Infrared harmonic approximation

The last step in the calculation of the frequencies of molecular vibrations, as observed in the infrared spectra, is carried out by combining Eqs. (54) and (55). The vibrational energy of a polyatomic molecule is then given in this, the harmonic approximation, by... 

For such molecules, all of the vibrations are active in both the infrared and Raman spectra. Usually, certain of the vibrations give very weak bands or lines, others overlap, and some are difficult to measure, as they occur at very low wavenumber values.40 Because the vibrations cannot always be observed, a model of the molecule is needed, in order to describe the normal modes. In this model, the nuclei are considered to be point masses, and the forces between them, springs that obey Hooke s law. Furthermore, the harmonic approximation is applied, in which any motion of the molecule is resolved in a sum of displacements parallel to the Cartesian coordinates, and these are called fundamental, normal modes of vibration. If the bond between two atoms having masses M, and M2 obeys Hooke s law, with a stiffness / of the spring, the frequency of vibration v is given by... 

Common spectroscopic techniques test small portions of the ground and/or excited state PES either around the gs minimum (IR and non-resonant Raman spectra, electronic absorption spectra.) or in the proximity of the excited state minimum (steady-state fluorescence). These spectra are then satisfactorily described in the best harmonic approximation, a local harmonic approach that approximates the PES with parabolas whose curvatures match the exact curvatures calculated at the specific position of interest [78]. Anharmonicity in this approach manifests itself with the dependence of harmonic frequencies and relaxation energies on the actual nuclear configuration [79]. Along these lines we predicted softened (hardened) vibrational frequencies for the ground (excited) state [74], amplified and p-dependent infrared and Raman intensities [68, 74], different Frank-Condon... 

These operators are the algebraic counterpart of the single-bond electric dipole operator in the rigid-rotor, harmonic approximation, which corresponds only to An = 1 infrared transitions. In this elementary... 

We consider, here, the case of infrared absorption spectroscopy. Raman intensities have been treated in a parallel way [6]. In the double harmonic approximation (where the forces and the changes in the total molecular dipole moment are linear with atomic splacements) the integrated intensity of the i-th normal mode is related to molecular properties by the relation ... 

While it is common knowledge that isotopic substitution alters the vibration frequencies of normal modes, it may be less well known that the Intensity of the corresponding infrared absorption band must be affected as well. Crawford (lj ) has shown that within the harmonic approximation the integrated intensity of an infrared absorption band, F., is given by... 

In Section II of this review we discuss the different forms of classical lattice dynamical treatments which have been applied to molecular solids. The applications to specific systems and comparison of results with experiment will then be taken up. In Section III we give a short treatment of quantum lattice dynamics, which has been developed to deal with quantum solids as helium and hydrogen. Classical approaches in the harmonic approximation fail for these systems. In Section IV, intensities of infrared and Raman spectra in the lattice vibration region are discussed. A group theoretical appendix has been added for the reader who is not familiar with this aspect. 

An alternative approach to the dynamics of a protein or one of its constituent elements (e.g., an a-helix) is to assume that the harmonic approximation is valid. Early attempts to examine dynamical properties of proteins or their fragments used the harmonic approximation. They were motivated by vibrational spectroscopic studies [24], where the calculation of normal mode frequencies from empirical potential functions has long been a standard step in the assignment of infrared spectra [25]. In calculating the normal vibrational modes of a molecule, one assumes that the vibrational displacements of the atoms from their equilibrium positions are... 

With optical techniques, vibrational dynamics are probed on spatial scales much greater than molecular sizes, or unit cell dimensions in crystals, commonly encountered. The scale is directly related to the wavelength of the incident radiation (in the range fl om 1 to 10(X) p,m in the infrared or about 0.5 p,m for Raman). Oscillators at very short distances, compared to the wavelength, are excited exclusively in phase. For molecules, only overall variations of the dipole moment or polarizability tensor can be probed. In crystals, only a very thin slice of reciprocal space about the centte of the BriUouin zone (k 0) can be probed. This corresponds to in-phase vibrations of a virtuaUy infinite number of unit cells. With optical techniques, band intensities are largely determined by symmetry-related selection rules, although these rales hold only in the harmonic approximation. 

To the extent that we can trust the harmonic approximation, each level of vibrational excitation (each increment in vibrational quantum number v) costs one vibrational constant in energy. As Table 8.2 shows, the vibrational constants of typical stretching motions place vibrational excitation energies in the infrared region of the spectrum. We can measure vibrational transitions that occur by absorption or emission or by scattering. Section 6.3 introduced the concept of Raman scattering, which in principle can be applied to the spectroscopy of any degree of freedom, but which is most commonly used for spectroscopy of vibrational states. 

Harmonic vibrational frequencies [3, 8-10] Infrared harmonic absorption intensities [11] Anharmonic vibrational frequencies [12] Raman intensities (harmonic approximation) [13, 14]... 

The derivation of equations for the transition moments at the anharmonic level is made difficult by the need to account for both the anharmonicity of the potential energy surface (PES) and of the property of interest. Owing to the complexity of such a treatment, various approximations have been employed, in particular, by considering independently the wave function and the property, so that different levels of theory can be apphed to each term and only one of them is treated beyond the harmonic approximation [244,245]. Following the first complete derivation by Handy and coworkers [243], Barone and Bloino adopted the alternative approach presented by Vazquez and Stanton [240] and proposed a general formulation for any property function of the normal coordinates or their associated momenta, which can be expanded in the form of a polynomial truncated at the third order. In this work, we will follow the latter approach, as applied to the infrared (IR) and Raman spectra. 

Marquardt R and Quack M 1989 Infrared-multlphoton excitation and wave packet motion of the harmonic and anharmonic oscillators exact solutions and quasiresonant approximation J. Chem. Phys. 90 6320-7... 

In a second kind of infrared ellipsometer a dynamic retarder, consisting of a photoelastic modulator (PEM), replaces the static one. The PEM produces a sinusoidal phase shift of approximately 40 kHz and supplies the detector exit with signals of the ground frequency and the second harmonic. From these two frequencies and two settings of the polarizer and PEM the ellipsometric spectra are determined [4.316]. This ellipsometer system is mainly used for rapid and relative measurements. 

Molecules possess discrete levels of rotational and vibrational energy. Transitions between vibrational levels occur by absorption of photons with frequencies v in the infrared range (wavelength 1-1000 p,m, wavenumbers 10,000-10 cm , energy differences 1240-1.24 meV). The C-0 stretch vibration, for example, is at 2143 cm . For small deviations of the atoms in a vibrating diatomic molecule from their equilibrium positions, the potential energy V(r) can be approximated by that of the harmonic oscillator ... 

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