194
6 Charge and Energy Transfer Processes
so here again the only nonvanishing contribution is −K XY,K K L L . The same holds
for the reverse process of triplet–triplet annihilation. So:
Ψ X,S 1 Ψ Y,S 0
ˆ
H el
Ψ XY, 1 (T T )
= −n X n Y
×
Ψ X,S 1 Ψ Y,S 0
[r (x 1 , y 1 )]
−1 ˆ
P XY
Ψ X,T 1,1 Ψ Y,T 1,−1 − Ψ X,T 1,0 Ψ Y,T 1,0 + Ψ X,T 1,−1 Ψ Y,T 1,1
.
(6.45)
Even choosing a simplified description based on four MOs, here we need more than
two Slater determinants. The singlet excitation localized on X is
ˆ
A Ψ X,S 1 Ψ Y,S 0 = 2
−1/2
|abcc| − |abcc|
.
(6.46)
If the local triplets are approximated by single excitations a → b and c → d, the
expression (6.44) becomes
Ψ XY, 1 (T T ) = 12
−1/2
2|abcd| − |abcd| − |abcd| − |abcd| − |abcd| + 2|abcd|
.
(6.47)
Then,
Ψ X,S 1 Ψ Y,S 0
ˆ
H el
Ψ XY, 1 (T T )
=
3
2
1/2
bb
r
−1
12
cd
−
aa
r
−1
12
cd
. (6.48)
Other interactions, in particular through charge transfer states, may contribute significantly to coupling the initial and final state [12]. All of them are short-range
interactions, as in the case of triplet sensitization.
6.4.3 Singlet-to-Singlet Excitation Energy Transfer: Dexter
and Förster Mechanisms
In the case of excitation transfer within the singlet manifold, both integrals in
Eq. (6.40) are generally nonvanishing. For the same reasons already seen in the previous section when discussing the triplet sensitization and singlet fission processes,
the K XY,K K L L integral decreases much faster than J XY,K K L L when X and Y get far
apart. When the exchange interaction is important, i.e., in the short range (a few Å),
we say that the “Dexter mechanism” is acting [13]. A different regime, the “Förster
mechanism,” is reached when the closest atoms of X and Y are ≈10 Å apart or more
[14]. Then, the exchange interaction is negligible and
K , L
ˆ
H el
K
, L
J XY,K K L L =
ρ X,K K (r 1 ) ρ Y,L L (r 2 )
r 12
dr
3
1 dr
3
2 . (6.49)
Here r 1 and r 2 are the positions of the electrons with space and spin coordinates x 1
and y 1 , respectively, r 12 = |r 2 − r 1 | and
6 Charge and Energy Transfer Processes
so here again the only nonvanishing contribution is −K XY,K K L L . The same holds
for the reverse process of triplet–triplet annihilation. So:
Ψ X,S 1 Ψ Y,S 0
ˆ
H el
Ψ XY, 1 (T T )
= −n X n Y
×
Ψ X,S 1 Ψ Y,S 0
[r (x 1 , y 1 )]
−1 ˆ
P XY
Ψ X,T 1,1 Ψ Y,T 1,−1 − Ψ X,T 1,0 Ψ Y,T 1,0 + Ψ X,T 1,−1 Ψ Y,T 1,1
.
(6.45)
Even choosing a simplified description based on four MOs, here we need more than
two Slater determinants. The singlet excitation localized on X is
ˆ
A Ψ X,S 1 Ψ Y,S 0 = 2
−1/2
|abcc| − |abcc|
.
(6.46)
If the local triplets are approximated by single excitations a → b and c → d, the
expression (6.44) becomes
Ψ XY, 1 (T T ) = 12
−1/2
2|abcd| − |abcd| − |abcd| − |abcd| − |abcd| + 2|abcd|
.
(6.47)
Then,
Ψ X,S 1 Ψ Y,S 0
ˆ
H el
Ψ XY, 1 (T T )
=
3
2
1/2
bb
r
−1
12
cd
−
aa
r
−1
12
cd
. (6.48)
Other interactions, in particular through charge transfer states, may contribute significantly to coupling the initial and final state [12]. All of them are short-range
interactions, as in the case of triplet sensitization.
6.4.3 Singlet-to-Singlet Excitation Energy Transfer: Dexter
and Förster Mechanisms
In the case of excitation transfer within the singlet manifold, both integrals in
Eq. (6.40) are generally nonvanishing. For the same reasons already seen in the previous section when discussing the triplet sensitization and singlet fission processes,
the K XY,K K L L integral decreases much faster than J XY,K K L L when X and Y get far
apart. When the exchange interaction is important, i.e., in the short range (a few Å),
we say that the “Dexter mechanism” is acting [13]. A different regime, the “Förster
mechanism,” is reached when the closest atoms of X and Y are ≈10 Å apart or more
[14]. Then, the exchange interaction is negligible and
K , L
ˆ
H el
K
, L
J XY,K K L L =
ρ X,K K (r 1 ) ρ Y,L L (r 2 )
r 12
dr
3
1 dr
3
2 . (6.49)
Here r 1 and r 2 are the positions of the electrons with space and spin coordinates x 1
and y 1 , respectively, r 12 = |r 2 − r 1 | and