Density unrelaxed

Figure 4.6 Left STM image of a stoichiometric 1 x 1 Ti02(l 1 0) surface, 14A x 14 A. Sample bias + 1.6 V, tunneling current 0.38 nA. The inset shows a ball-and-stick model of the unrelaxed 1 x 1 Ti02(l 1 0) surface. Rows of bridging oxygen atoms are labeled A and rows of fivefold coordinated titaniums B . Right contour plots of [0 1 l]-averaged charge densities associated with electron states within...

Fig. 5. Contour plot of the adiabatic potential-energy surface of an H atom in the (110) plane for the neutral H—B pair from a local-density pseudopotential calculation. The boron atom is at the center. For every hydrogen position, the B and Si atoms are allowed to relax, but only unrelaxed positions are indicated in the figure (Reprinted with permission from the American Physical Society, Denteneer, P.J.H., Van de Walle, C.G., and Pantelides, S.T. (1989). Phys. Rev. B 39, 10809.)...

Fig. 7. (a) Contour plot of the charge density in the (110) plane through the atoms for neutral H at the bond center. The Si atoms in their relaxed positions are indicated with black dots and connected with solid lines. Dashed lines connect the unrelaxed atomic positions. The contour interval is 50 units are electrons per unit cell (for a supercell containing 1 H and 32 Si atoms), (b) Contour plot of the difference between spin-up and spin-down densities in the (110) plane through the atoms for neutral H at the bond center. The contour interval is 2.5 electrons/(unit cell). (Reprinted with permission from the American Physical Society, Van de Walle et al., 1989.)... 

The electron density distribution of a known surface structure can be calculated from first-principles. Thus, the He diffraction data can be compared with theoretical results, in particular, to verify different structural models. Hamann (1981) performed first-principles calculations of the charge-density distributions of the GaAs(llO) surface, for both relaxed and unrelaxed configurations. The He diffraction data are in excellent agreement with the calculated charge-density distributions of the relaxed GaAs(llO) surface, and are clearly distinguished from the unrelaxed ones (Hamann, 1981). 

In Equation 1, R and V refer to the relaxed (low frequency) and unrelaxed (high frequency) dielectric constants, and AH is the measured activation energy for the y process. The latter was nearly independent of blend composition an average value of 8.7 kcal/mole was used. The integral in Equation 1 was found to be approximately independent of frequency in the range studied. The loss peak in absolute terms is rather weak, and values of eR — V were of the order of 10"2 and less. From these values, it was also possible to calculate the apparent dipolar density, Np2, using the Onsager relation (9) ... 

If the hierarchy of low-temperature/high-frequency (unrelaxed) G values corresponds more or less to the hierarchy of cohesive energy densities, it is completely modified at ambient temperature/low frequency - i.e., in the usual mechanical testing conditions - owing to the importance of the modulus gap at Tp (Fig. 11.5). 

As in the case of G, the unrelaxed tensile modulus will be found more or less proportional to the cohesive energy density, whereas the relaxed modulus will depend sharply on the activity of local motions. 

All the above observations seem to justify Porter s approach (Eq. 11.11)), according to which the Poisson s ratio should depend only on the cumulative loss tangent. It was found that the unrelaxed Poisson s ratio determined from ultrasound (5 MHz) propagation rate, for 12 of amine-crosslinked epoxy stoichiometric networks, displays only small variations (Av < 0.01), in spite of the relatively large variations of the cohesive energy density (0.59 < CED <0.66 GPa) and the crosslink density (2.0 5.9 mol kg 1)-... 

Unrelaxed and relaxed moduli for each transition. Let us recall that, generally, the ratio E (unrelaxed)/ E (relaxed) is higher than 10 for the a transition (it is a decreasing function of the crosslink density), whereas it is generally lower than 2 for secondary transitions where it does not depend directly on crosslink density but rather on the mechanical activity of the corresponding molecular motions. 

Since the unrelaxed bulk modulus, Ku e.g., determined by ultrasonic propagation velocity measurements, is a good measure of the cohesive energy density, CED (Ku 11 CED Chapter 10), and CED gives a good indication of the overall material s polarity, one can expect a correlation between Ku and W. This is shown in Fig. 14.3 for the amine-epoxy and styrene-vinyl ester networks. The following relationship is found ... 

A further issue arises in the Cl solvation models, because Cl wavefunction is not completely variational (the orbital variational parameter have a fixed value during the Cl coefficient optimization). In contrast with completely variational methods (HF/MFSCF), the Cl approach presents two nonequivalent ways of evaluating the value of a first-order observable, such as the electronic density of the nonlinear term of the effective Hamiltonian (Equation 1.107). The first approach (the so called unrelaxed density method) evaluates the electronic density as an expectation value using the Cl wavefunction coefficients. In contrast, the second approach, the so-called relaxed density method, evaluates the electronic density as a derivative of the free-energy functional [18], As a consequence, there should be two nonequivalent approaches to the calculation of the solvent reaction field induced by the molecular solute. The unrelaxed density approach is by far the simplest to implement and all the Cl solvation models described above have been based on this method. 

All the alternative variants of the MPn may be implemented using a relaxed density matrix or a unrelaxed density matrix, in analogy with the Cl solvation methods. In the first case the correlated electronic density is computed as a first derivatives of the free energy, while in the second case only the MPn perturbative wavefunction amplitudes are necessary. 

Other MP2 based solvent methods consist of the Onsager MP2-SCRF [19], within a relaxed density scheme analogous to the PTDE scheme, and a multipole MP2-SCRF model [28], based on a iterative unrelaxed approach. The analytical gradients and Hessian of the free energy at MP2-PTE level, has been developed within the PCM framework [29],... 

Therefore, experiments are performed on immobilized liquids , or in other words on amorphous water (also called vitreous water or glassy water). Currently, three structurally distinct amorphous states of water are known low- (LDA) , high- (HDA) and very high- (VHDA) density amorphous ice We emphasize that HDA is not a well defined state but rather comprises a number of substates. It has been suggested to use the nomenclature uHDA ( unrelaxed HDA ) ", eHDA ( expanded HDA ) " and/or rHDA ( relaxed HDA ) to account for this. Even though no signs of micro-crystallinity have been found in neutron or X-ray diffraction studies, it is unclear whether... 

The number of unrelaxed modes per chain at time t is equal to the mode index p. Each unrelaxed mode contributes energy of order kT to the stress relaxation modulus. The stress relaxation modulus at time t is proportional to the thermal energy kT and the number density of sections with Njp monomers, 4>/(b N/p) ... 

In a crystal, perturbations can be classified as internal and external. The internal perturbations are disturbances from an equilibrium condition, taken as an ideal uniform distribution of impurities or defects which do not modify the crystal lattice and the average electronic density. Mechanical perturbations can be microscopic, like those introduced by impurities or defects producing large local volume changes, which reflect on crystal lattice spacings when their concentration is large, or macroscopic due to residual or accidental stresses. Permanent perturbations can also be produced by unrelaxed stresses... 

Figure Bl.26.19. The variation of the electron density (A) from an unrelaxed surface and (B) showing the smoothing of the electron density to lower the kinetic density.

Melt-textured materials (Y-123, Nd-123) with U admixtures represent a highly promising system for high-Jc applications due to the possibihty of forming randomly oriented fission tracks (cf. sect. 6.3). Results on the critical current densities (taken by the flux-profile technique, i.e. in the nearly unrelaxed state) are shown in fig. 40 (Weinstein et al. 1998, Eisterer et al. 1998). 7c following thermal neutron irradiation to the optimal... 

After initial relaxation for a period t > cross-linking traps both unrelaxed and relaxed entanglements to produce the following densities respectively ... 

Ferry and co-workers [7], on the basis of the molecular theory of viscoelasticity, proposed that superposition should incorporate a small vertical shift factor Fopo/TP, where p is the density at the experimental temperature T and po relates to the reference temperature Tq. Further corrections have been suggested by McCrum and Morris [8] to deal with the changes in unrelaxed and relaxed compliances with temperature. 

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