Exchange-correlation term

DFT methods compute electron correlation via general functionals of the electron density (see Appendix A for details). DFT functionals partition the electronic energy into several components which are computed separately the kinetic energy, the electron-nuclear interaction, the Coulomb repulsion, and an exchange-correlation term accounting for the remainder of the electron-electron interaction (which is itself... 

We often split the exchange-correlation term into a sum of one part for exchange effects and one part for correlation effects. 

This fitting does not reduce the formal scaling, since the exchange-correlation term already is of order M. ... 

Goh, S. K., Gallant, R. T., St-Amant, A., 1998, Towards Linear Scaling for the Fits of the Exchange-Correlation Terms in the LCGTO-DF Method via a Divide-and-Conquer Approach , Int. J. Quant. Chem., 69, 405. 

Finally, in the third generation of DFT schemes, a portion of the exact Hartree-Fock (HF) exchange energy is mixed to the DFT exchange-correlation term, using the adiabatic con-... 

Once the coefficients for the expansion of the exchange-correlation term have been evaluated, all matrix elements can be calculated analytically. The Obara and Saika [47] recursive scheme has been used for the evaluation of the one and the two electron integrals. The total energy is therefore expressed in terms of the fitting coefficients for the electronic density and the exchange-correlation potential. 

An alternative procedure consists in using a numerical integration scheme to evaluate the exchange-correlation contribution. In this case, no auxiliary basis set is needed for the exchange-correlation terms, and numerically more reliable results can be obtained. 

The density functional theory (DFT) [32] represents the major alternative to methods based on the Hartree-Fock formalism. In DFT, the focus is not in the wavefunction, but in the electron density. The total energy of an n-electron system can in all generality be expressed as a summation of four terms (equation 4). The first three terms, making reference to the noninteracting kinetic energy, the electron-nucleus Coulomb attraction and the electron-electron Coulomb repulsion, can be computed in a straightforward way. The practical problem of this method is the calculation of the fourth term Exc, the exchange-correlation term, for which the exact expression is not known. 

As described in Chapter 1, the first term on the left-hand side describes the kinetic energy of the electron, V is the potential energy of an electron interacting with the nuclei, VH is the Flartree electron-electron repulsion potential, and Vxc is the exchange-correlation potential. This approach divides electron-electron interactions into a classical part, defined by the Flartree term, and everything else, which is lumped into the exchange-correlation term. The Flartree potential describes the Coulomb repulsion between the electron and the system s total electron density ... 

Equation (9.12) implies the assumption that the kinetic energy and exchange-correlation terms in Eq. (9.1) are the same for the crystal and the assembly of isolated ions. 

The crudeness of Skyrme forces has certain consequences. For example, the Skyrme functional has no any exchange-correlation term since the relevant effects are supposed to be already included into numerous Skyrme fitting parameters. Besides, the Skyrme functional may accept a diverse set of T-even and T-dd densities and currents. One may say that T-odd densities appear in the Skyrme functional partly because of its specific construction. Indeed, other effective nuclear forces (Gogny [40], Landau-Migdal [41]) do not exploit T-odd densities and currents for description of nuclear dynamics. 

Vc, an exchange-correlation term, Exc(p), and an external potential, [V , which arises primarily from nuclear-electron attraction but could include extramolecular perturbations, such as electric and magnetic fields. If the electronic wave function were expressed as a determinantal wave function, as in HF theory, then a set of equations functionally equivalent to the HF equations (A.40) emerges [324]. Thus... 

The wave eigenfunctions and energy eigenvalues were obtained by real space electronic state calculations, which were performed by the use of the program code SCAT of the DV-Xx molecular orbital (MO) cluster method with the Hartree-Fock-Slater approximation [8,9]. In the method, the exchange-correlation term Vxc in the one-electron Hamiltonian was expressed in terms of the statistical local potential (1),... 

Perdew has described a hierarchy of approximate treatments of the exchange-correlation term as Jacob s Ladder. The ladder is grounded in HF theory here on earth, and reaches to heaven, where the exact functional is found. Along the way are five rungs, each defining a set of assumptions made in creating an exchange-correlation expression. 

Kozuch and Martin have proposed combining the double hybrid method with the SCS-MP2 treatment along with a dispersion correction. This so-called DSD-DFT method (dispersion corrected, spin-component-scaled double hybrid) has the exchange-correlation term... 

The electronic structures of 1-D, 2-D, and 3-D silicon clusters were calculated using DV-Xa MO method . In the Xa method, the exchange-correlation term K,(r) in the one-electron Hamiltonian is written in terms of... 

In Fig. 2 we compare the 6s orbitals obtained for the two different couplings of the ion core. The difference in the calculations for these two orbitals is that the A a-coefficient for the exchange-correlation term in the Hartree-Slater Hamiltonian is varied to shift the calculated orbital energy to agree with the respective binding energy. The Hartree-Slater orbital for the 6s [ Fi/2 core] is also shown in Fig. 2. The inner nodes in this orbital are removed to obtain the 6s pseudoorbital. 

The major problem in DFT is deriving suitable formulas for the exchange-correlation term. Assume for the moment that such a functional is available, the problem is then -similar to that encountered in wave mechanics HF theory determine a set of orthogonal... 

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