Density Functional Electronegativity

The essence of the density functional theory consists in writing the total energy of an iV-electronic system as a function of electronic density of the fundamental state associated to the system 

The univocity of the relation between the external potential applied to the electronic system and the electronic density is provided by the Hohenberg-Kohn Theorems, see the Volume I of this five-volume book dedicated to quantum nanochemistry (Putz, 2016a). Moreover, one of the theorems also states the inequality relation between a density functional energy of any electronic state, E[p and the true functional energy of the fundamental electronic state of the system, i [p, namely as 

For the true/optimized density of the fundamental state, the multiplier works as the chemical potential associated to the system and is 

Quantum Nanochemistry—Volume III Quantum Molecules and Reactivity 

Applying the finite-difference approach for the system states variation by an electron acquirement (i.e., producingE -EA withEA the electronic affinity) and respectively towards the ionized one (having E + IP with IP the ionization potential) respecting the neutral system energy E, the so called Mulliken electronegativity expression is obtained for the chemical neutral (in equilibrium) systems 

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After such an extensive calculation, required for the determination of the density functional electronegativity values, one has in general sufficient detailed information about the system and its observables, that the electronegativity, useful for qualitative considerations, does not seem to be needed anymore. 

As it was derived in the previous Section 4.6.5 the Mulliken density functional electronegativity requires the knowledge of the electronic density under the external potential influence. Being exposed all the ingredients for the analytical expression for the partition function with only the external potential dependence, the electronic density computed through out of Feymnan-Kleinert path integral algorithm takes the form, see also Eq. (2.11) ... 

Putz, M. V., Russo, N., Sicilia, E. (2003). Atomic radii scale and related size properties from density functional electronegativity formulation. J. Phys. Chem. A 107(28), 5461-5465(001 10.1021/jp027492h). 

TABLE 3.3 Hardness Values for the Atomic Species Used Computed Both By Considering (Putz) Density Functional Electronegativity (DFT) (Ihitz et al., 2003) and Ghosh Biswas (2002) (GB) Atomic Radii for the Putz-DFT Chemical Hardness Considered, i.e.. Based on Density Functional Electronegativity (DFE)... 

FIGURE 3.5 Graphical correlation between the values of the molecular chemical hardness t] and the maximum hardness index Y for the Lewis acids of Table 3.1 employing the softness-, the second- and the fourth- order density functional electronegativity -DFE and Ghosh-Biswas-GB based atomic radii values of chemical hardness of Tables 3.4-3.7, from the top to bottom, in left and right sides on the draws, respectively (Putz, 2008c). 

The second observation is that unlike the acids previous cases the actual bases ordering looks quite similar for all employed finite difference, Ghosh-Biswas and density functional electronegativity recipes. Therefore, the present suggested soft-to-hard Lewis bases classification is that recommended by DFEP hierarchy - rooting on the most complex conceptual-computational containing algorithm. 

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. ... 

There have, however, been attempts to correlate Q-e values and hence reactivity ratios to, for example, c NMR chemical shifts 50 or the results of MO calculations 51153 and to provide a better theoretical basis for the parameters. Most recently, Zhan and Dixon153 applied density functional theory to demonstrate that Q values could be correlated to calculated values of the relative free energy for the radical monomer reaction (PA + Mn — PA ). The e values were correlated to values of the electronegativities of monomer and radical. 

Allen GC, Warren KD (1974) The Electronic Spectra of the Hexafluoro Complexes of the Second and Third Transition Series. 19 105-165 Alonso JA, Baibas LC (1993) Hardness of Metallic Clusters. 80 229-258 Alonso JA, Baibas LC (1987) Simple Density Functional Theory of the Electronegativity and Other Related Properties of Atoms and Ions. 66 41-78 Andersson LA, Dawson JH (1991) EXAFS Spectroscopy of Heme-Containing Oxygenases and Peroxidases. 74 1-40 Antanaitis BC, see Doi K (1988) 70 1-26... 

Alonso, J. A., Baibas, L. C. Simple Density Functional Theory of the Electronegativity and Other Related Properties of Atoms and Ions. Vol. 66, pp. 41-78. 

De Proft, F., Geerlings, P, 1997, Calculation of Ionization Energies, Electron Affinities, Electronegativities and Hardnesses Using Density Functional Methods , 7. Chem. Phys., 106, 3270. 

Nalewajski RF (1998) On the chemical potential/electronegativity equalization in density functional theory. Polish J Chem 72(7) 1763-1778... 

Parr RG, Donnelly RA, Levy M, Palke WE (1978) Electronegativity the density functional viewpoint. J Chem Phys 68(8) 3801-3807... 

Yang ZZ, Wang CS (2003) Atom-bond electronegativity equalization method and its applications based on density functional theory. J Theor Comput Chem 2(2) 273-299... 

As we have seen, an atom under pressure changes its electron structure drastically and consequently, its chemical reactivity is also modified. In this direction we can use the significant chemical concepts such as the electronegativity and hardness, which have foundations in the density functional theory [9]. The intuition tells us that the polarizability of an atom must be reduced when it is confined, because the electron density has less possibility to be extended. Furthermore, it is known that the polarizability is related directly with the softness of a system [14], Thus, we expect atoms to be harder than usual when they are confined by rigid walls. Estimates of the electronegativity, x and die hardness, tj, can be obtained from [9]... 

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