THE EVALUATION OF PROTONATION ENERGY OF MOLECULES IN TERMS OF QUANTUM THEORETICAL DESCRIPTORS - Theoretical and Computational Research in the 21st Century - page 219

10.9 MATHEMATICAL FORMULAE OF THE GLOBAL
REACTIVITY DESCRIPTORS INVOKED IN THE STUDY

According to Koopmans' theorem, the ionization potential (I) and the elec-
tron affinity (A) are computed as follows:

I = - ε HOMO (15)

A = - ε LUMO (16)

where ε HOMO and ε LUMO are the orbital energies of the highest occupied and
the lowest unoccupied orbitals.
Parr et al. [89,90] defi ned the chemical potential, μ , electronegativity,
χ , and hardness, η , in the framework of DFT [91] as follows:

μ = (∂E/∂N) v(r) = -χ = (I+A)/2 (17)

η = ½ [∂μ/∂N] v(r) = ½ [∂ 2 E/∂N 2 ] v(r) = ½ (I-A) (18)

where E , N , v (r), I, and A are the energy, the number of electrons, the exter-
nal potential, the ionization energy, and the electron affinity of an atomic
or molecular system, respectively.
Softness is a reactivity index and is defi ned as the reciprocal of hard-
ness:

S = (1/ η ) (19)

Parr et al. [92] defined electrophilicity index (ω) as follows:

ω = ( μ ) 2 /(2 η ) (20)

In this study, we have taken some hydrocarbons as Set 1; some alcohols,
carbonyls, carboxylic acids and esters as Set 2; some aliphatic amines
as Set 3; some aromatic amines as Set 4; some pyridine derivatives as
Set 5; and some amino acids as Set 6 for which the experimental proton-
ation [18,93-96] energy are known. The PQS Mol 1.2-20-win software
(PQSMol) have been used to calculate the global descriptors by using the
ab initio HF SCF method with the 6-31g basis set. The geometry optimiza-
tion technique is adopted. The ionization energy, I , the electronegativity,