Reviews in Computational Chemistry Volume 18

where ‘‘þ’’ and ‘‘ ’’ correspond to i ¼ 1 and i ¼ 2, respectively. Here, we outline
the procedure of building the CT free energy surfaces in the diabatic representation
and then discuss advantages of using the adiabatic representation.
When the donor–acceptor complex is placed in a solvent, its Hamiltonian
changes due to the solute–solvent interaction
Hint ¼ ^
E P ½30Š
Here, the dot product of two calligraphic letters stands for an integral over the
solvent volume V
ð
E^ P¼ ^E P dr ½31Š
V
and ^ E is the electric field operator of the transferred electron coupled to the
polarizability of the solvent P. The system Hamiltonian then becomes
H ¼ HB þ X
ðIiEiPÞa þ i ai þ ðHab Eab PÞ
a þ
b aa þ a þ a ab ½32Š
i¼a;b
where HB refers to the Hamiltonian of the solvent (thermal bath); Ei ¼
hfij ^ Ejfii and Eab ¼hfaj ^ Ejfbi. The solvent Hamiltonian HB includes two components. The first one is
an intrinsically quantum part that describes polarization of the electronic
clouds of the solvent molecules. This polarization is given by the electronic solvent
polarization, Pe. The second part is due to thermal nuclear motions that
can be classical or quantum in character. Here, to simplify the discussion, we
consider only the classical spectrum of nuclear fluctuations resulting in
the classical field of nuclear polarization, Pn. Fluctuations of the solvent
polarization field are usually well described within the Gaussian approximation,
35 leading to the quadratic solvent Hamiltonian
HB ¼ HB½PnŠþHB½PeŠ ¼1 2 Pn w 1
n
Paradigm of Free Energy Surfaces 161
Pn þ 1
2
o 2
e
Pe _ _
Pe þPe w 1
e Pe ½33Š
Here, we and wn are the Gaussian response functions of the electronic and
nuclear solvent polarization, respectively; Pe _ is the time derivative of the electronic
polarization field entering the corresponding kinetic energy term.
In terms of the Gaussian solvent model, 35 the nuclear response function is
defined through the correlator of corresponding polarization fluctuations
(high-temperature limit of the fluctuation–dissipation theorem 36 )
w nðr r 0 Þ¼b hdPnðrÞ dPnðr 0 Þi ½34Š
In Eq. [33], oe denotes a characteristic frequency of the optical excitations of
the solvent. The kinetic energy of the nuclear polarization Pn is left out in