Vibronic molecular energy levels

For a diatomic molecule, modeled as a harmonic oscillator, quantum mechanics reveals that the vibronic molecular energy levels are restricted to discrete values given by Equation 2.1... 

A more realistic approach towards atomic vibrations is the anharmonic oscillator (see Figure 2.1). Here, the vibronic molecular energy levels are given by Equation 2.2. 

The first attempt to explain the characteristic properties of molecular spectra in terms of the quantum mechanical equation of motion was undertaken by Born and Oppenheimer. The method presented in their famous paper of 1927 forms the theoretical background of the present analysis. The discussion of vibronic spectra is based on a model that reflects the discovered hierarchy of molecular energy levels. In most cases for molecules, there is a pattern followed in which each electronic state has an infrastructure built of vibrational energy levels, and in turn each vibrational state consists of rotational levels. In accordance with this scheme the total energy, has three distinct components of different orders of magnitude,... 

For molecules, the spectroscopic nomenclature for molecular energy levels and their vibronic and rotational sublevels is messy and very specialized. Already for homonuclear or heteronuclear diatomic molecules a new quantum number shows up, which quantifies the angular momentum along the internuclear axis, but the reader need not be burdened with the associated nomenclature. 

Fig. 1. The molecular energy level model used to discuss radiationless transitions in polyatomic molecules. 0O, s, and S0,S are vibronic components of the ground, an excited, and a third electronic state, respectively, in the Born-Oppenheimer approximation. 0S and 0 and 0j are assumed to be allowed, while transitions between j0,j and the thermally accessible 00 are assumed to be forbidden. The f 0n are the molecular eigenstates...

Fig. 4. Molecular energy level diagram used to discuss radiationless processes in polyatomic molecules. Q is the ground electronic state, and <>ox denotes a thermally accessible vibronic component of this state. Electric dipole transitions from to the electronic state are allowed (or vibronically induced), while those to <(), are forbidden. ,i designates a vibronic component of and <, j is a component of The electronic states o> >, 4>/ re obtained from the adiabatic Bom-Oppenheimer approximation.

In the present paper we assume that the molecule has the icosahedral symmetry. If one wants to consider a distortion of C 0+ or Cb0. the energy levels and their eigenvectors obtained here can be used as a starting point for the description of the Jahn-Teller effect in these systems. Indeed, the electron-phonon (or vibronic) coupling occurs if [.Tei]2 contains Fvib [19]. (Here Fd is the symmetry of an electronic molecular term, while J b is the symmetry of a vibrational normal mode.) Calculations using the terms in scheme of Ref. [4] have been performed in Ref. [20]. 

Ammonia was the first molecule for which the effect of the molecular inversion was studied experimentally and theoretically. Inversion in ammonia was subsequently found to be so important that this molecule played an important role in the history of molecular spectroscopy. Let us recall, for example that microwave spectroscopy started its era with the measurements " of the frequencies of transitions between the energy levels in the ground vibronic state of NH3 split by the inversion effect. Furthermore, the first proposal and realization of a molecular beam maser in 1955 was based on the inversion splittings of the energy levels in NH3. The Nobel Prize which Townes, Basov and Prochorov were awarded in 1964 clearly shows how important this discovery was. Another example of the role which the inversion of ammonia played in the extension of human knowledge is the discovery of NH3 in the interstellar space by Cheung and his co-workers in 1968, by measuring the... 

In this section we consider the role of molecular vibrations in resonant L NLO properties. Tliese properties govern the intensity of light absorption (or emission) accompanying tlie transition between two vibronic energy levels. Of interest here is one- and two-photon absorption (OPA and TPA). Applications of TPA, in particular, include three-dimensional optical data storage and photodynamic therapy. 

In Fig. 3.1, illustrating the general idea of the molecular photo transformation A—>B the vibronic energy levels of molecirles A and B are sketched and the arrows show optical excitation and some of the possible radiative and nonradiative transitions between the levels. 

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