The vibrational wave equation

We can therefore substitute for Erot(R) in the vibrational wave equation (2.146) to give 

The main difficulty in solving (2.153) lies in the evaluation of the potential energy term (2.154). Even in the case ofH2, calculation of V from the electronic wavefunctions for different values of is no easy matter. Usually, therefore, the vibrational wave equation is solved by inserting a restricted form of the potential experimental data on the rovibrational levels are then expressed in terms of constants introduced semiempirically, as we shall show. 

In a classic paper, Dunham [10] introduced a dimensionless vibrational variable defined by 

The simplest approximation for V is to assume that the vibration is harmonic, in which case the Hamiltonian becomes  

This form of the harmonic oscillator equation is particularly convenient for solution  

This form of the harmonic oscillator equation is particularly convenient for solution by the methods of matrix mechanics, based on the commutation relationships  

The eigenvalues and eigenfunctions of the simple harmonic oscillator are well known. A detailed account of the solution of the wave equation in (2.157) is given by Pauling and Wilson [11], The solution of equation (2.163) using creation and annihilation operators is described in the book by Bunker and Jensen [12]. The energy levels of the harmonic oscillator are given by 

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The vibrational wave function pv is a function of the internal coordinates (the normal coordinates are usually used) and is a solution of the vibrational wave equation. In Sec. 2-4 it was shown that the kinetic and potential energies of vibration arc given by the expressions... 

Substitution of the potential energy for this harmonic oscillator into the Schrodinger wave equation gives the allowed vibrational energy levels, which are quantified and have energies Ev given by... 

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...

Solution of the Schrddinger wave equation for the motion of the nuclei for this potential energy function leads to the following expression for the vibrational energy of the molecule ... 

Consider either the electronic wave functions or the vibrational wave functions of a molecule. These functions satisfy the Schrodinger equation... 

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. 

The Helmholtz equation resembles the spatial part of the classical wave equation for matter waves (waves in ocean, sound waves, vibrations of a string, electromagnetic waves in vacuum, etc.) of amplitude F = F(r, f) ... 

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

Clearly, if the matrix elements in this equation, and the vibrational wave functions, are known, the dipole moment function M(R) can be obtained. The square of the matrix... 

The vibrational wave functions can be calculated from an RKR representation of the potential well, which is based on the rotational constants and vibrational levels of the molecule concerned. This leaves the problem of determining the signs of the matrix elements. From equation (8.339) we may write... 

The approximation involved in factorization of the total wave function of a molecule into electronic, vibrational and rotational parts is known as the Bom-Oppenheimer approximation. Furthermore, the Schrodinger equation for the vibrational wave function (which is the only part considered here), transformed to the normal coordinates Qi (which are linear functions of the "infinitesimal displacements q yields equations of the harmonic oscillator t5q>e. For these reasons Lifson and Warshel have stressed that the force-field calculations should not be considered as classical-me-... 

Electronic motion with a typical frequency of 3 x 10 s" (i>= 10 cm ) is much faster than vibrational motion with a typical frequency of 3 x 10 s (v = 10 cm" ). As a result of this, the electric vector of light of frequencies appropriate for electronic excitation oscillates far too fast for the nuclei to follow it faithfully, so the wave function for the nuclear motion is still nearly the same immediately after the transition as before. The vibrational level of the excited state whose vibrational wave function is the most similar to this one has the largest transition moment and yields the most intense transition (is the easiest to reach). As the overlap of the vibrational wave function of a selected vibrational level of the excited state with the vibrational wave function of the initial state decreases, the transition moment into it decreases cf. Equation (1.36). Absorption intensity is proportional to the square of the overlap of the two nuclear wave functions, and drops to zero if they are orthogonal. This statement is known as the Franck-Condon principle (Franck, 1926 Condon, 1928 cf. also Schwartz, 1973) ... 

Equation (6-4) implies that excited vibrational states in local high-frequency modes characterized by the vibrational quantum numbers v and w in the initial and final electronic states, respectively become important when AG° I > The vibrational energies are and and the vibrational wave... 

The function y/mx is the solution of the rotational problem. The vibrational part, is a function of the normal coordinates and is the vibrational wave function. Substituting Eq. (4.24) in Eq. (4.23) and ignoring the rotational and translational contributions, the Schrddinger equation for the vibrational wave function will be ... 

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