Linear Combination Of Atomic Orbitals Method Research Articles

Self-consistent Hartree energy bands of MnO

A self-consistent Hartree energy-band calculation has been done for MnO using the linear combination of atomic-orbitals (LCAO) method. Gaussian-type atomic orbitals were used in the LCAO method. This calculation was done for paramagnetic MnO with the NaCl lattice structure. The results show that the energy bands around the Fermi level of MnO are usually flat, meaning that the electrons in this region are strongly localized. Therefore short-range correlation was added to the results of this band calculation. The short-range correlation effects were added by calculating atomic-type corrections to the original band structure. The results of this correlation calculation show that a large amount of energy is required to excite an electron from the Mn $3d$ band. Therefore the lowest excitation (the one that requires the least energy) is an excitation from the top of the O $2p$ band to the Fermi level. This yields a fundamental energy gap of 4.9 eV, which is in good agreement with optical absorption experiments. This fundamental energy gap of 4.9 eV implies that MnO is an insulator, in agreement with conductivity experiments. Also, our Hartree results for the valence bands of MnO agree very well with the results of photoemission experiments.

InAs-GaSb superlattice energy structure and its semiconductor-semimetal transition

We performed a band calculation, based on the LCAO (linear combination of atomic orbitals) method, for the InAs-GaSb superlattice. Of possible semiconductor combinations the InAs-GaSb is of special interest because of indications that at a heterojunction interface the bottom of the conduction band of InAs lies below the top of the valence band of GaSb. Our results show that the InAs-GaSb superlattice is a semiconductor when the layers of the constituent materials are thin, and becomes a semimetal when the layer thicknesses are increased. The critical InAs thickness for this transition is approximately 115 \AA{}, over which the electrons from the valence band of GaSb flood the conduction band of InAs. For the thick-layer limit, we treat the problem in a Fermi-Thomas approximation.

Application of cubic harmonic expansion in electronic structure calculation of defect using LCAO method

A new scheme for calculating the electronic structure of a defect in a solid by the method of linear combinations of atomic orbitals (LCAO) is presented. Instead of evaluating the multi-center integrals individually, the energy matrix elements are calculated en bloc by expanding the crystal potential and the LCAO wave functions in lattice harmonics around the defect centre. A novel procedure is described in which the atom-centered s-, p- etc. character of original atomic orbitals is partly lost, which enhance the variational freedom considerably. As an example, results of a preliminary work on F-centre in LiF are reported using a Gaussian atomic orbital basis.

Optical properties of hexagonal boron nitride

Optical absorption, reflectivity, and photoconductivity in the near-uv range (1950-3200 \AA{}) of a thin film of hexagonal boron nitride were measured. The main absorption peak was observed at 6.2 eV. A sharp fall at about 5.8 eV was attributed to the direct band gap. The temperature dependence of the band gap was found to be less than 4 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}5}$ eV/\ifmmode^\circ\else\textdegree\fi{}K. Self-consistent tight-binding band-structure calculations were performed on a two-dimensional hexagonal crystal model, using Hamiltonian matrix elements calculated by semiempirical LCAO (linear combination of atomic orbitals) methods. The calculated value for the band gap of hexagonal BN was in reasonably good agreement with the experimental value obtained in the present work, as well as with values reported earlier from electron-energy-loss and photoelectron-emission measurements. The calculations also predicted a very small change in the band gap with temperature, in agreement with the experimental observations.

Self-Consistent-FieldXαCluster Method for Polyatomic Molecules and Solids

This paper describes a practical self-consistent-field (SCF) method of calculating electronic energy levels and eigenfunctions, adapted for polyatomic molecules and solids. The one-electron Schr\"odinger equation is set up for a so-called "muffin-tin" approximation to the true potential, spherically symmetrical within spheres surrounding the various nuclei, constant in the region between the spheres, spherically symmetrical outside a sphere surrounding the molecule. The method of solving this equation is a multiple-scattering method, equivalent to the Korringa-Kohn-Rostoker (KKR) method often used for crystals. Once the eigenfunctions and eigenvalues of this problem are determined, one assumes that the orbitals of lowest eigenvalue are occupied, up to a Fermi level. From the resulting charge densities, one can compute a total energy, using a statistical approximation for the exchange correlation. This approximation has an undetermined factor $\ensuremath{\alpha}$ (whence the name $X\ensuremath{\alpha}$ method). The spin orbitals and occupation numbers are varied to minimize this total energy, resulting in one-electron equations. The value of $\ensuremath{\alpha}$ for an isolated atom is determined by requiring that the total energy, using the statistical approximation, should equal the precise Hartree-Fock energy. This leads to very accurate spin orbitals. In a molecule or crystal, one uses the $\ensuremath{\alpha}'\mathrm{s}$ characteristic of the various atoms within the atomic spheres, and a suitable average in the region between. The computer programs for making these self-consistent calculations, for such radicals and polyatomic molecules as S${\mathrm{O}}_{4}^{\ensuremath{-}2}$, Cl${\mathrm{O}}_{4}^{\ensuremath{-}}$, Mn${\mathrm{O}}_{4}^{\ensuremath{-}}$, and S${\mathrm{F}}_{6}$ have been worked out and calculations made. They are more than 100 times as fast as comparable programs using the LCAO (linear-combination-of-atomic-orbitals) method, and the results appear to be in better agreement with experiment than such LCAO results. For calculating the frequencies of optical transitions, one must make a self-consistent calculation, not for the initial or final state, but for what we call the transition state, in which occupation numbers are halfway between the initial and final states. Then it can be proved that the differences of eigenvalues of the $X\ensuremath{\alpha}$ method are more accurate than Hartree-Fock energy values, in that they take account of the modification or relaxation of the orbitals in going from the initial to the final states. These transition states, for a crystal, involve a localized perturbation at the site of the excited atom. The multiple-scattering method is adapted to the use of such perturbed crystals, as well as to isolated molecules, and to perfect crystals. It results, in such problems as x-ray absorption, in the use of localized orbitals rather than bandlike functions. The method is adapted to the calculation of magnetic problems, by use of a spin-polarized version of the method. The method can also be used for calculations of cohesive energy of crystals, and has been used successfully for several types of metals. Unlike most other SCF methods, long-range correlation is automatically included, so that the energy of a system as a function of internuclear positions automatically reduces to the proper values at infinite internuclear distances.

Study on corrosion inhibition from aspect of quantum chemistry

The physico-chemical properties of corrosion inhibitors are one of the principal criteria for the determination and selection of inhibitors, and it is apparent that such quantities as the dipole moment or electron density play an important role. Issuing from these and other assumptions, an attempt has been made to apply quantum-chemical values to the selection and evaluation of corrosion inhibitors. The calculation was carried out by the LCAO (Linear Combination of Atomic Orbitals) method for the homologous series of substituted pyridine derivatives. Molecular diagrams and other quantum values were obtained and correlated with the inhibition of corrosion produced by these substances as evaluated by polarization curves.

Crystal-Field Effects in the Tight-Binding Approximation: ReO3and Perovskite Structures

The augmented-plane-wave (APW) band structure of Re${\mathrm{O}}_{3}$ is analyzed in terms of the Slater-Koster linear-combination-of-atomic-orbitals (LCAO) interpolation scheme with nonorthogonal orbitals. This approach has several advantages over an earlier treatment involving orthonormal basis functions. First, it provides insight into the physical origin of the crystal-field splittings in Re${\mathrm{O}}_{3}$ and other transition-metal compounds. Second, it leads to more physically meaningful LCAO parameters. Finally, it provides a direct relationship between the crystal-field levels of an isolated transition-metal ion or molecular complex and the band structure of the periodic crystal. In the case of Re${\mathrm{O}}_{3}$, it is shown that the crystal-field effects are due to overlap and covalency between the rhenium $5d$ orbitals and the $2s$, $2p\ensuremath{\sigma}$, and $2p\ensuremath{\pi}$ orbitals of the neighboring oxygen ligands. The splitting between the ${e}_{g}$ and ${t}_{2g}$ bands is due to the $2s$ contribution ${\ensuremath{\Delta}}_{s}$. The difference between the $2s$ and $2p\ensuremath{\sigma}$ contributions ${\ensuremath{\Delta}}_{\ensuremath{\sigma}}\ensuremath{-}{\ensuremath{\Delta}}_{s}$ is responsible for the ${e}_{g}$ bandwidth. This same difference is proportional to the effective transfer integral $b$ in Anderson's theory of superexchange. The ${t}_{2g}$ bandwidth is due to ${\ensuremath{\Delta}}_{\ensuremath{\pi}}$, the $2p\ensuremath{\pi}$ overlap-covalency parameter. This LCAO method is applied to KNi${\mathrm{F}}_{3}$, using LCAO integrals determined from the Sugano-Shulman molecular-orbital calculation for the ${(\mathrm{N}\mathrm{i}{\mathrm{F}}_{6})}^{4\ensuremath{-}}$ complex. The resulting KNi${\mathrm{F}}_{3}$ band structure is qualitatively similar to the APW results for Re${\mathrm{O}}_{3}$, except the bandwidths are narrower by about a factor of 4. In the limit where the Coulomb energy $U$ is large compared with the ${e}_{g}$ and ${t}_{2g}$ bandwidths so that the electrons localize, it is shown that the crystal-field splitting between the localized ${e}_{g}$ and ${t}_{2g}$ Wannier functions is identical with that obtained by Sugano and Shulman via the molecular-orbital method.

An optimized LCAO method for crystals

A modification is proposed to the linear combination of atomic orbitals (LCAO) method for calculating electronic wave functions in crystals. Instead of using fixed atomic orbitals, atomic orbitals characterized by an internal parameter, such as scaling factor are used. This is optimized by applying a variational principle for Wannier functions, using as a trial function a localized orbital constructed from the atomic orbitals by a symmetric orthogonalization procedure. In this way, both the accuracy and range of applicability of the LCAO method for crystals are greatly increased. As shown for the example of the one-dimensional Mathieu problem, the optimum atomic orbital tends to expand as the interatomic distance decreases, and the optimized LCAO orbital tends to expand as the interatomic distance decreases, and the optimized LCAO method describes accurately the change in the energy bands and wave functions as the lattice parameter decreases and the electron states change from tight-binding to free-electron like. The application of the method to crystals containing defects and to amorphous materials is discussed.

Ionic Surface States from a Band-Edge Method

The properties of surface states on ionic or partially ionic crystals such as NaCl, CdS, and GaAs have been computed using the method of linear combination of atomic orbitals (LCAO) in a novel way. The method consists of simulating an ionic crystal by assigning $s$ and $p$ orbitals on alternate sites in a one-dimensional crystal so that the resonance integral alternates in sign along a chain. This feature causes the band gap to lie at the center of the Brillouin zone, as required for these crystals. Also, the unperturbed reference energies are conveniently chosen to be the band-edge energies, rather than the separated-atom energies as in the more usual LCAO method. In an $\mathrm{MX}$ crystal, $M$-like states appear somewhat below the conduction band when the lattice is terminated at an $M$ atom. Similarly, $X$-like states appear somewhat above the valence band when the lattice ends on an $X$ atom. The results complement and support the previous Madelung and Mathieu theories for ionic surface states, and have the proper qualitative behavior to agree with experiment.

Electronic Energy Bands in Strontium Titanate

A theoretical examination of the electronic energy bands of cubic strontium titanate has been performed by application of the LCAO (linear combination of atomic orbitals) method. Diagonal energies were determined from ionization potentials and crystalline fields based on a point-charge model for the ions. The ionic charges were adjusted to give agreement with the observed energy gap. Overlap integrals were estimated from free-ion wave functions. For the crystals studied, the calculations led to filled valence bands derived primarily from oxygen $2p$ orbitals and empty conduction bands derived predominantly from titanium $3d$ orbitals. In cubic strontium and barium titanates, there are six lowest-conduction-band ellipsoids lying along $〈100〉$ directions of $k$ space with minima probably at the edges of the Brillouin zone. The longitudinal mass is about $20\ensuremath{-}50 {m}_{0}$ and the transverse mass about $1 {m}_{0}$. Spin-orbit splitting removes degeneracy at $k=0$ and leads to additional conduction bands several hundredths of an eV above the lowest conduction band. Comparison is made with experimental data on conductivity, Hall effect, thermoelectric power, reflectivity, and soft x-ray emission. Results are in substantial agreement with experiment.

Exciton Models in the Alkali Halides

The common models for excitons in the alkali halides are criticized as to their goodness as zero-order models. Some new calculations on energies, oscillator strengths, and charge distributions are made with the models as they are commonly used. It is shown that the customary use of an LCAO (linear combination of atomic orbitals) method with the electron-transfer model gives qualitatively incorrect results, particularly for the wave functions and charge distributions, results which are not even internally consistent with the model. It is argued that the other model, an excitation model, gives at least semiquantitatively correct results, and is a satisfactory zero-order approximation. Finally, it is shown that in the alkali halides the LCAO method must inevitably be an inconvenient one for finding accurate charge distributions and wave functions for localized centers.