new accesslbillty of the lowest quintet state of organic … · 2012. 1. 18. · (s,) and...
TRANSCRIPT
Chemical Physics 78 (1983) 1-16 North-Holland Publishing Company
ACCESSlBILlTY OF THE LOWEST QUINTET STATE OF ORGANIC MOLECULES THROUGH TRIPLET-TRIPLET ANNIHILATION; AN INDO Cl STUDY
and
Received 19 January 1983
By the spin-allowed annihilation of two metastable triplet states (triplet-triplet annihilation - TT.4) one ekctronic ground state (S,) and one electronically excited singlet (S,) or triplet (T,) or quintet (QA) state are crearsd. pro\?ded the Sum oi :he excitation energies of the two metastable triplet states is sufficient for the creation of ths particular excited state. On the basis of semi-empirical calculations of the excitation energies of T; and Q, of forty-six conjugated or_ganic compounds it is show1 that Q, of benzene and some other compounds should be accessible through annihilation of like triplets (homo-lT.4). and th.tt Q, of many compounds should be accessible through annihilation of unlike triplets (hstsro-TT.4). The population of Q,. competing with that of S,. should cause an unusual magnetic-field drpendencr of the delayed fluorescence S, -S,,. In favourable cases. the population of Q, should lead to an inverse (positive) magnetic high-field effect on the dela\ed fluorescence.
1. Introduction
Organic molecules with an even number of elec- trons usually have singlet electronic ground states (S,) and electronically excited states of singlet. triplet. quintet,. _ _ multiplicity. Nearly nothing is known on electronically excited states with a mul- tiplicity higher than triplet. In particular. in no case is a lowest electronically excited quintet state (Q,) known (molecules with a quintet electronic ground state Q. are known [l-4]. but we do not refer to such molecules in the present paper). This lack of knowledge is a consequence of the fact that Q, always lies above the lowest excited singlet state (S,) and above the lowest excited triplet state (T,). These facts imply that the efficiency of inter- system crossing Sj -+ Ti -+ Q, must be very low
” Presrnt address: Department of Chemistry/D% Universit> of Pennsylvania. Philadelphia, PA 19101. USA.
because of the competing and in general \-cry fasr internal conversions of the upper excited states S, and T, into S, and T,. respectively (Kasha‘s rule). Hence. compared to the population of S, and T, by optical excitation. no general and efficient method for thz population of Q, is available. In contrast to T,. Qt in general cannot be mctastable because of intersysrem crossing Q, --) TA to lower-lying triplet states T,. Hence detection tnethods that depend on the metastability of the excited state to detect. cannot be used for the detection of Q,_ In particular. it should be very difficult to detect Q, of any compound by the phosphorescence Q, ---) SO or by electron spin reso- nance.
A spin-allowed process which in energetically favourable cases can lead to Q, is triplet-triplet annihilation (TTA). A pair of interacting mole- cules in their lowest and merastable triplet states T, and T; may have singlet. triple: and quintet
0301-0104/83/0000-0000/$03.00 8 1983 North-Holland
character [5]. Therefore. the creation of excited singlet states S, or S; or excited triplet states T1. or
T; or excited quintet states Q,,, or Q:, by TTA,
‘-‘-‘(T, .__T;) E ‘(S,...S~) or ‘(S,...S;) (la)
3(T,...S~)or’(S0...T;) (lb)
5(Q,,z-.- Sh) ors(S,...Q:,).
(lc)
is spin allowed and will take place. provided the excimtion energy of the particular excited state does not exceed the available excitation energy E(T,) + E(T;). The TTA of like molecules (T, = T;) is called homo-TTA. and TTA of unlike mole- cules (T, t T;) is called hetero-TTA. In the pre- sent paper. the simple term TTA will either mean homo-ITA or imply that the distinction between homo-ITA and hetero-TTA is not essential in the particular context.
The singlet reaction channel (la) of TTA can be monitored directly through observation of the SO-
called delayed fluorescence S, -Se [6]. If in the primary process an upper excited singlet state Si (i > 1) is populated, also a delayed fluorescence S, --t Se can be observed [7]. which. however, in general is very weak because of very fast internal conversion S, -+ S,. The existence of at least one additional reaction channel can be inferred from the fact that the probability of creation of one
excited singlet state by the annihilation of two triplet states is in general less than unity (8 131. Of the two remaining TTA channels, the triplet chan- nel (lb) must always contribute to the total TTA rate because there are always triplet states accessi- ble by TTA. Indirect information on the possible contribution of the quintet channel (lc) to the total TTA rate can be obtained from the magnetic-field dependence of the delayed fluores- cence [5,14-241 (see section 3). There is no doubt that in anthracene single crystals, that is, in the most thoroughly investigated system [ 14,15,24], Q, is not accessible through TTA.
The question, whether the lowest excited quin- tet state of any organic molecule can be populated by homo-TTA, has been answered differently. Stemlicht et al. [25] concluded from the results of
a theoretical calculation of Parr et al. [26] that Q, of benzene (‘A,,) is a possible final state of TTA. Suna [15] stated: “In practice. we shall always neglect XQ (that is, the quintet channel; see section 3). as there is no known system where double the triplet energy even remotely approaches the esti- mated energy of the lowest quintet state”. Lendi et al. [ 191 concluded from the observed magnetic-field dependence of the delayed fluorescence from liquid solutions of pyrene. 1,2-benzanthracene. phenan- threne, and 3.4benzpyrene [27] that with these compounds the energy of the lowest quintet state is smaller than twice the triplet energy. The only aromatic compound for which we found calculated quintet energies in the literature. is benzene [26.28.29]. If one assumes that the ab initio calcu- lations of Peyerimhoff and Buenker [29] yield the correct order of states also for states of different multiplicity. then Q,(‘A,,) of benzene lies below the lowest ‘E,, state. The excitation energy of the lowest ’ E , u state, however, is less than twice the triplet energy. Hence at least the case of benzene contradicts Suna’s statement. Wirz [30] concluded from simple HMO calculations that, apart from benzene. the lowest quintet state of other aromatic compounds (e.g. of triphenylene. pentaphene, 1.2.3,4.5,6,7,8-tetrabenzanthracene) also might be accessible by TTA.
The present investigation is an attempt to achieve some progress in the quintet-TTA prob- lem. First. we describe briefly a semi-empirical calcu:ational procedure, which allows to calculate singlet states and triplet states with a single set of parameters. Second, from the satisfactory agree- ment of calculated and experimental excitation energies of S, and T, for a variety of compounds we shall conclude that our method yields the exci- tation energy of Q, with about the same accuracy as those of S, and T,. Third, on the basis of the calculated excitation energies of T, and Q, and the experimental ones of T,, we shall discuss with which compounds Q, is likely to be accessible through homo-?TA or hetero-TTA. Fourth, we shall investigate how the magnetic high-field effect on the delayed fluorescence is changed when the quintet channel of TTA is also effective.
2. Calculation of the escitation energies of the first excited singlet, triplet and quintet states
-7.1. Computational method and parameters
The application of ab initio methods to the calculation of multiconfiguration wavefunctions and energies of excited states is limited to small molecules_ For molecules larger than benzene. and that means. for most of the molecules of interest here, semi-empirical model hamiltonians have to be used in order to reduce the computational problem to a manageable size. There exist a great variety of semi-empirical methods which differ in the choice of the model hamiltonian and the parameters and in taking into account all valence electrons or r electrons only.
In the well-known PPP T [3 I-341 and CNDO/S’ [35-401 variants most often cited in the literature. parameter sets have been optimized to yield the best agreement with selected experimental excita- tion energies_ In this sense not only semi-empirical values for certain integrals have to be regarded as parameters, but also the selection procedure for the configurations included in the final configura- tion interaction calculation. This optimization has been done separately for singlet states [40.41] and for triplet states [42-451, and hence the parameter set is applicable only to the multiplicity for which it has been optimized. Application of a parameter set to the “wrong” multiplicity gives much worse results. On the other hand, the use of different parameters for singlet states and triplet states may lead to inconsistent results. e.g. a singlet state may be calculated lower in energy than its triplet coun- terpart [46]. It is. therefore. to be expected that neither of the parameter sets will be adequate for quintet-state calculations. A quintet-optimized procedure, however. would be useless since no experimental data on quintet states are available for adjusting the parameters.
This dilemma would not exist which a calcula- tional procedure giving results of comparable ac-
i Standard abbreviations: PPP: Parker-Parr-Poplc. CNDO: complete neglect of differential ovcrlnp. INDO: intcrmcdiatc neglect of differential overltlp. S: spectroscopx. Cl: confie- oration interaction. hl0: moleculsr orbital. SCF: self-con-
sistent field.
curacy both for singlet and triplet energies for in this case it would be justified to espect calculated quintet-state energies to be of about the same accuracy as singlet- and triplet-state energies_ A method with these properties has been developed by one of us [47]. Here we restrict ourselves to a description of the essential features of the method.
The mode1 hamiltonian is of the INDO/S type. This allows for the splitting of singlet and triplet levels resulting from n + Z* excitations. In addi- tion. the interaction between ‘ZT* configurations and -‘aa* configurations is taken into account - in contrast to CNDO/S where these excitations do not mix [4S]. The parameter K which scales the r-overlap relative to the u-overlap for the calcula- tion of core-hamiltonian matrix elements. and all one-center integrals are taken from the CNDO/S procedure [40]. Electron-electron repulsion in- tegrals are approximated by the Pariser-Parr for- mula [32.33.49.50]. . Energies and wavefunctions of excited states are obtained in a CI calculation based on ground- state SCF molecular orbit&. In order LO have for the three multiplicities Cl bases of approximately equal quality-. doubly excited configurations are included also for singlets and triplets. If one takes into account only singly and doubly excited con- figurations. the ratio of the numbers of configura- tions to be taken into account in the singlet. triplet and quintet cases. respectively. is 2: 3: 1. An en- ergy-selected basis of 200 configurations \vas found
to be sufficient in the singlet case in this type of model [51]. Therefore we used 200 singlet. 300 triplet and 100 quintet configurations. respectively_ in order to maintain the relation given above.
With the values of the 0, parameters taken from the CNDO/S procedure [40]. a good correla- tion was found for singlet and triplet states of a variety of compounds: the calculated excitation energies were = 10% higher than the experimental ones. This discrepancy was reduced by changing the & parameter by = 10% from - 17.5 eV to a new value of - 16.0 eV.
2.?. Calculated excitation energies of St. Tl OJIC~ 0,:
comparison with experimental cahres
It is not possible to estimate in advance which class of organic compounds most likely contains
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molecules with the property E(Q,) G 2E(T,). Our choice of compounds for the present calculations was guided in part by experimental considerations (easily observable delayed fluorescence) and in part it was rather arbitrary. In fig. 1 the structural formulae of the selected compounds are shown, and in table 1 calculated excitation energies of S,. T, and Q, are compared with available experimen- tal values.
The calculated excitation energies refer (a) to isolated molecules in the gas phase and (b) to vertical transitions (nuclear distances remain un- changed). Experimental excitation energies refer (a) in most cases to molectdes in solution and (b) to 0,O transitions. Discrepancies between calcu- lated and experimental excitation energies up to a few 1000 cm-‘. may result from differences with respect to (a) or(b). An example for (a) is perylene (no. 14): the transition S, + S, is strongly allowed, and the red-shift of this transition resulting from solute-solvent interaction is of the order of 2000
cm-‘. An example for (b) is fluoranthene (no. 17): the 0.0 transitions of the S, =+ S, absorption band and the S, +-SO fluorescence band are weak, and the fhtorescence exhibits an unusually large Stokes shift of = 4000 cm-’ [57, graph 119C]; in this case the vertical-transition energy is expected to be considerably larger than the 0,0-transition energy. (In the phosphorescence spectrum of fluoranthene the 0,O transition is the strongest band: hence the agreement of the calculated and experimental tri- plet excitation energies is not surprising.) Triplet excitation energies do not much depend on the solvent.
The calculational error also increases somewhat with the size of the molecule due to the fact that, with increasing number of electrons in the mole- cule, a decreasing fraction of the total number of configurations is taken into account. This is equiv- alent to a decrease of the energy limit up to which configurations are taken into account.
In general calculated and experimental exci-
t .,..I .,..j .*..: . . . . : . . 10 15 20 25 30 35
S,.T, : Eexpt, /I1000crn“)
Fig. 2. Correlation plot of the calculated excitation energies E,(S,) and E,(T,) versus the experimental excitation energies E,(S,) and E,(T,). respectively. of the compounds in table 1. Benzene and the azabenzenes were not included in the correla- tion plot for reasons given in the text. A few other compounds were also omitted because no reliable values of E,(T,) were available.
tation energies agree quite well, both for S, and T,. With some compounds. however. the agreement is rather poor, the worst cases being benzene (no. 1) and its aza-analogues pyrazine (no. 30). pyridazine (no. 32) and s-tetrazine (no. 34). This fact de- mands some explanation_ It is known that the convergence of transition energies is rather poor for benzene in the CNDO CI scheme [51]. The CI
basis of the size applied here accounts already for a large fraction of the correlation energy in the S, state, while the corresponding contribution to the ground state is provided by higher-lying configura- tions not included_ This is exemplified by reduc- tion of the number of configurations to 40%. that is, to 80 singlets. 120 triplets and 40 quintets (table 1, row 2). This removes the doubly excited con- figurations coupling most strongly to S,. (The corresponding energy cut-off criterion is 12.5 eV. the same as in the larger basis for naphthalene.) The T, and Q, excitation energies are nearly unaf- fected by this truncation. in accordance with their generally lower sensitivity to correlation effects
(sre below). The calculated excitation energies in the first row of table 1 are then best explained by the assumption that the error is made for the ground state lvhich is lacking in = 5000-6000 cm -1 of correlation energy.
In the case of pyrazine and pyridazine our results can be compared to those of Chen and Hedges [SO]. These authors a!so calculated singlet and triplet excitation energies \vith a single param- eter set in INDO CL Their parameterization scheme. however. contains much more adjustable parameters than ours. and the parameters \vrre
adjusted in order to get best results for six-mem- bered N-heterocycles. (Note. that in our procedure
the change of the &. parameter is the only devia- tion from the CNDO/S standard parameteriza- tion.) In spite of this. their calculated excitation energies for pyrazine and pyridazine deviate from the experimental ones by as much as 5000 cm-‘. Obviously the excited states of these compounds cannot be properly described in the framsw-ork of a semi-empirical method restricted to valence orbital excitation. but diffuse virtual orbitals and Rydberg orbitals are required.
A correlation plot of the calculated excitation energies ( 17,) against the rxperimental ones (E,) is shown in fig. 2. The correlation lines are E,(S,) = 2140 cm-’ -i 0.963 E,(S,) and E,(T,) = 1070 cm-’ + 0.929 E,(T, ): the average differences J E,,
= (E, - EC),, are AE,_,(S,) = 1130 cm-’ and dE,,(T,) = - 320 cm -‘: the standard deviations cr of the EC values from the regression lines are a(S,)= 1960 cm-’ and a(T,)= 1300 cm-‘. The
deviations of the slopes )>I of the regression lines
from unity are not vet significant. because for both regression lines (1 - >>I )[( E,),,,, - (EC),,,] is still smaller than the respective standard deviation. The comparatively high value of rl E,,(S,) and the higher value of a(S,) (both relative to the T, correlation) result at least in part from the stronger solvent dependence of the E,(S, ) values.
For a number of compounds we have ascertained that our method yields also higher excited singlet and triplet states in the right order and with satisfactory accuracy_ As esarnples we compare calculated and experimental excitation energies of naphthalene in table 2 and of azulene in table 3. As far as reliable assignments of higher
8 B. Dick. B. Nickel / Accessibility of :he lowesr quinrer srare of organic ndecules
Table 2
Electronically excited states of naphthalene. Calculated (c) and experimental (gas. sol) excitation energies (E) refer to the singlet
ground state and are given in units of 1000 cm -‘_ The symmetry notation (Sym.) is based on x-axis = long axis. x-axis = short axis and z-axis perpendicular IO the molecular plane. Calculated oscillator strengths / refer to S, + S, and T, .- T,
No. Sym. E, / E gas E %‘d
0 I’Ag 0.00
1 I’&, 33.01 5x 10-s 32.14 a’ 32.2 h.L’ 2 1’Bz” 36.1 I 0.066 36.4 h’ 3 1’B,, 42.08 42.1 cd’ 4 Z’A, 43.79 44 5 c.d1
5 2’%” 47.82 0.235 6 2’9, 48.23 1.196 47.5 =’ 45.3 h’ 7 3’A, 48.35 48.8 =’
11 3’%u 53.59 0.043 52.5 I’ 13 4’A, 5450 55.7 Z’
25 4’ % 61.99 0.475 62.1 =’ 59-s “
I I”a_, 20.83 21.40 h’ 71.3 j’ 2 l’B,” 27.99 30.s i’ 7 2’AA, 41.70 0.005 38.7 J’
II 33B,, 46.83 0.093 45.5 J’ 26 43B,, 59.00 0.011 27 43A, 59.08 0.057 58.9”
34 PB,, 64.83 0.158 63.7”
I 15A, 48.05 2 t% 55.61 3 25A, 58.67 4 tsBJ” 59.24 5 2%” 61.96
a) Ref. [SS]. b, Ref. 1521. ‘I Ref. [Sl]. ‘) Ref. [SZ]. <) Ref. [83]. r’ Ref. (841. s’ Ref. [85]. h’ Ref. [56]. i’ Ref. [86].j’ Ref. [87]. ” Fake origin.
excited states can be made with these compounds. the agreement of calculated and experimental val- ues is good and extends beyond E,(Q,).
For the assessment of the reliability of the calculated quintet excitation energies. the follow- ing observation is important: The contribution of doubly excited configurations to the lowest triplet state was much smaller than that to the lowest excited singlet state, although the triplet calcula- tion included much more doubly excited config- urations than the singlet calculation. Thus the lowest triplet states showed in general only 0.5 to 3.0% doubly excited character (relative to the ground state), ;;rhereas the corresponding singlet states had 3 to 20% doubly excited character. In the singlet case the inclusion of doubly excited configurations is essential for obtaining the correct order of states, in the triplet case it seems to mainly correct for the depression of the ground
state. This is in line with the observation that standard CNDO/S calculations with Pariser in- tegrals give already satisfactory results for the low-lying triplet states [42,43]. Of course, doubly excited configurations may have a strong effect on higher triplet states [90]_
One can rationalize the different effect of dou- bly excited configurations on singlet states and triplet states in terms of the different correlation mechanisms involved. Triplet states always have at least two electrons in open shells, and therefore the correlation energy for these electrons is of intershell character. Singlet states. however, though expressed in open-shell configurations, could be of mainly closed-shell character when expressed in their natural orbitals. This involves the intrashell correlation energy, which is known to be larger than the intershell energy [90,91] and which, in the picture of the ground-state MO Cl approach, will
Table 3 Electronically excited staies of azulene. Calculated (EC) and experimental (E,) excitation energies refer to the singlet ground state and are given in units of 1000 cm-‘. The symmetry notation (Sym.) is based on z-axis = in-plane twofold rotation axis. .+axis in the molecular plane and y-axis perpendicular to
it. Calculated oscillator strengths f refer IO S, - S,. T, - T, and T,, + T?. EJS,) refers IO azulene in perfluorohexane at room temperature
No. Sym. EC EC / n-0
0 l’A, 0.00 1 I’B, 15.40 2 Z’A, 26.01 3 2’9, 32.5 1 4 3’A, 36.02 5 3’B, 38.24 6 4’ 9, 42.11 7 4’A, 42.16
1 PB, 14.45 2 13A, 14.91
3 23AAl 19.46 4 2”B, 21.23 5 3’B, 32.99 6 33A, 34.26 7 43B, 35.94 8 53B, 36.72
12 63B, 42.42
16 73-A I 48.63 17 7’B, 49.41
1 15A, 34.27 2 PB, 44.94
3 25B, 47.34 4 25A, 48.75
0.0 14.3 28.6 34.0 36.4
41.3
13.9 =’
?C)
38. b.F t 41.7 b.d’
0.005
0.019
0.036 0.358 0.001
0.036 0.295
f “.-I L-2
o.ooo3 0.0004 0.0001 o.OoOo 0.0322 0.0110
0.0012 0.0022 0.0001 0.0011 0.0019 0.0021 0.0063 0.0100 0.0000 0.0249 0.0299 0.0007
” 0,0-transidon of phosphorescence from azulene in a phena- zinc host crystal at 77 K 1641.
b’ Estimated from triplet-triplet absorption [88]. =) Shoulder.
” Maximum. =’ Not observed [89].
be accounted for by doubly and higher excited configurations. If one restricts the CI basis to singly excited configurations. then the correlation effects of the singlet states can be partly accounted for by the choice of the parameters. leading to different parameter sets for singlet and triplet states.
We conclude from these observations that the influence of the higher excited configurntions on the lowest excited states decreases with increasing multiplicity. We estimate that the effect of trip&
excited configurations on the calculated excitation energies of the lowest quintet states would be less than 1%. Therefore we further conclude that the present method yields the excitation energies of the lowest quintet sIates with about the sarnt
accuracy as those of the lotvest excited singlet and triplet states.
2.3. Accessibilir_r of rhe lowesr quinrer srare b_t
rriplet-triplet annihilariotl
The energy condition for the population of the lowest quintet state QI by triplet-triplet annihila- tion is
dE=E(Q,)-E(T,)-E(T;)<O. (2)
With homo-TTA T, = T; and with hetero-TTA
T, = T;. For E(Q, ) only calculated values are available. For E(T)) and E(Tj) in general both. experimental and calculated values are available. Hence. there are three possible ways for the evaluation of eq. (3) in the case of home-TT.4. The best choice is to take for E(T,) one calculated value, E,(T, )_ and one experimental value. E,(T, ). and to define
JEQT = k(Q,) - CO-,). (3) A EQTT = E,(Q,)--c(T,)-&t-T,)
=AEoT-E,(T,). (1)
The evaluation of the quintet-IT_% condition according to (4) has NO advantages: First. if the calculated excitation energies of S, and T, strongly deviate from the experimental values. this may result from a large sysrsmaric error affecting the
calculated excitation energies of all states in the same way. Esamples for this case are benzene (no. 1). p_vrazin+ [no. 30). and s-tetrazine (no. _74)_ With these compounds the agreement between the en- ergy differences EJS,) - E,(T,) and E,.(S,) - E,(T,) is much better than that between E,(S,) and E,(S,) or E,(T,) and E,(T,). Hence. by the definition (4) any s?;stematic error affecting E,_(Q,) and E,(T,) in the same way is automatically cancelled. Second. in the case of hetero-TT.L\ one would always take the experimental value for E(T;). and not the calculated one. The definition (4) is consistent with this practice.
10 B. Dick. B. Nickel / Accessihilig oj the lowest quintet stare of oganic tnolecule.~
In the following discussion we treat homo-TTA (I) and hetero-TTA (II) separately, and we restrict ourselves to those compounds for which it is known or likely that a delayed fluorescence can be observed.
.and the three annulenes (nos. 43-45). Of these compounds at least the first three should be suita- ble for an experimental investigation (with azulene a delayed fluorescence S, --, S, resulting from het- ero-TTA can be observed [92.93]).
(I) In table 1 five compounds have negative values of AEu,-,-: benzene (no. I), pyridine (no. 29). pyrazine (no. 30) pyrimidine (no_ 31) and s-triazine (no. 33). Of these compounds benzene is of particular interest. because it should be suitable for an experimental investigation, and because our calculation supports the early conjecture of Stern- licht et al. [25] that Q, of benzene might be populated by home-TTA. The absolute value of
A&-t for benzene, 4100 cm-‘, is still not large
3. Influence of quintet triplet-triplet on the magnetic-field dependence of fluorescence
annihilation the delayed
enough to predict with certainty that Q, of ben- zene is accessible through TTA. On the other hand, if we allow for an error in A EaT,- as large as *4000 cm-‘, some other compounds should also be considered as possible candidates for quintet- TTA: triphenylene (no. 11). pentaphene (no. 12).
fluorene (no. 21). carbazole (no. 22). benzotriazole (no. 37). and xanthone (no. 41). From these exam- ples follows that Suna’s [15] general statement of
the inaccessibility of the lowest quintet state through home-TTA is untenable.
In this section we investigate which magnetic- field dependence of the delayed fluorescence should be observed when quintet-TTA is energeti- cally possible and indeed takes place. There are four cases of interest: homo-TTA in molecular crystals. homo-TTA in liquid solutions. hetero- TTA in molecular crystals. and hetero-TTA in liquid solutions. Here we only treat the high-field effect in the first, the second. and the fourth case.
3.1. Homo-triplet-triplet ar~nil~ilation in molecular
ctytals
For phenanthrene (no. 7) 1,2-benzanthracene The theory of the magnetic-field dependence of
(no. 8) and pyrene (no. 13) the A EQTT values are the delayed fluorescence of pure (undoped) molec- 5300. 7700 and 10300 cm-‘, respectively_ It is very ular crystals has been developed by Merrifield unlikely that quintet-TTA can take place with one [5,16], Johnson and Merrifield [ 141. Suna [ 151. and
of these compounds, in contrast to the conjecture Sibani and Pedersen [24]. Here we follow the
of Lendi et al. [19]. (In their second theoretical simpler theory of Johnson and Merrifield [ 141. The paper on TTA [20], these authors state that the kinetic model of this theory is shown in the follow-
magnetic-field dependence of the delayed fluores- ing reaction scheme, which differs from the origi- cence of these compounds at high temperatures nal one by the explicit inclusion of the quintet does not yield conclusive evidence for quintet- channel and by allowing for the population of TTA.) upper excited states through TTA.
(II) The energy condition for quintet hetero- TTA is A EQT < E(T;). If we take as second com- pound benzene with E(T;) = 29500 cm-‘, then table 1 shows that with many compounds Q, should be accessible by hetero-TTA. Even if we allow for a large systematic error in all AEQT values of table 1, e.g. that all A Ea.,. values are too small by 4000 cm-‘, then still several compounds are left that are suitable for quintet hetero-TTA, e.g. 1,2_benzanthracene (no. 8), pentaphene (no. 12) azulene (no. 18), pentalenoheptalene (no. 19)
T, *T, $ '='IT ,... TJp 1
Mobile and uncorrelated triplet excitons T, form
pairs of correlated triplet excitons in a diffusion- controlled reaction (second-order rate constant k, )_ A triplet pair can either dissociate again (first-order rate constant k_,) or react. the reaction product being one excited singlet or triplet or quintet state. The creation of singlet excitons S, leads to an observable delayed fluorescence (DF). The basic assumptions of the theory are:
(a) The spin-lattice relaxation times are much shorter than the lifetime of the triplet excitons. and the temperature is high (e.g. > 100 K). Hence. for the present purpose. the concentrations of tri- plet excitons in the three triplet substates can be assumed to be equal_
(b) The spin-lattice relaxation times are still much longer than the lifetime of a triplet pair.
(c) Though triplet interaction is necessary for the correlation of the triplet excitons in a pair. the triplet-interaction energy is assumed to be zero.
(d) There are nine possible pair states (p = 1.2,.... 9). The pair states are eigenstates of the pair spin-Hamilton operator. but in general not eigenstates of the total-spin operator_ That means. a pair state (p) is in general a singlet-triplet- quintet mixture with amplitude factors Cl. Cf. and Cd (referring to the appropriate eigenfunc- [ions of total spin) and ]C,“]’ + ]Cf]‘+ ]CQp]’ = 1.
(e) For a given triplet pair. the effective rate constants for the three TTA channels (5) to (7) are ]CoP]‘ho, ]Cf]‘X,-, and ICSp]‘Xs. respectively. where X,, X,. and X, are multiplicity-specific first-order rate constants.
The magnetic-field dependence of TTA. and hence of the observable delayed fluorescence. re- sults from the interplay between spin-spin interac- tion in the individual triplet excitons (char- acterized by the zero-field splitting parameters D and E) and Zeeman interaction on the one hand. and from the relative magnitudes of k_ ,. Xc,, XI.. and h, on the other hand.
The second-order rate constant k, for the total singlet-TTA takes a particularly simple form in zero magnetic field and in the high-field limit. The basic equation for ks is the same in both cases and follows directly from the reaction scheme (s)-(7):
(9)
In zero field and in the high-field limit. the pair states are either singlet-quintet mixtures or pure quintet states or pure triplet states [14]_ In zero field there are three pair states with singlet char- acter_ each with l/3 singlet and 2/3 quintet char- acter. Hence from (9) follows
k,(O) = $k, x 3 k $A,
__, i fh, t $A,, -
In the high-field limit. \vith the magnetic field in off-resonance direction (see brlo\v). there are only two pair states with singlet character_ one with l/3 singlet and 2/3 quintet character_ and one with 2/3 singlet ancl l/3 quintet character [ 141. Hence from (9) follow5
k,(-r,)=$,
(With the magnetic field in resonance direction. the t\vo pair states \vith singlet character are degenerate and split into a pure single: pair state and a pure quintet pair state [ 141.)
At low triplet esciton concentmtions. most tri- plet cscitons decay by other processes than TTA. The intensity Iutz of the obscmable delayed fluo- rescence is then proportional KO k,. and the rcls- tive change of It,,.. in the high-field limit is
Obviously_ a positive high-field effect would imply X, > X, and hence would mean that quintet-TTA
takes place. (In an actual esprritnental investiga- tion it would be necessary to ascertain that TTA is the only magnetic-field sensitive process. because other magnetic-field sensitive processes. for exam-
ple quenching of triplet excitons by free radicals [94]. might also lead to a positive high-field effect.)
3.2. Homo-triplet-triplet annihilation in fluid solutions
Theoretical models for the magnetic-field dependence of the delayed fluorescence of liquid solutions of aromatic compounds were developed by Avakian et al. [ 171, Atkins and Evans [ 181. and Lendi et al. [19,20]. Here we follow the perturba- tion-theoretical approach of Lendi et al. [20]. in which the possibility of quintet-TTA is already included, and which leads to closed formulae in the limit of low viscosity (where rotational relaxa- tion of the interacting triplet molecules can be assumed to be much faster than =A). According to ref. [20], eq. (3.31). the relative change of I,,,-in the high-field limit is given by
I,,(~) - ~rxm - 64( D*)’
J&O) = 45(X,+k_,)(XQ+k_,)
x (As -X,)(X,+A,+2k_,)
[(Xs+X,+2k_,)‘+4J;] ’ (13)
where A,, ho and k_, have the same meaning as before; Jo is the singlet-quintet -splitting of the triplet pair in zero field, and D* is defined by
D* = (D’ + 3EZ)“Z, (14)
with D and E the zero-field splitting parameters of the triplet state. Again, as in the case of homo-mA in molecular crystals, a positive high-field effect would imply that quintet-‘ITA takes place.
3.3. Hetero-triplet-triplet annihilation in fluid solarions
The magnetic-field dependence of hetero-TTA in molecular crystals has been investigated experi- mentally and theoretically [95,96]. Examples of delayed fluorescence resulting from hetero-TTA in liquid solutions are known (from S, [97] and from upper excited singlet states [92,98]). However, the magnetic-field dependence of hetero-TTA in liquid
solutions has not yet been investigated, neither experimentally nor theoretically_ In the appendix we show that the perturbation-theoretical ap- proach of Lendi .et al. [20] can be easily adapted to the case of hetero-TTA. Eq. (13) remains valid in the case of hetero-TTA, if for (D*)’ the arithmetic mean is taken for the two triplet states (T,), and
CT,),,
(D*)‘= [(D;)‘+ (D,f)‘]/2. (15)
Hence the qualitative conclusion to be drawn from a positive high-field effect would be the same as in the case of homo-TTA.
3.4. Discussion
A general theory of TTA has to tackle three problems: first, to calculate for a triplet pair with definite relative orientation and distance and defi- nite pair spin state the probabilities of transitions to all possible final states [25,99.100] (final-states problem); second, to treat the effect of an external magnetic field [5.13-241 (spin problem); third. to average over all possible relative orientations and distances and over all final states (averaging prob- lem). In all theories of the magnetic-field depen- dence of ?TA [5,13-241 the tacit assumption is made that it is possible to separate the final-states problem from the spin problem and to treat it phenomenologically by introducing the spin- specific rate constants A,, h, and ho. In the case of molecular crystals this assumption is justified when TTA is essentially restricted to nearest neighbours and only one relative orientation of molecules has to be taken into account. In the case of liquid solutions it is questionable whether the separation of the final-states problem from the spin problem can be justified in the general theory of ref. [19]. Lendi et al. [19] simply state that they treat the magnetic-field dependence of TTA as a problem in spin space alone, without giving any justification. We believe that at least one has to be very cautious in deriving any conclusions from the quantitative agreement between theory and experi- ment in the case of liquid solutions with high viscosity.
If we assume that the theories in sections 3-l-3.3
are essentially correct. then the qualitative proof of quintet-TTA will depend on the existence of com- pounds (home-TTA) or pairs of compounds (het- ero-TTA) with the property ho/As > 1. One may assume that the variation of X,/X, (when quintet- TTA is energetically possible) should be about the same as that of h,/X,. When quintet-TTA cannot take place, the ratio X,/X-,- is closely related to the excited-singlet yield qs in TTA. From the fact [ 131, that in liquid solutions under similar experimental conditions qs = 0.56 for naphthalene and qs = 0.08 for anthracene. one may conclude that in general each of the three rate constants X,. h, and ho will substantially depend on the initial triplet states and on the accessible final excited states of the respective multiplicity. Moreover, since in the case of liquid solutions the A are average values with respect to relative orientation and distance, one should expect a greater variation of the h in the case of molecular crystals. Hence there is some hope that systems with the property X0/A, 3> 1 can be found.
If X, is smaller than X,. but still of the same order of magnitude, then one can try to prove X, > 0 by a quantitative evaluation of the ob- served magnetic-field dependence of the delayed fluorescence. However, at least in the case of liquid solutions [20] there is, apart from the necessary caution mentioned above, the additional problem that the theory contains only three independent combinations of the five parameters k_,, Xs, A,, Ja and D*. D* can be obtained from an indepen- dent ESR experiment. k_, cannot be measured independently_ One can only estimate the order of magnitude of k_ ,, and one may expect that in the small viscosity range, in which the theory [20] is valid, k_, is the only temperature- and viscosity- dependent parameter and has the temperature de- pendence of a diffusion coefficient, that is k_, a
T/q (T is the temperature and q is the viscosity).
4. Summary and conclusions
(1) Using an INDO CI procedure including singly and doubly excited configurations, we have
calculated the excitation energies of the lowest excited singlet states (S,). triplet states (T,) and quintet states (Q, ) for forty-six conjugated organic compounds.
(2) With most compounds calculated values (EC) and experimental values (E,) of the excitation energies of S, and T, agree satisfactorily_ Within the error limits of the method. the correlation of EC-values with EC-values is about the same for S, and T,.
(3) With compounds where the unambiguous assignment of upper excited states is possible
(naphthalene. azulene). the satisfactory agreement between calculated and experimental values of singlet and triplet states extends beyond E,(Q,).
(4) From (2) and (3) and from the consideration that triply and higher excited configurations should only little contribute to E,(Q,). Eve conclude that our INDO CI procedure yields E(Q,) with about the same accuracy as E(S,) and E(T,).
(5) On the basis of the calculated excitation energies of Q, and T, and the known esperimenrnl excitation energies of T,. we conclude that Q, of benzene is likely to be accessible through homo- TTA. and that Q, of many compounds should be accessible through hetero-TTA.
(6) If quintet-TTA takes place. and if for the multiplicity-specific rate constants 1,. X, the rela- tion X, > X, holds. then a positive magnetic high- field effect on the intensity of the delayed fluores- cence should be observed with homo-TTX in molecular crystals and with homo- and hetem-T’TX in liquid solutions of low viscosity.
Acknowledgement
BD gratefully acknowledges a research fellow-- ship of the Deutsche Forschungsgcmeinschaft and thanks Professor G. Hohlneicher for support. BN thanks Dr. J. Wit-z for his interest in the quinter- TTA problem and for helpful HMO calculations in an early stage of this investigation. We thank the Regionales Rechenzentrum Kijln for providing the necessary computer time.
14 B. Dick. B. Nickel / Accessibiliry of the lowsr quinrer sum of organic nde~~l~~
Appendix: Modification of the Lendi-Gerher- Labhart (LGL) theory for the case of hetero-tri- plet-triplet annihilation
= ~&m,“,~-“,jrrr: (_ I)“‘i--r”l
- 2j, + 1 -
We use the same nomenclature as LGL in their papers I [19] and II [20] and also refer to their formula numbers. To incorporate the case of het- ero-TTA we have to consider two sets of zero-field splitting parameters, D,. E, and D2,E,. The tri- plet-triplet interaction hamiltonian X”’ (II, 2.7) then takes the form
(A-5) and (A-6). imply that in eq. (A-4) all terms with i f k vanish. Therefore
fd’r= $ i: t: [T-(i)],,[(Tp”‘(i)],, i-1 m--2 n=-I
Xc’“(O,, szz) z z I
= C C x g2n(i)~~~~“(ai)T=‘(i). i-l nr=-2n=-I
(A-1)
xg2,*tik-,,,(i)(- 1)“‘/5. (A-7)
Taking the matrix elements of Tp from table A4.1 in ref. 1191 and taking into account (A-2), we finally get
fd’r=$c [g:(i) +2gz(i)] i=l
The only difference between (A-1) and (II, 2.7) is that gin in (II, 2.7) is replaced by g,,,(i) in (A.l) with
=&(Df+D;+3Ef+3E;)=$(D*)‘. (A-8)
g,(‘)=Di/3
and
Therefore, the extension of the LGLII theory to the case of the hetero-nA only requires the sub- stitution of D’ and E’ by the average values (Df+ D:)/2 and (Ef+ E:)/2_
g,(i)=g_$)=6-“‘Ei, i= 1.2. (A-2) References
In the perturbational treatment of the magnetic- field dependence of TTA [20], the only matrix
(‘r elements required are X ,, ( 5 d I < 9), whose abso- lute square (II, 3.21)
f”‘(.n,,JZz)=I-JCI:‘(D,,S22)IZ, 5=~!<9, (A-3)
has to be isotropically averaged:
fo”‘=(3c::‘(Q,, nz,lZ
=; ; i kml,, i.k= 1 M,&= -2 n.&= - 1
X Er;t’<k>lrlg,n(j)gzn.(k) x q,i’n(s2i)qy&&2k). (A-4)
According to eqs. (II, A3.1) and (II, A3.2)
%)A+, (Q) = $JS,,os~*~o, (A-5)
qgp)q~;,,(52)
111 PI
I31 141
PI WI
171 VI
I91
1101
IllI [I21
1131 iI41
tw iI61 1171
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