Singlet Fission - Department of Chemistry

J Chemical Reviews, XXXX, Vol. xxx, No. xx Smith and Michl
Figure 8. Jablonski diagram for singlet states of chromophores
of classes I-III, with the most common mode of population of the
S1 state shown (in rare cases of chromophores of classes II and III,
the h f l excited state might be S3 or even higher).
fission. 57,58 The procedure is also simple when there are only
two chromophores, A and B, and we will discuss this case
in some detail. To organize the material, we shall adopt a
simple model that has been used with small variations many
times before for many purposes (only a few examples are
listed 57-62 ). It was also used in a recent study of singlet fission
in which the time evolution of the S1 A S0 B state under the
effect of the one-electron part of Hint alone was followed. 33
By analogy to the cases of singlet and triplet energy
transfer, 62 in which this type of term dominates, this may be
an acceptable approximation.
In the model, the eigenstates of Hel A and Hel B are approximated
by single configurations built from Hartree-
Fock MOs of the chromophores A and B, respectively. The
MOs on A and B are assumed to be mutually Löwdin
orthogonalized, and the orthogonalization is assumed to have
almost fully preserved the localization. A further simplification
is introduced by limiting the MOs to hA and lA on A
and hB and lB on B. In spite of these drastic assumptions,
the model captures the fundamental physics of the most
important two of the three main classes of π-electron
chromophores, which include almost all of those that have
been of interest in studies of singlet fission. We refer to these
chromophore classes as classes I, II, and III (Figure 8).
Our purpose is to organize the material and provide
qualitative insight into the terms that need to be considered
when contemplating the origins of structural dependence of
the frequency factor A[S1S0 f 1 (TT)]. For the purposes of
an actual calculation, more rigor would be needed. In
particular, it would undoubtedly be necessary to treat overlap
in a manner similar to the now standard treatment of energy
transfer. 62 This would involve an initial diagonalization of a
subspace that contains local and charge transfer excitations
and dealing with overlap explicitly. Although such a
formulation appears quite straightforward, it has not yet been
published for singlet fission, and we will not use it here.
Chromophores of Class I. This class comprises those π
chromophores whose first excited singlet state S1 is reasonably
described as a result of an h f l (HOMO to LUMO)
electron promotion from the ground state and is separated
from the next higher singlet S2 by a significant energy gap.
Class I chromophores are exemplified by anthracene (1),
tetracene (2), perylene (9), and 1,3-diphenylisobenzofuran
(8). In compounds formally derived from a (4N + 2)-electron
perimeter, such as these, the S0 f S1 transition terminates
in Platt 63 and Moffitt’s 64 La state, carries considerable
oscillator strength, and represents the usual point of entry
into the excited singlet manifold when the ground state
absorbs light. In the less common chromophores derived
from a 4N-electron perimeter, the h f l transition has low
intensity and is often symmetry forbidden. 65,66 However, to
our knowledge, no such chromophores have been examined
for singlet fission.
Chromophores of Class II. In the less numerous chromophores
of class II, the strongly allowed h f l excitation
leads to the S2 state and still represents the usual entry point
into the excited singlet manifold by absorption of light. The
S1 state, which is often only slightly lower in energy and is
reached from the higher singlet states by fast internal
conversion, is quite well described as arising from an outof-phase
combination of h - 1 f l and h f l + 1 electron
promotions from the ground state, where h - 1 is the next
MO below h and l + 1 is the next MO above l in energy. Its
perimeter model label is Lb, and it carries low oscillator
strength. Typical examples of class II chromophores are those
in which the energy difference between the MOs h and h -
1 and that between the MOs l and l + 1 are small. Benzene,
where the degeneracy of the h and h - 1 and of the l and l
+ 1 MOs is exact, making the description a little more
complicated, is the best known case. Others are naphthalene,
phenanthrene, and pyrene (12). Since the simple model
adopted for our discussion does not include the MOs h - 1
and l + 1, it is incapable of describing the properties of the
S1 state of chromophores of class II. The required generalization
is straightforward, but very little work has been done
on singlet fission involving chromophores of this class, and
we will not generalize the model here.
Chromophores of Class III. Similarly as in chromophores
of class II, in chromophores of class III the strongly allowed
h f l excitation provides a good description of the nature
of the lowest excited state carrying a large oscillator strength
from the ground state (usually the S2 state), and again
represents a typical entry point into the excited singlet
manifold by absorption of light. Now, however, the S1 state,
which is usually only slightly lower in energy and can again
be reached from the higher singlets by fast internal conversion,
contains a large weight of a configuration in which both
electrons that occupy h in the ground state have been
promoted, h,h f l,l. Although commonly used, the usual
designation “doubly excited state” is often quite inaccurate
because the description of this S1 state may also contain large
contributions from additional singly excited configurations,
especially h - 1 f l and h f l + 1. Our model permits a
description of the doubly excited configuration, but not of
these additional configurations. The results will therefore be
even less accurate than those for chromophores of class I.
An alternative description of the doubly excited singlet state
is as two local triplet excitations intramolecularly coupled
into an overall singlet (e.g., in butadiene, it corresponds to
a singlet coupled pair of local triplets, one on each
ethylene 9-11 ).
The best known examples of chromophores of class III
are polyenes and related linearly conjugated polymers, but
very recently it has been proposed on the basis of ab initio
calculations 40 that pentacene (3)issimilarinthatitsS1 state,
which is of the h f l type, and its S2 state, which is of the
h,h f l,l type, are calculated to be almost exactly degenerate.
The well studied absorption and fluorescence spectra of
isolated molecules exclude the possibility that the doubly