Classical vibrational/rotational motion polyatomic

In the classical vibrational spectroscopy, the subject of investigation is the vibrational-rotational motion of polyatomic molecules near the very bottom of their potential-energy surface in the electronic ground state. In this case, the normal-mode approximation proves quite applicable. Indeed, one can expand as a Taylor series the potential energy of the molecule near the equilibrium position (the potential-energy minimum) and write down the molecular Hamiltonian in the form... 

The molecular model of the previous section can move as a whole, rotate about its center of mass, and vibrate. The translational motion does not ordinarily give rise to radiation. Classically, this follows because acceleration of charges is required for radiation. The rotational motion causes practically observable radiation if, and only if, the molecule has an electric (dipole) moment. The vibrational motions of the atoms within the molecule may also be associated nuth radiation if these motions alter the electric moment. A diatomic molecule has only one fundameiita] frequency of vibration so that if it has an electric moment its infrared emission spectrum will consist of a series of bands, the lowest of which in frequency corresponds to the distribution of rotational fre-c)uciicies for nonvibrating molecules. The other bands arise from combined rotation and vibration their centers correspond to the fundamental vibration frequency and its overtones. A polyatomic molecule has more than one fundamental frequency of vibration so that its spectrum is correspondingly richer. 

This chapter begins with a classical treatment of vibrational motion, because most of the important concepts that are specific to vibrations in polyatomics carry over naturally from the classical to the quantum mechanical description. In molecules with harmonic potential energy functions, vibrational motion occurs in normal modes that are mutually uncoupled. Coupling between vibrational modes inevitably occurs in the presence of anharmonic potentials (potentials exhibiting cubic and/or higher order terms in the nuclear coordinates). In molecules with sufficient symmetry, the use of group theory simplifies the procedure of obtaining the normal mode frequencies and coordinates. We obtain El selection rules for vibrational transitions in polyatomics, and consider the rotational fine structure of vibrational bands. We finally treat breakdown of the normal mode approximation in real molecules, and discuss the local mode formulation of vibrational motion in polyatomics. 

To consider the quantum mechanics of rotation of a polyatomic molecule, we first need the classical-mechanical expression for the rotational energy. We are considering the molecule to be a rigid rotor, with dimensions obtained by averaging over the vibrational motions. The classical mechanics of rotation of a rigid body in three dimensions is involved, and we shall simply summarize the results.2... 

It is quite straightforward to perform quasiclassical trajectory computations (QCT) on the reactions of polyatomic molecules providing a smooth global potential energy surface is available from which derivatives can be obtained with respect to the atomic coordinates. This method is described in detail in Classical Trajectory Simulations Final Conditions. Hamilton s equations are solved to follow the motion of the individual atoms as a function of time and the reactant and product vibrational and rotational states can be set or boxed to quantum mechanical energies. The method does not treat purely quantum mechanical effects such as tunneling, resonances. or interference but it can treat the full state-to-state, eneigy-resolved dynamics of a reaction and also produces rate constants. Numerous applications to polyatomic reactions have been reported. ... 

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