Review of Dynamics of a crystal lattice with defects

1990-1998. Spectra of local phonons were calculated for various model states of electron centers and the possibilities for comparison of these calculations with the experiment were analyzed. It was found that for various models of impurity centers, the spectrum of the ground state presents an infinite set of narrow bands lower than the maximum frequency of optical phonons. The influence of weak dispersion of phonons on the frequencies of local oscillations was studied and demonstrated to give rise to an additional resonance peak in the density of local oscillations.
The spectrum of local phonons was investigated for excited self-consistent states. It was found that for such states there exist frequencies which are higher than the maximum frequency of optical phonons and whose number is equal to the number of electron energy levels lying below the self-consistent state. The dependence of the spectrum of local oscillations on the effective charge was studied for the F-center. The spectrum of local phonons for the exciton was investigated. It was shown that the oscillation spectrum of a nonautolocalized exciton is similar to the oscillation spectrum of a weakly coupled F-center. The oscillation spectrum of an autolocalized exciton was shown to be similar, over the whole frequency range, to that of a strongly coupled polaron.
Pekar equation for a polaron and the equations for the F-center (describing electron behaviour in polar media) were studied numerically. In the former case we wanted to find solutions to the polaron equation different from spherically-symmetric ones (which are known). In the latter case our aim was to find spherically-symmetric solutions to the F-center equation on rather a large interval (so that we might study the asymptotical behaviour of the solutions at large distances from the coulomb center).    We suggested a procedure for approximation of non-spherically symmetric solutions to the polaron equation (small-dimension approximation – expansion of solutions in terms of spherical functions with distance dependent coefficients). Then the finding of solutions is reduced to the system (theoretically – infinite) of ordinary differential equations. Finite systems were solved with the help of the software package COLCON which makes possible rather an exact approximation of several non-spherically symmetrical solutions. A finite dimensional approximation (less satisfactory) was also considered. As regards the F-center equation, we revealed that the corresponding boundary problem can be solved with the use of the same software package COLCON. In this case both the solutions and their asymptotics can be found rather exactly.

1998-2005. For crystals with ion coupling we obtained the lowest values of the ground state energy in the whole range of existence of one-center two-electron systems, such as bipolaron and D^–-centers. Account was taken not only of the interactions with optical phonons but also of the interation with the acoustical branches of the phonon spectrum (covalent crystals and crystals with mixed ion-covalent binding).    We calculated the dependence of the energy of the ground state of exchange – bound pairs and the energy of the exchange interation of two paramagnetic centers on the distance between them with dew regard tj electron corrlations and electron-phonon interaction. It was shown that the electron-phonon interaction can play a decisive role (unlike the traditional Coulomb exchange) in antiferromagnetic interaction between two paramagnetic centers. Some particular examples of the calculated energy and exchange interation with ion and acoustical phonons was taken into account. The probe wave function for the calculation of the energy of a two-center electron system was chosen as a sum of Gaussians:

, where r₁, r₂ – are coordinates of the first and second electrons, R is the distance between paramagnetic centers, C_i, a_1i, a_2i, a_3i, b_1i, b_2i are variational parameters, P₁₂ is the operator for transform of electron coordinates. Control calculation of the energy of an H₂ molecule ground state performed with the use of the above system of functions gave exactly the experimental value (-1,173) of this quantity.