Polarized fluorescence spectroscopy of two-ground- and two-excited-state systems in solutions

22 November t996

CHEMICAL PHYSICS LETTERS ELSEVIER

Chemical Physics Letters 262 (1996) 495-506

Polarized fluorescence spectroscopy of two-ground- and two-excited-state systems in solutions J.J. F i s z

Institute of Physics, N. Copernicus University,uL Grudziqdzka5/7, PL 87-100 Toruh, Poland Received 8 March 1996; in final form 12 July 1996

Abstract

The polarized fluorescence spectroscopy of two-ground and two-excited state systems in solution is discussed. A statedependent diffusion model for the rotational motion of an asymmetric, symmetric and spherical rotor is considered. Analytic expressions for the polarized fluorescence components in the case of symmetric and spherical diffusors are derived. The absorption and emission transition dipole moments have different directions in the molecule-fixed frames in both groundand excited-state species. The case of double absorption bands with different directions of absorption dipole moments, in both ground-states species, is discussed. The problem of convolution between the instrument response function with a model decay is also discussed. Two methods for recovering the instrument response function are considered; namely, the scatter method and the method of a monoexponential reference compound. The theory discussed in this work can be employed to many problems important from an experimental point of view e.g., different kinds of excited-state reactions or excitation energy transfer in rigid double-chromophore complexes. Computer generated synthetic decays of polarized fluorescence and emission anisotropy for the one-ground- and two-excited-state problem are exemplified.

To the memory of my dear daughter Karolina. 1. Introduction

In this work we intend to discuss in more detail the theory of fluorescence depolarization experiments on two-ground- and two-excited-state systems in solutions. We consider the situation displayed in Scheme 1. Kinetic and dynamic processes for molecular systems undergoing processes relevant to Scheme 1 are described by the equation of motion a

at p(Oo, O I /2, t) = --,h, p(Oo,O [ /2, t) ,

(1)

where

p(Oo,O I/2,t) = ('p-.~(Oo,O I Y2, t) ) \ p__.2(a2o, 0 I /2, t)

,~=(i2I(~)+kl+k~2 '

--k12

-k2~ ) f/(2) d- kl q- k21 "

(2)

p--.((Oo,0 I /2, t) (sc = 1,2) are Green functions which describe the time-evolution (state-to-state kinetic relaxation and reorientational motion of molecules) of both excited states (see Refs. [ 1 ] and [2] for details). 0009-2614/96/$12.00 Copyright t~) 1996 Published by Elsevier Science B.V. All rights reserved. PH S0009-2614(96)01130-X

496

J.J. Fisz/Chemical Physics Letters 262 (1996) 495-506 excited states

k~

~

ground states

Scheme 1 ,~ is the matrix operator and/-1~() ( ( = 1,2) are the state-dependent time-development operators describing the rotational dynamics of the molecules. The equation of motion ( 1) has a more general applicability. It relates to solutions as well as to ordered molecular systems [ 1]. In the diffusion model for an asymmetric rotor in solution the time-development operator/q{~} has the form/~ff) = ~ : ] O~~) L~z, where the operators Li are identical to the quantum mechanical orbital momentum operators (i = x, y, z ) and D} () are the components of the diffusion tensor. For the symmetric rotor problem Dx(~) = D~,~) = D ~ ) , D~~) = DIE). Furthermore, if Dx{~) = D~.~) '= D.(~) = D {~), the problem reduces to the spherical rotor case. In the case of ordered systems (e.g., liquid crystals, membranes etc.), the time-development operator/2/has the form/~/{~) = ~ = l D~~) { L2 + Li (Li v(( s2) ) /kBT}, where V((J2) is the state-dependent aligning potential. The problem of the state-dependent rotational motion of fluorophores in solutions was considered earlier by Cross et al. [3]. In Ref. [3] explicit expressions for the time-evolution of polarized fluorescence components have been derived for spherical diffusors and for restricted orientations between absorption and emission dipole moments. The many-excited state problem for fluorescence depolarization was also discussed by Szabo [4]. He considered the case of ordered systems with the state-independent rotational motion and state-independent ordering of the fluorophores, but with a general assumption that all the dipoles (absorption and emission) have arbitrary directions. Szabo [4] showed that his expressions reduce to those obtained by Cross et al. [3] after assuming that the absorption end emission dipole moments for the same species are parallel to each other and assuming that the diffusion rates for the rotational motion of spherical rotors in the expressions derived in Ref. [ 3 ] are state-independent. In this work we wish to derive expressions for the polarized fluorescence components in the case of asymmetric, symmetric and spherical diffusors with state-dependent rotational motion under the assumption that all the transition dipole moments involved (absorption and emission) have arbitrary directions, in general. The expressions for the polarized fluorescence components for symmetric and spherical rotors are shown explicitly. In many experimental cases the ground-state species can be excited to two lowest excited states. For this reason, we also consider the case of double absorption bands with different directions of the absorption transitions dipole moments. To make our considerations more close to the experimental needs, we consider the problem of convolution between the instrument response function on the exciting pulse of light and the model decay describing the time evolution of polarized fluorescence components after a 6-pulse excitation. Two methods for recovering the instrument response function are accounted for in our discussion namely, the scatter method [5] and the reference convolution method [6,7]. We consider the case of an asymmetric diffusor in more detail, although it has less experimental importance, because this case can clarify the general rules for constructing a numerical algorithm for the analysis of relevant problems in the case of ordered systems and also in the case of symmetric or spherical rotors in solution when more excited states are involved. The theory discussed in this work can be employed to many problems important from an experimental point of view e.g. different kinds of excited-state reactions or incoherent excitation energy transfer in rigid double-chromophore complexes.

J.J. FidChemical

2. Polarized fluorescence

491

Physics Letters 262 (1996) 495-506

decay surface

The intensity of the polarized following formula [ 21,

fluorescence

emission

for the case indicated

in Scheme

1 is described

by the

(3)

where

includes all instrumental excitation and emission wavelength-dependent factors. Ok ( A,, ) and the absorbance of the kth ground state at A,, and the intensity of the emission band of state fluorescence decay and +(tvk) (t) is the correlation 5 at (km), respectively. Ph (6Vk)(t) is the polarization-free functions describing the rotational dynamics of molecules and which is coupled with the state-to-state kinetic relaxation of molecules. 8if is the angle between the polarization of the incident light, e^i, and the polarization of detected emission, 2,. The parallel component of the emission II] (t, AeX,A,,) is obtained for 0if = 0” where C ( Aex, A,,) K[( &,,,)

are

Pz(O”) = 1, and the perpendicular

one II( t, Aex, &,-,,) for @if = 90”, P2(90°)

= -l/2.

&,

(Bik’, spik’) and

C2,q2( r&‘), cpa’) are the modified spherical harmonics [9] parameterized by the polar angles defining the directions of absorption and emission transition dipole moments in the molecule-fixed frames in the ground state k and in the excited state f, respectively. R;,f;$, (t) , appearing in (4)) are the elements of the kinetics-dynamics relaxation matrix R(t) . The form of R(t) can be derived by solving the equation of motion ( 1) , transformed into the system of coupled differential equations for the expansion coefficients C(t) appearing in the Green functions expanded in the basis set of Wigner rotation matrices [9], i.e. [ 1,2] &C(r)

= -MC(t)

.

M is the matrix representation

(5) of A in the basis set of Wigner rotation matrices and

(6) U(t), K12 and K2t have the elements

(7) and

and where

H?) ,n,n,,fn,+,f = IQ_? + 1)/877?

s 0

498

J.J. Fisz/Chemical Physics Letters 262 (1996) 49.5-506

The solution

of the matrix equation

(5) is

C(r) =R(t) C(O) , R(t)

where

(9)

= V exp (-At)

V-‘.

respectively. When the explicit elements of R(t,k’ (t) i .e .

V and A = [A~~] are the eigenvector and eigenvalue matrices of M, form for the time-development operator A(r) (5 = 1,2) is considered, the

[(V [exp (-A44 t)] V-l)‘f’k’]j~,,,j,m,n, >

R$,:b,,,t,+ (t) =

(10)

can be calculated and the time-dependent terms Ph’6gk’(t) and 4 (6Sk)(t) in (4) can be reconstructed. For more details we refer to Refs. [ 1,2]. In (4) the index m in the elements of R(t) has been neglected because, as was shown in Ref. [ 21 by employing group theory methods, the matrix elements R!(, ) ,mnj,nr,n,( t) do not depend on m for isotropic molecular systems (solutions), independently of the model for rotational motion.

3. Diffusion model for rotational motion In the diffusion

model for an asymmetric

rotor

(11) Taking into account that

e,D$,,!, * e Y

[j’(j’+ j’(j’

D'j') *=i m'n'

H!f) J"'"~J'"'"' =

1)

112D(.f) * d,d+l

1

+ 1) - n’(n’ - 1)

j’(j’ [

* m',n'-I '

1 112

112 Dcj“ 1

112D(j')

+ 1) - n’(n’ - 1)

* m'.n'-I

+ 1) - n’(n’ + 1) 1

(j’) * , D,,,,,,,

i z D$‘,! * = -n’ one can calculate

1) -n’(n’+

the matrix elements

0

of the time-development

Ay,r (Dif’ + D:!))

+(Bj;!

where Ajt”J = i[j’(j’+l)

+ d2 Dif)

operator fi given by ( 1 I), i.e.

~5~~) 1

&,,,,(+2+

-n’2],

Bi’,;! 4,,n~_2) (Din - D$f) B,‘,:? = i [(j’(j’+l)-n’(n’*l))

)>

ajjt 9 Smd

,

(13)

(j’(j’+l)-(n’fl)(n’f2))]‘/2.From

the form of (13) it follows that within a given block U(f), the elements H$L j,m,n, do not couple the equations in (5) in which they correspond to different values of j and j’. For a given value of j = j’, there is no coupling between the equations for the expansion coefficients C,!:!(t) with even and odd values of n. Furthermore, the matrix representation of Z?f) does not depend on m. For this reason the index m will be omitted in all the expressions, i .e . C!f) will be denoted by C,‘n” (t), I$$,, , H$fi,,, and pm (t) ' Utn Jmn.j'm'n" HjLb,j,,,,,,, and R.$“b,,,, Rj,f;f$. Below we show the matrix equation decomposed into two independent subsystems

(5) for j = 2 (the case of second-rank correlation of equations, i.e. for even and odd values of n:

functions),

J.J. F i s z / C h e m i c a l / g~(l) ~22 (t)

/ ?l(l) '-' 22,22 r40) **22,20

H(X) /-/(1) Ll(l) 20,22 "-"20,20 **20,2 --2

C'~ol)(t)

o

C2(1_)2( t ) t~(2) "-22 (t)

t~<2)(t ) ~20

k21 0

0 k2]

0 0

0

0

k21

\

(1) i4(1) U21,21 "'21,2 -1 H(')l,2, U~121,2_l k21 0

C'2(2_)1(t)

0

I=

0

k21

k12

0

0

0

k12

0

0 0 k12 ,,(2) H (2) 0 1.,'22,22 22,20 ,,(2) H 20,22 (2) i1(2) "20,20 n20,2 -2 0 H~222,20 U(2) 2 -2,2 -2

kl2

499

P h y s i c s Letters 2 6 2 ( 1 9 9 6 ) 4 9 5 - 5 0 6

0

0 k12 m(2) r4(2) "21,21 "'21,2 -l H(22)_l,21U2(2-)L2 _l

C~)(t)

c 22(t) ~(2) t ~22 ( )

'

(14)

~(2) t w20 ( )

c 2(t)

{

i~kC~2_)1(')

(15)

Note that ~4(¢) = ~4(() ~4(() according to (13). We do not show here the matrix "*2n,2n' = ~4(~) *'2-n,2-n' **2n',2n = * * 2 - n , 2 - n ~' equation for j = 0 because it is exactly the same as in the case of a symmetric and spherical diffusor. It is worth citing here the symmetry relations for the matrix elements R~n¢i~,(t) in the case of an asymmetric

-(~,k) t ) = "'2-n,2-.'~ ~(~,k) tt~J' "'2n,2.'~'J ta(~,k) t.~ = R(~ ,k) tt~ 8.,.'+2 where n,n' = 0 , + 1 -4-2. They diffusor, which are: R2.,zn,( 2n,z.'~'J come out from the symmetry properties of I4(~) • *2n,2n t" They can also be found from group theory considerations in which the asymmetric rotor is treated as an asymmetric molecule (CI symmetry point group, in general) behaving in the sense of rotational dynamics as if it were an object with D2(s) statistical (hydrodynamical) symmetry. This problem has been considered in Ref. [ 10] in the case of ordered systems. The same method was demonstrated in Ref. [2] in the case of a symmetric rotor to which the statistical (hydrodynamical) symmetry group D ~ ) was ascribed. "2.,2.'~'J When the matrix elements •°(¢'k) t,~ are calculated, the time-evolution of the polarized fluorescence decays can be reconstructed, according to (3) and (4). When the problem is reduced to the one-excited state case, analytic expressions describing fluorescence depolarization for an asymmetric diffusor can be easily obtained. For a symmetric diffusor the matrix representation o f / ~ is diagonal, i.e.

Hj(mn.j'm'n' ~) = [j(j

+ 1) O ~ ) + n2 (DIE) - O~))] 6j.,.,/.,,., ,

(16)

and consequently, all the off-diagonal matrix elements •~4(O *2n,2n t (with n v~ n') disappear in equations (14) and (15). Therefore, only the equations for C2. 'O) (t) and for ~2.' t~(2) ~ tt~, with the same values of n and n' (i.e, n = n') are coupled. Hence, the systems of equations (14) and (15) can be split into five subsystems for two coupled differential equations. In order to simplify the notation, the matrix elements 11(¢) "jn.jn and o(~,k) x'.jn,j n will be denoted now by U)n~) and RJ~). The mentioned subsystems of two coupled differential equations read

C2n (t) "(2)

C2n (t)

U~n =

k12 (1) t'(2) ]~k C(22)(t) k21 V2n

'

(17)

for n 0, +1,-4-2. However, because u(¢) = H2(~., only three subsystems of equations are different, i.e. for n --- 0, 1,2. After solving the eigenvalues and eigenvectors, one finds that

./.J. Fisz/ChemicalPhysicsLetters262 (1996)495-506

500

(1 1)(t) R)q Rjq' (1,2)(t) Rjq( t) = ~ R)q (2A)(t) (2,2) Rjq (t)

)

A j[q e x p ( - L j q (1) t) - Bjq exp(-L)q2)t)] A/t~--'- k21

= (

(2)'" (l k12[ exp ( - Ljq t) -- e x p ( - L jqt)j/Vtajq

(2) [exp(-Ljq(2)t)exp(-L)q(1) t)]/~jq~(1) mjq exp( - L j q t) - Bjq exp( - L j q t) /

,

(18)

where q = 0 for j = 0 and q = O, 1,2 for j = 2, and where t j q(1) = ~l (U~,) q + Uj.q2)+ ~ j q ) ,

jq

2V/-~q u)(lq) - U)q )

,

2v/_~q

Bjq=

UI = kl + kl2 ,

0"2 = k2 + k21 ,

Rjq(t) = Rj - q ( t ) ,

c'b(('k)(t) = Z

,

"~ 4W,

(19)

(20)

(21)

U)~)=j(j+I)DI1)+q2(DO)-DII))+U1,

Since

) ,

O)q

Ajq= mjq =

Ljq( 2 )m1 ~ ((U)ql)"~- U)q( 2 ) -V ~ j q

U)~)=j(j+I)DI2)+q2(D~)-DI2))+U2,

W = k12 k21 •

(22) (23)

for symmetric rotor the correlation functions ~b(('k) (t) in (4) simplify to

° ( ( ' k )I,,] : " Tq(''i') , "'2q

(24)

q=0,1,2

where

T~ok,()=P2(O~Ai))p2(O(E()), T(k,(, = 3 T(2k,()= 43_sin 20(Ak) sin 2 0(E¢:) COS2(~PA-(k) _

sin20CAk) sin20(E(, COS(q/Ak) _ qCE()) , ~0(E()) .

(25)

The matrix elements of Roo(t) in the polarization-free term Ph (('k) (t) and the matrix elements of R2q(t) in the correlation function (24) are calculated from (18) for j = q = 0 and j = 2 with q = 0, 1,2, respectively. Hence, the explicit form for the polarized fluorescence decay for the symmetric rotor can be obtained. The matrix R0o(t) calculated from (18) is exactly the same as R0o,00(t) for an asymmetric rotor. A similar result for a spherical diffusor can be obtained without any further calculations. It can be obtained immediately from the expressions derived for a symmetric diffusor by substituting D ~ ) = DIIO = ~sphn((),where D(() sph is the rate for the rotational motion of a sphere. Note that in this case Rzo(t) = R21 (t) = R22(t) = Rsph(t) and ~q C2q(O(Ak) , ~O(Ak) ) t'~..,2q*t',O(O,. ~'E , WE"(()) = P2 (Oi~ () ), where VAEa(~'()is the angle between the absorption dipole moment in state k and the emission dipole moment of the same molecule in state (. Consequently, one obtains K~((,k) rt~ r~ to(k,()', c,b(('k) (t) = "'sph ' , ' / r21,~'AE 1 •

(26)

4. Double absorption bands

We discuss here an important case from the experimental point of view, i.e. when each species can be excited to two excited states, as indicated in Scheme 2. We assume that the absorption transition dipole

JJ. Fisz/ Chemical Physics Letters 262 (1996) 495-506

I~(1 b:

t ,|~ab."(2b)

k'

L. k " i ~12 ,

kl

k21

1"

Pab

,";(I,) r'-alo

501

k21Il ~( a2b)

I

2 Scheme

2

~(ka) and Pab ~,(kb) (where k = 1,2) have different orientations in the molecule-fixed frame, in general. moments I.~ ab k' and k" are the rates for the internal conversion process. It is worth mentioning here an interesting work by Herbich et al. [ 11 ] in which the directions of the absorption dipole moments for p-cyano-N,N-dimethylaniline (CDMA) and a number of its derivatives have been investigated in stretched polyethylene films by means of the linear dichroism (LD) method. CDMA and its derivatives show double emission in polar solvents and their photophysical properties can be explained in terms of the TICT (twisted intramolecular charge transfer) model [ 12]. As shown clearly in Ref. [ 11 ], the absorption transition dipole moments to the two lowest excited states have different orientations. Therefore, when studying similar compounds by means of polarized fluorescence spectroscopy, this possibility has to be reflected in the relations used in the analysis of the polarized fluorescence decays. The intensity of the polarized fluorescence decay, related to the present case, has the form 2 e ° ~ i ' ( ' ~ e x ) K ~ : ( ~ e m ) [ P h ( ( ' k ' ( t ) ÷ 45 P2(Oif)¢(,,ki)(/)]}

/~,~;(t,~ex,'~ern) = C(~ex, Aem) Z ( Z

,

i=a,b ~,k=-I

(27) where 2

¢(~,k,) (t) =

(£,k) ' ( t ) C2,q, (0 (ki), ~(kD ) I" * [ l~(() .,~(~)) R2q2,2q "'2.q2 ~"t'E ~Vr'E '

y~

(28)

ql,q2=-2

Introducing the relative spectral contribution coefficients ca(a~x) =

°J(ka)(a~)

Cb(a~x) =

o)(a)(,~ex ) ÷ O)(kb)(,~x) '

o ) ( b ) •" -

k t~)

O)(ka)(,~ex) ÷ (o~b)(l~ex) '

(29)

where ca(,~ex) + Cb(Aex) = ], formula (27) can be rewritten in the form 2

w (b)'"

"

[Pb(~:'k)(t)

"~("~) (t)] , (30)

where 2

~)(l~,k)(t ) = (a.b)

~

I~(~,k) [.'t r ,~(ka) ' ~_(ka), "'2q2,2ql ~'ty (Ca(/~ex) "2.qlkvA A ,. ÷ Cb(~ex) C2,q, (O(Akb),~o(kb))) TM

qhq2=--2

* (o E(e) ' ~(ze)) X c 2,q2

(3])

J.J. Fisz/ Chemical Physics Letters 262 (1996) 495-506

502

In the case of a symmetric diffusor v'(a,0) •'(~'~) (t) reads

q~((,k) ( t'~ = E (a,b)"'"

p((,k) "'2q (t) (ca(Aex) T(ka,sC) +Cb(hex) T(qkb,~))

(32)

q=0,1,2

where

T(Oki'() = P2(o(ki)) P2(O(()) , T(ki'() = 3 sin20(ki) sin20(E~) cos(~o(ki) T2(~i,¢)

3 sin20 (ki) sin20(¢) cos2Qo(ki)

--

~(~')) ,

~o(¢))

(33)

and where i = a, b. The same considerations for a spherical diffusor lead to

~)(',k) ]~(,~,k)ft, ( Ca(aex) /"2[t)'AE' -..~(kal~).~, nt- C0(~.ex) p2(o(kb,())) (a,b)(t) = "%ph '~1

(34)

The problem considered here can be extended to a greater number of higher excited states.

5. Convolution of the model decay with the instrument response function If the instrument response function is determined from the scatter method, one has to calculate the convolution integral between the instrument response function to the exciting laser pulse which is given by the profile of the scattered light F(t, aex) and the model decay which is a 8-pulse excitation response function [5], i.e. t

i(conv),,

~.em) = F(t, hex) * l&~/(t,,~ex,,~em)

= / F(t',hex) lo,~:(t-t',Aex,hem)

dt'

(35)

0

To calculate this convolution integral one has to evaluate the corresponding integrals for the elements of the relax((,k) ation matrix R (~:'k)(t) (for j = 0, 2). Taking into account that Rjql (~'k) ,jq2x(.x l ) = [E jp Vjql,JPexp ( - i~jp,jp t) Vj;,~q2] , L

J

according to (10), one finds t

R)q2,jq, (t) - (,,k)

=

F(t, hex) * R(('k?jq2,jq,(t) = [ E. ( f

F(t', Aex) e -a~''i' (t-t') dt' )Vjql'jpVj;,~q2] ((,k)

(36)

JP 0 In this method the instrument response function and the emission of the sample are collected at different wavelengths. Therefore, the time-shift between both signals is accounted for during the data analysis. To eliminate the wavelength dependence of the instrument response function, the reference convolution method is employed [6-8]. In this method integral (35) is replaced by t

1~,6 (cony)(, t , aex, Aem) = Fr(t, aex, aem) * h~&(t, hex, Aem) = / Fr(t', aex, aem) h~& (t - t', Aex, Aem) dt'

,I

,/ 0

(37) where

1 Aem) [l~,~:(t=O, Aex,Aem) 8(t) h~,~: ( t, &x, &m) - ar(Aex,

1 1~&(t, aex, Aem)] + ~d I~,~:(t, Aex,hem)+ -~r (38)

J.J.

Fisz/ Chemical Physics Letters 262 (1996) 495-506

503

and where 6(t) is the Dirac delta function. Fr(t, hex, hem) is the measured polarization-free (e.g. at the magic angle 54.7 °) monoexponential fluorescence decay of a reference compound, where Fr(t, hex, hem) = F(t, hex) * f r ( t , hex, hem) and f r ( t , hex, hem) = ar(hex, hem) exp(--t/rr). In the case considered here one has to calculate t

~((,k!jq2.jq,(t) = Fr(t,

hex, hem) *

h~,~i(t, hex, ~.em)

Fr(t', hex, hem) h~,~l ( t -

=

t', hex, hem) dt'

(39)

0

where

h~,~j(t, hex, hem) =

1 hem) ar (,,[ex,

1

~ (~:,k) ~

(t~jqt,jq2~sC,k~(t)+[~-~(1/'l'r--Ajp,jp)Vjq,,jpe-AJe'JetVjp, jq2]

~.

(40)

Jv

The reference convolution method requires the same experimental conditions when collecting the fluorescence decays of the sample and the decay of the reference compound. Therefore, both substances should be dissolved in solvents of similar indices of refraction, to avoid a possible time-shift between the decays of the sample and reference compound. (This shift may result from different optical paths within cuvettes with both solvents, for both the exciting light and the emitted fluorescence.) Otherwise, this time-shift must be accounted for when analyzing the experimental data.

6. Synthetic decays for the one-ground and two-excited state problem We consider here the case shown in Scheme 2, with the assumption that there is only one ground state species. Hence, the second excited state (state 2* in Scheme 2) is populated via the state-to-state kinetic relaxation process alone. It is assumed that the system can be excited to two excited states, i.e. to states 1" and 1"*. The fluorescence is detected only from excited states 1" and 2*. The decays of the polarized fluorescence components Ill(t, hex, hem) and l±(t, hex, hem) are calculated from the expressions derived in Section 4. The absorbances to(a)(hex) and w(b)(hex) used in the calculations are represented by bands A1 and A2 shown in Fig. la, respectively. The emission bands E1 and E2, in Fig. la, represent the shapes of the emission bands K1(hem) and x2(hem) of states 1" and 2*, correspondingly. For simplicity of the calculations, bands A1, A2, E1 and E2 have been normalized to unity (the intensity of A1 has been decreased to emphasize a lower probability of the excitation of the molecules to state 1"*). Furthermore, we assume that the rotational dynamics of molecules is described by the symmetric diffusor model. We assume (although this is not necessary) that the electronic transition dipole moments of bands A1 and E2 are directed perpendicular to the long molecular axis and the dipole moments corresponding to bands A2 and E1 are parallel to the long molecular axis. To further simplify the calculations, we also assume that the transition dipole moments of bands A1 and E2 are parallel to each other. In Fig. lb are shown the steady-state emission bands IE1 and IE2 of the states 1" and 2*. They can be calculated from the integrals f~°(lll(t, hex, hem) + 21±(t, hex, ~em))dt. The steady-state bands shown in Fig. lb have been calculated for the rate constants 1/kl = 4 ns, 1/k12 = 0.5 ns, k2 = 5 ns and k21 = 1 ns, and afterwards, they have been normalized with respect to the maximum intensity of the effective steadystate band. The polarized fluorescence decays that can be detected experimentally (i.e. lllC°nv)(t, hex, hem) and I~C°")(t, hex, hem)) have been obtained from the model polarized decays (a t~-pulse excitation fluorescence response) convoluted with the instrument response function S(t) (see Fig. lc), representing the instrument response function Fr(t', hex, hem) in (39). In our example, S(t) is the monoexponential fluorescence decay of Xanthione in hexane (with fluorescence lifetime 25 ps) and the channel width is 10 ps. To make the the convoluted polarized fluorescence decays similar to those which can be collected experimentally, Poisson

504

J.J. Fiszl Chemical Physics Letters 262 (1996) 495-506

1.0

E v

,,

,,

1.0

(o)

,,

\ l

E ~!

'~

IA1 ~!

>, 0.5

I

I I

,' ~,A2 '~'i', II

_c

!' I

0.0 200

%X

300

,.o ~

I'

1%, /

x

E1

'\

~. d

E2

3,

I' •

v

"~

',

\

II,

"\'\

c

x

400 500 Wavelength (nrn)

>, 0.5

600

0.0 200

300

400 500 Wavelength (nm)

600

. . . . . . . . . . . (C)'-

'0.8

:~0.4 ~' 0.2 o.o

~

?!t~. 2

. . . . . . . . . . . . . .

4 6 Time (ns)

8

10

Fig. 1. The absorption and emission bands employed in the calculations (a). The steady-state emission bands /El and IE2 (b). The simulated isotropic fluorescence decays at the emission wavelengths ( 1) 380 nm, (2) 440 nm and (3) 5 l0 nm (c). S(t) is the instrument response function used in the calculations of the convolution integrals.

noise has been added to the simulated polarized decays. In Fig. lc we show the synthetic polarization-free fluorescence decays calculated for the emission wavelengths: (1) 380 nm, (2) 440 nm and (3) 510 nm. They have been obtained from the relation lllC°"V)(t,Aex, Aem) + 21(tc°nv)(t, hex,Aem)- In Figs. 2 and 3 we show computer generated histograms of the polarized fluorescence and emission anisotropy decays for two excitation wavelengths (270 and 290 rim) and three emission wavelengths (380, 440 and 510 rim). In the calculations the values of the state-dependent diffusion tensor components 1/Dll) = 3 ns, 1 / D ~ ' = 5 ns, lIDS2)= 6 ns and 1/D~ ) = 10 ns, have been assumed. As seen in Figs. 2 and 3, the character of the decays of the polarized fluorescence components and emission anisotropy depends on the excitation and emission wavelengths. This is due to the presence of a greater number of absorption and emission overlapping bands, to which there correspond differently oriented electronic transition dipole moments. This also results from the fact that the system undergoes a photochemical reaction. Therefore, to obtain reliable information on the kinetic and dynamic processes and on the composition of an effective absorption and fluorescence band, the excitation- and emission-multiwavelength surface of the experimental polarized decays has to be analyzed globally. Although the rotational dynamics of the fluorophores in both states (assumed in the computer simulations) is slow, the information content on the rotational dynamics is evidently reduced due to relatively fast forward and backward kinetic relaxation of the system (see relations (22) and (23)). Therefore, for systems undergoing fast excited-state reactions (as compared to the rotational dynamics of the fluorophores) the excitation- and emission-multiwavelength surfaces of the polarized decays for two or more different time windows of the experiment have to be analyzed globally. This is necessary in order to obtain more precise information on the rotational dynamics of the fluorophores as well as on the orientation of the electronic transition dipole moments corresponding to all the absorption and emission bands involved. All the calculations and graphics presented in this section have been done in IDL (Interactive Data Language, Research Systems, Inc., Boulder).

505

J.J. Fisz/ Chemical Physics Letters 262 (1996) 495-506

1.0

ex: 2 7 0 ......

"~. 0.8

I,(t) .

.

.

nm

era: 3 8 0

nm 1.0

.

.

.

.

.

.

.

.

.

(ail

.

ex: 2 7 0 n m .............

era: 4 4 0

7~ o.~ ~t)

vc° 0.6

=o o.6

~ 0.4

-~

-

_c 0.2

nrn

ex: 2 7 0

(~)

0

....

nm

e m : 5 1 0 nrn

" . . . . . .

'

" "

'

" '

~'0.8

Ix(t) ~

0.4.

"~ 0.4 iv

0.2 0.01 . . . . . . . . . . . . 2 4 6 Time (ns)

0.0 8

1o

2

4

6

ex: 2 7 0 n m em: 380 nm o.o5 . . . . . . . . . . . . . (b)

ex: 2 7 0 n m 0.10

8

2

10

(ns)

Time

ex: 4 4 0

...........

ex:

nm

270

0.40

(d)

0.00

nm

8

10

em: 510 nm

(f)

.........

0.30

o

"E - 0 . 0 5

0.00 : ~ ' ~ ' - t

o

g

4 6 Time (ns)

" ":'.-L~, ";'--~:

r(t)

8

-0.10

i~

-0.10

"£o

0.20

g

O.lO

E -0.15

0.00

-0.20

-0.20 2

4 Time (ns)

6

,

,

8

,

,

2

,

,

,

,

,

4 T~me (ns)

,

,

.

.

.

.

.

-0.11 .

.

.

.

6

,

2

,

,

.

4 Time (ms)

,

,

,

,~=

6

8

Fig. 2. The polarized fluorescence decays and emission anisotropy at the excitation wavelength 270 nm and the emission wavelengths 380, 440 and 510 nm.

ex: 2 9 0

nm

em:

380

nm

ex: 2 9 0 n m

,,(t)

":'. 0.8

(o)

=o 0.6 ~>, '~ 0.4 ¢, _c 0.2

"~ 0 . 4

c 0.2 0.0

0.0

. . . . . . . . . . . . . . . . . . .

2

4 6 Time (ns)

ex: 2 9 0

nm

8

10

e m : ,580 n m

nm

(c)

1.0

. . . . . . . . . . 2 4 6 Time (ns)

0.0

,.., 8

ex: 4 4 0

2

1o

nm

ex: 2 9 0

nm

8

10

era: 5 1 0 n m

(r)

0.30 o

"~ -0.05

o

o.oo

"E o

r(t)

"E o

r(t)

-0.10

0.20 ~

_

_

.........

0.10

i_~ - O l O E -0.15 -0.20

4 6 Time (ns)

o,o ............

0.00

c

era: 5 1 0 n m

c 0.2

0.10

(b)

ex: 2 9 0 n m ...............

~ 0.6 :~0.4

I±(t)

ex: 2 9 0 n m

0.05 . . . . . . . . . . .

em: 440

lO ...............

1.0

0.00 .

.

.

.

.

2

.

.

.

.

.

.

4 Time (ns)

.

.

.

6

.

.

-0.20

T-

-.

-0.10

2

4 T;me (ns)

6

8

2

4 Time (ns)

6

8

Fig. 3. The polarized fluorescence decays and emission anisotropy at the excitation wavelength 290 nm and the emission wavelengths 380, 440 and 510 nm.

506

J.J. Fisz/ Chemical Physics Letters 262 (1996) 495-506

Acknowledgement I w o u l d like to thank one o f the referees for helpful comments. This w o r k was supported by the Polish C o m m i t t e e for Scientific Research ( K B N ) , within the Project 2.P303.089.04.

References J.J. Fisz, Chem. Phys. 181 (1994) 417. J.J. Fisz, Chem. Phys. Letters xxx, CPL 8770. A.J. Cross, D.W. Waldeck and G.R. Fleming, J. Chem. Phys. 78 (1983) 6455. A. Szabo, J. Chem. Phys. 81 (1984) 150. D.V. O'Connor and D. Phillips, Time-resolved fluorescence spectroscopy in biochemistry and biology (Plenum Press, New York, 1983). [6] P. Gauduchon and Ph. Wahl, Biophys. Chem. 8 (1978) 87. 171 L.J. Libertini and E.W. Small, Anal. Biochem. 138 (1984) 314. [8] M. Zuker, A.G. Szabo, L. Bramall, D.T. Krajcarski and 13. Selinger, Rev. Sci. Instrum. 56 (1985) 14. [9] D.M. Brink and G.R. Satchler, Angular momentum (Clarendon Press, Oxford, 1968). [10] J.J. Fisz, Chem. Phys. 114 (1987) 165. [ 11 ] J. Herbich, K. Rotkiewicz, J. Waluk, B. Andresen and E.W. Thulstrup, Chem. Phys. 138 (1989) 105. [ 12] Z.R Grabowski, K. Rotkiewicz, A. Siemiarczuk, D.J. Cowley and W. Baumann, Nouv. J. Chim. 3 (1979) 443. [ll [2] [31 [4] [5]