Solid state molecules crystal orbital methods

The SCF method for molecules has been extended into the Crystal Orbital (CO) method for systems with ID- or 3D- translational periodicityiMi). The CO method is in fact the band theory method of solid state theory applied in the spirit of molecular orbital methods. It is used to obtain the band structure as a means to explain the conductivity in these materials, and we have done so in our study of polyacetylene. There are however some difficulties associated with the use of the CO method to describe impurities or defects in polymers. The periodicity assumed in the CO formalism implies that impurities have the same periodicity. Thus the unit cell on which the translational periodicity is applied must be chosen carefully in such a way that the repeating impurities do not interact. In general this requirement implies that the unit cell be very large, a feature which results in extremely demanding computations and thus hinders the use of the CO method for the study of impurities. 

Generally, as we shall see in this Chapter, there is only one conformation of a molecule in any one crystal structure. One of the most common questions asked by solution chemists about the results of a crystal structure analysis is how can one be sure that the solid-state conformation is the same as that observed in solution The conformation found for a flexible molecule in the crystalline state is that of one of the various conformers found in solution. This has been verified by other physical methods such as nuclear magnetic resonance. If, however, a molecule is found to have the same conformation in several different crystal structures, it is reasonable to assume that this conformation has a low (although not necessarily the lowest) energy. This assumption can often be tested by calculation (by ah initio molecular orbital calculations, for example) of the appropriate theoretical potential energy curve. 

The prototype molecule (or cluster ) approach to the quantum chemical treatment of extended systems has proved to be a valuable tool, especially if the chemical phenomenon of interest is mainly of local character. The prototype molecule models were used e.g. in the quantum chemistry of silicates [58], zeolites [59] and enzymes [14]. In the solid state, quantum chemistry calculations on prototype molecules [60] represent an important alternative to crystal orbital type methods. Although these latter calculations may be very important as starting points even for the description of local phenomena which violate the exact translational symmetry, the cluster approaches have the advantage of providing a direct space representation of the wave function. 

In the second part (applications) we discuss some recent applications of LCAO methods to calculations of various crystalline properties. We consider, as is traditional for such books the results of some recent band-structure calculations and also the ways of local properties of electronic- structure description with the use of LCAO or Wannier-type orbitals. This approach allows chemical bonds in periodic systems to be analyzed, using the well-known concepts developed for molecules (atomic charge, bond order, atomic covalency and total valency). The analysis of models used in LCAO calculations for crystals with point defects and surfaces and illustrations of their applications for actual systems demonstrate the eflSciency of LCAO approach in the solid-state theory. A brief discussion about the existing LCAO computer codes is given in Appendix C. 

It is well known that one of the standard approaches to Eq. (2.1) is the linear combination of atomic orbitals (LCAO) method it consists in expanding the states of the solid in linear combination of atomic (or molecular) orbitals of the composing atoms (or molecules). This method, when not applied in oversimplified form, provides an accurate description of core and valence bands in any type of crystal (metals, semiconductors, and insulators). Applied with some caution, the method also provides precious information on lowest lying conduction States, replacing whenever necessary atomic orbitals with appropriate localized orbitals. ... 

The LMTO method has the computational speed and flexibility needed to perform calculations of electron states in molecules and compounds. Therefore in the present chapter we shall generalise the LMTO formalism purely within the atomic-sphere approximation to include the case of many inequivalent atoms per cell. The LMTO method is based on the variational principle in conjunction with energy-independent muffin-tin orbitals but, in addition to this approach, we have also considered the tail-cancellation principle which led to the KKR-ASA condition (2.8). Since the latter has conceptual advantages, we apply the tail-cancellation principle to the simplest possible case of more than one atom, namely the diatomic molecule. After that, we turn to crystalline solids and generalise or sometimes rederive the important equations of LMTO formalism. Hence, in addition to giving the LMTO equations for many atoms per cell, the present chapter may also serve as a short and compact presentation of the crystal-structure-dependent part of LMTO formalism. The potential-dependent part is treated in Chap.3. In the final sections are listed the modifications needed to calculate ground-state properties for materials with several atoms per cell. 

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... 

---