Linear combination atomic orbital method

The theoretical calculation of the surface states is considered by the linear combination atomic orbital (LCAO) method however, in some cases the relation between the electrode potential and the distance from the electrode surface is used. These electric potentials obey the Schrodinger equation and the surface states are obtained as eigenvalues. Inside the crystal lattice, the equation is reduced to the Mathieu differential equation with known solutions. Using the LCAO method, the interaction between the surface states and the electron is avoided. In this case, the proposed wave function is of the form ... 

Feynman model. The Feynman approach, or LCAO (hnear combination of atomic orbitals) method, assumes that a wavefunction of valence electrons i// in a metal is a linear combination of atomic functions ... 

The present article is an attempt to review those studies of pyridinelike heterocycles (mono-azines) and, to a lesser extent, their analogues and derivatives that have interpreted the behavior and estimated various physico-chemical properties of the compounds by the use of data calculated by the simplest version of the MO LCAO (molecular orbital, linear combination of atomic orbitals) method (both molecular orbital energies and expansion coefficients). In this review, attention is focused upon the use of the simple method because it has been applied to quite extensive sets of compounds and to the calculation of the most diverse properties. On the other hand, many fewer compounds and physico-chemical properties have been investigated by the more sophisticated methods. Such studies are referred to without being discussed in detail. In a couple of years, we believe, the extent of the applications of such methods will also be wide enough to warrant a detailed review. 

Linear combination of atomic orbitals (LCAO theory) A method for combining atomic orbitals to approximately compute molecular orbitals. 

Ditchfield, R. Self-consistent perturbation theory of diamagnetism. I. A gage-invariant LCAO(linear combination of atomic orbitals) method for NMR chemical shifts, Mol. Phys. 1974, 27, 789-807. 

There are a number of band-structure methods that make varying approximations in the solution of the Kohn-Sham equations. They are described in detail by Godwal et al. (1983) and Srivastava and Weaire (1987), and we shall discuss them only briefly. For each method, one must eon-struct Bloch functions delocalized by symmetry over all the unit cells of the solid. The methods may be conveniently divided into (1) pesudopo-tential methods, (2) linear combination of atomic orbital (LCAO) methods (3) muffin-tin methods, and (4) linear band-structure methods. The pseudopotential method is described in detail by Yin and Cohen (1982) the linear muffin-tin orbital method (LMTO) is described by Skriver (1984) the most advanced of the linear methods, the full-potential linearized augmented-plane-wave (FLAPW) method, is described by Jansen... 

Chelikowsky, J. R., and S. G. Louie (1984). First principles linear combination of atomic orbitals methods for the cohesive and structural properties of solids Application to diamond. Phys. Rev. B29, 3470-81. 

Summary Comments for the Linear Combination of Atomic Orbitals Method and Diatomic Molecules... 

The question now arises, why does the molecular orbital method afford four possibilities, three with spins paired, whereas the atomic orbital method seemed to yield only two possibilities, one with spins paired The answer comes from looking at a way of approximating ag and wave functions on the two atoms, which you used before. This way of approximating molecular orbitals is often called the LCAO approximation the linear combination of atomic orbitals. Fig. 6.8 shows cross-sections of A and 5. Clearly A + B looks like og (Fig. 6.7) and A — B looks like <7U. By using these approximations, the four functions of Table 6.1 can be rewritten into the forms shown in Table 6.2. When the products in Table 6.2 are expanded, and the results are compared with (6.1), the four functions take the forms shown in Table 6.3. 

This method is called the LCAO (Linear Combination of Atomic Orbitals) method. The electron density p(7) becomes ... 

What we have just described has its counterpart in a mathematical treatment called the LCAO (linear combination of atomic orbitals) method. In the LCAO treatment, wave functions for the atomic orbitals are combined in a linear fashion (by addition or subtraction) in order to obtain new wave functions for the molecular orbitals. 

This is called the linear combination of atomic orbitals method for creating molecular orbitals (the LC AO-MO method). As long as each AO is represented by a single mathematical function (corresponding to a minimal basis set see below), the calculations produce the same number of t/s as 0 s included. This corresponds to the statement that one gets as many molecular orbitals for a molecule as there are atomic orbitals on the individual atoms. Note that LCAO-MO is just one of many possible ways to computationally develop MOs. It is computationally expedient, and it is consistent with our notion that molecules are built up from combinations of atomic orbitals, a conceptual advantage over other possible ways of building up MOs. 

Reactivity indices for each of the ring carbons of phenylsulfide acetic acid and eleven chlorinated derivatives have been calculated by the linear combination of atomic orbitals method. Cocordano has shown that the auxinic activities of these compounds correlated well with the sum of the reactivity indices of the 3- and 6- carbon atoms.In addition, Mainster and Memory have proposed that alternative electronic indices, founded upon the Hu ckel molecular orbital theory, may be used to characterize chemical carcinogens.29 Based upon the use of superdelocalizability to estimate relative chemical reactivity, these indices have been Ugad to characterize molecular carcinogenicity accurately.29... 

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 ... 

The simplest MOs are constructed using the LCAO (linear combination of atomic orbitals) method introduced by Eq. 4.25, adding atomic orbital wave-functions centered on different nuclei in different proportions to obtain one MO wavefunction, often with the electron density delocalized across several nuclei. Extended to several nuclei, Eq. 4.25 becomes... 

Calculated by using the first-principles orthogonalized linear combination of atomic orbitals method in the local density approximation. The effective masses were found to be highly anisotropic and the values in the table are for F-A. Other values are 0.37 (F—K) and 0.32 (F—M) for the electron... 

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