Exchange-correlation potential, effect

LDA, these effects are modelled by the exchange-correlation potential In order to more accurately... 

The first three terms in Eq. (10-26), the election kinetic energy, the nucleus-election Coulombic attraction, and the repulsion term between charge distributions at points Ti and V2, are classical terms. All of the quantum effects are included in the exchange-correlation potential... 

The effective potential is written as a sum of the Coulomb potential and the exchange-correlation potential ... 

The third term of Eq (54) is the electronic Hartree potential, whereas the fourth one represents the exchange-correlation potential. This last term is usually obtained from a model exchange-correlation energy functional xc[pl To a first order approximation, the effective KS potential compatible with the electron density p f) given in Eq (51) may be written as ... 

The exchange-correlation potential v c reflects the atomic shell structure (see below). The shell structure of v arises from that of the pair-correlation function g. The shell structure of the latter, in its turn, originates from the antisymmetry property of the wavefunction of the system and is mainly an exchange effect. We will therefore consider first the exchange-only case. The influence of correlation effects on the atomic shell structure will be discussed at the end of this section. 

Let us now discuss the correlation effects on the atomic shell structure. We plot in Fig. 7 some of the described potentials for the case of the beryllium atom. The exact exchange-correlation potential v c is calculated from an accurate Cl (Configuration Interaction) density using the procedure described in [20]. The potentials Vx, and u" , are calculated within the optimized potential model (OPM) [21,40,41] and are probably very close to their exact values which can be obtained from the solution for of the OPM integral equation [21,40,41] by insertion of the exact Kohn-Sham orbitals instead of the OPM... 

In practical applications the term in the exchange-correlation potential involving this function turns out to be very small for atomic systems. Its main effect is an improved description of the peaks at the atomic shell boundaries. Neglect of this term still leads to an accurate approximation for the atomic exchange-correlation potential. 

In plain language, this means that a single electron feels the effect of other electrons only as an average, rather than feeling the instantaneous repulsive forces generated as electrons become close in space. If you compare Eq. (1.14) with the Kohn-Sham equations, Eq. (1.5), you will notice that the only difference between the two sets of equations is the additional exchange-correlation potential that appears in the Kohn-Sham equations. 

The effective Time Dependent Kohn-Sham (TDKS) potential vks p (r>0 is decomposed into several pieces. The external source field vext(r,0 characterizes the excitation mechanism, namely the electromagentic pulse as delivered by a by passing ion or a laser pulse. The term vlon(r,/) accounts for the effect of ions on electrons (the time dependence reflects here the fact that ions are allowed to move). Finally, appear the Coulomb (direct part) potential of the total electron density p, and the exchange correlation potential vxc[p](r,/). The latter xc potential is expressed as a functional of the electronic density, which is at the heart of the DFT description. In practice, the functional form of the potential has to be approximated. The simplest choice consists in the Time Dependent Local Density Approximation (TDLDA). This latter approximation approximation to express vxc[p(r, /)]... 

Fig. 2 Influence of the choice of the exchange-correlation potentials and of relativistic effects on the 31p-31p two-bond coupling constants in cis- and fra s-M(CO)4(PH3)2, M is Cr, Mo or W. (Graphics courtesy of Kaupp [97])...

As was mentioned above, in KS-TDDFT the effects of electron exchange and Coulomb correlation are incorporated in the exchange-correlation potential vxaJ and kernel fxl- While the potential determines the KS orbitals (j)ia and the zero-order TDDFRT estimate (35) for excitation energies, the kernel determines the change of vxca with Eqs. 21, 22, 24. Though both vxca and are well defined in the theory, their exact explicit form as functionals of the density is not known. Rather accurate vxca potentials can be constructed numerically from the ab initio densities p for atoms [35-38] and molecules [39-42]. However, this requires tedious correlated ab initio calculations, usually with some type of configuration interaction (Cl) method. Therefore, approximations to vxcn and are to be used in TDDFT. 

The computational effort of solving orbital Euler-Lagrange (OEL) equations is significantly reduced if the generally nonlocal exchange-correlation potential vxc can be replaced or approximated by a local potential vxc(r). A variationally defined optimal local potential is determined using the optimized effective potential (OEP) method [380, 398]. This method can be applied to any theory in which the model... 

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