Anisotropic harmonic motion

A trivariate normal distribution describes the probability distribution for anisotropic harmonic motion in three-dimensional space. In tensor notation (see appendix A for the notation, and appendix B for the treatment of symmetry and symmetry restrictions of tensor elements), with j and k (= 1, 3) indicating the axial directions,... 

The terms involving the subscript j represents the contribution of atom j to the computed structure factor, where nj is the occupancy, fj is the atomic scattering factor, and Ris the coordinate of atom i. In Eq. (13-4) the thermal effects are treated as anisotropic harmonic vibrational motion and U =< U U. > is the mean-square atomic displacement tensor when the thermal motion is treated as isotropic, Eq. (13-4) reduces to ... 

Of interest also are the results concerning deviations of the atomic fluctuations from simple isotropic and harmonic motion. As discussed in Chapt. XI, most X-ray refinements of proteins assume (out of necessity, because of the limited data set) that the motions are isotropic and harmonic. Simulations have shown that the fluctuations of protein atoms are highly anisotropic and for some atoms, strongly anharmonic. The anisotropy and anharmonicity of the atomic distribution functions in molecular dynamics simulations of proteins have been studied in considerable detail.193"197 To illustrate these aspects of the motions, we present some results for lysozyme196 and myoglobin.197 If Ux, Uy, and Uz are the fluctuations from the mean positions along the principal X, Y, and Z axes for the motion of a given atom and the mean-square fluctuations are... 

The great potential of the X-ray data for obtaining motional information has recently led to a molecular dynamics test197 of the standard refinement techniques that assume isotropic and harmonic motion. Since simulations have shown that the atomic fluctuations are highly anisotropic and, in some cases, anharmonic (see Chapt. VI.A.1), it is important to determine the errors introduced in the refinement process by their neglect. A direct experimental estimate of the errors resulting from the assumption of isotropic, harmonic temperature factors is difficult because sufficient data are not yet available for protein crystals. Moreover, any data set includes other errors that would obscure the analysis, and the specific correlation of temperature factors and motion is complicated by the need to account for static disorder in the crystal. As an alternative to an experimental analysis of the errors in the refinement of proteins, a purely theoretical approach has been used.197 The basic idea is to generate X-ray data from a molecular dynamics simulation... 

Fig. 14.13 Anisotropic harmonic lithium vibration shown as the green thermal ellipsoids with 95% probability refined by Rietveld analysis for room-temperature neutron diffraction data measured for LiFeP04. Expected curved one-dimensional continuous chains of lithium motion were drawn by the dashed lines to show how the motions of Li atoms evolve from vibrations to diffusion...

The model described above can be generalised to treat anisotropic diffusive motions which might be relevant in particular systems. For an anisotropic overdamped harmonic oscillator, with independent motions... 

To relate Dff to the anisotropic motion of a molecule in a liquid crystalline solvent, we employ the function P(d, < ), defined as the probability per unit solid angle of a molecular orientation specified by the angles 6 and <3>, the polar coordinates of the applied magnetic field direction relative to a molecule-fixed Cartesian coordinate system. We expand P(0, ) in real spherical harmonics ... 

It is well known from small molecule crystallography that the effects of thermal motion must be included in the interpretation of the X-ray data to obtain accurate structural results. Detailed models have been introduced to take account of anisotropic and anharmonic motions of the atoms and these models have been applied to high-resolution measurements for small molecules.413 In protein crystallography, the limited data available relative to the large number of parameters that have to be determined have made it necessary in most cases to assume that the atomic motions are isotropic and harmonic. With this assumption the structure factor F(Q), which is related to the measured intensity by 7(Q) = F(Q) 2, is given by... 

In the case of water, the situation is complicated because of the anisotropic nature of the potential. Thus, we have effective harmonic potential for translation, rotation, and librational motions. Each is characterized by a force constant and contributes to the partition function, free energy, and entropy. Furthermore, a water molecule can be categorized by the number of HBs it forms. Since these quantities can be considered as thermodynamic, they make a contribution as the entropy of mixing, also known as the cratic contribution. 

Lattice dynamics studies of the disordered j -phase are more scarce because, obviously, the standard harmonic method and the SCP method cannot be applied to this phase (although in some studies the harmonic method has still been used for the translational phonons, while neglecting the anisotropy of the potential.) Most calculations on have been made by classical Monte Carlo or Molecular Dynamics methods, using semiempirical atom-atom or quadrupole-quadrupole potentials. In our group [50, 52] we have investigated the motions in and the a — jS order/ disorder phase transition by means of the MF, RPA and TDH methods, using the same spherically expanded anisotropic ab initio potential which yields accurate properties for a-N2. 

In a crystal, displacements of atomic nuclei from equilibrium occur under the joint influence of the intramolecular and intermolecular force fields. X-ray structure analysis encodes this thermal motion information in the so-called anisotropic atomic displacement parameters (ADPs), a refinement of the simple isotropic Debye-Waller treatment (equation 5.33), whereby the isotropic parameter B is substituted by six parameters that describe a libration ellipsoid for each atom. When these ellipsoids are plotted [5], a nice representation of atomic and molecular motion is obtained at a glance (Fig. 11.3), and a collective examination sometimes suggests the characteristics of rigid-body molecular motion in the crystal, like rotation in the molecular plane for flat molecules. Lattice vibrations can be simulated by the static simulation methods of harmonic lattice dynamics described in Section 6.3, and, from them, ADPs can also be estimated [6]. 

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