177
J. Phys. Chem. 1982, 86, 177-181
lished about the behavior of the semiconductor/liquid lization of lower energy radiation. A number of examples interface and many new materials have been investigated. of sensitized processes a t semiconductors have been deParticulate systems have been devised and a number of scribed, but so far the observed efficiencies have been different photocatalytic and photoelectrosyntheticmethods rather low because thin sensitizer layers do not absorb have been developed. While the goal of an efficient and sufficient quantities of light and thicker layers tend to be stable system for the direct solar production of fuels reresistive. In some cases, recombination processes within mains an elusive one, semiconductor-based systems remain the sensitizer layers and dye stability also appear to be the most efficient chemical systems described so far for problems. However, this approach, particularly with particle systems, is still under active i n v e ~ t i g a t i o n . ~ ~ ? ~such ~ reactions. Moreover, research in photoelectrochemistry has led to an improved understanding of the semiConclusions conductor/liquid interface and the photoprocesses which About a decade has passed since Honda and FujishimaS occur there. It has also provided new insight into a variety suggested that PEC cells based on single-crystal n-type of processes (electrochemical, photographic, catalytic, Ti02 might be used for the photodecomposition of water photolytic). to H2 and 02.During this period much has been estabAcknowledgment. The support of the National Science (CHE 8000682), the Robert A. Welch FounFoundation (38)F-R. and A. J. Bard, J. Am. Chem. SOC.,101,6139(1979). dation, the Office of Naval Research, and the Solar Energy (39)A. Fujkhima and K. Honda, Bull. Chem. SOC.Jpn., 44, 1148 Research Institute is greatly appreciated. (1971);Nature (London),238, 37 (1972).
ARTICLES Excitation Energy Dependence of the Phosphorescence to Fluorescence Quantum Yield Ratio of NaNO, Crystal. An Interplay between the Energy-Dependent Intersystem Crossing and Vibrational Relaxation Fumlo Kokal and Tohru Azuml' W r t m e n t of W m k W , Facuity of Science, Tohoko University, Sendai980, Japan (Received July 14, 1981; In Final Form: September 29, 198 1)
The fluorescence and the phosphorescence of neat NaNOz crystal is observed at 4.2 K. The potential energy surfaces of the 'Bl(n?r*)and 3Bl(ns*)states are determined on the basis of the vibronic structures. The O-N-O bond angle differs from that of the ground state by 9.6' for 'Bl(n?r*) and 8.7O for 3Bl(n~*).The N-0 bond length differs by 0.05 A for either state. The excitation energy dependenceof the phosphorescence to fluorescence quantum yield ratios &/&) are determined at various excitation energies between 26 000 cm-' and 34000 cm-'. As the excitation energy increases,the &/& ratio increases. Especially,a remarkable increase at -28000 cm-' is noted. It is shown that none of the previously suggested mechanisms account for the sharp increase of the 4p/4F ratio at excitation energy as low as -28000 cm-'. From a model calculation we conclude that this excitation energy dependence of the &/& ratio is ascribed to an interplay between the energy-dependent intersystem crossing and vibrational relaxation.
-
1. Introduction
-
-
The quantum yield of the 'Bl(n?r*) 'Al fluorescence' and the 3Bl(na*) 'Al phosphorescence2 of NO, (the x axis being perpendicular to the O-N-0 plane) has been shown to depend on the excitation energy.'~~" It is now (1)s.Makishima,T. Tomotsu, M. Hirata, S. Hayakawa, K. Hasegawa, R. Kambe, and S.Shionoya in 'Luminescence of Organic and Inorganic Materials", H. P. Kallman and G. M. Spruch, Ma., Wiley, New York, 1962,p 443. (2)H. J. Maria, A. T. h t r o n g , and S. P. McGlynn, J. Chem. Phys., 48,4694 (1968). (3)K.K.Rebane,R. A. Avarmaa, and L. A. Rebane, Zzu. Akad. Nauk SSR, Ser. Az.,32,1381 (1968)(Bull. Acad. Sei. USSR, Phys. Ser. (Engl. Traml.), 32, 1281 (1968)). (4)R. M. Hochetraseer and A. P. Marchetti, J . Chem. Phys., 50,1727 (1969). ( 5 ) S.E.Clark and D. S. Tinti, Chem. Phys., 53, 403 (1980). 0022-3654/82/2086-0177$01.25/0
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understood that this excitation energy dependence of the quantum yield is due to the fact that the rate of the intersystem crossing from the highly excited (vibronic or electronic) singlet state to triplet states depends on the energy of the initially prepared singlet state. As to the mechanism of this energy-dependent intersystem crossing, however, a variety of interpretations have been given previously, and it appears that no unambiguous interpretation has yet been established. In 1962 Makishima et al.' reported that the fluorescence quantum yield of NaNOz crystal decreases when excitation is carried out to higher vibrational levels of the first excited singlet state, 'Bl(n7*). Similar observation was also reported by Rebane, Avarmaa, and Rebane3 for NO2- in alkali halide (KC1, KBr, and KI) hosts. This observation indicates the existence of an energy-dependent nonradia0 1982 American Chemical Society
178
Kokai and Azumi
The Journal of Physical Chemistry, Vol. 86, No. 2, 1982
tive deactivation process. However, as to the nature of the nonradiative process, no interpretation has been given by the two sets of authors mentioned above. This question was partly solved when Hochstrasser and Marchetti4 observed the excitation energy dependence of the phosphorescence intensity. They observed that the phosphorescence to fluorecence intensity ratio (referred hereafter as P/F ratio) increases from "'/50 exciting at 360 nm in the region of lBl(na*) to exciting at 290 nm. In view of the e~pectation'~~J that the absorption to the second excited singlet state lA,(n?r*) exists at around 290 nm, Hochstrasser and Marchetti interpreted that the excitation at 290 nm opens up the strongly allowed inter3Bl(n~*). system crossing route 'A,(n?r*) Somewhat different interpretation was given by Clark and Tinti! who studied the phenomenon in detail for NO2in the NaHC02host. They noted that no direct evidence was found in their spectra for the existence and the location of the 'A,(n?r*) state and that the P/F ratio starts increasing a t excitation energies below the anticipated location of the 'Az(na*) state. In view of their experimental results* carried out for NOz- perturbed by Ag+ ion that the intersystem crossing preferentially populates the T, spin sublevel and that the line width of the 'B,(n?r*) 'Al absorption spectrum increases as the energy increases! Clark and Tinti concluded that the enhancement of the intersystem crossing at higher energy is due to the strongly allowed process 'Bl(n?r*) 3A2(n?r*).From the energy at which the line width of the 'Bl(n?r*) 'A, increases, they estimated that the 3Az(nr*)state is located -2000 cm-' above the 'Bl(n?r*) state. This estimate, however, is questionable. In view of the theoretical expectation9J0that the singlet-triplet separation is only of the order of 0.1 eV for the A2(n?r*)configuration, together with the experimentally known locati~n'?~*~ of the lA,(n?r*) state, we believe that the 3A2(n~*) state should be located at much higher energy, at least -6000 cm-l above the 'Bl(n?r*) state. Thus, we believe that the enhancement of the P/F ratio at the excitation energy -2000 cm-l above the lB,(n?r*) state should be governed by a mechanism in which no higher singlet or triplet states play an important role. Recent suggestion by Sildos, Rebane, and Peet'l is illuminating in this respect. They noted that the energy at which the P/F ratio significantly increases corresponds to the third quantum state of the bending vibration, v2, in the 'Bl(n?r*) state. In view of this observation, they suggested that the potential energy surfaces of 'Bl(n?r*) and 3Bl(n?r*)cross at the third quantum state of the v2 vibration and that, because of this potential crossing, the efficiency of the 'Bl(na*) 3Bl(n7r*)intersystem crossing is enhanced significantly. Even though this process is made allowed only by vibronic perturbation involving b2 vibration, if the potential surfaces ever cross, the significant increase of the Franck-Condon factor may account for the enhancement of the intersystem crossing. It appears, however, that the potential crossing is unlikely to take place. The crossing would not occur unless the potential minima for the 'Bl(n?r*) and 3Bl(n?r*)states largely differ, which we believe to be unrealistic. Discussion outlined above impells us to believe that
-
-
-
-
-
(6)S.J. Strickler and M. Kasha, J. Am. Chem. SOC.,85, 2899 (1963). (7)W.G.Trawick and W. H. Everhardt, J. Chem. Phys. 22, 1462 (1954). (8)S. E. Clark and D. S. Tinti, Chem. Phys. Lett., 60, 292 (1979). (9)K. L. McEwen, J. Chem. Phys., 34, 547 (1961). (10)P.A. Benioff, J. Chem. Phys., 68,3405 (1978). (11) I. R.Sildos, L. A. Rebane, and V. E. Peet, J. Mol. Struct., 61,67 (1980).
FI uorescence
- o=- -
ON
a
"
I-
i
w
Z
3 0 w
m
Ce
2
n
i
*
n
-
e
ON
m
m
I
(bl Phosphorescence
> -
c a
-I
w
(z
15
16
17
I8
19
20
21
WAVENUMBER/
22
23
24
25
26
1 0 3 c ~ l
ngUe 1. Fluorescence (a) and Phosphorescence(b) of NaN02crystal at 4.2 K, the N2laser being the excitation source. The figures shown near the peaks denote the separations from the 0-0 band in cm-'and the Vibrational assignments. The 0-0 bands of the fluorescence and the phosphorescence are located at 25977 and 18958 cm-', r e spectiveb. he accuracy of the band tocation is f4 cm-' around the onset of the fluorescence and f3 cm-' around the onset of the phosphorescence. The ordinate scale of the phosphorescence is expanded to 20 tlmes.
further investigation is necessary to establish the mechanism of the energy-dependent intersystem crossing. In view of this understanding, in the present paper, we try to study the excitation energy dependence of the P/F ratio in more detail. Further, noting the importance of obtaining exact geometries of lBl(n?r*) and 3Bl(n?r*)states, we endeavor to determine the shapes of potential energy surfaces for these excited states from the vibrational structures of the fluorescence and the phosphorescence. The present paper is exclusively concerned with neat single crystals of NaN02. For this system a lot of studies have been carried out in the past on the nature and the locations of the excited ~ i n g l e t ~ , ~and * ~ Jtriplet4J5J6 ~-'~ states and the vibrational frequencies in the ground12 and excited4J2states. 2. Experimental Section Sodium nitrite was purified by recrystallization from aqueous solution followed by zone melting. A single crystal was grown by the conventional Bridgeman technique. The fluorescence and the phosphorescence were observed by a Spex 1702 monochromator equipped with an EM1 62565 photomultiplier tube. The exciting light from either a high-pressure mercury lamp or a xenon lamp was passed through either a Spex minimate or a Nikon G250 monochromator. Since the lifetime of the phosphorescence is on the order of microseconds, the phosphorescence cannot be separated from the fluorescence by means of a conventional phosphorwope. Therefore, a Par 160 boxcar integrator was used to isolate the phosphorescence by time-resolved spectroscopy, a Molectron UV 400 nitrogen laser being used as the excitation source. All of the measurements were carried out at 4.2 K. The wave(12)J. W.Sidman, J. Am. Chem. SOC.,79,2669,2675 (1957). (13)H.Yamaahita and R. Kato, J. Phys. Soc. Jpn., 29, 1557 (1970); M. Kamada, M. Yoshikawa, and R. Kato, ibid., 39,1004 (1975). (14)M. Kamada, T.Yoshimura, and R. Kato, J . Phys. SOC.Jpn., 42, 1660 (1977). (15)W.C. Allen and R. N. Dixon, Trans. Faraday SOC.,65, 1168 (1968). (16)W.Dietrich and D. Schmid, Phys. S t a t u Solidi B, 74,609(1976); J. U.von Schiitz and W. Dietrich, Chem. Phys. Lett., 51, 418 (1977).
Excitation-Energy-Dependent 4 p/4 of NaNO,
The Journal of Physical Chemistry, Vol. 86, No. 2, 7982 179
length-dependent sensitivity of the detector system was calibrated by means of a tungsten lamp of known intensity distribution.
3. Results Structures of the Fluorescence and the Phosphorescence. The fluorescence spectrum observed at 4.2 K is shown in Figure la. The general feature of the spectrum agrees with that reported previously.'J2J4 The spectrum consists of long progression of the totally symmetric bending vibration (v2). Each member of progression is accompanied by one quantum of the totally symmetric stretching vibration (vl) and some phonon side bands. The 0-0 band disappears completely because of the reabsorption;14 however, its location may be estimated from the locations of the vibronic bands and the phonon side bands. The estimated location of the 0-0 band is 25977 cm-l, which exactly coincides with the onset of the fluorescence excitation spectrum and with the 0-0 band of the 'Bl(na*) 'A, abs0rpti0n.l~The locations of the vibronic bands measured from the estimated 0-0 band are analyzed in terms of the following formula:
-
Y
= u/nl"
+ upn/
- x1/nl"n2/1 -
(1)
+ i 0X
0
'1
i
4
0
26
1
1
27
28
I
I
I
1
1
I
29
30
31
32
33
34
E X C I T A T I O N E N E R G Y / 10~cm-l
Figure 2. Excitation energy dependence of the phosphorescenceto fluorescence quantum yield ratio, 4 p/q5 F. The data are based on the number of quanta emitted in the spectral region shown in F w e 1. The vertical line at each point indicates the error limit.
4. Discussion
Bond Angles at 'B1(n?r*)and 3Bl(n?r*)States. In order where n? and &'' are the quantum numbers corresponding to understand the dynamical behavior of the excited states, to the u1 and v2 vibrations. Double primes refer, as usual, it is important to obtain information about the geometries to the ground electronic state. From the least-squares fits of the excited states. As to the 0-N-O bond angle change of the experimental data, we obtain the following results: brought about by the excitation, previously published rev; = 1330.3 cm-', Y/ = 831.5 cm-', x12/1 = 9.3 cm-', and sults appear to be inconsistent with the spectral characx22/1 = 0.73 cm-'. teristics shown in Figure 1. As is seen in Figure 1, the The phosphorescence spectrum, separated from the Franck-Condon maximum in the v2 progression lies at the fluorescence spectrum by time-resolved spectroscopy, is 2: band in the fluorescence, whereas the maximum lies at shown in Figure lb. The general feature of the spectrum 2; band in the phosphorescence. It is therefore exthe again agrees with that reported previ~usly.~ The phospected that the bond-angle difference is larger for the phorescence starts at the location between the 2: and 28 'Bl(na*) state than for the 3Bl(n?r*)state. However, in bands of the fluorescence. The 0-0 band is located at the previous literature, the bond-angle difference for the 18958 cm-' which coincides with the 0-0 band of the 3Bl(na*)state (14 f 2O as determined by Hochstrasser and 3Bl(n~*) IAl absorption spectrum observed by HoMarchetti4) is estimated to be larger than that for the chstrasser and M a r ~ h e t t i .The ~ phosphorescence is ob'B,(na*) state (9 f 4O or 12O as determined by Sidman12 served only up to the 2 j vibronic band; the lower energy and Kamada, Yoshimura, and Kato,14respectively). region of the spectrum cannot be obtained accurately In view of this descrepancy between the expectation and because of the significant decrease of the sensitivity. The the reported results, we have endeavored to reestimate the vibronic structure is quite analogous to that of the geometries of the 'Bl(n?r*) and 3Bl(na*)states by analyzing fluorescence except that the intensity distribution in the the spectral distribution of the fluorescence and the u2 progression is somewhat different (vide infra). Since phosphorescence. We follow the procedure described by only small numbers of the bands are observed, the anCoon, DeWames, and Loyd.17 harmonicity correction factors cannot be determined from The totally symmetric symmetry coordinates rl and r2 the phosphorescence spectrum; however, the location of (where rl and r2 refer to the N-0 stretching and 0-N-0 the vibronic bands measured from the 0-0 band of the bending, respectively) are related to the normal coordinates phosphorescence is consistent with eq 1 with a set of paq1 and q2 by the 2 by 2 matrix L as follows: rameters obtained above from the fluorescence spectrum. Excitation Energy Dependence of the Phosphorescence to Fluorescence Quantum Yield Ratio. The phosphores= cence to fluorescence quantum yield ratios ((fJp/C$F) obtained a t various excitation energies are shown in Figure The L matrix is evaluated in terms of the vibrational 2. The data are based on the number of quanta emitted frequencies12of l4NO< of and 16N02-and the equilibrium in the spectral region shown in Figure 1. Since the bond angle in the ground state in the following way: phosphorescence spectrum is recorded only to the 2: vibronic band, the obtained ratio is nearly two-thirds of the 5.486 X lo9 1.139 X lo9 true f#Jp/@F ratio. L= (3) -9.293 X lo9 1.055 X l o 9 Figure 2 reveals, as has been pointed out previou~ly,~*~ that the ratio tends to increase as the excitation By analysis of the spectral distributions in the fluorescence, energy increases. The increase is only slight below 28 OOO the displacements of the potential minimum are obtained cm-l, a t which significant increase of the $p/& ratio is for the 'Bl(n?r*) state as follows: observed. As the excitation energy increases further, the ratio only gradually increases, and finally, at the 34 000-cm-' excitation, the ratio is approximately 7 times (17)J. B. Coon, R. E.DeWames, and C.M. Loyd, J. Mol. Spectrosc., as large as the ratio at the low energy limit. 8, 285 (1962).
-
I:[ I;:[.
[
-
-
]
180
The Journal of Physical Chemlstry, Val. 86, No. 2, 1982
Os2
1 ialfluorescence
Kokai and Azumi l
J-
o observed
the ( b P / ~ Fratio starting at the excitation energy around 28000 cm-'. An alternative mechanism which appears energetically accessible is the one which involves the intersystem crossing route of 'Bl(na*) 3B2(a7r*). If this process takes place, the T, triplet sublevel should be preferentially populated. Our effort to determine the populating rates by ODMR experiments has been unsuccessful for neat NaNO, crystal. We therefore tentatively assume that the populating rates determined by Clark and Tinti5 for the Ag+ perturbed NaN02,for which the T, sublevel is preferentially populated, are applied to neat NaNO, as well. Then, the mechanism involving the lBl(na*) 3 B 2 ( ~ a * ) route is discarded. We thus conclude that, at least in the excitation energy region shown in Figure 2, the higher singlet and triplet states do not act as the initial or final states of the intersystem crossing. The excitation energy dependence is therefore considered to be reflected by the nature of the initially prepared vibronic state. In an isolated molecule case the excitation energy dependence of the nonradiative transition probability is commonly o b s e r ~ e d .In ~ the ~~~~ present case, the system under investigation is far from the isolated molecule case. However, as is revealed by the existence of the fluorescence from higher vibrational levels of l B , ( n ~ * ) ? ~the - ~ vibrational ~ relaxations from lower vibronic states of 'Bl(na*) are somewhat inhibited. Thus, the initially prepared vibronic states should be partially preserved in the time scale of the intersystem crossing. In this case, the interplay between the energy-dependent vibrational relaxation and intersystem crossing processes should lead to an excitation-energy-dependent&./& ratio. In what follows we examine the validity of this mechanism in terms of the available data. Since practically nothing is known about the dynamical behavior of the higher vibronic states, we ought to set out a somewhat simplified model. First, we assume that only the 1 9 " 9 vibronic states of 'B,(na*) are initially prepared. Second, the vibrational relaxation from the 1°2"30 state only results in the lo2"-'3O state as is expected from the harmonic approximation. Third, the intersystem crossing from the 1°2"30 state of 'Bl(nr*) goes only to the 1°2n+931 state of 3Bl(na*). In the 'Bl(na*) 3Bl(na*)intersystem crossing, the us vibrational mode should act as a promoting mode. Therefore, the number of the u3 vibrational quanta should differ by one between the initial and the final states. The above indicated final state is only one energetically accessible state. Fourth, radiative and nonradiative deactivation from the higher vibrational states of either 'Bl(na*) or 3Bl(na*) to the ground state is neglected.24 The rate constant for the vibrational relaxation from the lo2"3O vibronic state is denoted by Similarly, the rate constant for the intersystem crossing from this vibronic state is denoted by kit. Provided that the fluorescence and the phosphorescence come from the lo2O3O states of the 'B,(nn*) and 3Bl(na*),respectively, the phosphorescence
-
I calculated
-
2:,2y, 2Yo 2:
2:
207 2:
2:
2:
2:
2;
2:
g
Figure 3. Observed and calculated Franck-Condon factors associated with the up progressions of the fluorescence (a) and the phosphorescence (b). The potential displacements between the ground and excited states obtained from these results are, in the dimensionless coordinate, 2.78 for 'B,(nr*) and 2.58 for 3B,(na*). X
kg1/2m
Aq2 = 2.285 X
kg1l2 m
Aq, = 3.757
Similarly, by analysis of the spectral distributions in the phosphorescence, the displacements for the 3Bl(na*) state are obtained as follows: Aq, = 3.757 X kg1i2m Aq, = 2.102 X
kg1/2m
The calculated Franck-Condon distributions are compared with the experimental results in Figure 3. The bond-angle and the bond-length differences brought about by the excitation are obtained in terms of the Aq's and the L matrix. The final results for the bond-angle differences are 9.6 f 0.4' for the lBl(n?r*) state and 8.7 f 0.4' for the 3Bl(na*) state. For either of these states, the bond-length difference is estimated to be 0.05 A. Whether the bond angle increases or decreases upon the excitation cannot be determined from the spectral distribution alone. However, in view of the theoretical predication of Walsh,'s we expect that the bond angle is larger in the excited states. Mechanism of the Excitation Energy Dependence on the Phosphorescence to Fluorescence Quantum Yield Ratio. Having determined the geometries of the 'Bl(na*) and 3Bl(na*) states, we are now prepared to discuss the excitation energy dependence of the phosphorescence to fluorescence quantum yield ratio. We first examine the three mechanisms previously s ~ g g e s t e d . ~ f As i J ~is discussed in section 1, the mechanism involving the intersystem crossing route of 'A2(na*) 3Bl(n?r*) or 'Bl(nx*) 3A2(na*)is unlikely in view of the theoretically expected locations of '4(na*) and 3A,(na*) states. Then what about the mechanism that is due to the potential crossing between the 'Bl(na*) and 3Bl(na*) states? As is revealed from the excited-state geometries determined above, in which the bond angle differs only by 1' between the two states, the potential crossing never takes place. (Here we assume that potential displacements with respect to the ground state are of the same sign between the two states. The displacements to the opposite direction make the potential surfaces cross, but such a possibility is unrealistic.) We thus conclude that none of the previously suggested mechanisms accounts for the sharp increase of
-
(18)S. D.Walsh, J. Chem. Soc., 8,2266 (1953).
-
-
(19)S.F. Fischer, A. L. Stanford, and E. C. Lim, J. Chem. Phys., 61, 582 (1974). (20)K. F. Freed, Top. Appl. Phys., 15, 23 (1976). (21)P.Saari and K.Rebane, Solid State Commun., 7,887 (1969);R. Avarmaa and P. Saari, Phys. Status Solidi, 36, K177 (1969);K.K.Rebane, R. A. Avarmaa, L. A. Rebane, and P. M. Saari, 'Proceedings of the International Conference on Light Scattering in Solids", M. Balkanski, Ed., Flammarion Sciences, Paris, 1971,p 72. (22)Ya. Yu. Aaviksoo, A. 0. Aniyalg, P. M. Saari, and T. A. Soovik, Sou. Phys.-Solid State (Engl. Transl.), 19,477 (1977). (23)M.Hangyo, H. Yamanaka, and R. Kato, J. Phys. SOC.Jpn., 50, 895 (1981). (24)The quantum yield of the fluorescence from higher vibrational levels of lBl(nw*) is less than lo", which is in our treatment certainly neglected.
Excitation-Energy-Dependent
4 ,,/@I of NaNO,
The Journal of Physical Chemistty, Vol. 86,No. 2, 1982 181
TABLE I : Calculated Franck-Condon Factors F" Apociated with the Intersystem Crossing from the 1°2n30 Vibronic Vibronic State of 'B,(nn*) to the 1°2n+931 State of 'BB,(nn*)=
n 0 1 2 3 4 5
Fn/10-36 9.99 9.89 x 5.39 x 2.13 x 6.86 X 1.90 x
10 102 103
lo3
n
6 7 8 9 10 11
'I 14
calculated
O Observed
Fn/10-J6 4.70 x 1.06 X 2.24 x 4.43 x 8.33 x 1.49 X
0
lo4 10' 105 105 105
0 0
lo4 lo6 a The calculation is based on the displaced and distorted harmonic oscillator model in which the geometris and the force constants determined in this paper are used. 26
to fluoreecence quantum yield ratio at the 1"2n30excitation, ( $ ~ p / & ) ~ ,is
expressed as
1
I
,
I
,
I
I
I
27
28
29
30
31
32
33
34
E x c I TAT LON E N E R G Y / i o%m-'
Flgure 4. The calculated excitation energy dependence of 4 p/r$ ratb based on a model discussed in the text. Experimental data are also plotted.
where
rn=
fi Ym
m=l
Ym = k3b/(&
k!&)
(6)
and ~p is the fluorescence lifetime. In order to calculate eq 4, one should know the m dependence of Y ~ Each . term on the right-hand side of eq 6 is estimated as follows. As to the vibrational relaxation, we assume the relationship kzb = mk&b (7) as is expected in the harmonic approximation. As to the intersystem crossing we assume the relationship & / k L = Fm/P (8) where Fm refers to the calculated Franck-Condon factor associated with the intersystem crossing from the 1°2m30 of 'Bl(n?r*) to the 1°2m+g31of 3Bl(n~*).Further, k;, is expressed as ktc = ( ~ P / ~ F ) & F / ~ P ~ P (9) where T~ is the phosphorescence lifetime and kF and k p are the radiative rate constants for the fluorescence and the phosphorescence, respectively. We thus obtain
In this way, eq 4 is now expressed in terms of the calculated ratios of the Franck-Condon factors and the experimentally known parameters. The Franck-Condon factors calculated on the basis of the geometries and force constants determined above are shown in Table I. k!&
has been estimated by Aaviksoo, Aniyalg, Saari, and Soovik22to be -1O'O s-'. The other parameters are deO kF = lo6 s-'; termined anew as follows: ( c # J ~ / ~ F )= kp = 30 S-'; 7F = lo-* S; and 7p lo4 9. The relative c # J ~ /ratios &~ thus calculated are plotted in Figure 4 together with the experimental data. As is shown in Figure 4, the calculated 4p/& ratios exactly agree with the experimental data in the excitation energy region below 29 OOO cm-'. We thus believe that the excitation energy dependence of the 4P/4Fratio is ascribed to the mechanism discussed here. Above the excitation energy of 29 000 cm-', the discrepancy between the calculated and observed c#+/& ratios becomes significant. The experimental c#+/& ratios do not increase significantly above 29 000 cm-l, whereas the calculated 4p/& ratios continue to increase. This discrepancy appears to be due to the violation of the simple model adopted in the calculation. At the excitation energy of -29 OOO cm-', the uz vibrational mode is excited to its fifth quantum state, at which anharmonic effect may be sufficiently great to induce vibrational relaxation other than the one mentioned above. Thus, at higher energy, k$b is expected to increase more rapidly than the relation of eq 7. This may account for the observed near constancy of the 4p/& ratios at higher excitation energy.
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Acknowledgment. T.A. has realized the importance of the problem discussed in this paper through conversations with Professor S. P. McGlynn of Louisiana State University, to whom our thanks are due. We also acknowledge the benefit of discussions with Professor R. Kato of Kyoto University, Professor S. Shionoya of the University of Tokyo, and Professor T. Nakajima and Dr. Y. Fujimura of this Department.
