J . Phys. Chem. 1984,88, 1321-1323
1321
Electron-Exchange Reaction in Aqueous Solution Sven Larsson Department of Chemistry, University of Lund, Lund, Sweden (Received: September 14, 1982; In Final Form: August 25, 1983)
Electron tunneling in a water solvent is studied by using Landau-Zener and extended Hiickel (EH) theory. An equation for long-distance tunneling in a solvent is derived.
Introduction
9
Due to experimental and theoretical progress, electron selfexchange reactions in solution are now rather well A remaining problem is electronic nonadiabaticity. In the closest possible approach of two electron-exchanging metal ions with their first coordination layer intact, the electronic matrix element has been calculated to 10-3-10-4 au in a number of This should imply nonadiabatic behavior, but there is still no experimental evidence for this. In fact, experiments are well explained by theory if adiabaticity is assumed. Another theoretical problem is the difference in electron-transfer capability between a molecular bridge and the solvent itself. Experiments of Taube’ and others suggest that bridges may permit adiabatic transfer at more than 10 A between the metal ions. The tunneling method has sometimes been used to obtain rough estimates of the electronic matrix element. An obvious problem in this method is to calculate the barrier sufficiently correctly and solve the tunneling equations for an arbitrary shape of the barrier. This has not been done so far. Recently, I have instead suggested the use of the extended Hiickel (EH) method8 in order to understand the variation of the electronic matrix elements in various forms of matter.gJO One then avoids the use of potential barriers for the tunneling electron. An advantage over the ab initio methods is that calculations are feasible for systems which are larger by an order of magnitude in number of electrons and atoms. On the other hand, the method is crude and a careful assessment of its accuracy for simple systems is badly needed. A comparison to the results of Newton for water clusters will be made in the present paper. The background theory is provided in ref 1-6 and will not be repeated here.
9
3 1.
i 2. EE3
connected via the edges of the octahedra denoted EEn. n = 2 thus corresponds to the smallest outer-sphere complex. A third series of calculations were done on side-on complexes (3) denoted SSn,
f
P
Calculations One series of calculations were carried out on complexes connected by corners (1) denoted CCn where n denotes the number of layers of water molecules between the Fe ions; n = 1 gives an inner-sphere complex, and n = 2, 3... outer-sphere complexes. Another series of calculations were carried out on complexes (2)
(1) B. Brunschwig, J. Logan, M. D. Newton, and N. Sutin, J . Am. Chem. SOC..102. 5978 11980). (2) N.’Sutin a‘nd B.’Brunschwig, ACS Symp. Ser., No. 198, 105 (1982);
B. Brunschwig, C. Creutz, D. H. Macartney, T. K. Sham, and N. Sutin, Faraday Discuss. Chem. Soc., 74, 113 (1982). (3) M. D. Newton, Int. J . Quantum Chem., 514, 363 (1980). (4) B. L. Tembe, H. L. Friedman, and M. D. Newton, J. Chem. Phys., 76, 1490 (1982). (5) H. Friedman and M. D. Newton, Faraday Discuss. Chem. Soc., 74, 73 (1982). (6) J. Logan and M. D. Newton, J . Chem. Phys., 78,4086 (1983). (7) H. Taube, “Electron Transfer Reactions of Complex Ions in Solution”, Academic Press, New York, 1970. (8) R. Hoffmann, J. Chem. Phys., 39, 1397 (1963). For parameters see R. H. Summerville and R. Hoffmann, J. Am. Chem. Soc., 98,7240 (1976) (Appendix 2). (9) S. Larsson, J . Am. Chem. Soc., 103, 4034 (1981). (10) S. Larsson, J . Chem. Soc., Faraday Trans. 2, 79, 1375-88 (1983).
0022-3654/84/2088-1321$01.50/0
cc2
/ 3.
9
b
ss3
with the waters in a staggered geometry. SS2 thus corresponds to the complex originally studied by N e ~ t o n . ~ The calculations were done by using the standard version of the E H method. One set of calculations was carried out with inclusion of the overlap matrix S in the secular equation IH - tS( = 0
(1)
and another by using S = 1, which is often done in applications of semiempirical methods. In the E H method the Mulliken approximation is used for the nondiagonal matrix elements: Hij = KSij(Hii+ Hjj)/ 2 0 1984 American Chemical Society
(2)
1322 The Journal of Physical Chemistry, Vol. 88, No. 7, 1984
LOG A 1
R(A1
-8 2
I
8
8
10
12
Larsson steeper slope. EE2 in fact appears to be much more favorable than SS2 at R = 6 A. Since the screening of the positively charged metal ions is not less in EE2, it may be energetically as favorable as SS2. The behavior of IAal when the distance the metal ions is increased and new molecules of water are inserted into the empty space is of great principal interest. In Figure 1 we find that SS3 increases slightly above the value for SS2 at the same R, EE3 increases very much compared to EE2, whereas CC3 decreases and CC4 increases compared to CC2. The latter two are well in the nonadiabatic region, whereas EE3 is close to the adiabatic limit for R = 8 A. Apparently some conformations give a larger value of 1 41than others but IAal is below au for most of them. To obtain an estimation of Aa for a large distance between the metal ions we may use the equationgJO
Figure 1. Calculated values of A (auj as a function of distance between (b) direct interaction the metal ions: (a) results of Newton, ref 13 of 3d,2 functions * .); (c) SS2 (-); (dj EE2 (e) CC2 (---); ( f ) SS3 (0);(g) EE3 (v);(h) CC3 (+); (i) CC4 (A); (j)CC2 of ref 6 (---); (k) SS2 of ref 6 (--).
N-1
N
v=l
v-1
n n (b - a)-’
14
A = 2 ~ ~ 1 7 P,~
(4)
(a*.);
(e
(--e);
where K is here given the value 1.75. It was found necessary to use a double-{ expansion of the transition-metal 3d orbital. The water molecuies are kept in positions appropriate for Th symmetry for each Fe(H20)6 unit. Since this corresponds approximately to o h , we may use o h notation. Due to the thermal motions, either of the three tzs orbitals may be doubly occupied for Fe2+,even if the nuclear configuration under study may split these obitals by a small amount. Since for small A the adiabatic contribution is proportional to A2, it is reasonable to use the average
tl and q2 are matrix elements between a metal ion and the adjacent water molecule. P, are matrix elements between two consecutive water molecules, a is the orbital energy of the metal ion 3d-t,, orbital, and b is the orbital energy of an MO of water overlapping with the other molecules in the chain of molecules along the shortest path between the metal ions. Since we will find an exponential decrease of Aa with distance, we may limit ourselves to this path. 2pvis equal to A for electron exchange between two molecules of water. Ab initio calculations with large basis sets were carried out on the latter system in different orientations. If R is the 0. -0 distance, p,, may be approximated as
-
P = eoe-dR-Ro) (5) where eo = 1 au of energy; a = 2.3 A-’ and Ro = 0.8 8, is an average for different orientations. Inserting another water molecule between the first two gives, using eq 4 ( R = R1 R2) p’ = e 0 ~ - ~ ( R ~ - R o ) ~ o ~ - ~ ( R Ib 2-R -oal) /
+
+
where An is the energy splitting between the orbitals ti:!” and t$Zn- ti,!.. Since all e,+ and no e,i orbitals are occupied, no transfer takes place for Fe2+/Fe3+via the Fe-OH2 u orbitals.
Results and Discussion In Figure 1 the calculated values of log (Aal, using S = 1 in eq 1, are given as functions of the distance R between the metal ions. (Using the actual S in eq 1 leads to values of IAa( which, almost uniformly, are smaller by a factor of 3.) The results for SS2 are in very good agreement with the recent results of Logan and Newton, who included wave functions only on the water molecules in the region between the metal ions. Comparing the results of this work to those of Newton without inclusion of wave functions on the water molecules (dotted in Figure l ) , we find the slope to be considerably steeper in the E H results. This is partly due to the use of a double-{ basis. Improving on the description of the 3d orbitals (using the five-exponential approximation of Clementi) leads to roughly the same slope as that obtained by Newton but still with lower absolute values of !Aa]. The accuracy in the approximation used by Newton3 is hard to estimate. In the author’s opinion the use of Gaussian basis sets may introduce errors at long range, even if off-center functions are used. When the wave functions of the water molecules are included, the direct interactions are overshadowed by the interactions through the water molecules. The comparison between different methods is then more meaningful. The good agreement between the ab initio and E H methods for that case is therefore encouraging. Concerning the apex-to-apex approach of the metal complexes (CCZ), there is an increase of 141,but the slope is about the same as for SS2. The E H method gives a somewhat larger increase than ab initio. However, the shortest possible apex-to-apex distance is at least 7 A and at that distance 1A.J is already below the limit au. The latter may be considered as the limit of nonadiabatic behavior.’s2 The EE2 configurations which have been studied only by E H 41in between those of SS2 and CC2 with a slightly give values of 1
eoe-dR-Ro)ex x = aRo - In (Ib - al/eo)
(6) Comparing to eq 5 we find that every new water molecule contributes a factor ex to A. Each new H 2 0 extends the chain by a distance d (=3.0 A). We may now generalize eq 6 and replace it by a continuous function p’ ( R ) = eOe-dR-Ro)eXR/d=eoe-a’(R-Ro’) (7) Equation 4 may now be written as A = 21112p’
(8)
and v2 may be absorbed into the factor ea’Ra’. The final expression for A is thus A = 2eoe-dR-Ro’) (9) The exponential decay factor corresponding to solvent molecules between the outer-sphere complexes is thus a’ and it is different from the decay factor for direct interaction between these complexes. The difference is a’
-a
= -x/d =
E
- aRO- In ’b-a‘]/d
(10)
e0
Using a - b = 0.2 au and d = 3.0 8,we obtain a‘ - a = -1.1 A, Le., a’ = 1.2 A-’ for water, in agreement with our directly calculated results (Figure l). The negative correction term (a’- a ) is a measure of the difference in exponential decay with distance between outer-sphere complexes in “vacuum” and in the actual solvent environment. It is small if Ib - a1 is large (but Ib - a1 5 1 au) and if Ro/d is small. This is apparently the case for water where Ro/d = 0.27. A larger organic solvent molecule is likely to have a larger value of R0/d and a smaller b - a, thus making tunneling over a larger range possible.” If a single bridge molecule connects the metal
J . Phys. Chem. 1984,88, 1323-1324
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1323
ions, eq 4 cannot be used, but it corresponds to the case R o / d 1. The results obtained suggest strongly that long-range electron transfer is not possible in a water solution. Transfer along hydrogen bonds is likely to be favored but is still unlikely to occur over large distances. In the future it may be possible to improve
the accuracy of the extended Huckel-type models and examine this question in more detail. It would also be desirable to apply the ab initio methods to increasingly larger clusters, particularly some of those studied here which in the EH treatment has a rather large value of JAJ. With the background of the equation for A presented here, numerical studies of the transfer properties in solvents other than water would be very interesting.
(11) J. R. Miller, Science, 189, 221 (1975); J. Phys. Chem., 79, 1070 (1975); 82, 767 (1978).
Acknowledgment. I was supported by funds from the Swedish Natural Science Research Council (NFR).
Nuclear Magnetic Relaxation Dispersion In Protein Solutions. A Test of Proton-Exchange Coupling . Robert G. Bryant* and Marie Jarvis Chemistry Department, University of Minnesota, Minneapolis, Minnesota 55455 (Received: July 21, 1983)
The nuclear magnetic relaxation rates for solvent protons have been measured for water and dimethyl sulfoxide in 15% lysozyme solutions over a wide frequency range. The observation of a clear dispersion in the relaxation of the methyl protons of the dimethyl sulfoxide demonstrates that coupling the rotational motion of the protein to the solvent magnetic relaxation may be observed in a system where chemical exchange of solvent protons with macromolecule protons cannot contribute to the relaxation mechanism.
Introduction The dynamical coupling between globular protein molecules and the solvent proton or deutron nuclear magnetic relaxation rate is a well-established fact.’ Several features are central to understanding the nature of this dynamical effect: (1) The magnitude of the macromolecule-induced relaxation experienced by the solvent nuclei is small, though readily observable. (2) The frequency dependence of the nuclear magnetic relaxation provides a report of the hydrodynamic volume of the macromolecule in the sense that the inflection frequency of the dispersion plot scales as the hydrodynamic volume of the protein molecule over a very wide range of molecular weights. (3) The three observable solvent nuclei report very similar correlation times though the correlation times characterize the reorientation of vectors that have different orientations in the molecule fixed axis system.’ The simplest and most often used model for the interpretation of water magnetic relaxation in complex systems involves the rapid exchange of nuclei or whole solvent molecules between two environments that differ substantially in relaxation properties. In the protein solution one hypothesis has been that the dominant site is the bulk aqueous environment of the solution while the position of efficient relaxation is associated with ‘‘bound’! water molecule a t the protein surface. The properties of the strongly interacting or bound water molecule have been discussed by several author^.^ To summarize, if the water molecules at the protein surface are assumed to be rigidly fixed such that they rotate with the protein as some hydrodynamic measures of macromolecular size might be taken to suggest,” the size of the magnetic relaxation effect predicted exceeds that observed by about a factor of 100. Thus, such a limiting model fails.2 Inclusion of local rapid motion about a unique hydrogen bond axis may be invoked to decrease (1) Hallenga, K.; Koenig, S.H. Biochemisrry 1976, 25, 4255-64.
(2) Koenig, S.H.; Hallenga, K.; Shporer, M. Proc. Natl. Acad. Sci. U.S.A. 1975, 72, 2667-71. (3) Bryant, R. G. Annu. Rev. Phys. Chem. 1978,29, 167-187. Halle, B.; Wennerstrom, H. J . Chem. Phys. 1981, 75, 1928-43. (4) Marshall, A. G . “Biophysical Chemistry”; Wiley: New York, 1978;
pp 199-205.
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the apparent strength of the interaction driving nuclear relaxat i ~ n . ~However, ,~ the reduction falls far short of that required for the observation, though anisotropic motion at the protein surface appears to be well documented.’ In addition, analysis of solvent proton, deuterium, and oxygen relaxation, which have very different effective time scales for the averaging processes involved, leads to a contradiction in the exchange lifetimes of the solvent molecules at the protein surface. Koenig and co-workers have used this argument to dismiss the elementary exchange model, and variations of it, as fundamentally inadequate to account for the experimental observations.2 Recently, there has been renewed interest in a relaxation mechanism that involves the exchange of solvent protons rather than whole water molecules as a possible mechanism for coupling the rotational motion of the macromolecule to the solvent relaxation.* The number of exchangeable protons on a protein is a small fraction of the total number of water molecules that may interact at the surface of the protein. Therefore, the magnitude of the effect on relaxation may be much smaller than that predicted from the bound water models, though early arguments have suggested that the proton exchange of ionizable protein residues cannot be fast enough based on what is known about the proton-exchange rate constants for the functional groups i n ~ o l v e d . ~ We report here an experiment that is designed to test critically the proton-exchange mechanism as a means of coupling the macromolecule motion to the solvent proton magnetic relaxation. The strategy of the experiment is to study a solution where the protons of the solvent are known not to exchange with the functional groups of the protein. The basic result is that a nuclear magnetic relaxation dispersion spectrum is observed when proton
( 5 ) Koenig, S.H.; Schillinger, W. E. J . Bioi. Chem. 1969, 244, 3283-9. (6) Woessner, D. E. J . Chem. Phys. 1962, 36, 1-4. Woessner, D. E.; Simmerman, J. R. J . Phys. Chem. 1963, 67, 1590-600. (7) Shirley, W. M.; Bryant, R. G. J . Am. Chem. SOC.1982,104, 291C-8. (8) Bryant, R. G.; Halle, B. In “Biophysics of Water”; Franks, F., Mathias, S.F., Eds.; Wiley: New York, 1982; pp 389-93.
0 1984 American Chemical Society
