- Email: [email protected]
Non-condon effects on radiationless triplet decay in benzene and naphthalene
Volume 71, number 3
PHYSICS LJX-l-ERS
CHEMICAL
NON-CONDON EFFECTS ON RADIATIONLESS IN BENZEKE AND NAPHTHALENE * Wlllem SIEBRAND Dn ~SKMZof Chetnrstry Otrarra Canada Kid
and Marek Z. ZGIERSKI #a~~onaI OR6
Research
Comml
TRIPLET
15 June I980
DECAY
’ of Canada.
Recewed 10 hIarch 1980
Model calcuhtlons that avoId the Condon approwmation are reported for T,‘++ So mtersystem perdrurerated benzene and naphthalcne In both benzenes and III naphthalenexfg, skeletal bendmg to T:! are found to be more efficient as acceptmg modes than CHjtretchmg wbratlons.
crossing in normal wbmtions couphng
and T,
Recently Scharf [I] has argued that nonradtatwe decay of trlplrt benzene is dommated by energy transfer to ezg bendmg modes and not to CH-stretchmg modes, as usually assumed. He based his conclusion on the observation that some of these bendmg modes undergo large frequency changes between the trlplet and ground state [21, thereby glvmg rise to relatively large vIbratIona overlap mtegrals and hence to efficient decay channels. These new channels would greatly increase the calculated decay rate constant [3--51, brmgmg It closer to the experimental value Smce these modes mvolve mamly CC bending, they are relatively insensitive to deutermm substitution. Th2s could help in explammg the relative weak effect of deuteratlon on the triplet lifetime of benzene compared to other small aromatic hydrocarbons [6] In a recent paper 171, we have mvesttgated the effect of modes couplmg two excited states on radiationless decay of the lower state. These calculations are relevant to the benzene problem smce the ezg modes couple the states T,(3Blu) and T,(3Eluj. Such couphng modes contrlbute not only through their mcreased overlap integrals, but also by mcreasmg the effective vlbromc coupling between Tt and the ground state S,(‘At,). This is a nonCondon effect in that It IS due to the dependence of the eiectrontc transitton matrix eiement between T, and SO on the coordtnates of modes couphng T, to T2. In this note we extend calculations of the type reported earlier [7,8] to a model system simulating triplet decay UI benzene. The present calculattons are more general tn that they tnclude another effect indticed by the coupling, namely tnduect decay of T, via T, through a second-order matrix element. To keep the calculations tractable, we consider only four modes, each of which plays a different part in the transIttons. These modes are: Q,(b[,,), promotmg T, ‘L+=So radlatlonless transition, Qb(e2&, couphng T, to T,, Qc(etu), promoting T,%-+S, transitions and ps(aIg), a representatwe accepting mode. For slmpliclty we treat all modes as non-degenerate harmonic osc&tors with the same dlabatic frequency III all three states. Except for Qb, the couplings in which they arc involved are assumed to be so weak that the dlabattc frequencies equal the adiabatic frequencies_ The promoting modes Qa and Qc can therefore accept only one quantum of energy durmg the transition, the remainder being distributed among Qb and Q,. We assume Q, not to be displaced between T, and T_, , so that this mode does not modulate the couphng through Qb_ The displacement between T; and So IS chosen such that Qs mimics the properties of a si..fold degenerate anharmomc local CH-stretchmg mode [9] in the energy range of interest. The coupling through * Issued as NRCC 18213 * NRCC Research Associate.
Volume
72, number
3
CHEMICAL
PHYSICS
LE’I-I-ERS
15June
1980
Qb IS allowed to be strong enough to account for the large frequency drfferences of the e zg modes m So and Tt , but the degeneracy of Qb and its Jahn-Teller activtty m T2 are neglected. The vibromc couphngs are Introduced through lmear adiabatrc coupling operators K,Q,, where x = a. b and c [IO] _ The weak K and KC couphngs are Included m the usual fist-order form. The resultmg first-order corrected wavefunctrons Ttf , i$ and Sb are then used to dragonahze the i T, , Tz} matrt.\, wherem the two trrplet states are strongly coupled through Qb We e\pand all electromc wavefuncttoris m terms of a drabatrc (I.e., uncoupled) basrs set +E, # and d$ such that [I I]
7:=+,q, +q&?,ql~,,cQ~>.rf=a: +&Q,~~~/~g(Qs>. $,= +_f- &Qa+:,/ti,4(Qs> - I";cQci~:/~g(Q,h where E &(Q,)= f$(Q) - E!(Q) is a diabatrc potentral energy {Q,. Qb, QC. Q,} transformatton
From these first-order
7-t = 7-i cos Q + T-f sm @. where
corrected
Tz = -T:
wavefunctions.
sm Q + T$ cos $,
(1) drfference, Q bemg the column vector we form adtabattc wavefuncttons by the umtary
qJ =s;.
(3
[IO]
tan 70 = -2K,Q,IE,S),&,)_ The correspondmg
r,,,(Q)
adtab.rttc
(3) potenttals,
wfrrch m general
= E!?s(Q) + :EIS),,JQb) - f[$ E,s),,,(Qb)]2
E,,(Q) = E,s)(@) - f E:,,,
(QbI+ I[) E:,,,
(Qb
WIII not be harmonrc,
+ K;Q;
)I ’ + K;Q;
are
I”‘, } “‘, (4)
(5)
Because of the sunphfymg assumpttons Introduced, all drdbatrc frequencres C&, wtth x = a, b, c, s and _V= ~7, II. g can be replaced by the correspondmg adtabnttc frequenctes wx except 5?,, and C&,,. To calculate the rate constant of radrattonless decay for the lowest vtbratronal level of the trrplet mamfold, we assume. as usual, that the unttal state T, IO,, 06, 0,. OJ IS an adrabatrc Born-Oppenhermer state. The spur-orbrt couphng to ‘err* and lrro*’ states IS mcluded m thts descrrptton and direct spur-orbtt couphng between T, and So ts neglected [I 2] The two components Ti and Ti of T, combtne wtth dtfferent ground-state levels, so that the radtattonless decay rate constant can be wrrtten as a sum of two terms k, = k,, + kq2. In the usual golden rule formulatton, tfrese terms are proporttonal to the square of a matrL\ element of the nuclear kmettcenergy operator TN and a denstty of final states p
(6) The densttres of states p. and p, mcorpornte the energy conservatron condttrons trons U. u and u’. u’, respectrvely. They are chosen to have the “smooth” form 412
determmmg
the allowed
combma-
Volume 72, number 3 P&Q,
usI=
r-l
CHEMICAL exp[-+
+(~,~,0000
-
PHYSICS
15 June 1980
LETTERS
4,1uOu)‘l,
The linewidth parameter r IS assumed to include Eg 1lCOUIS the ~lbroruc energy of the state SoI I,ubO&. the effec; of couphng the modes Qa. Qb, Q, and Qs to aU other modes in the molecule. The following parameter values are chosen to represent benzene [ 13.14]- wa = Rb,,, = S2b,I = o &Z 1°C = 1580cm-1.w,=2w,=3160cm-1,r=~o,= 790cm-‘,E,(O)= 18Sw,=29200cm-‘.E,, -4.5~~ = 7 100 cm-‘. To study the effect of deuterlum substltutlon, additlonal calculations have been carried out with Since no unambiguous spectroscoprc data are avarlable for the coupling constants, % D = +GJ,” = 2370 cm-l. calculations have been carned out for three Kb values, namely (I) Kb = 0, corresponding to a model III which bending modes do not act as accepting modes: (il) Kb = 2.05, corresponding to the assignment of Burland and to the alternatwe assignment proposed by Scharf [I 1. Robinson [3,133, and (in) K, = 1.68, correspondmg To emphasize that our results for Kb = 0 include non-Condon effects, we also list a rate constant A,(FC) obtamed UI the conventional manner with the coupling through Q0 caIculated via Herzberg-Teller theory so that the acceptmg mode enters the expression for the rate constant III the form of a Franck-Condon factor [3-S] _ Since we are only calculatmg relative rate constants, the results are mdependent of K,, but depend on the ratio Q!= Ki/Kz which governs the ratio k,,/kqz between the duect and induecr channels. The results are presented for CY= I but can easdy be modified to fit other values of (lr. The values of B are chosen to match the calculated Franck-Condon envelope calculated for a &u-fold degenerate local CH-stretchmg mode [9] _ Smce this IS not possible with a smgle value of B, we report calculations for the values B = I-S,? and 2.5, apploprlate to the range eg,oooU = 10 000-20 000 cm-‘, where u IS the number of CMstretch quanta accepted by Qs_ The larger u, the larger the value of B requ:red to mlmlc the overlap properties OF the anharmomc CH oscdlator. The range of values chosen IS thus suitable for monitoring the change from a weakly accepting to a dommant Q, mode. For the deuterated molecule, B should be adJusted such that the actual displacemcnt Qsg - QDn IS mvarlant. Smce B = 21/2u~/“(~sg - &), ths means that BD differs from BH by a factor (w,” /GJH)3’2 _ The tesults of the calcularlons are collected m table 1. The values m the first row denote results obtained by the conventlonal Franck-Condon approach for the model parameters chosen. They therefore provide reference values with which the new results are to be compared. The second row corresponding to Kb = 0 shows an increase of the calculated rate constant by a factor of about 30 If non-Condon effects associated with ps are included [i 1 I_ where
Table 1 Calculated relauve rate constants k,(Kb) for TIU So III benzene-/l6 and -de and naphthaleneAs and -c& for dlffecent displacement parameters BH and BD kq(FC) refers to a conventlonal calculatton III which acceptmg modes appear as Fran&-Condon factors The notation 2 9(-15) stands for 2 9 X lo-” Na hthatene B cp/EP = 2 OO/L.30
Benzene BH/BD = 1 50 IO.97
BH,BD = 2.00/l 30
2.9(-IS),4
4(-20)
3 6(-13)/l
8 7(-14)/S
l(-19)
1.1(-l
BH,BD = 2 50/1.62
9(-17)
15(-11)/2.0(-15)
1),3 7(-16)
4.9(-10),4.3(-14)
l-1(-5),9.8(-IL)
L.3(-9),4
l-5(-7)/1.06-9)
2 9(-10),3.4(-10)
3 7(-10)/4.1(-10)
l.O(-10),7.7(-11)
1 3(-IO)/6
3 4(-10)/6.9(-11)
Z6(-9),2.L:-LO)
1 0(-9)/l 3 4(-lo),2
1 4(-9)/l-4(-9)
5 O(-9),1.7(-9)
3.8<-7)/6.8(-9)
5 3(-lo)/2
2.0(-9),2.4(-LO)
l-5(-8)/1.3(-9)
2(-9) 6(-10)
a) Based on o = Ki/Kz = 1. These rate constants
ye proportional
6(-11) 3(-10)
8(-LO)
7.9(-9),8.L(-17)
to LI.
413
CHEMICAL
Volume 72, number 3
PHYSICS
LETTERS
15 June
1980
The deuterlum effect predlcted by these two rows amounts to a reductton by a factor of the order IO”-IO”. The remainmg four rows show how Qb affects these predIctIons_ Broadly speakmg, It further mcreases the rate constant and sharply reduces the deutertum effect. Expenmentally, the rate constant IS found to be much larger than predtcted by the first row calculattons. To account for the elperlmental results, an mcrease by a factor IO’-IO6 IS required [3--51. For 01~ 1. the results of rows 4-7 predict Increases in the range lo’---106, dependmg on B and Kh. relntlve to the results of the first row. If B IS restrxted to the range I S- 2, the mcrease matches the required factor 104-106. The experImentall> observed deuterlum effect on the trlplet decay rate constant IS given by the ratlo [ 15-191
where Ii,“/kp_ the ratro of rndlatlve decay rate constants. IS in the range l-1 2 [15,16]. The ratio (8) IS much smaller than m other small JromatlL hydrocxbons e g., m naphthalene, k H /k D = 8-10 [6,1.5]. In the latter molecule. CH out-of-plane bendmg modes. mductng spin-orbtt couphng, are responstble for part of the deutertum effect on namely for a factor of about 1 8 [ 12,201 A slmllar effect IS expected for benzene. Smce our model does Qliq”. not Include such modes, the e\permxntal results have to be corrected for thrs effect before they can be compared wtth the model predicttons. Thus requires mformatlon on the phosphorescence quantum ylcld k,/k. Clearly. the rJtto (8) can only be used to determine the ratio kt/ky tf At 2 kp. Dl rect measurement [ 151. as well as the observdtlon of snmlar solvent eifects on liH and kD [ 16- 191, indtcate that this condltton IS indeed met and that x-gH/ligD= I-2, so that. after correctlon for out-of-plane mducmg modes, L m-plane bendmg modes play a more modest part m other small aromatic hydrocarbons, as evtdenced by thetr Ixgcr dcuternun effects We have shown prevtously [7] that non-Condon effects become less important for smaller energy gaps between the electronic states m transttlon Since benzene has the largest T, -So gap of any dromattc hydrocarbon. Its rehance on CC bending vtbrJtlons as acceptmg modes IS expected to be excephonally strong This 1s confirmed by the results hsted tn the fmal column of table 1 based on parameter values appropriate to naphthalenc [21] E,,,(O) = 13 5 wa = ?I 100 cm-‘. E,s),,, = 5 2 wa = 9600 cm-‘_ For BH = 2 and all other pJrametcrs as m the benzene calculations. these results predict much larger tsotope effects, as Indeed observed [6.15]. although they remam smaller than predtL.ted by the conventional Franck-Condon model In naphthalene118, the ob contrtbutton IS found to rematn modest. namely 0 1-O 3, dependmg on K, and 01. but m naphthalenedg. it IS found to become dommant (0 6-0.7) We conclude therefore that for non-deuterated aromatlc hb drocarbons other than benzene. the usual ratlonahLatlon of radtattonless trtplet decay rates based on CH-stretch vlbratlons ds tk dommant acccptmg modes remams vahd.
References [ 1I B
SChJf,
Chrm
PII) s Letters 68 (1979)
112
121 D St Butland, G Castro and C w Robmson, J Chem Phys 51 (1970) 131 D hf BtMmd and G \V Robmson. J Chcm Phys 5 1 (1969) 4548 141 A N~tzart and J Jortner, Theoret Chum Acta 30 (1973) 217 [51 I- hletz, Chem Ph>s 18 (1976) 385 [6] W. Stcbrand. J Chem Phys 47 (1967) 241 1 171 W St:brmd and bl 2 Zgterskt. J Chem Phys 72 (1980) 1611 [8] A P Penner. W Stebrand and hl Z ZgershI, J Chem Phys 69 (1978) 41-l
4100
5196
Volume 191 [ 101 [ 111 [ 121 [ 131 [ 141 [15] [ 161 [ 171 1181 [19] 1201 [21]
72, number
3
CHEMICAL
PHYSICS
15 June E980
LETTERS
V Lawetz, W Slebrand and G Orlandi, Chem. Phys Letters 16 (1972) 448. W H Henneker. A-P Penner, W Siebranri and bl 2 Zglerski, J. Chem. Phys. 69 (1978) 1884 W Slebrand and hl Z. Zgxxski. Chem. Phys. Letters 58 (1978) 8. B R. Henry and W SIrbrand, J. Chem Phys 54 (1971) 1072. J van Cgmond. D bl. Burland and J H van der WaaIs. Chem. Phys. Letters I2 (1971) 206 J H ran der Waals, A hI D Berghu~s and MS. de Groat. Mel Phys 13 (1967) 301. R Ltand EC Ltm. J Chcm. Phys 57 (1972)60X P hI Johnson and L Ziegler, J. Chem Phys 56 (1972) 2169. G r Hatch hl D. Erhtz and C.C Norman, ~1 hlolecttlv lumtnescence, ed. E C Lun (Beqamin, G W Robmson and R P. rrosch, J. Chem Phys 38 (1963) 1187 T E blartm and A H Kalantar, J Chem Phys 50 (1969) 1486. V Lawetz, G Orlandi and W Slebrand, J. Chem. Phys. 56 (1972) 4058. J B Buhs, Pbotophyslcs of aromatic molecules (Wtiey-lntersaence. New York, 1970) p_ 284
New York,
1969)
p- 21.
PHYSICS LJX-l-ERS
CHEMICAL
NON-CONDON EFFECTS ON RADIATIONLESS IN BENZEKE AND NAPHTHALENE * Wlllem SIEBRAND Dn ~SKMZof Chetnrstry Otrarra Canada Kid
and Marek Z. ZGIERSKI #a~~onaI OR6
Research
Comml
TRIPLET
15 June I980
DECAY
’ of Canada.
Recewed 10 hIarch 1980
Model calcuhtlons that avoId the Condon approwmation are reported for T,‘++ So mtersystem perdrurerated benzene and naphthalcne In both benzenes and III naphthalenexfg, skeletal bendmg to T:! are found to be more efficient as acceptmg modes than CHjtretchmg wbratlons.
crossing in normal wbmtions couphng
and T,
Recently Scharf [I] has argued that nonradtatwe decay of trlplrt benzene is dommated by energy transfer to ezg bendmg modes and not to CH-stretchmg modes, as usually assumed. He based his conclusion on the observation that some of these bendmg modes undergo large frequency changes between the trlplet and ground state [21, thereby glvmg rise to relatively large vIbratIona overlap mtegrals and hence to efficient decay channels. These new channels would greatly increase the calculated decay rate constant [3--51, brmgmg It closer to the experimental value Smce these modes mvolve mamly CC bending, they are relatively insensitive to deutermm substitution. Th2s could help in explammg the relative weak effect of deuteratlon on the triplet lifetime of benzene compared to other small aromatic hydrocarbons [6] In a recent paper 171, we have mvesttgated the effect of modes couplmg two excited states on radiationless decay of the lower state. These calculations are relevant to the benzene problem smce the ezg modes couple the states T,(3Blu) and T,(3Eluj. Such couphng modes contrlbute not only through their mcreased overlap integrals, but also by mcreasmg the effective vlbromc coupling between Tt and the ground state S,(‘At,). This is a nonCondon effect in that It IS due to the dependence of the eiectrontc transitton matrix eiement between T, and SO on the coordtnates of modes couphng T, to T2. In this note we extend calculations of the type reported earlier [7,8] to a model system simulating triplet decay UI benzene. The present calculattons are more general tn that they tnclude another effect indticed by the coupling, namely tnduect decay of T, via T, through a second-order matrix element. To keep the calculations tractable, we consider only four modes, each of which plays a different part in the transIttons. These modes are: Q,(b[,,), promotmg T, ‘L+=So radlatlonless transition, Qb(e2&, couphng T, to T,, Qc(etu), promoting T,%-+S, transitions and ps(aIg), a representatwe accepting mode. For slmpliclty we treat all modes as non-degenerate harmonic osc&tors with the same dlabatic frequency III all three states. Except for Qb, the couplings in which they arc involved are assumed to be so weak that the dlabattc frequencies equal the adiabatic frequencies_ The promoting modes Qa and Qc can therefore accept only one quantum of energy durmg the transition, the remainder being distributed among Qb and Q,. We assume Q, not to be displaced between T, and T_, , so that this mode does not modulate the couphng through Qb_ The displacement between T; and So IS chosen such that Qs mimics the properties of a si..fold degenerate anharmomc local CH-stretchmg mode [9] in the energy range of interest. The coupling through * Issued as NRCC 18213 * NRCC Research Associate.
Volume
72, number
3
CHEMICAL
PHYSICS
LE’I-I-ERS
15June
1980
Qb IS allowed to be strong enough to account for the large frequency drfferences of the e zg modes m So and Tt , but the degeneracy of Qb and its Jahn-Teller activtty m T2 are neglected. The vibromc couphngs are Introduced through lmear adiabatrc coupling operators K,Q,, where x = a. b and c [IO] _ The weak K and KC couphngs are Included m the usual fist-order form. The resultmg first-order corrected wavefunctrons Ttf , i$ and Sb are then used to dragonahze the i T, , Tz} matrt.\, wherem the two trrplet states are strongly coupled through Qb We e\pand all electromc wavefuncttoris m terms of a drabatrc (I.e., uncoupled) basrs set +E, # and d$ such that [I I]
7:=+,q, +q&?,ql~,,cQ~>.rf=a: +&Q,~~~/~g(Qs>. $,= +_f- &Qa+:,/ti,4(Qs> - I";cQci~:/~g(Q,h where E &(Q,)= f$(Q) - E!(Q) is a diabatrc potentral energy {Q,. Qb, QC. Q,} transformatton
From these first-order
7-t = 7-i cos Q + T-f sm @. where
corrected
Tz = -T:
wavefunctions.
sm Q + T$ cos $,
(1) drfference, Q bemg the column vector we form adtabattc wavefuncttons by the umtary
qJ =s;.
(3
[IO]
tan 70 = -2K,Q,IE,S),&,)_ The correspondmg
r,,,(Q)
adtab.rttc
(3) potenttals,
wfrrch m general
= E!?s(Q) + :EIS),,JQb) - f[$ E,s),,,(Qb)]2
E,,(Q) = E,s)(@) - f E:,,,
(QbI+ I[) E:,,,
(Qb
WIII not be harmonrc,
+ K;Q;
)I ’ + K;Q;
are
I”‘, } “‘, (4)
(5)
Because of the sunphfymg assumpttons Introduced, all drdbatrc frequencres C&, wtth x = a, b, c, s and _V= ~7, II. g can be replaced by the correspondmg adtabnttc frequenctes wx except 5?,, and C&,,. To calculate the rate constant of radrattonless decay for the lowest vtbratronal level of the trrplet mamfold, we assume. as usual, that the unttal state T, IO,, 06, 0,. OJ IS an adrabatrc Born-Oppenhermer state. The spur-orbrt couphng to ‘err* and lrro*’ states IS mcluded m thts descrrptton and direct spur-orbtt couphng between T, and So ts neglected [I 2] The two components Ti and Ti of T, combtne wtth dtfferent ground-state levels, so that the radtattonless decay rate constant can be wrrtten as a sum of two terms k, = k,, + kq2. In the usual golden rule formulatton, tfrese terms are proporttonal to the square of a matrL\ element of the nuclear kmettcenergy operator TN and a denstty of final states p
(6) The densttres of states p. and p, mcorpornte the energy conservatron condttrons trons U. u and u’. u’, respectrvely. They are chosen to have the “smooth” form 412
determmmg
the allowed
combma-
Volume 72, number 3 P&Q,
usI=
r-l
CHEMICAL exp[-+
+(~,~,0000
-
PHYSICS
15 June 1980
LETTERS
4,1uOu)‘l,
The linewidth parameter r IS assumed to include Eg 1lCOUIS the ~lbroruc energy of the state SoI I,ubO&. the effec; of couphng the modes Qa. Qb, Q, and Qs to aU other modes in the molecule. The following parameter values are chosen to represent benzene [ 13.14]- wa = Rb,,, = S2b,I = o &Z 1°C = 1580cm-1.w,=2w,=3160cm-1,r=~o,= 790cm-‘,E,(O)= 18Sw,=29200cm-‘.E,, -4.5~~ = 7 100 cm-‘. To study the effect of deuterlum substltutlon, additlonal calculations have been carried out with Since no unambiguous spectroscoprc data are avarlable for the coupling constants, % D = +GJ,” = 2370 cm-l. calculations have been carned out for three Kb values, namely (I) Kb = 0, corresponding to a model III which bending modes do not act as accepting modes: (il) Kb = 2.05, corresponding to the assignment of Burland and to the alternatwe assignment proposed by Scharf [I 1. Robinson [3,133, and (in) K, = 1.68, correspondmg To emphasize that our results for Kb = 0 include non-Condon effects, we also list a rate constant A,(FC) obtamed UI the conventional manner with the coupling through Q0 caIculated via Herzberg-Teller theory so that the acceptmg mode enters the expression for the rate constant III the form of a Franck-Condon factor [3-S] _ Since we are only calculatmg relative rate constants, the results are mdependent of K,, but depend on the ratio Q!= Ki/Kz which governs the ratio k,,/kqz between the duect and induecr channels. The results are presented for CY= I but can easdy be modified to fit other values of (lr. The values of B are chosen to match the calculated Franck-Condon envelope calculated for a &u-fold degenerate local CH-stretchmg mode [9] _ Smce this IS not possible with a smgle value of B, we report calculations for the values B = I-S,? and 2.5, apploprlate to the range eg,oooU = 10 000-20 000 cm-‘, where u IS the number of CMstretch quanta accepted by Qs_ The larger u, the larger the value of B requ:red to mlmlc the overlap properties OF the anharmomc CH oscdlator. The range of values chosen IS thus suitable for monitoring the change from a weakly accepting to a dommant Q, mode. For the deuterated molecule, B should be adJusted such that the actual displacemcnt Qsg - QDn IS mvarlant. Smce B = 21/2u~/“(~sg - &), ths means that BD differs from BH by a factor (w,” /GJH)3’2 _ The tesults of the calcularlons are collected m table 1. The values m the first row denote results obtained by the conventlonal Franck-Condon approach for the model parameters chosen. They therefore provide reference values with which the new results are to be compared. The second row corresponding to Kb = 0 shows an increase of the calculated rate constant by a factor of about 30 If non-Condon effects associated with ps are included [i 1 I_ where
Table 1 Calculated relauve rate constants k,(Kb) for TIU So III benzene-/l6 and -de and naphthaleneAs and -c& for dlffecent displacement parameters BH and BD kq(FC) refers to a conventlonal calculatton III which acceptmg modes appear as Fran&-Condon factors The notation 2 9(-15) stands for 2 9 X lo-” Na hthatene B cp/EP = 2 OO/L.30
Benzene BH/BD = 1 50 IO.97
BH,BD = 2.00/l 30
2.9(-IS),4
4(-20)
3 6(-13)/l
8 7(-14)/S
l(-19)
1.1(-l
BH,BD = 2 50/1.62
9(-17)
15(-11)/2.0(-15)
1),3 7(-16)
4.9(-10),4.3(-14)
l-1(-5),9.8(-IL)
L.3(-9),4
l-5(-7)/1.06-9)
2 9(-10),3.4(-10)
3 7(-10)/4.1(-10)
l.O(-10),7.7(-11)
1 3(-IO)/6
3 4(-10)/6.9(-11)
Z6(-9),2.L:-LO)
1 0(-9)/l 3 4(-lo),2
1 4(-9)/l-4(-9)
5 O(-9),1.7(-9)
3.8<-7)/6.8(-9)
5 3(-lo)/2
2.0(-9),2.4(-LO)
l-5(-8)/1.3(-9)
2(-9) 6(-10)
a) Based on o = Ki/Kz = 1. These rate constants
ye proportional
6(-11) 3(-10)
8(-LO)
7.9(-9),8.L(-17)
to LI.
413
CHEMICAL
Volume 72, number 3
PHYSICS
LETTERS
15 June
1980
The deuterlum effect predlcted by these two rows amounts to a reductton by a factor of the order IO”-IO”. The remainmg four rows show how Qb affects these predIctIons_ Broadly speakmg, It further mcreases the rate constant and sharply reduces the deutertum effect. Expenmentally, the rate constant IS found to be much larger than predtcted by the first row calculattons. To account for the elperlmental results, an mcrease by a factor IO’-IO6 IS required [3--51. For 01~ 1. the results of rows 4-7 predict Increases in the range lo’---106, dependmg on B and Kh. relntlve to the results of the first row. If B IS restrxted to the range I S- 2, the mcrease matches the required factor 104-106. The experImentall> observed deuterlum effect on the trlplet decay rate constant IS given by the ratlo [ 15-191
where Ii,“/kp_ the ratro of rndlatlve decay rate constants. IS in the range l-1 2 [15,16]. The ratio (8) IS much smaller than m other small JromatlL hydrocxbons e g., m naphthalene, k H /k D = 8-10 [6,1.5]. In the latter molecule. CH out-of-plane bendmg modes. mductng spin-orbtt couphng, are responstble for part of the deutertum effect on namely for a factor of about 1 8 [ 12,201 A slmllar effect IS expected for benzene. Smce our model does Qliq”. not Include such modes, the e\permxntal results have to be corrected for thrs effect before they can be compared wtth the model predicttons. Thus requires mformatlon on the phosphorescence quantum ylcld k,/k. Clearly. the rJtto (8) can only be used to determine the ratio kt/ky tf At 2 kp. Dl rect measurement [ 151. as well as the observdtlon of snmlar solvent eifects on liH and kD [ 16- 191, indtcate that this condltton IS indeed met and that x-gH/ligD= I-2, so that. after correctlon for out-of-plane mducmg modes, L m-plane bendmg modes play a more modest part m other small aromatic hydrocarbons, as evtdenced by thetr Ixgcr dcuternun effects We have shown prevtously [7] that non-Condon effects become less important for smaller energy gaps between the electronic states m transttlon Since benzene has the largest T, -So gap of any dromattc hydrocarbon. Its rehance on CC bending vtbrJtlons as acceptmg modes IS expected to be excephonally strong This 1s confirmed by the results hsted tn the fmal column of table 1 based on parameter values appropriate to naphthalenc [21] E,,,(O) = 13 5 wa = ?I 100 cm-‘. E,s),,, = 5 2 wa = 9600 cm-‘_ For BH = 2 and all other pJrametcrs as m the benzene calculations. these results predict much larger tsotope effects, as Indeed observed [6.15]. although they remam smaller than predtL.ted by the conventional Franck-Condon model In naphthalene118, the ob contrtbutton IS found to rematn modest. namely 0 1-O 3, dependmg on K, and 01. but m naphthalenedg. it IS found to become dommant (0 6-0.7) We conclude therefore that for non-deuterated aromatlc hb drocarbons other than benzene. the usual ratlonahLatlon of radtattonless trtplet decay rates based on CH-stretch vlbratlons ds tk dommant acccptmg modes remams vahd.
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112
121 D St Butland, G Castro and C w Robmson, J Chem Phys 51 (1970) 131 D hf BtMmd and G \V Robmson. J Chcm Phys 5 1 (1969) 4548 141 A N~tzart and J Jortner, Theoret Chum Acta 30 (1973) 217 [51 I- hletz, Chem Ph>s 18 (1976) 385 [6] W. Stcbrand. J Chem Phys 47 (1967) 241 1 171 W St:brmd and bl 2 Zgterskt. J Chem Phys 72 (1980) 1611 [8] A P Penner. W Stebrand and hl Z ZgershI, J Chem Phys 69 (1978) 41-l
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Volume 191 [ 101 [ 111 [ 121 [ 131 [ 141 [15] [ 161 [ 171 1181 [19] 1201 [21]
72, number
3
CHEMICAL
PHYSICS
15 June E980
LETTERS
V Lawetz, W Slebrand and G Orlandi, Chem. Phys Letters 16 (1972) 448. W H Henneker. A-P Penner, W Siebranri and bl 2 Zglerski, J. Chem. Phys. 69 (1978) 1884 W Slebrand and hl Z. Zgxxski. Chem. Phys. Letters 58 (1978) 8. B R. Henry and W SIrbrand, J. Chem Phys 54 (1971) 1072. J van Cgmond. D bl. Burland and J H van der WaaIs. Chem. Phys. Letters I2 (1971) 206 J H ran der Waals, A hI D Berghu~s and MS. de Groat. Mel Phys 13 (1967) 301. R Ltand EC Ltm. J Chcm. Phys 57 (1972)60X P hI Johnson and L Ziegler, J. Chem Phys 56 (1972) 2169. G r Hatch hl D. Erhtz and C.C Norman, ~1 hlolecttlv lumtnescence, ed. E C Lun (Beqamin, G W Robmson and R P. rrosch, J. Chem Phys 38 (1963) 1187 T E blartm and A H Kalantar, J Chem Phys 50 (1969) 1486. V Lawetz, G Orlandi and W Slebrand, J. Chem. Phys. 56 (1972) 4058. J B Buhs, Pbotophyslcs of aromatic molecules (Wtiey-lntersaence. New York, 1970) p_ 284
New York,
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