Coulomb theory

Dahlen, F. A., Suppe, J., and Davis, D. (1984). Mechanics of fold-and-thrust belts and accretionary wedges Cohesive Coulomb theory. /. Geophys. Res. 89,10087-10101. 

Dahlen, F. A., J. Suppe, and D. Davis (1984). Mechanics of fold-and-thrust belts and accretionary wedges Cohesive Coulomb theory. /. Geophys. Res. 89, 10,087-10,101. Davis, D., J. Suppe, and F. A. Dahlen (1983). Mechanics of fold-and-thrust belts and accretionary wedges. /. Geophys. Res. 88, 1153-1172. 

C.A. Coulomb, Theorie des machines simples, en ayant regard au frottement de leurs parties, et a la roideur des cordages, Mem. Math. Rhys. (Paris), 10, 161-342, 1785. 

Dirac-Coulomb theory within the mean field approximation (see Chapter 8) is routinely applied to molecules and allows us to estimate the relativistic effects even for large molecules. In the computer era. this means that there are computer programs available that allow anybody to perform relativistic calculations. 

Compared to typical chemical phenomena, the relativistic effects remain of marginal significance in almost all instances for the biomolecules or molecules typical in traditional organic chemistry. In inorganic chemistry, however, these effects could be much more important. Probably the Dirac-Coulomb theory combined with the mean field approach will remain a satisfactory standard for the vast majority of researchers, at least for the next few decades. At the same time, there... 

There is a formal similarity between (10) and the Coulomb theory of yield mentioned above for isotropic materials. On the latter theory yield occurs on the plane which makes an angle a with the tensile axis, for which the shear component of the applied stress reaches a critical value. This shear component at yield decreases linearly with the normal component of stress on that plane. For an axial stress [Pg.379]

The shear plane is the one which maximises the value of sin a cos a +ksin a, and it is easily shown that this plane makes an angle = itan" fc with the plane of maximum shear stress. On the Coulomb theory a is uniquely defined by a material constant, whereas in (10) 0 is an independent variable defining the specimen orientation. 

Crystal field theory assumes that all M-L interactions are purely electrostatic in nature. More specifically, it considers the electrostatic effect of a field of ligands on the energies of a metals valence-shell orbitals. To discuss CFT, we need only be aware of two fundamental concepts (1) the coulombic theory of electrostatic interactions and (2) the shapes of the valence orbitals of transition metals—that is, the nd orbitals ( = 3 for the first row of transition metals, etc.). The first concept involves only the familiar ideas of the repulsion of like and the attraction of dislike electrical charges. Quantitatively, Coulomb s law states that the potential energy of two charges Qj and Q2 separated by a distance r is given by the formula shown in Equation (4.3) ... 

A consideration of CFT starts with two fundamental concepts the coulombic theory of electrostatic interactions and a detailed knowledge of the shapes of d orbitals. Two-dimensional cross-sectional diagrams of hydrogen-like orbitals show them as being cut out of circular pieces of cloth by various nodes. The sum of the electron probabilities in a given subshell is a sphere. The five d orbitals appear to be composed of four similar and one, the d, special orbital. To see that the dj. is not unique in shape or energy, a set of six dependent d orbitals is first visualized. The dj. orbital turns out to be just a linear combination of two of these dependent orbitals that look exactly like the other four. 

According to the Coulomb theory for linear backfill surfaces with no surcharge loading, the static component acts at a point located H/3 above the height of the wall. On the other hand, the dynamic part can be taken to act at a point approximately 0.6H above the base of the wall (Seed and Whitman 1970). Therefore, the total lateral earth pressure is applied at a height ... 

If a compression test is conducted on a brittle glassy amorphous polymer, at what angle would the fracture plane be oriented (Review the Mohr Coulomb theory of failure.)... 

Debye-Hiickel theory The activity coefficient of an electrolyte depends markedly upon concentration. Jn dilute solutions, due to the Coulombic forces of attraction and repulsion, the ions tend to surround themselves with an atmosphere of oppositely charged ions. Debye and Hiickel showed that it was possible to explain the abnormal activity coefficients at least for very dilute solutions of electrolytes. 

The miderstanding of the quantum mechanics of atoms was pioneered by Bohr, in his theory of the hydrogen atom. This combined the classical ideas on planetary motion—applicable to the atom because of the fomial similarity of tlie gravitational potential to tlie Coulomb potential between an electron and nucleus—with the quantum ideas that had recently been introduced by Planck and Einstein. This led eventually to the fomial theory of quaiitum mechanics, first discovered by Heisenberg, and most conveniently expressed by Schrodinger in the wave equation that bears his name. 

The theory of strong electrolytes due to Debye and Htickel derives the exact limiting laws for low valence electrolytes and introduces the idea that the Coulomb interactions between ions are screened at finite ion concentrations. 

Coulomb C A 1785 Theorie des maohines simples Memoire de Mathematique et de Physique de I AcademIe Royale 161-342... 

Pisani [169] has used the density of states from periodic FIP (see B3.2.2.4) slab calculations to describe the host in which the cluster is embedded, where the applications have been primarily to ionic crystals such as LiE. The original calculation to derive the external Coulomb and exchange fields is usually done on a finite cluster and at a low level of ab initio theory (typically minimum basis set FIP, one electron only per atom treated explicitly). 

Many problems in force field investigations arise from the calculation of Coulomb interactions with fixed charges, thereby neglecting possible mutual polarization. With that obvious drawback in mind, Ulrich Sternberg developed the COSMOS (Computer Simulation of Molecular Structures) force field [30], which extends a classical molecular mechanics force field by serai-empirical charge calculation based on bond polarization theory [31, 32]. This approach has the advantage that the atomic charges depend on the three-dimensional structure of the molecule. Parts of the functional form of COSMOS were taken from the PIMM force field of Lindner et al., which combines self-consistent field theory for r-orbitals ( nr-SCF) with molecular mechanics [33, 34]. 

The problem of electrophilic substitution into the anilinium ion has been examined by the methods of m.o. theory. Attempts to simulate the --inductive effect in Hiickel M.o. theory by varying the Coulomb integral of C(j) (the carbon atom to which the NH3+ group is attached) remove 7r-electrons from the o- and -positions and add them to the... 

The early pioneers also include Benjamin Franklin and Charles de Coulomb. Franklin studied the effect of point electrodes in drawing electric currents. Coulomb discovered that a charged object gradually loses its charge i.e., he actually discovered the electrical conductivity of air. Coulomb s importance for the development of electrostatic air-cleaning methods is great, mainly because the present theories about electric charges and electric fields are based on his work. 

The correlation functions of the partly quenched system satisfy a set of replica Ornstein-Zernike equations (21)-(23). Each of them is a 2 x 2 matrix equation for the model in question. As in previous studies of ionic systems (see, e.g.. Refs. 69, 70), we denote the long-range terms of the pair correlation functions in ROZ equations by qij. Here we apply a linearized theory and assume that the long-range terms of the direct correlation functions are equal to the Coulomb potentials which are given by Eqs. (53)-(55). This assumption represents the mean spherical approximation for the model in question. Most importantly, (r) = 0 as mentioned before, the particles from different replicas do not interact. However, q]f r) 7 0 these functions describe screening effects of the ion-ion interactions between ions from different replicas mediated by the presence of charged obstacles, i.e., via the matrix. The functions q j (r) need to be obtained to apply them for proper renormalization of the ROZ equations for systems made of nonpoint ions. 

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