Hartree-Fock method crystal orbitals

Hartree-Fock LCAO Crystal Orbital Method... 

What we expect to obtain finally in the Hartree-Fock method for an infinite crystal are the molecular orbitals, which in this context will be called the crystal orbitals (COs). As usual, we will plan to expand the CO as linear combinations of atomic orbitals (cf. 427). Which atomic orbitals Well, those that we consider appropriate for a satisfactory description of the crystal e.g., the atomic orbitals of all the atoms of the crystal. We feel, however, that we are going to have a big problem trying to perform this task. 

The unrestricted and restricted open-sheU Hartree-Fock Methods (UHF and ROHF) for crystals use a single-determinant wavefunction of type (4.40) introduced for molecules. The differences appearing are common with those examined for the RHF LCAO method use of Bloch functions for crystalline orbitals, the dependence of the Fock matrix elements on the lattice sums over the direct lattice and the Brillouin-zone summation in the density matrix calculation. The use of one-determinant approaches is the only possibility of the first-principles wavefunction-based calculations for crystals as the many-determinant wavefunction approach (used for molecules) is practically unrealizable for the periodic systems. The UHF LCAO method allowed calculation of the bulk properties of different transition-metal compounds (oxides, perovskites) the qrstems with open shells due to the transition-metal atom. We discuss the results of these calculations in Chap. 9. The point defects in crystals in many cases form the open-sheU systems and also are interesting objects for UHF LCAO calculations (see Chap. 10). 

The formalism to incorporate translational symmetry into the usual Hartree-Fock approach, the crystal orbital technique, is not new at all 74,75). Reviews of recent devel-opements and applications of the Hartree-Fock crystal orbital method may be found in refs. 76 79). However, only few investigations on the evaluation of equilibrium geometries and other properties derived from computed potential surfaces of one-dimensional infinite crystals or polymers have been reported. 

In Section 2 we briefly summarize the basic mathematical expressions of the LCAO Hartree-Fock crystal orbital method both in its closed-shell and DODS (different orbitals for different spin) forms and describe the difficulties encountered in evaluating lattice sums in configuration space. Various possibilities for calculating optimally localised Wannier functions are also presented. They can be efficiently used in the calculation of excited states and correlation effects discussed in Section 3. 

The electronic structure of (CH) has been studied using various approaches within the framework of the one-dimensional tight-binding crystal orbital (CO) method, that is, from the Hiickel to the ab initio Hartree-Fock level (see, e.g., Kertesz, 1982). Some of the calculated results of the energetic stability of the (CH), isomers in Fig. 1 are listed in Table I. 

Transition metals are important materials with intriguing properties and they have been studied with ever improved methods. A major difficulty is posed by the standard one-electron models where the tight-binding model seems appropriate for the narrow, so-called d-bands while near-plane-wave crystal orbitals are adequate for the conduction bands. Canonical Hartree-Fock solutions are awkward starting points for the description of magnetic structures and the use of spin-polarized versions destroys basic symmetry properties. 

The electronic structures of poiy(fluoroacetylene) and poly(difluoroacetylene) have been investigated previously using the ab initio Hartree-Fock crystal orbital method with a minimum basis set (42). Only the cis and trans isomers with assumed, planar geometries were studied. The trans isomer was calculated to be more stable in both cases, and the trans compounds were predicted to be better intrinsic semiconductors and more conductive upon reductive doping than trans polyacetylene. However, our results show that head-to-tail poly(fluoroacetylene) prefers the cis structure and that the trans structure for poly(difluoroacetylene) will not be stable. Thus the conclusions reached previously need to be re-evaluated based on our new structural information. Furthermore, as noted above, addition of electrons to these polymers may lead to structural deformations that could significantly change the conductive nature of the materials. 

The self-consistent-field (SCF) ab initio Hartree-Fock crystal orbital method is applied with success to polysulfur nitride (SN)X, chains using non-local exchange and evaluating all integrals over atomic orbitals within 5 atomic neighbours accurately. 

We have applied the a. i. Hartree-Fock crystal orbital method to three nuclear configurations I was an equidistant chain with rg =1.62 K and all bond angles taken as 115° II was a chain taken from Boudelle s structure (4) chain III corresponded to the "Penn" structure (5). 

It turns out that another method exists where the approach has been to calculate so-called "free-ion" levels, so that the spin-orbit coupling factor, and the crystal field parameter, 4f, are not adjusted to give a "satisfactory" fit to the experimental data. ASfyboume (1963) and others have accomplished this task by using a Hartree-Fock approach (and a high speed computer) to calculate the energy levels of aU of the trivalent rare earth ions. 

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