View Online
Theory for Time Resolved Emission Spectra
Downloaded by National Chiao Tung University on 26 August 2011
Published on 01 January 1975 on http://pubs.rsc.org | doi:10.1039/F29757100773
BY GRAHAM
R. FLEMING
AND ONNOL. J. GIJZEMAN*
Davy Faraday Research Laboratory of the Royal Institution,
21, Albermarle Street, London WIX 4BS
KARLF. FREED
The James Franck Institute and the Department of Chemistry,
The University of Chicago, Chicago, Illinois 60637, U.S.A.
AND
SHENGH. LIN
Department of Chemistry, Arizona State University,
Tempe, Arizona 85281, U.S.A.
Received 31st July, 1974
Explicit equations for the calculation of time resolved emission spectra from molecules undergoing
collisional vibrational relaxation are presented. The equations describe the changes of the emission
spectrum with time, after a delta puke excitation. Alternatively, they can be used to obtain the
steady state (frequency resolved) emission spectrum as a function of the pressure of added buffer gas.
Some model cases are discussed in the light of available experimental data. Non-monotonic dependences of the emission intensity on time and pressure are exhibited which are related in appearance to
those expected from quantum mechanical interference effects.
1. INTRODUCTION
In a previous paper a theory for time resolved absorption spectra of molecules,
undergoing vibrational relaxation, has been developed. The time and frequency
dependence of the observed absorbance Aab(Cr), t ) can be expressed as :
where k,,. is the absorption coefficient for transitions from the state lad) (a labels
the electronic state and v’ its associated vibrational quantum number) and Xaoris
the concentration of molecules in the state lad). If, as has been supposed previously,
no other decay channels are available to molecules in the state lad) other than
vibrational relaxation, we have :
C x,,. = const.
0’
and a closed form expression for kiab(Cr), t ) can be obtained.’
For time resolved emission spectra, eqn 1 must be modified to :
where $b,,Ir is the emission intensity distribution function of the vibronic state Ibu”).
Since molecules will now decay via other channels from the state Ibu”) as well, the
is more complicated. The function Ybq(w,t ) represents the time
calculation of XbVfJ
773
View Online
Downloaded by National Chiao Tung University on 26 August 2011
Published on 01 January 1975 on http://pubs.rsc.org | doi:10.1039/F29757100773
774
T H E O R Y F O R T I M E R E S O L V E D EMISSION S P E C T R A
and frequency resolved emission spectrum. If the decay is uniform over the whole
spectrum, i.e., if Yba(wl,t ) K 9 b a (t )~
for2all
, times and all frequencies wl and w 2 ,
the shape of the emission will not change with time. If this is not the case, a measurement of the decay curve at different frequencies will provide a time resolved emission
spectrum. Alternatively, the single photon counting technique, used at a fixed time
delay and variable frequency can be employed to obtain the emission spectrum at a
fixed time after the excitation pulse.
In the present formalism a steady state measurement corresponds to the determination of
/:
dt $ba(m, f ) as a function of frequency. The steady state intensity at a
particular u) will be determined by the rate of interconversion of the various levels
Ibu"). This rate will be a function of the pressure of added buffer gas, or equivalently,
of the rate of vibrational redistribution between the optically active levels Ibu").
2. THEORY
The emission intensity distribution function
Ibv") is given by :
jbopp(cf-))
for a single vibronic state
where (Obv".au' is the frequency of the transition Ibu") + lad) and
transition moment.
is the
Rbo~~,av*
Here 0's represent vibrational wavefunctions (products of single mode harmonic
oscillator functions) and &a is the electronic transition moment. In the following
we consider only symmetry-allowed transitions (i.e., Rba independent of the nuclear
coordinates) and thus regard the electronic transition moment as a constant.
Introducing an average vibrational frequency G for all modes and transforming
to reduced quantities as has been done previously,l we obtain :
nj and the set v;, v $ . . . ul
Here S is the total displacement of all N modes, n =
j
denotes the number of quanta excited in the modes 1 , 2 . . . N. m is to be determined
From the relation
Am*
=
-
= m+n-k.
cr)
Thus, Aw* is the energy above the 0-0 band measured in terms of the average
number of quanta excited.
If each of the N modes, which contribute to the observed intensity is assumed to
relax independently we have :
The calculation of Xb,r(f) has been accomplished by Freed and Heller for the following model case. (i) Collisions change the vibrational quantum number by 1.
*
View Online
Downloaded by National Chiao Tung University on 26 August 2011
Published on 01 January 1975 on http://pubs.rsc.org | doi:10.1039/F29757100773
G . R . F L E M I N G , 0. L . J . G I J Z E M A N , K . F . FREED, S . H . LIN
775
(ii) The radiative and non-radiative decay rates of the oscillator change linearly with
quantum number, i.e., the decay rate of level 0‘’ is d+ Id’, where d is the decay rate of
the lowest level and Z the incremental decay rate. (iii) The spacing of the levels
(Ameff) represents the average amount of energy removed per collision. (iv) The
density of state function of the real molecule is taken into account by considering
a(D-fold) degenerate oscillator, where D is an empirical parameter.
Since in general Ameff(100-6OO cm-’) is less than one quantum of an optical mode
( A w l 21 1000-1500 cm-l) the calculated population of the effective levels has to be
converted into the population of the optical modes by specifying the ratio of their
frequencies. The explicit equations for Xbu”(t)are derived in the Appendix.
Thus, in order to calculate time resolved emission spectra, the following parameters
have to be specified :
V, the rate of collisional relaxation from level 1 to level 0 ;
hueff, the average amount of energy removed per collision ;
d and Z, the decay rate of the lowest level and the incremental decay rate;
D, the degeneracy of the oscillator ;
the ratio of hmeff and AG;
S, the displacement of the optical modes.
It has been shown previously that the mathematical expressions for vibrational
redistribution are identical to those for collisional vibrational relaxation, provided
that the system under consideration is sufficiently large. The net result of this process
will be the transformation of one optical quantum into other mode(s). It is thus
expected that the present theory can be applied to experimental results of this kind.
if we interpret V as the rate of vibrational redistribution.
3. RESULTS AND DISCUSSION
After substitution of eqn (2.6) and (2.4) into eqn (1.3), we are finally in a position
to calculate time resolved emission spectra. The spectra at t = 0 are of course
independent of d, 1 and V, and are identical to those published ear1ier.l In general,
the calculated time resolved emission spectra look very similar to the corresponding
time resolved absorption spectra,’ except for an overall decay superimposed on the
whole spectrum. The high energy side of the spectrum (Am* < 0) decays more
rapidly (cf. fig. 2), since emission in this region originates from levels with a shorter
lifetime.
A typical time resolved emission spectrum is shown in fig. 1. Since it can be
concluded from the parameters used that at t = 0 the optical mode is excited with
3 quanta, there are three minima in the t = 0 spectrum. At later times the three
distinct peaks merge into two bands, and finally (at t 2 0.5) a decaying Boltzmann
spectrum is obtained. The rate at which the spectra change into one another is
determined by V, the collision frequency ; a higher value of V produces a Boltzmann
spectrum at an earlier time.
The intensity of the emission is time and frequency dependent and is in general
not given by the sum of two exponentials, although this may give an excellent approximation to the observed decay (vide infra). At long times, when a Boltzmann
equilibrium is reached in the excited state the decay is exponential in all regions of
the spectrum with a rate constant :
The decay of the t
=
0 spectrum in fig. 1 is plotted in fig. 2 for three frequencies,
View Online
Downloaded by National Chiao Tung University on 26 August 2011
Published on 01 January 1975 on http://pubs.rsc.org | doi:10.1039/F29757100773
776
THEORY FOR TIME RESOLVED EMISSION SPECTRA
10
8
6
4
2
0 - 2 - 1
A&
Fro. 1.-Emission spectra for a molecule, consisting of one optical mode with displacement S = 2.
Other parameters : O/weff = 6, d = 1, I = 0.2, Y = 10, D = 2, 8 = 1.5. The initially excited level
is &ff(initial) = 20, corresponding to 3 quanta in the optical mode. -: t = 0 ; , . . . . : t = 0.1
(xl);
--- :t
= 0.5 ( x 20).
t
FIG.2.-Decay of the t = 0 spectrum of fig. 1 at the three maxima. V = 5, all other parameters as
in fig. 1.
View Online
Downloaded by National Chiao Tung University on 26 August 2011
Published on 01 January 1975 on http://pubs.rsc.org | doi:10.1039/F29757100773
G . R . FLEMING, 0 . L. J . GIJZEMAN, K . F . F R E E D , S. H . L I N
777
corresponding to the three maxima in the t = 0 spectrum. The collision frequency
was decreased in order to demonstrate more clearly the non-exponential decay. The
curve for Ao* = 1 is seen to exhibit a minimum. This is due to the fact that during
relaxation molecules are at first removed from levels which emit in this spectral region,
whereas at later times they reach levels which do emit at this frequency. This effect
is of course quite separate from the occurrence of quantum
which is a
purely intramolecular phenomenon. Experimentally, however, the effects look very
similar. It has been argued before that the parameter Vmay also be taken to be the
rate of vibrational redistribution, provided that the system is sufficiently large. Thus,
if this is the case, the occurrence of beats in the decay curve is not necessarily a
purely " quantum phenomenon ", even if the molecule is " isolated " during its
lifetime.
A situation similar to the one shown in fig. 2 has in fact been reported by Formosinho et a1.' in a study of the time and frequency resolved T-T absorption spectrum
of anthracene. The " decay '' observed in this case will correspond to the short time
behaviour in fig. 2.
IF.
Intensity
0 30
0 -20
h
Y
.
.
I
3
Y
*@
0 .I0
8
6
4
2
0
-
2
-
4
Am*
FIG.3.-Time inte traged emission spectra as a function of the collision frequency V. All parameters
as in fig. 1. The spectra have been nonnalised to unit area. - V = 1 ;-A-A-: v = 5 ;-*-*-:
V = 10 ; - - - - : V = 100. The inset shows the relative dependence of the intensity at different fre* Am* = -2 ;
quencies on pressure, which is proportional to V. .A. A, Am* = - 3 ; *
- -0--0--,A a * = -1.
-
a,
The spectra discussed so far require time and frequency resolution of the emitted
luminescence. An experimentally more easily accessible situation will be the
measurement of steady state emission spectra as a function of added buffer gas.
In the present formalism one would then determine the time integral of Yba(co,t )
as a function of frequency, for different values of V. Time integrated spectra,
with the parameters used in fig. 1, are shown in fig. 3 for different values of the
View Online
Downloaded by National Chiao Tung University on 26 August 2011
Published on 01 January 1975 on http://pubs.rsc.org | doi:10.1039/F29757100773
778
THEORY FOR TIME RESOLVED EMISSION S P E C T R A
collision frequency. All spectra have been normalised to unit area, in order to
facilitate comparison. At higher pressure (large V ) the spectra resemble more and
more the Boltzmann spectrum (see fig. 1). Some structure on the high frequency
side remains discernible even at Y = 100, i.e., Y = 90kB (t -+ co) or about 90
effective collisions per lifetime. Since one would not expect a unit efficiency for
collisions, this may correspond to fairly high pressures of added gas. Except for the
lowest values of Y (V = 1 and of course Y = 0) there is little resemblance to the time
resolved spectra shown in fig. 1. Evidently, the integration over t (steady state
measurement) " smoothes " the initial fast decay.
The insert of fig. 3 shows the relative intensity of the emission at a particular
frequency as a function of V. Maxima and minima in this plot occur only at
frequencies A.o* < 0, i.e., emission from hot levels. In the rest of the spectral region
the intensity appears to be a monotonically increasing (or decreasing) function of
pressure. The same holds for the total (frequency integrated) emission intensity.
1-
0.1.
.-x
Y
ic
Y
.
I
0 .or
\
0,001
0.1
0.2
0-3 0.4
0.5
0.6
0.7
0.8
0.9
t
FIG.4.-Decay curve for the simulated pyrene emission spectrum. Parameters : three optical modes
with total displacementS = 2.4, d/weff = 10, d = 1, I = 0.03,Y = 10, D = 3 , 6 = 0.5. Theinitially
excited levels C,m(initial) are 15, 0, 0 respectively for the three (independent) oscillators. curve A :
total decay curve, decomposed into a long and short (B) component. curve C : decay of the Boltzmann ( Y + 00) spectrum.
Drent et aZ.* have reported a minimum in the quantum yield of biacetyl as a
function of added gas pressure. They attributed this effect to a purely quantum
mechanical interference phenomenon. From the discussion given here it appears
that a simple collisional (or even intramolecular) process may also be responsible
for their observations, provided that they monitored only a part of the spectral
emission region.
View Online
Downloaded by National Chiao Tung University on 26 August 2011
Published on 01 January 1975 on http://pubs.rsc.org | doi:10.1039/F29757100773
G . R . FLEMING, 0. L . J . GIJZEMAN, K . F . FREED, S . H . L I N
779
Structure, similar to that shown in the inset of fig. 3, has in fact been observed
in the fluorescenceof benzene single vibronic level^,^ and can be understood completely
on the basis of collisional deacti~ation.~
It has been remarked before that the parameter Y may also be taken to be the
rate of vibrational redistribution. As an application of this we consider in some
detail the non-exponential decay of isolated pyrene, O which has been attributed to
slow vibrational redistribution.1° A spectrum was simulated using the parameters
shown in fig. 4 and the decay of the maximum of the broad and structureless
fluorescence was determined. Fig. 4 shows this decay and the decay of the Boltzmann
spectrum, obtained by taking a large value for V. As can be seen from the figure,
the actual intensity curve can be reasonably decomposed into two exponentials,
with lifetimes of -0.25 and -0.07 respectively. The Boltzmann lifetime is 0.29.
Normalising the latter value to the one obtained experimentally l o (z = 350 ns),
we find zlonO= 300 ns (expt. 290 ns) and Zshort = 85 ns (expt. 70 ns). The value of
V then corresponds to a redistribution rate of 0.83 x lo7 s-l, in agreement with the
experimental value of 0.8 - 1.4 x lo7 s-l. The experimental value, however, was
obtained from a completely different kinetic analysis.l o
In summary, the theory, presented in this paper provides explicit equations for
the calculation of time resolved emission spectra. The results appear to be in good
agreement with the available experimental data and provide a convenient basis for
the discussion of inter- and intra-molecular relaxation in electronically excited molecules.
APPENDIX
In the model proposed by Freed and Heller the probability of finding the molecule in
level n, if at t = 0 the excitation is in level rn, is given by :
s=o
where G,,,(s, t ) is the generating function
Here the basic parameters of the model are : D , the degeneracy of the oscillator ;d, the total
decay rate of the lowest level; Z, the incremental decay rate; p = exp(hcoeff/kT)= exp 8 ;
u, the collision frequency for the transition u = 0 -+ u = 1 (the collision frequency V, used
in the text is related to u by V = pu.). All other quantities are defined in terms of these
parameters as follows :
b =$ 6 ~
y = l+p+Z/v,
p = 2 - 4p,
(-43)
a, = d+oD(l--y/2),
y = tanh(bt), z = (y-2s)/8.
Carrying out the differentiationand putting s = 0, we obtain after some algebra :
where :
N
n
+
+
m ! (rn D - 1 i) !
Y'
View Online
780
THEORY FOR T I M E RESOLVED EMISSION SPECTRA
A recursion relation can be obtained for pp(Y) to facilitate numerical computations.
P;(Y) = F;+(Y)-
m
+ 1-n + Y ( m+ n +D - 1) +F;.n
2(
n+D-2y
Y)
n
~
Downloaded by National Chiao Tung University on 26 August 2011
Published on 01 January 1975 on http://pubs.rsc.org | doi:10.1039/F29757100773
In order to allow for the fact that in general licoeff is smaller than one quantum of an
optical mode, we define :
il
x,Jt) = i C
Pi(m, t )
=io
(A61
the frequency ratio of the optical and effective modes, and
where il- io+ 1 = 6/coeff,
- io/(il- io+ 1). This means that, say the first ten levels of the effective oscillator are
counted as the zero'th level of the optical mode, the next ten levels as the first level of the
optical mode etc. Eqn (A6), (A5),and (A4) finally provide the explicit expressions for
uIt
Xbu'j(t).
We thank the S.R.C. for the award of a Studentship to G. R. F. as well as the
Royal Society European Exchange Programme for the award of a Fellowship to
0. L. J. G.
G . R. Fleming, 0. L. J. Gijzeman and S . H. Lin, J.C.S. Furuday IZ, 1974, 70, 1074.
S. H. Lin, L. Coangelo and H. Eyring, Proc. Nat. Acud. Sci. U.S.A., 1971, 68, 2135; S. H.
Lin, J. Chem. Phys., 1973, 58, 5760.
K. F. Freed and D. F. Heller, J. Chem. Phys., 1974, 61, 3942.
G. R. Fleming, 0. L. J. Gijzeman and S. H. Lin,J.C.S. Furaduy ZI, 1974, 70, 37.
C. A. Langhoff and G . Wilse Robinson, Mol. Phys., 1973,26, 249.
J. Jortner and R. S . Berry, J. Chem. Phys., 1968,48,2757.
S . J. Fromosinho, G . Porter and M. A. West, Chem. Phys. Letters, 1970, 6 , 7.
* E. Drent, J. Konimandeur, A. Nitzan and J. Jortner, Chem. Phys. Letters, 1971,9,273.
M. Stockburger, in Organic Molecular Photophysics, ed. J. B. Birks (Wiley, London, 1973),
VOl. I, p. 57.
l o T. Deinum, C. J. Werkhoven, J. Langelaar, R. P. H. Rettschnick, and J. D. W. van Voorsf,
Chem. Phys. Letters, 1971, 11,478 ; 1973, 18, 171.