Reviews in Computational Chemistry Volume 18

160 Charge-Transfer Reactions in Condensed Phases
that leads to Eq. [24] provided the integrals over the segments ð i 1;
i þ1Þ and ð 1; þ1Þ are equal. This happens when G1ðx; XÞ is analytic
in x inside the closed contour with the two segments as its boundaries. The
linear relation between F2ðXÞ and F1ðXÞ breaks down when the generating
function is not analytic inside this contour.
Two-State Model
The two-state model (TSM) provides a very basic description of quantum
transitions in condensed-phase media. It limits the manifold of the electronic
states of the donor–acceptor complex to only two states participating in
the transition. In this section, the TSM will be explored analytically in order
to reveal several important properties of ET and CT reactions. The gas-phase
Hamiltonian of the TSM reads
H0 ¼ X
Iia þ i ai þ Hab a þ a ab þ a þ
b aa
½26Š
i ¼ a;b
where Ii are diagonal gas-phase energies, and Hab is the off-diagonal Hamiltonian
matrix element usually called the ET matrix element. 7 In Eq. [26], aþ i , ai
are the fermionic creation and annihilation operators in the states i ¼ a; b.
The Hamiltonian in Eq. [26] is usually referred to as the diabatic representation,
employing the diabatic basis set ffa; fbg in which the Hamiltonian
matrix is not diagonal. There is, of course, no unique diabatic basis as any pair
f ~ fa; ~ fbg obtained from ffa; fbg by a unitary transformation can define a new
basis. A unitary transformation defines a linear combination of fa and fb which, for a two-state system, can be represented as a rotation of the
ffa; fbg basis on the angle c
~f a ¼ cos cf a þ sin cf b
~f a ¼ sin cf a þ cos cf b
One such rotation is usually singled out. A unitary transformation ffa; fbg! ff1; f2g diagonalizing the Hamiltonian matrix
H0 ¼ X
½28Š
i¼1;2
Eia þ i ai
generates the adiabatic basis set ff 1; f 2g. The adiabatic gas-phase energies are
then given as
Ei ¼ 1
2 ðIa þ I bÞ
1
2
½27Š
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðIb IaÞ 2 þ 4H2 q
ab;
E12 ¼ E2 E1 ½29Š