Separation of the vibrational and rotational wave equations

Integrating (—ij /2/i)V over electronic coordinates and using (2.135) we obtain 

Note that, because is independent of the Euler angles 6 and 4 , there is no contribution 

The third, fourth and fifth terms in braces in equation (2.141) represent small adiabatic corrections to the potential energy function. They all have a reduced mass dependence, unlike Tnuci(l ), and so are the origin of the isotopic shifts in the electronic energy [5]. 

The corresponding rotation-vibration wave equation in the Born-Oppenheimer approximation, in which all coupling of electronic and nuclear motions is neglected, is 

therefore, still necessary to solve equation (2.127) for the electronic energies and wave functions. 

Note that, because // is independent of the Euler angles 0 and (f , there is no contribution from the term in J2. Substituting (2.138) into (2.136), we obtain the wave equation for the rotation vibration wave functions in the Born adiabatic approximation  

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The logical way to begin the mathematical treatment of the vibration and rotation of a molecule is to set up the classical expressions for the kinetic and potential energies of the molecule in terms of the coordinates of the atoms, and then to use these expressions to obtain the wave equa-lion for vibration, rotation, and translation. Following this, it should bo proved that when the proper coordinate system is used, the complete wave equation can be approximately separated into three equations, one for translation, one for rotation, and one for vibration. Unfortunately, this procedure is not a very simple one and utilizes more quantum-mechanical technique than is required for the discussion of the vibrational ( ((nation itself. Consequently, the actual carrying out of the separation ill be deferred until Chap. 11, and only a summary of the results thus (ibi allied will be presented at this point. The reader who prefers to follow the more logical order may turn to Chap. 11 before continuing with the present sections. 

Such a separation is exact for atoms. For molecules, only the translational motion of the whole system can be rigorously separated, while their kinetic energy includes all kinds of motion, vibration and rotation as well as translation. First, as in the case of atoms, the translational motion of the molecule is isolated. Then a two-step approximation can be introduced. The first is the separation of the rotation of the molecule as a whole, and thus the remaining equation describes only the internal motion of the system. The second step is the application of the Born-Oppenheimer approximation, in order to separate the electronic and the nuclear motion. Since the relatively heavy nuclei move much more slowly than the electrons, the latter can be assumed to move about a fixed nuclear arrangement. Accordingly, not only the translation and rotation of the whole molecular system but also the internal motion of the nuclei is ignored. The molecular wave function is written as a product of the nuclear and electronic wave functions. The electronic wave function depends on the positions of both nuclei and electrons but it is solved for the motion of the electrons only. 

Our treatment of the nuclear Schrodinger equation for diatomic molecules has shown that the wave function for nuclear motion can be separated into rotational, vibrational, and translational wave functions ... 

The two degrees of freedom associated with the ring puckering are, therefore, an ordinary vibration and a type of one-dimensional rotation in which the phase of the puckering moves around the ring the latter is not, however, a true rotation since there in no angular momentum about the axis of rotation, and so is described as a pseudo-rotation . This separation of the wave equation is not exact, but it has been stated that exact separation is possible and, on the assumptions of harmonic oscillations and small amplitudes of vibrations, leads to the same results as those given. 

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 vibrational structure may be explained as follows For each state of a molecule there is a wave function that depends on time, as well as on the internal space and spin coordinates of all electrons and all nuclei, assuming that the overall translational and rotational motions of the molecule have been separated from internal motion. A set of stationary states exists whose observable properties, such as energy, charge density, etc., do not change in time. These states may be described by the time-independent part of their wave functions alone. Their wave functions are the solutions of the time-independent Schrddinger equation and depend only on the internal coordinates q = 9, Qz,. . . of all electrons and the internal coordinates Q = Q, Qz, of all nuclei. 

With the set of coordinates best adapted for one of the equilibrium configurations, the wave equation can be set up, approximately separated into translational, rotational, and vibrational parts, and then solved. The same procedure can be carried through for the other equilibrium configuration. The two wave functions thus obtained will be different, and general quantum mechanical arguments indicate that a better solution can be obtained by taking a linear combination of the two original ones. 

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