Kinetic energy, classical rotational

As an example we take the case of the rotating molecule which we treat as a rotator (i.e. as a rigid body which can rotate about a fixed axis). If A is its moment of inertia about this axis, then its kinetic energy when rotating with angular velocity co is, according to classical mechanics,... 

Then, the classical expression for the kinetic energy of rotation takes the form ... 

Therefore, the simplest classical treatment in which the propagator exp(it (T+V) ) is approximated in the product form exp(it (T) ) exp(it (V)/fc) and die nuclear kinetic energy T is conserved during the transition produces a nonsensical approximation to the non BO rate. This should not be surprising because (a) In the photon absorption case, the photon induces a transition in the electronic degrees of freedom which subsequently cause changes in the vibration-rotation energy, while (b) in the non BO case, the electronic and vibration-... 

We next apply these classical relationships to the rigid diatomic molecule. Since the molecule is rotating freely about its center of mass, the potential energy is zero and the classical-mechanical Hamiltonian function H is just the kinetic energy of the two particles,... 

The first term on the right is the translational kinetic energy of the molecule as a whole this simply adds a constant to the total energy, and we shall omit this term. The second and third terms are ihe rotational and vibrational kinetic energies of the molecule. The final term is the energy of interaction between rotation and vibration. To get the classical-mechanical Hamiltonian function, we add the potential energy V to (5.2), where U is a function of the relative positions of the nuclei. 

A complete treatment of this derivation can be found in Ref. [19]. The first three terms in the kinetic energy operator indicates the presence of a 3D harmonic oscillator, and the final two terms indicate the presence of a 2D rotator (as for the hydrogen atom). A similar conclusion was made by Auberbach et al. [22] where they use a semi-classical quantisation method and the molecule is said to undergo a unimodal distortion and then the semi-classical Hamiltonian is found to be separated into two parts - a harmonic oscillator part with three vibrational coordinates and a rotational part with two rotational coordinates. However, more progress in terms of specifying the wavefunctions of the system can be made by following a different approach. 

In the case of U and Cy, the question of accounting for the kinetic energy of the intramolecular degrees of freedom arises. Usually this is done by assuming that rotational degrees of freedom can be treated classically and that the intramolecular vibrational modes make no contribution. However, other assumptions have been made. This question is important when detailed agreement with an experiment is desired. The difference between the measured and calculated quantities can vary by 2-3 kcal/mol depending on the way in which the kinetic component is estimated. 

Figure 11. Rotational energy , dependence of the cross section for O + HCl and O + DCl reactions at E uk = 10 kcal/mol. Compared are the results of the classical trajectory calculations ( ) [47], kinematic mass model results which only include the energetic effects of the reactant rotation (o). kinematic mass model results which include rotational effects on the distribution of collisions with the barrier in addition to the rotational energetic effects ( ), and the kinematic mass model results for rotationally unexcited reactants 0 = 0) and the translational kinetic energy increased by the amoimt of (A) [62].

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