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Magnetic field effects on chemical reaction yields arising from avoided level-crossings in molecular triplet states
5 May 1995
CHEMICAL PHYSICS LETTERS Chemical Physics Letters 237 (1995) 183-188
ELSEVIER
Magnetic field effects on chemical reaction yields arising from avoided level-crossings in molecular triplet states J.K. Hicks, P.J. H o r e * Physical and Theoretical Chemistry Laboratory, Oxford Unicersity, South Parks Road, Oxford OX1 3QZ, UK
Received 28 February 1995
Abstract
The unexpectedly large magnetic field effect found for continuously illuminated reaction centres of the photosynthetic bacterium Rhodobacter sphaeroides at cryogenic temperatures is discussed in terms of the magnetic field-induced mixing of sub-levels mechanism. The pronounced maximum in the steady state concentration of the triplet state of the primary donor at a field strength close to 20 mT is shown to arise from the avoided level-crossing of the triplet spin states when the magnetic field is almost along the Z axis of the zero-field splitting of the triplet. The dramatic change in the spin eigenstates as the field is varied through the avoided crossing leads to changes in triplet-to-ground-state intersystem crossing rates, and hence the triplet concentration. The predicted 350% change in triplet concentration is similar to that observed experimentally.
1. Introduction
Probably the most common and certainly the best understood way in which magnetic fields can alter the yields of chemical reactions is the so-called radical pair mechanism (RPM) [1-4]. Its origin lies in the coherent mixing, by electron-nuclear hyperfine interactions (hfi), of electronic singlet and triplet states of a radical pair intermediate, and the inhibition of this mixing by the Zeeman interactions of the two unpaired electrons. In fields that are weak compared to the hfi, the three triplet sub-levels are degenerate and are mixed equally with the singlet state S; in fields much stronger than the hfi, by contrast, the T± 1 states are well removed in energy, and only T O mixes with S. If radical pairs are able to
react from both singlet and triplet states, this magnetic field effect (MFE) on the efficiency of radical pair singlet-triplet interconversion in the radical pair translates into changes in the relative yields of singlet and triplet reaction products. A magnetic field thus inhibits the formation of triplet product from a radical pair initially formed in a singlet state, by reducing from three to one the number of 'pathways' available for production of the triplet radical pair. A second type of magnetic field effect is possible for reactions involving a molecular triplet, provided it is selectively depopulated by anisotropic intersystem crossing (ISC) [5-7]. The effect exists because each of the three spin eigenstates I i) (i = 1, 2, 3) is a field-dependent linear combination of the zero field states l X ) , I Y), I Z ) [8]. Writing
]i)= ~ C q ( B o ) l q ) * C o r r e s p o n d i n g author.
0009-2614/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSDI 0009-2614(95)00289-8
q
(q=X,Y,Z),
(1)
184
J.K. Hicks, P.J. Hore / Chemical Physics Letters 237 (1995) 183-188
where the field-dependent coefficients Ciq(Bo) are obtained by diagonalizing the triplet spin Hamiltonian, the ISC rate constants at non-zero field are given by [5,6]
ki(Bo)=~_~lciq(no)12kq
(q=S,Y,Z).
P*----.~ lr +1-1 3r ÷ -1 , [P / ~ [ P I l
(2)
kI
q
For example, in a magnetic field that is strong compared to the triplet zero field splitting (ZFS), and parallel to its Z axis, the eigenstates and corresponding ISC rates are [8] [1) = 2 - 1 / 2 { I X ) + i] Y ) } ,
k~ =½(k. +kv),
12) = 2 - 1 / 2 { I X ) - i l Y)},
k2 =
13) = - I z ) ,
k 3 = k z.
l(k
+
.
13)
Fig. 2. Steps involved in charge separation, and triplet formation and decay, in bacterial reaction centres in which further electron transport is blocked.
kv),
(3) At zero field, by contrast, the rates are just k x, kv, and k z. As originally recognized by Lous and Hoff [5,6], such variation in the ISC rates leads directly to a magnetic field-dependent concentration of triplets. This mechanism has been dubbed MIMS (magnetic field-induced mixing of sublevels) [7]. An experimental magnetic field effect that appears to be largely dominated by MIMS is shown in
Fig. 1. This linear dichroic (LD) field effect on the optical absorption of quinone-reduced reaction centres of the photosynthetic bacterium Rhodobacter (Rb.) sphaeroides at 25 K, was measured by van der Vos et al. [9]. The appearance of features at magnetic field strengths (-~ 20 mT) much larger than the magnetic interactions in the primary donor-acceptor radical pair but comparable to those in the donor triplet suggests that MIMS, as well as the RPM, is at work, as concluded by van der Vos et al. [9]. Our purpose here is to elaborate the general ideas put forward in Ref. [9], to clarify the source of the MIMS effect, and to explain more fully the general shape of the signal in Fig. 1. Our approach is based on the original theoretical description of the experiment by Lous and Hoff [5,6].
2. Theory
LD-MFE
d(LD-MFE)/dBo i
0
i
i
l0
,
2O
i
__~
30
40
magnetic field B0/mT Fig. 1. The derivative of the linear dichroic magnetic field effect at 25 K for reaction centres of Rb. sphaeroides (solid line) and its numerically integrated form (dashed line). These data were kindly supplied, prior to publication, by van der Vos and Hoff [9].
Our discussion of the MIMS effect centres on the initial steps of the electron transport chain in photosynthetic bacteria (Fig. 2). P is the primary electron donor (a bacteriochlorophyll dimer) and I the first electron acceptor (a bacteriopheophytin). 3 p and P * are respectively triplet and excited singlet states of P. 1[P+I-] and 3[p+I-] are singlet and triplet radical pairs. Further electron transport is prevented by removal or, as in the case of Fig. 1, pre-reduction of the secondary electron acceptor, a quinone. As indicated in Fig. 2, the three spin states 1i) (i = 1, 2, 3) of 3 p (henceforth referred to as T) are populated f r o m 3[p+ I - ] at r a t e s g i , and depopulated by intersystem crossing with first order rate con-
J.K. Hicks, P.J. Hore / Chemical Physics Letters 237 (1995) 183-188
stants k i. Under steady state conditions, ignoring spin-lattice relaxation, one has K i = ki[Ti], where [T~] is the population of 1i). The total amount of T is therefore 3
Ki(Bo) k~(Bo),
3
[T] = Y'~ [T~] = ~ i=1
(4)
i=1
in which the ISC rate constants k~(Bo) are given by Eq. (2). This expression not only shows that anisotropic ISC is a fundamental requirement for the MIMS effect, for otherwise the three rates constants k~ are identical and field-independent (see Eq. (2)), but also includes the RPM contribution to IT] via the field-dependence of the rates K v Note that differential population of the triplet eigenstates by the RPM is not a prerequisite for a MIMS magnetic field effect. Ki(B o) can be found using
g~(no)
= kT[RP] I(i]
~bRP>I 2,
(5)
in which k T is the first order rate constant for formation of T from the triplet radical pair; [RP] is the steady state radical pair concentration; and (il ~RP> is the projection of I ~bRP>, the electron spin state of the radical pair, onto I i). Writing ]qt RP) =
~_.bp(Bo) lp>
(p=S,
T+I, T 0,
T_l),
P
(6)
radical pair density operator, and the sums run over p, p ' = T + l , To, T_ 1 (but not S because all ( S I q ) = 0), and q, q ' = X, Y, Z. A rigorous calculation of pRP would involve the solution of the stochastic Liouville equation for the radical pair including Zeeman, hyperfine, exchange and dipolar interactions, formation of the pair in its singlet state, and reaction from its singlet and triplet states. However, since we wish to focus on the magnetic field effect originating in the triplet, we avoid this complexity, but retain the qualitative features of the RPM-MFE, by introducing a few simple approximations. At zero field, where all three triplet radical pair states are mixed with IS) by the hyperfine interaction, we take (Trl pRP I T r ) = 7 1 ( r ~ + 1, 0, - 1 ) . At magnetic fields much larger than the exchange, dipolar and hyperfine interactions in the radical pair, but not necessarily large compared to the zero field splitting in the triplet, we suppose that the only non-zero element of pRP is ( T 0 [ p R P I T o ) = I , the IT+a) and I T l ) radical pair states being too far removed in energy from IS> at such field strengths to be significantly populated. Hence approximate expressions for the three populating rates, at zero field K izF, and at high field K i"F , are K zF=lkT[RP ]
(i=1,2,3),
(9)
KHF(Bo)=kT[RP]Y'~ Y'~[cq(Bo)]*[Cq,(Bo) ] q
it follows that
(il I~RP) =
185
q'
x ( q I T 0 ) ( T 0 I q')
~bp(Bo)(il p)
( i = 1, 2, 3). (10)
P
= ~ ~bp(Bo)(ilq)(qlP) P
The projections of the radical pair ITo) state onto the triplet zero field states are
q
= ~_~Ebp(Bo)[Ciq(Bo)]'(q]P>, P
( X ITo) = sin 0 cos ~b,
q
(7)
(YpT o ) = s i n 0 sin ~b, ( Z I T 0 ) = cos 0,
and hence
in which the angles 0 and ~b specify the direction of the magnetic field with respect to the (X, Y, Z) axes of the triplet. The variation of K i with the field B 0 between these limits is taken to be
Ki=kT[RP]Y'~ Y'~ E Y'.ppRP'(Bo)[Ciq(Bo)]* p
p'
q
(11)
q'
X [Cq,(Bo) ] ( q l p > ( p ' l q ' > ,
(s) in w h i c h ,mp,(B v- R-e o) --bpb;,,* is a matrix element of the
KZFA2+ KyFB2 Ki(Bo) =
A2 +Bo2
(i = 1, 2, 3),
(12)
186
J.K. Hicks, P.J. Hore / Chemical Physics Letters 237 (1995) 183-188
where A is of the order of the root-mean-square hyperfine interaction in the radical pair. Thus the mid-point (Ki = "/L=-iirk'ZF+ K/nF]) of the RPM field effect occurs at B 0 = A. Note that KiNF is not independent of B 0 because of the field dependence of the C~q(Bo) coefficients (the HF superscript refers to the radical pair interactions, not the triplet). We stress that the exact form of Eqs. (9), (10) and (12) is not crucial: it is sufficient that they reproduce qualitatively the RPM-MFE. The dependence of K i ( B o) on B 0 is suggested by the Rabi formula for a two-level system subject to a time-dependent perturbation [10]: on average, the fraction of radical pairs present as triplets is proportional to A 2 / ( A 2 + B2). Finally, the linear dichroic magnetic field effect, detected as the difference in absorption of light polarized parallel (par) and perpendicular (per) to the magnetic field, for a randomly oriented sample of reaction centres [5,6], is calculated as LD-MFE( B 0) ot
fo2~fo~[T](B0)((/z -(~.v~
Vpar) 2
r
(a)
2-
.3
[-
O
/
e~
,/ 2
iI
8 "6 0
E !
sin 0 d 0 d 4 ) ,
.......
(c)
/i,
I / -Z
•=
I
(13) where [T] is given by Eqs. (2), (4), and (9)-(12). In Eq. (13), ~ is the optical transition dipole moment, and Vp~r and Vp~ are the electric vectors of the polarized probe light.
(b)
\ \\
I
, /j/ ii
k
\
/
I
\, /
.q 0
0
10
20
30
magnetic field B 0 / mT
3. Results and discussion Fig. 3a shows the LD-MFE calculated according to Eq. (13). The values used for k x , k v, k z and the zero field splitting parameters, D(3p) and E(3p), are those appropriate for 3 p in Rb. sphaeroides [11,12] (see caption). The optical transition moment /t is taken to be3Parallel to the Y axis of the zero field splitting in P. (In practice [5,6], the triplet concentration is monitored via the optical absorption of ground state P whose transition dipole moment is approximately in this direction [13,14]; the very low concentrations of P * and P +I- mean that [P] + [ 3p] is constant.) Despite the crudity of the approximations used to describe the RPM-MFE, the calculated LD signal bears a striking resemblance to the experi-
Fig. 3. (a) Calculated linear dichroic magnetic field effect (dashed line) and its derivative (solid line). Parameter values: k x = 9.0 × 103 s - l ; k r = 8 . 0 × 1 0 3 s-a; k z = l . 4 × 1 0 3 s -1 [11,12]; ] D ( 3 p ) I = 2 0 . 0 mT; I E ( a P ) i = 3 . 5 mT [11]; A = 2 . 0 roT. (b) Magnetic field effect on the calculated triplet concentrations when the field is parallel to the X (short dashed), Y (solid), or Z (long dashed) axis of the triplet zero field splitting. (c) As (b) except that the field has been rotated away from each of the zero field axes by 5°. ~ X, ~ Y, ~ Z correspond, respectively, to (0, 4)) = (90 °, 5°), (90 °, 85°), (5 °, 0°).
mental spectrum, Fig. 1. As the magnetic field is increased, the signal initially drops at a rate determined by the value of the radical pair hyperfine parameter A, before rising to a prominent maximum (at = 20 mT) whose position is sensitive only to the value of D(3p). Also noticeable is a very shallow
J.K. Hicks, P.J. Hore / Chemical Physics Letters 237 (1995) 183-188
I1)
187
off-diagonal elements of the spin Hamiltonian, giving new eigenstates and rate constants (approximately): [2')=l{[x)-ily)} 1
1
+ 2-1/2 [ Z ) , 1
k 2, = -~kx + g k y + $kz,
Iv) Ix)
[3') = ½{ Ix) - i] Y)} - 2-1/2 [ z ) , ky
0,
Iz)
........ il3) 0
10 20 magnetic field B0/mT
30
Fig. 4. Energy levels of 3p when the magnetic field is parallel to
the Z axis (solid lines) and rotated 5° away from the Z axis (dashed lines). Drawn for D( 3P)= + 20.0 roT, E(3p)= + 3.5 roT.
depression in the positive part of the derivative LD signal around 13 mT. A more pronounced dip is found in the experimental signal (Fig. 1) at similar field strengths. The zero crossings of the calculated derivative LD signal correspond reasonably well to those in the experimental trace. At first sight, the maximum at B o = D(3P) would appear to be related to the crossing of two of the three triplet sub-levels, which occurs at just this field strength when the magnetic field is oriented along the Z axis of the zero field splitting (Fig. 4). However, the two eigenstates involved in this level crossing are to a good approximation (Zeeman interaction >> IE(3p)I, see e.g. Ref. [8]) independent of B 0, and given by [2) and 13) in Eq. (3). From Eq. (2), therefore, the ISC rate constants k i should also be field-independent around 20 mT, and no special MFE would be expected. This conclusion is corroborated by the calculated field-dependence of [T] (Fig. 3b). The key to understanding the maximum LDMFE at B 0 --~D(3p) is the fact that the level crossing becomes an avoided crossing as soon as the field direction deviates even slightly from the Z axis (Fig. 4) [15]. For a 20 mT field, almost parallel to the Z axis, the states 12) and 13) in Eq. (3) are mixed by
=
1 x + ~k 1 r + ~k 1 z. ~k
(14)
As one moves away from the avoided crossing, to fields lower or higher than 20 mT, the eigenstates and rate constants revert to those in Eq. (3). This analysis is supported by Fig. 3c which shows calculated triplet concentrations for field directions almost parallel to X, Y and Z. Now there is a very pronounced minimum in [T] around 20 mT when the field direction is close to the Z axis. Also visible in Fig. 3c is a small hump around 13 mT when the field is nearly parallel to the Y axis of the zero field splitting. This arises from the only other avoided crossing of the triplet sub-levels, and is smaller than the first because of the relatively minor deviation from axial symmetry of both the zero field splitting and the intersystem crossing rates. It is this field effect on [T] that gives rise to the slight depression at 13 mT in Fig. 3a, referred to above. This argument can easily be extended to obtain an estimate of the size of the magnetic field effect at B 0 = D(3p). Using the ISC rates and triplet eigenstates in Eqs. (3) and (14), and the populating rates in Eq. (10), the triplet concentrations for a 20 mT field exactly parallel (par) and nearly parallel ( = par) to the Z axis of 3p are in the approximate ratio
[T]par
z
[T]=par z
k x + kv + 2kz 4kz
(15)
which using the ISC rate constants appropriate for Rb. sphaeroides (Fig. 3 caption), is approximately 3.5. Even larger effects can be anticipated for triplets with a larger anisotropy in their intersystem crossing rates.
Acknowledgement This work was made possible by a Twinning Grant from the European Commission. PJH thanks
188
J.K. Hicks, P,J. Hore / Chemical Physics Letters 237 (1995) 183-188
Stephen Sexton, Robert van der Vos and Professor Arnold Hoff for advice and enlightening discussions on many occasions, and is pleased to acknowledge an informative conversation with Professor J.H. van der Waals.
References [1] H.-J. Werner, K. Schulten and A. Weller, Biochim. Biophys. Acta 502 (1978) 255. [2] R. Haberkorn and M.E. Michel-Beyerle, Biophys. J. 26 (1979) 489. [3] A.J. Hoff, Quart. Rev. Biophys. 14 (1981) 599. [4] U.E. Steiner and T. Ulrich, Chem. Rev. 89 (1989) 51. [5] E.J. Lous, Doctoral Thesis, Leiden University (1988). [6] E.J. Lous and A.J. Hoff, in: The photosynthetic bacterial reaction centre - structure and dynamics, eds. J. Breton and A. Vermeglio (Plenum Press, New York, 1988).
[7] A.J. Hoff, P. Gast, R. van der Vos, J. Vrieze, E.M. Franken and E.J. Lous, Proceedings of the 2nd Conference on Magnetic field and spin effects in chemistry and related phenomena, Konstanz, Germany, 1992. [8] A. Carrington and A.D. McLachlan, Introduction to magnetic resonance (Harper, New York, 1967) pp. 118-121. [9] R. van der Vos, E.M. Franken, S.J. Sexton, S. Shochat, P. Gast, P.J. Hore and A.J. Hoff, Biochim. Biophys. Acta, in press. [10] P.W. Atkins, Molecular quantum mechanics (Oxford Univ. Press, Oxford, 1983) p. 190. [11] H.J. den Blanken, A.D.J.M. Jongenelis and A.J. Hoff, Biochim. Biophys. Acta 725 (1983) 472. [12] A.J. Hoff, Biochim. Biophys. Acta 440 (1976) 765. [13] E.J. Lous and A.J. Hoff, Proc. Natl. Acad. Sci. US 84 (1987) 6147. [14] S. Shochat, T. Arlt, C. Francke, P. Gast, P.I. van Noort, S.C.M. Otte, J.P.M. Schelvis, S. Schmidt, E. Vijgenboom, J. Vrieze, W. Zinth and A.J. Hoff, Photosynth. Res. 40 (1994) 55. [15] W.S. Veeman and J.H. van der Waals, Chem. Phys. Letters 7 (1970) 65.
CHEMICAL PHYSICS LETTERS Chemical Physics Letters 237 (1995) 183-188
ELSEVIER
Magnetic field effects on chemical reaction yields arising from avoided level-crossings in molecular triplet states J.K. Hicks, P.J. H o r e * Physical and Theoretical Chemistry Laboratory, Oxford Unicersity, South Parks Road, Oxford OX1 3QZ, UK
Received 28 February 1995
Abstract
The unexpectedly large magnetic field effect found for continuously illuminated reaction centres of the photosynthetic bacterium Rhodobacter sphaeroides at cryogenic temperatures is discussed in terms of the magnetic field-induced mixing of sub-levels mechanism. The pronounced maximum in the steady state concentration of the triplet state of the primary donor at a field strength close to 20 mT is shown to arise from the avoided level-crossing of the triplet spin states when the magnetic field is almost along the Z axis of the zero-field splitting of the triplet. The dramatic change in the spin eigenstates as the field is varied through the avoided crossing leads to changes in triplet-to-ground-state intersystem crossing rates, and hence the triplet concentration. The predicted 350% change in triplet concentration is similar to that observed experimentally.
1. Introduction
Probably the most common and certainly the best understood way in which magnetic fields can alter the yields of chemical reactions is the so-called radical pair mechanism (RPM) [1-4]. Its origin lies in the coherent mixing, by electron-nuclear hyperfine interactions (hfi), of electronic singlet and triplet states of a radical pair intermediate, and the inhibition of this mixing by the Zeeman interactions of the two unpaired electrons. In fields that are weak compared to the hfi, the three triplet sub-levels are degenerate and are mixed equally with the singlet state S; in fields much stronger than the hfi, by contrast, the T± 1 states are well removed in energy, and only T O mixes with S. If radical pairs are able to
react from both singlet and triplet states, this magnetic field effect (MFE) on the efficiency of radical pair singlet-triplet interconversion in the radical pair translates into changes in the relative yields of singlet and triplet reaction products. A magnetic field thus inhibits the formation of triplet product from a radical pair initially formed in a singlet state, by reducing from three to one the number of 'pathways' available for production of the triplet radical pair. A second type of magnetic field effect is possible for reactions involving a molecular triplet, provided it is selectively depopulated by anisotropic intersystem crossing (ISC) [5-7]. The effect exists because each of the three spin eigenstates I i) (i = 1, 2, 3) is a field-dependent linear combination of the zero field states l X ) , I Y), I Z ) [8]. Writing
]i)= ~ C q ( B o ) l q ) * C o r r e s p o n d i n g author.
0009-2614/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSDI 0009-2614(95)00289-8
q
(q=X,Y,Z),
(1)
184
J.K. Hicks, P.J. Hore / Chemical Physics Letters 237 (1995) 183-188
where the field-dependent coefficients Ciq(Bo) are obtained by diagonalizing the triplet spin Hamiltonian, the ISC rate constants at non-zero field are given by [5,6]
ki(Bo)=~_~lciq(no)12kq
(q=S,Y,Z).
P*----.~ lr +1-1 3r ÷ -1 , [P / ~ [ P I l
(2)
kI
q
For example, in a magnetic field that is strong compared to the triplet zero field splitting (ZFS), and parallel to its Z axis, the eigenstates and corresponding ISC rates are [8] [1) = 2 - 1 / 2 { I X ) + i] Y ) } ,
k~ =½(k. +kv),
12) = 2 - 1 / 2 { I X ) - i l Y)},
k2 =
13) = - I z ) ,
k 3 = k z.
l(k
+
.
13)
Fig. 2. Steps involved in charge separation, and triplet formation and decay, in bacterial reaction centres in which further electron transport is blocked.
kv),
(3) At zero field, by contrast, the rates are just k x, kv, and k z. As originally recognized by Lous and Hoff [5,6], such variation in the ISC rates leads directly to a magnetic field-dependent concentration of triplets. This mechanism has been dubbed MIMS (magnetic field-induced mixing of sublevels) [7]. An experimental magnetic field effect that appears to be largely dominated by MIMS is shown in
Fig. 1. This linear dichroic (LD) field effect on the optical absorption of quinone-reduced reaction centres of the photosynthetic bacterium Rhodobacter (Rb.) sphaeroides at 25 K, was measured by van der Vos et al. [9]. The appearance of features at magnetic field strengths (-~ 20 mT) much larger than the magnetic interactions in the primary donor-acceptor radical pair but comparable to those in the donor triplet suggests that MIMS, as well as the RPM, is at work, as concluded by van der Vos et al. [9]. Our purpose here is to elaborate the general ideas put forward in Ref. [9], to clarify the source of the MIMS effect, and to explain more fully the general shape of the signal in Fig. 1. Our approach is based on the original theoretical description of the experiment by Lous and Hoff [5,6].
2. Theory
LD-MFE
d(LD-MFE)/dBo i
0
i
i
l0
,
2O
i
__~
30
40
magnetic field B0/mT Fig. 1. The derivative of the linear dichroic magnetic field effect at 25 K for reaction centres of Rb. sphaeroides (solid line) and its numerically integrated form (dashed line). These data were kindly supplied, prior to publication, by van der Vos and Hoff [9].
Our discussion of the MIMS effect centres on the initial steps of the electron transport chain in photosynthetic bacteria (Fig. 2). P is the primary electron donor (a bacteriochlorophyll dimer) and I the first electron acceptor (a bacteriopheophytin). 3 p and P * are respectively triplet and excited singlet states of P. 1[P+I-] and 3[p+I-] are singlet and triplet radical pairs. Further electron transport is prevented by removal or, as in the case of Fig. 1, pre-reduction of the secondary electron acceptor, a quinone. As indicated in Fig. 2, the three spin states 1i) (i = 1, 2, 3) of 3 p (henceforth referred to as T) are populated f r o m 3[p+ I - ] at r a t e s g i , and depopulated by intersystem crossing with first order rate con-
J.K. Hicks, P.J. Hore / Chemical Physics Letters 237 (1995) 183-188
stants k i. Under steady state conditions, ignoring spin-lattice relaxation, one has K i = ki[Ti], where [T~] is the population of 1i). The total amount of T is therefore 3
Ki(Bo) k~(Bo),
3
[T] = Y'~ [T~] = ~ i=1
(4)
i=1
in which the ISC rate constants k~(Bo) are given by Eq. (2). This expression not only shows that anisotropic ISC is a fundamental requirement for the MIMS effect, for otherwise the three rates constants k~ are identical and field-independent (see Eq. (2)), but also includes the RPM contribution to IT] via the field-dependence of the rates K v Note that differential population of the triplet eigenstates by the RPM is not a prerequisite for a MIMS magnetic field effect. Ki(B o) can be found using
g~(no)
= kT[RP] I(i]
~bRP>I 2,
(5)
in which k T is the first order rate constant for formation of T from the triplet radical pair; [RP] is the steady state radical pair concentration; and (il ~RP> is the projection of I ~bRP>, the electron spin state of the radical pair, onto I i). Writing ]qt RP) =
~_.bp(Bo) lp>
(p=S,
T+I, T 0,
T_l),
P
(6)
radical pair density operator, and the sums run over p, p ' = T + l , To, T_ 1 (but not S because all ( S I q ) = 0), and q, q ' = X, Y, Z. A rigorous calculation of pRP would involve the solution of the stochastic Liouville equation for the radical pair including Zeeman, hyperfine, exchange and dipolar interactions, formation of the pair in its singlet state, and reaction from its singlet and triplet states. However, since we wish to focus on the magnetic field effect originating in the triplet, we avoid this complexity, but retain the qualitative features of the RPM-MFE, by introducing a few simple approximations. At zero field, where all three triplet radical pair states are mixed with IS) by the hyperfine interaction, we take (Trl pRP I T r ) = 7 1 ( r ~ + 1, 0, - 1 ) . At magnetic fields much larger than the exchange, dipolar and hyperfine interactions in the radical pair, but not necessarily large compared to the zero field splitting in the triplet, we suppose that the only non-zero element of pRP is ( T 0 [ p R P I T o ) = I , the IT+a) and I T l ) radical pair states being too far removed in energy from IS> at such field strengths to be significantly populated. Hence approximate expressions for the three populating rates, at zero field K izF, and at high field K i"F , are K zF=lkT[RP ]
(i=1,2,3),
(9)
KHF(Bo)=kT[RP]Y'~ Y'~[cq(Bo)]*[Cq,(Bo) ] q
it follows that
(il I~RP) =
185
q'
x ( q I T 0 ) ( T 0 I q')
~bp(Bo)(il p)
( i = 1, 2, 3). (10)
P
= ~ ~bp(Bo)(ilq)(qlP) P
The projections of the radical pair ITo) state onto the triplet zero field states are
q
= ~_~Ebp(Bo)[Ciq(Bo)]'(q]P>, P
( X ITo) = sin 0 cos ~b,
q
(7)
(YpT o ) = s i n 0 sin ~b, ( Z I T 0 ) = cos 0,
and hence
in which the angles 0 and ~b specify the direction of the magnetic field with respect to the (X, Y, Z) axes of the triplet. The variation of K i with the field B 0 between these limits is taken to be
Ki=kT[RP]Y'~ Y'~ E Y'.ppRP'(Bo)[Ciq(Bo)]* p
p'
q
(11)
q'
X [Cq,(Bo) ] ( q l p > ( p ' l q ' > ,
(s) in w h i c h ,mp,(B v- R-e o) --bpb;,,* is a matrix element of the
KZFA2+ KyFB2 Ki(Bo) =
A2 +Bo2
(i = 1, 2, 3),
(12)
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J.K. Hicks, P.J. Hore / Chemical Physics Letters 237 (1995) 183-188
where A is of the order of the root-mean-square hyperfine interaction in the radical pair. Thus the mid-point (Ki = "/L=-iirk'ZF+ K/nF]) of the RPM field effect occurs at B 0 = A. Note that KiNF is not independent of B 0 because of the field dependence of the C~q(Bo) coefficients (the HF superscript refers to the radical pair interactions, not the triplet). We stress that the exact form of Eqs. (9), (10) and (12) is not crucial: it is sufficient that they reproduce qualitatively the RPM-MFE. The dependence of K i ( B o) on B 0 is suggested by the Rabi formula for a two-level system subject to a time-dependent perturbation [10]: on average, the fraction of radical pairs present as triplets is proportional to A 2 / ( A 2 + B2). Finally, the linear dichroic magnetic field effect, detected as the difference in absorption of light polarized parallel (par) and perpendicular (per) to the magnetic field, for a randomly oriented sample of reaction centres [5,6], is calculated as LD-MFE( B 0) ot
fo2~fo~[T](B0)((/z -(~.v~
Vpar) 2
r
(a)
2-
.3
[-
O
/
e~
,/ 2
iI
8 "6 0
E !
sin 0 d 0 d 4 ) ,
.......
(c)
/i,
I / -Z
•=
I
(13) where [T] is given by Eqs. (2), (4), and (9)-(12). In Eq. (13), ~ is the optical transition dipole moment, and Vp~r and Vp~ are the electric vectors of the polarized probe light.
(b)
\ \\
I
, /j/ ii
k
\
/
I
\, /
.q 0
0
10
20
30
magnetic field B 0 / mT
3. Results and discussion Fig. 3a shows the LD-MFE calculated according to Eq. (13). The values used for k x , k v, k z and the zero field splitting parameters, D(3p) and E(3p), are those appropriate for 3 p in Rb. sphaeroides [11,12] (see caption). The optical transition moment /t is taken to be3Parallel to the Y axis of the zero field splitting in P. (In practice [5,6], the triplet concentration is monitored via the optical absorption of ground state P whose transition dipole moment is approximately in this direction [13,14]; the very low concentrations of P * and P +I- mean that [P] + [ 3p] is constant.) Despite the crudity of the approximations used to describe the RPM-MFE, the calculated LD signal bears a striking resemblance to the experi-
Fig. 3. (a) Calculated linear dichroic magnetic field effect (dashed line) and its derivative (solid line). Parameter values: k x = 9.0 × 103 s - l ; k r = 8 . 0 × 1 0 3 s-a; k z = l . 4 × 1 0 3 s -1 [11,12]; ] D ( 3 p ) I = 2 0 . 0 mT; I E ( a P ) i = 3 . 5 mT [11]; A = 2 . 0 roT. (b) Magnetic field effect on the calculated triplet concentrations when the field is parallel to the X (short dashed), Y (solid), or Z (long dashed) axis of the triplet zero field splitting. (c) As (b) except that the field has been rotated away from each of the zero field axes by 5°. ~ X, ~ Y, ~ Z correspond, respectively, to (0, 4)) = (90 °, 5°), (90 °, 85°), (5 °, 0°).
mental spectrum, Fig. 1. As the magnetic field is increased, the signal initially drops at a rate determined by the value of the radical pair hyperfine parameter A, before rising to a prominent maximum (at = 20 mT) whose position is sensitive only to the value of D(3p). Also noticeable is a very shallow
J.K. Hicks, P.J. Hore / Chemical Physics Letters 237 (1995) 183-188
I1)
187
off-diagonal elements of the spin Hamiltonian, giving new eigenstates and rate constants (approximately): [2')=l{[x)-ily)} 1
1
+ 2-1/2 [ Z ) , 1
k 2, = -~kx + g k y + $kz,
Iv) Ix)
[3') = ½{ Ix) - i] Y)} - 2-1/2 [ z ) , ky
0,
Iz)
........ il3) 0
10 20 magnetic field B0/mT
30
Fig. 4. Energy levels of 3p when the magnetic field is parallel to
the Z axis (solid lines) and rotated 5° away from the Z axis (dashed lines). Drawn for D( 3P)= + 20.0 roT, E(3p)= + 3.5 roT.
depression in the positive part of the derivative LD signal around 13 mT. A more pronounced dip is found in the experimental signal (Fig. 1) at similar field strengths. The zero crossings of the calculated derivative LD signal correspond reasonably well to those in the experimental trace. At first sight, the maximum at B o = D(3P) would appear to be related to the crossing of two of the three triplet sub-levels, which occurs at just this field strength when the magnetic field is oriented along the Z axis of the zero field splitting (Fig. 4). However, the two eigenstates involved in this level crossing are to a good approximation (Zeeman interaction >> IE(3p)I, see e.g. Ref. [8]) independent of B 0, and given by [2) and 13) in Eq. (3). From Eq. (2), therefore, the ISC rate constants k i should also be field-independent around 20 mT, and no special MFE would be expected. This conclusion is corroborated by the calculated field-dependence of [T] (Fig. 3b). The key to understanding the maximum LDMFE at B 0 --~D(3p) is the fact that the level crossing becomes an avoided crossing as soon as the field direction deviates even slightly from the Z axis (Fig. 4) [15]. For a 20 mT field, almost parallel to the Z axis, the states 12) and 13) in Eq. (3) are mixed by
=
1 x + ~k 1 r + ~k 1 z. ~k
(14)
As one moves away from the avoided crossing, to fields lower or higher than 20 mT, the eigenstates and rate constants revert to those in Eq. (3). This analysis is supported by Fig. 3c which shows calculated triplet concentrations for field directions almost parallel to X, Y and Z. Now there is a very pronounced minimum in [T] around 20 mT when the field direction is close to the Z axis. Also visible in Fig. 3c is a small hump around 13 mT when the field is nearly parallel to the Y axis of the zero field splitting. This arises from the only other avoided crossing of the triplet sub-levels, and is smaller than the first because of the relatively minor deviation from axial symmetry of both the zero field splitting and the intersystem crossing rates. It is this field effect on [T] that gives rise to the slight depression at 13 mT in Fig. 3a, referred to above. This argument can easily be extended to obtain an estimate of the size of the magnetic field effect at B 0 = D(3p). Using the ISC rates and triplet eigenstates in Eqs. (3) and (14), and the populating rates in Eq. (10), the triplet concentrations for a 20 mT field exactly parallel (par) and nearly parallel ( = par) to the Z axis of 3p are in the approximate ratio
[T]par
z
[T]=par z
k x + kv + 2kz 4kz
(15)
which using the ISC rate constants appropriate for Rb. sphaeroides (Fig. 3 caption), is approximately 3.5. Even larger effects can be anticipated for triplets with a larger anisotropy in their intersystem crossing rates.
Acknowledgement This work was made possible by a Twinning Grant from the European Commission. PJH thanks
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J.K. Hicks, P,J. Hore / Chemical Physics Letters 237 (1995) 183-188
Stephen Sexton, Robert van der Vos and Professor Arnold Hoff for advice and enlightening discussions on many occasions, and is pleased to acknowledge an informative conversation with Professor J.H. van der Waals.
References [1] H.-J. Werner, K. Schulten and A. Weller, Biochim. Biophys. Acta 502 (1978) 255. [2] R. Haberkorn and M.E. Michel-Beyerle, Biophys. J. 26 (1979) 489. [3] A.J. Hoff, Quart. Rev. Biophys. 14 (1981) 599. [4] U.E. Steiner and T. Ulrich, Chem. Rev. 89 (1989) 51. [5] E.J. Lous, Doctoral Thesis, Leiden University (1988). [6] E.J. Lous and A.J. Hoff, in: The photosynthetic bacterial reaction centre - structure and dynamics, eds. J. Breton and A. Vermeglio (Plenum Press, New York, 1988).
[7] A.J. Hoff, P. Gast, R. van der Vos, J. Vrieze, E.M. Franken and E.J. Lous, Proceedings of the 2nd Conference on Magnetic field and spin effects in chemistry and related phenomena, Konstanz, Germany, 1992. [8] A. Carrington and A.D. McLachlan, Introduction to magnetic resonance (Harper, New York, 1967) pp. 118-121. [9] R. van der Vos, E.M. Franken, S.J. Sexton, S. Shochat, P. Gast, P.J. Hore and A.J. Hoff, Biochim. Biophys. Acta, in press. [10] P.W. Atkins, Molecular quantum mechanics (Oxford Univ. Press, Oxford, 1983) p. 190. [11] H.J. den Blanken, A.D.J.M. Jongenelis and A.J. Hoff, Biochim. Biophys. Acta 725 (1983) 472. [12] A.J. Hoff, Biochim. Biophys. Acta 440 (1976) 765. [13] E.J. Lous and A.J. Hoff, Proc. Natl. Acad. Sci. US 84 (1987) 6147. [14] S. Shochat, T. Arlt, C. Francke, P. Gast, P.I. van Noort, S.C.M. Otte, J.P.M. Schelvis, S. Schmidt, E. Vijgenboom, J. Vrieze, W. Zinth and A.J. Hoff, Photosynth. Res. 40 (1994) 55. [15] W.S. Veeman and J.H. van der Waals, Chem. Phys. Letters 7 (1970) 65.