LCAO crystal orbital method

Hartree-Fock LCAO Crystal Orbital Method... 

ABSTRACT, - After a short review of the ab initio SCF LCAO crystal orbital method the negative factor technique for the determination of the density of states in a disordered chain is presented. After this the problem of electron correlation in polymers is discussed and the generation of correlated (quasiparticle) valence and conduction bands is reviewed. 

The value A Eqp 2.98 e l is still by 1 eV larger, than the experimental one /30/. This discrepancy can be explained, ho i/ever, by 1.) the missing part of correlation ( i/hich would reduce according to the estimate of Suhai /22/ the theoretical gap of trans PA to 2.5 eV), by 2.) the neglect of relaxation effects (the N+1 and N-1 particle states were calculated in the formalism given in 2.3 with the help of wave functions obtained for the N particle state) and by 3.) the neglect of phonon polaron effects /22/. It should be mentioned that to take into account relaxation effects one could use the so-called open-shell SCF LCAO crystal orbital method published already in 1975 /lO/. 

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 ab initio SCF LCAO crystal orbital (CO) method (which applies a non-local exchange and keeps all the occurring three-and four center integrals if the number of neighbours to be taken into account has been chosen) was developed about twelve years ago (1 ). In this theory one has to solve the generalized eigenvalue equation... 

The solution of Eq. (1.1.20) with the hamiltonian of Eq. (1.1.7) and under the Bloch conditions (1.1.22) is conceptually similar to the solution under the LCAO-MO assumption (1.1.12). A major difference is that while for the molecular problem one solution is sufficient, the crystal problem is a function of A and therefore the variational problem must be solved a number of times equal to the number of sampling points in the independent part of A-space (the first Brillouin zone). These computational difficulties limit the applicability of the crystal orbital method to rather small molecules and unit cells, the urea crystal being probably the limit. Moreover, the crystal orbital method cannot be applied with a proper consideration of correlation energy because of computational limitations. 

The School started with a review by Andre and Delhalle of the SCF LCAO crystal orbital theory in its initio form with special emphasis on fast computational methods for polymers with... 

A crystalline solid can be considered as a huge, single molecule subsequently, the electronic wave functions of this giant molecule can be constructed with the help of the molecular orbital (MO) methodology [19]. That is, the electrons are introduced into crystal orbitals, which are extended along the entire crystal, where each crystal orbital can accommodate two electrons with opposite spins. A good approximation for the construction of a crystal MO is the linear combination of atomic orbitals (LCAO) method, where the MOs are constructed as a LCAO of the atoms composing the crystal [19]. 

Bom von Karman Boundary Condition in 3-D Crystal Orbitals from Bloch Functions (LCAO CO Method)... 

Any CO will be a linear combination of such Bloch functions, each corresponding to a given /. This is equivalent to the LCAO expansion for molecular orbitals, the only difference is that we have cleverly preorganized the atomic orbitals (of one type) into symmetry orbitals (Bloch functions). Hence, it is indeed appropriate to call this approach as the LCAO CO method (Linear Combination of Atomic Orbitals — Crystal Orbitals), analogous to the LCAO MO (cf. p. 429). There is, however, a problem. Each CO should be a linear combination of the for various types of x and for various k. Only then would we have the full analogy a molecular orbital is a linear combination of all the atomic orhitals belonging to the atomic basis seL ... 

Self-Consistent Field An iterative method of solving the Fock equation. Self-Consistent Field Linear Combination of Atomic Orbitals - Crystal Orbitals An iterative method of solving the Fock equation for crystals (in the LCAO CO approximation). 

Crystal orbitals from Bloch functions (LCAO CO method)... 

I he set of SCb LCAO t O equations will be very similar to the set tor the molecular orbital method (SCF LCAO MO). In principle, the only difference will be that, in the crystal case, we will consequently use symmetry orbitals (Bloch functions) instead of atomic orbitals. 

In Sect. 4.1.5 the Hartree-Fock LCAO approximation for periodic qrstems was considered. The main difference of the CO LCAO method (crystalline orbitals as linear combination of atomic orbitals) from that used in molecular quantum chemistry, the MO LCAO (molecular orbitals as hnear combination of atomic orbitals) method was explained. In the CO LCAO approximation the one-electron wavefunction of a crystal (CO - ifih R)) is expanded in Bloch sums Xt kiR) of AOs ... 

We do not give here a derivation of the ab initio SCF LCAO CO method (for this see /I,10/), only the final expressions are written down in the ID case. Let us write down a crystal orbital in the form of a linear combination of Bloch orbitals. 

If the basic set is chosen to consist of atomic orbitals, this relation forms the fundament for the MO-LCAO method in molecular and crystal theory. In its SCF form this approach was first used by Coulson (1938), and later it has been systematized by Roothaan (1951). More details about the SCF results within molecular theory will be given later in a special section. 

The HF CO method is especially efficient if the Bloch orbitals are calculated in the form of a linear combination of atomic orbitals (LCAO)1 2 since in this case the large amount of experience collected in the field of molecular quantum mechanics can be used in crystal HF studies. The atomic basis orbitals applied for the above mentioned expansion are usually optimized in atoms and molecules. They can be Slater-type exponential functions if the integrals are evaluated in momentum space3 or Gaussian orbitals if one prefers to work in configuration space. The specific computational problems arising from the infinite periodic crystal potential will be discussed later. 

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