Molecular dynamics study of orientational order and rotational melting in clusters of TeF6

Molecular dynamics simulations of the behavior of molecules in crystalline clusters of TeF₆ were carried out on systems of 100, 150, 250, and 350 molecules. Several diagnostic functions were applied to investigate whether rotational melting occurred before translational melting. These functions included the coefficient of rotational diffusionD _θ(T), the “orientational Lindemann index” δ_θ(T), the “orientational angular distribution function”Q(θ,T), and the “orientational pair-correlation function”g _θ(r, T). All indicators implied that rotational melting occurred before translational melting, that it began with the outermost molecules, and that its onset for smaller clusters was at lower temperatures than for larger clusters. Results also showed that the rotational transition coincided with the transition from a lower symmetry phase (monoclinic) to cubic, a phenomenon that had been noted by others to occur with some regularity for systems of globular molecules.     

On inferences of bond character from bond length; the important role of nonbonded interactions

Although experimental molecular structure is one of the oldest and most frequently applied diagnostic tools in studies of the nature of chemical bonds, it is concluded that its application to all but the grossest details of bonds such as carbon-carbon bonds remains entirely speculative. The concept of bond length is discussed to emphasize the need for careful consideration of ambiguities associated with the natural indeterminacy of atomic positions. A theoretical basis is given to the empirical Schomaker-Stevenson rule relating bond length to electronegativity. The divergent views of Pauling and Walsh on hybridization, ionic character, and bond strength are reconciled to some extent with the aid of a simple model. It is shown that structural effects commonly attributed to conjugation, hyperconjugation, hybridization, and partial ionic character can be rationalized to a remarkable extent in terms of nonbonded interactions. It is suggested that these factors are not fundamentally as distinct from each other as they are often assumed to be, and that much more serious attention should be given to the role of nonbonded interactions.     

Error matrices in gas-electron diffraction: II. Influence of weight matrix

The properties of three different forms of error matrices in electron diffraction are investigated, assuming the presence of stationary, Gaussian, Markovian noise in the primary data. The error matrices studied are M_x^p based on the optimum weight matrix P the bona fide error matrix M_x^w based on the nonoptimum weight matrix W, and the false error matrix M_x^o commonly calculated by diffractionists using the formula for the optimum error matrix while incorporating a nonoptimum weighting. Simple formulae relating the elements of the various matrices are derived in the case where W is the best diagonal weight matrix and where geometric constraints are not imposed on parameters. The influence of geometric constraints is tested. Calculations indicate that diagonal weight matrices in ordinary circumstances give results imperceptibly inferior to the results obtained with the best nondiagonal weight matrices. Elements of M_x^w closely approach those of M_x^p whereas elements of the false error matrix, taken alone, may be very misleading.     

Nuclear acoustic resonance in metals

A comprehensive quantum theoretical treatment of nuclear acoustic resonance (NAR) in metals is presented for the first time. Basic equations describing the NAR-absorption and NAR-dispersion are derived from the sound induced perturbation Hamiltonian Ih(t) by applying a generalized form of the Kubo susceptibility. It is shown that in metals, where a sound wave may induce nuclear magnetic dipole and nuclear electric quadrupole transitions simultaneously, the appearance of interference terms enables one to determine not only the absolute values but also the signs of the gradient-elastic tensor components. Explicit expressions are displayed for the dipolar, quadrupolar and interference contributions to the generalized NAR susceptibility in cubic metals. As an example the derivative of the expected⁹³Nb NAR-absorption line (|m|=1) is calculated for different signs of the gradient elastic tensor componentS ₄₄.     

Molecular structure of F2POPF2: an electron diffraction study

The most important geometric parameters and associated uncertainties (2σ) determined for F₂POPF₂ are the distances (r_g) P-O = 1.631 ± 0.010 Å, P-F = 1.568 ± 0.004 Å, and angles POP = 135.2 ± 1.8°, OPF = 97.6 ± 1.2°, and FPF = 99.2 ± 2.4°. Amplitudes of vibration were also found. The large POP angle and relatively short P-O bond length are consistent with a significant degree of pπ-dπ bonding. Our structure interpretation differs from an earlier one reported by Arnold and Rankin in the relative P-O and P-F bond lengths and in the conclusion that the molecule exists in a distribution of not very rigid, probably staggered, conformers instead of one fairly rigid structure.     

Molecular dynamics simulations of the freezing of gold nanoparticles

A set of molten gold clusters, each with 1157 gold atoms, was studied by molecular dynamics simulations as the clusters underwent freezing at three different temperatures. Most of the clusters attained an icosahedral structure upon freezing, a structure found to be stable to mild annealing. Other structures observed were imperfect truncated decahedral, truncated octahedral and hexagonal close packed structures. The role of kinetics in the process of cluster solidification is discussed. Received 6 November 2000     

Effects of anharmonicity of molecular vibrations on the diffraction of electrons: Part III. Predictive models

It has long been known how to relate anharmonic vibrational distribution functions to scattered electron intensities when deriving molecular parameters from gas-phase electron diffraction patterns. What has been lacking is a convenient procedure for estimating the characteristic asymmetry parameters of radial distribution functions for polyatomic molecules, particularly in the case of non-bonded internuclear distances. In the present work alternative models of bond distribution functions are discussed briefly and a plausible model is proposed for geminal non-bonded distances. Numerical examples are worked out for the ground-states of CO₂, CS₂, H₂O, and D₂O. Variations of the asymmetry parameters of SO₂ and SF₆ with temperature are examined. It is shown that the effect of asymmetry can become quite large at elevated temperatures.     

Hydrodynamic tensor density functional theory with correct susceptibility

In a previous work the authors developed a family of orbital-free tensor equations for the density functional theory [J. Chem. Phys. 124, 024105 (2006)]. The theory is a combination of the coupled hydrodynamic moment equation hierarchy with a cumulant truncation of the one-body electron density matrix. A basic ingredient in the theory is how to truncate the series of equation of motion for the moments. In the original work the authors assumed that the cumulants vanish above a certain order (N). Here the authors show how to modify this assumption to obtain the correct susceptibilities. This is done for N=3, a level above the previous study. At the desired truncation level a few relevant terms are added, which, with the right combination of coefficients, lead to excellent agreement with the Kohn-Sham Lindhard susceptibilities for an uninteracting system. The approach is also powerful away from linear response, as demonstrated in a nonperturbative study of a jellium with a repulsive core, where excellent matching with Kohn-Sham simulations is obtained, while the Thomas-Fermi and von Weiszacker methods show significant deviations. In addition, time-dependent linear response studies at the new N=3 level demonstrate the author's previous assertion that as the order of the theory is increased new additional transverse sound modes appear mimicking the random phase approximation transverse dispersion region.     

Molecular dynamics simulations of solid state recrystallization I: Observation of grain growth in annealed iron nanoparticles

Molecular dynamics simulations of solid state recrystallization and grain growth in iron nanoparticles containing 1436 atoms were carried out. During the period of relaxation of supercooled liquid drops and during thermal annealing of the solids they froze to, changes in disorder were followed by monitoring changes in energy and the migration of grain boundaries. All 27 polycrystalline nanoparticles, which were generated with different grain boundaries, were observed to recystallize into single crystals during annealing. Larger grains consumed the smaller ones. In particular, two sets of solid particles, designated as A and B, each with two grains, were treated to generate 18 members of each set with different thermal histories. This provided small ensembles (of 18 members each) from which rates at which the larger grain engulfed the smaller one, could be determined. The rate was higher, the smaller the degree of misorientation between the grains, a result contrary to the general rule based on published experiments, but the reason was clear. Crystal A, which happened to have a somewhat lower angle of misorientation, also had a higher population of defects, as confirmed by its higher energy. Accordingly, its driving force to recrystallize was greater. Although the mechanism of recrystallization is commonly called nucleation, our results, which probe the system on an atomic scale, were not able to identify nuclei unequivocally. By contrast, our technique can and does reveal nuclei in the freezing of liquids and in transformations from one solid phase to another. An alternative rationale for a nucleation-like process in our results is proposed.     

A study of early career teachers’ practices related to language and language diversity during mathematics instruction

The role of language in mathematics teaching and learning is increasingly highlighted by standards and reform movements in the US. However, little is known about teachers’, and especially early career teachers’ (ECTs) practices and understandings related to language in mathematics instruction. This multiple case study explored the language-related understandings and practices of six ECTs in diverse elementary classrooms. Using iterative cycles of analysis, we found that all ECTs regularly attended to students’ mathematical vocabulary use and development. Yet, there was variability in ECTs’ focus on how to teach mathematical vocabulary, expectations for students’ precise use of mathematical terminology, and the use of multiple languages during instruction. These findings indicate that ECTs need more targeted support during teacher preparation and early career teaching in order to better support all students’ language development in the mathematics classroom.