Vibrational transitions Schrodinger equation

The Application of the Schrodinger Equation to the Motions of Electrons and Nuclei in a Molecule Lead to the Chemists Picture of Electronic Energy Surfaces on Which Vibration and Rotation Occurs and Among Which Transitions Take Place. 

In 1925, before the development of the Schrodinger equation, Franck put forward qualitative arguments to explain the various types of intensity distributions found in vibronic transitions. His conclusions were based on an appreciation of the fact that an electronic transition in a molecule takes place much more rapidly than a vibrational transition so that, in a vibronic transition, the nuclei have very nearly the same position and velocity before and after the transition. 

Figure 9.7 Vibrational energy levels determined from solution of the one-dimensional Schrodinger equation for some arbitrary variable 6 (some higher levels not shown). In addition to the energy levels (horizontal lines across the potential curve), the vibrational wave functions are shown for levels 0 and 3. Conventionally, the wave functions are plotted in units of (probability) with the same abscissa as the potential curve and an individual ordinate having its zero at the same height as the location of the vibrational level on the energy ordinate - those coordinate systems are explicitly represented here. Note that the absorption frequency typically measured by infrared spectroscopy is associated with the 0 —> 1 transition, as indicated on the plot. For the harmonic oscillator potential, all energy levels are separated by the same amount, but this is not necessarily the case for a more general potential...

In one quantum mechanical approach based on the diabatic approximation , the electron is assumed to be confined initially at one of the reactant sites and electron transfer is treated as a transition between the vibrational levels of the reactants to those of the products. The quantum mechanical treatment begins with the time dependent Schrodinger equation, Hip = -ihSiplSt, where the wavefunction tj/ is written as a sum of the initial (reactant) and final (product) states. In the limit that the Bom-Oppenheimer approximation for the separation of electronic and nuclear motion is valid, the time dependent Schrodinger equation eventually leads to the Golden Rule result in equation (25). 

We notice that the electronic transition moment has been multiplied with a vibrational-overlap integral. In the solution of the vibrational problem, the vibrational wave functions will depend only upon the geometry and the force constants of the molecule. Therefore, only if all these parameters are identical in the two electronic states 1 and 2 will the two sets of vibrational wave functions be the solutions to the same Schrodinger equation. 

As was already mentioned in Section 3.4, we can calculate the vibration—inversion-rotation energy levels of ammonia by solving the Schrodinger equation [Eq. (3.46)]. We are of course primarily interested in the determination of the potential function of ammonia from the experimental frequencies of transitions between these levels (Fig. 11), Le. we must solve the inverse eigenvalue problem [Eq. (3.46)]. 

In a second set of calculations we have studied two kinds of transitions in Habsorption from the ground ro-vibrational state of X to all discrete levels of the first excited electronic state B +, and the emission from the ground ro-vibrational state of B 1Ej to all discrete levels of the ground electronic state X 1E+. The potentials and the transition moments have been taken from the numerically exact solutions of the electronic Schrodinger equation for Hi obtained by Wolniewicz (21). 

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