Approximate separation of rotations and vibrations

Vibrations cannot be exactly separated from rotations for a very simple reason, during vibrations the length R of the molecule changes, this makes the momentum of inertia I = pF change and influences the rotation of the molecule according to eq. (6.25), p. 231. 

The separation is feasible only when making an approximation, e.g., when assuming the mean value of the momentum of inertia instead of the momentum itself. Sueh a mean value is elose to / = pl, where Re stands for the position of the minimum of the potential energy Vko- So, we may decide to accept the potential (6.25) for the oseillations in the form  

Sinee the last term is a eonstant, this immediately gives the separation of the rotations from the vibrational equation (6.24) 

we may always write the potential Uk(R) as a number UkiRe) plus the sfisi SV rest labelled by bosc(..R)  

it is appropriate to call Uk(Re.) the electronic energy Egi (corresponding electronic to the equilibrium internuclear distance in electronic state k), while the function energy b sc(-R) stands, therefore, for the oscillation potential satisfying Vosc(Re) = 0. After introducing this into eq. (6.26) we obtain the equation for oscillations (in general anharmonic) 

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Quasi-Rigid Model-Simplifying by Eckart Conditions Approximation Decoupling of Rotations and Vibration Spherical, Sjfminetric, and Asymmetric Tops Separation of Translational, Rotational, and Vibrational Motions... 

The close-coupling equations are also applicable to electron-molecule collision but severe computational difficulties arise due to the large number of rotational and vibrational channels that must be retained in the expansion for the system wavefiinction. In the fixed nuclei approximation, the Bom-Oppenlieimer separation of electronic and nuclear motion pennits electronic motion and scattering amplitudes f, (R) to be detemiined at fixed intemuclear separations R. Then in the adiabatic nuclear approximation the scattering amplitude for ... 

The nuclear function %a(R) is usually expanded in terms of a wave function describing the vibrational motion of the nuclei, and a rotational wave function [36, 37]. Analysis of the vibrational part of the wave function usually assumes that the vibrational motion is harmonic, such that a normal mode analysis can be applied [36, 38]. The breakdown of this approximation leads to vibrational coupling, commonly termed intramolecular vibrational energy redistribution, IVR. The rotational basis is usually taken as the rigid rotor basis [36, 38 -0]. This separation between vibrational and rotational motions neglects centrifugal and Coriolis coupling of rotation and vibration [36, 38—401. Next, we will write the wave packet prepared by the pump laser in terms of the zeroth-order BO basis as... 

This results from the mathematical fact that an exponential of a sum is a product of the exponentials of each of the terms in the sum, e.g., 6 + " = e e .) We should keep in mind that Equations 11.41 and 11.42 correspond to an approximation. For instance, vibrational-rotational coupling prevents strict separation of the vibrational and rotational parts of a molecular Hamiltonian. For now, we will use Equation 11.42 to obtain q via obtaining each of the elements in the product. We shall also rely on model problems presented in Chapters 7 and 8 as idealizations of rotation and vibration of a diatomic molecule. 

For a RRKM calculation without any approximations, the complete vibrational/rotational Flamiltonian for the imimolecular system is used to calculate the reactant density and transition state s sum of states. No approximations are made regarding the coupling between vibration and rotation. Flowever, for many molecules the exact nature of the coupling between vibration and rotation is uncertain, particularly at high energies, and a model in which rotation and vibration are assumed separable is widely used to calculate the quantum RRKM k(E,J) [4,16]. To illustrate this model, first consider a linear polyatomic molecule which decomposes via a linear transition state. The rotational energy for tire reactant is assumed to be that for a rigid rotor, i.e. 

In order to calculate q (Q) all possible quantum states are needed. It is usually assumed that the energy of a molecule can be approximated as a sum of terms involving translational, rotational, vibrational and electronical states. Except for a few cases this is a good approximation. For linear, floppy (soft bending potential), molecules the separation of the rotational and vibrational modes may be problematic. If two energy surfaces come close together (avoided crossing), the separability of the electronic and vibrational modes may be a poor approximation (breakdown of the Bom-Oppenheimer approximation. Section 3.1). 

The complete wave function of a molecule is called the rovibronic wave function. In the simplest approximation, the rovibronic function is a product of rotational, vibrational, and electronic functions. For certain applications, the rotational motion is first neglected, and the vibrational and electronic motions are treated together. The rotational motion is then taken into account. The wave function for electronic and vibrational motion is called the vibronic wave function. Just as we separately classified the electronic and vibrational wave functions according to their symmetries, we can do the same for the vibronic functions. In the simplest approximation, the vibronic wave function is a product of electronic and vibrational wave functions, and we can thus readily determine its symmetry. For example, if the electronic state is an e2 state and the vibrational state is a state, then the vibronic wave function is... 

The adiabatic Bom-Gppenheimer (BO) approximation is a fundamental starting point for the theory of molecules and solids. The separation of the electronic, vibrational, and rotational manifolds, and, furthermore, the form of the electronic wavefunction are directly related to this approximation. 

In the calculation of the thermodynamic properties of the ideal gas, the approximation is made that the energies can be separated into independent contributions from the various degrees of freedom. Translational and electronic energy levels are present in the ideal monatomic gas.ww For the molecular gas, rotational and vibrational energy levels are added. For some molecules, internal rotational energy levels are also present. The equations that relate these energy levels to the mass, moments of inertia, and vibrational frequencies are summarized in Appendix 6. 

It will be recalled that our use of the Bom adiabatic approximation in section 2.6 enabled us to separate the nuclear and electronic parts of the total wave function. This separation led to wave equations for the rotational and vibrational motions of the nuclei. We now briefly reconsider this approximation, with the promise that we shall study it at greater length in chapters 6 and 7. 

The procedure described above, as for diatomic molecules, is based on the approximation that rotational and vibrational energies are separable, and that the oscillations are simple harmonic in character. The allowance for interaction, etc., has been made in a number of cases by utilizing actual energy levels derived from spectroscopic measurements. The results are, however, not greatly different from those obtained by the approximate method that has been given here. 

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