Derivation Kohn-Sham model

In standard DFT an energy functional, E[p, is minimized under the constraint that the density p(r) is normalized to the number of electrons, which requires that the functional is differentiable with respect to the density at the minimum. In the Kohn-Sham model, an interacting system is simulated by a system of noninteracting electrons moving in a local potential, which requires that the derivative of the functionals involved can be represented by a local function. 

Fh p) = Ec p) + Fxc p) + FM + FEAPhFT p) (3.15) (with subscripts C, XC, eN, Ext, and T denoting Coulomb, exchange-correlation, electron-nuclear attraction, external, and kinetic energies respectively). It is CTucial to remark that (3,15) is not the Kohn-Sham decomposition familiar in conventional presentations of DFT. There is no reference, model, nor auxiliary system involved in (3.15). From the construction presented above it is clear that in order to maintain consistency and to define functional derivatives properly all... 

Theoretical considerations leading to a density functional theory (DFT) formulation of the reaction field (RF) approach to solvent effects are discussed. The first model is based upon isolelectronic processes that take place at the nucleus of the host system. The energy variations are derived from the nuclear transition state (ZTS) model. The solvation energy is expressed in terms of the electrostatic potential at the nucleus of a pseudo atom having a fractional nuclear charge. This procedure avoids the introduction of arbitrary ionic radii in the calculation of insertion energy, since all integrations involved are performed over [O.ooJ The quality of the approximations made are discussed within the frame of the Kohn-Sham formulation of density functional theory. 

Analysis and Modelling of Atomic and Molecular Kohn-Sham Potentials derivation see [26,27]) the variational problem can be rewritten as... 

Assuming the existence of such a Frechet functional derivative [26,102] constitutes the locality hypothesis. If this hypothesis were valid, the OEL and Kohn-Sham equations would be equivalent, determining the same model or reference state. 

The first derivative of the density matrix with respect to the magnetic induction (dPfiv/dBi) is obtained by solving the coupled-perturbed Hartree-Fock (or Kohn-Sham) equations to which the first derivative of the effective Fock (or Kohn-Sham) operator with respect to the magnetic induction contributes. Due to the use of GIAOs, specific corrections arising from the effective operator Hcnv describing the environment effects will appear. We refer to Ref. [28] for the PCM model and to Ref. [29] for the DPM within either a HF or DFT description of the solute molecule. 

Mathematical expressions for the functionals which are found in the Kohn-Sham operator are usually derived either from the model of a uniform electron gas or from a fitting procedure to calculated electron densities of noble gas atoms. Two different functionals are then derived. One is the exchange functional Fx and the other the correlation functional Fc, which are related to the exchange and correlation energies in ab initio theory. We point out, however, that the definition of the two terms in DFT is slightly different from ab initio theory, which means that the corresponding energies cannot be directly compared between the two methods. 

Pa and pB correspond to the monomers forming the complex. The pa,Pb overlap is small but not negligible in this model system. Moreover, forming a complex affects the electron density of each monomer. Especially, electron density of H2 becomes polarized due to the large dipole moment of NCH. Zeroth-order GEA, leads to a reasonably good approximation to — sp PB as indicated by the similar qualitative behavior of the electron density derived from the reference calculations (Kohn-Sham - not depending on approximations to T ad pA, Pb]) and Eqs. 31-32. The second-order contribution to T ad pA, Pb worsens qualitatively the results. 

In Kohn-Sham (KS) density functional theory (DFT), the occupied orbital functions of a model state are derived by minimizing the ground-state energy functionals of Hohenberg and Kohn. It has been assumed for some time that effective potentials in the orbital KS equations are always equivalent to local potential functions. When tested by accurate model calculations, this locality assumption is found to fail for more than two electrons. Here this failure is explored in detail. The sources of the locality hypothesis in current DFT thinking are examined, and it is shown how the theory can be extended to an orbital functional theory (OFT) that removes the inconsistencies and paradoxes. 

Since hybrid functionals, Meta-GGAs, SIC, the Fock term and all other orbital functionals depend on the density only implicitly, via the orbitals i[n, it is not possible to directly calculate the functional derivative vxc = 5Exc/5n. Instead one must use indirect approaches to minimize E[n and obtain vxc. In the case of the kinetic-energy functional Ts[ 0j[rr] ] this indirect approach is simply the Kohn-Sham scheme, described in Sec. 4. In the case of orbital expressions for Exc the corresponding indirect scheme is known as the optimized effective potential (OEP) [120] or, equivalently, the optimized-potential model (OPM) [121]. The minimization of the orbital functional with respect to the density is achieved by repeated application of the chain rule for functional derivatives,... 

A very important point is that, contrary to methods based on a Hartree-Fock zero-order wave function, those rooted in the Kohn-Sham approach appear equally reliable for closed- and open-shell systems across the periodic table. Coupling the reliability of the results with the speed of computations and the availability of analytical first and second derivatives paves the route for the characterization of the most significant parts of complex potential energy surfaces retaining the cleaness and ease of interpretation of a single determinant formalism. This is at the heart of more dynamically based models of physico-chemical properties and reactivity. 

We make now use of our numerical results for the self-energy operator in C and in Si to derive a sin jle, analytic model. It will in particular elucidate how an intrinsically long-ranged and non-local e-h polarization giving rise to the dynamical correlation in a non-metal can still approximately be cast into a local effect. We then use the relation (2.22), which expresses the Kohn-Sham potential v in terms of Z, to derive a model expression... 

M. E. Casida, Pl ys. Rev. B, 59,4694-4698 (1999). Correlated Optimized Effective-Potential Treatment of the Derivative Discontinuity and of the Highest Occupied Kohn-Sham Eigenvalue A Janak-Type Theorem for the Optimized Effective-Potential Model. 

The density functional (DF) method has been successful and quite useful in correlating experimental results when model densities are used in the calculations. In fact, the equations characteristic of the DF method can be derived from a variational approach as Kohn and Sham showed some time ago. In this approach, when model densities are introduced, it is not always possible to relate such densities to corresponding wave functions this is the N-representability problem. Fortunately, for any normalized well behaved density there exists a Slater single determinant this type of density is then N-representable. The problem of approximately N-representable density functional density matrices has been recently discussed by Soirat et al. [118], In spite... 

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