Normal-mode limit basis

The fact that both the local- and the normal-mode limits are contained within the algebraic approach allows one to study in a straightforward way the transition from one to the other. It is convenient to use, for this study, the local basis [Eq. (4.17)] and diagonalize the Hamiltonian for two identical bonds... 

The normal modes associated with these frequencies are characterized by motion limited to the hydrogen atom in question. The values of the frequencies are in reasonable agreement with observations which place this peak in the range 2745-2710 cm S given our knowledge of basis set effects from the carbonyl stretch frequencies. ... 

With this definition, due to Child and Halonen (1984), local-mode molecules are near to the = 0 limit, normal mode molecules have —> 1. The correlation diagram for the spectrum is shown in Figure 4.3, for the multiplet P = va + vb = 4. It has become customary to denote the local basis not by the quantum numbers va, vh, but by the combinations... 

This value of electron affinity, calculated from the bottom of the potential energy well, must be corrected for the zero-point energy (ZPE) effects. It may be expected that the normal modes of the anion will be softer than in SF6 and the ZPE correction will increase the computed value of electron affinity. In order to establish the magnitude of ZPE and determine the character of the stationary point for the anion, harmonic vibrational analysis was performed. Due to limited computer resources, only the DF1 calculations were done using the extended, (3dl/, 3dlf) polarized basis set. Zero-point energies were found to be 0.51 eV and 0.32 eV for SF6 and SFg, respectively the ZPE correction to the computed value of electron affinity is thus about 0.2 eV, and the ZPE-corrected value of electron affinity is 1.6 eV. 

An excellent example of non-accidental intramolecular resonance is provided by a polyatomic molecule with two chemically identical bonds. The spectrum and dynamics of such a molecule may equally well be described by an effective Hamiltonian expressed in basis sets corresponding to either of two opposite limiting cases normal mode (H -,RMAL) and local mode (Hlocal)-... 

The large-dimension limit has recently resolved at least some of the difficulties of the molecular model. The molecule-like structure falls out quite naturally from the rigid bent triatomic Lewis configuration obtained in the limit D — oo [5], and the Langmuir vibrations at finite D can be analyzed in terms of normal modes, which provide a set of approximate quantum numbers [6,7]. These results are obtained directly from the Schrodinger equation, in contrast to the phenomenological basis of some of the earlier studies. When coupled with an analysis of the rotations of the Lewis structure, this approach provides an excellent alternative classification scheme for the doubly-excited spectrum [8]. Furthermore, an analysis [7] of the normal modes offers a simple explanation of the connection between the explicitly molecular approaches of Herrick and of Briggs on the one hand, and the hyperspherical approach, which is rather different in its formulation and basic philosophy. 

The intramolecular dynamics is usually described on the basis of vibrational normal modes, V it) where n index individuates the molecule and v the vibrational mode. Limiting the mode expansion to the first order, we can write the molecular polarizability as... 

The interaction between particle and applied field determines the concentration profile distribution inside the channel. Theory assumes that the particles do not interact with each other, that is, the particle concentration is so low that they can be considered in a condition of infinite dilution and thus with an ideal behavior. Under these assumptions, the field-particle interaction has been classified into three major elution modes Brownian (or normal), steric, and hyperlayer (or focusing). These elution modes, which are limiting cases, correspond to different mechanisms of separation and, theoretically, they can explain FFF retention on the basis of particle properties such as the... 

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