Reviews in Computational Chemistry Volume 18

166 Charge-Transfer Reactions in Condensed Phases
redistribution, no screening of the electron field by rearrangement of the electrolyte
ions occurs, and the electron field includes the field of the image charge
on the metal surface
ð
DeðrÞ ¼e j eðr 0 Þj 2 r
1
jr r 0 j
1
jr r 0 im
j dr0
where r 0 im is the mirror image of the electron at the point r0 relative to the electrode
plane, eðrÞ is the wave function of the localized electron, and e is the
electron charge. (In Eq. [54], e appears because we are not using atomic units.
Thoughout this chapter, the energies are generally in electron volts.) The offdiagonal
solute–solvent coupling is dropped in the off-diagonal part of the system
Hamiltonian in Eq. [53] as no experimental or theoretical information is
currently available about the strength of the off-diagonal solute field in the
near-to-electrode region.
The free energy surface for the electron heterogeneous discharge can be
directly written as
½54Š
e bFðYd Þ ¼ðbQBÞ 1 TrnTr el½dðY d De PnÞ^rŠ ½55Š
where QB refers to the partition function of the pure solvent and the Dirac delta
function is invoked. In electrochemical discharge, the reactant is coupled to
a macroscopic bath of metal electrons. The total number of electrons in the
system is thus not conserved, and the grand canonical ensemble should be considered
for the electronic subsystem. The density matrix in Eq. [55] then reads
^r ¼ e bðm eN HÞ
Here, me is the chemical potential of the electronic subsystem containing
N ¼ c þ c þ X
½57Š
k
c þ k ck
electrons.
The path-integral formulation of the trace in Eq. [55] allows us to take it
exactly. This leads to the following expression for the free energy surface 46
FðY d Þ¼ ðYdÞ 2
4l d þ EðYdÞ 2 þ b 1 2
ln4
b ~
2p
i bEðYd Þ
2p
! 2
½56Š
3
5 ½58Š
Here, Y d is the classical reaction coordinate, ðxÞ is the gamma function, and
EðY d Þ¼E m e Y d ¼ l d þ eZ Y d
½59Š