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 Up-Pumping 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
expression for the density of vibrational energy states:' (E, + aE,)-l (s - l)!nhui R = (1 -
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 1 as e 0; Le., dynamics of the collision event. Clearly, R 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
)-I(
1
+ El +e aElr) - I
(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
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
E2 + aE2,
(4)
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
