Fukui function electronic

Electronegativity, Hardness, Softness and the Fukui Function Electron Density Reactivity Indexes... 

We have used transferable atom equivalent (TAE) descriptors [116,117] that encode the distributions of electron density based molecular properties, such as kinetic energy densities, local average ionization potentials, Fukui functions, electron density gradients, and second derivatives as well as the density itself. In addition autocorrelation descriptors (RAD) were used and represent the molecular geometry characteristics of the molecules, while they are also canonical and independent of 3D coordinates. The 2D descriptors alone or in combination with the latter 3D descriptors were calculated for 26 data sets collated by us from numerous publications. These data sets encompass various ADME/TOX-related enzymes, transporters, and ion channels as... 

A reaction will in general involve a change in the electron density, which may be quantified in terms of the Fukui function. ... 

The Fukui function indicates the change in the electron density at a given position when the number of electrons is changed. We may define two finite difference versions of the function, corresponding to addition or removal of an electron. 

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. ... 

Yang, W., and R. G. Parr. 1985. Hardness, softness and the Fukui function in the electronic theory of metals and clusters. Proc. Natl. Acad. Sci. USA 82, 6723. 

It is important to mention that the chemical potential and the hardness, p, and 17, are global-type response functions that characterize the molecule as a whole, while the electronic density p(r), the Fukui function fir), and the dual descriptor A/(r) are local-type response functions whose values depend upon the position within the molecule. 

A common simplification of the Fukui function is to condense its values to individual atoms in the molecule [33]. That is, through the use of a particular population analysis, one can determine the number of electrons associated with every atom in the molecule. The condensed Fukui functions is then determined... 

For the second derivative of the electronic density with respect to the number of electrons, the dual descriptor, one can proceed as in the case of the energy. That is, the Fukui function using the Heaviside function [25] is written as... 

Now, the Fukui function is closely related to the frontier orbitals. This can be seen from Equations 2.29 and 2.30, together with Equation 2.42, because if one determines the electron densities of the iV0 — 1- and the N() + 1-electron systems with the orbitals set corresponding to the Mj-electron system, then... 

The Fukui function is primarily associated with the response of the density function of a system to a change in number of electrons (N) under the constraint of a constant external potential [v(r)]. To probe the more global reactivity, indicators in the grand canonical ensemble are often obtained by replacing derivatives with respect to N, by derivatives with respect to the chemical potential /x. As a consequence, in the grand canonical ensemble, the local softness sir) replaces the Fukui function/(r). Both quantities are thus mutually related and can be written as follows ... 

Once again, due to the discontinuity of the electron density with respect to N, finite difference approximation leads to three types of Fukui function for a system, namely (l)/+(r) for nucleophilic attack measured by the electron density change following addition of an electron, (2)/ (r) for electrophilic attack measured by the electron density change upon removal of an electron, and (3)/°(r) for radical attack approximated as the average of both previous terms. They are defined as follows ... 

Using one-electron orbital picture, Fukui functions can be approximately defined as... 

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] ... 

One possible solution of this problem is to differentiate a radical first as electrophilic or nucleophilic with respect to its partner, depending upon its tendency to gain or lose electron. Then the relevant atomic Fukui function (/+ or / ) or softness f.v+ or s ) should be used. Using this approach, regiochemistry of radical addition to heteratom C=X double bond (aldehydes, nitrones, imines, etc.) and heteronuclear ring compounds (such as uracil, thymine, furan, pyridine, etc.) could be explained [34], A more rigorous approach will be to define the Fukui function for radical attack in such a way that it takes care of the inherent nature of a radical and thus differentiates one radical from the other. 

The Fukui function, denoted by fir), is defined as the differential change in electron density due to an infinitesimal change in the number of electrons [1], That is,... 

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. 

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. 

Why—and when—does the Fukui function work The first restriction—already noted in the original 1984 paper—is that the Fukui function predicts favorable interactions between molecules that are far apart. This can be understood because when one uses the perturbation expansion about the separated reagent limit to approximate the interaction energy between reagents, one of the terms that arises is the Coulomb interaction between the Fukui functions of the electron-donor and the electron-acceptor [59,60],... 

This term can only control regioselectivity if the transition state occurs relatively early along the reaction path (so that the asymptotic expansion about the separated reagent limit is still relevant) and if the extent of electron-transfer is large compared to the electrostatic interactions between the reagents. The importance of Equation 18.27 for explaining the utility and scope of the Fukui function was first noted by Berkowitz in 1987 [59]. 

The utility of the Fukui function for predicting chemical reactivity can also be described using the variational principle for the Fukui function [61,62], The Fukui function from the above discussion, /v (r), represents the best way to add an infinitesimal fraction of an electron to a system in the sense that the electron density pv/v(r) I has lower energy than any other N I -electron density... 

This quantity is trivially computed from the Fukui function, / (r) [78-80], and the shape function, and it has a simple interpretation the shape Fukui function measures where the relative abundance of electrons increases or decreases when electrons are added to (or removed from) a system. In our experience, plotting r) often provides a simpler and easier way to interpret picture of chemical reactivity than the Fukui function itself. Perhaps this is because ofr) is the local density approximation (LDA) to the Fukui function [81]. Since the numerator in Equation 19.27, / (r) — olr), is the post-LDA correction to the Fukui function [81], the shape... 

Fukui function represents the deviation of the system s response to adding or subtracting electrons from electron gas behavior. 

In a second approach of the reactivity, one fragment A is represented by its electronic density and the other, B, by some reactivity probe of A. In the usual approach, which permits to define chemical hardness, softness, Fukui functions, etc., the probe is simply a change in the total number of electrons of A. [5,6,8] More realistic probes are an electrostatic potential cf>, a pseudopotential (as in Equation 24.102), or an electric field E. For instance, let us consider a homogeneous electric field E applied to a fragment A. How does this field modify the intermolecular forces in A Again, the Hellman-Feynman theorem [22,23] tells us that for an instantaneous nuclear configuration, the force on each atom changes by... 

Polarizabilities are responses to a potential (the gradient of which is a field). On the contrary, Fukui functions, chemical hardness and softness are responses to a transfer or removal of an integer number of electrons. Both responses are DFT descriptors but the responses which involve a change in the number of... 

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