10504
J. Phys. Chem. 1994, 98, 10504-10513
Energy Transfer in the Inverted Region: Calculation of Relative Rate Constants by Emission Spectral Fitting Zakir Murtaza, Darla K. Graff, Arden P. Zipp, Laura A. Worl, Wayne E. Jones, Jr., W. Douglas Bates, and Thomas J. Meyer* Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27599-3290 Received: April 21, 1994; In Final Form: July 25, 1994@
Bimolecular energy transfer quenching of the metal-to-ligand-charge-transfer (MLCT) excited states of a series of 2,2'-bipyridine (bpy) complexes of Osn by anthracene or 2,3-benzanthracene (tetracene) has been studied in 3:l (v/v) acetonitrile-benzene at room temperature. Energy transfer rate constants (k,) vary from 6.3 x lo7 to 5.1 x lo9 M-' s-', which is below the diffusion-controlled limit of 9.1 x lo9 M-' s-l. For anthracene and the first triplet state (TI) of 2,3-benzanthracene, k, reaches a maximum at a driving force (-AG ") of +OS2 eV. With a further increase in -AG O, k, decreases, falling as low as 3.2 x lo8 M-' s-' with 2,3-benzanthracene as quencher, consistent with inverted behavior. Energy transfer occurs via the triplettriplet exchange (Dexter) mechanism. There is evidence for quenching by the second triplet state of 2,3benzanthracene (Tz) at even higher driving force. Application of a Franck-Condon analysis to the emission spectral profiles of the complexes and phosphorescence from anthracene has provided kinetic parameters for calculating relative energy transfer rate constants based on the usual Golden Rule formalism. When combined with an appropriate quenching model (including TZfor 2,3-benzanthracene), it is possible to account for the driving force dependence of k, quantitatively, thus demonstrating the use of spectral fitting parameters to calculate relative rate constants for energy transfer. On the basis of this analysis and comparison between experimental and calculated k, values, VI = 2.5 cm-' for quenching by anthracene and TI of 2,3-benzanthracene, and VZ = 8 cm-' for quenching by TZof 2,3-benzanthracene.
Introduction More than 30 years ago, Marcus predicted that there should be an inverted region for electron transfer.' Electron transfer theory, formulated classically or by time-dependent perturbation theory, results in the electron transfer rate constant being the product of three factors: electronic coupling, changes in vibrational displacement, and a temperature-dependent solvent distribution function. In the classical limit, the vibrational and solvent terms have a quadratic dependence on the free energy change, which leads to the prediction of "normal" and "inverted' regions. In the former, the rate constant increases with an increase in driving force (-AG O), and in the latter, it decreases with increased driving force. Since the time of Marcus's initial theoretical prediction, the inverted region has been observed experimentally in many electron transfer Triplet-triplet energy transfer reactions can be discussed with the same formalism, and in the classical limit are predicted to possess the same quadratic free energy dependence with normal and inverted regions.8 Many exothermic triplet-triplet energy transfer rate constants for organic donor/acceptor pairs have been reported in the l i t e r a t ~ r e . ~These , ' ~ rate constants are near the diffusion-controlled limit and do not fall appreciably below that limit as the driving force is increased. Wilkinson et al. studied" the effect of driving force on energy transfer quenching of organic triplets by coordination compounds. These studies were marked by the presence of multiple high-lying quencher states which masked inverted behavior as the driving force of the reaction (energy of the donors) was increased. Earlier work of Steel and co-workers'* on exothermic singlet-singlet energy transfer reactions of azo compounds revealed examples of Marcus inverted behavior. More recently, the characteristic free energy dependence of the inverted region has been observed @
Abstract published in Advance ACS Abstracts, September 1, 1994.
0022-365419412098- 10504$04.SO10
for energy transfer in a series of exergonic gas-phase rea~tions,'~%~ as well as in two different series of exergonic intramolecular rea~tions.~~,~~~ We report here bimolecular energy transfer reactions in solution which display inverted behavior. We also develop a general methodology for using emission spectroscopic parameters to evaluate vibrational overlap factors and calculate relative rate constants as a function of the driving force, -AG O. The reactions involve the quenching of the metal-to-ligandcharge-transfer (MLCT) excited states of Os"-bpy complexes (bpy = 2,2'-bipyridine) by anthracene (An) and 2,3-benzanthracene (tetracene, tet), e.g. reaction 1.
These results have been presented, in part, in a preliminary cornm~nication.~~
Experimental Data Materials. The OsII-bpy complexes used in this study are listed in Table 1 along with a labeling scheme. They were available from previous studies as PF6- salt^.'^,'^^ Structures of the ligands and quenchers are illustrated in Figure 1. The quenchers were obtained from Aldrich and used without further purification. Spectral grade solvents (Burdick and Jackson) were used in all spectral and kinetic work. Initial experiments were conducted in acetonitrile as solvent; however, the limited solubility of the organic quenchers restricted the number of donorlacceptor pairs which could be studied. For this reason, quenching studies were extended to a 3:l (vlv) mixture of acetonitrile and benzene. Measurements. Emission spectra were corrected for detector response and recorded by using a Model F212 Spex Fluorolog-2
0 1994 American Chemical Society
J. Phys. Chem., Vol. 98, No. 41, 1994 10505
Energy Transfer in the Inverted Region TABLE 1: Complexes and Quencher9 label complex label
quencher
1 2 3
Structures are shown in Figure 1.
was 20%, although the percent quenching was much higher for most reactions. Quencher concentrations typically ranged from 0.0019 to 0.037 M. In eq 2, IOand to are the emission intensity and excited-state lifetime, respectively, measured in the absence of quencher, and k+ is the quenching rate constant. Experimental quenching data were fit to eq 2 with a standard least squares regression. A typical data set included five data points that were the average of several measurements at a given concentration and yielded slopes with f8.0% error. When combined with lifetime values that were accurate to -5%, k, values were obtained to within -9.5%.
ZdZ = 1
+ k,t[Q]
(24
+ k,[Q]
(2b)
zdz = 1 das
anthracene (An)
2.3.benzanthrdcene (tetracenc. le0
Figure 1. Structures of ligands and quenchers.
series spectrophotometer. Emission intensities (Z)were obtained by either integrating the spectral profiles over their entire frequency range or measuring the intensity at the emission maximum; the two methods were shown to give equivalent results for several cases. Reaction mixtures were flushed with solvent-saturated nitrogen for '/2 h before the spectra were recorded. Time-resolved emission measurements were made by using a PRA LN1000/LN102 nitrogen laser/dye laser combination for excitation at A = 532 nm (< 100 @/pulse, -600 ps). Emission was monitored at right angles by using a PRA B204-3 monochromator and a cooled, 10-stage, Hamamatsu R928 photomultiplier. The output from the PMT was terminated through 50 G? to a LeCroy 6880 GHz digitizer with a LeCroy 350 MHz amplifier, or a LeCroy 9400 125 MHz digitizing oscilloscope. Either digitizer was interfaced to a personal computer, with software of our own design, for data analysis and workup. Transient absorption measurements were performed with a system partially described previ~usly.~'The system incorporates a Quanta Ray DCR-2A Nd:YAG laser, in this case with 532 nm-second harmonic output at 4 @/pulse. The excitation beam was delivered coaxial by an Applied Photophysics laser kinetic spectrometer with a 250 W pulsed xenon lamp, f3.4 monochromator, and a Hamamatsu R446 photomultiplier tube by using appropriate dichroic optics from CIV East. The output of the photomultiplier was coupled to either the LeCroy 9400 or 6880 digitizing oscilloscope, which had been interfaced to a personal computer. Electronic synchronization and control of the experiment was achieved through electronics of our own design. When possible, energy transfer rate constants were measured by standard Stem-Volmer techniques. The decrease in emission intensity (Z)and/or lifetime (z) of the excited-state donor was measured as a function of the concentration of added quencher. Stem-Volmer plots, eq 2, were linear over the range of quencher concentrations used, and the intercepts were unity within experimental error. The minimum percent quenching
The use of 2,3-benzanthracene (tetracene, tet) as quencher was complicated by its absorption at A < 500 nm, which overlaps with the absorption spectra of the Os" complexes. For the reaction pairs involving high-energy Osn absorbers, it was not possible to excite the Osn complexes selectively. The resulting tetracene fluorescence masked the MLCT emission, obviating the use of standard emission quenching techniques. For these cases, time-resolved emission was used to determine the quenching rate constants. The measurements of Osn*-based emission were initiated 40 ns after excitation. At early times, residual fluorescence due to direct excitation of tetracene dominated, but Stem-Volmer analysis of tdt at longer times gave reproducible quenching rate constants (*lo%). Transient absorption measurements were conducted on the energy transfer pairs la, lb, 4a, and 4b (Table l), in order to obtain direct evidence that energy transfer was occurring from the MLCT excited state to the quencher. In these experiments, the appearance and decay of the organic triplets were monitored at their A, (tetracene, 470 nm; anthracene, 430 nm) and timeresolved which gave k, values comparable to those obtained by the other techniques.
Analysis Emission Spectral Profiles. Emission spectral profiles were analyzed by comparing experimental spectra with spectra generated by using eq 3.18 This equation results from a standard
""=$[( ~(E0-0)
Eo Eovhiw
I-.
3sv
V!
( E - Eo
[ [
exp -4 In 2
+vho)
AYO,l/2
i;)l
(3)
Franck-Condon analysis and expresses the energy dependence of the emission intensity (in cm-', relative to the intensity of the 0-0 transition) in terms of four parameters: EO,ho,S, and AY0,ln. Vibronic contributions are included as a single, averaged mode of quantum spacing ho and electron-vibrational coupling constant S. The electron-vibrational coupling constant is related to the change in equilibrium displacement between states, AQq, and the reduced mass, p, by S = (1/2)(p~/h)(AQ,)~. The summation in eq 3 was performed over the first five quantum levels. The full width at half-maximum, ACOJ/Z,includes contributions from low-frequency modes treated classically and the solvent. (Temperature effects, in the form of "hot bands" are not specifically included in this equation or in our analysis, but have been included elsewhere.18a) The energy quantity, EO, is the energy difference between the v* = 0 v = 0 vibrational levels in the excited and ground state. Figure 2 illustrates the
-
Murtaza et al.
10506 J. Phys. Chem., Vol. 98, No. 41, 1994
11500
12500
13500
14500
15500
("1)
B: anthracene
Energy
Figure 2. Graphical depiction of parameters used in emission spectral fitting showing deconvolution into four vibronic components.
spectral parameters EO,hw, and AVO,~/Z. The magnitude of S determines the relative intensities of the individual components in the vibrational progression. The emission profile of Os(bpy)32+is shown in Figure 3a with a "best fit" spectrum calculated from eq 3 by using the parameters listed in-Table 2. The parameters for the series of 0s" complexes are listed in Table 2. Because the room-temperature phosphorescence of anthracene in fluid solution is difficult to obtain,lga the anthracene parameters used in our rate calculations were estimated by fitting the low-temperature spectrum in Figure 3b. The frozen ethanol-methanol matrix used for this purpose shows a weak emission feature centered near 14000 cm-'. This was subtracted to obtain the spectrum used in the spectral fitting of Figure 3b. Previous studies on polypyridyl complexes18a*z0 have shown the parameters Sj and hcu,to be largely insensitive to temperature changes, while the full width at half-maximum, AV0,1/2, varies with temperature according to eq 4:
(AYo,1/2)z= 16 In 2kbTx',
(4)
In this equation, xl0 is the sum of the solvent reorganization energy ko)and low-frequency vibrational modes treated clasThe best fit to the low-temperature anthracene sically ki,~). spectrum was obtained by using the six C-C stretching frequenciesz1 observed in the time-resolved resonance Raman spectrum of the anthracene tripletlgb and consistent with the progressions observed in single-crystal and low-temperature measurements of anthracene phosphorescence.2z The room-
1d o 0
12800
13800
14800
(CUI")
Figure 3. (A) Spectral fitting results for room-temperature Os(bpy)32+ emission in CH3CN. See complex 1 in Table 2 for the parameters. (B) Spectral fitting results for low-temperature (77K) phosphorescence of anthracene in 4:l EtOH-MeOH with 10% Et1 added. See Table 3 for the parameters.
temperature phosphorescence reported by Grellmanlga is very weak, only 1600 cm-' wide, and reveals only the lowest frequency progression (380 cm-l). Our spectrum and band shape fitting parameters are supported by other literature-cited anthracene phosphorescence spectra.z2 Spectral parameters are listed in Table 2, along with estimated room-temperature parameters obtained by applying eq 4. In the low-temperature fit for anthracene, AYoJ/~is related to the by eq 4. We assumed xo solvent reorganizational energy, to be temperature independent; the bandwidth of the Grellman phosphorescence at room temperature in cyclohexane gives an identical value. The low-frequency vibration at 385 cm-' from L 150 cm-', was treated our low-temperature fit, XL = S L * ~ W = classically and added to xo to give xl0. The low-temperature value of 240 cm-' was used in eq 4 with T = 77 K to obtain xo = 97 cm-'. At room temperature, the total reorganizational energy from the low-frequency vibration and the solvent is %lo = x0 XL = 251 cm-I, giving AVOJZ = 755 cm-'. A correction to the low-temperature EOvalue of 14 890 cm-' was
xo,
+
TABLE 2: Emission SDectral Fitting Parameters at Room TemDerature in CHqCN" complex Os(bpy)?+ Os(bPY)2(PYz)z2+ Os(bp~)ddas)~+ W~P ) zY (d~pm)~+ Os(bpy)z(dppe)2+ Os(bPY)z(CO)(PYr)z+ Os(b~y)(das)2~+ Os(bpy)2(CO)(MeCN)2+ O~(~PY)(~PPY)~~+ averaged
EO,cm-I
t i o D , cm-I
13 600 13 900 14 700 15 700 15 900 16 950 17 100 18 500 19 000
1350 1300 1350 1325 1325 1300 1325 1300 1350 1325
SO
0.68 0.70 0.85 1 .oo 1.05 1.36 1.10 1S O 1.20 1.05
A%.I/Z,cm-' 1550 1570 1940 1975 1940 2010 2130 2190 2030 1940
,fo,
cm-'
1040 1075 1640 1700 1640 1760 1975 2090 1795 1635
'Based on a four-parameter fit by using eq 3. The estimated uncertainties are EO (f5%), SO (&lo%), AVOJ~(f5%). The uncertainties in the parameters are relatively large, and some of the parameters are highly correlated. A combination of parameters that gave an acceptable fit falls within the ranges given above.
J. Phys. Chem., Vol. 98, No. 41, 1994 10507
Energy Transfer in the Inverted Region
TABLE 3: Multimode Spectral Parameters for Fitting the Low-Tem erature Anthracene Phosphorescence and Averaged Room-Temperature Values
r
temp = 77 K (multimode) EO(cm-') = 14 890
temp = 298 K (average mode) EO(cm-') = 14 750
h o t (cm-I) s,. 1565 0.2 1470 0.15 1360 0.30 1180 0.20 740 0.04 385 0.40 AV0.112 (cm-') = 240 x0 (cm-l) = 97
(cm-I)
RoA
1360
SA
0.9
AVOJ~(cm-I) = 770
x ' ~(cm-') = 260
Phosphorescence was measured in 4: 1 ethanol-methanol glass at 77 K; 10% Et1 was added to enhance emission by the heavy atom effect!* See text for averaging procedure. Frequencies obtained from a
Raman data.lgb
made by subtracting XL, resulting in a room-temperature EOof 14 750 cm-'. In order to develop a consistent, simplified treatment of the contributions from the Os donors and organic acceptors, the other five frequencies of anthracene were represented as a single "averaged" mode16aof hw and S calculated by eq 5 and listed in Table 3. In eq 5 , S, and hwj are electron-vibrational coupling constants and quantum spacings of the contributing modes. These relationships are well-known in the time-dependent formalism as the MIME or missing-mode effectz3and have been discussed for polycyclic aromatics in particular, in terms of the Franck-Condon principle.24
TABLE 4: Free Energy Changes and Quenching Rate Constants in 3:l (v/v) Acetonitrile-Benzene at 298 f 2 K reaction pair" kq, M-' s-' x low8 log(kq) methodb AG ', eVc la 0.63 7.80 e,ta 0.05 lb 27.9 9.45 e,ta -0.52 2a 1.5 8.19 e 0.01 2b 25.1 9.40 e -0.56 9.04 e -0.17 3a 11.0 3b 8.85 8.95 te -0.73 4a 15.5 9.19 ta -0.30 4b 3.25 8.51 te -0.86 5a 12.3 9.09 e -0.31 5b 1.67 8.60 te -0.88 6a 27.1 9.43 e -0.46 6b 50.0 9.70 te -1.02 7b 34.0 9.53 e - 1.07 8a 30.7 9.49 te -0.69 8b 51.9 9.71 te - 1.26 9a 12.7 9.10 te -0.72 9b 25.6 9.41 te - 1.28
-
The reaction labels follow from Table 1. Thus, l a refers to the + 3An. The code to the reaction Os(bpy)3*+* An O~(bpy)3~+ experimental techniques used to measure the quenching rate constants is as follows: e, emission quenching; te, transient emission (lifetime); ta, transient absorption. The error associated with each is as follows: e, f9.5%; te, f6%; and ta, f3%. Calculated from eq 6 by using the data in Table 2 and the reported energy values for the lowest quencher triplet states at room temperature.
+
kinetics. The MLCT excited states are largely triplets with considerable singlet character due to spin-orbit coupling. The organic acceptors are nearly pure triplets. In these reactions, triplet-triplet energy transfer occurs via the Dexter mechanism. The general rate law for the kinetic scheme is given by27
.N
N
The fitting parameters for anthracene were used for 2,3benzanthracene with the exception of EO. The triplet excitedstate energies of the organic quenchers are 14700 cm-' (anthracene) and 10 200 cm-I (2,3-benzanthra~ene).~~.~~ The reported energy of the second triplet state of 2,3-benzanthracene is 20 600 cm-', estimated from the T1 T2 absorption. The free energy change for energy transfer was calculated from eq 6 by using the data in Table 2 where Eo(AIA*) and Eo(D/D*) are the energies of the acceptor and donor excited states and xt0 was calculated by eq 4 as described above.25
-
ket
1
=
_--
K A(;¶
?):I(
In these equations the diffusional rate constants (kd, k-d) are related to the equilibrium constant for formation of the association complex between reactants (KA)by
(9)
KA = kd1k-d
Free energy values for the various donor/acceptor pairs are listed in Table 4. Kinetic Analysis. The kinetic events that lead to energy transfer are illustrated in Scheme 1 for the particular case of quenching of [0s(b)3l2+* (b is bpy), by anthracene, An.
SCHEME 1 Os(b)32+
+ An -Os(b)32+*+ An
Os(b):+*,
hv
An
2Os(b);+, k-et
kd k-d
3An
K*
Os(b)?
+ 3An
In the scheme, quenching is assumed to be diffusional in character, consistent with the appearance of Stem-Volmer
The rate constants for forward and reverse energy transfer are related to AG O by
-AGO
k,jk-,, = K,, = exp(-) RT
A value of Ka = 2.3 M-' was calculated from the Fuoss equation.z8 The diffusion-controlled limit in CH3CN is (1-2) x 1O'O M-' s-'.l0 A value of kd for the 3:l solvent mixture was determined by measuring the rate constant for reductive quenching of [Ru(bpy)3l2+*by tetramethyl-p-phenylenediamine (TMPD), R~(bpy)3~+*TMPD Ru(bpy)?+ TMPD+, in the mixed solvent, which occurs at or near the diffusioncontrolled limit.z9 The rate constant for this reaction in the 3: 1
+
-
+
10508 J. Phys. Chem., Vol. 98, No. 41, 1994
Murtaza et al.
acetonitrile-benzene mixture was 9.1 x lo9 M-' s-l, and this was taken to be kd.
Rate Calculations Application of time-dependent perturbation theory to energy transfer between molecules shows that the rate constant can be written as the product of the squared electron exchange matrix element, V 2 , and a Franck-Condon vibrational overlap term, A similar formalism is used to describe electron transfer reacti0ns.3~ The resulting expression for the rate constant for energy transfer33 is given by eq 11. F.8910,30931
2nv
ket,cdc = (&dC
Evaluation of the vibrational overlap terms assuming harmonic oscillators gives
TABLE 5: Results of Vibrational Overlap and Energy Transfer Rate Calculations, Two-State Model reaction Pair
FCdC(l, XlOS
la lb 2a 2b 3a 3b 4a 4b 5a 5b 6a 6b 7b 8a 8b 9a 9b
0.85 17.65 2.00 15.79 7.67 12.24 13.98 8.29 14.86 7.74 17.58 5.79 4.85 17.21 7.29 15.20 10.32
kt.falcb,s-1
x 10-8
1odka.cdc)e
0.63 13.06 1.48 11.68 5.67 9.05 10.34 6.18 10.99 5.82 13.00 6.76 9.07 12.72 36.31 1 1.24 68.98
8.12 9.35 8.50 9.31 9.06 8.92 9.28 8.82 9.30 8.81 9.35 9.09 9.22 9.35 9.65 9.30 9.76
Fcdc calculated by using eq 1 lb. For the reason pairs involving calculated 2,3-benzanthracene, FFdc= FCdc,' Fcdc,2;see eq 12. kt,cdc by using eq l l a and V = 2.5 cm-'. For the reason pairs involving 2,3-benzanthracene,krCdc= kt,l kt,2with V I = 2.5 cm-l and VZ= 8 cm-'. k, calculated by using eq 7. For the reaction pairs involving 2,3-benzanthracene, eq 12 was used. In these calculations, K A = 2.3 M-' and kd = 9.1 x 10" M-' S-I.
+ +
Here, SD and SA are the electron-vibrational coupling constants, and h w and ~ h w the ~ donor and acceptor quantum spacings. kB is the Boltzmann constant and xl0,mthe total solvent reorganizational energy
10
'I
I
?
'4b
The m and n* indices in eq l l b are vibrational quantum numbers for the donor and acceptor. The Franck-Condon calculation, eq l l b , involves a summation over ground state vibrational levels of the Osn* donor (m) and excited state vibrational levels of the organic acceptor (n*). Inclusion of vibrational levels over m = n* = 7 made a negligible contribution to the calculated overlap (less than 0.1%). The importance of eqs 11 and 6 is that they explicitly define the vibrational overlap factor in terms of parameters obtainable from Raman spectroscopy or emission spectral fitting. The quantity X ' ~ , A D is the sum of the individual donor and acceptor reorganizational energies, x'~,A and x'~,D, arising from the solvent and low-frequency vibrational modes treated classically. These quantities were calculated from eq 4 and the bandwidth parameter, APOJ~, obtained by spectral fitting. In the preliminary communication describing this work, the quantity x t 0 ,was ~ mistakenly taken as the average (x'~,A x'~,D)I~ rather than the sum x'~,A x t O , ~ . l 4 Two types of rate calculations were performed by using eq 11 and the spectral fitting parameters in Tables 2 and 3. In the first, calculations of vibrational overlap factors were made for each donor/acceptor pair. Results of these calculations involving anthracene and 2,3-benzanthracene are presented in Table 5 . The electron exchange matrix element, V, was assumed to be constant for all reacting pairs and taken as the "best fit" value for agreement between the experimental values in Table 4 and the values in Table 5 calculated by using eqs 7 and 11. This fitting procedure involved plotting experimental versus calculated rate constants and the use of least squares analysis to solve for the best value of V. In the second type of calculation, averaged parameters SD, h w ~and , AP0,1/2, given at the bottom of Table 2, were used to represent the homologous series of Osn MLCT donors. With
+
+
I
I
i
.. -1 5
2a
~~
1
1
I
-0 5
.
0
05
AGO
Figure 4. Experimental kg values plotted as log 4 versus AG (data from Table 4) in 3:l (v/v) acetonitrile-benzene. The labeling scheme O
follows from Table 1.
the average parameters in hand, it was possible to parametrize eq 7b with AG O as the sole parameter and thus compare the experimental dependence of kq on AG O with the calculated dependence from eqs 7b and 11. The use of averaged values for SD, h w ~ etc., , although common in the electron transfer literature, is an approximation. For the Os"*(bpy) MLCT excited states it is known that SD varies linearly with EOover an extended range of excited-state energies.16a This added dependence of k,, on AG O was not included in our calculations and is largely responsible for the deviation that exists between the average and pairwise calculations.
Results For the reactions listed in Table 4,experimental rate constants in 3:l (v/v) CH3CN-benzene at 295 i 2 K are given, along with AG values calculated from eq 6. The technique(s) used in each measurement is indicated. Reasonable agreement between rate constants was found for those cases in which it was possible to use multiple methods of measurement. In Figure 4 is shown a plot of log kq versus AG '. The reactions are indicated by using the labeling scheme in Table 1.
Energy Transfer in the Inverted Region
J. Phys. Chem., Vol. 98, No. 41, 1994 10509
averaged fit
1
I I
'1,
I
\
\ \
4
Figure 7. Diagram of two-state model for energy transfer to 2,3benzanthracene. Decay from Osn*to Osn is accompaniedby excitation from Tet to TI or Tz.
1, .1 5
1
.o
5
0
calcuiatod
I
0 5
AGO
Figure 6. As in Figure 5, experimental rate constants (0) for 2,3benzanthracene as quencher. Calculated points (B) and the function in eq 7 calculatedby using the "averaged parameters for Osn* from Table 2 and the anthracene parameters in Table 3.
Data in Figure 4 with anthracene as quencher are plotted as log kq versus AG O in Figure 5. The experimental points are overlaid with the results of the point-by-point calculation of bt by using eq l l a with V = 2.5 cm-', the vibrational overlap factors in Table 5, and eq 7 with kd = 9.1 x lo9 M-' s-' and K A = 2.3 M-' for the calculation of k,. The dotted line is the fit calculated by using the average donor parameters (bottom row, Table 2) and the same V, k d , and K A as for the pairwise calculations. Emission spectra were measured in CH3CN. Representative complexes, e.g. 1,4, and 6, were also measured in the benzeneCH3CN solvent mixture and found to be virtually the same. The anthracene phosphorescence was measured in 4:l (v/v) ethanol-methanol at 77 K. The change in solvent means that the value of x'~,A (eq 4), which includes the solvent contribution to the spectral bandwidth, is not strictly correct for application in the rate constant calculation. However, the difference in this quantity between the two media is expected to be small,34and in the sum x l O = , ~x ' ~ , A x'~,D (eq 1IC),x ' ~ , A is less than 30% of the total (see also ref 35). A second subset of data from Figure 4, in which 2,3benzanthracene was used as the quencher, is likewise replotted in Figure 6. In this figure is shown the variation of log kq with AG O calculated by using EO= 10 200 cm-' for the lowest triplet excited state and the averaged acceptor parameters for anthracene in Table 3. The pairwise calculated points and the fit to eq 7 are overlaid with the experimental points by using V = 2.5 cm-', as for the anthracene data.
+
It is apparent from the data in Figure 6 that rate constants for reactions 6b, 7b, 8b, and 9b deviate significantly from the diffusional energy transfer model. The second lowest triplet state of 2,3-benzanthracene (T2) is reported26to lie at -20 600 cm-' (Table 3), which is -10 OOO cm-I above the lowest triplet (TI). As illustrated in Figure 7, this state could contribute to energy transfer for the higher energy donors. The participation of higher lying states in energy transfer quenching has been predicted and observed in many energy transfer reaction^.^^,^^^^^,^^ Our analysis was extended to include transfer to either of the acceptor's two lowest lying triplet states, as shown by the model in Figure 7. For each reaction pair involving 2,3benzanthracene, the energy transfer rate constant to each of the two lowest triplet states was calculated and summed. To keep the problem tractable and minimize the number of parameters required, SD,h w ~and , x'~,D for the anthracene triplet were used to calculate the rate constants to both T1 and TZ of 2,3benzanthracene. The two pathways were distinguished by different EOvalues, electron exchange matrix elements, VI and V2, and calculated rate constants, ket,l and ket,z. The general rate law for the two-state model (analogous to eq 7 for the onestate model) is given in eq 12: kd
kq=(l
+-+ ket,tot k-d
'-et,'
+ k-et,Z
(12)
ket,tot
The back energy transfer rate constants, Let,' and k-et,~,were calculated from eq 10 and eq 12 by using the relation ke,tot = kt,l ket,z. The results of these calculations are listed for the 2,3-benzanthracene data in Table 5. In order to obtain a satisfactory fit to the point-by-point data, it was necessary to use 18 500 cm-' as the energy of Tz, Figure 8. The values of kt,Fcac, and log kq from the two-state model with T2 = 18 500 cm-' and Vz = 8 cm-' are included in Table 5 . The combined results of the two-state model for 2,3-benzanthracene and the one-state model for anthracene are shown plotted in Figure 8 with VI = 2.5 cm-' for the lowest triplet state and VZ = 8 cm-' for Tz in 2,3-benzanthracene. The same plot but with Tz = 19 500 cm-' is also shown for comparison.
+
Discussion The most important feature of the data in Figure 4 is the experimental evidence for inverted behavior in triplet-triplet energy transfer. For anthracene as quencher, k, increases with increasing driving force (-AG ") from 6.3 x lo7 M-' s-' for O~(bpy)3~+* (AG O = 0.04 eV) to reach a maximum of 3.1 x lo9 M-' s-I for 0~(bpy)2(CO)(MeCN)~+* (AG O = -0.69 eV), which is well below the diffusion-controlled limit of 9.1 x lo9
Murtaza et al.
10510 J. Phys. Chem., Vol. 98, No. 41, 1994 10 1
I
I
I
w g
Y
M 0
L1
I
calculated
o
experimental
'til I
7 -1.5
1
-0.5
0
averaged fit (TZ = 18,500cm-1 I -----averaged fit ITZ=l9.500cm.ll
I
0.5
AGO
Figure 8. Experimental (0)and calculated (W) rate constants by using eq 7 for anthracene as quencher, and eq 12 (the two-state quencher model) for 2.3-benzanthracene as quencher. Averaged fits are plotted with Tz = 18 500 cm-’, VI = 2.5 cm-l, and VZ= 8 cm-’ (heavy dashed line), and with Tz = 19 500 cm-’, Vl = 2.5 cm-’, and VZ= 8 cm-l
(light dashed line). M-’ s-l. For 2,3-benzanthraceneas quencher, a further increase in driving force causes kq to decrease with -AG O to as low as 3.2 x lo8 M-’ s-’ for Os(bpy)z(dppm)2+*(AG O = -0.86 eV). As the driving force is increased still further, kq increases rather than continuing to decrease. This behavior can be attributed to participation of the second triplet of the acceptor, Tz, as discussed below. Because of the utilization of medium-frequency ring-stretching modes as energy acceptors, the dependence of ket on AG O is not simply the parabolic dependence predicted by Marcus theory in the classical limit. The theoretical formalism for the inclusion of quantum vibrations in state to state transitions, eq 11, developed by Ulstrup and J ~ r t n e rstill , ~ ~predicts ~ a region of inverted behavior, although the rate of change of k, with AG O is less steep than in the classical case. This is illustrated by the plot of log k, versus AGO in Figure 5 , which was calculated by using eqs 7 and 11 for ket. The “roll over” of the experimental points in Figure 4 indicates the entry of the energy transfer reactions into the inverted region. As is observable in our calculations of Fcdc, the decrease in rate constant is due to a decrease in the FranckCondon overlap factor as -AGO increases and the energy difference between donor and acceptor levels becomes greater. This appears to be the first example of bimolecular triplettriplet energy transfer in the inverted region in solution. Earlier examples of singlet-singlet transfer were mentioned in the Introduction. By comparison, there are few examples of inverted region behavior in diffusional, outer-sphere electron transfer even though there have been many attempts (at least by us) to observe it.37 One difference between the two is the magnitude of the solvent reorganizational energy. For electron transfer, the dipole change associated with the reaction is usually much greater than that for energy transfer, causing the reorganization of the solvent to be greater. Since the onset of the inverted region occurs at AG O = -lo,a greater driving force is required for inverted behavior to occur. Another difference is in the magnitude of the electronic matrix element for the two processes. The electron transfer matrix element (H&)involves overlap between donor and acceptor wave functions, possesses an exponential dependence on the donorlacceptor distance,38aand has been shown to possess orientational dependences for “through bond” intramolecular t r a n ~ f e r . ~For ~ , ~energy ~ ~ , ~transfer by the Dexter mechanism, the electron exchange matrix element (V) also varies exponentially with donor/acceptor d i s t a n ~ e ~and ~ , ~possesses ’ an orien-
tational d e p e n d e n ~ e . ~Because ~ . ~ ~ of the two-electron nature of energy transfer, there is a requirement for two overlaps between donor and acceptor. There is a squared exponential dependence, and the magnitude of V is decreased relative to If V is sufficiently small, the limiting value of the rate constant will be below the diffusion-controlled limit. From the experimental data for anthracene quenching, the maximum value of kq reaches 3.1 x lo9 M-’ s-l at -0.69 eV, which is still a factor of -3 below the diffusion-controlled limit. Rate constants well below the diffusion-controlled limit have been observed in the quenching of organic donor excited states by transition metal complexes of Crm, Fern, Rum, and AIm, for example.” From the plot in Figure 5, which uses the average spectral fitting parameters in Table 2, the calculated value of ket reaches a maximum of 1.4 x lo9 s-l at AG O = -0.50 eV and V = 2.5 cm-’ . In order for electron or energy transfer to be observed at high driving force, it is necessary that ket not greatly exceed k-d, which is -4.0 x lo9 s-l for the cases studied here. Typical values39for H a b range from -20 cm-’ to -1000 cm-’. For V = 20 cm-‘, k,, would increase to 9.0 x 1O1O M-’ s-l, well in excess of 4.0 x lo9 s-’. Thus, slightly larger values of H a b compared to V would be sufficient to mask the appearance of inverted electron transfer. Quenching of multiple excited states has been especially welldocumented in diffusional quenching reactions between organic donor excited states and transition metal complexes by Wilkinson et al.” In plots of log k versus AGO, there are several examples of well-defined breaks signalling the onset of more rapid quenching by upper excited states. There is no clear evidence in their data for inverted behavior. The data were interpreted by using an expression for the activation barrier derived by Agmon and L e ~ i n e(Marcus ~~~.~ derived a similar expression with specific application to proton transfer36b).This expression predicts an exponential dependence on free energy change and an asymptotic approach to a limiting rate. In retrospect, the inverted region may be hidden in the Wilkinson data because of a limited data set and contributions to quenching by both lower and upper states.
The Energy Transfer Process From the data of Crosby et al?Iaqb and the analysis of Kober and Meyer,41c the MLCT excited “state” of [Osn(bpy)3Iz+ actually consists of three low-lying MLCT states, all having appreciable triplet character and all significantly populated at room temperature. The lowest state, which is a pure triplet, lies -30 and -105 cm-I below the other two states. Spinorbit coupling causes significant mixing of the upper two states with low-lying “singlets”, the lowest of which is -5000 cm-I higher in energy. Given the mixed spin character of the Osn* complexes, they are capable of undergoing either singlet (Forster or Dexter) or triplet (Dexter) t r a n ~ f e r . ~Energy ~ , ~ ~ transfer to the organic acceptors is by electron exchange and the Dexter mechanism since the acceptor states are nearly pure triplets. On the basis of this analysis the lowest state of [Os(bpy)3I2+*,which plays a negligible role in nonradiative decay at room temperature, should be the major contributor to energy transfer. The other two states also contribute, but to a lesser degree given their Boltzmann populations (36 and 21% at 298 K) and lower triplet character , The Dexter mechanism requires direct contact of the reacting pairs and occurs via spatial overlap of donor and acceptor orbitals. The magnitudes of the electron exchange matrix elements, V = 2.5 and 8 cm-’, obtained from the kinetic fits
Energy Transfer in the Inverted Region
J. Phys. Chem., Vol. 98, No. 41, 1994 10511
are consistent with the range of values (1-50 cm-’) which have been r e p ~ r t e dfor ~ ~energy , ~ ~ transfer involving triplet states in aromatic molecules, as well as with values (2.5-60 cm-’) reported for energy transfer in a series of rigidly spaced donor/ acceptor pairs.13cg39
Calculation of Relative Rate Constants for Energy Transfer by Emission Spectral Fitting The second important feature demonstrated in this work is the use of emission spectral fitting to calculate relative k‘s for energy transfer. In the pioneering work of Forster and Dexter,31s32expressions for k were derived from time-dependent perturbation theory and the Golden Rule,
PF = (2n/h)(Y1*lklYF)2
(13)
In this equation, PIFis the probability that the transition will occur^from the initial excited state, ‘PI*,to the final state, YF, and H is the operator describing the perturbation that causes the states to mix. Energy transfer is inhTrently a two-electron process. In the Forster mechanism, H , is the electrostatic dipole-dipole interaction between the electrons in the initial and final states. The perturbation for the Dexter mechanism is the electrostatic interaction between electrons (requiring an account of the quantum mechanical indistinguishability of like particles). The Born-Oppenheimer approximation allows the electronic (4) and vibrational 01) parts of the wave function (Y) to be separated. If the perturbation operator for each of the mechanisms acts only on the electronic part of the wave function, eq 13 may be rewritten as
The final rate expressions for the Forster and Dexter mechanisms can be written to coincide with this latter form: P{d*a,da*}(Forster) =2 ~ (3h5c4Qa } f ( y f Q d E 8dR ‘n4xd (154 (Dexter) = F{ 2n y Ye4 e~p(?]}f(~:(~dE ‘{d*a,da*} K
Ro
E ( 15b)
The parameters in eq 15a include n, the refractive index of the solvent; R, the donorlacceptor distance of separation; Qa, the integrated area of the acceptor absorption spectrum; rd, the lifetime of the excited-state donor; K, the solvent dielectric constant; L, the sum of the average Bohr radii for the donor and acceptor; and Y, a dimensionless quantity,
