J. Phys. Chem. 1995, 99, 17680-17690
17680
ARTICLES Nonradiative Decay in Rhenium(1) Monometallic Complexes of 2,3-Di(2-pyridyl)pyrazine and 2,3-Di(2-pyridyl)quinoxaline John A. Baiano,i3* Robert J. Kessler,g Richard S. Lumpkin," Michael J. Munley,? and W. Rorer Murphy, Jr.*>+ Department of Chemistry, Seton Hall University, South Orange, New Jersey 07079-2694, Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27599-3290, and Department of Chemistry and Materials Science Graduate Program, University of Alabama in Huntsville, Huntsville, Alabama 35899 Received: June 7, 1995; In Final Form: September 15, 1995@
The photophysical properties of the low-lying, emissive metal to ligand charge transfer (MLCT) excited states of two series of complexes of the type [Re(BL)(CO)3L]S.(BL = 2,3-di(2-pyridyl)pyrazine (dpp) and 2,3-di(2-pyridyl)quinoxaline(dpq); L = N-methylimidazole, trimethylphosphine, acetonitrile, and substituted pyridines) have been investigated. These systems have been studied by emission spectroscopy, electronic absorption spectroscopy, infrared absorption spectroscopy, excited state lifetime measurements, electrochemistry, and resonance Raman spectroscopy of the emissive states. The results show that the rate constant for nonradiative decay (knr) is the dominant mode of excited state relaxation and that the "energy gap law" for radiationless decay in the weak coupling limit is obeyed by these two series of complexes. The relative ~ from onenonradiative decay rates have been evaluated utilizing the parameters EO,S,, and A Y , ,obtained mode emission band-shape analysis and ho,. The hornvalues were determined from a quantitative analysis of resonance Raman data of the emissive state. The hornvalues for both classes of complexes include contributions from a C E O stretching acceptor mode at ca. 2020-2040 cm-', which is observed in the resonance Raman of all complexes. The results of this analysis show that the CO modes make a small contribution to the overall nonradiative decay rate. The greater degree of complex solvent interactions, as evidenced by large Av1,2values, is the major factor in the large values of k,,, relative to non-carbonyl-containing complexes.
Introduction The photochemistry and photophysics of rhenium(1) complexes of mixed polypyridyl and carbonyl ligand fields, such as Re(bpy)(CO)3Cl (bpy = 2,2'-bipyridine), have been topics of scrutiny for over two decades. Early studies by Wrighton and co-workers established the rich excited state features of these complexes.',2 Of particular interest in many Re(1) complexes are the relatively intense metal to ligand charge transfer (MLCT) absorptions occurring at ca. 400 nm and the long-lived luminescent excited states observed in fluid solution at room temperature. The donor-acceptor chromophore in these complexes is a charge transfer from a low spin Re(1) d6 orbital to a polypyridyl-centered px* orbital. These excited states have often been compared with those of [Ru(bpy)312', which have a similar orbital composition. The [Re(polypyridyl)(CO)3L]+ (L = C1-, P(CH3)3, l-methylimidazole, pyridine, and substituted pyridines) series offers a number of characteristics which make them particularly useful (and in some ways superior to ruthenium(I1) polypyridyl complexes) in the study of charge transfer excited states. Two valuable characteristics are straightforward synthetic and the spectroscopic and electrochemical simplification of having a single polypyridyl acceptor ligand.'
* To whom correspondence should be addressed. Seton Hall University. Present address: Hoffmann-La Roche, Inc., Nutley, NJ 071 10. 8 University of North Carolina. l 1 University of Alabama in Huntsville. Abstract published in Advance ACS Abstracts, November 1, 1995. @
0022-365419512099-17680$09.00/0
As interest in the photophysics of Re1-tricarbonylpolypyridine complexes containing stronger n acceptor ligands has i n c r e a ~ e d , ~ - ~ . ~there , ~ . ' have ~ - ~ "been reports of Re' complexes containing the complex polypyridyl bridging ligands (BL, Figure 1) 2,3-di(2-pyridyl)pyrazine (dpp),3.4.'5.'62,3-di(2-pyridyl)quinoxaline ( d ~ q )and , ~ 2,3-di(2-pyridyl)benzo(g)quinoxaline (dpb).4 The ligands dpp, dpq, and dpb were often used for the colorimetric analysis of metals3' and are remarkable in that dpp and dpq yield bimetallic complexes of Ru(1I) having relatively long-lived, luminescent excited state^.^^-^^ Extensions of the nuclearity of such BL systems have led to a variety of supramolecularcomplexes with lifetimes and emission quantum yields similar to the bimetallic analogs.39 The series dpp, dpq, and dpb offer a number of advantages for studying the photophysical and electrochemical properties of metal complexes. These advantages include a structural relationship to bpy, relatively low lying pn* acceptor, and enhanced "environmental sensitivity" of the unbound nitrogens in monometallic complexes. This is particularly apparent in [Re(BL)(CO),L]+ due to the implicit localization of the MLCT excited state on the single BL ligand. This latter feature is a potentially useful property for application in luminescent structural probes of double-helical nucleic acids.42 The series also allows the study of the effects of systematic variations in the ligand x system on complex properties. Most of the early photophysical studies have concentrated on the complexes containing dpp, utilizing transient absorbance, electron paramagnetic resonance, and resonance Raman spec-
0 1995 American Chemical Society
Rhenium(1) Monometallic Complexes
N,
J. Phys. Chem., Vol. 99, No. 50, 1995 17681 [Re(dpp)(CO)3(L)]ClO4 (L = P(CH3)3, 1-methylimidazole, pyridine, and substituted pyridine^)^ included parallel studies on the dpq anal0gs.4~The nonradiative decay rates of this latter series of complexes have an exceptionally strong dependence on the energy gap. This report attempts to account for this dependence and extends our previous work on the dpp complexes.
Experimental Section Figure 1. Structures of the acceptor ligands.
troscopies. These studies showed that many of the properties critical to the photochemistry and electrochemistry observed in Ru'I-bpy are conserved in complexes containing dpp as the chromophoric ligand,4O but there are some distinct differences in the dpq complexes, including the additional vibrations associated with the quinoxaline portion of d ~ q . ~ I Another approach to the study of the photophysics of luminescent metal complexes utilized the energy gap law (EGL). The energy gap law (EGL) shows that the rate of nonradiative decay (knr)is exponentially dependent on the energy difference between the ground state and the .emissive excited state (the energy gap). Previous work has demonstrated that the excited state lifetimes of the MLCT excited states of Ru(II), Os(II), and Re(1) complexes are controlled by knr, so the energy gap law is a useful framework within which to study such molecules. Since the lowest energy excited state of Re'-polypyridyl complexes arises from a d(Re(1)) to pn*(polypyridyl) charge transfer, the energy gap of this transition systematically varies with the Re(1) d orbital energy levels. The energies of the d orbitals are controlled by the combined ligand field strengths of the polypyridyl ligand, the three carbonyls, and the "nonchromophoric" ligand L. The EGL has been used in the study of the nonradiative decay processes in a series of [Re(bpy)(C0)3Llnf complexes, where the energy gap was controlled by varying L3,6 and varying the chromophoric l i g a ~ ~ d Using .~,~ this method to analyze nonradiative decay in the [Re(dpp)(CO)3LIn+(L = C1-, n = 0; L = N-methylimidazole, pyridine, 4-phenylpyridine, 4-methylpyridine, trimethylphosphine, acetonitrile, (n = l)) series, we demonstrated that dpp and bpy are vibrationally similar and differ only in the relative energies of the acceptor orbital^.^ Our results were consistent with those previously reported in which the excited states of the Re1-polypyridyl complexes follow the EGL but significantly differ in slope and intercept from Os"- or Rut[-polypyridyl complexes not containing carbonyl ligands.lO,llThis difference is ostensibly due to participation of the carbonyl ligand(s) in the decay of the excited state of the Re c o m p l e x e ~ . This ~~~ participation has also been cited as a reason for the lack of vibrational structure in the emission spectra of these complexes in low-temperature glassy matrices? although structured emission under such conditions has been observed for [Os(bpy)n(C0)Xln+(X = H-, D-, C1-, n = 1; X = py, 4,4'-bipyridine, P(C6H5)3, pyrazine, n = 2).I23l3 This lack of vibrational structure has inhibited attempts to model the emission spectra of rhenium polypyridyl c o m p l e x e ~ , ~since ~ - ~ accurate energies for the medium-frequency acceptor modes are not directly available from emission data. The observation of bands in the region 2020-2040 cm-' in the resonance Raman spectra of Re(4,4'bis(ethoxycarbonyl)-2,2'-bpy)(C0)3Cl assignable to v(C0) transitions led one group to use an estimated value of 1450 cm-I for Franck-Condon analysis of the emission ~ p e c t r a . ~The participation of this mode in the acceptance of excited state energy was also cited as a major factor in the relatively short lifetimes observed for such c o m p l e ~ e s . ~ Our earlier work on the dpp complexes Re(dpp)(C0)3Cl and
Materials. All preparative solvents were of reagent grade and used without further purification. Solvents for electrochemical, laser-induced time-resolved luminescence and emission experiments were high-purity acetonitrile and methylene chloride. For electrochemicalexperiments the solvent was dried over activated 4 A molecular sieves. Tetrabutylammonium hexafluorophosphate (TBAH) was prepared from tetrabutylammonium bromide (Fluka) and hexafluorophosphoric acid (AlfaVentron). The product was triply recrystallized from ethanol and dried in uucuo at 100 "C for 6 h. All starting compounds used in the syntheses below were obtained commercially with the exception of dpq, which was synthesized from 2,2'-pyridil (Aldrich) and o-phenylenediamine (Aldrich) using literature
Physical Measurements. Elemental analyses for the complexes were performed by the elemental analysis group at Merck and Co., Rahway, NJ. Infrared spectra were recorded in methylene chloride solution on a Mattson Instruments Inc. Cygnus 25 FTIR spectrometer in a liquid cell with NaCl windows at a resolution of 2 cm-'. Electronic absorption spectra were obtained in acetonitrile on a Hewlett-Packard Model 8452A diode array spectrophotometer. Electrochemical measurements were carried out in 0.1 M TBAWacetonitrile solution with a platinum disk working electrode, a platinum wire auxiliary electrode, and a 3 M NaCl Ag/AgCl reference electrode (+0.20 V vs the normal hydrogen electrode). All cyclic (CV) and Osteryoung square wave voltammograms (OSWV) were obtained using a Bioanalytical Systems BAS-100 electrochemical analyzer. The scan rate for cyclic voltammograms was 100 mV/s. The conditions for OSWV were as follows: square wave amplitude, 25 mV; sampling points, 256; frequency, 15 Hz; step potential, 4 mV. The E1/2 values for the BLw- process were calculated from cyclic voltammetry using the equation
E,,, = (E;
+EJ/2
EPaand Epc are the anodic and cathodic peak potentials from the cyclic voltammograms, respectively. Raman spectra were measured using an ISA UlOOO 1.0 m scanning double monochromator with a spectral slit width of 5 cm-I . Photon-countingdetection employed a thermoelectrically cooled Hamamatsu R943-02 photomultiplier. All complexes were excited at 406.7 nm with radiant power in the range of ca. 60- 100 mW using a 5 W Kr+ laser (Coherent Model 190K). This excitation wavelength is in resonance with the lowest energy MLCT absorption of all complexes examined. Scattered light was collected at 135" from incidence with spinning to minimize local heating and photodegradation. Electronic absorption spectra measured before and after the Raman experiment were compared to assure sample integrity throughout the sample illumination. Sample concentrations were typically 1-5 mM and were adjusted to maximize the scattering intensity relative to the luminescence of these strongly emitting compounds. Raman spectra of most of the complexes were performed in both methylene chloride and acetonitrile solution.
17682 J. Phys. Chem., Vol. 99, No. 50, 1995 Generally, Raman spectra in methylene chloride were used for determination of ho, to maintain environmental consistency with the other photophysical experiments. In the cases where the Raman bands of a complex were obscured by methylene chloride bands, the intensities of the corresponding bands in acetonitrile were used. In these instances an intensity for the band obscured in methylene chloride was calculated by ratioing its unobscured intensity in acetonitrile to another unobscured peak found in both the acetonitrile and methylene chloride spectra. The ratio obtained from the following equation is considered solvent independent: I(CH,Cl,) =
Z(reference,CH,Cl,) x I(CH3CN) Z(reference,CH,CN)
where I(CH2C12) is the calculated intensity of the obscured peak in the CH2C12 spectrum, Z(reference,CH,Clz) is the intensity of the common reference peak in the CH2Clz spectrum, I(CH3CN) is the intensity of the corresponding peak in the CH3CN spectrum, and Z(reference,CHsCN) is the intensity of the common reference peak in the CH3CN spectrum. This calculation must be used to convert the intensity of the obscured band into the same basis as the other methylene chloride peaks to prevent compounded errors in the calculation of ho,. The equipment and techniques for measuring emission spectra, quantum yields, and excited state lifetimes have been described previ~usly.~ To validate the accuracy of the emission energies obtained on the Spex Model F222A spectrofluorimeter, the emission spectra of a number of previously reported compounds, including the low-energy luminophores [Ru(dpq)3I2+ and [0s(bpy)3l2+,were obtained and their emission maxima were compared to the literature values. The emission energies obtained on the Spex F222A were in excellent agreement with literature values even in the low-energy regime, where the correction for system response is significant. The close agreement of the [Ru(dpq)312+and [O~(bpy)3]~'emission maxima with literature particularly establishes the accuracy of the emission energies for the dpq series of complexes whose emission energies also fall in this range.40 Emission spectra were corrected for system response, smoothed, converted to wavenumber scale using the Parker and Rees method,4" and normalized using a locally developed BASIC program. Modeling of the emission spectra was performed in a separate BASIC program by calculation of an emission spectrum based on either one or two vibrational modes.33 The algorithm in this program impliments the following equations: 3.46.47
I,( E ) -= 10-0
i 2 ([(
z,,=Ovl=-b
- v,hw, - v , h c ~35'; , Eo-0
,j
Baiano et al. m, where A, is the dimensionless displacement of mode m in the excited state relative to the ground-state, b = number of hot bands included in the calculation, and Avl,? = full width at half-maximum of a single vibrational component (cm-I), where an array of emission intensities due to each individual vibrational component relative to the intensity of the 0-0 component (I&!?)/ lo-0) at each frequency is calculated. This equation includes solvent broadening of the vibrational components as a function of temperature. The contributions due to low-frequency modes and hot bands were not included. The ho, values were calculated from an intensity-weighted sum of the resonance Raman bands observed via eq 8 (see the Discussion section). Goodness of fit of the emission spectral modeling was evaluated by visually comparing the calculated spectrum with the experimental spectrum. Syntheses. Literature methods were used for the synthesis of [Re(bpy)(CO)3(L)]C104 (L = MIm, C1-, py, CH3CN),6" [Ru(bpy131(PF6)2,4' [Ru(dpp)d(PFd2,49and [Ru(dpq)31(PFd2.50Pure [Os(bpy)3](PF& was provided by Professor Karen Brewer of Virginia Polytechnic Institute and State University. The starting complex fac-[Re(dpq)(CO)3(CH,CN)]ClO4 was prepared in a fashion analogous to [Re(dpp)(CO)3(CH3CN)]C1043except Re(dpq)(CO)3C14was used as the starting material (yield 70%). The fac-[Re(dpq)(CO)3(L)]C104(L = pyridine, 4-methylpyridine, 4-phenylpyridine, 1-methylimidazole, trimethylphosphine) series of complexes were synthesized in a fashion analogous to the [Re(dpp)(CO)3(L)]ClO, (ref 3) series except for the use of chromatographed Re(dpq)(CO)3(CH3CN)]ClO4 as the starting material. All transition metal complexes used in spectroscopic or electrochemical experiments were purified before use by gradient elution (85% CHzC12/15% CH3CN to 100% CH3CN) gravity-feed liquid chromatography on Fisher A-540-500 alumina.
Results Elemental Analyses and Sample Purity. The elemental analyses for all of the Re(1) complexes used in this study are listed in Supplemental Table 1. These data verify the formulation and purity of the complexes. Electrochemistry. Table 1 contains the half-wave potentials for the first (most positive) reductions as well as the peak potentials from OSWV for the first (least positive) oxidations of all complexes examined. The most positive reductions are assigned as reversible or quasi-reversible BL localized reduct i o ~The~ least positive oxidations are assigned as Re-centered oxidations, on the basis of analogous complexes.4 The Re"" process is irreversible on the cyclic voltammetric time scale, requiring E,, values to be taken from OSWV peaks. Electronic Absorption Spectroscopy. The A,,, and 6 values for the complexes studied are listed in Table 1. The UV-vis spectra of the dpq complexes, with the exception of L = C1-, show two structured bands with maxima in the region from 260 to 380 nm. On the basis of calculations analogous to those previously described for BL = dpp," the spectra of the dpq C1complex, and those of the dpq free ligand," the lowest energy band (ca. 380 nm) in assigned to heavily overlapping n n* and MLCT transitions. The low-energy bands also display a slight solvent dependence, shifting to longer wavelengths in media of lower dielectric constant, consistent with MLCT character in these transition^.^ As noted for the dpp series of spectra, when the relatively weak field C1- ligand is replaced with stronger field ligands, the MLCT maxima occur at higher energies. These data also support the MLCT assignment. Infrared Spectroscopy. The infrared spectra of the complexes show strong absorptions in the carbonyl stretching region
-
EO-0 = energy of the 0-0 transition (cm-I), E = independent variable (cm-I), U,JI = vibrational quantum number for the "averaged" medium (m) or low (1) frequency deactivating mode, ho,, hwl = energy quantum spacing of the m or 1 deactivating mode (cm-I), S, = (Am2/2)is the Huang-Rhys factor for mode
J. Phys. Chem., Vol. 99, No. 50, 1995 17683
Rhenium(1) Monometallic Complexes
TABLE 1: Physical Properties of Re(BL)(C0)3Lfa complex
A,,/nm 400
410
[Re(dpq)(CO)dMepy)I+ [R~(~Ps)(CO)~(P~PY)I+
[Re(dpq)(CWAN)I+
254 286 400 332 264 408 332 262 398 284 394 342 286 25 8 330 282 254 449 370 266 376 264 3 80 264 380 264 3 80 276 378 266 376 264
EN-] cm-l 3600 2770 sh 16 500 16 900 3130 sh 10 300 sh 20 700 2600 sh 9640 sh 20 900 3400 sh 32 300 3730 sh 9030 sh 17 900 18 400 10 600 17 200 17 200 3410 10 600 27 000 9130 26 500 10 900 31 700 11 300 32 700 11 300 45 600 10 800 28 500 12 000 29 100
Eoa (VI
Eli2
(VI
v(C0) (cm-l)
1.48 1.76
-0.97 -0.83
2025, 1927, 1903 2034, 1938, 1926
1.94
-0.79
2038, 1945, 1930
1.94
-0.82
2038, 1945, 1930
2.01
-0.77
2038, 1945, 1929
2.02
-0.86
2040, 1959, 1930
2.05
-0.84
2042,2030~ 1946, 1938
1.55
-0.70
2026, 1927, 1903
1.87
-0.61
2035, 1941, 1934
2.02
-0.57
2040, 1943, 1934
2.01
-0.54
2038, 1943, 1934
2.01
-0.56
2038, 1943, 1934
2.05
--0.57
2041, 1958, 1926
2.05
-0.59
2044, 1950, 1945
"All A,,, E , EI12. and E$ data obtained in acetonitrile. Epavalues were measured via OSWV. Ell2 values were measured from cyclic voltammograms. The infrared data were obtained from second-derivative analysis of the spectra as methylene chloride solutions.
(1800-2100 cm-I, Table 1). The complexes show two types of IR absorption patterns based on the L-ligand donor atom. Three carbonyl bands are observed in the complexes where L is a non-nitrogen donor atom ligand (L = C1-, P(CH3)3). One band is centered at ca. 2040 cm-', with two poorly resolved bands centered at ca. 1935 cm-I. The IR of Re(dpp)(CO)sCl is typical of this pattern of three CO stretches (see Supplemental Figure la). In this case (L is a non-nitrogen donor atom) the site symmetry is C,, for which standard group theory predicts three fundamental modes: two of A' symmetry and one of A". Using the symmetry of the donor atoms (e.g. the site symmetry) to predict the CO vibrational pattern is justified since the carbonyls respond directly to the electron density on the metal center and this electron density is largely controlled by the BLand L-donor atoms. When L is a nitrogen donor, the site symmetry is essentially C3u,so the complex should display two infrared active vibrations (A, and E). Two carbonyl bands are in fact observed for the complexes where L is a nitrogen donor ligand (L = MIm, CH3CN, or pyridine and substituted pyridines). The observation of two CO stretches in the IR of C3vRe(1) has been previously observed for the [Re(bpy)(C0)3Llf series.6a This pattern is illustrated by [Re(dpp)(C0)3(py)]C104 (see supplemental Figure lb). These two bands are centered at ca. 2030-2040 and 1935 cm-I. The lower energy band in this case is broad with respect to the higher energy absorption. The second derivative absorbance spectrum with respect to wavenumber resolves the 1935 cm-I band into two components, while the higher energy band contains only one pure component.43 The values for v(CO) reported in Table 1 were obtained from second-derivative spectra. The coalescence of the low-energy peak for nitrogen
donor L-ligands is interpreted as the extent to which the structure of each complex approximates a C3v site symmetry. Emission Spectroscopy. All Re' complexes exhibit broad unstructured luminescence in both fluid solution at room temperature and glassy ethanol at 77 K. In contrast, the emission spectra of [ R ~ ( d p p ) 3 ] ~ [+R, ~~~( d p q ) 3 1 ~ + and , ~ ~[Ru(bpy)3I2+ l o display well-resolved vibronic progressions when observed at 77 K in glassy ethanol. From this observation, it is clear that unstructured emission is not due to the acceptor ligand. An emission spectrum representative of the Rer-dpq series in methylene chloride at room temperature is presented in Figure 2A. Table 2 contains the emission energies and radiative quantum efficiencies in methylene chloride at room temperature for all Re' complexes examined. Correlation of Emission Energies and Reduction Potentials. The linear correlation of the room temperature emission maximum (Eem, eV) with the difference (AE112) between the first reduction (most positive reduction; E112 (BLo/-) in V) and first oxidation (least positive oxidation; Epa (Re") in V) potentials has been used as supporting evidence for the MLCT nature of an emissive ~ t a t e . ~ , ~A. "plot for the dpq series of complexes was found to be lir1ear.4~ This observation is consistent with the assignments of the electrochemical data above as ligand-centered reductions and metal-centered oxidations. Bivariate regression statistics have been employed in analyses to establish the degree of correlation between Eemand AE112 since they are dependent variables (presumably effects of a common cause) and thus subject to experimental uncerThe best fit line had a slope of 0.34 eVN, an intercept of 0.87 eV, and a correlation coefficient of 0.985. Slopes of less than 1 are believed to arise from the differing
Baiano et al.
17684 J. Phys. Chem., Vol. 99, No. 50, 1995
Excited State Lifetimes. The excited state lifetimes t can be found in Table 2 and were obtained from two-parameter fitting of the exponential decay curves using the Gauss-Newton (GN) a l g ~ r i t h r n . ~ ~The - ~ ~rate constants for excited state radiative and nonradiative decay, k, and k,,, respectively, are given in Table 2. The quantities k, and k, were calculated using eqs 3 and 4 assuming that the quantum efficiency for intersystem crossing is equal to unity (@iqc
1000 800 X
'$ 600 +I 2 400 5 1200 2
(3)
0 13
I5
14
17
16
18
19
(4)
loo0 800
.& 2 600
.f
3 400 a
z"€
200 0 13
14
15
EnergykK
16
17
Figure 2. Experimental room temperature and calculated emission spectra of (A) [Re(dpq)(CO)3(P(CH3),)1C104 and (B) [Re(dpp)(CO)3(MIm)](C104). See text for details.
Coulombic attraction of the ( d . 7 ~ ) core ~ and the (pn*)' ligand in the excited state emission and the (d.@ core and the reduced (px*)I ligand in the ground state electrochemical mea~urements.~ The linearity of these correlations indicate that for each chromophoric ligand series the same metal and ligand orbitals are involved in both electrochemical and emission processes. The emission energies in the dpq series are consistent with the relative ligand field strengths of the nonchromophonc ligands (L). Table 2 illustrates that the lowest MLCT emission energies are associated with the weaker field n-donating C1- and MIm groups and shift to higher energy as stronger field x-accepting ligands are substituted. Ligands which possess x donor bonding characteristics decrease the energy of the MLCT emission by destabilizing the filled, predominantly metal based djt acceptor orbitals. Conversely, these orbitals are stabilized by x accepting ligands due to back-bonding interactions yielding higher emission energies.6a-'
'
It should also be noted that a common approximation for the nonradiative decay constant is k,, = l/tfor weakly luminescent molecules (a,
