Principal Axis Frame Research Articles

Polycrystalline ENDOR patterns from centers with axial EPR spectra. General formulas and simple analytic expressions for deriving geometric information from dipolar couplings

We present a formulation for rapid simulations of polycrystalline ENDOR patterns for systems whose EPR spectra exhibit axial-resolved magnetic interactions (g tensor and “central”-nucleus hyperfine couplings). The ENDOR response of interest is from a nucleus with nonaxial hyperfine couplings and unresolved splittings. The equations simplify when the hyperfine tensor of the nucleus being observed in ENDOR has axial symmetry in its principal axis frame, as would occur for an I = 1 2 nucleus ( 1H or 19F) interacting by a through-space, electron-nuclear dipolar interaction. In such cases the field dependence of the ENDOR frequencies can be described by simple analytic functions, well suited to least-squares determination of metrical parameters that define the nuclear position. Sample calculations are given for a proton interacting with a nitroxide radical.

Symmetry properties of the lattice dynamics of nonrigid molecular crystals

The symmetry properties of the lattice dynamics of nonrigid molecular crystals are discussed on mass-weighted translational coordinates and inertia moment-weighted rotational coordinates as well as molecular internal normal coordinates in molecular principal axes frame. An equivalent reducible representation of the wavevector group and the classification of molecular crystal normal modes are provided. A detailed illustrative example is presented.

Anisotropy and anharmonicity of atomic fluctuations in proteins: analysis of a molecular dynamics simulation.

Positional probability density functions (pdf) for the atomic fluctuations are determined from a molecular dynamics simulation for hen egg-white lysozyme. Most atoms are found to have motions that are highly anisotropic but only slightly anharmonic. The largest deviations from harmonic motion are in the direction of the largest rms fluctuations in the local principal axis frame. Backbone atoms tend to be more nearly harmonic than sidechain atoms. The atoms with the largest anharmonicities tend to have pdfs with multiple peaks, each of which is close to harmonic. Several model pdfs are evaluated on the basis of how well they fit probability densities from the dynamics simulations when parameterized in terms of the moments of the distribution. Gram-Charlier and Edgeworth perturbation expansions, which have been successful in describing the motions of small molecules in crystals, are shown to be inadequate for the distributions found in the dynamics of proteins. Multipeaked distribution functions are found to be more appropriate.

Rotating and principal-axis frames in multipole NMR

The use of a spherical tensor operator basis to describe the nuclear magnetic state makes it ideal for considering rotations and transformations of the density operator to different frames. In this paper it is shown how all polarizations can be transformed to different frames which display the greatest reduction of coupling. These correspond to various rotating and/or principal-axis frames.

Proton shift tensors in hydrogen molybdenum bronze

Proton N.M.R. powder pattern lineshapes for a series of hydrogen molybdenum bronzes prepared using a wet chemistry method are presented and compared with published results for bronzes prepared using the hydrogen spillover method. The results demonstrate that a bronze is formed which exhibits the same unusual proton N.M.R. spectra as found for H1.6MoO3 prepared by the hydrogen spillover method. Single crystals prepared by the wet chemistry method exhibit an obvious visual anisotropy. A single crystal 1H N.M.R. rotation study of H1.6MoO3 is reported. Although the detailed structure of the bronze is not known, the principal axis frame of the tensor can be related simply to the crystal frame of the bronze at room temperature. Unusual temperature dependence of the tensor components is reported and discussed in terms of electronic properties of the solid.

Nuclear wobbling motion

A microscopic model of the nuclear wobbling motion at high spin in the small-oscillation approximation is derived within the framework of time-dependent self-consistent field theory. The physically important effects arise from a term −Ω(t) · J , similar to the cranking model, but here, Ω(t) is a time-dependent angular velocity. The formulation makes it possible to relate the parameters of the classical asymmetric rotor model, in particular, the three moments of inertia, to microscopic quantities. It also provides a new technique for separating the “spurious” degree of freedom associated with this branch of excitations. In this formulation, the components of angular momentum along both the laboratory and principal-axis frames can readily be derived. Some mathematical peculiarities arising from the description relative to a nonuniformly rotating frame are discussed in detail.

Homonuclear dipolar-modulated chemical shift spectra in polycrystalline solids: The proton chemical shift tensor in CCl3COOH

A single resonance, multiple pulse nuclear magnetic resonance experiment is proposed and demonstrated which produces a homonuclear dipolar modulation of a chemical shift powder pattern. The correlation generated between the homonuclear dipolar Hamiltonian and the chemical shift Hamiltonian can be used to determine the relative orientation of the chemical shift principal axis frame to the molecular frame. The experimental scheme is applied to determine the orientation of the proton–proton vector in a CCl3COOH dimer relative to the chemical shift tensor of the hydrogen-bonded proton.

On the selection of sign for the observable dipole moment derivatives

The main criteria used in relation to the task of selecting a unique sign set for the experimental dipole moment derivatives, such as isotopic invariance of atomic polar tensors and effective charges ξα, are discussed. It is shown that the selection of a principal axis frame is particularly relevant to this problem. Conclusions are illustrated with a complete discussion of sign sets for CH3X and CD3X (X=F, Cl, Br, I) molecules.

On the Attitude Dynamics of Spinning Deformable Systems

A general method for attitude dynamics investigation of spinning deformable systems is presented. The stability of given equilibrium configurations is determined by use of the Liapunov technique. Necessary and sufficient conditions are obtained in most cases. The choice of the body reference frame is considered, and the motion of various frames, such as principal axes frame, mean frame, and Tisserand's frame, is investigated.