Lattice vibrations, Hamiltonian system

The OOA, also known as Kugel-Khomskii approach, is based on the partitioning of a coupled electron-phonon system into an electron spin-orbital system and crystal lattice vibrations. Correspondingly, Hilbert space of vibronic wave functions is partitioned into two subspaces, spin-orbital electron states and crystal-lattice phonon states. A similar partitioning procedure has been applied in many areas of atomic, molecular, and nuclear physics with widespread success. It s most important advantage is the limited (finite) manifold of orbital and spin electron states in which the effective Hamiltonian operates. For the complex problem of cooperative JT effect, this partitioning simplifies its solution a lot. 

We may list differences between the liquid water system and the FPU model the latter will be examined in the next section as a representative system in the study of many-dimensional Hamiltonian systems. The most important difference would be that the FPU model describes a lattice vibration around an equilibrium point and the potential energy function possesses a single minimum, whereas there are infinitely many local potential minima and the potential energy landscape generally becomes ragged in the case of the liquid water system. The reason why the character of the potential landscape could be so important is that the raggedness is considered as an origin of slow motions in liquid water or supercooled liquids. 

A scheme as described here is indispensable for a quantum dynamical treatment of strongly delocalized systems, such as solid hydrogen (van Kranendonk, 1983) or the plastic phases of other molecular crystals. We have shown, however (Jansen et al., 1984), that it is also very suitable to treat the anharmonic librations in ordered phases. Moreover, the RPA method yields the exact result in the limit of a harmonic crystal Hamiltonian, which makes it appropriate to describe the weakly anharmonic translational vibrations, too. We have extended the theory (Briels et al., 1984) in order to include these translational motions, as well as the coupled rotational-translational lattice vibrations. In this section, we outline the general theory and present the relevant formulas for the coupled... 

There is, however, another possibility which was first investigated by Sussmann (1964). The elementary tunnelling process which we have considered neglects the fact that each one-proton system is embedded in a crystal at finite temperature and thus may interact with lattice vibrations. These vibrations may be thought of quite simply as periodically modifying the 0-0 distances in the crystal and, from the amplitudes given in table 6.2, these changes may be up to +o-iA. The perturbations which these vibrations introduce may be included in the Hamiltonian and, by (9.66), then contribute to AW and so, by (9.77), increase the mobility at finite temperatures. 

In what follows we shall take A to be the minority constituent (x < 0.5), and choose 5 > 0. Except when explicitly noted, we restrict Vb to have a constant value V, independent of x, and connect nearest nei bors only. Although in this section we consider electrons in an alloy, a very similar Hamiltonian is used to describe lattice vibrations in the presence of mass disorder. The results of this section may be extended by a simple transformation to describe such phonon systems (Economou, 1971b). 

Interpretation and systemization of the magnetic properties of lanthanide compounds are based on crystal field theory which has been rqjeatedly discussed in literature, in particular by Morrison and Leavitt (1982) in volume 5 of this Handbook. So we begin our chapter with a short account of crystal field theory in a comparatively simple form with a minimal number of initial parameters with a clear physical meaning. This immediately provides the interaction hamiltonians of 4f electrons with deformations and vibrations of the crystal lattice. Within the framework of this theory one can easily calculate the distortions of the crystal field near impurity ions. A clear idea of the nature of magnetic phenomena in simple dielectric lanthanide compounds is certainly useful for consideration of systems with a more complicated electron structirre. 

We present a derivation of the broadening due to the solvent according to a system/ bath quantum approach, originally worked out in the field of solid-state physics to treat the effect of electron/phonon couplings in the electronic transitions of electron traps in crystals [67, 68]. This approach has the advantage to treat all the nuclear degrees of freedom of the system solute/medium on the same foot, namely as coupled oscillators. The same type of approach has been adopted by Jortner and co-workers [69] to derive a quantum theory of thermal electron transfer in polar solvents. In that case, the solvent outside the first solvation shell was treated as a dielectric continuum and, in the frame of the polaron theory, the vibrational modes of the outer medium, that is, the polar modes, play the same role as the lattice optical modes of the crystal investigated elsewhere [67,68]. The total Hamiltonian of the solute (5) and the medium (m) can be formally written as... 

In the adiabatic approximation, the Hamiltonian of the system H can be presented as a superposition of the lattice part H, Hamiltonian Hp, describing the adsorbed molecule vibrations in the mean potential of the surface phonons (V(x, x and the interaction Hamiltonian Hjjjp defined by ... 

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