Classical diatom line shapes

For classical line shape calculations one needs the induced dipole moment as function of time, p(R(t)), averaged over angular momenta and speeds of relative motion. In other words, one solves Newton s equation of motion, or one of its integrals, of the two-particle system. After suitable averaging, one obtains the spectral profile by Fourier transform. 

Suppose we have two particles with masses mi and m2, and position vectors R1 and R2 in the laboratory frame, respectively, that interact through the isotropic intermolecular potentential V(R), with R = R2--RiI, with no other forces acting on them. For each particle, an equation of motion may be written down, 

Since the direction of the interaction force, -VF, is along the relative position vector, R, the vector product of R with R, Eq. 5.68, vanishes, RxR = 0. In other words, RxmR = Rxmv = L, the angular momentum of relative motion, is a constant of motion. Its magnitude may be expressed in polar coordinates R, 3, according to 

The equation of relative motion, Eq. 5.68, may also be integrated with respect to position, dR, to give the law of energy conservation, 

In this expression, Rq designates the greatest root of the argument of the square root appearing in the denominator Ro is the outermost classical turning point. Equation 5.70 represents the inverse function, t(R), of the desired solution R(vo,b t). With the knowledge of the time dependence of 

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The classical Maxwell Boltzmann distribution law may be used to approximate the distribution of population among the quantized energy levels of a gas-phase diatomic molecule. For most diatomic molecules (such as CO, but not NO), each separate line shape observed in the infrared spectrum corresponds to a simultaneous change in vibrational... 

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