Reviews in Computational Chemistry Volume 18

is, however, small for the usual conditions of CT reactions and will be
neglected throughout the discussion below.
The energies E ½PnŠ in Eq. [35] depend on the nuclear solvent polarization
that serves as a three-dimensional (3D) nuclear reaction coordinate
driving electronic transitions. The two-state model actually sets up two directions:
the vector of the differential field E ab and the off-diagonal field E ab.
Therefore, only two projections of Pn need to be considered: the longitudinal
field parallel to E ab and the transverse field perpendicular to E ab. In the case
when the directions of the differential and off-diagonal fields coincide, one
needs to consider only the longitudinal field, and the theory can be formulated
in terms of the scalar reaction coordinate
Y d ¼ E ab Pn ½39Š
The superscript ‘‘d’’ in the above equations refers to ‘‘diabatic’’ since the diabatic
basis set is used to define the electric field difference E ab. The corresponding
free energy profile is obtained by projecting the nuclear
polarization Pn on the direction of the solute field difference
e bF ðYd Þ ¼
ð
DPndðY d
E ab PnÞe
bE ½PnŠ
where DPn denotes a functional integral 41 over the field PnðrÞ.
The integration in Eq. [40] generates the upper and lower CT free energy
surfaces that, after the shift in the reaction coordinate Y d ! Y d þ
E ab w n Eav, take the following form 42
with
and
F ðY d Þ¼ ðYd Þ 2
Paradigm of Free Energy Surfaces 163
4l d
½40Š
EðYdÞ þ C ½41Š
2
EðY d Þ¼½ð F d 0 Yd Þ 2 þ 4ðH ab þ a abð F d s Y d ÞÞ 2 Š 1=2
C ¼ Fd 0a þ Fd 0b
2
þ ld
4
The constant a ab in Eq. [42] represents the ratio of the collinear difference
and off-diagonal fields of the solute
½42Š
½43Š
a ab ¼E ab= E ab ½44Š