Nonorthogonality Effects Research Articles

On code verification of RANS solvers

This article discusses Code Verification of Reynolds-Averaged Navier Stokes (RANS) solvers that rely on face based finite volume discretizations for volumes of arbitrary shape. The study includes test cases with known analytical solutions (generated with the method of manufactured solutions) corresponding to laminar and turbulent flow, with the latter using eddy-viscosity turbulence models. The procedure to perform Code Verification based on grid refinement studies is discussed and the requirements for its correct application are illustrated in a simple one-dimensional problem. It is shown that geometrically similar grids are recommended for proper Code Verification and so the data should not have scatter making the use of least square fits unnecessary. Results show that it may be advantageous to determine the extrapolated error to cell size/time step zero instead of assuming that it is zero, especially when it is hard to determine the asymptotic order of grid convergence. In the RANS examples, several of the features of the ReFRESCO solver are checked including the effects of the available turbulence models in the convergence properties of the code. It is shown that it is required to account for non-orthogonality effects in the discretization of the diffusion terms and that the turbulence quantities transport equations can deteriorate the order of grid convergence of mean flow quantities.

Part and whole in wavefunction/DFT embedding

Density functional theory (DFT)-based subsystem and embedding methods have found a widespread use in Quantum Chemistry. The combination of correlated wavefunction (WF) methods and density-based embedding methods is particularly promising for accurate descriptions of complex systems. Here, we address some conceptual issues in such WF/DFT methods concerning (1) the interpretation of wavefunctions and densities in these frameworks, (2) the interpretation of subsystem and supersystem density changes as a polarization of the electronic system, and (3) non-orthogonality effects between wavefunctions of ground and excited states in WF/DFT embedding making use of state-specific embedding potentials. Illustrative examples are provided to analyze the significance of these issues in practical calculations. We find that physically reasonable subsystem densities and subsystem dipole moments can often be obtained from subsystem calculations making use of typical setups in practice, in spite of formal issues that are in principle in conflict with such interpretations. For excited-state WF/DFT calculations, we demonstrate that orthogonality violations due to state-specific embedding potentials are usually small.

Nonlinear two-point flux approximation for modeling full-tensor effects in subsurface flow simulations

Subsurface flow models can exhibit strong full-tensor anisotropy due to either permeability or grid nonorthogonality effects. Upscaling procedures, for example, generate full-tensor effects on the coarse scale even for cases in which the underlying fine-scale permeability is isotropic. A multipoint flux approximation (MPFA) is often needed to accurately simulate flow for such systems. In this paper, we present and apply a different approach, nonlinear two-point flux approximation (NTPFA), for modeling systems with full-tensor effects. In NTPFA, transmissibility (which provides interblock connections) is determined from reference global flux and pressure fields for a specific flow problem. These fields can be generated using either fully resolved or approximate global simulations. The use of fully resolved simulations leads to an NTPFA method that corresponds to global upscaling procedures, while the use of approximate simulations gives a method corresponding to recently developed local–global techniques. For both approaches, NTPFA algorithms applicable to both single-scale full-tensor permeability systems and two-scale systems are described. A unified framework is introduced, which enables single-scale and two-scale problems to be viewed in a consistent manner. Extensive numerical results demonstrate that the global and local–global NTPFA techniques provide accurate flow predictions over wide parameter ranges for both single-scale and two-scale systems, though the global procedure is more accurate overall. The applicability of NTPFA to the simulation of two-phase flow in upscaled models is also demonstrated.

Relativistic configuration interaction Fe III 3d6 J = 4→ 3d54p J = 3, 4, 5 f-values

Electric dipole f-values have been obtained between the seven 3d6 J = 4 even states, and all the 3d54p J = 3, 4, 5 odd states. Seventy-two of these exceed 0.01 and are published here. The average agreement between the length and velocity gauges is 1.8%. Due to the small size of the f-values, non-orthogonality effects may be significant in shell jump cases, as we have here. Many of the odd levels are so nearly degenerate that the introduction of small semi-empirical corrections is necessary. We compare favourably (19% on average) to the most modern of the three available semi-empirical data sets.

Magnetometer Autocalibration Leveraging Measurement Locus Constraints

An autocalibration algorithm for calibration of magnetometer triads used to determine heading in small aerial vehicles is developed and its performance is analyzed. The algorithm estimates the 12 calibration parameters, which are elements of an affine transformation mapping measured, and thus, erroneous magnetometer measurements into corrected field measurements. The mapping corrects magnetometer output errors due to hard-iron biases, scale factor deviations, and nonorthogonality effects caused by soft irons or physical arrangement of the sensor axes. The correction mapping is determined by leveraging the fact that the locus of Earth's magnetic field vector as measured by a triad of magnetometers is an ellipsoid. The coefficients defining the ellipsoid are computed using a total least-squares estimator. It is shown that, generally, in such autocalibration algorithms, soft-iron errors and sensing axes nonorthogonality can result in an error mode that is unobservable. These conditions are identified and the algorithm incorporates a method for detecting them. In these instances, even though autocalibration can provide an estimate of only 9 of the 12 calibration parameters, it is shown that how external measurements from a triad of accelerometers and a global positioning system receiver can provide the information needed to estimate the unobservable error mode.

On the Calculation and Modeling of Magnetic Exchange Interactions in Weakly Bonded Systems: The Case of the Ferromagnetic Copper(II) &mgr;(2)-Azido Bridged Complexes.

The magnetic exchange interactions in copper(II) &mgr;(2)-azido bridged complexes have been studied using several density functionals and both GTO and STO basis sets. From a methodological point of view, we have taken into proper account nonorthogonality effects in the framework of the broken symmetry approach. A remarkable agreement with experimental data has been obtained using the crystallographic geometry and the new MPW1PW functional. However, modeling of the true ligands by ammonia molecules and complete optimization of the geometry of the isolated complex significantly deteriorate the results. While this can lead to limitations on quantitative studies, general trends and magnetostructural correlations remain very significant. These results are, furthermore, not very sensitive to technical details, like the form of the functional or the type of basis set used.

Environment-dependent tight-binding model for molybdenum

A transferable orthogonal tight-binding model is developed for molybdenum, with special emphasis on applications in molecular-dynamics studies. The elements of the Hamiltonian matrix and the repulsive potential are allowed to depend on the environment in order to account for the effects of the neglected three-center matrix elements, for the neglected nonorthogonality effects and for the variation of the finite set of basis orbitals in different configurations as well. To check the accuracy of the model, the structural energy differences, the elastic constants, the phonon spectrum along high-symmetry lines in the Brillouin zone, the formation and migration energy of a vacancy, the formation energy of an octahedral interstitial atom, surface energies, and relaxations, as well as reconstructions of a (100) surface, are calculated and compared with ab initio data and experimental results.

Orbital nonorthogonality effects in band structure calculations within the tight-binding scheme

The consequences of orbital nonorthogonality on the structure of energy bands in the one-electron approximation are studied for an infinite one-dimensional monoatomic chain with nearest-neighbor interactions in the tight-binding scheme. Nearest-neighbor overlap introduces modifications into the band structure which affect the functional form of the band, in such a way that symmetry with respect to the corrected atomic level is lost. It is found that effective masses at the top of the band decrease while those at the bottom increase. It is shown explicitly how the nearest-neighbor nonorthogonality effects can be simulated by introducing more-distant-neighbor interactions while the orbital orthogonality requirement is maintained.

Simulation of the band structure of liquids: Some calculation considerations and a test of the mean-spherical approximation

The discovery that simple classical liquid theories such as the mean-spherical approximation (MSA) provide a route for calculating electronic band structure opens up some interesting possibilities for investigating the interplay between liquid structure and electronic behavior. However, the MSA is better known for providing qualitative insights than it is for the accuracy of its quantitative predictions. In this paper we compare the MSA-computed band structure of a hard-sphere liquid with the exact band structure obtained from direct numerical diagonalization of the Hamiltonian at Monte Carlo–generated liquid configurations. In the medium dense to dense liquid range, the agreement is qualitatively reasonable with both s and p orbital basis sets, regardless of whether one includes nonorthogonality effects. Lower densities, as expected, are not as well treated. In the course of this comparison we also found it necessary to be aware of the limitations on computer-simulated band structures which are imposed by finite system size.

Pseudostate description of breakup in the coupled-reaction-channel method: Numerical study of nonorthogonality effects.

To simulate the effects of the breakup channel on rearrangement amplitudes, the conventional coupled-reaction-channel (CRC) expansion is augmented by pseudoreaction channels. The construction of the projector for the extended CRC space is discussed, and transition-operator equations on this space are given. By solving the full and post-approximation forms of the CRC equations for a model three-particle problem, the crucial role played by the nonorthogonality terms is demonstrated.

Simple model for structural properties and crystal stability of sp-bonded solids.

A simple, universal model for structural properties of sp-bonded semiconductors and insulators is presented. The model elucidates the physical mechanisms determining the chemical trends and predicts semiquantitatively the stable crystal structures, bond lengths, bulk moduli, transition pressures of structural phase transformations, long-wavelength transverse optical phonons, and band structures for binary nonmetals in the rocksalt, cesium chloride, and zinc-blende phases. The theory explains the puzzling strong cation and weak anion dependence of the observed structural transition pressures. It predicts, as a function of pressure, a universal sequence of structural phase transformations among the cubic phases of binary solids. A drastic softening of the transverse optical phonons across the pressure-induced phase transition from the zinc-blende to the rocksalt structure in II-VI compounds is also predicted. The physical origin of this softening is shown to be closely related to ferroelectricity. It is shown that the chemical trends in the structural properties of semiconductors and insulators are governed by a counterbalance of attractive and repulsive short-range interactions, whereas long-range interactions play only a minor role, in contrast to the classical point-charge models of ionic crystals. The theory is based on the semiempirical tight-binding method and includes charge transfer and nonorthogonality effects. Only properties of the neutral atoms are used as input for a given crystal. The total energy is explicitly minimized as a function of volume in order to find the static and dynamic equilibrium crystal properties.

Channel non-orthogonality effects on the heavy-ion transfer polarization potential

The channel-non-orthogonality correction to the heavy-ion transfer polarization potential is discussed.

Excitation of 17O and 18O scattered by 208Pb in the Coulomb-Nuclear interference region

Inelastic scattering of 17O and 18O by 208Pb has been measured at 78 MeV, and compared with theoretical predictions. In both cases the nuclear amplitude is dominated by the valence nucleon contribution. Direct excitation is the principal mechanism in 18O scattering, while for 17O excitation, two-step transfer is more important. Recoil and non-orthogonality effects must be taken into account.

Non-Orthogonality Effects in Quasi-Elastic Reactions

Calculations both of the Pauli Principle effects in elastic scattering and of the nucleon-transfer channels involve channel states that are not mutually orthogonal and normalised. A unified treatment is given for the non-orthogonalities from either source, resulting in a non-local residual potential that is explicitly Hermitian, independent of the incident kinetic energy, and cumulative in the number of exchange configurations.

Examination of direct radiative capture formalism

Within the direct reaction framework the formalism and approximations used to describe particle radiative capture processes at low to medium energies are reviewed. Survey calculations documenting sizeable cross-section sensitivity to different forms of the electromagnetic interaction (Siegert's theorem versus J · A ) that preserve overall gauge invariance are reported. The source of this large difference is the inconsistent use of different initial and final state model hamiltonians. This defect, common to most reaction theories using optical-model wave functions, generates wave function nonorthogonality effects which are found to be significant for M1 transitions. The importance of the spin-flip contribution to both electric and magnetic transitions is also investigated.

A note on non-orthogonality effects in coupled channel methods for rearrangement reactions

Arguments are given that the presence of non-orthogonality terms in coupled channel formulations of rearrangement scattering is the consequence of inconsistent treatment of multichannel dynamics.

Effective interactions and the coupled reaction channel formalism

Many-body scattering theory is used to derive coupled reaction channel-type equations. With the choice of the Alt, Grassberger, and Sandhas off-shell extension for the rearrangement transition operators, the effective interactions which enter these equations are shown to be free of all relevant two-cluster channel unitarity cuts and satisfy well-behaved dynamical equations which possess an explicit multiple scattering structure. Several approximations to these equations are presented and the importance of various nonorthogonality effects is discussed. Symmetrized forms of these equations which take into account the effects of particle identity are found. This permits the inclusion of the Pauli principle in coupled reaction channel calculations in a simple and practical manner.NUCLEAR REACTIONS Generalized coupled reaction channel and resonating group formalisms. Dynamical connected-kernel integral equations for effective potentials. Pauli principle and channel nonorthogonality effects.

Non-orthogonality effects in heavy-ion transfer reactions

Calculations are made of the effects arising from the non-orthogonality of channel states in heavy-ion neutron transfer reactions, as well as the associated elastic-scattering processes. A single iteration is performed upon the integral equations associated with the coupled-channel treatment of the transfer process. The reactions considered are 13C( 16O, 17O) 12C leading to the ground state and first excited state of 17O and the 29Si( 16O, 17O) 28Si reaction leading to the first excited state of 17O. Corrections are found to be as large as 10 % at angles where the cross sections are sizeable. Similar results are obtained for the elastic-scattering processes. The choice of reactions was made so as to emphasise the importance of these corrections. Consideration of the errors associated with the depiction of bound states in terms of a sum of Hankel functions and the use of a local-momentum approximation for the channel motion suggests that the figure for the correction represents an upper bound.

Neutron densities and the single particle structure of several even-even nuclei fromCa40toPb208

Previously developed techniques which sum the squares of proton single particle wave functions to obtain nuclear charge densities are applied to the study of neutron distribution in $^{40,48}\mathrm{Ca}$, $^{58,64}\mathrm{Ni}$, $^{116,124}\mathrm{Sn}$, and $^{208}\mathrm{Pb}$ by comparing to those neutron densities deduced from 800 MeV proton elastic scattering data. The proton and neutron single particle wave functions are derived from a one-body, nonlocal Woods-Saxon binding potential whose parameters are adjusted to give the experimental single particle energies. Empirical spectroscopic factors determine the appropriate occupation probabilities for the single particle levels near the Fermi surface. Proper attention is given to nonorthogonality problems and to the removal of the spurious center-of-mass motion. These semiphenomenological neutron densities are compared to the predictions of the density matrix expansion variant of Hartree-Fock theory and to densities which are empirically deduced from recent 800 MeV polarized proton elastic scattering data. These "experimental" neutron distributions are obtained from approximate second order Kerman, McManus, and Thaler optical potential analyses using essentially "model independent" neutron densities. Qualitatively good agreement is obtained between the semiphenomenological neutron densities computed here, the density matrix expansion predictions, and the empirical results.NUCLEAR STRUCTURE Sums of squares of single particle wave functions; nonlocal potentials; nonorthogonality effects; computed proton and neutron one-body densities for $^{40,48}\mathrm{Ca}$, $^{58,64}\mathrm{Ni}$, $^{116,124}\mathrm{Sn}$, and $^{208}\mathrm{Pb}$.

Semiclassical description of two-nucleon transfer between superfluid nuclei

Two-nucleon transfer reactions between superfluid nuclei are treated in the formalism of Dietrich and Hara starting from a microscopic Hamiltonian in second quantization. Amplitudes for simultaneous and successive transfer of neutron pairs between tin isotopes are calculated in perturbation theory. Nonorthogonality effects are numerically estimated. The possibility of a nuclear Josephson effect is predicted just below the barrier.