Plane motion

Here, we discuss the motion of a system of three identical nuclei in the vicinity of the D3/, configuration. The conventional coordinates for the in-plane motion are employed, as shown in Figure 5. The noraial coordinates Qx, Qy, Qz), the plane polar coordinates (p,(p,z), and the Cartesian displacement coordinates (xi,yhZi of the three nuclei (t = 1,2,3) are related by [20,94]... 

The widths of the narrow Lorentzians representing slow motions in the plane and perpendicular to the plane of the bilayer are compared in Ligures 11a and 11b, respectively. Lor the in-plane motion, the MD values for Q = 0.5 A agree well with the experimental results, but the increase with Q is significantly overestimated in the simulation compared to the experimental values. This suggests that the slower component of the in-plane motion in the simulation is too fast at short distances. On the other hand, the MD line widths for the slower component of the out-of-plane motion agree well with the experimental results at 30% hydration. As in the case of the LISL, the simulation predicts a slight anisotropy not seen in the experimental data. 

Analysis of neutron data in terms of models that include lipid center-of-mass diffusion in a cylinder has led to estimates of the amplitudes of the lateral and out-of-plane motion and their corresponding diffusion constants. It is important to keep in mind that these diffusion constants are not derived from a Brownian dynamics model and are therefore not comparable to diffusion constants computed from simulations via the Einstein relation. Our comparison in the previous section of the Lorentzian line widths from simulation and neutron data has provided a direct, model-independent assessment of the integrity of the time scales of the dynamic processes predicted by the simulation. We estimate the amplimdes within the cylindrical diffusion model, i.e., the length (twice the out-of-plane amplitude) L and the radius (in-plane amplitude) R of the cylinder, respectively, as follows ... 

The similarity in the behaviour of coupling constants as a function of e in both radicals allows to discuss vibrational averaging effects simply in terms of the potential governing the out-of-plane motion. 

Fig. 5.19 Low-frequency Fe modes of Fe(TPP)(NO) predicted on the basis of B3LYP calculations. The modes mainly involve porphyrin core translation, Fe-NO torsion, Fe-N-O bending, and Fe out-of-plane motion coupled to doming of the porphyrin core. Arrows representing mass-weighted atomic displacements are 100(my/mFe) longer than the zero-point vibrational amplitude of atom j. Color scheme as in Fig. 5.15 (taken from [101])...

FIG. 2 Molecular orientation angles at liquid interfaces for rodlike molecules. The out-of-plane motion is a rotation away from the OZ axis, whereas the in-plane motion is performed with the OX, OY) plane. 

Simple shear (also known as planar Couette flow) is achieved when fluid is contained between two plane parallel plates in relative in-plane motion. If the velocity direction is taken to be x, one has x = y, all other xa 3 zero and... 

The ab initio molecular dynamics study by Hudock et al. discussed above for uracil included thymine as well [126], Similarly to uracil, it was found that the first ultrafast component of the photoelectron spectra corresponds to relaxation on the S2 minimum. Subsequently a barrier exists on the S2 surface leading to the conical intersection between S2 and Si. The barrier involves out-of-plane motion of the methyl group attached to C5 in thymine or out-of-plane motion of H5 in uracil. Because of the difference of masses between these two molecules, kinematic factors will lead to a slower rate (longer lifetime) in thymine compared to uracil. Experimentally there are three components for the lifetimes of these systems, a subpicosecond, a picosecond and a nanosecond component. The picosecond component, which is suggested to correspond to the nonadiabatic S2/S1 transition, is 2.4 ps in uracil and 6.4 ps in thymine. This difference in the lifetimes could be explained by the barrier described above. 

Problems arise in two situations firstly, if the molecule is linear, no complete set of 3N — 6 internal coordinates is possible. For this case, a method for constructing PES in terms of Cartesian coordinates could be used.56 Secondly, if the molecule is planar, atom-atom distances (or their reciprocals) cannot provide a complete set of internal coordinates, since they cannot describe out-of-plane motion. However, we have found the coordinates Zn so useful that we retain these coordinates and avoid planar geometries (except for three atoms, when only linear geometries are taboo). That is,... 

Fig. 28. Schematic of potential energy surfaces of the vinoxy radical system. All energies are in eV, include zero-point energy, and are relative to CH2CHO (X2A//). Calculated energies are compared with experimentally-determined values in parentheses. Transition states 1—5 are labelled, along with the rate constant definitions from RRKM calculations. The solid potential curves to the left of vinoxy retain Cs symmetry. The avoided crossing (dotted lines) which forms TS5 arises when Cs symmetry is broken by out-of-plane motion. (From Osborn et al.67)...

DISLOCATION. In crystallography, a type of lattice imperfection whose existence in metals is postulated in order to account for the phenomenon uf crystal growth and of slip, particularly for the low value of shear stress required lo initiate slip. One section of the crystal adjacent to the slip plane is assumed to contain one mure atomic plane that the section on the opposite side of the slip plane. Motion of the dislocation results in displacement of one of the sections with respect to another. 

In order to attain maximum clarity, we will carry out below a complete proof of the anti-dynamo theorem for the case of plane motion with vx = 0 in Cartesian coordinates (cf. [6]). Then we will discuss the general case more briefly. 

As shown in Figure 2.2, the isotropic g tensor shift (Agiso) is almost linearly dependent on the NO bond length, whereas it does not display any regular trend with respect to out-of-plane motion. 

This intermolecular potential for ADN ionic crystal has further been developed to describe the lowest phase of ammonium nitrate (phase V) [150]. The intermolecular potential contains similar potential terms as for the ADN crystal. This potential was extended to include intramolecular potential terms for bond stretches, bond bending and torsional motions. The corresponding set of force constants used in the intramolecular part of the potential was parameterized based on the ab initio calculated vibrational frequencies of the isolated ammonium and nitrate ions. The temperature dependence of the structural parameters indicate that experimental unit cell dimensions can be well reproduced, with little translational and rotational disorder of the ions in the crystal over the temperature range 4.2-250 K. Moreover, the anisotropic expansion of the lattice dimensions, predominantly along a and b axes were also found in agreement with experimental data. These were interpreted as being due to the out-of-plane motions of the nitrate ions which are positions perpendicular on both these axes. 

Langer and Doltsinis [41, 42] find that the nonadiabatic transition parameter (10-10) is correlated to variations in the C(5)C(6) bond length as well as to out-of-plane motions. The importance of this degree of freedom for radiationless decay has been pointed out previously by Zgierski et al. [103],... 

The secondary, slower process with an exponential coefficient of 293 fs is concerned more with nonadiabatic transitions out of the CT state, where out-of-plane structural fluctuations play a more significant role. Indeed, this time-scale seems appropriate, as the time for a phase cycle of these out-of-plane motions (0 and ) is of the order of 100 to 200 fs, and P10 is observed to have the largest magnitude when both time-derivatives (d0/dt and d

[Pg.295]

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