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3.9: Rotation of Linear Molecules

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  • The rotational motion of a linear polyatomic molecule can be treated as an extension of the diatomic molecule case. One obtains the \(Y_{J,M} (\theta,\phi)\) as rotational wavefunctions and, within the approximation in which the centrifugal potential is approximated at the equilibrium geometry of the molecule \((R_e)\), the energy levels are:

    \[ E^0_J = \hbar^2 \dfrac{J(J+1)}{2I}. \]

    Here the total moment of inertia I of the molecule takes the place of \(\mu R_e^2\) in the diatomic molecule case

    \[ I = \sum\limits_a m_a (R_a - R_{CofM})^2; \]

    \(m_a\) is the mass of atom a whose distance from the center of mass of the molecule is \((R_a - R_{CofM}).\) The rotational level with quantum number J is (2J+1)-fold degenerate again because there are (2J+1) M- values.

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