Introducing “Colored” Molecular Topology by Reactivity Indices of Electronegativity and Chemical Hardness

Abstract

Within the context of conceptual density functional theory chemical reactivity definitions of electronegativity (EN) and chemical hardness (HD), nine forms of their finite difference are expressed in order to consider the global “coloring” of the molecular topology with respect to their symmetry centers (atomic centers or bonding centers), according to the so-called Timişoara–Parma rule. The resulting parabolic-reactive energy in terms of EN and HD is compared with the bond topological Wiener index for short list of PAH (poly-aromatic hydrocarbons) selected as paradigmatic structures for validating the new reactivity descriptors based on topological quantities.

Dedicated to Professor Ante Graovac, coauthor of this chapter, who prematurely passed away soon after the manuscript was completed

Appendix 9. A Compact Finite Differences for Electronegativity and Chemical Hardness

Given the values of a function f(n) on a set of nodes $\left\{ {\ldots, n{-}3,n-2,n-1,n,n{+}1,} \right.$ $\left.{n+2,n+3,\ldots } \right\}$, the finite difference approximations of the first ${{f^{\prime}}_n}$ and second ${{f^{\prime\prime}}_n}$ derivatives in the node n will spectrally depend on all the nodal values. However, the compact finite differences, or Padé, schemes that mimic this global dependence is written as (Lele 1992)

$$ \begin{array}{llll}[b] {\beta_1}{{{f^{\prime}}}_{n-2 }}& + {\alpha_1}{{{f^{\prime}}}_{n-1 }}+{{{f^{\prime}}}_n}+{\alpha_1}{{{f^{\prime}}}_{i+1 }}+{\beta_1}{{{f^{\prime}}}_{n+2 }} \\ & = {c_1}\frac{{{f_{n+3 }}-{f_{n-3 }}}}{6}+{b_1}\frac{{{f_{n+2 }}-{f_{n-2 }}}}{4}+{a_1}\frac{{{f_{n+1 }}-{f_{n-1 }}}}{2} \end{array} $$

(9.A.1)

$$ \begin{array}{llll} {\beta_2}\,{{{f^{\prime\prime}}}_{n-2 }} &+ {\alpha_2}\,{{{f^{\prime\prime}}}_{n-1 }}+{{{f^{\prime\prime}}}_n}+{\alpha_2}\,{{{f^{\prime\prime}}}_{i+1 }}+{\beta_2}\,{{{f^{\prime\prime}}}_{n+2 }} \\ &= {c_2}\frac{{{f_{n+3 }}{-}2{f_n}{+}{f_{n-3 }}}}{9}\,{+}\,{b_2}\frac{{{f_{n+2 }}{-}2{f_n}{+}{f_{n-2 }}}}{4}\,{+}\,{a_2}\left( {{f_{n+1}}{-}2{f_n}{+}{f_{n-1 }}}\right)\!. \end{array} $$

(9.A.2)

The involved sets of coefficients, $\left\{ {{a_1},{b_1},{c_1},{\alpha_1},{\beta_1}} \right\}$ and $\left\{ {{a_2},{b_2},{c_2},{\alpha_2},{\beta_2}} \right\}$, are derived by matching Taylor series coefficients of various orders. In this way, their particularizations can be reached as the second (2C)-, fourth (4C)-, and sixth (6C)-order central differences; standard Pade (SP) schemes; sixth (6T)- and eight (8T)-order tridiagonal schemes; and eighth (8P)- and tenth (10P)-order pentadiagonal schemes up to spectral-like resolution (SLR) ones; see Table 9.1.

Assuming that the function f(n) is the total energy E(N) in the actual node that corresponds to the number of electrons and the compact finite difference, the derivatives of Eqs. (9.2) and (9.5) may be accurately evaluated through considering the states with N − 3, N − 2, N − 1, N + 1, N + 2, and N + 3 electrons, whereas the derivatives in the neighbor states will be taken only as their most neighboring dependency. This way, the working formulas for electronegativity will be (Putz 2010)

$$ \begin{array}{llll}[b] -\chi & = \frac{{\partial E}}{{\partial N}}\left| {_{{\left| N \right\rangle }}} \right. \\ & \cong {a_1}\frac{{{E_{N+1 }}-{E_{N-1 }}}}{2}+{b_1}\frac{{{E_{N+2 }}-{E_{N-2 }}}}{4}+{c_1}\frac{{{E_{N+3 }}-{E_{N-3 }}}}{6} \\ & \quad - {\alpha_1}\left( {\frac{{\partial E}}{{\partial N}}\left| {_{{\left| {N-1} \right\rangle }}} \right.+\frac{{\partial E}}{{\partial N}}\left| {_{{\left| {N+1} \right\rangle }}} \right.}\! \right)-{\beta_1}\left( {\frac{{\partial E}}{{\partial N}}\left| {_{\left| {N-2} \right\rangle}} \right.+\frac{{\partial E}}{{\partial N}}\left| {_{\left| {N+2} \right\rangle}} \right.} \!\right) \\ & = {a_1}\frac{{{E_{N+1 }}-{E_{N-1 }}}}{2}+{b_1}\frac{{{E_{N+2 }}-{E_{N-2 }}}}{4}+{c_1}\frac{{{E_{N+3 }}-{E_{N-3 }}}}{6} \\ & \quad - {\alpha_1}\left( {{a_1}\frac{{{E_N}-{E_{N-2 }}}}{2}+{a_1}\frac{{{E_{N+2 }}-{E_N}}}{2}} \right)\\ & \quad -{\beta_1}\left( {{a_1}\frac{{{E_{N-1 }}-{E_{N-3 }}}}{2}+{a_1}\frac{{{E_{N+3 }}-{E_{N+1 }}}}{2}} \right) \\ & = {a_1}\left( {1+{\beta_1}\,} \right)\frac{{{E_{N+1 }}-{E_{N-1 }}}}{2}+\left( {{b_1}-2{a_1}{\alpha_1}} \right)\frac{{{E_{N+2 }}-{E_{N-2 }}}}{4}\\ & \quad +\left( {{c_1}-3{a_1}{\beta_1}} \right)\frac{{{E_{N+3 }}-{E_{N-3 }}}}{6} \end{array} $$

(9.A.3)

and respectively for the chemical hardness as (Putz et al. 2004)

$$ \begin{array}{llll}[b] 2\eta & = \frac{{{\partial^2}E}}{{\partial {N^2}}}\left| {_{{\left| N \right\rangle }}} \right. \\ & \cong 2{a_2}\frac{{{E_{N+1 }}{-}2{E_N}{+}{E_{N-1 }}}}{2}\,{+}\,{b_2}\frac{{{E_{N+2 }}{-}2{E_N}{+}{E_{N-2 }}}}{4}{+}{c_2}\frac{{{E_{N+3 }}{-}2{E_N}{+}{E_{N-3 }}}}{9} \\ & \quad - {\alpha_2}\left( {\frac{{{\partial^2}E}}{{\partial {N^2}}}\left| {_{{\left| {N-1} \right\rangle }}} \right.+\frac{{{\partial^2}E}}{{\partial {N^2}}}\left| {_{{\left| {N+1} \right\rangle }}} \right.} \! \right)-{\beta_2}\left( {\frac{{{\partial^2}E}}{{\partial {N^2}}}\left| {_{{\left| {N-2} \right\rangle }}} \right.+\frac{{{\partial^2}E}}{{\partial {N^2}}}\left| {_{{\left| {N+2} \right\rangle }}} \right.} \!\right) \\ \end{array} $$

$$ \begin{array}{llll} {} & = 2{a_2}\frac{{{E_{N+1 }}{-}2{E_N}{+}{E_{N-1 }}}}{2}\,{+}\,{b_2}\frac{{{E_{N+2 }}{-}2{E_N}{+}{E_{N-2 }}}}{4}{+}{c_2}\frac{{{E_{N+3 }}{-}2{E_N}{+}{E_{N-3 }}}}{9} \\ & \quad - {\alpha_2}\left( {2{a_2}\frac{{{E_N}-2{E_{N-1 }}+{E_{N-2 }}}}{2}+2{a_2}\frac{{{E_{N+2 }}-2{E_{N+1 }}+{E_N}}}{2}} \right) \\ & \quad - {\beta_2}\left( {2{a_2}\frac{{{E_{N-1 }}-2{E_{N-2 }}+{E_{N-3 }}}}{2}+2{a_2}\frac{{{E_{N+3 }}-2{E_{N+2 }}+{E_{N+1 }}}}{2}} \right) \\ & = 2{a_2}\left( {1+2{\alpha_2}-{\beta_2}} \right)\frac{{{E_{N+1 }}+{E_{N-1 }}}}{2}{+}\left( {8{a_2}{\beta_2}+{b_2}-4{a_2}{\alpha_2}} \right)\frac{{{E_{N+2 }}+{E_{N-2 }}}}{4} \\ & \quad + \left( {{c_2}-9{a_2}{\beta_2}} \right)\frac{{{E_{N+3 }}+{E_{N-3 }}}}{9}-\left( {2{a_2}+\frac{1}{2}{b_2}+\frac{2}{9}{c_2}+2{a_2}{\alpha_2}} \right){E_N}\end{array} $$

(9.A.4)

where the involved parameters discriminate between various schemes of computations and the spectral-like resolution – SLR (Lele 1992).

Next, Eqs. (9.A.3) and (9.A.4) may be rewritten in terms of the observational quantities, as the ionization energy and electronic affinity are with the aid of their basic definitions from the involved eigen-energies of ith (i = 1,2,3) order:

$$ {I_i}={E_{N-i }}-{E_{N-i+1 }} $$

(9.A.5)

$$ {A_i}={E_{N+i-1 }}-{E_{N+i }} $$

(9.A.6)

As such they allow the energetic equivalents for the differences (Putz 2010)

$$ {E_{N+1 }}-{E_{N-1 }}=-\left( {{I_1}+{A_1}} \right) $$

(9.A.7)

$$ {E_{N+2 }}-{E_{N-2 }}=-\left( {{I_1}+{A_1}} \right)-\left( {{I_2}+{A_2}} \right) $$

(9.A.8)

$$ {E_{N+3 }}-{E_{N-3 }}=-\left( {{I_1}+{A_1}} \right)-\left( {{I_2}+{A_2}} \right)-\left( {{I_3}+{A_3}} \right) $$

(9.A.9)

and for the respective sums (Putz et al. 2004)

$$ {E_{N+1 }}+{E_{N-1 }}=\left( {{I_1}-{A_1}} \right)+2{E_N} $$

(9.A.10)

$$ {E_{N+2 }}+{E_{N-2 }}=\left( {{I_1}-{A_1}} \right)+\left( {{I_2}-{A_2}} \right)+2{E_N} $$

(9.A.11)

$$ {E_{N+3 }}+{E_{N-3 }}=\left( {{I_1}-{A_1}} \right)+\left( {{I_2}-{A_2}} \right)+\left( {{I_3}-{A_3}} \right)+2{E_N} $$

(9.A.12)

being then implemented to provide the associate “spectral” molecular analytical forms of electronegativity (Putz 2010)

$$ \begin{array}{llll}[b] {\chi_{\mathrm{CFD}}}&=\left[ {{a_1}\left( {1-{\alpha_1}} \right)+\frac{1}{2}{b_1}+\frac{1}{3}{c_1}} \right]\frac{{{I_1}+{A_1}}}{2} \\ & \quad +\left[ {{b_1}+\frac{2}{3}{c_1}-2{a_1}\left( {{\alpha_1}+{\beta_1}} \right)} \right]\frac{{{I_2}+{A_2}}}{4} \\ & \quad +\left( {{c_1}-3{a_1}{\beta_1}} \right)\frac{{{I_3}+{A_3}}}{6} \end{array} $$

(9.A.13)

and for chemical hardness (Putz et al. 2004)

$$ \begin{array}{llll}[b] {\eta_{\mathrm{CFD}}} & =\left[ {{a_2}\left( {1-{\alpha_2}+2{\beta_2}} \right)+\frac{1}{4}{b_2}+\frac{1}{9}{c_2}} \right]\frac{{{I_1}-{A_1}}}{2} \\& \quad +\left[ {\frac{1}{2}{b_2}+\frac{2}{9}{c_2}+2{a_2}\left( {{\beta_2}-{\alpha_2}} \right)} \right]\frac{{{I_2}-{A_2}}}{4} \\ & \quad +\left[ {\frac{1}{3}{c_2}-3{a_2}{\beta_2}} \right]\frac{{{I_3}-{A_3}}}{6} \end{array} $$

(9.A.14)

It is worth remarking that when particularizing these formulas for the fashioned two-point central finite difference, i.e., when having ${a_1}=1,{b_1}={c_1}={\alpha_1}={\beta_1}=0$ and ${a_2}=1,\,{b_2}={c_2}={\alpha_2}={\beta_2}=0$ of Table 9.1, one recovers the consecrated Mulliken (spectral) electronegativity (Mulliken 1934)

$$ {\chi_{2\mathrm{C}}}=\frac{{{I_1}+{A_1}}}{2} $$

(9.A.15)

and the chemical hardness basic form relating with the celebrated Pearson nucleophilic–electrophilic reactivity gap (Pearson 1997; Parr and Yang 1989)

$$ {\eta_{2\mathrm{C}}}=\frac{{{I_1}-{A_1}}}{2} $$

(9.A.16)

Finally, for computational purposes, formulas (9.A13) and (9.A14) may be once more reconsidered within the Koopmans’ frozen core approximation (Koopmans 1934), in which various orders of ionization potentials and electronic affinities are replaced by the corresponding frontier energies

$$ {I_i}=-{\varepsilon_{{\mathrm{HOMO}\;(i)}}} $$

(9.A.17)

$$ {A_i}=-{\varepsilon_{{\mathrm{LUMO}\;(i)}}} $$

(9.A.18)

so that the actual working compact finite difference (CFD) orbital molecular electronegativity and chemical hardness unfold as given in Eqs. (9.6) and (9.7), respectively.

Note that the actual CFD electronegativity and chemical hardness expressions do not distinguish for the atoms-in-molecule contributions, while providing post-bonding information and values, i.e., for characterizing the already stabilized/optimized molecular structure towards its further reactive encountering.