Symmetric anharmonic oscillator

For a symmetric molecule and large perturbation, the electrons also experience anharmonic restoring forces. For a symmetric molecule, the potential energy as a function of electronic displacement (symmetric anharmonic oscillator) can be represented by the graph shown in Fig. 3. 

FIGURE 3 Potential energy as a function of displacement for a symmetric anharmonic oscillator potential. 

Anharmonicify of CO2 Vibrations, Generaiization of Symmetric Modes. Compare general and simplified expressions for energy of a CO2 molecule as an anharmonic oscillator (5-14) and (5-16). Establish a relationship between the simplified coefficient of anharmonicity, Xas, generalizing the symmetric modes with the conventional ones. 

The Fourier spectrum of these oscillations (H a ), shown in Fig. 2, consists of a sharp symmetric peak centered at a frequency of about 600 T, and a smaller peak at around 1200 T, obviously the second harmonic in the spectrum of an anharmonic oscillation. The oscillations being thus periodic in 1/H, we are confident that we observe in fact the Shubnikov-de Haas effect. An observed frequency F is then related to an extremal cross-section S of the Fermi Surface normal to the magnetic field direction by S=(2iTe/tic)F /13/, and thus geometric information about the Fermi Surface can be obtained from the angular dependence F(0). The result for the fundamental peak frequency in ET2Cu(NCS)2 is shown in Fig. 3. 

For the reflection symmetric two-level electron-phonon models with linear coupling to one phonon mode (exciton, dimer) Shore et al. [4] introduced variational wave function in a form of linear combination of the harmonic oscillator wave functions related with two levels. Two asymmetric minima of elfective polaron potential turn coupled by a variational parameter (VP) respecting its anharmonism by assuming two-center variational phonon wave function. This approach was shown to yield the lowest ground state energy for the two-level models [4,5]. 

Although the multidimensional PES for the totally symmetric modes are harmonic oscillators, we emphasize that (pronounced) anharmonicity of the adiabatic PES comes into play as soon as non-totally symmetric modes are included [6]. The minima of the diabatic PES can be determined by retaining only the totally... 

Just as we corrected the expressions for the rigid rotor to allow for the centrifugal effect and an interaction with the vibration, we also must adjust the expression for the harmonic oscillator to account for the anharmonicity in the oscillation. The potential energy surface for the molecule is not symmetrical (Fig. 25.2). The parabola (dotted figure) represents the potential energy of the harmonic oscillator. The correct potential energy is shown by the full lines the vibration is anharmonic. The vibrational energy levels for such a system can be approximated by a series ... 

In the harmonic oscillator, approximation combination tones are forbidden as well as overtones. Darling and Dennison (33) in their classic paper gave the anharmonicity and the potential constants for the water molecule. vi is the symmetrical stretching... 

Let us now examine the circumstances under which the various terms given by Eqs. (62)-(65) contribute to the SERS intensity. Most generally, we expect either molecule-to-metal transfer, in which case Af and B must be considered, or metal-to-molecule transfer, in which case Aj, and C must be considered. It is unlikely that charge transfer in both directions would occur simultaneously. In either case, notice that due to the term i k) k f) in A or these terms should only contribute to Raman transitions (i— /) which are totally symmetric (assuming / is totally symmetric). However, intense overtones are possible. The terms B and C have a factor (/ ( /), which enables both totally symmetric and non-totally symmetric vibrations. In the harmonic oscillator approximation, we expect no overtones to be allowed, although they would be weakly allowed if slight anharmonicities are included. 

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