Computational methods
All the calculations were carried out with the Gaussian 16 program package,22 employing the M06-2X and the 6-31G(d,p) basis set, all the geometries were fully optimized, and their frequency calculations were carried out.23 Pentylethanoate was considered as the lipid medium to mimic the membrane cell environment, using (SMD) the continuum solvation model density.24The M06-2X functional has been recommended by its developers to perform kinetic calculations and is reported to be one of the best performing functionals for kinetic calculations in solutions.25,26 SMD is a solvation model that could be employed to any charged or uncharged solute in a liquid medium.24 The geometry optimization and frequency calculations, including the solvent effect by the first excited-state singlet were performed employing the TD-DFT calculations at the TD-M06-2X/6-31G(d,p) level. The energy values were improved by a single point calculation at the M06-2X/6-311++G(d,p) level.
The conventional transition state theory (TST) was used to estimate the reaction rate constant, according to the following:27,28
\(k=\frac{k_{B}T}{h}e^{-G^{\neq}/RT}\) (1)
where kB , h, and R are the Boltzmann, Planck, and gas constants, respectively; additionally,T and ΔG are the temperature and the Gibbs free energy of activation, respectively.
The single electron transfer mechanism (SET) was considered in all the reactions, and the Gibbs free energy of reaction (∆G) was calculated from separate reactants and products of each SET reaction; as well as, the Gibbs free energy of activation (∆G) was obtained employing Marcus theory.29–31 The Gibbs free energy of activation was described based on the Gibbs free energy of reaction (∆G) and the nuclear reorganization energy (λ).
\(G^{\neq}=\frac{\lambda}{4}\left(1+\frac{\Delta G}{\lambda}\right)^{2}\)(2)
The nuclear reorganization energy (λ ) was obtained according to the following:
\(\lambda=\Delta E-\Delta G\) (3)
thus, ∆E was the difference of no adiabatic energies between reactants and vertical products. This estimate is comparable to that suggested for intramolecular electron exchange reactions.32
The apparent rate constant cannot be directly obtained from TST calculations when calculated reaction rate constants values are closer to, or within the diffusion-limit regime; hence, Collins-Kimbal theory was used:33
\(k_{\text{app}}=\frac{k_{D}k}{k_{D}+k}\) (3)
where k is the thermal reaction rate constant and was obtained from transition state theory calculations and kDis the steady-state Smoluchowski rate constant in an irreversible bimolecular diffusion-controlled reaction,34 and was calculated according to the following equation:
\(k_{D}=4\pi RD_{\text{AB}}N_{A}\) (4)
where R is the reaction distance, DAB is the mutual diffusion coefficient of the reactants A and B, andNA is the Avogadro number.DAB has been calculated fromDA andDB .35 The Stoke-Einstein approach was employed to estimate DA andDB ,36,37 according to the following equation:
\(D=\frac{k_{B}T}{6\pi\eta\mathcal{a}}\) (5)
where k B and T are the Boltzmann constant and temperature, respectively, while the η is the viscosity of the solvent, in this case, pentylethanoate (η = 8.62 x 10-4 Pa s), and the radius of the solute\(\mathcal{a}\).
In the kinetic study, the endergonic reaction paths were not included because such reactions would be reversible; as a consequence, the corresponding products would not be observed. The methodology employed in this study is in line with the quantum mechanics-based test for overall free radical scavenging activity (QM-ORSA); besides, this methodology has been verified by contrast with experimental results.38