Cluster embedding in a crystal

Most of existing embedding schemes are intuitive. In a number of papers [B18,B19,B21] (see also reviews in [A1-A3]), the Embedding Molecular Cluster (EMC) method has been proposed. It illustrates the idea of how the embedding problem can be solved, at least in principle. Based on the Theory of Electronic Separability, a crystal is split into a set of structural elements (ions, molecules, bonds, etc.). Each structural element (including the cluster region itself) is described by a many-electron wave function called a group function. Assuming weak overlap between neighbouring group functions (the so-called strong orthogonality approximation), general equations for both the cluster and (polarisable) outside region have been derived following consistent quantum-mechanical principles. It was shown that both regions must be considered in a self-consistent way and, at the same time, the outside region may be simulated by means of a semi-classical model which was found to be similar, but not exactly equal, to the well-known "shell model". It was shown that the corresponding force-constant matrix for the outside region changes due to the defect, and the problem turns out to be non-linear with respect to the cluster electronic density matrix (as, in fact, is the case with any polaron problem).

In the EMC method, the cluster is described by an effective Hamiltonian, containing along with the polarisation potential also some other terms of short-range nature. These latter contributions have non-local character and arise due to exchange and polarisation-exchange interactions of the cluster atoms with adjacent atoms of the outside region. Since both regions in the EMC method are considered within the same theoretical framework, it was possible to show that both the short-range potential and the force-constant matrix of the outside region are expressed via one and the same set of parameters. Therefore, it is possible to obtain the short-range crystal potential by just studying the perfect crystal lattice dynamic problem. Note that the consistency between the outside region and the cluster is fulfilled automatically in the parametrisation scheme. The EMC method gives a consistent microscopic physical basis for a number of existing cluster embedding models based on intuitive arguments.

As a logical extension of the EMC method, a more general approach has been developed [A3], in which the overlap of the group functions of the structure elements is rigorously taken into account. This generalisation is based on the recently developed Expanded Arrow Diagram (EAD) technique [B26,B27,B33a,B57] used as a convenient mathematical tool to deal with strongly overlapping electronic group functions. It was shown that the total energy of the system (and even more generally, the system density matrices of arbitrary order) can be represented as a EAD expansion, which allows one to gather all terms up to a desired order with respect to overlap quite easily. Simple and convenient rules for calculating contributions from any diagram have also been given in [B27,B57]. Using the EAD technique, we have obtained the corresponding diagram expansions for the crystal force-constant matrix, polarisation and the short-range potentials [B63]. A general method has been developed in this paper which can be used for deriving atomistic models of inter-atomic interactions with built-in dipole, quadrupole, etc. polarisation and overlap effects taken into account explicitly. This work is being done in collaboration with St. Petersburg University (I.V.Abarenkov, I.Tupitsin).

7/2/1999