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# 3.4: Rotation and Vibration of Diatomic Molecules

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

For a diatomic species, the vibration-rotation $$\left(\dfrac{V}{R}\right)$$ kinetic energy operator can be expressed as follows in terms of the bond length R and the angles $$\theta \text{ and } \phi$$ that describe the orientation of the bond axis relative to a laboratory-fixed coordinate system:

$T_{V/R} = \dfrac{-\hbar^2}{2\mu}\left[ \dfrac{1}{R^2}\dfrac{\partial}{\partial R}\left( R^2\dfrac{\partial}{\partial R} \right) - \left( \dfrac{L}{R\hbar}\right)^2 \right],$

where the square of the rotational angular momentum of the diatomic species is

$L^2 = \hbar^2 \left[ \dfrac{1}{sin \theta} \dfrac{\partial}{\partial \theta} \left( sin \theta \dfrac{\partial}{\partial \theta} \right) + \dfrac{1}{sin^2 \theta} \dfrac{\partial^2}{\partial \phi^2} \right].$

Because the potential $$E_j (R)$$ depends on R but not on $$\theta \text{ or } \phi \text{ , the } \dfrac{V}{R} \text{ function } Xi^0_{j,m}$$ can be written as a product of an angular part and an R-dependent part; moreover, because $$L^2$$ contains the full angle-dependence of $$T_{V/R} , Xi^0_{j,n}$$ can be written as

$\Xi^0_{j,n} = Y_{J,M}(\theta,\phi)F_{j,J,v}(R).$

The general subscript n, which had represented the state in the full set of 3M-3 R-space coordinates, is replaced by the three quantum numbers J,M, and v (i.e., once one focuses on the three specific coordinates $$R,\theta, \text{ and } \phi$$ , a total of three quantum numbers arise in place of the symbol n).

Substituting this product form for $$\Xi^0_{j,n}$$ into the $$\dfrac{V}{R}$$ equation gives:

$\dfrac{-\hbar^2}{2\mu}\left[ \dfrac{1}{R^2} \dfrac{\partial}{\partial R}\left( R^2\dfrac{\partial}{\partial R} \right) - \dfrac{J(J+1)}{R^2\hbar^2} \right] F_{j,J,v}(R) + E_j(R) F_{j,J,v}(R) = E^0_{j,J,v} F_{j,J,v}$

as the equation for the vibrational (i.e., R-dependent) wavefunction within electronic state j and with the species rotating with $$J(J+1) \hbar^2$$ as the square of the total angular momentum and a projection along the laboratory-fixed Z-axis of $$M\hbar.$$ The fact that the $$F_{j,J,v}$$ functions do not depend on the M quantum number derives from the fact that the $$T_{V/R}$$ kinetic energy operator does not explicitly contain $$J_Z$$; only $$J^2$$ appears in $$T_{V/R}.$$

The solutions for which J=0 correspond to vibrational states in which the species has no rotational energy; they obey

$\dfrac{-\hbar^2}{2\mu} \left[ \dfrac{1}{R^2} \dfrac{\partial}{\partial R}\left( R^2 \dfrac{\partial}{\partial R} \right) \right] F_{j,0,v}(R) + E_j(R)F_{j,0,v}(R) = E^0_{j,0,v}F_{j,0,v}.$

The differential-operator parts of this equation can be simplified somewhat by substituting $$F= \dfrac{\chi}{R}$$ and thus obtaining the following equation for the new function $$\chi:$$

$\dfrac{-\hbar^2}{2\mu} \dfrac{\partial}{\partial R} \dfrac{\partial}{\partial R} \chi_{j,0,v}(R) + E_j(R) \chi_{j,0,v}(R) = E^0_{j,0,v}\chi_{j,0,v}.$

Solutions for which $$J\neq 0$$ require the vibrational wavefunction and energy to respond to the presence of the 'centrifugal potential' given by $$\frac{\hbar^2 J(J+1)}{2\mu R^2}$$; these solutions obey the full coupled V/R equations given above.