Atom motions anharmonicity

In low-spin transition metal complexes, the preferential occupancy of the d orbitals in the crystal field tends to create excess density in the voids between the bonds, which means that anharmonicity tends to reinforce the electron density asphericity. We will discuss, in the following sections, to what extent the two effects can be separated by combined use of aspherical atom and anharmonic thermal motion formalisms. 

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 effects of anisotropy and anharmonicity in the atomic motions on the refinement of X-ray data for protein crystals are described in Chapt. XI. 

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... 

A normal mode calculation is based upon the assumption that the energy surface is quadratic in the vicinity of the energy minimum (the harmonic approximation). Deviations from the harmonic model can require corrections to calculated thermodynamic properties. One way to estimate anharmonic corrections is to calculate a force constant matrix using the atomic motions obtained from a molecular d)namics simulation such simulations are not restricted to movements on a harmonic energy surface. The eigenvalues and eigenvectors are then calculated for this quasi-harmonic force-constant matrix in the normal way, giving a model which implicitly incorporates the anharmonic effects. 

The body of experimental data for clusters assembled from a molecule of HF or HCl with one or more Ar atoms presents a challenge and a testing ground for theoretical predictions. The vibrations of these weakly bound floppy clusters are difficult to determine by analytical methods because of large-amplitude motions, anharmonic motions, multiple minima, and high number of dimensions. Two sets of QMC calculations for the Ar HF n = 1-4) system have been reported recently one by Lewerenz ° and the other by Niyaz et al. Both are successful in predicting energies, structures, and hydrofluoric acid frequency shifts for these species. 

Figure 4 Temperature dependence of selected atomic displacement parameters for the Mg, Si, Al and O atoms in pyrope garnet. Analysis of the displacement parameters obtained from accurate single-crystal diffraction data allow evaluation of the zero-point energy contribution, separation of static and dynamic effects, and detection of anharmonic vibrational contributions to the atomic motion.

Here the anharmonic nature appears only in the equilibrium distances a changed by thermal expansion, see (5.52), and the altered vibrational frequencies w corresponding to the renormalized force constant f, see (5.55,57). We therefore approximate the atomic motion by a system of uncoupled normal vibrations but with different equilibrium positions and vibrational frequencies. This is, of course, an approximation since the anharmonic terms of the potential energy really lead to interactions of the normal vibrations. From (5.68,69), we obtain... 

Among the main theoretical methods of investigation of the dynamic properties of macromolecules are molecular dynamics (MD) simulations and harmonic analysis. MD simulation is a technique in which the classical equation of motion for all atoms of a molecule is integrated over a finite period of time. Harmonic analysis is a direct way of analyzing vibrational motions. Harmonicity of the potential function is a basic assumption in the normal mode approximation used in harmonic analysis. This is known to be inadequate in the case of biological macromolecules, such as proteins, because anharmonic effects, which MD has shown to be important in protein motion, are neglected [1, 2, 3]. 

Vibrational energy, which is associated with the alternate extension and compression of die chemical bonds. For small displacements from the low-temperature equilibrium distance, the vibrational properties are those of simple harmonic motion, but at higher levels of vibrational energy, an anharmonic effect appears which plays an important role in the way in which atoms separate from tire molecule. The vibrational energy of a molecule is described in tire quantum theory by the equation... 

A dynamic transition in the internal motions of proteins is seen with increasing temperamre [22]. The basic elements of this transition are reproduced by MD simulation [23]. As the temperature is increased, a transition from harmonic to anharmonic motion is seen, evidenced by a rapid increase in the atomic mean-square displacements. Comparison of simulation with quasielastic neutron scattering experiment has led to an interpretation of the dynamics involved in terms of rigid-body motions of the side chain atoms, in a way analogous to that shown above for the X-ray diffuse scattering [24]. 

According to Bartell (1961a), the relative motion of the interacting non-bonded atoms is described by means of a harmonic oscillator when the two atoms are bonded to the same atom, and by means of two superimposed harmonic oscillators when the atoms are linked to each other via more than one intervening atom. It is the second case which is of interest in connection with the biphenyl inversion transition state. The non-bonded interaction will of course introduce anharmonicity, but since a first-order perturbation calculation of the energy only implies an... 

With data averaged in point group m, the first refinements were carried out to estimate the atomic coordinates and anisotropic thermal motion parameters IP s. We have started with the atomic coordinates and equivalent isotropic thermal parameters of Joswig et al. [14] determined by neutron diffraction at room temperature. The high order X-ray data (0.9 < s < 1.28A-1) were used in this case in order not to alter these parameters by the valence electron density contributing to low order structure factors. Hydrogen atoms of the water molecules were refined isotropically with all data and the distance O-H were kept fixed at 0.95 A until the end of the multipolar refinement. The inspection of the residual Fourier maps has revealed anharmonic thermal motion features around the Ca2+ cation. Therefore, the coefficients up to order 6 of the Gram-Charlier expansion [15] were refined for the calcium cation in the scolecite. 

The model fundamental to all analyses of vibrational motion requires that the atoms in the system oscillate with small amplitude about some defined set of equilibrium positions. The Hamiltonian describing this motion is customarily taken to be quadratic in the atomic displacements, hence in principle a set of normal modes can be found in terms of these normal modes both the kinetic energy and the potential energy of the system are diagonal. The interaction of the system with electromagnetic radiation, i.e. excitation of specific normal modes of vibration, is then governed by selection rules which depend on features of the microscopic symmetry. It is well known that this model can be worked out in detail for small molecules and for crystalline solids. In some very favorable simple cases the effects of anharmonicity can be accounted for, provided they are not too large. 

Oscillations of atoms and their groups, typically occurring on a time scale of 10 12-10 14 s, are much faster motions. However, there is evidently no sharp distinction between these motions and other slower motions. When damping and anharmonicity arise, the oscillations become diffusive and have the properties of transitions between microstates. It is natural to suppose (as... 

One result of studying nonlinear optical phenomena is, for instance, the determination of this susceptibility tensor, which supplies information about the anharmonicity of the potential between atoms in a crystal lattice. A simple electrodynamic model which relates the anharmonic motion of the bond charge to the higher-order nonlinear susceptibilities has been proposed by Levine The application of his theory to calculations of the nonlinearities in a-quarz yields excellent agreement with experimental data. 

Bis(pyridine)(meso-tetraphenylporphinato)iron(II), discussed in section 10.5.2, was reanalyzed to evaluate the importance of anharmonic motion in nitrogen-temperature transition metal studies. A number of different refinements are summarized in Table 10.13. It is striking that treating the Fe atom as spherical... 

As first shown by Dawson (1967), Eq. (11.3) can be generalized by inclusion of anharmonicity of the thermal motion, which becomes pronounced at higher temperatures. We express the anharmonic temperature factor of the diamond-type structure [Chapter 2, Eq. (2.45)] as 71(H) = TC(H) -f iX(H), in analogy with the description of the atomic scattering factors. Incorporation of the temperature... 

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