Valence deformation

The exact Eq. (4.2.17) takes into account the effect of the reservoir (the condensed phase) on the spectral line shape through the parameter 77. Consideration of a concrete microscopic model of the valence-deformation vibrations makes it possible to estimate the basic parameters y and 77 of the theory and to introduce the exchange mode anharmonicity caused by a reorientation barrier of the deformation vibrations thereby, one can fully take advantage of the GF representation in the form (4.2.11) which allows summation over a finite number of states. 

Valence-deformation vibrations of a molecular subsystem in condensed phase... 

The usual practice in modeling the valence deformation of H-atoms is to terminate the expansion at the dipolar level (/max = U with a bond directed dipole) and to fix the radial exponents. In addition, H-atoms bonding to the same type of atoms are often constrained to be identical, regardless of their involvement in non-bonded interactions. Model studies on radial functions, obtained by projection of theoretical densities onto nucleus-centered spherical harmonics, show that (a) second neighbors have significant effects on H-atoms,... 

Correlations have been found between certain absorption patterns in the infrared and the concentrations of aromatic and paraffinic carbons given by the ndA/method (see article 3.1.3.). The absorptions at 1600 cm due to vibrations of valence electrons in carbon-carbon bonds in aromatic rings and at 720 cm (see the spectrum in Figure 3.8) due to paraffinic chain deformations are directly related to the aromatic and paraffinic carbon concentrations, respectively. )... 

Dislocation theory as a portion of the subject of solid-state physics is somewhat beyond the scope of this book, but it is desirable to examine the subject briefly in terms of its implications in surface chemistry. Perhaps the most elementary type of defect is that of an extra or interstitial atom—Frenkel defect [110]—or a missing atom or vacancy—Schottky defect [111]. Such point defects play an important role in the treatment of diffusion and electrical conductivities in solids and the solubility of a salt in the host lattice of another or different valence type [112]. Point defects have a thermodynamic basis for their existence in terms of the energy and entropy of their formation, the situation is similar to the formation of isolated holes and erratic atoms on a surface. Dislocations, on the other hand, may be viewed as an organized concentration of point defects they are lattice defects and play an important role in the mechanism of the plastic deformation of solids. Lattice defects or dislocations are not thermodynamic in the sense of the point defects their formation is intimately connected with the mechanism of nucleation and crystal growth (see Section IX-4), and they constitute an important source of surface imperfection. 

A is a parameter that can be varied to give the correct amount of ionic character. Another way to view the valence bond picture is that the incorporation of ionic character corrects the overemphasis that the valence bond treatment places on electron correlation. The molecular orbital wavefimction underestimates electron correlation and requires methods such as configuration interaction to correct for it. Although the presence of ionic structures in species such as H2 appears coimterintuitive to many chemists, such species are widely used to explain certain other phenomena such as the ortho/para or meta directing properties of substituted benzene compounds imder electrophilic attack. Moverover, it has been shown that the ionic structures correspond to the deformation of the atomic orbitals when daey are involved in chemical bonds. 

As the most notable contribution of ab initio studies, it was revealed that the different modes of molecular deformation (i.e. bond stretching, valence angle bending and internal rotation) are excited simultaneously and not sequentially at different levels of stress. Intuitive arguments, implied by molecular mechanics and other semi-empirical procedures, lead to the erroneous assumption that the relative extent of deformation under stress of covalent bonds, valence angles and internal rotation angles (Ar A0 AO) should be inversely proportional to the relative stiffness of the deformation modes which, for a typical polyolefin, are 100 10 1 [15]. A completly different picture emerged from the Hartree-Fock calculations where the determined values of Ar A0 AO actually vary in the ratio of 1 2.4 9 [91]. 

According to the transition state theory, the pre-exponential factor A is related to the frequency at which the reactants arrange into an adequate configuration for reaction to occur. For an homolytic bond scission, A is the vibrational frequency of the reacting bond along the reaction coordinates, which is of the order of 1013 to 1014 s 1. In reaction theory, this frequency is diffusion dependent, and therefore, should be inversely proportional to the medium viscosity. Also, since the applied stress deforms the valence geometry and changes the force constants, it is expected... 

In some respects arenediazonium ions show analogies to acetylene. Acetylene has two deformation vibrations, v4 at 613.5 cm-1 and v6 at 729.6 cm-1, as shown in Figure 7-1 (Feldmann et al., 1956). The fact that the symmetrical vibration v4 has a lower frequency than v6 can be understood from BartelPs valence-shell electron-pair repulsion (VSEPR) model (1968) on the basis of a <pseudo-Jahn-Teller> effect. 

Extra radial flexibility has been proved necessary in order to model the valence charge density of metal atoms, in minerals [6,11], and coordination complexes [5], and similar evidence of the inability of single-exponential deformation functions to account for all the information present in the observations have also been found in studies of organic [12, 13] and inorganic [14] molecular crystals. 

Figure 2. L-alanine. Dynamic deformation density in the COO plane, (a) Model dynamic deformation density A Modei. (b) MaxEnt dynamic deformation density (Agj, (x)) map obtained with a non-uniform prior of spherical-valence shells. Map size 6.0A x 6.0A Contour levels from -1.0 to 1.0 eA 3, step 0.075 e A-f...

When the reconstruction of the density is carried out by modulation of a prior prejudice of spherical atoms, only the deformation features have to be accommodated this can be accomplished relatively easily, and the Lagrange multipliers are usually below 0.01 in modulus, or even smaller for valence-only runs. No aliasing problems occur in the synthesis of (x). 

Figure 6(b) shows the difference between the MaxEnt valence density and the reference density, in the COO- plane. The error peaks in the bonding and lone-pair regions, where the deformation features are systematically lower than the reference map (negative contours). The deviation from the reference is largest in the region around the Cl atom valence shell, and reaches -0.406 e A 3. 

The (< ME(x)) map is of course less noisy than any of the individual noisy maps the deviation from the reference model map shows the same systematic underestimation of the deformation features as observed in density A, with a maximum negative error of -0.362 e A-3, again in the region of the valence shell of the Cl atom. 

Sections of the density from one of these fits, which we will refer to as calculation B, are shown in Figure 7 the MaxEnt deformation density in the COO- plane is shown in Figure 7(a) Figure 7(b) is the difference between the MaxEnt valence density and the reference density in the same plane. The lower noise content of the data is clearly visible, when the map is compared with the one for calculation A in particular, the lone pairs on the oxygen atoms are better defined. The rms deviation from the reference is as low as 0.023 e A 3. 

Figures, l-Alanine.Fits to noisy data Calculations A (experimental noise) and B (10% experimental noise). MaxEnt, deformation and error density profiles along the Cl-01 bond. Solid line Model valence density. Dashed line MaxEnt density A. Dot-dashed line MaxEnt density B. Dotted line valence-shells non-uniform prior.

According to the aspherical-atom formalism proposed by Stewart [12], the one-electron density function is represented by an expansion in terms of rigid pseudoatoms, each formed by a core-invariant part and a deformable valence part. Spherical surface harmonics (multipoles) are employed to describe the directional properties of the deformable part. Our model consisted of two monopole (three for the sulfur atom), three dipole, five quadrupole, and seven octopole functions for each non-H atom. The generalised scattering factors (GSF) for the monopoles of these species were computed from the Hartree-Fockatomic functions tabulated by Clementi [14]. 

The interactions between bulky phenyl substituents in the polymer chain can give more steric hindrance than the deformation of the valency angles in the four membered ring. Similar interactions prevent the polymerization of 1,1-diphenylethylene and 2,2-diphenyloxirane (16). Thus, octaphenylcyclotetrasilane can be thermodynamically more stable than linear perphenylpolysilane and no initiator exists capable of converting this cycle to the linear polymer. 

Systems involving more mass points are capable of more complex vibrations, since the vibrational modes may involve several to many atoms and all three dimensions are available for vibrational movements. Vibrations where primarily the distances along the bond axis between the involved atoms change during the vibration are called valence vibrations. Vibrations causing a deformation of a bond angle are referred to as deformation vibrations. Deformation movements can also rock , wag or twist a molecular (sub-) structure (Figure 1). 

As a simple model which takes into account valence and deformation vibrations of a molecule imbedded in the condensed phase, we consider a diatomic molecule with two degrees of freedom corresponding to valence (in the radial variable r) and torsional (in the angular variable [Pg.94]

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