Kohn-Sham functionals

According to the variational principle, the ground state of the system is described by those electronic wavefunctions which minimize the Kohn-Sham functional. The presence of an external perturbation is represented by a perturbation functional, Ep, that is added to the unperturbed Kohn-Sham functional ... 

The calculation itself is somewhat lengthy, since it involves second derivatives of the Kohn-Sham functional with respect to the orbitals, and does not provide much insight into the physics of the problem. We therefore refer the interested reader to related references [13, 91]. The final stationarity equation reads ... 

The minimum of the Kohn-Sham functional is attained for the transformed wavefunctions e and e namely, when the... 

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 ... 

In the Flartree-Fock (FIF) method, the spurious self-interaction energy in the Flartree potential is exactly cancelled by the contributions to the energy from exchange. This would also occur in DFT if we knew the exact Kohn-Sham functional. In any approximate DFT functional, however, a systematic error arises due to incomplete cancellation of the self-interaction energy. 

Kohn-Sham functional for atomic clusters reads... 

Comparison of the Kohn-Sham and Skyrme functionals leads to a natural question why these two functionals exploit, for the time-dependent problem, so different sets of basic densities and currents If the Kohn-Sham functional is content with one density, the Skyrme forces operate with a diverse set of densities and currents, both T-even and T-odd. Then, should we consider T-odd densities as genuine for the description of dynamics of finite many-body systems or they are a pequliarity of nuclear forces This question is very nontrivial and still poorly studied. We present below some comments which, at least partly, clarify this point. 

As compared with the Kohn-Sham functional for electronic systems, the nuclear Skyrme functional is less genuine. The main (Coulomb) interaction in the Kohn-Sham problem is well known and only exchange and corellations should be modeled. Instead, in the nuclear case, even the basic interaction is unknown and should be approximated, e.g. by the simple contact interaction in Skyrme forces. 

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. 

Kohn-Sham functional makes the task of approximating the total energy functional easier but also offers a possibility to perform practical calculations. Euler-Lagrange minimization of the functional EKS[(pi, (p2,. [Pg.7]

This energy is an explicit functional of a set of the auxiliary functions, namely the Kohn-Sham orbitals The first term in the Kohn-Sham functional Eq. 

Kohn-Sham functional written in terms of the density itself (3), rather than writing the density in terms of one-electron wavefunctions (10). 

Application of the variational principle, 8 /8p(r) = 0, to the Kohn-Sham functional... 

The basis of the Kohn-Sham functional is an approximation to exchange-correlation energy of electrons. The approximations should include all many-body contributions to energy that is beyond the Hartree theory. The most common choices for the exchangeenergy functionals are local density approximation and generalized density approximation (Section 8.6). However, application of these functionals reveals a poor fitness for treatment systems that are bonded by the van der Waals forces. 

The Hohenberg-Kohn principles provide the theoretical basis of Density Functional Theory, specifically that the total energy of a quantum mechanical system is determined by the electron density through the Kohn-Sham functional. In order to make use of this very important theoretical finding, Kohn-Sham equations are derived, and these can be used to determine the electronic ground state of atomic systems. 

The Kohn-Sham functional employing the actual KS-statistical average ... 

---