Kohn-Sham Fukui functions

We call the Fukui function / (r) the HOMO response. Equation 24.39 is demonstrated as follows. The PhomoW is the so-called Kohn-Sham Fukui function denoted as f (r) [32]. According to the first-order perturbation theory, one has... 

We demonstrate now that six Fukui functions at constant electron number can be defined for an isolated molecule. The two Kohn-Sham Fukui functions are [32]... 

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... 

Inutility of the Softnesses and Discrepancy Between the Ensemble and the Kohn-Sham Fukui Functions for Localized Systems. .. 154... 

Fuentealba P, Cedillo A (1999) The variations of the hardness and the Kohn-Sham Fukui function under an external perturbation. J Chem Phys 110(20) 9807-9811... 

Gazquez JL, Vela A, Galvan M (1987) Fukui Function, Electronegativity and Hardness in the Kohn-Sham Theory. 66 79-98... 

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] ... 

Based on the foregoing discussion, one might suppose that the Fukui function is nothing more than a DFT-inspired restatement of frontier molecular orbital (FMO) theory. This is not quite true. Because DFT is, in principle, exact, the Fukui function includes effects—notably electron correlation and orbital relaxation—that are a priori neglected in an FMO approach. This is most clear when the electron density is expressed in terms of the occupied Kohn-Sham spin-orbitals [16],... 

In most cases, the orbital relaxation contribution is negligible and the Fukui function and the FMO reactivity indicators give the same results. For example, the Fukui functions and the FMO densities both predict that electrophilic attack on propylene occurs on the double bond (Figure 18.1) and that nucleophilic attack on BF3 occurs at the Boron center (Figure 18.2). The rare cases where orbital relaxation effects are nonnegligible are precisely the cases where the Fukui functions should be preferred over the FMO reactivity indicators [19-22], In short, while FMO theory is based on orbitals from an independent electron approximation like Hartree-Fock or Kohn-Sham, the Fukui function is based on the true many-electron density. 

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]. 

Korchowiec, J. and T. Uchimaru. 1998. The charge transfer Fukui function Extension of the hnite-difference approach to reactive systems. J. Phys. Chem. A 102 10167-10172. Michalak, A., De Proft, F., Geerlings, P., and R. F. Nalewajski. 1999. Fukui functions from the relaxed Kohn-Sham orbitals. J. Phys. Chem. A 103 762-771. 

Yang, W., Parr, R. G., and R. Pucci. 1984. Electron density, Kohn-Sham frontier orbitals, and Fukui functions. J. Chem. Phys. 81 2862-2863. 

Equation (27d) states that the kernel Jf(r,r ) is asymptotically equal to the Hartree-Kohn-Sham static dielectric function. Thus the expression in Eq. (20) for the Fukui function is just a short-range linear mapping of the frontier density, and the expression in Eq. (21) for the local softness is the same mapping of the local DOS. It is the frontier-orbital density which drives the chemical response measured by the Fukui function, and the local DOS which drives that measured by the local softness. 

On the other hand, there is a discrepancy between the predictions of the Kohn-Sham theory of Sect. 2 and of the direct ensemble approach of the present section for the electrophilic and nucleophilic Fukui functions and for the chemical potential. Direct use of the ensemble of Eqs. (28) yields results for f (r), Eq. (34a), and for n, Eq. (30a), which are independent of JT in the range... 

It was pointed out in [2,3] that nuclear-configuration changes define chemical reactions so that nuclear reactivities should be defined and set on equal footing with the corresponding electronic reactivities. Thus nuclear Fukui functions , Eq. (59), and nuclear softnesses a Eq. (60), were defined in [2], and explicit Kohn-Sham expressions were found for them, Eqs. (61H64), as reviewed in Sect. 5. These are electron-transfer reactivities and are valid only for extended systems, leaving open the question of nuclear electron-transfer reactivities for localized systems and nuclear isoelectronic reactivities for all systems. 

Density functional theory (DFT) provides an efficient method to include correlation energy in electronic structure calculations, namely the Kohn-Sham method 1 in addition, it constitutes a solid support to reactivity models.2 DFT framework has been used to formalize empirical reactivity descriptors, such as electronegativity,3 hardness4 and electrophilicity index.5 The frontier orbital theory was generalized by the introduction of Fukui function,6 and new reactivity parameters have also been proposed.7,8 Moreover, relationships between those parameters have been found, and general methods to relate new quantities exist.9... 

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] ... 

Still another way of using DFT, which does not depend directly on approximate solution of Kohn-Sham equations, is the quantification and clarification of traditional chemical concepts, such as electronegativity [6], hardness, softness, Fukui functions, and other reactivity indices [6, 175], or aromaticity [176]. The true potential of DFT for this kind of investigation is only beginning to be explored, but holds much promise. 

This says that /(r) is the functional derivative (section 7.2.3.2, The Kohn-Sham equations) of the chemical potential with respect to the external potential (i.e. the potential caused by the nuclear framework), at constant electron number and that it is also the derivative of the electron density with respect to electron number at constant external potential. The second equality shows /(r) to be the sensitivity of p(r) to a change in N, at constant geometry. A change in electron density should be primarily electron withdrawal from or addition to the HOMO or LUMO, the frontier orbitals of Fukui [114], hence the name bestowed on the function by Parr and Yang. Since p(r) varies from point to point in a molecule, so does the Fukui function. Parr and Yang argue that a large value of fix) at a site favors reactivity at that site, but to apply the concept to specific reactions they define three Fukui functions ( condensed Fukui functions [80]) ... 

The Kohn-Sham (KS) theory for sybsystems (10,11,14, 15), a direct analog of the corresponding Hartree-Fock approach (16,17), which can be used to define the chemical potential, hardness/softness or the Fukui function (FF) characteristics of reactants (10,11), involves the so called nonadditive kinetic energy functional of the noninteracting system (14), the structure of which is still little understood. Similar nonadditive contributions to the exchange-correlation energy may play a vital role in the DFT treatment of the reactive systems, van der Waals... 

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