J . Phys. Chem. 1989, 93, 5065-5070
5065
Selective Solvation Effects on Mixed-Valency Species in Mixed Solvents J. Katriel*.+ Department of Chemistry, Northwestern University, Evanston, Illinois 60208
and M. A. Ratner* Department of Chemistry and Materials Research Center, Northwestern University, Evanston. Illinois 60208 (Received: September 23, 1988; In Final Form: February 24, 1989)
A simple binding-site model is developed to describe the behavior and optical response of binuclear mixed-valent metal complexes in mixed solvents. A free energy expression is constructed, including energy terms due to electron delocalization, to solvent molecules in the first solvation shell, and to interactions between solvent molecules, both in the bulk and in the first shell. An entropy term includes effects of parititioning the solvent species between bulk solvent and solvation shells. Minimization of the free energy yields an equilibrium condition; the resulting equations are precisely those of the two-sublattice model for magnetism. Conditions for symmetric solvation or for asymmetric solvation (with a corresponding blue shift in the intervalence-transferabsorption band) are developed, as a function of solvent composition and temperature. A phase diagram is given; at low temperatures, asymmetric solvation will almost always occur.
I. Introduction More than two decades ago, Robin and Day' classified mixed-valent molecules and solids as belonging to one of three classes. A great deal of recent attention has been focused on optical electron-transfer and charge-localization effects in formally symmetric mixed-valent ruthenium complexes of the general type [LSRuBRuLs]+s,where B is a bridging ligand such as pyrazine, dinitrogen, 4,4'-bipyridyl, or related ligands and Ls represents a 5-fold nitrogenous coordination environment around the ruthenium, for example, L = NH3.2-'Z In these species, the optical electron-transfer band, intervalence-transfer band, or metal-tometal charge-transfer band is an extremely important observable. In the Robin-Day 111 case, corresponding to complete delocalization, the energy of this intervalence-transfer band is approximately twice the bridge-assisted effective-tunneling integral between the two ruthenium sitesS4 In the Robin-Day 11, partly localized case, the intervalence-transfer energy is given in the harmonic approximation as12~13 E,, = Xi + A0 + AE A E f (1)
Within a dielectric continuum m ~ d e I , ' ~one * ' ~expects the behavior of the intervalence-transfer band (IVT) in the mixed-valent species to scale with l / D s - l / D m ,where D, and D , are the static and optical dielectric constants, respectively. Hupp and his collaborators have found, however, effects apparently due to specific asymmetric solvation in mixed solvent^.'^-'^ For example, Figure 1 shows the optical electron-transfer energies for the system with B = 4,4'-bipyridyl; note the very large effect of the addition of a small amount of dimethyl sulfoxide to pure acetonitrile, resulting in the 20%blue shift of the IVT frequen~y.'~ Hupp and Weydert convincingly attribute this effect to asymmetric solvation of the complex, with the dimethyl sulfoxide preferentially solvating the higher charge site. This selective solvation breaks the symmetry of the formal chemical structure, so that AE in eq 1 no longer
(1) Robin, M.; Day, P. Adv. Inorg. Chem. Radiochem. 1967, 10, 247. (2) Prog. Inorg. Chem. 1983, 30. (3) Richardson, D . E.; Taube, H. Coord. Chem. Rev. 1984, 60, 107. (4) Mikkelsen, K. V.; Ratner, M. A. Chem. Reb. 1987, 87, 113. (5) Creutz, C.; Taube, H. J. Am. Chem. SOC.1969, 91, 3988; Ibid. 1973, 95, 1086. (6) Brown, D. B., Ed. Mixed-Valence Compounds; Reidel: Dordrecht, The where Xi and A,, are, respectively, the inner-sphere and outer-sphere Netherlands, 1985. (solvent) reorganization energies,I4AE is the free energy difference (7) Hush, N . S. Chem. Phys. 1975, 10, 361. between the relaxed initial and final states, and AE'corresponds (8) Creutz, C. Prog. Inorg. Chem. 1983, 30, 1. (9) Wong, K. Y.; Schatz, R. N. Prog. Inorg. Chem. 1981, 28, 370. to any additional contributions from electronic effects such as (10) Furholz, U.; Burgi, H. B.; Wagner, F. E.; Stahler, A.; Ammeter, J. spin-orbit coupling. For symmetric mixed-valence species, such H.; Krausz, E.; Clark, R. J. H.; Stead, M. J.; Ludi, A. J. Am. Chem. SOC. as we will be concerned with here, AE = 0. An important aspect 1984, 106, 121. of the spectroscopy of these systems has been the assignment of (11) Mayoh, B.; Day, P. J. Chem. SOC.,Dalton Trans. 1974, 846. (12) Hupp, J. T.; Meyer, T. J. Inorg. Chem. 1987, 26, 2332. the complex as Robin-Day I1 or Robin-Day 111, based on the (13) Hupp, J. T.; Weydert, J. Inorg. Chem. 1987, 26, 2657. characteristics of the optical electron-transfer band.1-3,8v9JsJ6 (14) Newton, M. D.; Sutin, N. Annu. Rev. Phys. Chem. 1984, 35, 437. For highly charged metal complexes such as [LsRuBRuLS]+s, (15) Kober, E. M.; Goldsby, K. A.; Narayan, D. N.-S.; Meyer, T. J. J. Am. solvent and counterion effects are expected to be very important. Chem. SOC.1983, 105, 4303. (16) Richardson, D. E.; Taube, H. Coord. Chem. Rev. 1984, 60, 107. Recent work has shown that, as might be expected, counterion (17) Lowery, M. D.; Hammack, W. S.; Drickamer, H. G.; Hendrickson, interactions can lead to substantially different intervalence-transfer D.N. J . Am. Chem. SOC.1987, 109, 8019. bands and charge can be localized by ion-pairing One (18) Blackbourn, R. L.; Hupp, J. T. Chem. Phys. Lett. 1988, 150, 399. effect of solvent interactions, and the dominant one for many (19) Marcus, R. A. Annu. Rev. Phys. Chem. 1964, 15, 155; J. Chem. Phys. 1956, 24, 960, 979; Ibid. 1965, 43, 679. considerations, can be described solely on the basis of the dielectric (20) Hush, N. S. Electrochim. Acta 1968, 13, 1005; Trans. Faraday SOC. continuum model, as pioneered by Marcus19 and by Hush.20 1961, 57, 557; Prog. Inorg. Chem. 1967, 8, 391. Recent work has focused very extensively on solvent interactions, (21) Van der Zwan, G.; Hynes, J. T. J . Chem. Phys. 1982, 76, 2993. and such effects as solvent relaxation, solvent reorganization (22) Zusman, L. D. Chem. Phys. 1980, 49, 295. (23) Sumi, H.; Marcus, R. A. J. Chem. Phys. 1986,84,4894. Nadler, W.; dynamic^,^'-^^ the molecular nature of the solvent,2s change of Marcus, R. A. J. Chem. Phys. 1987, 86, 3906. electronic wave function due to solvent interactions,26and others (24) Wolynes, P. G. J. Chem. Phys. 1987, 86, 1957. have been extensively discussed theoretically and in some cases (25) Calef, D.; Wolynes, P. G. J. Phys. Chem. 1983, 87, 3387; J. Chem. mapped e~perimentally.~' Phys. 1983, 78, 470. (26) Mikkelsen, K. V.; Ratner, M. A. Int. J . Quantum Chem. 1988,522, Electron transfer or optical intervalence transfer in mixed solvents has very recently been studied in Hupp's g r o ~ p . ' ~ , ~ 707; ~ J. Phys. Chem. 1989, 93, 1759. (27) Gennet, F.; Milner, D.; Weaver, M. J. J. Phys. Chem. 1985,89, 2287. (28) Blackbourn, R. L.; Hupp, J. T. J. Phys. Chem. 1988, 92, 2817. McGuire, M.; McLendon, G. B. J . Phys. Chem. 1986, 90, 2449. Kosower, 'Permanent address: Department of Chemistry, Technion, Israel Institute E. M. J. Am. Chem. SOC.1985, 107, 1114. of Technology, 32000 Haifa, Israel.
+
0022-3654/89/2093-5065$01.50/0
0 1989 American Chemical Society
,
,
Katriel and Ratner
The Journal of Physical Chemistry, Vol. 93, No. 13, 1989
’1
W l e fraction CH3CN
ip
0;”
Op
Of
0.;
0
0.2
0.4
0.6
0.8
8
of two molecular species, called s and t. Within a given volume, we denote by So and To the total number of s and t molecules. (In the experiments of Hupp’s g r o ~ p , ’ ~for , ’ ~example, s might be DMSO and t acetonitrile). Within this volume, there is one binuclear complex. We now denote by A and B the two ligand and metal sites within the binuclear complex. Then, we take S A and S B .respectively, as the number of s molecules attached to sites A and B, respectively. Likewise, T Aand TBare the number of t-type molecules attached at sites A and B. The number of free (outside the first coordination sphere) s-type and t-type molecules are denoted s and t , respectively. Particle conservation then requires
+ SB + s = SO TA + TB + t = To
SA
.o
1
mole fraction DMSO
Figure 1. Optical electron-transfer energies (IVT band maxima) for [(NH3)SRu]2(4,4’-bipyridyl)]s+ in a mixed solvent of DMSO and ace-
tonitrile. Note the blue shift, due to asymmetric solution, at small DMSO mole fraction. From Hupp and Weydert.I4 vanishes. This is responsible for the blue shift with the addition of a small amount of DMSO; as more DMSO is added, the IVT settles down to its value of roughly 8000 cm-’ in pure DMSO. Hupp and Weydert attribute this selective solvation to the formation of hydrogen bonds “between the acidic hydrogens of the coordinated ammonia ligands and the basic functionalities of discrete solvent molecules”. Such hydrogen bonding would be stronger at a more highly charged ruthenium site and would account for the selective solvation. Selective solvation, such as that discussed by Hupp and coworkers, clearly involves a partial ordering of the solvent around the metal complex. It thus corresponds to a reduction in entropy, and one might suspect that, at higher temperatures, such an entropy reduction would be more difficult to attain. Although the temperature dependence of IVT absorption in mixed-solvent, mixed-valent systems has not yet been discussed, nevertheless, it is of interest to attempt a theoretical description of this process. The present paper addresses the issue of selective solvation in bicentric mixed-valent species. In particular, we study the dependence of the first coordination shell on temperature and solvent composition. The calculation is based on equilibrium statistical mechanics. The energetics of such interactions as solvent-ligand coordination, solvent-solvent intermolecular attractions, and intermolecular interactions among coordinated solvent species are all taken as parameters. Only dilute solutions are considered, so that solvation of each binuclear complex is independent of the presence of other complexes. Counterions are ignored. The treatment then proceeds by constructing a free energy function dependent upon the constituent parameters of the pure solvent and the first coordination shell and minimizes this free energy with respect to composition variables. The resulting equations yield relationships among the numbers of coordinated and free species of each of the two components of the mixed solvent. These equations can be solved analytically in certain limits and numerically over the whole range of the relevant parameters. They yield interesting predictions concerning the existence of a selective solvation phase; in particular, they predict that, at high temperatures, there should be no selective solvation. Below a certain critical temperature range, an unsymmetrical, selectively solvated species (one with AE # 0) will exist, just as has been invoked by Hupp’s group; this will generally result (following eq 1) in a blue shift of the IVT band. The model is defined, and the free energy expression is constructed, in section 11. In section 111, the equilibrium solution, corresponding to minimum free energy, is constructed and some limits are discussed. Section IV presents phase diagrams with variable composition and temperature. Some comments are presented in section V. 11. The Model: The Free Energy of Solvated Complexes
We take as our model an isolated bridged dinuclear metal complex, in a mixed solvent. We assume that the solvent consists
(2a) (2b)
We define Coas number of sites in the first solvent sphere around the ligated rutheniums on each site, so that
TA
+ SA = TB + SB = CO
(3a)
The effective charges on the two sites of the binuclear complex are denoted NA and NB, and then, charge conservation requires
NA
+ NB = 2No
(3b)
with 2N0 defining the total charge of the binuclear complex (2No = 5 in the ruthenium case). We have now defined the general physical situation, with the only real assumption being that the number of solvent molecules in the first solvent cage around the metal complex, 2C0, is a fixed number. There are eight variables (SA, SB, TA,TB, S, t , NA, NB) and five relationships (eq 2 and 3) among them. This leaves three independent variables whose values are to be determined. We determine the equilibrium condition, and therefore the solvation numbers and the values of the eight parameters, by minimizing the total free energy. The free energy functional can be written as a F = -[(NA 2 Y
-(TA’ 2
P + ( N B- No)2] + ?(SA’ + SB’) +
+ TB2)+ 6(SATA + SBTB) +
Yo + NBTB)+ Po -s’ + -t’ + 2 2 6ost + kflSA In SA + S B In S B + TA In TA + TB In TB + s In s + t In t - ( s + t ) In ( s + t ) ) (4a)
+
€(NASA NBSB) + {(NATA
The Greek letters correspond to energy values. The first term in (4a) describes the delocalization in the isolated metal complex: a is a positive quantity describing the energy cost involved in localizing electrons on one site or the other in the isolated binuclear species. (The more usual r e p r e ~ e n t a t i o nin ~ ~terms ~ ~ ’ ~of tunneling between sites A and B gives an identical term, if the energy origin is shifted by the constant aNo and if the constant a = 2I&I is associated with PH the Huckel tunneling integral between A and B. This identity follows easily either from the second-quantized form of the tunneling Hamiltonian or from examining the 2 X 2 matrix representations of the two operators.) Notice that this term arises from the raising of the kinetic energy when the electron is localized and occurs in the free (unsolvated) molecule. The binding energies of the s and t species to sites A and B depend on the local electron numbers NA and N B , and are, respectively denoted e and {. They enter into the fifth and sixth terms of the free energy; these energies in general are negative. The second, third, and fourth terms describe interactions within the first solvation shell. Thus, two s-type solvent species within the first solvent sphere about site A would interact with an energy p / 2 ; p, 7,and 6 are generally negative. Likewise, the free solvent species will interact, and Po, yo,and 6o describe the relative attractive interactions of s with s, t with t, and s with t: Once again, all of these parameters should be negative and correspond to attractive interactions among the bulk molecules. If PO = yo = fro, Raoult’s law should hold. The most important energy terms
The Journal of Physical Chemistry, Vol. 93, No. 13, 1989 5067
Mixed-Valency Species in Mixed Solvents
in (4a) are expected to be those proportional to a, t, and {. The logarithmic, entropy term in eq 4a describes the partial ordering of s-type and t-type solvent species around sites A and B. It is derived from taking the entire distribution of s-type and t-type species, which is s t , and distributing them in the three possible situations (on A, on B, and in bulk). Application of Stirling's formula then yields the logarithmic term in (4a). The situation now depends on fourteen parameters, of which nine describe the energetics of solvation and of particle-particle interaction (a,P, y, 6, t, {, Po, yo, a0). One is a statistical variable (the temperature T ) and the other four (No,Co, s, t ) are stoichiometric variables. Analysis of any given situation requires values to be taken for all these parameters; we will proceed with the analysis in general and find that only certain lumped combinations of the energy parameters in fact turn out to be relevant. The entropy will be maximized and the mixed-valent energy term proportional to a will be minimized for identical solvation on the two sites. Selective solvation will be due to interactions, in particular those proportional to t and {.
+
111. Equilibrium Condition and Limiting Cases The equilibrium state and the extent of selective solvation are found by minimizing the total free energy functional. There are three independent variables among the eight concentration variables SA,SB,TA, TB, s, t , NA, and NB. We choose them to be NA, SA,and SBand write the minimization conditions on the free energy as
dF/dNA = O
(4b)
Solution of these equations determines the equilibrium state. The simplest way to solve these equations it to rewrite the free energy functional in terms of the three independent variables. As a function of NA, the free energy F is a parabola with a minimum (a > 0) at
Substituting eq 5 in eq 4a, we obtain for the free energy as a function of SA and SBthe expression
F = U/2(S,4*
+ S B ~+)bsASB + C(SA+ SB)+ entropy term of (4a) (4e)
up to an irrelevant additive constant. We have, for notational simplification, defined the effective energy parameters a, b, and c by
+ (Po + y o - ZO)- ( t - n 2 / 2 a b E (00 + yo - 260) + ( t - { ) 2 / 2 a c E ( 6 - y)Co + - {)No + (60 - P0)So + a
= (P + y - 26)
S
G
m c .o
1 2co
Q
4i SO
Figure 2. Range of values of symmetric solvation shell occupancy S for different values of the bulk concentration S,. The vertical arrows indicate the variation of p from T = 0 to T- m. Conditions: (a) c > 0 (lower triangle) and c < -(a + b)Co(upper triangle); (b) -(a + b)Co < c i0. S increases for So > UT= -cUo/(a b)Coand decreases for So < UT.
+
entropic factor were not present, eq 6 would be isomorphic to those of a two-sublattice ferromagnet in an external magnetic field, with the field intensity proportional to the concentration c.*'*~O Actually, the second entropy factor can be simplified: If we assume that the solution is dilute in the binuclear complex but in neither s nor t, we have So and To much, much greater than Co so that the second entropy factor in eq 6 can be written
To - 2co
+ S A + SB
SO - ( S A
-k
SB)
Equations 4c and 4d yield
To
= -so
(8)
the ratio of the overall concentrations. Thus, its contribution to the total free energy is simply to add the term kT In (To/&,),which is independent of the solvation variables SAand SB.It corresponds, in the language of the king model, to an additive temperaturedependent term in the effective external magnetic field.
IV. Phase Diagrams: Solvation Behavior at Arbitrary Temperatures For finite temperature, one must solve the coupled eq 7 to obtain the solvation behavior. We first consider the symmetric situation, and then examine conditions under which asymmetric solvation might be observed. A. Symmetrical (Delocalized) Solution. Equations 5 and 7 always have a symmetrical solution, for which S A = S B E S. For this solution, eq 5 reduces to N A = NB = No and eq 7a and 7b both reduce to (a
1
+ b)S + c + kT In
(C
(Yo - 6o)(To - 2Co) ( 6 )
uo
USZC,
Ud2
=0
(9a)
Assuming equality of the partial molar volumes, we write the total number of solvent molecules as
Uo = So + TO
(10)
Requiring the argument of the logarithm in eq 9 to be positive, we find that the solution is bracketed by max(0, Co- T 0 / 2 ) 4 S Imin(C,, S0/2)
(11)
and n=
With Uo,the total number of solvent molecules, taken as a constant, the range of values of S , as a function of So, is given in Figure 2. The values of S will always lie within this parallelogram. In the high-temperature limit, the argument of the logarithmic function equals 1, i.e.
The solutions to eq 7 determine the equilibrium condition. The solvent concentration, defined by So and To, appears in c and is present in the second entropic factor. If the second
(29) Smart, J. S . Effectiue Field Theories of Magnetism; Saunders: Philadelphia, 1966. (30) Chakravarty, A. S. Introduction to The Magnetic Properties of Solids; Wiley: New York, 1980.
5068
The Journal of Physical Chemistry, Vol. 93, No. 13, I989
Katriel and Ratner
I This equality means that the distribution of the two bound ligand species is the same as in the bulk. To obtain the low-temperature limit, we note that (for mutually miscible solvents) u b > 0. We shall first consider the case in which Co is less than both S0/2and T0/2. Writing eq 9 in the form
+
s
To-2Co+ 2 s
co-s
so- 2s
-
we find the following three cases at T 0: (i) c > 0, in which case S = 0; (ii) -(a b)Co < c < 0, in which case S = - c / ( a b); (iii) c < -(a b)Co,in which case S = Co. Noting that the temperature T is a single-valued function of S, so that S is a monotonic function of T , we find that different behaviors of S as a function of Twill arise, depending on the value of c. The latter reflects the relative energetics of solvent-molecule interactions and of binding within the first solvent sphere. For c < -(a b)Co,S decreases from its low-temperature value Co to its high-temperature value, given by eq 12. Inspecting the parallelogram in Figure 2, we note that for So< 2C0 the above statement has to be modified, since the low-temperature value of S is S0/2. Thus, the unified statement should be that S decreases from its low-temperature value, min(Co, S0/2) to its highest temperature values SoCo/Uo. The decrease in S with increasing temperature reflects the fact that, in this case, s prefers to be closely coordinated to the ligated complex and will equally distribute, increasing its relative concentration to the bulk of solution, only when driven entropically at high temperatures. For c > 0, S increases from its low-temperature value. This corresponds to the situation of weak solvation, in which energetically the s-type species prefer not to bind within the first solvation shell. For -(a + b)Co < -c < 0, if So < - c / a b(Uo/Co),S will decrease from min(So/2, - c / ( a b ) ) to SoCo/Uoas the temperature increases from zero to infinity. If, however, So > -c/u + b(Uo/Co).S will increase from the low-temperature value max(-c/(a b ) , Co - T 0 / 2 ) to SoCo/Uoas the temperature increases from zero to infinity. B. Unsymmetrical (Localized) Solution. Thus, there exists over the entire temperature range a solution corresponding to symmetric solvation, and the actual numbers of s and t molecules present in the first solvation shell and then the bulk can be obtained by solving eq 9a. The more interesting situation, corresponding to that discovered at room temperature by Hupp and collaborators, is the unsymmetrical solution. In this case, by analogy with the two-sublattice ferromagnet, and in consonance with the thermodynamic expectation that homogeneity must be favored by the entropic term at high temperatures, we find that there exists a critical temperature Tc, above which the only solution is the symmetrical one, SA = SB.Below Tc, a stable, localized (asymmetric) solvation situation occurs. Figure 3 presents the behavior of SA and SBin a composition/ temperature phase diagram (note that if rotated by 90%, this resembles the phase diagram of the binary solution, with T , an upper consolute temperature). For temperatures below Tc, for any overall solvent concentration So and To, the symmetric solvation situation is unstable with respect to asymmetric, preferential solvation (just as the homogeneous solution becomes unstable with respect to two separated liquid phases, below an upper consolute temperature). This can be checked by observing that above Tc the symmetrical solution corresponds to the (only) minimum of the free energy. Below Tc, the symmetrical solution is a saddle point, the true minimum being the asymmetrical one. We can obtain an expression for the critical temperature Tc by requiring that
+
+
+
+
+
T
c4 IO
+
+
S
5 T
Figure 3. Temperature dependence of the composition of the solvation layer for the symmetrical and asymmetrical complexes. To the left of the curved line, asymmetrical solution occurs. Conditions: (a) c = 1 > 0; (b) -(a b)Co < c = -1 < 0, So = 50, UT = 5 ; (c) -(a + b)Co < c = -15 < 0, So = 50, UT = 75; (d) c = -25 < -(a + b)Co = 20.
+
In addition, at Tc, the solution must become the symmetric one, SA = SB, so that (9) must also be satisfied. The solution to (13)
is
The Journal of Physical Chemistry, Vol. 93, No. 13, 1989 5069
Mixed-Valency Species in Mixed Solvents
Simultaneous solution of (14), (15), and (9) yields both the critical temperature Tc and the composition value SCat the critical point. The extent of asymmetry in the solvation increases monotonically as the temperature drops. Figure 4 presents the dependence of the critical temperature, and the critical first-shell coordination Sc, on overall solvent composition. For pure t , TC vanishes as does SC (since there are no s-type molecules, there can be no solvation by s-type molecules). For pure s solvent, the critical temperature again vanishes and Sc = SA = SB= Co. Not surprisingly, the value of SCincreases monotonically with the mole fraction of s in the solvent. Generally, Tc will increase as the relative mole fractions approach 0.5. By solving eq 7 for SAand SBand any given temperature, we can calculate the relative solvation difference AS = SA- SBand its dependence on either solvent composition or temperature. Figure 5 gives an example for a particular choice of energetic and stoichiometric parameters. Notice that, not surprisingly, the extent of selective solvation decreases with an increase in temperature. As T Tc, these roughly parabolic curves will collapse, as AS = SA- SBgoes to zero at the critical temperature. At very low temperatures, the stable solution corresponding to asymmetric solvation becomes exponential
so Figure 4. Dependence of the critical temperature and of the critical value of S on the solvent composition.
lot
-
Co - S A
2
/J 3
SB2 cy‘e-p’lkT 0
=
co-To - co so - co
/J‘
cy’
-(c
=c
=C o -
+d o )
As 5
I I
+ bCo
so - co To - co
with the “chemical potentials” II and 1’ including driving force c and ihe relative energy terms a and b.
V. Comments The distinction between localized (Robin-Day I or 11) and delocalized (Robin-Day 111) behavior for mixed-valent species has been discussed extensively. Goodenough long ago3’ proposed the criterion involving a critical effective electron overlap integral between A and B sites, and a substantial amount of work on electronic structure has f o l l ~ w e d . * * ~ J The ~ , ~ importance ~-~~ of solvent interactions in localizing an otherwise delocalized, symmetric species is inherent in the early discussions of MarcusI9 and of involving solvent reorganization energies, and indeed, the solvent dependence of optical electron transfer can, in many situations, be reasonably understood by simple dielectric continuum arguments. More recently, the molecular nature of the solvent shell has been stressed and specific molecular interactions have been shown to lead to asymmetry and valence trapping. Work on localization and induced energy differences AE (eq 1) has been reported. For example, the C r e u t z - T a ~ b e * *ion, ~~~*~ with a pyrazine bridge, has been shown to adopt different crystal structures in the solid, depending on the nature of the counterion.” These differing structures are due to ion-pair interactions with the formally 5+ charged binuclear complex. Hendrickson and his collaborators have shown, in an elegant series of studies,37that solvation effects in mixed-valent solids can change rates of in(31) Goodenough, J. J . Appl. Phys. 1966, 37, 1415. (32) Ondrechen, M. J.; Ratner, M. A,; Ellis, D. E. Chem. Phys. Left. 1984, 109, 50.
(33) Larsson, S. J . Chem. SOC.,Faraday Trans. 2 1983, 79, 1375; Chem. Phys. Left. 1982, 90, 136; Inf. J . Quantum Chem., in press. (34) Zhang, L. T.; KO,J.; Ondrechen, M. J. J . Am. Chem. SOC.1987,109, 1666. (35) Newton, M. D. A C S S y m p . Ser. 1982, No. 189, 254. (36) Beattie, J. K.; Hush, N. C.; Taylor, P. R.; Ralston, C. L.; White, A. H. J . Chem. Soc., Dalton Tram. 1977, 1125. Hush, N. S.; Edgar, A,; Beattie, J. K. Chem. Phys. Lett. 1980, 69, 128. (37) For example: Woehler, S. E.; Wittebort, R. J.; Oh, S. M.; Hendrickson, D. N.; Inniss, D.; Strouse, C. E. J . Am. Chem. Soc. 1986, 108,2938.
~
100
so Figure 5. Dependence of the asymmetry parameter AS = SA- SBon the solvent composition.
tramolecular electron transfer and substantially alter the spectra. Most recently, it has been shown that ion-pair interactions in solution can lead to asymmetries and valence trappings.I7J8 In the case of mixed solvents, Blackbourn, Hupp, and Weydert’3,28 have shown that asymmetric, selective solvation by one of the two components in the mixed solvent can result in valence trapping, energy asymmetry with AE # 0, and a substantial blue shift in the IVT band. They also argue that most “solvent reorganization” occurs in the first few solvent shells. (This is supported by recent simulation studies by Mataga and collaborator^,^^ demonstrating that substantial reorganization occurs in the first solvent shell.) Their measurements were made at room temperature. The phenomenon of selectively solvation has not, to our knowledge, been discussed extensively in the literature and was the motivation for the present theoretical model. We have assumed a very dilute solution of binuclear complexes and have differentiated two regions in which solvent molecules may be found: a first solvation shell and a bulk range beyond that first solvation shell. This simplified picture is based both on energetic considerations and on the observation by Blackbourn and H ~ p p that * ~ it is possible to vary the primary solvation layer of the dimer largely independently of the predominant bulk solvent composition. Assuming, then, that solvent molecules are present either in the bulk or in the first-coordination sphere around the metal complex, we have constructed a free energy functional and, by minimization of the free energy, deduced the phase diagram as a function of temperature and solvent composition. While at all temperatures there exists a symmetrical solution, with charge delocalization and identical solvation on the two sides of the binuclear complex, below a given critical temperature Tc, (38) Hatano, Y.;Saito, M.; Kakitani, T.; Mataga, N . J . Phys. Chem. 1988, 92, 1008. In this paper, Monte-Carlo simulation is used to show that a very thin dipolar solvent shell, of width 1 . 1 A, is very strongly polarized by a central charge and that, for small-charge z, this polarization drops off quickly with the radius of the shell.
5070
J . Phys. Chem. 1989, 93, 5070-5078
the symmetric solvation is unstable, with respect to an asymmetric solvation. This asymmetric solvation results in a blue shifting of the IVT band, as AE in eq 1 no longer vanishes. Such blue shifting was the primary observation of Hupp and Weydert that motivated the present work. The model, although simplified, includes the dominant interactions one expects to see. In particular, the nonideality of the solvent mixture can be well described by the Po, yo, and bo parameters of the free energy (eq 4a) and interactions within the first solvent shell can differ from interactions in the bulk. There is no continuous dielectric term in the free energy expression of eq 4a. Indeed, if the specific terms of (4a) proportional to P,?, 6, t, and {were replaced by their continuous dielectric equivalents, the effect of selective solvation would be absent! Thus, selective solvation, like solvent-separated ion pairing, is apparently a result of the molecularity of the solvent. Terms such as Born solvation, which can be well described in a continuous dielectric picture, are absent in eq 4a but would not alter our conclusions. We expect the model to hold over a reasonable range of reasonable solutions. Stirling’s approximation is made in treating the entropic part of the free energy, but we do not expect substantial numerical errors to arise from this, even at relatively small values of the coordination number C,.
To compare the results of this model with experiment, one really requires values of the energetic parameters occurring in eq 4a as well as for the stoichiometric variables Co, No, So, and To. Stoichiometric variables are fairly clear for any given complex and solvent mixture. The energetic values can be estimated from the thermodynamics of solvation and from the Henry’s law and Raoult’s law constants of the solvent mixture. For comparison with the IVT band maximum, one requries the functional dependence of AE with solvation. This can be obtained for the two partners in the binuclear complex independently, as Hupp and his collaborators have ~ h o w n . ’ ~Thus, ~ ~ * from a calculated value one should be able to describe, with fair accuracy, of SAand SB, the band position in mixed solvents. More direct structural probes, such as NMR, should also be interpretable in terms of the calculated values of SA.
Acknowledgment. We are grateful to Professor J. T. Hupp and his group for inspiring this research and for a series of useful discussions. We thank the Chemistry Division of the National Science Foundation for support of this research. Further support was provided by the Technion Vice President for Research Fund-The E. and J. Bishop Research Fund. We thank the reviewer for several incisive and probing remarks.
Experimental and ab Initio Vibrational Spectra of 1,2-Dibromoethane, meso-1,2-Dideuterio-l,2-dibromoethane, and Chiral 1,2-Dideuterio-l,2-dibromoethane P. K. Bose, D. 0. Henderson, C. S. Ewig,* and P. L. Polavarapu* Department of Chemistry, Vanderbilt University, Nashville, Tennessee 37235 (Received: September 30, 1988; I n Final Form: January 20, 1989)
The vibrational spectra of 1,2-dibromoethane and of the m m and chiral forms of 1,2-dideuterio-l,2-dibromoethaneare presented. The experimental spectra were obtained for neat liquid samples as well as for the molecules isolated in low-temperaturematrices. The ab initio calculations of the vibrational frequencies and intensities were carried out with different basis sets.
introduction Vibrational spectra of 1,2-dibromoethane have been studied since the late 1940s from two different viewpoints. In the first, the observed spectral bands were analyzed as to the nature of the symmetry and vibrational assignments. Second, the observed band intensities and their temperature dependence were used to deduce the population ratio of the rotational isomers. The most recent vibrational analysis that we are aware of is that of Tanabe et al.,’ where the experimental infrared and Raman spectral data for liquid samples were presented and some revisions to the previous assignments suggested by Shimanowhi* were proposed. It is now established3” that 1,2-dibromoethane exists in both trans and gauche forms with their relative populations being different in different phases. In the vapor and solid phases, the trans isomer is considered to be predominant. In the liquid phase, on the other hand, both trans and gauche isomers are considered to be present in significant amounts. One of the valuable aspects of this molecule is that the trans isomer has C2, symmetry while the gauche isomer has Cz symmetry. Thus a comparison of infrared and Raman spectra of the solid sample reveals the symmetry nature of bands, since the mutual exclusion principle applies ( I ) Tanabe, K.; Hiraishi, J.; Tamura, T. J . Mol. Srrucr. 1976, 33, 19. (2) Shimanouchi, T. Tables of Molecular Vibrational Frequencies; Na-
tional Bureau of Standards: Washington, DC, 1967; Part I . (3) Fernholt, L.; Kveseth, K. Acfa Chem. Scand. 1978, ,432, 6 3 . (4) Kakimoto, M.; Fujiyama, T. Bull. Chem. SOC.Jpn. 1975, 48, 3003. ( 5 ) Hiraishi, J.; Shinoda, T. Bull. Chem. SOC.Jpn. 1975, 48, 2385. (6) Abraham, R. J.; Gatti, G. J Chem. SOC.B 1969, 961.
0022-3654/89/2093-5070$01.50/0
for the trans isomer. Similarly comparing the vibrational spectra of the sample in solid and liquid phases reveals the identities of the bands due to the gauche isomer. Since the trans isomer is predominant in the vapor phase, and the molecules isolated in inert low-temperature matrices are generally believed to have the same conformation as that in the vapor phase, vibrational spectra of matrix-isolated 1,2-dibromoethane are expected to be of significant value in determining the vibrational properties of the trans isomer. These spectra have not yet been reported. In the previous studies,’ the vibrational assignments of the observed bands were offered using symmetry arguments, comparison among similar molecules, and isotopic substitution studies. Theoretical vibrational analyses have also been carried out with the force constants that were either assumed or transferred from similar molecules. Recent emergence of the efficient ab initio programs7,*and development of contracted basis sets for the Br atom9 make it possible now to undertake an a b initio vibrational analysis of 1,2-dibromoethane and evaluate the predictive capa( 7 ) Frisch, M. J.; Binkley, J. S.; Schlegel, H. B.; Raghavachari, K.; Melius, C. F.; Martin, R. L.; Stewart, J. J. P.; Bobrowicz, F. W.; Rohlfing, C. M.: Kahn, L. R.;Defrees, D. J.; Seeger, R.; Whiteside, R. A,; Fox, D. J.; Fleuder, E. M.; Pople, J . A. Gaussian 86; Carnegie-Mellon Quantum Chemistry Publishing Unit: Pittsburgh, PA, 1984. (8) Amos, R. D.; Rice, J. E. CADPAC: The Cambridge Analyrical Derioariues Package; Issue 4.0, Cambridge, 1987. (9) del Conde, G.; Bagus, P. S . ; Bauschlicher, C. W., Jr. Theor. Chim. Acta 1977, 45, 12 1.
0 1989 American Chemical Society
