Model Potential Parameters Research Articles

Model potential calculations for second-row transition metal molecules within the local-spin-density method

The model potential (MP) method originally proposed by Huzinaga and Bonifacic is extended to spin-polarized local-spin-density calculations, including scalar relativistic effects. The theoretical justification of the MP method in this case is studied and the method of optimization of the basis functions and MP parameters is given. The validity of the frozen core approximation is studied for Mo2, Ru2, and Ag2. It is found that the MP can very accurately reproduce all-electron (AE) results if the 4p electrons of Ag and the 3d electrons of Mo are also considered as valence electrons, although inclusion of these electrons in the core still yields a useful level of accuracy. It is shown that the present MP results are not sensitive to basis set superposition errors (BSSE). Upon inclusion of the scalar relativistic effects the calculated bond length and vibrational frequency of Ag2 are in near perfect agreement with experiment, while the dissociation energy is overestimated by 23% with the ‘‘best’’ local potential (VWN). MP calculations have also been performed for AgH, AgO, and AgF. The same level of agreement with experiment as for Ag2 was found, with the exception of the bond length for AgO. Our calculated bond length is 0.05 Å shorter than the presently accepted experimental value. Since some uncertainty is associated with the spectroscopic assignments for AgO we believe an experimental reexamination would be in order.

Model potential method in As 2 and As 2 molecular calculations

Model potential parameters and a valence basis set are determined for the As atom. The Hartree-Fock-Roothaan calculation was performed for As 2 and As 2 using the model potential for the core electrons of the As atom. The molecular-orbital energies given by the model potential calculation agree very well with the reference all-electron calculation. The calculated values of the equilibrium bond length and spectroscopic constants (vibrational, rotational, and force constants) agree with experimental data.

Collisional transfer of rovibrational energy from quantum calculations. II. The case of LiH with He

AbstractThe collisional exchange of energy between He atoms and a strongly polar diatomic, the LiH target, has been studied theoretically over a range of relative energies that had been previously sampled by experiments with molecular beams. The relative importance of the rotational and/or vibrational channels is examined by studying in detail the effect of the interaction via a model potential coupling parameter. The different behavior exhibited by differential cross sections (total and partial inelastic) is also analyzed in terms of the strength and shape of the interaction anisotropy.

Upper bounds for the zeroth moment of the potential curve for some simple diatoms

An upper bound is computed for the zeroth moment of the potential curve for two=electron diatoms and also for diatoms with all electrons on one centre. Inequalities for the parameters of model potentials can then be found and examples are given for the cases of H2+, H2 and HHe+. Resulting constraints on the form of semi-empirical potentials are discussed.

Correlation effects in heavy multi-electron ions represented by 4-4 transitions in the Zn-like isoelectronic sequence

A model potential relativistic method is used to calculate the energies of the lower levels with configurations 4s4p, 4p2, 4s4d, 4p4d in the Zn I isoelectronic series for nuclear charges Z=36-50. The calculation is made in the first order of perturbation theory with the effective inclusion of higher corrections. The role of various correlation effects is evaluated: core polarisation due to outer electrons, superposition of configurations, mutual screening of the outer electrons. As a zeroth approximation the authors obtain the wavefunctions of the outer electron in the field of the core model potential; the model potential parameter being determined from the empirical energy levels of Cu-like ions. An extrapolation procedure for model potential parameters is used for the calculation of energy levels of Cu-like ions. Energy levels 4lj and 5lj of Cu-like ions for Z=36-60 are given with an accuracy of about 0.1%. The accuracy of the energies of Zn-like ions is within 1%.

Nonrelativistic and quasirelativistic model potential calculations on AgH and Ag2

AbstractThe major relativistic effects are included into the model potential (MP) method of Bonifacic and Huzinaga. The effects are incorporated on the level of Cowan and Griffin's relativistic Hartree–Fock (RHF) method. The model potential parameters are determined using the results of nonrelativistic Hartree–Fock (NHF) and RHF calculations. A new scheme of selection of the basis functions for use in atomic and molecular MP calculations is proposed. To obtain agreement with the Hartree–Fock calculations on AgH and Ag2, the 4p shell has to be included explicitly in the MP calculations. The explicit treatment of the 4p electrons and the resulting reduction of the core size are necessary in order to overcome difficulties with approximate representation of the large 4p–4d core‐valence interactions on the MP level.

Liquid crystals of disc-like molecules

Liquid crystals of disc-like molecules fall into two distinct structural types, the columnar and the nematic. The columnar phase in its simplest form consists of discs stacked aperiodically in columns, the different columns constituting a two-dimensional array, whereas the nematic phase is an orientationally ordered arrangement of discs without any long-range translational order. After a brief review of the structures of the various mesophases, two topics are considered in somewhat greater detail. (i) An extension of McMillan’s mean field model of smectic A to liquid crystals of disc-like molecules. The translational order is now assumed to be periodic in two dimensions. Calculations show that when the lattice is hexagonal, or departs from it only slightly, the transition from the columnar to the isotropic phase may take place either directly or via a nematic phase, depending on the model potential parameters Interpreting a to be a measure of the chain lengths as in McMillan's model, the theoretical phase diagram is shown to be in broad agreement with the experimental trends. As the asymmetry of the lattice is increased the theory predicts the occurrence of a smectic A phase as well. The new smectic A phase is biaxial. (ii) A theoretical study of fluctuations in the columnar liquid crystal. It is shown that the Frank elasticity of the liquid-like columns stabilizes the two-dimensional order, a result that was in fact envisaged by Peierls and Landau in the 1930s. The meansquare fluctuation of the lattice as well as the Debye—Waller factor show a certain dependence on the linear dimensions of the sample.

Pseudopotential band structure ofAl1−x−yGaxInyAs

The fundamental energy gaps ${\ensuremath{\Gamma}}_{15v}\ensuremath{-}{\ensuremath{\Gamma}}_{1c}$, ${\ensuremath{\Gamma}}_{15v}\ensuremath{-}{L}_{1c}$, ${\ensuremath{\Gamma}}_{15v}\ensuremath{-}{X}_{1c}$, of the quaternary solid solutions ${\mathrm{Al}}_{1\ensuremath{-}x\ensuremath{-}y}{\mathrm{Ga}}_{x}{\mathrm{In}}_{y}\mathrm{As}$ have been studied for a grid of values of the compositional parameters $x$ and $y$, with the use of a model version of the coherent-potential approximation (CPA). The pseudopotential of Aymerich, Meloni, and Mula has been applied to the limiting ternary alloys by taking the charge transfer, between anionic and cationic sites, and the core repulsive part of the potential as adjustable parameters to fit the experimental bowing coefficients of the fundamental optical transitions. The values of the energy surfaces ${E}_{i}(x, y)$ of quaternary solutions have been then obtained from the usual pseudopotential band-structure calculations with the use of, for each composition, geometrically interpolated values of the model potential parameters both for the virtual-crystal approximation term and for the CPA correction. The comparison of theoretical results to the energy-transition values obtained from the direct interpolation of the corresponding values of ternary alloys and to experimental measurements should allow us to clarify the law of combination of the nonlinear coefficients in multinary solid solutions of increasing complexity.

The use of model potentials in molecular calculations. I

The model potential method proposed by Bonifacic and Huzinaga in 1974 is unique among various effective core potential methods in that it is capable of producing the pseudovalence orbitals with the proper nodal structure. This unique feature of the method has now been fully exploited and implemented by improving the process of determination of the model potential parameters and the valence orbital basis functions. The present article includes a general exposition of the method and the application to a number of atoms: Li–Ne, Na–Ar, K, Ca, Ni, Cu, Br, and Pd.

New developments in the model potential method

Significant improvements are introduced in the process of determining the model potential parameters and truncating the basis sets in Bonifacic and Huzinaga’s model potential. The pseudovalence orbitals can simulate closely the shapes of the reference atomic valence orbitals given by all-electron calculations including the inner nodes. Test molecular calculations are carried out for CO, HCl, P2, and Cl2. The model potential developed by the present author yields molecular orbital energies, atomization energies, geometries, and spectroscopic constants in near perfect agreement with the result of all-electron reference calculations.

Molecular dynamics simulation of the structure of liquid rubidium along the saturated vapour-pressure curve

The structure and the thermal motion of ions in liquid rubidium are simulated for six states including the critical point along the saturated vapour-pressure curve using the molecular dynamics method. The effective pair potential between rubidium ions in expanded liquid states is calculated by extrapolating the optimised model potential parameters proposed by Cowley (1976) and the screening parameters of Singwi et al. (1970). It becomes deeper and softer at lower densities. The calculated structure factor, S(Q) and the pair distribution function, g(r) are found in better agreement with the measured ones than those calculated by Mountain (1978) using the empty-core pair potential of Price (1971). The calculated short-range orders for an expanded state close to the critical point are significantly different from those of liquid argon near the critical point, which is mostly due to softening of repulsion between rubidium ions. The self-diffusion constant is calculated and fits Tn law with n=1.65 along most of the saturated vapour-pressure curve, in contrast to the case of fluid argon where it fits the Arrhenius-type formula.

Model potential calculations of the Na-He system

The model potential method developed by Valiron et al. (1977, 1979) is applied to the calculation of the potential energies of the ground and excited states of the Na-He molecular system. The excellent agreement with the ab initio calculations and the stability of the results to variations in the model potential parameters establish the reliability of the method. Interesting features in the more highly excited states are observed. The calculated interaction energies are used to compute the profile of the fluorescence spectrum from optically thin Na vapour in the presence of He at moderate pressures. The theoretical results for the Na resonance line profile are compared with the experimental data of York et al.

Neutron elastic scattering from 92, 96, 98, 100Mo between En = 7 and MeV

Elastic scattering of 7, 9, 11, 20 and 26 MeV neutrons from 92, 96, 98, 100Mo has been measured using the Ohio University time-of-flight facility. The measured yields were normalized with respect to the incident flux at 0°, to obtain absolute differential cross sections. These values were corrected for dead time, detector efficiency, flux attenuation, multiple scattering, finite geometry, neutron source anisotropy and compound elastic contributions. Relative uncertainties are estimated to be between 5%–10%, and the uncertainty in the normalization is estimated to be less than 5%. The data were used to obtain neutron optical model potential parameters. The energy and isospin dependence of the potential depths is presented. Coupled channel calculations are presented and compared with standard optical model calculations.

Lattice dynamics of thorium in a resonance model pseudopotential theory

A microscopic theory of the lattice dynamics of the actinide metal thorium is developed based on the model potential method of Heine and Abarenkov (1965). The model potential parameters Al(E) for thorium are determined from the spectroscopic term values of the trebly ionised free atom in the spirit of the quantum defect method. The electron-phonon interaction is then represented in the rigid-ion approximation by an actinide model potential. The effect of the itinerant 5f electronic states in actinides is incorporated through an s-f hybridisation in the energy bands which simulates Ziman's resonance model (1965) in the l=3 phase shift. Physical effects associated with the nonlocality of the model potential are also included. Calculated phonon dispersion curves obtained for thorium are in good agreement with experimental results.

The calculation of model potential parameters; an additional comment

For original article see ibid., vol.7, p.1207 (1977). The Heine-Abarenkov model potential for an ion in a metal is a functional of the energy and conduction electron potential, ve(r). The nature of this functional relationship is investigated using the integral expression for the scattering phase shifts, and the weighting function required for the (ve)t average is derived in terms of the screened ion radial wavefunctions.

Calculation of the bandstructure of aluminium using the model potential method

The model potential theory of Heine and Abarenkov (1964) and Shaw (1968) has been applied to the calculation of the energy bandstructure and density-of-states for aluminium. The use of Shaw's optimised model potential parameters is found to produce energies which differ from those of the ab initio calculations by approximately 0.03 Ryd. The model potential procedure due to Bunyan (1977), however, leads to improved agreement with the ab initio calculations to within 0.01 Ryd, provided the potential is modelled to agree with that used in the ab initio calculation. The results indicate that basic bandstructure data for the simple metals can be obtained from the model potential theory to an accuracy comparable to that achieved in more complex first-principles calculations.

Hartree-Fock-Slater procedure for the calculation of optimised model potential parameters for aluminium

The procedure for obtaining optimised Heine-Abarenkov model potentials from the free ion term values is investigated by using Hartree-Fock-Slater calculations to model the ion in its metallic environment. The usual model potential approximation based on linear extrapolation of the model potential parameters from the term values into the conduction band and the r5 weighting factor used to determine an average potential (V)rest are shown to lead to appreciable error. The relationship between the bandstructure Fermi energy and the Fermi energy deduced from cohesive energy measurement is examined and improved agreement obtained.

Neutron elastic scattering at 11 MeV and the isospin dependence of the neutron-nucleus optical potential

Elastic scattering of 11 MeV neutrons from Mg, Al, S, Ca, V, Mn, Fe, Co, Ni, Nb, 92,96,98, 100Mo, In, 120Sn, Ho, Ta, 206Pb, Pb and Bi have been measured with the Ohio University tandem Van de Graaff accelerator. The required neutrons were produced by the 2H(d, n) 3He reaction. Standard pulsed beam time-of-flight techniques were employed with a multiple detector system which included five large area liquid scintillators and plastic scintillator monitor. Measurements of the incident flux at zero degree were used to normalize the differential cross sections. Energy resolution was approximately 360 keV (FWHM). The measured cross sections were corrected for dead time, detector efficiency, attenuation, multiple scattering, finite geometry, neutron source anisotropy, contaminants and compound elastic contribution. Relative and normalization uncertainties in the measured cross section are 3–5% and approximately 3.5 % respectively. A search for an average set of optical model potential parameters based solely on the measured (n, n) data for the isotopically pure samples and for which inelastic contributions to the elastic data are insignificant yield isopin dependent potential strengths, V R = [ 47.14−22.50(N−Z) A ] MeV and W D = [ 12.16−2.03(N−Z) A ] MeV .

An energy dependent model pseudopotential

A modification of the form of Shaw's optimised model potential (1968) is proposed so as (a) to decrease the energy dependence of the model potential parameters and (b) to increase the flexibility, and thereby the applicability, of the model potential. Calculations of the Fermi surface distortions of K, Rb and Cs metals using the present proposed model pseudopotential show significant improvements in the results over those obtained from Shaw's optimised model potential. The phonon dispersion relations of K, Rb, Cs and Pb are also calculated and the results obtained lend support to the necessity of the modification.

Spin-flip probability in the inelastic scattering of 14.2 MeV neutrons from the 2.23 MeV state of 32S

The spin-flip probability for 14.2 MeV neutrons scattered inelastically from the Q = −2.23 MeV first excited state of sulphur has been measured at scattering angles between 40° and 160° using a γ-correlated neutron time-of-flight method. The results of the measurements were compared to the predictions of a coupled-channels code. A vibrator model with a quadrupole deformation parameter β 2 of 0.37 was used, and searches on the optical model potential parameters were made. The calculations were able to account for elastic and inelastic scattering data, but did not provide a satisfactory fit to the spin-flip data.