Potential charge density

A cubic lattice is superimposed onto the solute(s) and the surrounding solvent. Values of the electrostatic potential, charge density, dielectric constant and ionic strength are assigned to each grid point. The atomic charges do not usually coincide with a grid point and so the... 

Loeb, AL Overbeek, JTG Wiersema, PH, The Electrical Double Layer Around a Spherical Colloid Particle, Computation of the Potential, Charge Density, and Free Energy of the Electrical Double Layer Around a sperical Colloid Particle M.I.T. Press Cambridge, MA, 1961. Lorentz, HA, Wied, Ann. 11, 70, 1880. 

At a phase boundary (or interface) the molecular species experience anisotropic forces, which vary with the distance from the interface. This causes a net orientation of solvent and other molecular dipoles and a net excess of ions near the phase boundary, on both sides of the solution. The term electrified interface means that there occur differences in potential, charge densities, dipole moments, and electric currents. 

Appelbaum, J. A., and Hamann, D. R. (1973a). Surface potential, charge density, and ionization potential of Si(lll) - a self-consistent calculation. Phys. Rev. Lett. 32, 225-228. 

Figure 28. a) Ion redistribution process at the contact MX/MX concentration effects.117 b) Variation of the potentials, charge densities, and dielectric displacements at the contact of two Frenkel defect ionic conductors.94 (Reprinted from J. Maier, Ionic Conduction in Space Charge Regions. Prog. Solid St. Chem. 171-263, 23, Copyright 1995 with permission from Elsevier.)... 

Figure 3.47 Potential-charge density transients from galvanostatic pulse experiments in the system Au(111)/10-3 M Bi(C104)3 + 1 M HCIO4 at T= 298 K [3.117], Initial underpotentials A i/mV= 110 (1) 190 (2) 220 (3) 223 (4) 225 (5), anodic pulse current densitiy i = 20 mA cm. ...

A software that provides visualization and display of molecular surfaces, orbitals, electrostatic potentials, charge densities and spin densities (http //www.cambridgesoft.com/)... 

To solve the PB equation for arbitrary geometries requires some type of discretization, to convert the partial differential equation into a set of difference equations. Finite difference methods divide space into a cubic lattice, with the potential, charge density, and ion accessibility defined at the lattice points (or grid points ) and the permittivity defined on the branches (or grid lines ). Equation [1] becomes a system of simultaneous equations referred to as the finite difference Poisson-Boltzmann (FDPB) equation ... 

There are mutual links between SHG at electrodes and electrochemistry, and the results of classical electrochemical measurements, simply because these methods probe different physical properties of the interfacial regime. Adsorption and potential, charge density, surface structure, and the formation of (ordered) adlayers have a significant impact on both the behavior of the electric double layer and the SH response. SHG was applied to study sulfate adsorption at Au(lll) [129]. 

The 4-3IG basis set is not exactly a double zeta basis since only the valence functions are doubled and a single function is still used for each inner shell orbital. It may be termed a split valence shell basis set. The inner shells contribute little to most chemical properties and usually vary only slightly from molecule to molecule. Not splitting the inner shell functions has some effect on the total energy, but little effect on dipole moments, valence ionization potentials, charge densities, dissociation energies, and most other calculated quantities of chemical interest. The 4-3IG basis thus consists of 2 functions for H and He, 9 functions for Li to Ne, 13 functions for Na to Ar,..., etc. For hydrogen the contractions are... 

The mathematics is completed by one additional theorem relating the divergence of the gradient of the electrical potential at a given point to the charge density at that point through Poisson s equation... 

One can write acid-base equilibrium constants for the species in the inner compact layer and ion pair association constants for the outer compact layer. In these constants, the concentration or activity of an ion is related to that in the bulk by a term e p(-erp/kT), where yp is the potential appropriate to the layer [25]. The charge density in both layers is given by the algebraic sum of the ions present per unit area, which is related to the number of ions removed from solution by, for example, a pH titration. If the capacity of the layers can be estimated, one has a relationship between the charge density and potential and thence to the experimentally measurable zeta potential [26]. 

The effect known either as electroosmosis or electroendosmosis is a complement to that of electrophoresis. In the latter case, when a field F is applied, the surface or particle is mobile and moves relative to the solvent, which is fixed (in laboratory coordinates). If, however, the surface is fixed, it is the mobile diffuse layer that moves under an applied field, carrying solution with it. If one has a tube of radius r whose walls possess a certain potential and charge density, then Eqs. V-35 and V-36 again apply, with v now being the velocity of the diffuse layer. For water at 25°C, a field of about 1500 V/cm is needed to produce a velocity of 1 cm/sec if f is 100 mV (see Problem V-14). 

This equation is usually solved self-consistently . An approximate charge is assumed to estimate the exchange-correlation potential and to detennine the Flartree potential from equation Al.3.16. These approximate potentials are inserted in the Kolm-Sham equation and the total charge density is obtained from equation A 1.3.14. The output charge density is used to construct new exchange-correlation and Flartree potentials. The process is repeated nntil the input and output charge densities or potentials are identical to within some prescribed tolerance. 

This ionic potential is periodic. A translation of r to r + R can be acconnnodated by simply reordering the sunnnation. Since the valence charge density is also periodic, the total potential is periodic as the Hartree and exchange-correlation potentials are fiinctions of the charge density. In this situation, it can be shown that the wavefiinctions for crystalline matter can be written as... 

The model used is the RPM. The average electrostatic potential ifr) at a distance r away from an ion / is related to tire charge density p.(r) by Poisson s equation... 

At finite positive and negative charge densities on the electrode, the counterion density profiles often exhibit significantly higher maxima, i.e. there is an overshoot, and the derived potential actually shows oscillations itself close to the electrode surface at concentrations above about 1 M. 

The most elementary mean-field models of electronic structure introduce a potential that an electron at r would experience if it were interacting with a spatially averaged electrostatic charge density arising from the N- 1 remaining electrons ... 

Frequent approximations made in TB teclmiques in the name of achieving a fast method are the use of a minimal basis set, the lack of a self-consistent charge density, the fitting of matrix elements of the potential. 

Most TB approaches are not charge self-consistent. This means that they do not ensure that the charge derived from the wavefiinctions yields the effective potential assumed in their calculation. Some methods have been developed which yield charge densities consistent with the electronic potential [14, H and 16]. 

In DFT, the electronic density rather than the wavefiinction is tire basic variable. Flohenberg and Kohn showed [24] that all the observable ground-state properties of a system of interacting electrons moving in an external potential are uniquely dependent on the charge density p(r) that minimizes the system s total... 

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