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Polarized fluorescence decay surface for many-ground- and many-excited-state species in solution
22 November 1996
CHEMICAL PHYSICS LETTERS ELSEVIER
Chemical Physics Letters 262 (1996) 507-518
Polarized fluorescence decay surface for many-ground- and many-excited-state species in solution J.J. F i s z Institute of Physics, N. Copernicus University, ul. Grudzi(utzka 5/7, PL 87-100 Toruh, Poland Received 8 March 1996; in final form 12 July 1996
Abstract
The theory of fluorescence depolarization experiments on many-ground- and many-excited-state species in solution, is presented. In the formalism discussed the state-dependent rotational dynamics of solutes is coupled with the state-to-state kinetic relaxation processes involved. The rotational dynamics of molecules can be different in different excited states due to state-dependent solvation effects (state-dependent solute-solvent interactions) which can modulate the hydrodynamic size and shape of rotating objects. Another reason for the state-dependence of rotational dynamics can be possible changes in the geometries of the solutes in different excited states. Two cases are distinguished: (a) state-dependent geometrical transformations of solutes which do not change the orientation of the principal axes diagonalizing the diffusion tensors in the excited states involved and (b) geometrical transformation of molecules which change the orientation of the principal axes (state-dependent choice for principal axes). It is assumed that in case (b) the change in the molecular geometry is not an extra source for the molecular reorientation or that this effect can be neglected to a first approximation. The case of rigid isotropic systems is considered. To the memory o f my dear daughter Karolina.
1. Introduction
In this work the description of polarized fluorescence experiments on photoreactive systems in solution will be discussed. This problem represents a particular case mentioned in briefly in a more general treatment discussed in our earlier work [ 1 ], where the many-excited-state problem for ordered systems has been considered. In this work we wish to discuss in more detail solutions and rigid isotropic systems. We consider the situation displayed in Scheme 1. Due to excitation at certain excitation wavelengths more ground-state species can be excited simultaneously (the absorption bands of the species in the ground state can overlap). Polarized emission detected at certain emission wavelengths can be contributed to by emission from more excited states. It is assumed that the primary-excited state species undergo photochemical transformations. One assumes that due to different solute-solvent interactions in different excited states or possible state-dependent geometrical transformations of molecules, the rotational dynamics of molecules is state-dependent. The description of the problem in terms of Green functions p(O0, 01~2, t) [2] and Wigner rotation matrices D~ 2) (~O) formalism [3] enables the consideration of a more general situation when the state-dependent time-development operators 0009-2614/96/$12.00 Copyright (~) 1996 Published by Elsevier Science B.V. All rights reserved. PII S0009-2614(96)01 1 31-1
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J.J. Fisz/Chemical Physics Letters 262 (1996) 507-518
k42
k31
1
~
k3
°°°
equilibration
~ ° °
°
Scheme 1
describing the rotational motion of molecules may have different forms (possible modifications of the diffusion model - e.g. diffusion in an aligning potential in the case of ordered systems). The only assumption made is that the time-development operators are not time-dependent. In our treatment the equation of motion, which is a set of coupled differential equations for state-dependent Green functions, is solved according to an algorithm [ 1 ] in which a method for solving the Fokker-Pianck equation is employed (see e.g. Ref. [2] ). The formula for the polarized fluorescence intensity in the case of isotropic systems (solutions) is derived and simplified to spherical symmetry for the system by employing group theory tools. The group theory methods are employed also for the symmetry reduction of the reorientational correlation functions for a cylindrically symmetric object (which corresponds to a symmetric diffusor in the diffusion model) and a spherically symmetric object (which corresponds to a spherical diffusor in the diffusion model). The same considerations can be repeated for bi-axial objects (asymmetric diffusors). By employing group theory methods we show that for proper simplification of the formula for polarized fluorescence decay no a priori information on the model for rotational motion is required. Exactly the same results are obtained after considering the diffusion model. The approach discussed in this work can be employed to many cases important from an experimental point of view, e.g different kinds of excited-state reactions (inter- and intramolecular light-induced processes) and incoherent excitation energy transfer in double-chromophore (or many-chromophore) rigid complexes. Depending on the particular experimental needs, the theory discussed in this Letter can be further developed. In the present version of the discussed formalism we assume the rates for state-to-state kinetic relaxation and the rates for rotational motion as time-independent quantities. This assumption has been made in order to simplify the problem. However, in all such cases in which the organization of the solvent molecules around the fluorophores has a basic meaning for their kinetic and dynamic properties, the timescale for the solvent relaxation can be important. In particular, in all such cases in which the timescale for the solvent relaxation is not negligible as compared to the excited-state lifetimes of the fluorophores, the rates for state-to-state kinetic relaxation as well as the rates for rotational dynamics should be treated as the time-dependent quantities. Formally, this assumption can be accounted for in our formalism. From a mathematical point of view, however, another method for solving the equation of motion must be employed. This problem will be the subject of independent discussion. The problem of state-dependent rotational dynamics coupled to state-to-state kinetic relaxation processes in solutions was considered earlier by Cross et al. [4]. In the treatment discussed in Ref. [4] the time-evolution of state-dependent distribution functions N(/2, t) of molecules in the excited states is considered. The equation of motion describing the rotational diffusion of molecules, aN(~2, t)/at = [I N(/2, t), can be solved by separating the time-dependence in the equation of motion. This leads to the eigenstate problem /~ ~'(/2) = E q~(/2) [5]. The matrix representation o f / ~ for an asymmetric diffusor is not diagonal in the basis set of Wigner rotation matrices (or normalized Wigner rotation matrices - symmetric diffusor eigenfunctions). Favro [ 6 ] has
J.J. Fisz/Cheraical Physics Letters 262 (1996) 507-518
509
derived such linear combinations of the symmetric rotor eigenfunctions for which the abovementioned eigenstate problem holds (i.e. the matrix representation o f / ~ is diagonal). Therefore, N(/2, t) can be expanded in a basis set of asymmetric rotor eigenfunctions ~,(/,) (/2) with the expansion coefficients decaying exponentially with the eigenvalues Ejmn [7]. This fact has been employed by many authors [5,8,9] in deriving the time-dependence of the polarized fluorescence and the emission anisotropy in solution. Cross et al. [4] have differentiated the asymmetric rotor eigenfunctions for different excited states because the coefficients in linear combinations of the symmetric rotor eigenfunctions are parameterized by the components of the state-dependent diffusion tensor. The case considered in the last section of this work, i.e. rigid isotropic systems, has been considered earlier in an interesting work by Tanaka et al. [ 10], who have discussed such problems as dual emission and energy transfer between unlike molecules. The problem of incoherent excitation energy transfer in rigid double-chromophore complexes, in rigid media, was also discussed earlier in Ref. [ 1 1].
2. Polarized fluorescence components The intensity of the polarized fluorescence emission for the case indicated in Scheme 1 is described by the following formula [ 1]
1~,~i ( t, Aex,'~em) = C (/~ex, ,~em) ×
oh
~ (-Ok( aex ) K( (Aem) (,k=l
1 pe(:~)(/20) W
pk_..f(/20, Oi/2, t)
Pemg'/2) d / ~ d/2
(1)
C(,~ex, ~.em) includes all instrumental excitation and emission wavelength-dependent factors, wk(,~ex) means the absorbance of the kth ground state at &x. K~(~em) represents the intensity of the emission band of state ~: at (,~em). K~'(Aem) = kr,~ g~(1~m), where kr,~ is the radiative rate constant and g~(,~em) is the normalized spectral distribution of the emitted photons. x,k (/20) and /'em,¢(/2) describe the angular dependence of the polarized excitation (absorption) and pe(O,) -(~i) emission probabilities, respectively. Pk~¢ (/'20, 01/2, t) is the Green function describing the conditional probability for finding an excited molecules at time t at angular orientation /2 in state ~, which at t = 0 was excited to the primary excited state k and its initial angular orientation was Do. These propagators describe both the kinetic and dynamic (reorientational) relaxation processes, over all excited states involved. The Euler angles /2 = (a,/3, y), appearing in the relation above describe the angular orientation of molecule-fixed frame and the laboratory-fixed one. It is assumed that the molecule-fixed frame corresponds to the principal axes of the diffusion tensors of molecules in different excited states. 1/8~ 2 in (1) means that the equilibrium orientation distribution of all the ground-state species is isotropic (solution). The expressions in ( 1 ) describing P~(~ (Do) n(~f) and t"em,((/2) take the following form: 2 pe(:'k)(Do) ~ 3 COS2(Oi,/2~b,k) = 1 + 2 E C2*p,(Oi'~pi) E O(p~?q;(DO)C2'ql(O(k"~°(k))' pl=--2 ql 2 t" * t a f t ) .~(~) ,-,(~I)(/2) ,~3 COS2( e f ' ]'~em,~") = 1 + 2 E (2) (/2),.-,2,q2~O'E ,WE ) /"e rn,,~ C2"p2(Of' ~Of) E D p2,q2 p2=-2 q2
(2)
(~i,/2ab,k ) means the angle between the unit vector ~i, representing the polarization direction of the exciting light, and the absorption dipole moment direction/2~b,k. (~f, ]~em,~:) means the same for the direction of polarization of the detected light, Of, and direction of the emission dipole moment, /2~m,g. The polar angles (Oi,~Oi)
J.J. Fisz/Chemical Physics Letters 262 (1996) 507-518
510
Zt
vt
Xt Scheme 2
and (Of, ~of) define the angular orientations of 0; and 0f in the laboratory frame, respectively. (O~Ak), ~,~Ak)) and .0(¢), t E q~E (~:)) describe the orientations of the absorption and emission transition dipole moments of the molecules in states i and ~ (see Scheme 2), respectively. Cj,.,(O, ~o) are the modified spherical harmonics [3] (Cj, m(O, q~) = [47r/(2j + 1) ]1/2 ~,,,,(0, ~o), where these latter are spherical harmonics). The formulae above have been obtained after taking into account the following relations: (a) 3 cos2(0) = 1 + 2/'2(0) where 0 is the angle between two vectors, (b) P2(0) = ~-'~. C~,n(Ol,~l)C2,n(02,~2) where (01,~pl) and ( 0 1 , ~ t ) l ) are the polar angles describing the orientations of both vectors in the same coordinate frame and (c) the components of C2,n(O2, ~2) can be transformed to another coordinate system (e.g., molecule-fixed frame) i.e. C2,,,(02,q~2) = ~--~., D ~,,, (2) *(/2) C2,m( 02,~02). ' t P2(0) is a second rank Legendre polynomial, To find the explicit form of the Green function pk--.~(/20, 0[/2, t) one can consider the time-dependence of the propagator p_.((O0, 0[/2, t), which describes the conditional probability of finding a molecule in the excited state s¢: at time t, which at t = 0 was excited to an arbitrary primary excited state, i.e. p-~(/20, 0[/2, t) = ~ , pk_.e(O0,0l/2, t) Ak,ex. Ak,ex = 1 if state k is the primary excited one and Ak,.~ = 0 otherwise. In (1) the Ak,ex function is represented by absorbance wk(Aex). The initial conditions for p--.e(O0,01/2, t) are such that p_.~(ao,01/2,o) = ~ pk-.~(Oo,0l/2,0) A~,~xBE,k, where pk__.((Oo,01/2o,0 ) = 6(/20 - / 2 ) . The state-to-state kinetic relaxation and reorientational molecular dynamics over all excited states, in Scheme 1, is described by the following system of differential equations for the propagators p-.~:(/20,0[/2, t) (s~ = 1 ..... s) at p--.l(Oo,0[/2, t) = -
~I (I) + kl +
kli p--.l(Oo, OI/2, t) i=2
Nil p--+i(Oo, O[/2, t)
,
i=2
(3) atp_.s(Oo,0[/2, t ) = -
-~--~ ki, p-.i( /2o,Ol/2, t) +
1~1~s) + k, + ~-~ ksi p ~ s ( /2o,O[O,t)
i=1
The solution to the equation of motion (3) has the form [ 1]
t=l
.
J.J. Fisz/Chemical Physics Letters 262 (1996) 507-518
--
k=l
87/-2
E
"'jmn,j'm'n' (t) O(('k)
511
D m'n' (j') (1"20) D(mJ)~*([2) Ak,ex
j/m/n /
=£pk--.e( f20,OlO, t)
(4)
ak,ex ,
k
and from its structure the explicit form of p k ~ ( g ~ , 01//, t) is immediately obvious•
2.1. The properties of the solution to the equation of motion The equation of motion (3) can be rewritten in matrix form: 3 Ot P(O°'O[ O,t) = - , ~ p(O0,O[ a , t ) .
(5)
After expanding the Green functions in the basis set of Wigner rotation matrices 2j+l
p-.¢(~.01 f2, t ) = E
8zfi C)(m~(t) D(/)*(Y2)'
(6)
jmn
the equation of motion (5) can be replaced by the set of coupled differential equations for the coefficients C)m,(t), [1] i.e. 0 at C(t) = - M C ( t ) .
(7)
The formal solution to the system of equations (7) is C(t) = R(t) C(O). R(t) describes all the relaxation processes (state-to-state kinetic relaxation and reorientational molecular dynamics) over all the excited states involved, where R(t) = V exp ( - A t) V -1. V and A = [t~qq] are the eigenvector and eigenvalue matrices of matrix M, respectively. M is the matrix representation, in the basis set of Wigner functions, of the matrix operator ~, in the equation of motion (5). The matrices M and R(t) have the following structures:
UIK2 ' 3 / iR2tR3RsI
[ Klz U (2) K32 M = ] KI3 K23 U c3) •
:
•
K~2 K~3 .
\ Kls K2s K3s
U is)
,
R(t) =
| R ( 2 a ) ( t ) R(2'2)(t) R(2'3)(t) R(3a)(t) W3'z)(t) W3'3)(t) :
•
:
\ R ( S a ) ( t ) R(S'2)(t) R(S'3)(t)
U (¢) are submatrices with elements [U (~)] P,q = H(¢) ~jmn,j'm'n' = t4(t:) "'jmn,fm'n'"~-( k( J t - E p =s l , p # ( ~,j'm','--
2j' + 1 f ~i
j
/.)(j) (~2) /t/(~) D(f) *tn~ dO mtn ~ \ a~,l -.,..
R(2'~)(t) R(3's)(t)
.
(8)
•
R(S'~)(t) k(p) ~mn,j'm'n%
where (9)
12
The submatrices Kxy are diagonal, i.e. [Kxy] jmn,j'n,'n' = - k x y ~ j m n ' j ' m ' n ' '
(lO)
where kxy is the rate for state-to-state kinetic relaxation from an excited state x to excited state y. The elements (¢,k) .n, (t) = [(V [exp (-•qq/)] V - I ) (so'k) ]jmn,j,mtn,• Index s¢ means the state from which of R (('k) (t) a r e Rjmn,j,m. the polarized emission is observed, while k means one of the primary excited states.
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J.J. Fisz/Cheraical Physics Letters 262 (1996) 507-518
When the explicit form of the the time-development operator /2/(~) is known, the expansion coefficients C(t) can be calculated from Eq. (7), and hence, the Green functions (6) can be reconstructed. The form of the reconstructed Green function is given by (4). Note that, in principle, only the elements of R(t) have to be calculated. The problem of determining the eigenvectors and eigenvalues of matrix M can be essentially simplified if the system of equations (7) can be split into independent subsystems of equations. This is equivalent to rearranging matrix M into a form of independent submatrices for which the eigenvectors and eigenvalues can be calculated separately. 2.2. S y m m e t ~ adapted form of the Green functions The propagators p ~ ( O o , 0[/'2if), t) must be projected into symmetry adapted forms due to the spherical symmetry of the system as a whole (solutions are spherically symmetric systems). The required invariance means that p__.f(O0,0112, t) must be invariant with respect to arbitrary changes in the orientation of the laboratory-fixed frame. If R(ST) represents a rotational transformation of the laboratory-fixed frame under the Euler angles ST, R(12') D(.]n) (12) = }-~m' D(j) ~j)n ~,(12) represents the transformation of the Wigner Blt nl* (ST) D nlt rotation matrices under R (/2'). The linear combinations of Wigner rotation matrices which are totally symmetric with respect to all symmetry operations in a proper rotation group of symmetry O~+)(3) can be generated by employing the projection operator i.e. Siso[ l = ~ 2
,/
R(ST)[ ]dsT,
.(2' and where ¢j') "{,120) - O(J) * (12) ] Siso [[D m'n' --ran
_ 2j - -1 + 1
~m.,'
~jj'
D u(J) n,(120)
Z
-]')(J) - qn * ( 1 2 ) "
q
By replacing each product basis set of Wigner matrices in (4) by the corresponding linear combination generated by Siso[ ], one obtains the symmetry adapted propagator p__.¢(.O0,0112, t ) = i=1
2j8,.//.2 + 1 ~(s¢,i) "'jn,jn' (t)
z
m
/7(J) (,1"20) D(J2 *(12) ~mn'
]
Aiex , '
(11)
where the elements R(('i)jn,jn'~ [ I)~ do not depend on the index m and the equality ~'~., Rjmn,jmn,(t )(('i) = (2j+" 1) R(¢'i)jn,jn,(t) has been employed. This result means that in the system of equations (7) the equations with different values of j are independent. Secondly, the equations are indistinguishable with respect to the index m. After inserting (11 ) and (2) into (1), performing the integrations over 12o and 12 and after employing the orthogonality properties for the Wigner matrices [3], the decay of the polarized fluorescence surface takes the final form I0'el (t'h'ex'~'em) -- C(s~ex'"~em) Z C°k('~ex)K((J~em)[Ph("k)(t)-t-~ sC,k
,
e2(oif)~((,k,(/)]
(12)
where 2 Ph(('k)(t) = R~', k)oo(t) '
dP(('k)(t) =
Z
2q2,2q~'J ~2,q~t~A ,~PA ) C2,q2
,
•
(13)
ql,q2=--2
Ph (('k) (t) means the polarization-free fluorescence decay. It describes state-to-state kinetic relaxation processes. The second term in (12) describes the polarization effects of the emitted fluorescence. Oif is the angle between Oi
J.J.
Fisz/Chemical Physics Letters 262 (1996) 507-518
513
and ~f. ~b(('k) (t) is the reorientational correlation function, describing the state-dependent rotational molecular motion of the system coupled with the state-to-state kinetic relaxation processes. The parallel fluorescence component Ill (t, Aex, Aem) (i.e. when ~ill~f) obtained for P2(0 °) = l, and the perpendicular one Ix(t, Aex, Aem) (i.e. when ~i±ef) for P2(90 °) = - 1 / 2 . The structure of ~b(¢,k)(t) can be simplified if the appropriate hydrodynamical shapes of the solutes are considered. Further simplifications can appear after accounting for real molecular symmetries (symmetry point groups) which restrict the possible orientations of electronic transition dipole moments. Let us consider the case of cylindrically symmetric objects. Hydrodynamical cylindrical symmetry of an object means that the Green function ( l l ) must be invariant: (a) with respect to arbitrary rotation of the molecule-fixed frame around its Z axis (i.e. C,~ (Z) symmetry operation) and (b) with respect to rotation by 7r around any axis perpendicular the Z axis of the molecule-fixed frame (i.e. C2(ZZ) symmetry operation). Both kinds of symmetry operations are performed simultaneously on the molecule-fixed frames at t = 0 and t > 0. In other words, the hydrodynamical (statistical) symmetry of a symmetric rotor is D ~ ) which contains purely rotational symmetry operations as in the symmetry point group Do, h, with the difference that for the D(S~ symmetry group these symmetry operations do not have exact meaning as in the point group Do~h. These kinds of symmetries have been considered in case of ordered systems [ 12 ]. It can be easily shown that due to the C,~ (Z) symmetry operation Dm.,(Oo) (J) DcJ) * (j,2) must be replaced in (11) by D(J)COo) m.. D(J.~) *(12) 6nn,. Secondly, due to the C 2 ( / Z ) symmetry operation, the product basis vectors D.(,J2(Do) D.(Jn) *(1"2) (for m,n 4= 0) must (j)
*
be replaced by linear combinations (1/2) [D.(/.) (/20) D(.L.) *(/2) + D(mJ).(Oo) Din_ n (12)], which are generated by the projection operator S[ ] = (1/2) [E, C2(-I-Z) ] [ ], where E is the identity symmetry operation. Finally, one obtains the following form of (1 3): 2
dp(~'k) (t) = ~
R(J'k)(t) T(pkg) ,
(14)
p=0
where
R(o£'k' = O(¢'k'' " ' 2 0 , 2 0 ~ t)
To(~g) = P2(0(Ak)) P2(0(E~, )
'
RI¢,,) = ~+ [~, \..z,.2,,., (~:,k, , t. , R~:.')
= ~, /~. ,(, 2(2. ,k2 2) ~t ,.,,1
RE_,.2_,(t)) ,(~:,k) -Jr- Rz_2.E_2(t)) (sO'k)
T(k.~) = 3 sin 2o(ak) sin 20(if) COS(~O(A k) _ q~(ff)) , ,
T2(kg)
= 3
sin20(Ak)sin20(E~:)
COS
2(~p(k)
--
~O(E ¢) )
.
(15)
According to group theory considerations the rotational dynamics of a cylindrically symmetric object is described by three correlation functions (in the case when second rank correlation functions are considered). The simplest case appears for a spherically symmetric rotor. By employing a similar projection operator like that when considering the spherical symmetry of solutions (spherical hydrodynamical symmetry of an object means that the molecule-fixed frame can be selected in an arbitrary way). one obtains 2 t~ (so'k) ( t )
=
D
( q~2.__ ~("k) P2(O(A~0 ) "'2q2,2qi tt)) *" ql,
-
=
R~spk(t)r2~VAE n r'~Ckg)) ,
(16)
2
n(¢) is the angle between absorption where Rsp h(~:'k),-Jr t~ describes the dynamic-kinetic relaxation for a sphere and ~AZ and emission dipole moments in states k and ~:, respectively. 3. Diffusion model for rotational motion
In the diffusion model for an asymmetric rotor the time-development operator/~(~) has the form
514
J.J. Fisz/ Chemical Physics Letters 262 (1996) 507-518 3
f/(¢) = Z D)¢) L/2'
(17)
i=l
where the operators Li are identical to the quantum mechanical orbital momentum operators (i = x , y , z ) a n d (() D i are the components of the diffusion tensor. For a symmetric rotor ~nroblem D (~)x = D(()r = D(¢)- --
Dl~e~rtheskr~
ifnlD~x~) = D~,~) = D~O = D(~), the problem reduces to the spherical rotor case"
"
• e e ts of the time-development operator for an anisotropic diffusor can be easily calculated by j '1 *.(/2)" One obtains taking into account explicit definitions for Lx D(J')m'#*(/2)' Lr. D(J')m'n'* (/2) and Lz D (m'n' mn,jtm'n' -~
Aj'n'
+
+ n'2 D~ ~)] ~nn,
8.,.,+2 + 8j,.,
-
)
jj,
(18)
where
Aj, n, = l [j' (j' + l ) - n '2] and
j'n' = 1
[(j ' ( j ' +,,
_
)) (j'(j' + , ) - ( n '
+
As seen from the form of (18), there is no coupling between the equations in the system (7) for which j 4: j'. Hence, the system of equations (7) can be decomposed into sets of independent equations, each for different values of j. Furthermore, all the equations are indistinguishable with respect to the index m. For a symmetric diffusor "'jmn,j'm'nt 74(¢) -~ [j (j + 1) D ~ ) + n 2 (DII~:) - D(A_() )] ~jmn,jtm'n'. Because the matrix representation of [Kxy] is also diagonal, the system of equations for a given j can be decomposed into subsystems, each for different values of n. Note that it is not important whether n takes a positive or negative value. This means that for j = 2 (the range of correlation functions in our formulae) the number of independent subsystems of differential equations reduces from 5 equations to 3 for each excited state (, and they are coupled over all the excited states involved. Another consequence of this fact is that R((,k). 2n,Zn[,t)- = "'2-n,2-n'.'J p<¢,k) r'~ in (15). For a spherical rotor D ¢O = Dx~:) = D~• ~) = D~(), and consequently "jmn,jtm'n' 14(() = J(J + I) D (() ¢~jj, 6rnm' 6nn'. In this case the matrix representation of the time-development operator does not depend on the index n. This means that for every excited state and given value of j, there exist only one independent equation, which is coupled with the corresponding (the same) equations for the remaining excited states.
4. State-dependent choice for principal axes of the state-dependent diffusion tensors In the case when the molecules in different excited states have different principal axes for the diffusion tensors (they are diagonal in these coordinate systems), the Euler angles corresponding, to different excited states are distinguished by proper indices. Therefore, in the formula for polarized fluorescence decay (1) the Euler angles /20 must be replaced by D0(k) and/2 by /2(~). As shown in Ref. [ 1 ] the blocks U (~:) are the same as in the usual case, while the elements of submatrices Kxy become in this case Kxy]
jmn,jr nVn~
FI(J')[AO(Y-'-*x)~ = - k x v ~n'n i~ ! ~jj' ~mm' .
(19)
J.J.
515
F i s z / Chemical Physics Letters 262 (1996) 5 0 7 - 5 1 8
where Af2(y-'x) are the Euler angles transforming the principal axes from state y to state x and it is assumed that A/2 (-~'-~x) A/2cx~y) -1 Hence, F I ( f ) f A ( ) ( Y ' - - ' x ) ' I = D (j') *(Al-I(x~Y)). It is important to note here that the ~k) ) and ( 0 e( 0 , ,,~(() angles ( o~k) ~ , ~o~ "re ) are defined in different molecule-fixed frames for k v~ sc, if the state-to-state kinetic relaxation is followed by geometrical transformation of the solutes. This has to be taken into account when considering relation (13). Note also that the blocks Kxy couple the equations in (7) for different values of n. We assume here that the change in the molecular geometry is not an extra source for the molecular reorientation or that this effect can be neglected to a first approximation.
5. Time-dependent and steady-state emission anisotropy surface The expression for the emission anisotropy decay surface can be easily obtained from the polarized fluorescence decays defined by (12), i.e. r(t, Aex,Aem)
=
I[l(,~ex,,~ern,t) -I_l_(,~ex, Aem, t) IIl(Aex, Aem,t) + 2I±(Aex, Aem,t)
=
0.4)-]~(,k tOk(~ex) K((Aem) (b(k'()(t) ~--]~(,k tOk(Aex) K(('~em) PhC~:'k)(t)
(20)
The steady-state emission anisotropy surface (r(a~b, ~.em)) can be obtained from similar calculations performed on the steady-state (continuous excitation) intensities of the polarized fluorescence i.e.
(I~, ~f(~ex, Aem))= C(Aex,~em) E
t°k(Aex) r((Aem)
[(Ph (('k)) + 34
P2(Oif)
(t~(sC'k))]j .
(21)
g,k
The symbol (...) means the integral f o "'" dt, where (Ph(g'k)) = /°(~'k)\ \"00,00/
'
(dP(('k)) =
2 Z /R(~'k) \ " ,,,(k) (k), p , ra(() ..(~) \ 2q2,2qt I t~2,ql~,ffA ,~0A ) W2,q2tVE ,WE ) , ql,q2=-2
(22)
and (~,(~,k) \
_
(23) J jn,jn'
-- L
a jn,jn'
Finally, one obtains the formula for the steady-state excitation- and emission-wavelength dependent emission anisotropy surface (lll(Aex, Aem)> - (I±(Aex,'~em)) E¢.k t°k(Aex) K~:(Aem) (q~(k,~:)) (r(Aex,Aem)) = (/'~A--~'x,Ae-e-~ ~- 2("~-l(-'~'ex,A-~m)} =0.4 . E~C,k tOk(~ex) K¢(aem) (Ph (~:'t))
(24)
6. Rigid isotropic systems By neglecting the angular dependence of the Green function in (1), one obtains immediately the formula for the polarized fluorescence decay surface for rigid molecular systems. The kinetic relaxation matrix for the propagators p__.¢(Olt) (~ = 1 ..... s) can be easily obtained from (3) by neglecting the state-dependent operators f/(¢), i.e. at P ~ l ( ° l t )
=-
kl +
kli i=2
p~l(Olt) -
kil p--i(OIt) i=2
,
516
J.J. Fiszl Chemical Physics Letters 262 (1996) 507-518
(25) s-I a p~s(0lt)
Ot
-
-
ki~ p--.i(Olt) i=l
The final result for the polarized fluorescence intensity decay surface can be obtained without any additional . . =_ ~,((,k) t,~ is obvious in this case, and where the elements ..oo,oo(t) ~,((,k).. calculations because relation R(g,k) 2q2,2ql(t) --oo,oo~-J are the same as those appearing in the polarization-free term Ph(('k)(t) already derived. Consequently, one obtains
l~,~f(t, aex, aem) = C(a~x,aem) ~
wk(aex) K¢(a~m)R(~'k)tt oo,oo, ~,
[1 _.]._4 g Pz(Oif) p2(0(~())] .
(26)
(,k The expressions for the time-dependent and steady-state emission anisotropy are
r(t,a.,aom) =O4 Ze, k o,k(aex) ,,e(aom) R i (t) e2(0k e)) Z ( , k °')k(aex) K((~.em)
R((,k) t t~ 00,00~, ,
(27)
and
(r(aex, ~-em)) = 0.4 ~"~(,k O~k(Aex) K((,Aem) \"00,00/ /l~(('k)\ P2 --AE (a(k'() ) (R(~,k)~ O)k(Aex) K((~.em) 00,001
~-~'~:,k
(28)
"AEtl(k'~)
where is the angle between the absorption dipole of a molecule in the ground state state k and the emission dipole moment of the same molecule in excited state ~r.
7. Discussion
The formalism discussed in this work is based on a number of simplifying assumptions mentioned in Section 1. The validity of these assumptions must always be verified for every particular experimental case under consideration. In its present form, this approach cannot be utilized in such experimental cases in which the timescale for the solvent rearrangement around the fluorophores (rotational and translational motion of the solvent molecules) after electronic excitation or after relaxation of the fluorophores to other excited states, is comparable with the timescale for state-to-state kinetic relaxation processes and rotational motion of the fluorophores. This problem relates to all processes controlled by the solvent organization and its changes around the fluorophores. Therefore, one can expect that in ultrafast laser spectroscopy (femto/picosecond timescale) the applicability of the formalism outlined in this work will be extremely restricted, because the timescale of such experiments is so short that the problem cannot be described in terms of time-independent kinetic rate constants and time-independent diffusion tensors. The solvent relaxation process and the spectrum relaxation have to be accounted for in the model. In picosecond spectroscopy one can deal with experimental cases in which the approach discussed here cannot be applied for the same reasons as" in the case of ultrafast spectroscopy. On the other hand, however, if the timescale for the solvent relaxation (and spectrum relaxation) is short compared to the timescale of the kinetics and rotational dynamics of the fluorophores, one can expect that the model described in this work can be applied in the experimental data analysis. Nevertheless, this should always be verified for any individual experimental case, e.g. by precise preliminary kinetic studies. From the kinetic studies one can conclude whether the kinetics of the system can be described by the discrete values of the rate constants. If this is the case, one can expect
J.J. Fisz/ Chemical Physics Letters 262 (1996) 507-518
517
that also the state-dependent rotational dynamics of the fluorophores can be described by discrete values of the rate constants (the components of the diffusion tensors). In nanosecond spectroscopy the timescale for solvent relaxation can be shorter by many orders as compared to the timescale for the state-to-state kinetics and rotational dynamics of fluorophores, both ranging in the nanosecond timescale. In such cases the approach described in this work can be applied, provided that no other complications appear and that the kinetics of the system is described by the discrete values of the rate constants. In some experimental cases the kinetics of the system must be described in terms of the distributed rate constants, although the system (as a whole) is in an equilibrium state, in a given excited state. Such situations can occur for systems in which the fluorescence signal detected is contributed to by signals from fluorophores having different configurations of the solvent molecules and the rates for the kinetic relaxation are strongly solvent-configuration dependent. In such cases, expressions describing the fluorescence signals can be obtained by integrating the expressions derived in this work over the distributed rate constants (assuming particular shapes for the distributions of the corresponding rate constants). The same can be considered for the components of the state-dependent diffusion tensors, because rotational motion of the fluorophores is also a solvent-configuration dependent process. The description of the many-level problem in terms of distributions for the kinetic rate constants and state-dependent diffusion tensors can be important for the analysis of excitation energy transfer processes for double-chromophore (or many-chromophore) complexes. The formalism discussed here can be employed in studies of rigid complexes (i.e. a single donor-acceptor configuration). In the case of a mixture of different donor-acceptor configurations, the relations describing polarized fluorescence decays should be replaced by the weighted sum over the decays corresponding to different donor-acceptor configurations. In a limiting case, when the distribution of the possible donor-acceptor configurations has a continuous character, the weighted sum of the decays has to be replaced by an integral over the distributed rate constants for the excitation energy transfer process. This description can be utilized in studies of conformational structures of the macromolecules (e.g. proteins) labeled by donor and acceptor systems. Assuming that donors and acceptors are attached rigidly in the macromolecules, the shape of the distribution for the energy transfer rate constants will reflect the distribution of the possible macromolecular conformers, in which the distance and angular orientation between the donors and the acceptors are different. The same problem can also be parameterized in terms of the multimodal distributions of the possible donor-acceptor configurations, where two or more possible donoracceptor configurational distributions centered at different most probable donor-acceptor configurations have to be assumed in the model. The applicability of our approach to the energy transfer problem is limited to the case of incoherent excitation of the system, i.e. the excitation is localized on one of the chromophores in a double- or many-chromophore complex. Also the same limitations as those discussed above and concerning the timescale for the solvent relaxation (for polar systems in polar solvents) have to be taken into account when analyzing real experimental cases.
Acknowledgement I would like to thank one of the referees for valuable discussions. This work was supported by the Polish Committee for Scientific Research (KBN), within the Project 2.P303.089.04.
References [1] J.J. Fisz, Chem. Phys. 181 (1994) 417. [2] H. Risken, in: The Fokker-Planckequation. Methods of solutions and applications (Springer, Berlin, 1984). [3] D.M. Brink and G.R. Satchler, Angular momentum(Clarendon Press, Oxford, 1968). [4] A.J. Cross, D.W. Waldeck and G.R. Fleming, J. Chem. Phys. 78 (1983) 6455.
518
[5] [6] [71 [8] [9] [ 10] [11] [121
J.J. Fisz/Chemical Physics Letters 262 (1996) 507-518
M. Ehrenberg and R. Rigler, Chem. Phys. Lett. 14 (1972) 539. D. Favro, Phys. Rev. 119 (1960) 53. W.T. Huntress, J. Chem. Phys. 48 (1968) 3524. T. Tao, Biopolymers 8 (1969) 609. T.J. Chuang and K.B. Eisenthal, J. Chem. Phys. 57 (1972) 5094. E Tanaka and N. Mataga, Photochem. Photobiol. 29 (1979) 1091. A.A. Demidov and D.L. Andrews, Chem. Phys. Lett. 235 (1995) 327. J.J. Fisz, Chem. Phys. 114 (1987) 165.
CHEMICAL PHYSICS LETTERS ELSEVIER
Chemical Physics Letters 262 (1996) 507-518
Polarized fluorescence decay surface for many-ground- and many-excited-state species in solution J.J. F i s z Institute of Physics, N. Copernicus University, ul. Grudzi(utzka 5/7, PL 87-100 Toruh, Poland Received 8 March 1996; in final form 12 July 1996
Abstract
The theory of fluorescence depolarization experiments on many-ground- and many-excited-state species in solution, is presented. In the formalism discussed the state-dependent rotational dynamics of solutes is coupled with the state-to-state kinetic relaxation processes involved. The rotational dynamics of molecules can be different in different excited states due to state-dependent solvation effects (state-dependent solute-solvent interactions) which can modulate the hydrodynamic size and shape of rotating objects. Another reason for the state-dependence of rotational dynamics can be possible changes in the geometries of the solutes in different excited states. Two cases are distinguished: (a) state-dependent geometrical transformations of solutes which do not change the orientation of the principal axes diagonalizing the diffusion tensors in the excited states involved and (b) geometrical transformation of molecules which change the orientation of the principal axes (state-dependent choice for principal axes). It is assumed that in case (b) the change in the molecular geometry is not an extra source for the molecular reorientation or that this effect can be neglected to a first approximation. The case of rigid isotropic systems is considered. To the memory o f my dear daughter Karolina.
1. Introduction
In this work the description of polarized fluorescence experiments on photoreactive systems in solution will be discussed. This problem represents a particular case mentioned in briefly in a more general treatment discussed in our earlier work [ 1 ], where the many-excited-state problem for ordered systems has been considered. In this work we wish to discuss in more detail solutions and rigid isotropic systems. We consider the situation displayed in Scheme 1. Due to excitation at certain excitation wavelengths more ground-state species can be excited simultaneously (the absorption bands of the species in the ground state can overlap). Polarized emission detected at certain emission wavelengths can be contributed to by emission from more excited states. It is assumed that the primary-excited state species undergo photochemical transformations. One assumes that due to different solute-solvent interactions in different excited states or possible state-dependent geometrical transformations of molecules, the rotational dynamics of molecules is state-dependent. The description of the problem in terms of Green functions p(O0, 01~2, t) [2] and Wigner rotation matrices D~ 2) (~O) formalism [3] enables the consideration of a more general situation when the state-dependent time-development operators 0009-2614/96/$12.00 Copyright (~) 1996 Published by Elsevier Science B.V. All rights reserved. PII S0009-2614(96)01 1 31-1
508
J.J. Fisz/Chemical Physics Letters 262 (1996) 507-518
k42
k31
1
~
k3
°°°
equilibration
~ ° °
°
Scheme 1
describing the rotational motion of molecules may have different forms (possible modifications of the diffusion model - e.g. diffusion in an aligning potential in the case of ordered systems). The only assumption made is that the time-development operators are not time-dependent. In our treatment the equation of motion, which is a set of coupled differential equations for state-dependent Green functions, is solved according to an algorithm [ 1 ] in which a method for solving the Fokker-Pianck equation is employed (see e.g. Ref. [2] ). The formula for the polarized fluorescence intensity in the case of isotropic systems (solutions) is derived and simplified to spherical symmetry for the system by employing group theory tools. The group theory methods are employed also for the symmetry reduction of the reorientational correlation functions for a cylindrically symmetric object (which corresponds to a symmetric diffusor in the diffusion model) and a spherically symmetric object (which corresponds to a spherical diffusor in the diffusion model). The same considerations can be repeated for bi-axial objects (asymmetric diffusors). By employing group theory methods we show that for proper simplification of the formula for polarized fluorescence decay no a priori information on the model for rotational motion is required. Exactly the same results are obtained after considering the diffusion model. The approach discussed in this work can be employed to many cases important from an experimental point of view, e.g different kinds of excited-state reactions (inter- and intramolecular light-induced processes) and incoherent excitation energy transfer in double-chromophore (or many-chromophore) rigid complexes. Depending on the particular experimental needs, the theory discussed in this Letter can be further developed. In the present version of the discussed formalism we assume the rates for state-to-state kinetic relaxation and the rates for rotational motion as time-independent quantities. This assumption has been made in order to simplify the problem. However, in all such cases in which the organization of the solvent molecules around the fluorophores has a basic meaning for their kinetic and dynamic properties, the timescale for the solvent relaxation can be important. In particular, in all such cases in which the timescale for the solvent relaxation is not negligible as compared to the excited-state lifetimes of the fluorophores, the rates for state-to-state kinetic relaxation as well as the rates for rotational dynamics should be treated as the time-dependent quantities. Formally, this assumption can be accounted for in our formalism. From a mathematical point of view, however, another method for solving the equation of motion must be employed. This problem will be the subject of independent discussion. The problem of state-dependent rotational dynamics coupled to state-to-state kinetic relaxation processes in solutions was considered earlier by Cross et al. [4]. In the treatment discussed in Ref. [4] the time-evolution of state-dependent distribution functions N(/2, t) of molecules in the excited states is considered. The equation of motion describing the rotational diffusion of molecules, aN(~2, t)/at = [I N(/2, t), can be solved by separating the time-dependence in the equation of motion. This leads to the eigenstate problem /~ ~'(/2) = E q~(/2) [5]. The matrix representation o f / ~ for an asymmetric diffusor is not diagonal in the basis set of Wigner rotation matrices (or normalized Wigner rotation matrices - symmetric diffusor eigenfunctions). Favro [ 6 ] has
J.J. Fisz/Cheraical Physics Letters 262 (1996) 507-518
509
derived such linear combinations of the symmetric rotor eigenfunctions for which the abovementioned eigenstate problem holds (i.e. the matrix representation o f / ~ is diagonal). Therefore, N(/2, t) can be expanded in a basis set of asymmetric rotor eigenfunctions ~,(/,) (/2) with the expansion coefficients decaying exponentially with the eigenvalues Ejmn [7]. This fact has been employed by many authors [5,8,9] in deriving the time-dependence of the polarized fluorescence and the emission anisotropy in solution. Cross et al. [4] have differentiated the asymmetric rotor eigenfunctions for different excited states because the coefficients in linear combinations of the symmetric rotor eigenfunctions are parameterized by the components of the state-dependent diffusion tensor. The case considered in the last section of this work, i.e. rigid isotropic systems, has been considered earlier in an interesting work by Tanaka et al. [ 10], who have discussed such problems as dual emission and energy transfer between unlike molecules. The problem of incoherent excitation energy transfer in rigid double-chromophore complexes, in rigid media, was also discussed earlier in Ref. [ 1 1].
2. Polarized fluorescence components The intensity of the polarized fluorescence emission for the case indicated in Scheme 1 is described by the following formula [ 1]
1~,~i ( t, Aex,'~em) = C (/~ex, ,~em) ×
oh
~ (-Ok( aex ) K( (Aem) (,k=l
1 pe(:~)(/20) W
pk_..f(/20, Oi/2, t)
Pemg'/2) d / ~ d/2
(1)
C(,~ex, ~.em) includes all instrumental excitation and emission wavelength-dependent factors, wk(,~ex) means the absorbance of the kth ground state at &x. K~(~em) represents the intensity of the emission band of state ~: at (,~em). K~'(Aem) = kr,~ g~(1~m), where kr,~ is the radiative rate constant and g~(,~em) is the normalized spectral distribution of the emitted photons. x,k (/20) and /'em,¢(/2) describe the angular dependence of the polarized excitation (absorption) and pe(O,) -(~i) emission probabilities, respectively. Pk~¢ (/'20, 01/2, t) is the Green function describing the conditional probability for finding an excited molecules at time t at angular orientation /2 in state ~, which at t = 0 was excited to the primary excited state k and its initial angular orientation was Do. These propagators describe both the kinetic and dynamic (reorientational) relaxation processes, over all excited states involved. The Euler angles /2 = (a,/3, y), appearing in the relation above describe the angular orientation of molecule-fixed frame and the laboratory-fixed one. It is assumed that the molecule-fixed frame corresponds to the principal axes of the diffusion tensors of molecules in different excited states. 1/8~ 2 in (1) means that the equilibrium orientation distribution of all the ground-state species is isotropic (solution). The expressions in ( 1 ) describing P~(~ (Do) n(~f) and t"em,((/2) take the following form: 2 pe(:'k)(Do) ~ 3 COS2(Oi,/2~b,k) = 1 + 2 E C2*p,(Oi'~pi) E O(p~?q;(DO)C2'ql(O(k"~°(k))' pl=--2 ql 2 t" * t a f t ) .~(~) ,-,(~I)(/2) ,~3 COS2( e f ' ]'~em,~") = 1 + 2 E (2) (/2),.-,2,q2~O'E ,WE ) /"e rn,,~ C2"p2(Of' ~Of) E D p2,q2 p2=-2 q2
(2)
(~i,/2ab,k ) means the angle between the unit vector ~i, representing the polarization direction of the exciting light, and the absorption dipole moment direction/2~b,k. (~f, ]~em,~:) means the same for the direction of polarization of the detected light, Of, and direction of the emission dipole moment, /2~m,g. The polar angles (Oi,~Oi)
J.J. Fisz/Chemical Physics Letters 262 (1996) 507-518
510
Zt
vt
Xt Scheme 2
and (Of, ~of) define the angular orientations of 0; and 0f in the laboratory frame, respectively. (O~Ak), ~,~Ak)) and .0(¢), t E q~E (~:)) describe the orientations of the absorption and emission transition dipole moments of the molecules in states i and ~ (see Scheme 2), respectively. Cj,.,(O, ~o) are the modified spherical harmonics [3] (Cj, m(O, q~) = [47r/(2j + 1) ]1/2 ~,,,,(0, ~o), where these latter are spherical harmonics). The formulae above have been obtained after taking into account the following relations: (a) 3 cos2(0) = 1 + 2/'2(0) where 0 is the angle between two vectors, (b) P2(0) = ~-'~. C~,n(Ol,~l)C2,n(02,~2) where (01,~pl) and ( 0 1 , ~ t ) l ) are the polar angles describing the orientations of both vectors in the same coordinate frame and (c) the components of C2,n(O2, ~2) can be transformed to another coordinate system (e.g., molecule-fixed frame) i.e. C2,,,(02,q~2) = ~--~., D ~,,, (2) *(/2) C2,m( 02,~02). ' t P2(0) is a second rank Legendre polynomial, To find the explicit form of the Green function pk--.~(/20, 0[/2, t) one can consider the time-dependence of the propagator p_.((O0, 0[/2, t), which describes the conditional probability of finding a molecule in the excited state s¢: at time t, which at t = 0 was excited to an arbitrary primary excited state, i.e. p-~(/20, 0[/2, t) = ~ , pk_.e(O0,0l/2, t) Ak,ex. Ak,ex = 1 if state k is the primary excited one and Ak,.~ = 0 otherwise. In (1) the Ak,ex function is represented by absorbance wk(Aex). The initial conditions for p--.e(O0,01/2, t) are such that p_.~(ao,01/2,o) = ~ pk-.~(Oo,0l/2,0) A~,~xBE,k, where pk__.((Oo,01/2o,0 ) = 6(/20 - / 2 ) . The state-to-state kinetic relaxation and reorientational molecular dynamics over all excited states, in Scheme 1, is described by the following system of differential equations for the propagators p-.~:(/20,0[/2, t) (s~ = 1 ..... s) at p--.l(Oo,0[/2, t) = -
~I (I) + kl +
kli p--.l(Oo, OI/2, t) i=2
Nil p--+i(Oo, O[/2, t)
,
i=2
(3) atp_.s(Oo,0[/2, t ) = -
-~--~ ki, p-.i( /2o,Ol/2, t) +
1~1~s) + k, + ~-~ ksi p ~ s ( /2o,O[O,t)
i=1
The solution to the equation of motion (3) has the form [ 1]
t=l
.
J.J. Fisz/Chemical Physics Letters 262 (1996) 507-518
--
k=l
87/-2
E
"'jmn,j'm'n' (t) O(('k)
511
D m'n' (j') (1"20) D(mJ)~*([2) Ak,ex
j/m/n /
=£pk--.e( f20,OlO, t)
(4)
ak,ex ,
k
and from its structure the explicit form of p k ~ ( g ~ , 01//, t) is immediately obvious•
2.1. The properties of the solution to the equation of motion The equation of motion (3) can be rewritten in matrix form: 3 Ot P(O°'O[ O,t) = - , ~ p(O0,O[ a , t ) .
(5)
After expanding the Green functions in the basis set of Wigner rotation matrices 2j+l
p-.¢(~.01 f2, t ) = E
8zfi C)(m~(t) D(/)*(Y2)'
(6)
jmn
the equation of motion (5) can be replaced by the set of coupled differential equations for the coefficients C)m,(t), [1] i.e. 0 at C(t) = - M C ( t ) .
(7)
The formal solution to the system of equations (7) is C(t) = R(t) C(O). R(t) describes all the relaxation processes (state-to-state kinetic relaxation and reorientational molecular dynamics) over all the excited states involved, where R(t) = V exp ( - A t) V -1. V and A = [t~qq] are the eigenvector and eigenvalue matrices of matrix M, respectively. M is the matrix representation, in the basis set of Wigner functions, of the matrix operator ~, in the equation of motion (5). The matrices M and R(t) have the following structures:
UIK2 ' 3 / iR2tR3RsI
[ Klz U (2) K32 M = ] KI3 K23 U c3) •
:
•
K~2 K~3 .
\ Kls K2s K3s
U is)
,
R(t) =
| R ( 2 a ) ( t ) R(2'2)(t) R(2'3)(t) R(3a)(t) W3'z)(t) W3'3)(t) :
•
:
\ R ( S a ) ( t ) R(S'2)(t) R(S'3)(t)
U (¢) are submatrices with elements [U (~)] P,q = H(¢) ~jmn,j'm'n' = t4(t:) "'jmn,fm'n'"~-( k( J t - E p =s l , p # ( ~,j'm','--
2j' + 1 f ~i
j
/.)(j) (~2) /t/(~) D(f) *tn~ dO mtn ~ \ a~,l -.,..
R(2'~)(t) R(3's)(t)
.
(8)
•
R(S'~)(t) k(p) ~mn,j'm'n%
where (9)
12
The submatrices Kxy are diagonal, i.e. [Kxy] jmn,j'n,'n' = - k x y ~ j m n ' j ' m ' n ' '
(lO)
where kxy is the rate for state-to-state kinetic relaxation from an excited state x to excited state y. The elements (¢,k) .n, (t) = [(V [exp (-•qq/)] V - I ) (so'k) ]jmn,j,mtn,• Index s¢ means the state from which of R (('k) (t) a r e Rjmn,j,m. the polarized emission is observed, while k means one of the primary excited states.
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J.J. Fisz/Cheraical Physics Letters 262 (1996) 507-518
When the explicit form of the the time-development operator /2/(~) is known, the expansion coefficients C(t) can be calculated from Eq. (7), and hence, the Green functions (6) can be reconstructed. The form of the reconstructed Green function is given by (4). Note that, in principle, only the elements of R(t) have to be calculated. The problem of determining the eigenvectors and eigenvalues of matrix M can be essentially simplified if the system of equations (7) can be split into independent subsystems of equations. This is equivalent to rearranging matrix M into a form of independent submatrices for which the eigenvectors and eigenvalues can be calculated separately. 2.2. S y m m e t ~ adapted form of the Green functions The propagators p ~ ( O o , 0[/'2if), t) must be projected into symmetry adapted forms due to the spherical symmetry of the system as a whole (solutions are spherically symmetric systems). The required invariance means that p__.f(O0,0112, t) must be invariant with respect to arbitrary changes in the orientation of the laboratory-fixed frame. If R(ST) represents a rotational transformation of the laboratory-fixed frame under the Euler angles ST, R(12') D(.]n) (12) = }-~m' D(j) ~j)n ~,(12) represents the transformation of the Wigner Blt nl* (ST) D nlt rotation matrices under R (/2'). The linear combinations of Wigner rotation matrices which are totally symmetric with respect to all symmetry operations in a proper rotation group of symmetry O~+)(3) can be generated by employing the projection operator i.e. Siso[ l = ~ 2
,/
R(ST)[ ]dsT,
.(2' and where ¢j') "{,120) - O(J) * (12) ] Siso [[D m'n' --ran
_ 2j - -1 + 1
~m.,'
~jj'
D u(J) n,(120)
Z
-]')(J) - qn * ( 1 2 ) "
q
By replacing each product basis set of Wigner matrices in (4) by the corresponding linear combination generated by Siso[ ], one obtains the symmetry adapted propagator p__.¢(.O0,0112, t ) = i=1
2j8,.//.2 + 1 ~(s¢,i) "'jn,jn' (t)
z
m
/7(J) (,1"20) D(J2 *(12) ~mn'
]
Aiex , '
(11)
where the elements R(('i)jn,jn'~ [ I)~ do not depend on the index m and the equality ~'~., Rjmn,jmn,(t )(('i) = (2j+" 1) R(¢'i)jn,jn,(t) has been employed. This result means that in the system of equations (7) the equations with different values of j are independent. Secondly, the equations are indistinguishable with respect to the index m. After inserting (11 ) and (2) into (1), performing the integrations over 12o and 12 and after employing the orthogonality properties for the Wigner matrices [3], the decay of the polarized fluorescence surface takes the final form I0'el (t'h'ex'~'em) -- C(s~ex'"~em) Z C°k('~ex)K((J~em)[Ph("k)(t)-t-~ sC,k
,
e2(oif)~((,k,(/)]
(12)
where 2 Ph(('k)(t) = R~', k)oo(t) '
dP(('k)(t) =
Z
2q2,2q~'J ~2,q~t~A ,~PA ) C2,q2
,
•
(13)
ql,q2=--2
Ph (('k) (t) means the polarization-free fluorescence decay. It describes state-to-state kinetic relaxation processes. The second term in (12) describes the polarization effects of the emitted fluorescence. Oif is the angle between Oi
J.J.
Fisz/Chemical Physics Letters 262 (1996) 507-518
513
and ~f. ~b(('k) (t) is the reorientational correlation function, describing the state-dependent rotational molecular motion of the system coupled with the state-to-state kinetic relaxation processes. The parallel fluorescence component Ill (t, Aex, Aem) (i.e. when ~ill~f) obtained for P2(0 °) = l, and the perpendicular one Ix(t, Aex, Aem) (i.e. when ~i±ef) for P2(90 °) = - 1 / 2 . The structure of ~b(¢,k)(t) can be simplified if the appropriate hydrodynamical shapes of the solutes are considered. Further simplifications can appear after accounting for real molecular symmetries (symmetry point groups) which restrict the possible orientations of electronic transition dipole moments. Let us consider the case of cylindrically symmetric objects. Hydrodynamical cylindrical symmetry of an object means that the Green function ( l l ) must be invariant: (a) with respect to arbitrary rotation of the molecule-fixed frame around its Z axis (i.e. C,~ (Z) symmetry operation) and (b) with respect to rotation by 7r around any axis perpendicular the Z axis of the molecule-fixed frame (i.e. C2(ZZ) symmetry operation). Both kinds of symmetry operations are performed simultaneously on the molecule-fixed frames at t = 0 and t > 0. In other words, the hydrodynamical (statistical) symmetry of a symmetric rotor is D ~ ) which contains purely rotational symmetry operations as in the symmetry point group Do, h, with the difference that for the D(S~ symmetry group these symmetry operations do not have exact meaning as in the point group Do~h. These kinds of symmetries have been considered in case of ordered systems [ 12 ]. It can be easily shown that due to the C,~ (Z) symmetry operation Dm.,(Oo) (J) DcJ) * (j,2) must be replaced in (11) by D(J)COo) m.. D(J.~) *(12) 6nn,. Secondly, due to the C 2 ( / Z ) symmetry operation, the product basis vectors D.(,J2(Do) D.(Jn) *(1"2) (for m,n 4= 0) must (j)
*
be replaced by linear combinations (1/2) [D.(/.) (/20) D(.L.) *(/2) + D(mJ).(Oo) Din_ n (12)], which are generated by the projection operator S[ ] = (1/2) [E, C2(-I-Z) ] [ ], where E is the identity symmetry operation. Finally, one obtains the following form of (1 3): 2
dp(~'k) (t) = ~
R(J'k)(t) T(pkg) ,
(14)
p=0
where
R(o£'k' = O(¢'k'' " ' 2 0 , 2 0 ~ t)
To(~g) = P2(0(Ak)) P2(0(E~, )
'
RI¢,,) = ~+ [~, \..z,.2,,., (~:,k, , t. , R~:.')
= ~, /~. ,(, 2(2. ,k2 2) ~t ,.,,1
RE_,.2_,(t)) ,(~:,k) -Jr- Rz_2.E_2(t)) (sO'k)
T(k.~) = 3 sin 2o(ak) sin 20(if) COS(~O(A k) _ q~(ff)) , ,
T2(kg)
= 3
sin20(Ak)sin20(E~:)
COS
2(~p(k)
--
~O(E ¢) )
.
(15)
According to group theory considerations the rotational dynamics of a cylindrically symmetric object is described by three correlation functions (in the case when second rank correlation functions are considered). The simplest case appears for a spherically symmetric rotor. By employing a similar projection operator like that when considering the spherical symmetry of solutions (spherical hydrodynamical symmetry of an object means that the molecule-fixed frame can be selected in an arbitrary way). one obtains 2 t~ (so'k) ( t )
=
D
( q~2.__ ~("k) P2(O(A~0 ) "'2q2,2qi tt)) *" ql,
-
=
R~spk(t)r2~VAE n r'~Ckg)) ,
(16)
2
n(¢) is the angle between absorption where Rsp h(~:'k),-Jr t~ describes the dynamic-kinetic relaxation for a sphere and ~AZ and emission dipole moments in states k and ~:, respectively. 3. Diffusion model for rotational motion
In the diffusion model for an asymmetric rotor the time-development operator/~(~) has the form
514
J.J. Fisz/ Chemical Physics Letters 262 (1996) 507-518 3
f/(¢) = Z D)¢) L/2'
(17)
i=l
where the operators Li are identical to the quantum mechanical orbital momentum operators (i = x , y , z ) a n d (() D i are the components of the diffusion tensor. For a symmetric rotor ~nroblem D (~)x = D(()r = D(¢)- --
Dl~e~rtheskr~
ifnlD~x~) = D~,~) = D~O = D(~), the problem reduces to the spherical rotor case"
"
• e e ts of the time-development operator for an anisotropic diffusor can be easily calculated by j '1 *.(/2)" One obtains taking into account explicit definitions for Lx D(J')m'#*(/2)' Lr. D(J')m'n'* (/2) and Lz D (m'n' mn,jtm'n' -~
Aj'n'
+
+ n'2 D~ ~)] ~nn,
8.,.,+2 + 8j,.,
-
)
jj,
(18)
where
Aj, n, = l [j' (j' + l ) - n '2] and
j'n' = 1
[(j ' ( j ' +,,
_
)) (j'(j' + , ) - ( n '
+
As seen from the form of (18), there is no coupling between the equations in the system (7) for which j 4: j'. Hence, the system of equations (7) can be decomposed into sets of independent equations, each for different values of j. Furthermore, all the equations are indistinguishable with respect to the index m. For a symmetric diffusor "'jmn,j'm'nt 74(¢) -~ [j (j + 1) D ~ ) + n 2 (DII~:) - D(A_() )] ~jmn,jtm'n'. Because the matrix representation of [Kxy] is also diagonal, the system of equations for a given j can be decomposed into subsystems, each for different values of n. Note that it is not important whether n takes a positive or negative value. This means that for j = 2 (the range of correlation functions in our formulae) the number of independent subsystems of differential equations reduces from 5 equations to 3 for each excited state (, and they are coupled over all the excited states involved. Another consequence of this fact is that R((,k). 2n,Zn[,t)- = "'2-n,2-n'.'J p<¢,k) r'~ in (15). For a spherical rotor D ¢O = Dx~:) = D~• ~) = D~(), and consequently "jmn,jtm'n' 14(() = J(J + I) D (() ¢~jj, 6rnm' 6nn'. In this case the matrix representation of the time-development operator does not depend on the index n. This means that for every excited state and given value of j, there exist only one independent equation, which is coupled with the corresponding (the same) equations for the remaining excited states.
4. State-dependent choice for principal axes of the state-dependent diffusion tensors In the case when the molecules in different excited states have different principal axes for the diffusion tensors (they are diagonal in these coordinate systems), the Euler angles corresponding, to different excited states are distinguished by proper indices. Therefore, in the formula for polarized fluorescence decay (1) the Euler angles /20 must be replaced by D0(k) and/2 by /2(~). As shown in Ref. [ 1 ] the blocks U (~:) are the same as in the usual case, while the elements of submatrices Kxy become in this case Kxy]
jmn,jr nVn~
FI(J')[AO(Y-'-*x)~ = - k x v ~n'n i~ ! ~jj' ~mm' .
(19)
J.J.
515
F i s z / Chemical Physics Letters 262 (1996) 5 0 7 - 5 1 8
where Af2(y-'x) are the Euler angles transforming the principal axes from state y to state x and it is assumed that A/2 (-~'-~x) A/2cx~y) -1 Hence, F I ( f ) f A ( ) ( Y ' - - ' x ) ' I = D (j') *(Al-I(x~Y)). It is important to note here that the ~k) ) and ( 0 e( 0 , ,,~(() angles ( o~k) ~ , ~o~ "re ) are defined in different molecule-fixed frames for k v~ sc, if the state-to-state kinetic relaxation is followed by geometrical transformation of the solutes. This has to be taken into account when considering relation (13). Note also that the blocks Kxy couple the equations in (7) for different values of n. We assume here that the change in the molecular geometry is not an extra source for the molecular reorientation or that this effect can be neglected to a first approximation.
5. Time-dependent and steady-state emission anisotropy surface The expression for the emission anisotropy decay surface can be easily obtained from the polarized fluorescence decays defined by (12), i.e. r(t, Aex,Aem)
=
I[l(,~ex,,~ern,t) -I_l_(,~ex, Aem, t) IIl(Aex, Aem,t) + 2I±(Aex, Aem,t)
=
0.4)-]~(,k tOk(~ex) K((Aem) (b(k'()(t) ~--]~(,k tOk(Aex) K(('~em) PhC~:'k)(t)
(20)
The steady-state emission anisotropy surface (r(a~b, ~.em)) can be obtained from similar calculations performed on the steady-state (continuous excitation) intensities of the polarized fluorescence i.e.
(I~, ~f(~ex, Aem))= C(Aex,~em) E
t°k(Aex) r((Aem)
[(Ph (('k)) + 34
P2(Oif)
(t~(sC'k))]j .
(21)
g,k
The symbol (...) means the integral f o "'" dt, where (Ph(g'k)) = /°(~'k)\ \"00,00/
'
(dP(('k)) =
2 Z /R(~'k) \ " ,,,(k) (k), p , ra(() ..(~) \ 2q2,2qt I t~2,ql~,ffA ,~0A ) W2,q2tVE ,WE ) , ql,q2=-2
(22)
and (~,(~,k) \
_
(23) J jn,jn'
-- L
a jn,jn'
Finally, one obtains the formula for the steady-state excitation- and emission-wavelength dependent emission anisotropy surface (lll(Aex, Aem)> - (I±(Aex,'~em)) E¢.k t°k(Aex) K~:(Aem) (q~(k,~:)) (r(Aex,Aem)) = (/'~A--~'x,Ae-e-~ ~- 2("~-l(-'~'ex,A-~m)} =0.4 . E~C,k tOk(~ex) K¢(aem) (Ph (~:'t))
(24)
6. Rigid isotropic systems By neglecting the angular dependence of the Green function in (1), one obtains immediately the formula for the polarized fluorescence decay surface for rigid molecular systems. The kinetic relaxation matrix for the propagators p__.¢(Olt) (~ = 1 ..... s) can be easily obtained from (3) by neglecting the state-dependent operators f/(¢), i.e. at P ~ l ( ° l t )
=-
kl +
kli i=2
p~l(Olt) -
kil p--i(OIt) i=2
,
516
J.J. Fiszl Chemical Physics Letters 262 (1996) 507-518
(25) s-I a p~s(0lt)
Ot
-
-
ki~ p--.i(Olt) i=l
The final result for the polarized fluorescence intensity decay surface can be obtained without any additional . . =_ ~,((,k) t,~ is obvious in this case, and where the elements ..oo,oo(t) ~,((,k).. calculations because relation R(g,k) 2q2,2ql(t) --oo,oo~-J are the same as those appearing in the polarization-free term Ph(('k)(t) already derived. Consequently, one obtains
l~,~f(t, aex, aem) = C(a~x,aem) ~
wk(aex) K¢(a~m)R(~'k)tt oo,oo, ~,
[1 _.]._4 g Pz(Oif) p2(0(~())] .
(26)
(,k The expressions for the time-dependent and steady-state emission anisotropy are
r(t,a.,aom) =O4 Ze, k o,k(aex) ,,e(aom) R i (t) e2(0k e)) Z ( , k °')k(aex) K((~.em)
R((,k) t t~ 00,00~, ,
(27)
and
(r(aex, ~-em)) = 0.4 ~"~(,k O~k(Aex) K((,Aem) \"00,00/ /l~(('k)\ P2 --AE (a(k'() ) (R(~,k)~ O)k(Aex) K((~.em) 00,001
~-~'~:,k
(28)
"AEtl(k'~)
where is the angle between the absorption dipole of a molecule in the ground state state k and the emission dipole moment of the same molecule in excited state ~r.
7. Discussion
The formalism discussed in this work is based on a number of simplifying assumptions mentioned in Section 1. The validity of these assumptions must always be verified for every particular experimental case under consideration. In its present form, this approach cannot be utilized in such experimental cases in which the timescale for the solvent rearrangement around the fluorophores (rotational and translational motion of the solvent molecules) after electronic excitation or after relaxation of the fluorophores to other excited states, is comparable with the timescale for state-to-state kinetic relaxation processes and rotational motion of the fluorophores. This problem relates to all processes controlled by the solvent organization and its changes around the fluorophores. Therefore, one can expect that in ultrafast laser spectroscopy (femto/picosecond timescale) the applicability of the formalism outlined in this work will be extremely restricted, because the timescale of such experiments is so short that the problem cannot be described in terms of time-independent kinetic rate constants and time-independent diffusion tensors. The solvent relaxation process and the spectrum relaxation have to be accounted for in the model. In picosecond spectroscopy one can deal with experimental cases in which the approach discussed here cannot be applied for the same reasons as" in the case of ultrafast spectroscopy. On the other hand, however, if the timescale for the solvent relaxation (and spectrum relaxation) is short compared to the timescale of the kinetics and rotational dynamics of the fluorophores, one can expect that the model described in this work can be applied in the experimental data analysis. Nevertheless, this should always be verified for any individual experimental case, e.g. by precise preliminary kinetic studies. From the kinetic studies one can conclude whether the kinetics of the system can be described by the discrete values of the rate constants. If this is the case, one can expect
J.J. Fisz/ Chemical Physics Letters 262 (1996) 507-518
517
that also the state-dependent rotational dynamics of the fluorophores can be described by discrete values of the rate constants (the components of the diffusion tensors). In nanosecond spectroscopy the timescale for solvent relaxation can be shorter by many orders as compared to the timescale for the state-to-state kinetics and rotational dynamics of fluorophores, both ranging in the nanosecond timescale. In such cases the approach described in this work can be applied, provided that no other complications appear and that the kinetics of the system is described by the discrete values of the rate constants. In some experimental cases the kinetics of the system must be described in terms of the distributed rate constants, although the system (as a whole) is in an equilibrium state, in a given excited state. Such situations can occur for systems in which the fluorescence signal detected is contributed to by signals from fluorophores having different configurations of the solvent molecules and the rates for the kinetic relaxation are strongly solvent-configuration dependent. In such cases, expressions describing the fluorescence signals can be obtained by integrating the expressions derived in this work over the distributed rate constants (assuming particular shapes for the distributions of the corresponding rate constants). The same can be considered for the components of the state-dependent diffusion tensors, because rotational motion of the fluorophores is also a solvent-configuration dependent process. The description of the many-level problem in terms of distributions for the kinetic rate constants and state-dependent diffusion tensors can be important for the analysis of excitation energy transfer processes for double-chromophore (or many-chromophore) complexes. The formalism discussed here can be employed in studies of rigid complexes (i.e. a single donor-acceptor configuration). In the case of a mixture of different donor-acceptor configurations, the relations describing polarized fluorescence decays should be replaced by the weighted sum over the decays corresponding to different donor-acceptor configurations. In a limiting case, when the distribution of the possible donor-acceptor configurations has a continuous character, the weighted sum of the decays has to be replaced by an integral over the distributed rate constants for the excitation energy transfer process. This description can be utilized in studies of conformational structures of the macromolecules (e.g. proteins) labeled by donor and acceptor systems. Assuming that donors and acceptors are attached rigidly in the macromolecules, the shape of the distribution for the energy transfer rate constants will reflect the distribution of the possible macromolecular conformers, in which the distance and angular orientation between the donors and the acceptors are different. The same problem can also be parameterized in terms of the multimodal distributions of the possible donor-acceptor configurations, where two or more possible donoracceptor configurational distributions centered at different most probable donor-acceptor configurations have to be assumed in the model. The applicability of our approach to the energy transfer problem is limited to the case of incoherent excitation of the system, i.e. the excitation is localized on one of the chromophores in a double- or many-chromophore complex. Also the same limitations as those discussed above and concerning the timescale for the solvent relaxation (for polar systems in polar solvents) have to be taken into account when analyzing real experimental cases.
Acknowledgement I would like to thank one of the referees for valuable discussions. This work was supported by the Polish Committee for Scientific Research (KBN), within the Project 2.P303.089.04.
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