An improved LCAO interpolation scheme for energy band structures. Application to four compounds (ScN,ScP, TiN,ZrN) crystallizing in the sodium chloride structure

An improved version of the LCAO interpolation scheme using metal s-, p-, d-, and non metal s-, and p-basis functions is presented for transition metal compounds with sodium chloride structure. This method enables us to interpolate with reasonable accuracy occupied bands as well as unoccupied energy bands up to 0.9 Rydberg above the Fermi level for the compounds ScN, ScP, TiN and ZrN. Due to the limited basis, problems arise however with bands of predominantly transition metal f or non metal d character lying in this energy range - as is the case for ScP.Optimized parameter sets are presented for the compounds ScN, ScP, TiN and ZrN. They were used for the calculation of the imaginary part of the complex dielectric function, ₂(), as will be shown in two forthcoming papers.     

Computation of LCAO wave functions for ground states of polymers and solids

The LCAO form of the Hartree–Fock method is discussed in its application to crystals. General formulae are given for obtaining Fourier coefficient of electronic density (in direct space) as well as of the band structure (in momentum space). Finally, it is shown that in its LCAO form, Slater–Hartree–Fock equations are very simple and that this method is of interest for numerical applications. Special integrals occurring in this formalism are evaluated for a Gaussian basis in the last part of this paper.     

Localized impurity states in the Hartree-Fock,LCAO approximation. I.

One-electron energies and wave functions for deep trap impurity electrons in a crystal are calculated by the Hartree-Fock, single determinant method. The interactions arising from a many-electron single determinant crystal wave function, with automatic inclusion of exchange effects, are those which determine the one-electron functions and energies. The crystal plus impurity system has no translational symmetry and hence the Bloch theorem is not applicable for the solution of the essentially infinite Hartree-Fock eigenvalue matrix. Thus we develop a technique in which the Hamiltonian and overlap matrices are written in terms of bordered matrices, with the interaction of the impurity functions with the rest of the crystal environment contained in the bordering rows and columns. The resulting secular equation explicitly includes the effects of orthogonalization of the entire basis set, including the impurity functions. This technique could be used in an iterative calculation of the electronic structure of a small number of electrons, assuming that the rest of the electrons in the environment are fixed according to an initial estimate.     

Electron densities and chemical bonding in TiC,TiN, and TiO derived from energy band calculations

The chemical bonding in the interesting class of refractory transition metal compounds is illustrated for TiC, TiN, and TiO. Self-consistent augmented plane wave (APW ) calculations are already available for these compounds. Using the respective potentials we have repeated the band calculations on a finer k grid with the linearized APW method to obtain accurate densities of states (DOS ). These DOS can be divided into local partial contributions to characterize the bonding. Further information can be obtained from a decomposition of the metal d DOS into t_2g and e_g symmetry components. These partial local DOS are compared with the LCAO counterpart and give a first picture of the chemical bonding in these compounds. The electron densities corresponding to the occupied valence states are obtained from the LAPW calculations. They provide further insight into characteristic trends in the series from TiC to TiO: around the nonmetal site the density shows increasing localization; around the metal site the deviation from spherical symmetry changes from e_g to t_2g. These effects can be traced back to the three types of valence bands. Electron density plots of characteristic band states (all energies of a selected k point in the Brillouin zone) will be shown. These plots can describe the different types of bonding occurring in these systems.     

First‐principles study of the TiN(111)/ZrN(111) interface

The TiN(111)/ZrN(111) interface was studied by first‐principles method to provide the theoretical basis for developing the TiN/ZrN coatings. Twelve geometry structures of TiN(111)/ZrN(111) interfaces were established. The calculated interfacial work of adhesion reveals that the N‐terminated TiN/N‐terminated ZrN interface with TL site shows the strongest stability. For this TiN(111)/ZrN(111) interface, the results of the partial density of state indicate that the chemical bonding at the interface appeals both ionic and covalent characteristic, which is same as that in the bulk materials. The partial density of states for Zr, Ti, and N atoms at the interface are very similar with those in the bulk, which reveals that the electronic structure transition at the interface is smooth. The results of charge density and charge density difference demonstrate that the lost charge of Ti atom is larger than that of Zr atom, indicating that TiN is more ionic than ZrN. Calculations of the work of fracture indicate that the mechanical failure of the ZrN(111)/TiN(111) interface will take place at the interface. Besides that, the calculation result of the TiN(111)/ZrN(111) interface implies that the TiZrN₂ phase might be formed at the interface because the contacting of the N―N bond is the most stable.     

Enthalpy distributions and densities of states in linear polymers

Using polypropylene as an example, we applied a method we have recently developed to calculate the probability distribution of enthalpy from the temperature dependence of the heat capacity. The method involves the use of local temperature expansions of the heat capacity to calculate a set of moments of the enthalpy distribution. Using the maximum‐entropy method, one can then construct the enthalpy distribution for the system. The method is completely model free. The enthalpy distribution so obtained is the analogue of the Maxwell–Boltzmann distribution of kinetic energies for a gas, and like that function, tells one the distribution of enthalpies that an average unit in the polymer chain can have, a quantity that is crucial to understanding the chemical and physical properties of a polymer. Given the enthalpy distribution, one can then calculate the Gibbs free energy and the density of states that correspond to a particular value of enthalpy, thus giving one an expanded thermodynamics of the system in addition to the usual average quantities. We illustrate the fact that the Gibbs free‐energy distribution for this system scales as a simple function of temperature and that the density‐of ‐states function yields a simple empirical partition function for the system giving both the average thermodynamics and the distribution functions. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 1513–1518, 2001     

Localized impurity states in the Hartree-Fock,LCAO approximation II. The F Center in KCI

We apply the techniques of a previous paper (I) to the F center in KCl. Our purpose is to place the application of Hartree-Fock methods to the F center on a firm theoretical basis by calculating in a consistent manner the magnitude and effect of approximations commonly made in less complete treatments. It is shown that the familiar point-ion approximations and crystal-field approximations with partial consideration of exchange effects are special cases of our results. We compute wave functions and energies step by step for each of the various levels of approximation possible with our model. It is found that the functions resulting from the point-ion model are not good approximations to the final wave functions. Our results show that exchange effects with at least the first two shells of nearest neighbors should be considered since they are of the same order of magnitude as terms in the point-ion model. Overlaps of the F-center function with ion functions out to sixth neighbors are considered. The absorption energy for the F center is calculated to be 0.1619 Ry as compared with the experimentally observed energy of 0.170 Ry. The magnetic hyperfine structure contact terms are calculated for the first two shells of nearest neighbor ions, using approximate orthogonalized functions, and found to be 29.7 Mc/h for the nearest neighbor K⁺ ions and 10.9 Mc/h for the next nearest neighbor Cl^? ions. The experimentally observed values are 21.6 and 7.0, respectively. Given these differences and the excessively low values of the one-electron energies, it is concluded that electronic and ionic polarization effects in the ionized crystal states must be considered to calculate accurate F-center wave functions and absolute energy levels.     

One and two body densities for excited states of the helium confined atom

A study of the first excited states of the helium atom confined under impenetrable spherical walls is carried out. Both single particle and two body, intracule and extracule, densities are constructed. Crossing levels and Hund's rule are analyzed in terms of the contribution to the total energy from kinetic, electron–nucleus, and electron–electron energies. A study about the behavior of the single particle and two body densities is carried out. The Multiconfiguration Parameterized Optimized Effective Potential method is employed with a cut-off factor to account for Dirichlet boundary conditions. Single particle density is analytically constructed whereas the Monte Carlo algorithm is used to calculate two body densities.     

Zur Mischkristallbildung in den quasibinären Systemen von UN und den Mononitriden der IV A-Gruppe TiN,ZrN und HfN

Zusammenfassung Röntgenographische Untersuchungen in den Systemen von UN mit den Nitriden der IV A-Übergangsmetalle: TiN, ZrN und HfN ergaben bei 2000° C vollkommene Mischbarkeit für die quasibinären Systeme UN–ZrN und UN–HfN sowie eine Löslichkeit von etwa 1 bis 4 Mol% TiN in UN bei 1700 bis 2400°C.

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X-Ray methods were used to investigate the systems UN with nitrides of the IV A-Group transition metals: TiN, ZrN and HfN. At 2000°C the quasibinary systems UN–ZrN and UN–HfN show complete solid solubility where as TiN is soluble in UN from about 1 Mol% at 1700°C to about 4 Mol% at 2400°C.

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Mit 3 Abbildungen     

A non-emperical LCAO MO SCF investigation of the ground and core ionized states of tetrahedrane and cyclobutadiene

Non-empirical LCAO MO SCF calculations have been carried out on the ground and localised core hole states of tetrahedrane and cyclobutadiene to investigate the suitability of ESCA as a technique for distinguishing between structurally related, isomeric hydrocarbons. Differences in energy are found to be considerably magnified in the core hole state manifold and shifts in core binding energies and shake-up structures are shown to be distinctive.     

Ab initio calculations of the ground and excited states of the ZrN molecule including spin‐orbit effects

The electronic structures with spin‐orbit effects of the zirconium nitride ZrN molecule are investigated by the methods of multireference single and double configuration interaction. The potential energy curves are calculated along with the spectroscopic constants for the lowest‐lying 34 spin‐orbit states Ω in ZrN. A good agreement is displayed by comparing the calculated spectroscopic constants with those available experimentally. The permanent dipole moments are calculated along with the vibrational energies. New results are obtained in this work for 29 spin‐orbit states and their spectroscopic constants calculated. © 2015 Wiley Periodicals, Inc.     

Generalized quantum mechanical two-centre problems. IV. On the accuracy of simple LCAO approximations for H 2 + states and the Eckart criterion

If x denotes an exact solution of the quantum mechanical two centre Coulomb problem, we optimize a normalized LCAO approximation by making the overlap S = (x¦) a maximum. In this context we study how a weight factor (r _a r _b )^–1 in the definition of the inner product changes the approximation and the expectation value of electronic energy. Finally we compare the lower bound given by the Eckart criterion with the exact overlap. Results are reported for H ₂ ⁺ states 1s_g and 2p_u.Dedicated to Professor Hermann Hartmann on occasion of his 70th birthday on May 4th, 1984     

Low-coordination phases of both crystalline and fluid states of alkali metals under extreme conditions of high and low densities,respectively

After a brief summary of early work, involving the present authors, relating to low coordination phases of some alkalis in either dense crystalline states at high pressure (e.g. Li) or low density metallic fluids near criticality (Cs and Rb), contact is made with the very recent density functional study by Pickard and Needs (Phys. Rev. Lett. 102, 146401 (2009)). Whereas these authors predict three- and four-fold coordination numbers for extremely high pressure crystalline phases of Li, we stress here the remarkable behaviour of the heavy alkali metallic fluids Cs and Rb along the liquid–vapour coexistence curve towards the critical point. Coordination numbers ～8?10 near melting then reduce, as the density is lowered, to 2 at or near the critical point.     

LCAO中轨道对称性匹配的两种判别方法

著名分子轨道理论的核心内容是,组成MO的AO必须满足三条原则即对称性一致原则,最大重叠原则和能量相近原则.在这三条原则中,对称性一致原则是首要的,它决定着AO是否能组合成MO.因此在应用LCAO-MO理论处理问题时,首先要判别相组合的AO对称性是否一致.目前文献[1～5]中,均提出了用对称轴和对称面判别组合轨道的对称性是否一致,但都没有给出完善的判别方法.因此对此问题很有必要进一步研究,以期得到更好的结果.本文在研究LCAO-MO三原则新推证方法[6]的基础上,提出了对称面和对称轴各自独立判据两接近AO的对称性是否一致的理论公式,又通过分析得出了完整的判别方法.该法简便实用,并对常见的s、p、d、f等AO的组合进行了判据,结果无一例外.     

The determination of SCF LCAO solutions for open shell configurations

A series of calculations has been carried out using the McWeeny and Roothaan Open Shell methods for determining LCAO-SCF solutions. The work reported here suggests that in order to obtain convergence to the absolute minimum with the McWeeny method it is often necessary to provide a very good initial approximation and that the rate of convergence could be improved by replacing the Steepest Descent by a more powerful minimization technique. Further the Roothaan method does not converge in all cases considered and it appears that Extrapolation techniques accelerate convergence but do not induce convergence in cases displaying oscillatory behaviour.

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Zusammenfassung LCAO-SCF-Rechnungen an offenen Schalen wurden nach den Methoden von Roothaan und McWeeny durchgeführt. Um bei dem McWeenyschen Verfahren Konvergenz zum absoluten Minimum zu erhalten, ist oft eine schon sehr gute Anfangsnäherung nötig. Die Methode des steilsten Abstiegs ist keineswegs optimal. Extrapolationsmethoden können wohl eine Konvergenz beschleunigen, nicht aber das beim Roothaan-Verfahren manchmal auftretende Oszillieren unterdrücken.

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Résumé On a exécuté des calculs LCAO SCF en couches ouvertes avec les méthodes de Roothaan et de McWeeny. Dans la méthode de McWeeny il est souvent nécessaire d'avoir une bonne approximation initiale, pour obtenir une convergence vers le minimum absolu. La méthode de descente le plus escarpé n'est pas optimale. Les méthodes d'extrapolation peuvent accélérer la convergence en effet, mais ils ne suppriment pas l'oscillation, qui arrive quelquefois dans la méthode de Roothaan.