The electronic mechanism of triplet-triplet annihilation resulting in the delayed fluorescence ... which lead to radiationless energy transfer between two triplet-ex-.
THEORETICAL B.
F.
MODEL
OF
TRIPLET--TRIPLET
ANNIHILATION
Minaev
UDC 535.37
The e l e c t r o n i c m e c h a n i s m of t r i p l e t - t r i p l e t annihilation r e s u l t i n g in the delayed f l u o r e s c e n c e of organic m o l e c u l e s in liquid solutions is investigated. T h e r e is a d i s c u s s i o n of the nature and strength of the interactions which lead to r a d i a t i o n l e s s e n e r g y t r a n s f e r between two t r i p l e t - e x cited m o l e c u l e s in a collision c o m p l e x , when the f i r s t excited singlet state of one of the m o l e cules is filled. The role of i n t e r m e d i a t e s t a t e s with charge t r a n s f e r and also of the unstable e x c i m e r state in the contact complex is noted. p - T y p e delayed f l u o r e s c e n c e (DF), investigated in detail b y P a r k e r [1], is one of the m o s t i n t e r e s t i n g photophysical p r o c e s s e s . Delayed f l u o r e s c e n c e a r i s e s as a r e s u l t of the collision of two t r i p l e t - e x c i t e d m o l e cules; all the excitation " c o l l e c t s " on one of the m o l e c u l e s , which e m i t s DF [1]. The annihilation p r o c e s s 9 c o n f o r m s to the s c h e m e [1-5] T, q-- T', PiK, ~.i K_, ( T...T)i~-~-> S, '-~ S O.
(i)
H e r e K i a n d K-1 a r e the r a t e constants of m o l e c u l a r collision and dissociation of the complex, r e s p e c t i v e l y ; the index i n u m b e r s the different spin states of the complex, t a k i n g into account the quantum n u m b e r s of the total spin (S). and its p r o j e c t i o n (Ms); P i i s the p r o b a b i l i t y of f o r m a t i o n of a complex in the i-th state at each collision; K~ is the r a t e constant f o r the f o r m a t i o n of the f i r s t excited singlet state in one of the m o l e c u l e s of the i - t h state in the complex. A contact complex f o r m e d by two t r i p l e t - e x c i t e d m o l e c u l e s has nine spin s t a t e s (one singlet, three t r i p l e t s , and five quintuplets) [2, 3]. To account f o r the effect of a m a g n e t i c field on the DF of a n t h r a c e n e c r y s t a l s , M e r r i f i e l d suggested that T - T annihilation is s e l e c t i v e with r e s p e c t to the spin, i . e . , only the singlet component of the complex m a k e s a contribution to the constant K 2 [2, 3]. B a s i c a l l y , .the M e r r i f i e l d model a s s u m e s that f r a c t i o n s of the singlet component a r e distributed o v e r all i s t a t e s , i . e . , K~ = K2Xi, w h e r e hi = I{S]*2i)l 2, and Xi depends on the e x t e r n a l m a g n e t i c field. This kinetic model does not d e s c r i b e the m e c h a n i s m of T - T annihilation itself, i . e . , the nature and strength of the interactions d e t e r m i n i n g the constant K 2. T h e s e questions a r e c o n s i d e r e d in the p r e s e n t work. The total annihilation r a t e constant T is [2, 3] 9
9
i - K l ~ i=, K~-~K-I
9 t
K~~-K-I '
(2)
which takes into account that photoexcitation produces two isolated molecules in T states with uncorrelated spins and, hence, the formation of any spin state in the collision complex is of the same probability (Pi=1/9) [2]. No a t t e m p t is made to take into account that the t r i p l e t p a i r s m a y b r e a k up and then r e f o r m , p e r f o r m i n g diffusionai motion. The p r e s e n t w o r k does not c o n s i d e r internal and e x t e r n a l m a g n e t i c p e r t u r b a t i o n s , i . e . , the mixing of spin m u l t i p l e t s is neglected, since these p r o b l e m s demand s e p a r a t e t r e a t m e n t . The a n a l y s i s is b a s e d on a p u r e l y static m o d e l of the p a i r , which is c h a r a c t e r i z e d by s o m e r a n d o m spatial s t r u c t u r e . To elucidate the m e c h a n i s m of the interactions d e t e r m i n i n g the constant K2, it is n e c e s s a r y to c o n s i d e r the r a d i a t i o n l e s s p r o c e s s between the initial r a t e l ~ g (the singlet state of the contact complex) and the final state Si + So. The two molecules participating in T - T annihilation are denoted by A and B. In the absence of molecular interactions (RAB = ~), the triplet-pair states ~ g are steady. They are all degenerate, since there are no internal transfer interactions, and consist of a simple combination of two states with ~A = ~B = 3, p = q = 1, from the set of eigenstates of the Hamiltonians for the individual molecules (HA and HB)
=
~~
~Atry"
.
"
(3)
K a r a g a n d i n s k State U n i v e r s i t y . T r a n s l a t e d f r o m I z v e s t i y a Yysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 1 2 - 1 7 , S e p t e m b e r , 1978. Original a r t i c l e submitted S e p t e m b e r 23, 1977.
1120
0 0 3 8 - 5 6 9 7 / 7 8 / 2 1 0 9 - 1120507.50 9 1979 Plenum Publishing C o r p o r a t i o n
H e r e ~A~A (p) and ~AEA (p) a r e the wave function and e n e r g y of the p - t h steady state of molecule A with m u l t i plicity ~A = 2SA + 1. The t r i p l e t - p a i r s t a t e s m a y be written using C l e b s c h - G o r d a n coefficients [7]
~+~,rf~= ~ ~,r~'~,,,,r~;,~< 1, m,; 1, .,~.IsM~ >.
(4)
?/zl, ,rn~
The Hamiltonian for the contact p a i r is of the f o r m H = HA + HB + HAB. When RAB = % t h e r e is no m o l e c u l a r interaction (HAB = 0) and the function in Eq. (4) s a t i s f i e s the equation H ~ g = (3EA(I) + 3EB(1))~g. The final state lq~f --~lq~A(1) l~'13(o) with e n e r g y 1Ef = IEA0 ) + 1EB(0) is also steady when tlA] B = 0% H o w e v e r , when RAB ~1t~ + I I ~ , w h e r e R~ is the van d e r W a a l s radius of m o l e c u l e A, the states *~g and lq~f a r e not s t e a d y , since they a r e not e i g e n s t a t e s of the Hamiltonian H. Hence, the s y s t e m , initially in state l ~ g as a r e s u l t of collision, u n d e r g o e s a r a d i a t i o n l e s s t r a n s i t i o n a f t e r a t i m e t to the state l~f. Using the theory of [6] f o r r a d i a tionless t r a n s i t i o n s , the s t a t e s lq~g and lq~f produced m a y be r e g a r d e d as of z e r o O r d e r . - T h e s e states m u s t be a l m o s t d e g e n e r a t e , that i s , it is n e c e s s a r y to choose a s e r i e s of vibronic final s t a t e s ~(n) _ ~a~isIying .... the q u a s i d e g e n e r a c y condition, with which the initial state lq/g is " m i x e d " as a r e s u l t of H [6]. I t this case the total p r o b a b i l i t y of a r a d i a t i o n l e s s t r a n s i t i o n l ~ g _ , ~ l ~ f ( w ) p e r unit time (or the r a t e constant K2) is [6] l < z ~.l z g > i ,
K , = o, tt) = 2 , : . _ _ Z I < , % I H I , % > I : , ~ t hN
= ~2~p - ~ -~ F fg,
(5)
w h e r e p is the density of final s t a t e s close to the d e g e n e r a c y point; N is the n u m b e r of m o l e c u l e s p e r unit volume (cma); 9 and • a r e the e l e c t r o n i c and nuclear wave functions; F f g is the F r a n c k - C o n d o n f a c t o r . To calculate the m a t r i x e l e m e n t s in Eq. (4) and to evaluate the r a t e constant K2, the wave functions in Eqs. (3) and (4) a r e w r i t t e n within the f r a m e w o r k of the m o l e c u l a r o r b i t a l (MO) method. To simplify the p r e s e n t a t i o n , filled e l e c t r o n shells a r e omitted. The solution r e q u i r e d for Eq. (3) is of the f o r m
H e r e the s t a n d a r d notation is used f o r the d e t e r m i n a n t s and the a - and fl-spin o r b i t a l s ; ~0j and ~0l (~oi and ~010 a r e the u p p e r filled and l o w e r e m p t y MO of the singlet ground state of molecule A (molecule B). The u p p e r sign in Eq. (6) c o r r e s p o n d s to the singlet and the l o w e r to the triplet. The initial singlet state a r i s i n g in the collision of two t r i p l e t m o l e c u l e s d e s c r i b e d by Eq. (6) m a y be obtained f r o m Eq. (4) in the f o r m [7, 8] {21 ,~-,..~,.~1 + 21
- -
I ~e,~l-
--
-
-
In w r i t i n g the final state for the homogeneous T - T annihilation, it is n e c e s s a r y to take into account the c o n figurational i n t e r a c t i o n (CI) for the two d e g e n e r a t e configurations 1 -~.a,~'+ _ =
~1
" 4( ,13tI.i
l.a,F/), .
(9)
F o r h e t e r o g e n e o u s T - T annihilation it is n e c e s s a r y to take one of the s t a t e s in Eq. (8) o r their m i x t u r e as a r e s u l t of CI, depending on the r e l a t i o n between the e n e r g i e s 3EA(i) and aEB(1). The m a t r i x e l e m e n t of the int e r a c t i o n between the s t a t e s ~q~g and l ~ f is* =
=
l(/~ I z~) - (]zlZe)l,
(to)
where 9
?t ( 2 ) ?x ( 2 ) ~_; ,j
d%d%.
(11)
~ rl,2 ]
F o r any collision g e o m e t r y fl -~ 0, since the two t e r m s in Eq. (10) a r e of s i m i l a r magnitude and cancel out. T h u s , m o l e c u l a r i n t e r a c t i o n s do not d i r e c t l y m i x the initial and final singlet s t a t e s in the contact p a i r on T - T annihilation, and this p r o c e s s is evidently a s s o c i a t e d with indirect mixing through " i n t e r m e d i a t e s . " Such an i n t e r m e d i a t e m a y be a state with charge t r a n s f e r (SCT). F o r h e t e r o g e n e o u s T - T annihilation with IA > IB and A B > AA , w h e r e I B and A B a r e the ionization potential and e l e c t r o n affinity of molecule B, the SCT wave function m a y be w r i t t e n in the f o r m *Since, in the collision complex, the o v e r l a p p i n g of the o r b i t a l s of m o l e c u l e s A and B is s m a l l , the functions ~i, Cj, q~, and q~ a r e a s s u m e d to b e o r t h o n o r m a l . 1121
1
(12) For homogendous T - T annihilation it is n e c e s s a r y to take into account CI for "prospective" SCT. Taking into account mixing of SCT with the final states gives
9
..
l/~..) ,~C T
CT
f '-}- ~AEcT]
CT.
(13)
Then the matrix element/3 in Eq. (5) takes the f o r m < " r . ' , , I H t " "~g->"
=
-
,
--
"
..
T] H
'~Fg>=
,,, ~..
(14)
C
The interaction integrals are
ii) + iii l uu) - 2 (]~r ire)l,
(15)
[htl-~ (ii]ti) -Jr-QcK]/i) + (j]] li) -- (]l I l])],
(16)
t~~-
(17)
~' = - ll~j, ~- (ii I/J) §
~" -- 1 / / ~ lr
where hji is a one-electron shell integral
S
,~ zAe
~,(1)dv, = (]lhl/).
If the kinetic energy is neglected in Eq. (17), the integrals of Eq. (15) and Eq. (16) may be given a simple physical interpretation in accordance with the arguments of [9], This involves isoiating in Eqs, (15) and (16) t e r m s of the f o r m
"'
~Zael ) r,-~- ] i +
V. (.4 +) = l t - ~
(JilZi).
(19)
Equation (18) is the potential energy of the electron with a charge distribution qlqi in the field of the neutral excited molecule B and Eq. (19) is the potential energy of the same charge distribution in the field of the ion A +. Neglecting the integral (ji]lj)in comparison with. (jj]li) and taking into account Eqs. (i8) and (19), Eq. (16) takes the f o r m
i/
+ v,(8o)l.
(16,)
Analogously, neglecting the integral (jR[ik)yields ~ ' = - [ t,'j, (B 0) + Vj, (A+)]
(15')
The t e r m s of type Vli(B~ may be neglected in comparison with Vli(Ad), which is approximated as the interaction energy of the c h a r a c t e r i s t i c density (-eS/i) with the positive charge (+e) at a distance of RAB/2; ( - e ) is the electron charge and Sli is the overlap integral for the MO ~! and qi [5, 9]*
s,,=
Z Z" c,.,c,,s,,,,,
(20)
*In deriving Eqs. (10), (15), and (16), all the MO were assumed to be 0rthonormal, which simplifies the d e r i v a tion. The integral Sli was evaluated using the accurate MO, which overlap. This inconsistency does not involve any significant e r r o r since, ff the accurate (orthonormal) MO are used throughout, this would lead to additional t e r m s in Eqs. (10), (15), and (16) but these t e r m s are small - onthe order of S~v.
1122
w h e r e ~A denotes s u m m a t i o n o v e r all the a t o m i c o r b i t a l s (AO) of the molecule A; Clt ~ a r e LCAO coefficients. The integral in Eq. (20) depends on the g e o m e t r y of the collision complex. When RAB = 3 A, St~u is on the o r d e r of 10 -3 for Slater AO and 10 -2 for H a r t r e e - F o c k A O [5]. F o r s y m m e t r i c collision (a s t r u c t u r e of " s a n d wich" type: the a t o m s of the two a r o m a t i c m o l e c u l e s a r e positioned one above the other) the integral S/i is close to z e r o , while Sji is close to S#v. Since an a r b i t r a r y g e o m e t r i c configuration of the complex is produced in r a n d o m collisions, it m a y be a s s u m e d that, f o r m e a n values of the o v e r l a p i n t e g r a l s , S/i -~ Sij -~ Stay/2. In that case
i,3' I.-. 1/
r--
i
'
' ---
2e ~ 2
Rat~"
T h e n , taking the e s t i m a t e S#~u = 10 -2, Ji0'l = 387 c m -1. The e n e r g y difference between the locally excited state and the SCT is a s s u m e d to be 1 eV. Then Eq. (14) g i v e s , finally, fl = 28 cm -1. In the case of homogeneous annihilation, a s i m i l a r e s t i m a t e is obtained f o r the i n t e g r a l (l~#~]H[l~g), taking into account SCT with i n v e r s e t r a n s f e r of the type A - B +. Now c o n s i d e r the configurattonal i n t e r a c t i o n of Eqs. (8) and (9); this is equivalent to taking into account the r e s o n a n c e d e t e r m i n i n g the stability of the e x c i m e r s and the shift in their f l u o r e s c e n c e . It is known that
=
'2 (t~- l iz) -
(WI
xz);
