Electron Correlations in Molecules and Crystals

1 Electron Correlations in Molecules Post-Hartree-Fock Methods 

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The localized molecular orbitals (LMO) are extensively used not only for the chemical-bonding analysis in molecules but also in the local correlation methods [41] (we consider the problem of electron correlation in molecules and crystals in Chap. 5). The LMO are generated from the canonical MO occupied by electrons and found in the Hartree-Fock or DFT calculations. This generation is based on one or other localization criteria [42]. 

As a result of the development of the x-ray method of studying the structure of crystals and the band-spectroscopic method and especially the electron-diffraction method of studying gas molecules, a large amount of information about interatomic distances in molecules and crystals has been collected. It has been found that the values of interatomic distances corresponding to covalent bonds can be correlated in a simple way in terms of a set of values of covalent bond radii of atoms, as described below.1... 

Electrons in molecules and crystals repel each other according to Coulomb s law, with the repulsion energy depending on the interelectron distance as This interaction creates a correlation hole around any electron, i.e. the probabihty to find any pair of electrons at the same point of spin-coordinate space is zero. Prom this point of view only the Hartree product of molecular or crystalline spin-orbitals... 

Table X gives an idea of the strength of the various expansion methods, and it shows that, by using the principal term only, one can hardly expect to reach even the above-mentioned chemical margin, even if the wave function W gO(D) is actually very close in the helium case. This means that one has to rely on expansions in complete sets, and the construction of the modern electronic computers has fortunately greatly facilitated the numerical solution of secular equations of high order and the calculation of the matrix elements involved. For atoms, the development will probably go very fast, but, for small molecules one has first to program the conventional Hartree-Fock scheme in a fully self-consistent way for the computers, before the next step can be taken. For large molecules and crystals, the entire situation is much more complicated, and it will hence probably take a rather long time before one can hope to get a detailed understanding of the correlation phenomena in these systems.

These properties of the d-shell chromophore (group) prove the necessity of the localized description of d-electrons of transition metal atom in TMCs with explicit account for effects of electron correlations in it. Incidentally, during the time of QC development (more than three quarters of century) there was a period when two directions based on two different approximate descriptions of electronic structure of molecular systems coexisted. This reproduced division of chemistry itself to organic and inorganic and took into account specificity of the molecules related to these classical fields. The organic QC was then limited by the Hiickel method, the elementary version of the HFR MO LCAO method. The description of inorganic compounds — mainly TMCs,— within the QC of that time was based on the crystal field... 

The AIMP method as a common strategy for effective core potential calculations in molecules and for embedded cluster calculations, has been detailed and reviewed [17]. In this paper, we will pay special attention to its applications in the field of structure and spectroscopy of crystal defects created by actinide element impurities, where relativistic effects are a determinant factor, electron correlation and host embedding effects are also key elements, and not only the ground state but also large manifolds of hundreds of excited states are involved in the chemical and physical processes of interest. 

The local MP2 electron-correlation method for nonconducting crystals [109] is an extension to crystalline solids of the local correlation MP2 method for molecules (see Sect. 5.1.5), starting from a local representation of the occupied and virtual HF subspaces. The localized HF crystalline orbitals of the occupied states are provided in the LCAO approximation by the CRYSTAL program [23] and based on a Boys localization criterion. The localization technique was considered in Sect. 3.3.3. The label im of the occupied localized Wannier functions (LWF) Wim = Wj(r — Rm) includes the type of LWF and translation vector Rm, indicating the primitive unit cell, in which the LWF is centered (m = 0 for the reference cell). The index i runs from 1 to A i, the number of filled electron bands used for the localization procedure the correlation calculation is restricted usually to valence bands LWFs. The latter are expressed as a linear combination of the Gaussian-type atomic orbitals (AOs) Xfiif Rn) = Xfin numbered by index = 1,..., M M is the number of AOs in the reference cell) and the cell n translation vector... 

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