Reviews in Computational Chemistry Volume 18

where Z is the electrode overpotential. Equation [58] presents the exact solution
for the free energy surface of an electrochemical system along the classical
reaction coordinate Y d . It includes the free energy of a classical Gaussian solvent
fluctuation (the first term) and the free energy of charge redistribution
between the localized reactant state and the continuum of delocalized conduction
states of the metal (the second and the third terms). Delocalization
effectively proceeds on the range of reaction coordinates given by the effective
width
built on the direct electron overlap
~ ¼ þ pb 1
¼ p X
rFjHkj 2
k
and the width of the thermal distribution of the conductance electrons on the
metal Fermi level (pb 1 ); r F is the electron density of states of the metal on its
Fermi level. In the limit
Eq. [58] reduces to the free energy
½60Š
½61Š
b ~ 1 ½62Š
FðY d Þ¼ ðYdÞ 2
4l d þ EðYdÞ p cot 1 EðYdÞ ~ þ ~
2p ln ½ðb ~ Þ 2 þðbEðY d ÞÞ 2 Š ½63Š
The overlap ~ can be replaced by when pb 1 . Equation [63] then leads
to the ground-state energy EðYdÞ (zero temperature for the electronic subsystem)
often used to describe adiabatic heterogeneous CT. 45
Equations [58] and [63] indicate an important point concerning the
instantaneous energies obtained by tracing out (integrating) the electronic
degrees of freedom of the system (Eq. [15]). When the separation of electronic
states is much higher than the thermal energy kBT, the free energies can be
replaced by energies. This does not happen for heterogeneous discharge
where thermal excitations of the conductance electrons lead to entropic effects
embodied in the temperature-dependent summand in ~ (Eq. [60]).
BEYOND THE PARABOLAS
Beyond the Parabolas 167
The paradigm of free energy surfaces provides a very convenient and
productive conceptual framework to analyze the thermodynamics and
dynamics of electronic transitions in condensed phases. It, in fact, replaces the
complex dynamics of a quantum subsystem interacting with a many-body