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and k3 together suggest that PDI--pep-Zn+ lies ∼1.2kT below
Zn*, which is consistent with the finding of a measurable
delayed fluorescence component. The value of k5 is consistent
with the finding of essentially no decay of PDI--pep-Zn+ over
the 3.5 ns time scale of the experiments. However, given the
time span of the measurements, it is more appropriate to use
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an upper limit k5 < (10 ns)-1
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Appendix B. Fo1rster Calculations
The Fo¨rster rate is calculated using the standard formulation
based on the point dipole-dipole approximation (eq B1), using
the program PhotochemCAD.45
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kFo¨rster ) (8.8 × 1023)κ2ΦfmJn-4R-6(τm)-1
(B1)
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Here, κ2 is the orientation factor (obtained by dynamic averaging
about the pigment-linker bonds), Φmf is the fluorescence yield
of the donor in the absence of the acceptor, J is the spectral
overlap term (in cm6 mmol-1) obtained from the optical spectra
of the monomers, n is the refractive index (1.496 for toluene),
R6 is the donor-acceptor center-to-center distance (in Å)
obtained from molecular modeling, and τm is the lifetime of
the donor in the absence of the acceptor. For other Fo¨rster
calculations of perylene-bis(imide) dyes, see ref 46.
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(36) One calculation of k3 in Figure 3 based on the Zn* lifetime in the
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same value is obtained solely from the Zn* lifetime in the dyad versus the
monomer of k3 ) (τ)-1 - (τm)-1 ) (0.4 ns)-1 - (2.4 ns)-1 ∼ (0.5 ns)-1
.
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lifetime data via the expression k3 ) [(Φmf /Φf) - 1]/τm, using values of
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good agreement with the value of k3 ∼ (0.5 ns)-1 obtained from the
equilibrium kinetic model for the decay of Zn* in Figure 3 using the lifetime
and yield data (see Appendix A).
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