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