Density functional theory Fukui function

In the frozen MO approximation the last terms are zero and the Fukui functions are given directly by the contributions from the HOMO and LUMO. The preferred site of attack is therefore at the atom(s) with the largest MO coefficients in the HOMO/LUMO, in exact agreement with FMO theory. The Fukui function(s) may be considered as the equivalent (or generalization) of FMO methods within Density Functional Theory (Chapter 6). 

Besides the already mentioned Fukui function, there are a couple of other commonly used concepts which can be connected with Density Functional Theory (Chapter 6). The electronic chemical potential p is given as the first derivative of the energy with respect to the number of electrons, which in a finite difference version is given as half the sum of the ionization potential and the electron affinity. Except for a difference in sign, this is exactly the Mulliken definition of electronegativity. ... 

Quantum mechanical approaches have been successfully used to predict hydrogen abstraction potentials and likely sites of metabolism of drug molecules [78-81]. AMI, Fukui functions, and density functional theory calculations could identify potential sites of metabolism. Activation energies for hydrogen abstraction were calculated by Olsen et al. [81] to be below 80 kj/mol, suggesting most CH groups can be metabolized which particular one depends on steric accessibility and intrinsic reactivities. 

When a molecule accepts electrons, the electrons tend to go to places where/1 (r) is large because it is at these locations that the molecule is most able to stabilize additional electrons. Therefore a molecule is susceptible to nucleophilic attack at sites where/ "(r) is large. Similarly, a molecule is susceptible to electrophilic attack at sites where f (r) is large, because these are the regions where electron removal destabilizes the molecule the least. In chemical density functional theory (DFT), the Fukui functions are the key regioselectivity indicators for electron-transfer controlled reactions. 

Conceptual density functional theory (DFT) [1-7] has been quite successful in explaining chemical bonding and reactivity through various global and local reactivity descriptors as described in the previous chapters. The Fukui function (FF) [4,5] is an important local reactivity descriptor that is used to describe the relative reactivity of the atomic sites in a molecule. The FF [4,5] is defined as... 

Fukui, K. 1987. Role of frontier orbitals in chemical reactions. Science 218 747-754. Geerlings, P., De Proft, F., and W. Langenaeker. 2003. Conceptual density functional theory. Chem. Rev. 103 1793-1873. 

Parr and collaborators [8-12] showed how Fukui s frontier-orbital concept could be grounded in a rigorous many-electron theory, density-functional theory (DFT) [8,16-18], They used the ensemble formulation of DFT to introduce the expectation value Jf of the total electron number as a continuous variable. They then defined the Fukui functions... 

We have now seen that the effort of Parr and collaborators [8-12] to put Fukui s frontier-orbital concept of chemical reactivity on sound footing in density-functional theory through the definition of the Fukui function and the local and global softness works only for extended systems. This restriction to extended systems raises a sixth issue. In both the local softness and the Fukui function, Eqs. (54) and (53a), the orbitals at the chemical potential represent both the LUMO and the HOMO in the Fukui sense. However, there is a continuum of unoccupied KS states above the chemical potential accessible even to weak chemical perturbations any linear combination of which could in principle be selected as the LUMO, and similarly for states below fi and the HOMO. This ambiguity in the frontier-orbital concept obviously applies as well to localized systems when there is more than one KS state significantly affected by a chemical perturbation. 

The Fukui function f(r) is a local electronic descriptor of reactivity which finds its origin within density-functional theory (DFT) and is defined as [Parr and Yang, 1989] ... 

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... 

None of the above results addresses the question of the position of reaction in the aromatic molecule. In DFT this is done by considering the Fukui function,/. It is interesting that this orientation problem was also the topic of the first paper on frontier orbital theoryReaction was predicted to occur at the position of highest frontier orbital (FO) electron density. The frontier orbital in electrophilic substitution would be the HOMO. If this orbital were written as the usual linear combination of atomic orbitals, then the density at each atom would simply be the square of the coefficient in the LCAO, or c where i indicates the atom. Since this is also one of the ways of approximating / the success of the FO method may also be claimed for DFT. However, the details of the Fukui function application will be postponed briefly to look at a method unique to density functional theory, and using the concept of hardness. ... 

DFT-based descriptors are those derived from the Density Functional Theory and the most important are chemical potential, molecular electronegativity, hardness and softness indices, and Fukui functions. 

The Fukui functions f(r) are local electronic descriptors of reactivity that find their origin within Density Functional Theory (DPT) and are defined as [Fukui, 1982 Parr and Yang, 1989]... 

Popular qualitative chemical concepts such as electronegativity [1] and hardness [2] have been widely used in understanding various aspects of chemical reactivity. A rigorous theoretical basis for these concepts has been provided by density functional theory (DFT). These reactivity indices are better appreciated in terms of the associated electronic structure principles such as electronegativity equalization principle (EEP), hard-soft acid-base principle, maximum hardness principle, minimum polarizability principle (MPP), etc. Local reactivity descriptors such as density, Fukui function, local softness, etc., have been used successfully in the studies of site selectivity in a molecule. Local variants of the structure principles have also been proposed. The importance of these structure principles in the study of different facets of medicinal chemistry has been highlighted. Because chemical reactions are actually dynamic processes, time-dependent profiles of these reactivity descriptors and the dynamic counterparts of the structure principles have been made use of in order to follow a chemical reaction from start to finish. 

In order to understand the importance of frontier orbitals in chemical reactivity, Berkowitz [213] studied the frontier-controlled reactions within the purview of density functional theory. It is evident that the directional characteristics of frontier orbitals determine the extent of charge transfer, and soft-soft interactions are frontier-con-trolled. A somewhat similar analysis showed that charge transfer would be facilitated at a place where the difference in local softness of two partners is large [87], It may be noted that Fukui function is obtainable from local softness but the reverse is not true. On the other hand, local hardness suffers from the drawback of ambiguity [87], which allows one to even consider it to be equal to global hardness without disturbing their... 

Thus, one can see that within the framework provided by density functional theory, the basic equations for the description of a chemical event, Eqs. (4) and (7), may be expressed in terms of basic variables such as the chemical potential (electronegativity), the chemical hardness and the fukui function (frontier orbitals). In fact, through this approach one may introduce a coherent quantitative language of hardness and softness functions which are nonlocal, local, and global [29]. The global softness is given by... 

Fukui Function, Local Reactivity, Cytochrome, Compound I, Metabolism, Conceptual Density Functional Theory, Fmodepside, Chlorpyrifos, Parathion... 

There has also been considerable interest in the theoretical analysis of the relative reactivity of the position on the indole ring. One of the quantities that can be calculated is the condensed Fukui function [f ], which, in the context of density function theory, provides a measure of respruise to an approaching electrophile [8]. For indole, the 1-, 2-, and 3-positi(Mis are calculated as 0.08, 0.05, and 0.18, consistent with the observed preference for substitution at C-3 [9]. 

The applications of local quantities start with the use of the Fukui function in the frontier-electron theory of chemical reactivity within a density functional framework [19]. In this approach there are three different types of Fukui functions, viz.,... 

Yet another series of MQSM can be derived from the field of study called conceptual density functional theory. In this field, many concepts are obtained that may be written as derivatives of the electron density. For a recent overview of this field, one may consult the review by Geerlings et al. or the classic textbook by Parr and Yang." One example of such a derivative is the Fukui function, defined as... 

---