Operator exchange-correlation

Godby R W, Schluter M and Sham L J 1988 Self-energy operators and exchange-correlation potentials in semiconductors Phys. Rev. B 37 10159-75... 

The first term is the familiar one-electron operator, the second term represents the Coulomb potential, and the third term is called exchange-correlation potential. HF and DFT differ only in this last term. In HF theory there is only a nonlocal exchange term, while in DFT the term is local and supposed to cover both exchange and correlation. It arises as a functional derivative with respect to the density ... 

Note that the operator fKS differs from the Fock operator f that we introduced in Section 1.3 in connection with the Hartree-Fock scheme only in the way the exchange and correlation potentials are treated. In the former, the non-classical contributions are expressed via the - in its exact form unknown - exchange-correlation potential Vxc, the functional derivative of Exc with respect to the charge density. In the latter, correlation is neglected... 

First, the self-energy operator is replaced by a local exchange-correlation potential, which is given by the functional derivative of the exchange-correlation energy with respect to the electron density ... 

Equation (96) shows that the effective KS potential may be simply obtained by adding to the standard KS potential of the isolated solute, an electrostatic correction which turns out to be the RE potential Or, and the exchange- correlation correction 8vxc. It is worth mentioning here, that Eq (96) is formally equivalent to the effective Fock operator correction bfteffi defined in the context of the self consistent reaction field (SCRF) theory [2,3,14] within the HF theory, the exchange contribution is exactly self-contained in Or, whereas correlation effects are completely neglected. As a result, within the HF theory 8v = Or, as expected. 

Spin-dependent operators are now introduced. The external potential can be an operator Vext acting on the two-component spinors. The exchange-correlation potential is defined as in Eq. [27], although Exc is now a functional Exc = Exc[pap] of the spin-density matrix. The exchange-correlation potential is then... 

Given an expression for the self-energy operator, equations (2) and (4) must he solved self-consistently. E(E) is also called the exchange-correlation potential, it is manifestly non-local and energy dependent. 

It follows from the definition of the functionals and Ti that the exchange-correlation functional is invariant under the rotation and translation of the electron density. This follows directly from the fact that the kinetic energy operator f and the two-electron operator W are invariant with respect to these transformations. This also has implications for the exchange-correlation functional. 

It is seen that the basic operator (65) and strength matrix (66) have now simple expressions via 6p f) from (62). The operator (65) has exchange-correlation and Coulomb terms. For electric multipole oscillations (dipole plasmon,. ..), the Coulomb term dominates. 

All electron calculations were carried out with the DFT program suite Turbomole (152,153). The clusters were treated as open-shell systems in the unrestricted Kohn-Sham framework. For the calculations we used the Becke-Perdew exchange-correlation functional dubbed BP86 (154,155) and the hybrid B3LYP functional (156,157). For BP86 we invoked the resolution-of-the-iden-tity (RI) approximation as implemented in Turbomole. For all atoms included in our models we employed Ahlrichs valence triple-C TZVP basis set with polarization functions on all atoms (158). If not noted otherwise, initial guess orbitals were obtained by extended Hiickel theory. Local spin analyses were performed with our local Turbomole version, where either Lowdin (131) or Mulliken (132) pseudo-projection operators were employed. Broken-symmetry determinants were obtained with our restrained optimization tool (136). Pictures of molecular structures were created with Pymol (159). 

Kohn-Sham orbitals (18)), Vn is the external, nuclear potential, and p is the electronic momentum operator. Hence, the first integral represents the kinetic and potential energy of a model system with the same density but without electron-electron interaction. The second term is the classical Coulomb interaction of the electron density with itself. Exc> the exchange-correlation (XC) energy, and ENR are functionals of the density. The exact functional form for Exc is unknown it is defined through equation 1 (79), and some suitable approximation has to be chosen in any practical application of... 

We have to consider the calculation of the fourth term, the problem term, in the KS operator of Eq. 7.23, the exchange-correlation potential vXc(r). This is defined as the functional derivative [36, 37] of the exchange-correlation energy functional, fsxc[p(r)], with respect to the electron density functional (Eq. 7.23). The exchange-correlation energy UX( lp(r)], a functional of the electron density function p(r), is a quantity which depends on the function p(r ) and on just what mathematical form the... 

Here Ho is the kinetic energy operator of valence electrons Vps is the pseudopotential [40,41] which defines the atomic core. V = eUn(r) is the Hartree energy which satisfies the Poisson equation ArUn(r) = —4nep(r) with proper boundary conditions as discussed in the previous subsection. The last term is the exchange-correlation potential Vxc [p which is a functional of the density. Many forms of 14c exist and we use the simplest one which is the local density approximation [42] (LDA). One may also consider the generalized gradient approximation (GGA) [43,44] which can be implemented for transport calculations without too much difficulty [45]. Importantly a self-consistent solution of Eq. (2) is necessary because Hks is a functional of the charge density p. One constructs p from the KS states Ts, p(r) = (r p r) = ns Fs(r) 2, where p is the density matrix,... 

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