Kohn-Sham derivation

To. solve the Kohn-Sham equations a self-consistent approach is taken. An initial guess of the density is fed into Equation (3.47) from which a set of orbitals can be derived, leading to an improved value for the density, which is then used in the second iteration, and so on until convergence is achieved. 

In addition to the energy terms for the exchange-correlation contribution (which enables the total energy to be determined) it is necessary to have corresponding terms for the potential, Vxc[p(i )]/ which are used to solve the Kohn-Sham equations. These are obtained as the appropriate first derivatives using Equation (3.52). 

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... 

Note that in all current implementations of TDDFT the so-called adiabatic approximation is employed. Here, the time-dependent exchange-correlation potential that occurs in the corresponding time-dependent Kohn-Sham equations and which is rigorously defined as the functional derivative of the exchange-correlation action Axc[p] with respect to the time-dependent electron-density is approximated as the functional derivative of the standard, time-independent Exc with respect to the charge density at time t, i. e.,... 

Up to this point the derivation has exactly paralleled the Hartree-Fock case, which only differs in using the corresponding Fock matrix, F rather than the Kohn-Sham counterpart, Fks. By expanding fKS into its components, the individual elements of the Kohn-Sham matrix become... 

The tensor of the static first hyperpolarizabilities P is defined as the third derivative of the energy with respect to the electric field components and hence involves one additional field differentiation compared to polarizabilities. Implementations employing analytic derivatives in the Kohn-Sham framework have been described by Colwell et al., 1993, and Lee and Colwell, 1994, for LDA and GGA functionals, respectively. If no analytic derivatives are available, some finite field approximation is used. In these cases the P tensor is preferably computed by numerically differentiating the analytically obtained polarizabilities. In this way only one non-analytical step, susceptible to numerical noise, is involved. Just as for polarizabilities, the individual tensor components are not regularly reported, but rather... 

Applying the variational principle to the energy given by Eq. 1, Kohn and Sham reformulated the density functional theory by deriving a set of one-electron Hartree-like equations leading to the Kohn-Sham orbitals v().(r) involved in the calculation of p(r)15. The Kohn-Sham (KS) equations are written as follows ... 

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 ... 

Because of the separation into a time-independent unperturbed wavefunction and a time-dependent perturbation correction, the time derivative on the right-hand side of the time-dependent Kohn-Sham equation will act only on the response orbitals. From this perturbed wavefunction the first-order response density follows as ... 

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. 

While the hydrodynamical scheme mentioned above involves the density quantities directly, an alternative second scheme based on their orbital partitioning along the lines of the Kohn-Sham [4] version of time-independent DFT has been derived by Ghosh and Dhara [14]. In this scheme, one obtains the exact densities p(r, t) and j(r, t) from the TD orbitals (///,(r, t) obtained by solving the effective one-particle TD Schrodinger-like equations given by... 

In this short review, a brief overview of the underlying principles of TDDFT has been presented. The formal aspects for TDDFT in the presence of scalar potentials with periodic time dependence as well as TD electric and magnetic fields with arbitrary time dependence are discussed. This formalism is suitable for treatment of interaction with radiation in atomic and molecular systems. The Kohn-Sham-like TD equations are derived, and it is shown that the basic picture of the original Kohn-Sham theory in terms of a fictitious system of noninteracting particles is retained and a suitable expression for the effective potential is derived. 

In the Kohn-Sham equation above, the Coulomb potential and the XC potential are obtained from their energy counterparts by taking the functional derivative of the latter with respect to the density. Thus... 

As is evident from the above, both the physics invoked to derive the potential of Equation 7.31 and the numerical results presented show that Wx gives an accurate exchange potential for the excited states. When the proposal was initially made, there was no mathematical proof of the existence of a Kohn-Sham equation for excited states. It is only during the past few years that DFT of excited states [34-37], akin to its ground-state counterpart, is being developed. 

Next, using the concept [2,64] of adiabatic connection, Kohn-Sham-like equations can be derived. We suppose the existence of a continuous path between the interacting and the noninteracting systems. The density , of the th electron state is the same along the path. 

By the way, through ensemble theory with unequal weights, Ref. [68] identifies an effective potential derivative discontinuity that links physical excitation energies to excited Kohn-Sham orbital energies from a ground-state calculation.)... 

Kohn-Sham equations is rather complicated for an arbitrarily selected set of weighting factors, and has to be derived separately for every different case of interest. For a spherically symmetric external potential and equal weighting factors, however, the Kohn-Sham equations have a very simple form, as shown in Ref. [72], In this case the noninteracting kinetic energy is given by... 

Fukui functions and other response properties can also be derived from the one-electron Kohn-Sham orbitals of the unperturbed system [14]. Following Equation 12.9, Fukui functions can be connected and estimated within the molecular orbital picture as well. Under frozen orbital approximation (FOA of Fukui) and neglecting the second-order variations in the electron density, the Fukui function can be approximated as follows [15] ... 

The derivatives of these Kohn-Sham quantities relative to the potentials are other reactivity descriptors ... 

All these functional derivatives are well defined and do not involve any actual derivative relative to the electron number. It is remarkable that the derivatives of the Kohn-Sham chemical potential /rs gives the so-called radical Fukui function [8] either in a frozen orbital approximation or by including the relaxation of the KS band structure. On the other hand, the derivative of the Kohn-Sham HOMO-FUMO gap (defined here as a positive quantity) is the so-called nonlinear Fukui function fir) [26,32,50] also called Fukui difference [51]. 

The frontier orbitals responses (or bare Fukui functions) f (r) and the Kohn-Sham Fukui functions (or screened Fukui functions)/, (r) are related by Dyson equations obtained by using the PRF and its inverse [32]. Indeed, by using Equation 24.57 and the chain rule for functional derivatives in Equation 24.36, one obtains... 

Higher-order derivatives with respect to external potential define xi(r, r1), Xi(r, r1, r"), etc., and their response with N define j(r, rJ), g2(r, r, / ), etc. This chain of derivatives is diagrammatically depicted in Figure 25.1 [22]. Thus, an exact one-electron formulation of all chemical responses (linear and nonlinear hardness, FF) in terms of Kohn-Sham orbital of the unperturbed system was derived [22b]. 

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