Hindered oscillation model

The above treatment of hindered rotors assumes that a given mode can be approximated as a one-dimensional rigid rotor, and studies for small systems have shown that this is generally a reasonable assumption in those cases (82). However, for larger molecules, the various motions become increasingly coupled, and a (considerably more complex) multidimensional treatment may be needed in those cases. When coupling is significant, the use of a one-dimensional hindered rotor model may actually introduce more error than the (fully decoupled) harmonic oscillator treatment. Hence, in these cases, the one-dimensional hindered rotor model should be used cautiously. 

The next refinement of the model takes into account that the shape of most molecular species differs from being rod-like typical nematogenic molecules are given in Table 4.6-1. The resulting behaviour of such a bi-axial molecule is often associated with hindered rotation, however it can also be understood from a rigid-body model where different moments of inertia lead to oscillations of different angular amplitudes in spite of identical (thermal) excitation and identical repulsive forces (Korte, 1983). This can be summarized by order parameters defined as above but referring to one of the two shorter. 

Since it became clear from various observations that the librational motions of the molecules, even in the ordered a and y phases of nitrogen at low temperature, have too large amplitudes to be described correctly by (quasi-) harmonic models, we have resorted to the alternative lattice dynamics theories that were described in Section IV. Most of these theories have been developed for large-amplitude rotational oscillations, hindered or even free rotations, and remain valid when the molecular orientations become more and more localized. 

Adsorption energies on metals calculated in a cluster approach often show considerable oscillations with size and shape of the cluster models because such (finite) clusters describe the surface electronic structure insufficiently [257-260]. These models may yield rather different results for the Pauli repulsion between adsorbate and substrate, depending on whether pertinent cluster orbitals localized at the adsorption site are occupied or empty. The discrete density of states is an inherent feature of clusters that may prevent a correct description of the polarizability of a metal surface and thus hinders cluster size convergence of adsorption energies [257]. Even embedding of metal clusters does not offer an easy way out of this dilemma [260,261]. Anyway, the form of conventional moderately large cluster models may be particularly crucial. Such models are inherently two-dimensional with substrate atoms from two or three crystal layers usually taken into accormt thus, a large fraction of atoms at the cluster boundaries lacks proper coordination. 

The vibrational and rotational components can be calculated from the harmonic oscillator and rigid rotor models, for example, whose expressions can be found in many textbooks of statistical thermodynamics [20]. If a more sophisticated correction is needed, vibrational anharmonic corrections and the hindered rotor are also valid models to be considered. The translational component can be calculated from the respective partition function or approximated, for example, by 3I2RT, the value found for an ideal monoatomic gas. 

The above mentioned and many other model potentials are discontinuous at the core radius. This discontinuity leads to long-range oscillations of their Fourier transforms, hindering their use in plane-wave calculations. A recently proposed model pseudo-potential overcomes this difficulty the evanescent core potential of Fiolhais et al. [41]... 

Concerning the proper treatment of torsional anharmonicity, which still represents a challenging aspect for accurate thermochemical calculations of complex molecules [257-265], a hindered-rotor anharmonic oscilattor (HRAO) model has been shown to provide accurate results[62, 72, 117, 204]. The HRAO model is based on a generalization to anharmonic force fields of the hindered-rotor harmonic oscillator (HRHO) model [257] that automatically identifies internal rotation modes and rotating groups during the normal-mode vibrational analysis. 

From 1954 to 1956, Lifshitz derived the theoretical description for the forces betv een two parallel plates of dielectric materials across a vacuum [19]. This theory was extended together with Dzyaloshinskii and Pitaevskii between 1959 and 1961 to include the effect of a third dielectric filling the gap between the plates [20]. However, the complicated structure of their solution hindered its widespread acceptance and initially caused doubt of its practical use [21]. A simplified derivation of the van der Waals forces between parallel plates was introduced by van Kampen et al. [22] based on a model in which the fiuctuations were represented by a sum of harmonic oscillators. Since the bulk modes are independent of distance between the surfaces, only surface modes contribute to the van der Waals force. Based on the van Kampen calculations, Parsegian and Ninham showed in a series of papers in 1970 that the van der Waals forces could be calculated based on available dielectric data [23]. This paned the way for a general quantitative description of van der Waals forces. 

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