumber$

We can approximate the centrifugal barrier term by setting it equal to its value at $$R = R_0$$, writing

$E_r = \frac{\hbar^2}{2\mu_{12} R^2_0} L(L + 1) \nonumber$

In this approximation we are treating the molecule as a rigid rotator. We can also approximate $$E^g (R)$$ by Taylor expanding about $$R = R_0$$. Because this point is a minimum, the first derivative is zero:

$E^g (R) \simeq E^g (R_0) + \frac{1}{2} k(R − R_0)^2 + \dots \nonumber$

where $$k$$ is the value of the second derivative of $$E^g$$ at $$R = R_0$$.

With these two approximations, the radial equation becomes

$\left[-\frac{\hbar^{2}}{2 \mu_{12}} \frac{\mathrm{d}^{2}}{\mathrm{d} R^{2}}+\frac{1}{2} k\left(R-R_{0}\right)^{2}-E_{N}\right] \mathcal{R}_{\mathcal{N} \mathcal{L}}=0 \nonumber$

where

$E_N = E − E^g (R_0) − E_r \nonumber$

This is the equation for a simple harmonic oscillator with energies

$E_N = \hbar \omega_0 (N + \frac{1}{2} ), \quad N = 0, 1, 2, \dots \nonumber$

where $$\omega_0 = \sqrt{k/\mu_{12}}$$. The vibrational energies are of the order of a few tenths of an eV, whereas the rotational energies are of the order of $$10^{−3}$$ eV. Both are much smaller than the spacing of the electronic levels. Transitions between these various levels give rise to molecular spectra. The pure rotational spectrum consists of closely-spaced lines in the infrared or microwave range. Transitions which also involve changes to the vibrational state give rise to vibrational-rotational band spectra

This page titled 7.4: Rotational and Vibrational Modes is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Graeme Ackland via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.