J. Phys. Chem. 1991, 95, 6741-6744
6741
Applkatlon of Electrostatic W e 1 to Redox Potentials of Tetranuclear Iron-Sulfur Clusters Masato K&ka,*
Takenori Tomohiro, and Hiroaki(Yohmei) Okuno
Biological Chemistry Division, National Chemical Laboratory for Industry, Tsukuba, Ibaraki 305, Japan (Received: December 10, 1990; In Final Form: March 5, 1991)
An electrostaticmodel for tetranuclear ironsulfur (Fe,S4) clusters is derived for the first time from Kirkwood's theory and is used to calculate differences (AE)in the redox potential between each redox step of the FeS, clusters. The calculated AE values are in generally good agreement with the experimental data. The theoretical dependency of AE on the dielectric constant of ligand moiety is also supported experimentally by using macrocyclic-ligand-type Fe4S4clusters.
Introduction Tetranuclear cubane-type ironsulfur clusters (FeS, clusters) have k e n mognizd as analogues for the active site of iron-sulfur proteins such as ferredoxin and high-potential protein,'+ and at the same time, applied to various chemical reactions as electroThe natural iron-sulfur proteins are known to catalyze the electron-transfer reactions in photosynthesis,and the corresponding model clusters serve to provide information on the fundamental functions of these proteins. One of the essential qualities of the Fe4S4model clusters may be the redox potentials, which reflect the relative stabilities of oxidation states of the clusters. Molecular orbital (MO)calculations, e.g. the extended Hockel (EH) method, the restricted HartreeFock-Slater-LCAO-frozen core (HFS-FC) method, etc., have been used for the theoretical investigation of Fe,S, clusters and have produced much valuable inf~rmation.'~Although the MO methods are powerful tools for detailed analyses of the electronic structures of the clusters, they afford little information about the macroscopic effects of environments (ligands and solvents) around the Fe4S4cubane core. Kirkwood's classical electrostatic model," on the other hand, is expected to offer theoretical hints concerning these environmental influences upon the electrostatic properties, although this model makes some simplifying assumptions. From the viewpoint of the electrostatic theory, the Fe4S4clusters may be represented merely by the assembly of point charges that correspond to the iron or the sulfur atoms in the cluster, located within a sphere surrounded by ligand and solvent phases. The electrostatic model is applicable to the calculation of the electrostatic energy of such a point-charge system. Furthermore, use of such a model allows one to easily and intuitively correlate the calculated results with the actual properties of the Fe4S4clusters. Thus, in the present study, the electrostatic model was applied for the first time to the Fe4.S4 clusters. Of special interest is the effect of the ligand moiety, wz., the polarity of the ligand, which is expected to be easily compared with experimental results. In order to substantiate the conclusion obtained from the model, we prepared Fe4S4clusters bearing nonpolar macrocyclic ligands. Experimental Section Synthesis of Macrocyclic Fe4S4Clusters. To study the effect of the ligand moiety, it is much more appropriate to adopt an alkyl-type ligand than an aryl-type one, since in the former case we can almost neglect the influence of the change of charge distribution in the cluster and can extract only the effect of polarity of the ligand layer. The finally selected structure of the cluster is shown in Figure 1. According to Corey-Pauling-Kolutan (CPK) model examinations, a pore size of at least 10 A was necessary to incorporate the Fe4S4core snugly inside the intramolecular nonpolar domain. We therefore synthesized macrocycles with a 28-membered ring (pore size ca. 11 A) and greater, that is, 28-, 32-, 36-, 40-, and 44-membered rings. The macro-
* To whom correspondence should be addressed. 0022-3654191/209S-6741$02.S0/0
cycles and the corresponding Fe4S4clusters were prepared in the previously described manner.' Measurement of Redox (Half-Wave) Potentials. Redox potentials were measured by cyclic voltammetry on a Hokuto Denko HA-SO1 instrument with a working electrode (R), a counter electrode (Pt), and a reference electrode (SCE). The scan rate was 100 mV s-l. The concentration of the cluster was 1 mmol dm-' in the presence of n-Bu4NBF4 (0.2 mol d m 3 .
Theory We consider an electrostatic model, which is compafed of point charges within a core (I, radius b, dielectric constant D,) surrounded by a shell (11; thickness a-b dielectric constant D,,,), as shown in Figure 2. The whole sphere is then immersed in a solution with a dielectric constant D containing electrolyte salt (concentration CE). Mdiscrete point charges, e l , e2, ...,eM,are distributed within the core (I). Applying the method developed by Kirkwood14 to the present system, the electrostatic energy of the point charge assembly can be derived as follows. By use of a polar coordinate system, the electrostatic potential (VI) at any point (r, 8, 4) inside the core (I) is given by eqs 1 and 2 where
r' and i k are, respectively, the vectors pointing from the origin to (1) DePamphilii, B. V.; Averill, B. A,; Herskovitz, T.; Que, L., Jr.; Holm, R. H. J. Am. Chem. Soc. 1974,96,4159. (2) Que, L.,Jr.; Bobrik, M.A.; Ibers, J. A.; Holm, R. H. J. Am. Chem. Soc. 1974,96,4168. ( 3 ) Hill, C.L.; Renaud, J.; Holm, R. H.; Mortenson, L. E. J. Am. Chem. Sa.1977,99,2549. (4)(a) Okuno, Y.; Uoto, K.; Sasaki, Y.; Yonemitsu, 0.; Tomohiro, T. J. Chem.Sa.,Chem. Commun. 1987,874.(b) Uoto, K.; Tomohiro, T.; Okuno, H(Y). Itwrg. Chim. Acta 1990, 170, 123. ( 5 ) Okuno, Y.; Uoto, K.; Yonemitsu, 0.; Tomohiro, T. J. Chem. Soc., Chem. Commun. 1987,1018. (6)Okuno, H(Y).; Uoto, K.; Tomohiro, T. Chem. Express 1990, 5, 37. (7) (a) Tomohiro, T.; Uoto, K.; Okuno H(Y). J . Chem. Soc., Dolton Trans. 1990,2459. (b) Okuno, H(Y); Uoto, K.; Tomohiro, T.; Youinou, M.-T. Ibid. 1990. 3375. (8) Kodaka, M.;Tomohiro, T.; Okuno,H(Y). Chem. Express 1990,5,97. (9)Kodaka, M.; Tomohiro, T.; Okuno,H(Y). Ch" Express 1990,5,117. (10)Kodaka,M.;Tomohiro, T.; Lee, A. L.; Okuno, H(Y). J. Chem. Soc., Chem. Commun. 1989, 1479. (1 1) Tomohiro. T.; Uoto. K.; Okuno. H(Y). J . Chem. Soc., Chem. Commun. 1990, 194. (12)Kodaka.M.;Lee, A. L.;Tomohiro. T.: Okuno. H(Y). . . Chem.Ex~ress i99'o,j,233. (13) Geurts, P. J. M.;Gospelink, J. W.; Van der Avoird, A,; Baerends, E. J.; Snijders, J. G. Chem. Phys. 1980, 46, 133. (14)Kirkwood, J. G.J. Chem. Phys. 1934,2, 351.
0 1991 American Chemical Society
6142 The Journal of Physical Chemistry, Vol. 95, No. 17, 1991
Kodaka et al. In the solution outside the sphere with radius c, the interaction between the point-charge system and the electrolyte ions leads to a mean space charge. Then, the potential V4 in this region is shown as
where K is the Debye-HGckel parameter, N is Avogadro's number, k is Boltzmann's constant, Tis the absolute temperature (298 K in this work), e is the electronic charge, co is the permittivity of a vacuum, Z , is the valence type of electrolyte ion i, and C, is the concentration of ion i. The constants A,, B,,,,,,C,,,,,, E,,,, F,, and G,,,,, are determined from the boundary conditions at r = b, r = a, and r = c. Consequently, the electrostaticenergy (W)of the point charges is given by
1: n = 6 2: n = l
3: n = 8 4: n - 9
5: n = l O
Figure 1. Structures of macrocyclic Fe4S4clusters.
k=l/=l
A =
+ 1)[(Dm - Dj)(nD, + (n + I)D)a2"+' + (D- 0), x (nD, + (n + l)Di)b2"+'](a/b)2"+'/D~"'(4n2- l)[((n +
n(n
1)D,
+ nD,)(nD, + (n + 1)D)a" + n(n + l)(D, - D,)(D, - D)b2"+'] (10)
B= D
Figwe 2. Schematicmodel of Fe,S4cluster. Points k and I represent sites of Fe, S or H atoms of [Fe4S,(SH)4]p (n = 1, 2, 3, 4). Dielectric constants are D, within radius 6 [sphere (I)], D, between radii b and a [shell (II)], and D in solution. Within the boundary of radius c, no
external ions can penetrate. the arbitrary point (r, 8,4) and to the point charge ek within the sphere (I); Pnm(cose) is the associated Legendre function. In the shell (11) the potential (V2)should also satisfy the Laplace equation and therefore V2
n-0 m=-n
(3
+
Enmp)pnm(cos e)dm'
(3)
In addition to the point charges, there are usually electrolyte ions in the solution. If the mean distance Of closest approach Of these ions to the surface of the shell (11) is C, there will be a third spherical b u n d a y of radius c9within which no ions can penetrate. In this region, the potential V, must have the form V, =
e 3 (5+
n-0 m i - n
G,,,/)pnm(cos
e)dmr
(4)
[n(n + 1)(D- D,)(D, - D,)a2"+'+ (nD+ (n + 1)D,) X (nD, + ( n + 1)Di)b2"+']/(n+ I)[n(D, - D,)(nD, + (n +
+ n(D - D,)(nD, + (n + 1)D,)b2"']
1)D)a"
(1 1 )
c = [(nD + ( n + l)D,)(D, - D,)b2"+' +
((n + 1)D, + nD,)(D - Dm)a2"+']/[((n + 1)D, + nD,)x [nD,+ (n + l)D)U"+' + n(n + l)(Di - D,)(D, - D)b2"+'] (12)
+
+
G = n[(nD + (n I)D,,,)(D,,, - D,)bzn+' ((n + 1)D, nD,)(D - D , p + ' ] / [ ( ( n 1)D, nD,)(nD, (n + 1)D)a2"+'+ n(n + l)(Di - D,)(D, - D)b2"+'](13)
+
+
+
+
where the Pn(cos e&/) is the ordinary Legendre function; 0, is the angle between the vectors ikand F!. The four parts of eq 8 have the following physical meanings. The first part represents the work of charging the point charges in an unbounded medium of dielectric constant D,,excluding the self-energy (k 3 r) of each point charge. The second part represents the self-energy. Point charges are known to have infinite self-energy, so in the calculation of the self-energy we adopt a rigid-body sphere with a radius (Rk) instead of a point charge in order to avoid mathematical breakdown. The third part of eq 8 refers to the modification arising
Redox Potentials of Tetranuclear Iron-Sulfur Clusters
The Journal of Physical Chemistry, Vol. 95, No. 17, 1991 6743
TABLE I: Atomic Charge#in (Fe&(SH)T ( B = l , f 3,4) Used in C8kuLtion of Redox PotcsthP atomic charge atom 12340.22 0.26 0.23 Fe 0.27 S'b
S H
-0.21 -0.53 0.22
-0.31 -0.60 0.15
-0.40 -0.63 0.05
I
-0.50 -0.70 -0.02
Macrocyc& Fe,S4 clusters
1
Reference 13. Inorganic sulfur bridging iron atoms.
0.5
from the fact that the point-charge assembly is a bounded cavity within media of dielectric constants Dm(shell (11)) and D (solvent). And, finally, the fourth part gives the interaction between the point charges and the extemal electrolyte ions in the solution. The redox potential (E) of the point-charge system is consequently approximated by
where Grd and Go, are the free energies of the reduced and the oxidized species, respectively, Wrd and Wo, represent the electrostatic energies of the corresponding species, n is the difference in the net charge between the reduced and oxidized species, and F is the Faraday constant.
0.o 0
20
60
40
80
D Figure 3. Dependence of calculated redox potential difference (AE,or AE2) on dielectric constant of solvent (D) in the case of a - b = 2.0 A and CE = 0.1 mol dm-': ( 0 )AEl, D,,, = 2; (0) AE,,D,,, = 4; (+) AE2, D,,, = 2; (A) AE2, D,,, = 4; ( 0 )observed data for [Fe4S4(S-r-Bu)4]z-; observed data for (m) 1, (X) 2, (m) 3, (A) 4, and (m) 5. *.O
I
Application of Electrostatic Model to Fe& Clusters We next apply the electrostatic theory to actual Fe4S4clusters. In this model each atom of the Fe4S4cluster is replaced by a point charge that has the net charge estimated from a molecular orbital calculation. Here, [Fe4S4(SH),]" (n = 1,2, 3,4) is adopted as a Fe4S4cluster, since the gross atomic charges can be estimated from the HFS-FC method for this type of iron-sulfur complex (Table I).13 The ligand moiety of Fe4S4clusters is, further, represented by the shell (11) in Figure 2. The actual numerical calculations were done for eqs 8-14. It was confirmed that the infinite series in eq 8 essentially converge by calculation up to n = 50. The Legendre function, P,(cos Okl) in eq 9, was easily calculated by using the following recurrence relations. Po(x) = 1
P,(x) = x
For the electrostatic energy of different oxidized form (1-, 2-, 3-, and 4-) of the Fe4S4cluster, the corresponding atomic charges in Table I were substituted for ekand el in eqs 8 and 9. The atomic radius ( R J , which is necessary for the calculation of the selfenergy in eq 8, is assumed to be a Van der Waals radius or crystalline radius: Fe, 0.69 A; S, 1.85 A; HI 1.20 The coordinates of the Fe4S4cluster are obtained from X-ray crystalline data.2 Other parameters to be fixed are a, b, c, D,,and Dm;thus D, was assumed to be 1 .O"; b was set to 5.0 A, which is a somewhat longer than average (4.55 A) length between the origin of the cluster and the S-H protons; c - a was assumed to be 5.0 A, which corresponds to the crystalline radius of n-Bu4N+,I6one of the popular supporting electrolytes in the measurement of cyclic voltammetry. The other parameters, a and D,,,, were varied within reasonable ranges as indicated below. The calculated differences in redox potential, AEl [E(1-/2-) - E(2-/3-)] and AE2 [E(2-/3-) - E(3-/4-)], are illustrated as a function of D, a - b, or CE (Figures 3-5). As shown in Figure 3, both AE, and AE2 depend little on D, while these values are more sensitive to 0,. The thickness of the ligand shell (a - b) also influences AEl and AE2 to some extent; that is, the potential difference values become slightly larger as the thickness increases (Figure 4). Here, the thickness is altered from 1.0 to 4.0 A in view of the actual dimensions of the ligands. The external ions (IS) Pauling, L. The " w e of the Chemicul Bond, Cornell University Press: Ithaca. New York, 1960. (16) Ohtaki, H.Solute-Solvent Interactions Viewed from Their Microscopic Behaulor, Syokabo: Tokyo, 1989.
0.5
t
0
1
3
2
a-b
4
5
(1)
Figure 4. Dependence of calculated redox potential differences (AE,or AE,) on thickness of ligand layer (a - b) in the case of D = 40 and CE = 0.1 mol dm-': ( 0 )AE,,Dm = 2; (0) AEl, 0, = 4; (+) AE,,D,,, = 2; (A) A E 2 , 0, = 4.
0-75.1
2
i 0.o 0.00 0.02 0.04 0.06
0.08 0.10
0.12
CE (mol d ~ n ' ~ )
Figure 5. Dependence of calculated redox potential differences (AE, or A&) on concentration of electrolyte salt (CE)in the case of u - b = 2.0 A and D = 40: ( 0 )AEl, D,,, = 2; ( 0 )AEl, D , = 4; (+) AE2, D,,, = 2; (A) A E 2 , D, = 4.
in the solution, on the other hand, have much less effect on AEl and AE2 at around 0.1 mol dm-' (Figure 5 ) . Of all the variable parameters, Dm seems to have the greatest influence on the potential differences, so its effect is discussed in more detail. The macroscopic dielectric constant is an experimentally derived property of bulk solvent that reflects the polarizability of solvent molecules. On the other hand, microscopic
J. Phys. Chem. 1991, 95, 6744-6745
6744
dielectric constant or "local dielectric constant" is very difficult to determine experimentally and/or theoretically, and it has been usually estimated quite empirically. For example, in molecular mechanics calculations such as "CHARM", a value around 2-5 has proved to be most adquate as the local dielectric constant in the interior of enzymes. Therefore, we treated D,,, as a variable parameter, so that the uncertainty about the local dielectric constant is buried in its range. The calculation was undertaken, therefore, in the case of D,,, = 2 and 4. As shown in Figures 3-5, the present model suggests that the potential differences (MI, AE2) grow larger with the decrease in D,,, and that AEl (range 1.2-1.7 V) is larger than AE3 (0.8-1.3 V). To substantiate this theoretical conclusion, we note the experimental results. In Figure 3 are also illustrated the observed AEl values of an unclad-type Fe4S, cluster, [Fe4S4(S-t-Bu),12-,and those of the macrocyclic Fe,S4 clusters (1,2, 3,4,5). In the case of [Fe4S4(S-r-Bu),12-, surely the experimental AEl (1.2-1.3 V) and AE2 (0.74 V in DMF) values are close to the calculated curves. Although not shown in Figure 3, another type of unclad cluster, [Fe4S4(S-iPr)4]2-, gives the similar tendency that M I (1.26 V in CH2C12) is larger than AE2 (0.72 V in DMF). On the other hand, the macrocyclic Fe4S4clusters with a less polar and thicker ligand layer show larger AEl than the unclad-type clusters, which is well compatible with the theoretical suggestion shown (Figure 3). As for the AE2 value, the macrocyclic Fe4S4clusters did not give the redox waves in the 3-/4- step, and thus an experimental AE2 is
not shown in Figure 3. Another possible reason for this tendency may be the steric distortion by the macrocycles bound to the FeS4 core. This is less likely, however, since the observed Ml values are essentially independent of the ring size of the macrocycles (28-44-membered). Although we can imagine another much simpler model in which the sphere is uniformly charged, this model may be inadquate for the following reasons. The electrostatic energy of such a sphere is generally known to be proportional to the square of total charge, and therefore the following relation holds in view of eq 14: E( 1 -/2-) :E(2-/3-) :E(3-/4-) = 3 :5:7 (16) Consequently, AEl [E(1-/2-) - E(2-/3-)] should be equal to AE2 [E(2-/3-) - E(3-/4-)], which does not agree with the experimental result. In this simple model, moreover, AEl and AE2 are decreased about one order of magnitude below the observed values. Finally the effect of dielectric saturation should be discussed briefly. This effect, caused by an extremely strong electric field around an ion, is known to decrease the dielectric constant of solution. Although this effect may influence the D value to some extent, the essential conclusion of the present study would be unchanged in view of the fact that AEl and AE2 are not so dependent upon D (Figure 3). In principle, the model used in this work should also be a p plicable to other metal complexes with various arbitrary shapes, although the mathematical derivation would be more complicated.
COMMENTS Rdatlve ProbaMmtly of Energy Pooling vs UpPumping for Energy Tramfor during Molecular Encounters
Sir: Generally, energy transfers occur during collisions between two molecules, one of which carries energy E2 and the other El ( E 2 > El). It is interesting to question what factors determine the relative probabilities that as a result of such an encounter the product states will be closer in energy (Le., energy pooling) vs farther apart (up-pumping). A straightforward analysis shows that the ratio of probabilities follows from the principle of detailed balance. Designate the probability for pooling per encounter by PPIand for up-pumping by P,. Then, at statistical equilibrium p,0122N2(E2) NI(EI) = P,.122N2(E2-4 NI(El+f) ( 1 ) where e is the amount of energy transferred and ul* is the mean collision cross section. Since ME) = & W D E / Q ( T ) (2) where p(E) is the density of energy states and Q( 7)is the partition function (3)
Note that the ratio of probabilities applies whether the collision partners are actually in thermal equilibrium or not. Also, relation 3 is valid whether the molecules 1 and 2 are the same or different species. At first glance this is unexpected since R is independent of temperature and, in particular, is independent of the molecular dynamics of the collision event. Clearly, R 1 as e 0; Le., pooling and uppumping are equally probable in the limit of very small energy transfers. To obtain an intuitive impression of the significance of (3), consider v-v energy transfers, using the Whitten-Rabinovitch
- -
0022-3654191 /2095-6744302.50/0
expression for the density of vibrational energy states:' (E, + aE,)-l (s - l)!nhui R = (1 -
E2 + aE2,
)-I(
1
+ El +e aElr) - I
(4)
(5)
Expand, and neglect terms in (e/(E2+ u E ~ , )and ) ~ higher
R = l -
- I)e
(SI - 1 ) e + E2 + UE,~ El + aE,,
(s2
For a single species r
R
=+
1
+
- l)e
1
1
aEll - E2 + aE2, El R > 1 by an amount that depends on =(s - 1)((E2- EI)/(E2EI)). Were one to use an empirical representation of the vibrational state density in the form2 (S
p(E,) = exp[aE, - bE:], with b
