Essentials of Computational Chemistry

414 11 IMPLICIT MODELS FOR CONDENSED PHASES
A +• (gas) + e – (gas)
A +• (sol) + e – (gas)
∆G o (gas)
A (gas)
A (sol)
A (gas) + e – (gas)
A (sol) + e – (gas)
∆G o (gas)
A –• (gas)
∆Go S(A +• ) ∆Go S(A –• ∆G )
o ∆G S(A)
o ∆G S(A)
o = 0 ∆Go = 0
∆G o (sol)
∆G o (sol)
A –• (sol)
Figure 11.10 Thermodynamic cycles for one-electron oxidation (left) and reduction (right) potentials
in solution
thermodynamic cycles perforce involve open-shell species and free electrons. Note that
the oxidation and reduction cycles in the gas phase correspond conceptually to ionization
potentials and electron affinities, respectively, except that IPs and EAs are enthalpies, not
free energies, so thermal and entropic terms must be included therein. For the free electron,
like the free proton, the electronic energy is zero, but the sum of the PV and translational
terms leads to a total gas-phase free energy at 298 K and 1 atm of −0.00001 a.u. (it is
a coincidence that for this standard state the free energy associated with the translational
entropy almost exactly cancels the enthalpy).
Another key feature of redox thermodynamic cycles is that the free energy change in
solution is still defined to involve a gas-phase electron, that is, the solvation free energy of
the electron is happily not an issue. And, once again, redox potentials in solution typically
assume 1 M standard states for all species (but not always; in this chapter’s case study, for
instance, all redox potentials were measured and computed for chloride ion concentrations
buffered to 0.001 M). So, free energy changes associated with concentration adjustments
must also be properly taken into account.
Once the free energy change in solution has been computed, the absolute redox potential
Eo may be computed as
E o =− Go
(11.29)
nF
where n is the number of electrons transferred and F is the Faraday constant equal to
23.061 kcal mol −1 V −1 . Note that while Figure 11.10 presents thermodynamic cycles for
one-electron processes, analogous cycles involving multiple electrons are readily constructed
and may sometimes be more amenable to experimental determination.
In practice, experimental redox potentials are reported relative to a standard electrode. If
the standard is the NHE, one subtracts 4.36 V from the absolute reduction potential (the ‘cost’
of the free electron) or adds 4.36 V to the absolute oxidation potential (the ‘return’ from the
removed electron) in order to determine the relative potential. Adjustment to other standard
electrodes is straightforward, since their potentials relative to the NHE are well known.
With respect to accuracy, it is again important to employ basis sets including diffuse
functions when anions are present as either reactants or products. With large well balanced
basis sets, B3LYP for gas-phase energetics, and a PB SCRF solvation model, Baik and