Molecules anharmonicity

The vibrational levels (6.54) are only approximate, because we neglected the anharmonic cubic, quartic,... terms in the potential energy (6.7). For a diatomic molecule, anharmonicity adds the correction - hvexe v + )2 to... 

Linear Triatomic Molecules Anharmonic (Fermi) Interactions Rotational Spectroscopy Electromagnetic Transition Intensities... 

In our discussion so far, we have assumed that the motions of atoms in a vibrating molecule are harmonic. Although making this assumption made the mathematics easier, it is not a realistic view of the motion of atoms in a real vibrating molecule. Anharmonic motion is the type of motion that really takes place in vibrating molecules. The energy levels of such an anharmonic oscillator are approximately given by... 

In polyatomic molecules, anharmonicity causes not only the appearance of overtones, but weak combination bands and difference bands as well. A combination band appears near the frequency of the sum of two (or sometimes three) fundamental bands. For example, CHCI3, which has a CH stretch band at 3019 cm" and a CH bend band at 1216 cm also has a weak band at 4217 cm" a little lower than the expected summation position at 4235 cm" because of anharmonicity. This results from a transition from the ground level to a new combination level not involved in fundamental transitions. A binary combination level involves two different normal coordinates with... 

AMBER A Program for Simulation of Biological and Organic Molecules Anharmonic Molecular Force Fields ... 

Biological and Organic Molecules Anharmonic Molecular Force Fields Carbocation Force Fields Carbohydrate Force Fields CHARMM The Energy Function and Its Parameterization Force Fields A Brief Introduction Force Fields A General Discussion Force Fields CFF Force Fields MM3 GROMOS Force Field and OPLS Force Fields). The multitudinous issues related to achieving convergence will be discussed below. 

Among the main theoretical methods of investigation of the dynamic properties of macromolecules are molecular dynamics (MD) simulations and harmonic analysis. MD simulation is a technique in which the classical equation of motion for all atoms of a molecule is integrated over a finite period of time. Harmonic analysis is a direct way of analyzing vibrational motions. Harmonicity of the potential function is a basic assumption in the normal mode approximation used in harmonic analysis. This is known to be inadequate in the case of biological macromolecules, such as proteins, because anharmonic effects, which MD has shown to be important in protein motion, are neglected [1, 2, 3]. 

It is possible to use computational techniques to gain insight into the vibrational motion of molecules. There are a number of computational methods available that have varying degrees of accuracy. These methods can be powerful tools if the user is aware of their strengths and weaknesses. The user is advised to use ah initio or DFT calculations with an appropriate scale factor if at all possible. Anharmonic corrections should be considered only if very-high-accuracy results are necessary. Semiempirical and molecular mechanics methods should be tried cautiously when the molecular system prevents using the other methods mentioned. 

Equations (6.5) and (6.12) contain terms in x to the second and higher powers. If the expressions for the dipole moment /i and the polarizability a were linear in x, then /i and ot would be said to vary harmonically with x. The effect of higher terms is known as anharmonicity and, because this particular kind of anharmonicity is concerned with electrical properties of a molecule, it is referred to as electrical anharmonicity. One effect of it is to cause the vibrational selection mle Au = 1 in infrared and Raman spectroscopy to be modified to Au = 1, 2, 3,. However, since electrical anharmonicity is usually small, the effect is to make only a very small contribution to the intensities of Av = 2, 3,. .. transitions, which are known as vibrational overtones. 

Figure 6.4 Potential energy curve and energy levels for a diatomic molecule behaving as an anharmonic oscillator compared with those for a harmonic oscillator (dashed curve)...

A molecule may show both electrical and mechanical anharmonicity, but the latter is generally much more important and it is usual to define a harmonic oscillator as one which is harmonic in the mechanical sense. It is possible, therefore, that a harmonic oscillator may show electrical anharmonicity. 

In a diatomic molecule one of the main effects of mechanical anharmonicity, the only type that concerns us in detail, is to cause the vibrational energy levels to close up smoothly with increasing v, as shown in Figure 6.4. The separation of the levels becomes zero at the limit of dissociation. 

Figure 6.35 illustrates how anharmonicity mixes the two vibrations Vj and V3 of CO2. If the molecule starts from the point X and proceeds to B it will tend to follow the line of maximum slope shown. In doing so it deviates considerably from the dashed line representing and so involves an admixture of Qi and Qj. 

The reason why the spacings are equal, and not the 1-0, 2-1, 3-2,... anharmonic intervals, is explained in Figure 9.21. The laser radiation of wavenumber Vg takes benzene molecules into the virtual state Fj from which they may drop down to the v = level. The resulting Stokes scattering is, as mentioned above, extremely intense in the forward direction with about 50 per cent of the incident radiation scattered at a wavenumber of Vg — Vj. This radiation is sufficiently intense to take other molecules into the virtual state V2, resulting in intense scattering at Vg — 2vj, and so on. 

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