Principles of Fluorescence Spectroscopy

210 SOLVENT AND ENVIRONMENTAL EFFECTS
6.2.1. Derivation of the Lippert Equation
The interactions responsible for general solvent effects are
best understood by derivation of the Lippert equation. This
equation can be written as
hc ∆ν hc( ν A ν F) 2∆f
a 3 (µ E µ G) 2 const
(6.2)
where ∆ν is the frequency shift (in cm –1 ) between absorption
and emission, ∆f is the orientation polarizability, and
µ E and µ G are the excited- and ground-state dipole
moments, respectively. The Lippert equation is derived by
consideration of the interaction of a fluorophore with the
solvent, and the timescale of these interactions. We need to
recall the Franck-Condon principle, which states that nuclei
do not move during an electronic transition (10 –15 s). In contrast,
the electrons of the solvent molecules can redistribute
around the new excited state dipole during this time span. In
addition, because of the relatively long lifetime of the excited
state (-10 –8 s), the solvent molecules can orientate to
their equilibrium position around the excited state of the
fluorophore prior to emission.
Derivation of the Lippert equation starts with the consideration
of a point dipole in a continuous dielectric medium
(Figure 6.4). The energy of the dipole in an electric field
given by
E dipole µR
(6.3)
where R is the electric field. 8 In our case the electric field is
the relative reactive field in the dielectric induced by the
dipole. The reactive field is parallel and opposite to the
Table 6.2. Stokes Shifts Expected from General Solvent Effects
ν A – ν F
Solvent ε n ∆f a (cm –1 ) b λ max c (nm)
Hexane 1.874 1.372 –0.0011d 35 350d Chloroform 4.98 1.447 0.1523 4697 418.9
Ethyl Acetate 6.09 1.372 0.201 6200 447.0
1-Octanol 10.3 1.427 0.2263 6979 463.1
1-Butanol 17.85 1.399 0.2644 8154 489.8
n-Propanol 21.65 1.385 0.2763 8522 498.8
Methanol 33.1 1.326 0.3098 9554 525.8
a∆f = [(ε – 1)/(2ε + 1)] – [(n2 – 1)/(2n2 + 1)].
bFrom eq. 6.1, assuming µG = 6D, µ E = 20D, µ E – µ G = 14D, and a cavity radius of 4 Å. 4.8D is the dipole moment
that results from a charge separation of 1 unit of charge (4.8 x 10 –10 esu) by 1 Å (10 –8 cm). 1 x 10 –18 esu cm is 1
Debye unit (1.0D).
cAssuming an absorption maximum of 350 nm.
dThe small negative value of ∆f was ignored.
direction of the dipole, and is proportional to the magnitude
of the dipole moment,
(6.4)
In this equation f is the polarizability of the solvent and a is
the cavity radius. The polarizability of the solvent is a result
of both the mobility of electrons in the solvent and the
dipole moment of the solvent molecules. Each of these
components has a different time dependence. Reorientation
of the electrons in the solvent is essentially instantaneous.
The high frequency polarizability f(n) is a function of the
refractive index:
(6.5)
The polarizability of the solvent also depends on the dielectric
constant, which includes the effect of molecular orientation
of the solvent molecules. Because of the slower
timescale of molecular reorientation, this component is
called the low frequency polarizability of the solvent, and is
given by
The difference between these two terms is
∆f
R 2µ
f 3 a
f(n) n2 1
2n 2 1
f(ε)
ε 1
2ε 1
ε 1
2ε 1 n2 1
2n 2 1
(6.6)
(6.7)