Effective Kohn-Sham potential

The presence of the Lagrange multiplier function o)(r) is fundamental in this variation, as it this multiplier that assumes the role of the effective Kohn-Sham potential ... 

Another interesting empirical solution was proposed recently by Lilienfeld et al.150 A special term taking the form of an atom-centered pseudopotential was added to the effective Kohn-Sham potential to correct the energies derived from the Kohn-Sham GGA calculations. The method was tested on noble-gas and benzene dimers. 

An iterative procedure can be set up in order to calculate the effective Kohn-Sham potential. The details of this procedure are given elsewhere [85, 86, 88]. Once this potential is obtained, we can analize it in terms of its components and extricate from it the Kohn-Sham exchange-correlation potential which, in turn, can be written as a sum of the exchange and the correlation potentials ... 

Note that, unlike the full Hohenberg-Kohn theorem, Kato s theorem does apply only to superpositions of Coulomb potentials, and can therefore not be applied directly to the effective Kohn-Sham potential. 

We have anployed the parametrized DFTB method of Porezag et al. [33,34]. The approximate DFTB method is based on the density-functional theory of Hohenberg and Kohn in the formulation of Kohn and Sham [43,44]. In this method, the single particle wave functions l (r) of the Kohn-Sham equations are expanded in a set of atomic-like basis functions < > , with m being a compound index that describes the atom on which the function is centered, the angular dependence of the function, as well as its radial dependence. These functions are obtained from self-consistent density functional calculations on the isolated atoms employing a large set of Slater-type basis functions. The effective Kohn-Sham potential Feff(r) is approximated as a simple superposition of the potentials of the neutral atoms... 

Up to this point we have discussed DFT in terms of the charge (or particle) density n r) as a fundamental variable. To reproduce the correct charge density of the interacting system in the noninteracting (Kohn-Sham) system, one must apply to the latter the effective Kohn-Sham potential Vs = w -f wh + Wxc, in which the last two terms simulate the effect of the electron-electron interaction on the charge density. This form of DFT, which is the one proposed originally, could also be called charge-only DFT. It is not the most widely used DFT in practical applications. Much more common is a formulation that employs one density for each spin, (r) and (r), i.e. works with two fundamental variables. To reproduce both of these... 

Within the Kohn-Sham formalism of DFT, V is substituted by the Kohn-Sham potential Vks Expanding equation 7 a little, we note that spin-orbit effects are implicitly included in the ZORA Hamiltonian ... 

The introduction in 1965 by Kohn and Sham7 of a practical computational scheme may, therefore, be considered to be the next major milestone in the development of formal DFT. The essential ingredient in this approach is the postulation of a reference system of N noninteracting electrons, moving in an effective external potential vs(r), the so-called Kohn-Sham potential, instead of the electrostatic potential v(r) of the nuclei ... 

The first term in brackets is the usual kinetic energy operator. The noninteracting reference system has the property that its one-determinantal wavefunction of the lowest N orbitals yields the exact density of the interacting system with external potential v(r) as a sum over densities of the occupied orbitals, that is, p(r) = Xl<)>,l2, and the corresponding exact energy E[p(r)]. The Kohn-Sham potential should account for all effects stemming from the electron-nuclear and electron-electron interactions. Not only does the Kohn-Sham potential contain the attractive potential v(r) of the nuclei and the classical Coulomb repulsion VCoul(r) within the electron density p(r), but it also accounts for all exchange and correlation effects, which have so to say been folded into a local potential vxc r) ... 

Norman, M.R. and Koelling, D.D. (1984). Towards a Kohn-Sham potential via the optimized effective potential method, Phys. Rev. B 30, 5530-5540. 

In the implementation of the QM/MM approach with the real-space method, the QM cell that contains the real-space grids is embedded in the MM cell. One should take care for the evaluation of the potential upc(r) defined as Eq. (17-20). When a point charge in MM region goes inside the QM cell, it makes a singularity in the effective Kohn-Sham Hamiltonian, which may give rise to a numerical instability. To circumvent the problem, we replace a point charge distribution... 

II. The density of the interacting system of interest can be obtained as the density of an auxiliary system of non-interacting particles moving in an effective local single-particle potential, the so-called Kohn Sham potential. 

Except in the ad-atom s vicinity, the metal screens out the effects of the ad-atom on the total charge density and on the Kohn-Sham potential, even though the disturbance in the individual Kohn-Sham single-particle wavefunctions is not short-ranged [38, p. 618]. That result is reproduced in Figure 2, together with the... 

The local Kohn-Sham potential uks in Eq. (2.3) is chosen in such a way that the electron density p (r, t) = I (r, 0 P obtained from the effective single particle orbitals -fi equals the true interacting many body density. According to... 

The constant force on each electron is positive, and the infinite potential barrier (needed to normalize the orbitals) is far away L oo. The chemical potential is fi = 0. Classically, electrons are confined to the half space z > 0, but in our quantum treatment they tunnel into the classically forbidden region z < 0. The linear variation of the effective potential in the region close to z = 0 can model the Kohn-Sham potential in the surface or edge region of jellium [23], and the evanescent decay of the electron density for large negative z models the tail of the electron density (which is not in principle described by any gradient expansion). 

If Ha is the density that is given by the lowest-lying orbital the Kohn-Sham potential for the whole system coincides with the effective potential in Eq. (1). No additional potential is needed (w" (r) = 0). 

Table 14 Fano parameters for different ionization resonances for some closed-shell atoms. Except for the calculations marked EXX, the Kohn-Sham single-particle eigenvalues have in all cases been shifted to match the ionization potential. Exact KS represents results with the exact Kohn-Sham potential, GGA those with a gradient-corrected density functional, EXX exact-exchange without correlation, EXX- -LDA the same but with the inclusion of correlation effects with a local-density approximation, and Exp. experimental results. The results are from ref. 91... 

In addition we have calculated the electron affinities within the jellium model. But, as well known from experience in atomic physics, the LDA functional has to be corrected for so-called self-interaction effects (SIE) as originally proposed by Perdew and Zunger [8]. As explained in detail below, this leads to an orbital-dependent Kohn-Sham potential by the replacement [31] ... 

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... 

Time-dependent density-functional theory (TDDFT) extends the basic ideas of ground-state density-functional theory (DFT) to the treatment of excitations and of more general time-dependent phenomena. TDDFT can be viewed as an alternative formulation of time-dependent quantum mechanics but, in contrast to the normal approach that relies on wave-functions and on the many-body Schrodinger equation, its basic variable is the one-body electron density, n(r,t). The advantages are clear The many-body wave-function, a function in a 3A-dimensional space (where N is the number of electrons in the system), is a very complex mathematical object, while the density is a simple function that depends solely on the 3-dimensional vector r. The standard way to obtain n r,t) is with the help of a fictitious system of noninteracting electrons, the Kohn-Sham system. The final equations are simple to tackle numerically, and are routinely solved for systems with a large number of atoms. These electrons feel an effective potential, the time-dependent Kohn-Sham potential. The exact form of this potential is unknown, and has therefore to be approximated. 

Prom such an action functional, one seeks to determine the local Kohn-Sham potential through a series of chain rules for functional derivatives. The procedure is called the optimized effective potential (OEP) or the optimized potential method (OPM) for historical reasons [15,16]. The derivation of the time-dependent version of the OEP equations is very similar to the ground-state case. Due to space limitations we will not present the derivation in this chapter. The interested reader is advised to consult the original paper [13], one of the more recent publications [17,18], or the chapter by E. Engel contained in this volume. The final form of the OEP equation that determines the EXX potential is... 

Mainly apply to TDDFT, where approximate exchange-correlation potentials are used in the effective Kohn-Sham Hamiltonian. 

Molecules by Time-Dependent Density Functional Theory Based on Effective Exact Exchange Kohn-Sham Potentials. 

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