Rotation-vibration wave equation

The corresponding rotation-vibration wave equation in the Born-Oppenheimer approximation, in which all coupling of electronic and nuclear motions is neglected, is... [Pg.61]

The simplest molecular problem, viz. H, can be attacked by numerical methods to jdeld accurate vibrational and rotational energy levels in the electronic ground state. This calculation was done by Wind (66a) using the electronic energy Eg(R) determined by him 66b) to a very high degree of accuracy and the rotation-vibration wave equation derived by Cohen, Hiskes and Riddell (76)... [Pg.236]

Note that, because 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 ... [Pg.61]

Figure 1.13 shows the potential function, vibrational wave functions and energy levels for a harmonic oscillator. Just as for rotation it is convenient to use term values instead of energy levels. Vibrational term values G(v) invariably have dimensions of wavenumber, so we have, from Equation (1.69),... [Pg.137]

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 ... [Pg.329]

Show that if the overlap between torsional-vibration wave functions corresponding to oscillation about different equilibrium configurations is neglected, the perturbation-theory secular equation (1.207) for internal rotation in ethane has the same form as the secular equation for the Hiickel MOs of the cyclopropenyl system, thereby justifying (5.96)-(5.98). Write down an expression (in terms of the Hamiltonian and the wave functions) for the energy splitting between sublevels of each torsional level. [Pg.371]

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. [Pg.67]

In chapter 2 we showed how the wave equation of a vibrating rotator was derived through a series of coordinate transformations. We discussed the solutions of this wave equation in section 2.8, and the particular problem of representing the potential in which the nuclei move. We outlined the relatively simple solutions obtained for a harmonic oscillator, the corrections which are introduced to take account of anharmonicity, and derived an expression for the rovibrational energies. Our treatment was relatively brief, so we now return to this subject in rather more detail. [Pg.233]

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... [Pg.506]

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-... [Pg.7]

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 ... [Pg.145]

This is seen by allowing k to become infinite, causing the third term to vanish (because >- — ). A rigid molecule would have no vibrational energy, so the first term would become an additive constant. The rigid rotator is often discussed as a separate problem, with the wave equation... [Pg.271]

The straightforward way to treat the rotational and vibrational motion of a polyatomic molecule would be to set up the wave equation for (Eq. 34-4), introducing for [/ ( ) an expres-... [Pg.275]

The relations are not quite so easily understandable for vibrating and rotating systems, but Schrodinger made the remarkable discovery that the wave equation is applicable quite generally if handled according to the following prescription. [Pg.125]

Big Chemical Encyclopedia

Chemical substances, components, reactions, process design ...