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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

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


    \[E_N = E − E^g (R_0) − E_r \nonumber\]

    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.

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