Exchange charge density

Gorling, A. (1999). New KS method for molecules based on an exchange charge density generating the exact KS exchange potential, Phys. Rev. Lett. 83,... 

Equation 2.49 looks similar to Equation 2.47, except that instead of the charge density p(f), we have an exchange charge density ... 

Where is the exchange charge density located, when the exchange operator acts on an occupied spin orbital We assnme that electron 1 is fixed in its position with spin up. Set cj) = c )]j and calcnlate at the point 2. The resnlting function is stiU complicated. First, we find the value of at the point fi where the first electron is located ... 

Exchange Charge Density Generating the Exact Local KS Exchange Potential. 

In a number of classic papers Hohenberg, Kohn and Sham established a theoretical framework for justifying the replacement of die many-body wavefiinction by one-electron orbitals [15, 20, 21]. In particular, they proposed that die charge density plays a central role in describing the electronic stnicture of matter. A key aspect of their work was the local density approximation (LDA). Within this approximation, one can express the exchange energy as... 

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 exchange-repulsion energy is approximately proportional to the overlap of the charge densities of the interacting molecules [71, 72 and 73]... 

Coulomb integrals Jij describe the coulombic interaction of one charge density (( )i2 above) with another charge density (c )j2 above) exchange integrals Kij describe the interaction of an overlap charge density (i.e., a density of the form ( )i( )j) with itself ((l)i(l)j with ( )i( )j in the above). 

Her workers to fit the exchange-correlation potential and the charge density (in the Coulomb potential) to a linear combination of Gaussian-typc functions. 

If the areas of the electrodes are assumed to be 1 cm, and taking the equilibrium exchange current density /g for the Ag /Ag equilibrium to be 10 A cm", then /g will be 10 A, which is a very high rate of charge transfer. A similar situation will prevail at electrode II, and rates of exchange of silver ions and the potential will be the same as for electrode I. 

Because of their lower charge densities, monovalent ions are exchanged less efficiently, so chloride and sodium ions in particular tend to slip through the resin beds. Where twin-bed DI is employed,... 

Presumably the most important kinetie parameter used in the deseription of the kineties of an eleetrode is the exchange current density or the almost equivalent rate constant. It indicates the speed of the heterogeneous process of charging or discharging species at the phase boundary, i.e. the charge transfer process. Its value is influenced by numerous factors of the investigated system. For both applied and fundamental aspects of electrochemical research a list of reported values should be helpful. It concludes this volume. 

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