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# 3.6: The Rigid Rotor and Harmonic Oscillator

• • Contributed by Jack Simons
• Professor Emeritus and Henry Eyring Scientist (Chemistry) at University of Utah

Treatment of the rotational motion at the zeroth-order level described above introduces the so-called 'rigid rotor' energy levels and wavefunctions: $$E_J = \hbar^2 \frac{J(J+1)}{2\mu R_e^2} \text{ and } Y_{J,M} (\theta, \phi)$$; these same quantities arise when the diatomic molecule is treated as a rigid rod of length $$R_e.$$ The spacings between successive rotational levels within this approximation are

$\Delta E_{J+1,J} = 2hcB(J+1),$

where the so-called rotational constant B is given in $$cm^{-1}$$ as

$B = \dfrac{h}{8\pi^2 c\mu R_e^2}.$

The rotational level J is (2J+1)-fold degenerate because the energy $$E_J$$ is independent of the M quantum number of which there are (2J+1) values for each J: M= -J, -J+1, -J+2, ... J-2, J-1, J.

The explicit form of the zeroth-order vibrational wavefunctions and energy levels, $$F^0_{j,v} \text{ and } E^0_{ j,v},$$ depends on the description used for the electronic potential energy surface $$E_j(R).$$ In the crudest useful approximation, $$E_j(R)$$ is taken to be a so-called harmonic potential

$E_j(R) \approx \dfrac{1}{2}k_j (R-R_e)^2 ;$

as a consequence, the wavefunctions and energy levels reduce to

$E^0_{j,v} = E_j(R_e) + \hbar \dfrac{\sqrt{k}}{\mu}\left( v + \dfrac{1}{2}\right), \text{ and }$

$F^0_{j,v}(R) = \dfrac{1}{\sqrt{2^v v!}} \sqrt{\dfrac{\alpha}{\pi}}e^{\dfrac{-\alpha (R-R_e)^2}{2}}H_v \sqrt{\alpha}(R-R_e),$

where $$\alpha = \dfrac{\sqrt{k_j \mu}}{\hbar} \text{ and } H_v(y)$$ denotes the Hermite polynomial defined by:

$H_v(y) = (-1)^v e^{y^2}\dfrac{d^v}{dy^v}e^{-y^2}.$

The solution of the vibrational differential equation

$\dfrac{-\hbar^2}{2\mu}\left[ \dfrac{1}{R^2}\dfrac{\partial}{\partial R}\left( R^2\dfrac{\partial}{\partial R} \right) \right] F_{j,v}(R) + E_j(R) F_{j,v}(R) = E_{j,v} F_{j,v}$

is treated in EWK, Atkins, and McQuarrie.

These harmonic-oscillator solutions predict evenly spaced energy levels (i.e., no anharmonicity) that persist for all $$v$$. It is, of course, known that molecular vibrations display anharmonicity (i.e., the energy levels move closer together as one moves to higher $$v$$) and that quantized vibrational motion ceases once the bond dissociation energy is reached.