Kohn-Sham theory derivative

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. 

Here the differentiation is shown as being with respect to p(r), but note that in Kohn-Sham theory p(r) is expressed in terms of Kohn-Sham orbitals (Eq. 7.22). Functional derivatives, which are akin to ordinary derivatives, are discussed by Parr and Yang [36] and outlined by Levine [37]. 

In Kohn-Sham theory, densities are postulated to be sums of orbital densities, for functions (pi in the orbital Hilbert space. This generates a Banach space [102] of density functions. Thomas-Fermi theory can be derived if an energy functional E[p] = I p + F [ p is postulated to exist, defined for all normalized ground-state... 

This argument shows that the locality hypothesis fails for more than two electrons because the assumed Frechet derivative must be generalized to a Gateaux derivative, equivalent in the context of OEL equations to a linear operator that acts on orbital wave functions. The conclusion is that the use by Kohn and Sham of Schrodinger s operator t is variationally correct, but no equivalent Thomas-Fermi theory exists for more than two electrons. Empirical evidence (atomic shell structure, chemical binding) supports the Kohn-Sham choice of the nonlocal kinetic energy operator, in comparison with Thomas-Fermi theory [288]. A further implication is that if an explicit approximate local density functional Exc is postulated, as in the local-density approximation (LDA) [205], the resulting Kohn-Sham theory is variation-ally correct. Typically, for Exc = f exc(p)p d3r, the density functional derivative is a Frechet derivative, the local potential function vxc = exc + p dexc/dp. 

H. A. Physical interpretation of Kohn-Sham theory exchange-correlation energy functional and its derivative... 

H. A. PHYSICAL INTERPRETATION OF KOHN-SHAM THEORY EXCHANGE-CORRELATION ENERGY FUNCTIONAL AND ITS DERIVATIVE... 

The electron-transfer reactivities are defined as derivatives of the electron-density p(r) with respect to total electron number Jf, Ufr), or chemical potential p, s r). The treatment of JT as a continuous variable [8-12] is justified by reference to the ensemble formulation of density-functional theory [8,18] and, in consequence, of the Kohn-Sham theory. We show in Sect. 4, in previously unpublished work [42], that this ensemble formulation yields either vanishing or infinite local and global softnesses for localized systems with... 

Because electron number can be continuous in the extended version of Kohn-Sham theory [42], Fukui functions may be determined as derivatives (Eq. (32)). The explicit forms for f+ and f can be given in this formalism as [82] ... 

Kohn-Sham theory in order to define the local potential representing electron correlations as well as other properties derived within its context. 

Within the context of Kohn-Sham theory, the assumption underlying the LDA is that each point of the nonuniform electron density is uniform but with a density corresponding to the local value. In the LDA for exchange, the wavefunction is therefore assumed to be a Slater determinant of plane waves at each electron position. The corresponding pair-correlation density g[ r, r p(r) is thus the expectation of Eq. (66) taken with res[ ct to this Slater determinant, with the resulting expression then assumed valid locally. (The superscript (0) indicates the result is derived from uniform electron gas theory.)... 

There is, of course, much that remains to be understood with regard to the physical interpretation. For example, the correlation-kinetic-energy field Z, (r) and potential W, (r) need to be investigated further. However, since accurate wavefunctions and the Kohn-Sham theory orbitals derived from the resulting density now exist for light atoms [40] and molecules [54], it is possible to determine, as for the Helium atom, the structure of the fields P(r), < P(r), and Zt (r), and the potentials WjP(r), W (r), W (r), and W (r) derived from them, respectively. A study of these results should lead to insights into the correlation and correlation-kinetic-energy components, and to the numerical determination of the asymptotic power-law structure of these fields and potentials. The analytical determination of the asymptotic structure of either [Z, (r), W, (r)] or [if (r), WP(r)] would then lead to the structure of the other. 

The field z(r [y]) thus defined is for the interacting system since the tensor involves the density matrix y(r,r ) of Eq. (6). With the field z(r [yj) derived similarly from the tensor tajj(r [ys])) written in terms of the idempotent Dirac density matrix > s( F) of Kohn-Sham theory, the field Z, (r) is then defined as... 

The electron-interaction component Wee(r) was originally derived, as noted previously, by Harbola and Sahni [9] via Coulomb s law. Since this component does not contain any correlation-kinetic-energy contributions, it does not [9,17,18] satisfy the Kohn-Sham theory sum rule relating the corresponding electron-correlation energy E [p] to its functional derivative (potential) vf (r). The sum rule, which is derived [19,20] from the virial theorem, and in which the correlation-kinetic-energy Tc[p] contribution is made explicit is... 

The pair-correlation density g,(r, r ) derived from the Kohn-Sham theory Slater determinant is... 

In standard Kohn-Sham theory, the exchange-correlation potential v o (r)is the sum of its exchange v (r) and correlation v (r) components which are in turn defined as the functional derivatives... 

As we have already observed above, within the hardness (interaction) representation (see Tables 1, 3) the FF indices provide important weighting factors in combination formulas which express the global CS and potentials in terms of the local properties, relevant for the resolution in question. The FF expressions from the EE equations are invalid in the MO resolution, since no equalization of the orbital potentials can take place, due to obvious constraints on the MO occupations in the Hartree-Fock (HF) theory [61]. Moreover, standard chain-rule transformations of derivatives are not applicable in the MO resolution since some of the derivatives involved are not properly defined. Various approaches to the local FF, f(f), have been proposed e.g., those expressing f(f), in terms of the frontier orbital densities [11, 25], or the spin densities [38]. Also the finite difference estimates of the chemical potential (electronegativity) and hardness have been proposed in the MO and Kohn-Sham theories for various electron configurations [10, 11, 19, 52, 61b, 62, 63]. 

In this work, an alternative strategy for deriving the spin-density of an embedded molecule is used. Instead of applying Kohn-Sham theory to the whole system, the spin/electron densities of different subsystems are treated separately. The two subsystems correspond to a) the embedded molecule the radical b) the embedding molecule(s) the atoms or molecules forming the complex with the radical. For a given electron-density of the embedding molecules. 

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