Kohn-Sham local potential

An interesting observation can be added here if the local orbitals are not constructed from current-conservation criterion as discussed below but instead are adjusted to fit an LDA calculation based on the Kohn-Sham local potential v, we can alter-natively calculate p(r,r ) and construct approximate HF bands. 

The Kohn-Sham local potential contains a term for the electron-nuclei coulombic attraction, a term for the electron-electron coulombic repulsion, and an exchange term, which incorporates all the exchange and electron correlation effects ... 

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, /)]... 

Nesbet, R.K. and Colle, R. (2000). Tests of the locality of exact Kohn-Sham exchange potentials, Phys. Rev. A 61, 012503. 

Provided the potential t) is local in r, in the limit that X - oo we will have p - p, independent of the choice of t). In this limit then, Equation (5) gives the Kohn-Sham orbitals and eigenvalues. The determinant formed from these orbitals is a wave function obtained from the density p,. 

However, one feature of the HF potential is that it is not a local potential. In the case of perfect data (i.e. zero experimental error), the fitted orbitals obtained are no longer Kohn-Sham orbitals, as they would have been if a local potential (for example, the local exchange approximation [27]) had been used. Since the fitted orbitals can be described as orbitals which minimise the HF energy and are constrained produce the real density , they are obviously quite closely related to the Kohn-Sham orbitals, which are orbitals which minimise the kinetic energy and produce the real density . In fact, Levy [16] has already considered these kind of orbitals within the context of hybrid density functional theories. 

Note that the Kohn-Sham Hamiltonian hKS [Eq. (4.1)] is a local operator, uniquely determined by electron density15. This is the main difference with respect to the Hartree-Fock equations which contain a nonlocal operator, namely the exchange part of the potential operator. In addition, the KS equations incorporate the correlation effects through Vxc whereas they are lacking in the Hartree-Fock SCF scheme. Nevertheless, though the latter model cannot be considered a special case of the KS equations, there are some similarities between the Hartree-Fock and the Kohn-Sham methods, as both lead to a set of one-electron equations allowing to describe an n-electron system. 

Whereas the classic Kohn-Sham (KS) formulation of DFT is restricted to the time-independent case, the formalism of TD-DFT generalizes KS theory to include the case of a time-dependent, local external potential w(t) [27]. 

In this review we will give an overview of the properties (asymptotics, shell-structure, bond midpoint peaks) of exact Kohn-Sham potentials in atomic and molecular systems. Reproduction of these properties is a much more severe test for approximate density functionals than the reproduction of global quantities such as energies. Moreover, as the local properties of the exchange-correlation potential such as the atomic shell structure and the molecular bond midpoint peaks are closely related to the behavior of the exchange-correlation hole in these shell and bond midpoint regions, one might be able to construct... 

Corresponding to the interacting N + co-electron system we now define a Kohn-Sham potential Vs(r N -f co). This is a local potential defined by the requirement that it must yield the same density piy+ , in a fictitious noninteract-... 

Calculation of Kohn-Sham Orbitals and Potentials by Local-Scaling Transformations... 

In the remainder of this section, we give a brief overview of some of the functionals that are most widely used in plane-wave DFT calculations by examining each of the different approaches identified in Fig. 10.2 in turn. The simplest approximation to the true Kohn-Sham functional is the local density approximation (LDA). In the LDA, the local exchange-correlation potential in the Kohn-Sham equations [Eq. (1.5)] is defined as the exchange potential for the spatially uniform electron gas with the same density as the local electron density ... 

The time-dependent density functional theory [38] for electronic systems is usually implemented at adiabatic local density approximation (ALDA) when density and single-particle potential are supposed to vary slowly both in time and space. Last years, the current-dependent Kohn-Sham functionals with a current density as a basic variable were introduced to treat the collective motion beyond ALDA (see e.g. [13]). These functionals are robust for a time-dependent linear response problem where the ordinary density functionals become strongly nonlocal. The theory is reformulated in terms of a vector potential for exchange and correlations, depending on the induced current density. So, T-odd variables appear in electronic functionals as well. 

Hohenberg-Kohn theorems, but use the Kohn-Sham construction and local approximations to such non-local potentials and often lump together the exchange and the correlation energies into an exchange-correlation energy Exc[n], This yields a local exchange-correlation potential vxc(t) in the Kohn-Sham equations that determine the Kohn-Sham spin orbitals j, i.e. 

In comparing Eq. (13) to the Kohn-Sham equations Eq. (3) one concludes that E(.r, x E), since it is derived from exact many-electron theory [22], is the exact Coulomb (direct) plus exchange-correlation potential. It is non-local and also energy-dependent. In view of this it is hard to see how the various forms of constructed local exchange correlation potentials that are in use today can ever capture the full details of the correlation problem. 

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