Reactivity index orbitals

Density functional orbital reactivity indices. Fundamentals and applications. 

Equations 29-37 provide a practical scheme to compute orbital reactivity indices, in vicinity of the system equilibrium point in the parametric space of the KS-eigenvalues and occupation numbers. 

TABLE 4.15 The Training (Calibration) and Test (Marked With Asterisk - to be chosen) Compounds Studied Along Their HOMA (Harmonic Oscillator Model of Aromatic) Index (Mosquera et al., 2007 Ciesielski et al., 2009) and of Associated Computed (Hypercube, 2002 (Semiempirical, AMI, Polak-Ribier optimization procedure)) Structural First, Second, and Third Order HOMO (Highest Occupied Molecular Orbital) and LUMO (Lowest Unoccupied Molecular Orbital) Reactivity Indices (Tarko Putz, 2010)... 

The first mixed derivative is the Fukui function f(r) [28], a frontier molecular orbital reactivity index ... 

The contribution of the frontier orbitals would be maximized in certain special donor-acceptor reactions. The stabilization energy is represented by Eqs. (3.25) and (3.26). Even in a less extreme case, the frontier orbital contribution maybe much more than in the expression of the superdelocalizability. If we adopt the approximation of Eq. (6.3), the intramolecular comparison of reactivity can be made only by the numerator value. In this way, it is understood that the frontier electron density, /r, is qualified to be an intramolecular reactivity index. The finding of the parallelism between fr and the experimental results has thus become the origin of the frontier-electron theory. The definition of fr is hence as follows ... 

Brown (1959) has presented a charge transfer model of the transition state for electrophilic reactions which differs appreciably from that proposed by Fukui and his collaborators and leads to the definition of a new reactivity index termed the Z value . The model is based on a more conventional formulation of the charge transfer mechanism, which avoids the complete transfer of electrons associated with v = 0,1,2 in Fukui s model. There is no dependence on the formation of a pseudo tt orbital in the transition state, nor is hyperconjugation invoked. A wave function for a charge transfer complex is written as a linear combination of a wave function < o describing the unperturbed ground state of the molecule under attack, and a function which differs from (Pq in the replacement... 

Graphical Models are introduced and illustrated in Chapter 4. Among other quantities, these include models for presentation and interpretation of electron distributions and electrostatic potentials as well as for the molecular orbitals themselves. Property maps, which typically combine the electron density (representing overall molecular size and shape) with the electrostatic potential, the local ionization potential, the spin density, or with the value of a particular molecular orbital (representing a property or a reactivity index where it can be accessed) are introduced and illustrated. 

The last step in this argument, the frontier orbital approximation , is evidently of doubtful validity, and it is not surprising that the frontier electron density is not a reliable reactivity index, though at least one example of its failure 76 >84 has been disputed85 as attributable to an unsuitable choice of parameters. Since neither author gave his parameters, it is difficult to judge the validity of the criticism. We shall, however, return to the question of the frontier orbital approximation later. 

An electronic parameter that often correlates with metabolic rates is the electrophilic (or nucleophilic) superdelocalizability. This quantity is a reactivity index formulated by Fukui and colleagues as an orbital-weighted electron density.145 The total electrophilic superdelocalizability, 2SE, summed over all atoms in a molecule, exhibits a parallelism with the hydrophobic parameter, log P, in several series of compounds such as PAHs and aliphatic amines, where it is probably approximating molecular volume. 

For benzo[. ]thiophene, there is a it-MO delocalization between the two aromatic rings. For benzo[r]thiophene, the orbitals of the five-membered rings are localized on the heteroatom, C-1 and C-3, and there is no it-MO delocalization on the heterocyclic five-membered rings. These results are in agreement with the theoretical aromaticity of these molecules as are theoretical results from the reactivity indexes. 

There are many ways to measure the reactivity index, and all of them are feasible in such a study. By finding one that works—i.e., gives significant statistics in Equation 1—statements may be formulated regarding possible mechanisms. In this type of work, it must be remembered that it is not valid to imply causality to a correlation. However, from a pragmatic point of view, it is possible to set up models that can be tested by further synthesis. The measurements used for the reactivity index in the following examples include hydrolysis constants and molecular orbital calculations. 

A reactivity index suitable for use in Equation 1 was calculated by using the simple molecular orbital techniques described by the Pullmans (14). Many indexes may be deduced from this type of procedure. The one that seemed to have the most significance for the correlation was the energy of the highest occupied molecular orbital (HOMO). This index is a relative measure of the ability of an electron to be transferred to an acceptor molecule. The calculations were performed on the substituted phenol in the imidazoline structure. This simplification was made since it could be assumed that any perturbation caused by the imidazole would be insulated from the rest of the molecule by the methylene group. 

Up to a few years ago chemical reactivity was discussed in term of reactivity indexes. These approaches, although valuable, will not be discussed here, since they have been frequently reviewed in the past40-44). Nor will we discuss the perturbation molecular orbital theory for reactants, which has been the subject of extensive reviews 45—47) Extensions of this method can be found in papers by Klopman 48 5°) and Dougherty 51). I shall now mention some methods which have not yet found wide popularity but seem very promising. I mean the criterion of maxi-... 

Abstract. The development of theories for interpreting the course of chemical reactions is one of the most important achievements of theoretical chemistry in the twentieth century. I selected the paper by Fukui et al. from 1952, proposing the frontier electron density as the reactivity index for the orientation of electrophilic substitution reactions. This paper may be regarded as a bridge between an older reactivity theory, the electronic theory of organic chemistry, and new ones predicting the stereochemical courses of reactions such as frontier orbital theory and the Woodward-Hoffmann rule. 

In many instances, the relaxation terms, <]>, can be ignored, and the frontier molecular orbital theory as first put forth by Fukui - is recovered. For particular molecules, the identification reactive sites using f(t) has been successful.A convenient way to visualize the reactive sites in a molecule using the above reactivity index is to first display an isosurface of the electron density that just encloses the van der JKials volumes of the individual atoms in the molecule. Typically, the value of this isosurface is between 0.002 and 0.005. Next, the values of the reactivity index are mapped upon this surface and color coded from blue (zero) to ted (most positive). 

An intrinsic reactivity index (IRI) has been developed, with a view to capturing electro-and nucleophilicity on a single scale, and using frontier molecular orbital data to access values. A correlation of IRI with Mayr s E and N parameters is also described. 

The electrophilic, ambiphilic and nucleophilic characters of a range of singlet carbenes have been compared to their intrinsic reactivity index This quantification is supposed to evaluate the nucleophilic and electrophilic properties of compounds and has been previously assessed for a range of organic compounds but appears unadapted to singlet carbene and presents no advantages compared to frontier molecular orbital and empirical evaluations. 

Next, we consider the first order mixed derivatives of the energy with respect to electron and spin number on one hand and the external potentials v and V5 on the other hand. These derivatives are generalizations of the Fukui function, the reactivity index used to probe frontier orbital-controlled reactions. The generalized Fukui function /jvjv can be considered as the spin-resolved extension of the regular Fukui function given in eqn (26)... 

Now, it seems natural to compare the two stages of a molecular bonding, for a given reactivity index, which through their difference should reveal the excess chemical information responsible for the stability of that molecular system. In other words, by subtracting the already formed molecular orbital (MO) information from that obtained by superposition of atomic information in bonding H the present discussion follows (Putz, 2010b)... 

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