Lattice vibrations crystal Hamiltonian

Here P(i) is linear momentum conjugated to the distortion coordinate 2(0-In the theory of the JT effect, the linear-coupling case E b described by the Hamiltonian (7) is the easiest one. Its matrix part includes just diagonal matrices. As distinguished from this simple case, the general JT case is a tough problem of complex dynamics of electrons coupled to crystal lattice vibrations. The... 

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. 

The stability of stagnant motions is a significant problem not only in the theoretical subject but also in the practical measurement. For instance, in the experiments of quartz oscillators the 1 // spectral fluctuations are frequently observed, which is considered to be good examples for the nonstationary motions generated in the Hamiltonian dynamics of crystal lattice vibrations [6-8],... 

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... 

As shown in Ch. 3, the crystal Hamiltonian, expressed in terms of the operators Pj-f and P/ (in the following the index / will be omitted) when the lattice vibrations are not taken into account, has the following form (see also eqn 3.39)... 

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. 

Another technique to obtain the effects of the anharmonic terms on the excitation frequencies and the properties of molecular crystals is the Self-Consistent Phonon (SCP) method [71]. This method is based on the thermodynamic variation principle, Eq. (14), for the exact Hamiltonian given in Eq. (10), with the internal coordinates not explicitly considered. As the approximate Hamiltonian one takes the harmonic Hamiltonian of Eq. (18). The force constants in Eq. (18) are not calculated at the equilibrium positions and orientations of the molecules as in Eq. (19), however. Instead, they are considered as variational parameters, to be optimized by minimization of the Helmholtz free energy according to Eq. (14). The optimized force constants are found to be the thermodynamic (and thus temperature dependent) averages of the second derivatives of the potential over the (harmonic) lattice vibrations ... 

As a result of different bonding properties (which arise from different interionic separations in these electronic states) in the ground and excited states of an impurity ion in a crystal, they may have different geometries, what is revealed in the shift of the potential energy surfaces of the considered electron states and their different curvature. The latter is defined by the differences of the vibrational frequencies in these states, and, since this difference rarely exceeds few percents, can be readily neglected. In order to perform a qualitative analysis of this phenomenon, we use the effective Hamiltonian Hyiq, which describes the interaction of the electron states with the lattice normal modes in the form... 

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... 

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