Spherical Top

Examples: methane $CH_4$, carbon tetrachloride $CCl_4$, sulfur hexafluoride $SF_6$.

For this case ( $I = I_A = I_B = I_C$) the rotational energy is given by:

$$E = \frac{J_A^2+J_B^2+J_C^2}{2I} = \frac{J^2}{2I}$$ (33)

We corresponding quantum mechanical expression for the rotational energy is given by:

$$E_J = \frac{\hbar^2}{2I}\cdot J(J+1),$$ (34)

because the eigenvalues of the operator ${\bf J}^2$ are ${\bf J}^2 \Psi = \hbar^2 J(J+1)\Psi$.

Usually in spectroscopy eq. (34) is presented in the form

$$E_J = h c B\cdot J(J+1),$$ (35)

where the rotational constant $B$ measured in $cm^{-1}$. The product $h c B$ in eq.(35) is equal to $\hbar^2/2I$. The rotational constant $B$ is as

$$B = \frac{\hbar}{4\pi c I}$$ (36)

The energy of a rotational state is normally expressed in rotational terms $F(J)$ measured in wavenumbers, $cm^{-1}$:

$$F(J) = BJ(J+1)$$ (37)

Then the separation between the neighbor energy levels is

$$F(J) - F(J-1) = 2BJ$$ (38)

As seen from eq. (36) the rotational constant $B$ decreases as the moment of inertia $I$ increases. Therefore, large molecules have closely spaced rotating energy levels.

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