Nita A. Lewis Stanford Unlverslty Stanford, CA 94305
II
Potential Energy Diagrams A conceptual tool in the study of electron transfer
Ever since Faradav oerformed his historic emeriments in 1834 showing that some solutions could condu& electricity, scientists have been fascinated by the properties of electrons. Not the least of this fascination, in more recent times, arises from the realization that electrons are "light ~articles"and hence are not necessarily subject to cla'sicai laws of motion. Tiny as an electron is, i t may have an enormous effect on the properties of metal ions. For example, Cr11(OH2)62+,which has seven d electrons, is an extremely "labile" ion and, in water, i t exchanges more than half its Hz0 ligands with the bulk solvent in sec ( I ) . However, Cr111(H20)63+,which has only six d electrons, is extremely stable or "inert." It exchanees half of its lieands with the hulk aaueous solvent an ---~ ~~-~ average of every eiiht days (2). Inorganic chemists have learned to take advantage of these differences in reactivity to probe the detailed nature of electron transfer reactions.' Althowh the theoretical desmintion of a svstem u n d e ~ o i w electron Gansfer may be difficuit for the average student t i com~rehend.it is not necessarv to understand the detailed mathemati&involved in the treatment to gain an appreciation of the processes involved. A rich conceptual tool exists in the form df the potential energy diagram,the use of which was early shown by the work of R. A. Marcus. However, before examining these diagrams, we have to understand some of the factors which are important in electron transfer reactions. Most of the soecial nro~ertiesof electron transfer reactions and other charge transfe; processes are a consequence of the Franck-Condon principle (3) which states that the energies of the orbitals participating in the reaction must be equalized before the electron mav transfer. This equalization is necessary since nuclear mdtion is much slower than electronic motion. Nuclear motion occurs on a time scale of 10-l3 see whereas electron motion occurs in about lW1'sec. Since the electron is moving approximately 100 times as fast as any nuclei, it appears m the electron that the nuclei are "frozen" into position during the time that the electron travels from the or1)itnl of one nucleus to that of another. The energies of the orbitals participating in the reaction can be changed by movement of the nuclei alone their internuclear bonds, but since this is a slow process,-it must occur before electron transfer. The first class of electron transfer reactions to lend itself to theoretical description was the outer-sphere tvpe.' This reaction is one of the simplest kinds of chemicai ieactions which can be envisaged since no bonds are made or broken during the course of the reaction. The symmetrical, homonuclear one-electron transfer between Fe(OH&2+ and Fe(OH2)63+ is an example of such a reaction2 The relative hond lengths and electronic configurations of these molecules are shown in Fieure 1. The electronlransfer may be regarded as being from one t l , orbital to another bv direct orbital interaction. This t&fer cannot occur with the molecules having their normal dimensions, however, since the energy of Fe(OH&" and Fe(OH2)e2+ orbitals will be quite different (recall the Franck-Condon principle). Fortunately, molecules do not have static dimensions. They are constantly stretching and compressing along their internuclear bonds. This means that, although most of the Fe(OH2)P molecules in solution wiU have normal Fe(II1) bond lengths and most Fe(OHz)6ZCmolecules will have Fe(I1) bond lengths a t any given temperature, some
-
~~
~
478 1 Journal of Chemical Education
-+
4i i
4-
'2.
+ Wd+ + .,.nra
f f
+
44-
4-
A
,rn.l.
Figure 1. Relativeband lmgms and electrmic anfigurations of Fe(OH&*+and Fe(Ct12)s3+.The activated cwrplex shw Fe--OH2 bondr. all equal and at some distance intermediate betweenUwse of Fe (1I)and Fe (Ill). The nuclesrdimemim of me reactants in me activated complex will be closer to mat of Fe(OHz1s1+ than Fe(OHl)e2+bemuse of the stronger forceconstants of the bonds in me former species. of the molecules will have considerahly longer than average hond lengths and others will have honds that are much shorter than average. When an FetOH1)6'+ which has bond lengths which areat some intermediate position between an Fe(1l) and an Fe(11I) encounters an Fe(OH?)&+with the same dimensions: the energies of their orbit& will be the same and electron transfer may occur. Because of the difference in force constants of the Fe(I1) and Fe(II1) bonds, the Fe(I1)-OH2 bonds will shorten to a greater extent in reaching this equal energy condition than the F~(III)-OHZ bonds will stretch. This close encounter combination of Fe(OH2)63+ and Fe(OH2)G2+having a donor and acceptor orbital of equal energy is called the activated complex (Fig. I), and much of the free energy of activation required for electron transfer is expended in reaching this state. The probability of two molecules with exactly these di1 For a discussion of
inner- and outer-sphere reactions, see Taube,
H., "Electron Transfer Reactions of Complex Ions in Solution," Academic Presa, New York, 1970, Chapter 11.
2 Symmetrical homonuclear exchange xeactions are measured by tracer studies using radioactive isotopes of the metal. Far a further discussion of these studies, see Sykes, A. G., Kinetics of Inorganic Reactions, Pergamon Press, 1966, Chapter 2. Tonfigurations of reactants in which all hond lengths are not the same can also result in the equal energy condition found in the activated homonuclear electron exchange -..-eomolex for the svmmetrical ~,. reactim. However, these configurationswill he tbund much less frequently than the totally symmctrienl arrangement shown in Figure I and we will ignore the unxymmetriral posxthilrtres.
~~~~.~ ~~
~
~
~
I NUCLEAR CONFIOURITION
Figure 2. (a) A general potential energy diagram fwms heteronuclear reaction red, +ox* 4 3 ox, + red2 R
P
(b) A general potenHal e n e r g y t i i fathe symnebicalhomanucb m a t i i redl 0x2 a 0x3 redr where red, has the same ligands as ox, and red2
+
+
simiwarly resembles ox2.
mensions colliding with one another is not as low as one might think.4 In even a fairly dilute solution, there will he many millions of collisions per second and a large number of these will be in the coiifiguration of the activated complex. When the Fe(OHz)@ expands (or Fe(OH2)62+contracts) along its intemuclear axes, the effective sue of the ion changes so that the solvent molecules surrounding the ion must reorganize themselves. The intemuclear bonds of the solvent molecules and the reacting species (in this case Fe(OH2)63+ and Fe(OH2)62+) can be envisaged as tiny springs of varying strengths. In a highly polarizable solvent such as water, these "s~rines" . " form a hiehlv connected network. Anv movement of the reacting species necessarily involves a collective reoreanization of the entire svstem (solvent and reactants). Thus. the influence of solvent on an electron transfer process can he enormous. The total change in the free energy of activation is, then (4)
-
d
-
-
AG* Act* + AGi* + AGO* (1) where Act* is the free energy required to bring the reactants toaether to their positions in the activated complex, AGi* (the in&sphere reu&anivation energy) is the free inergy required to equalize the energy ul'the donor and acceptur orbitals by a suitable fluctuation of the internuclear honds of the inner coordination shell and AGO* (the outer-sphere reorganization energy) is the free energy required t o reorganize the solvent molecules surrounding the reactants. Potential Energy Diagrams in Outer-Sphere Reactions The first to use potential energy diagrams to describe uuter-sphere electron transfer reactions was H. A. hlarcus (5). A general diagram is shown in Figure 2(a). I t consists of two overlabping potential energy wells, one for the reactants (ER),the other for the products (Ep). Each corresponds to the lowest electronic state (the ground state) of each chemical pair. The "nuclear configuration" is defined as the totality of all nuclear coordinates in the system. If a molecule has N atoms, there are 3 N nuclear coordinates (i.e., x , y: and z coordinates are needed for each nucleus to fix its oos~tionin soace). In statistical mechanical oarlance. there &e 3 N degrees of freedom. However, for e a c h k o l e c u ~there , are 3 translational and 3 rotational5 degrees of freedom. The remaining 3N-6 degrees of freedom are vibrational. The translational and rotational degrees of freedom leave all internuclear distances unchanged. - . so thev leave the potential energy unchanged.= Even though the translational and rotational deerees of freedom do not affect the potential energy, they cannot be separated mathematically from the problem without a knowledge of the numher of molecules interacting. This is impossible in a solvent svstem (which we viewed earlier as a network of "springs"). Therefore, Marcus (5) defines the ~
nuclear configuration as the totality of translational, rotational, and vibrational degrees of freedom of all the molecules of all the reacting species and of the molecules in the surrounding medium (a t o b l of 3 N degrees of freedom where N is the n~rmherof nuclei present). The potential energy diagrams shown here represent a "slice" through the 3 N dimensional surface along the lower region where the electron transfer reaction is uccurring. Where the K:'Rand E p surfaces "intersect" in Figure 2(a) there is formed a surfaces (represented by a pointon this diagram) of one less configurat&nal dimension (or one less degree of freedom). Our discussion of potential energy surfaces has so far been restricted to heteronuclear reactions such as Tlf + AgZ+ 9 TI2+ Ag+ (2) (red,) (0x2) (0x1) (re&) or unsymmetrical homonuclear reactions such as Fez++ Fe(bipyhc3 9 Fe3+ + Fe(bipy)~+~ (3) (red,) (oxz) (0x1) (red4 both of which are descriljed by the potential energy diagram in Figure %a). In these cases, red, has the same ligands as ox), and 0x2 similarly resembles redz. The potential energy diagram for a symmetrical homonuclear reaction such as the following reaction Fe(OH2)c2++ *Fe(OHds3+9 Fe(OHM+ + *Fe(OHz)s2+ (4) red, 0x2 0x1 red2 discussed earlier is somewhat simpler since the potential en) prod& (EP) are ergy surfaces for the reactants ( k ~and identical and have equal enerw minima ( F i r e 2(h)). Fur any symmetrical reaction, redl &d 0x1 have the same ligands as red2 and 0x2. Recall. in our discussion of the svmmetrical homonuclear reaction between Fe(OH&Z+ and ; F ~ ( o H Z ) ~ ~we + , said that the enerzies of the electron donatine and acceotine orbitals had to he-equal. This equalization w&accompli~hed6y having the Fe(1I) compress its internuclear honds and Fe(II1) stretch its internuclear honds until both Fe(OH2)G2+and *Fe(OHz)e3+ had equal internuclear bond distances. This equality applies only to the symmetrical homonuclear reaction (Fig. 2(h)).
+
The collision frequency between the two species A and H in aolution, ( % ~ n i~) ,about 1tS1M-'sec-' where the rate uf reaction is
5 Fbr a linear molecule, there are only two rotational degrees of freedom. Since we will be concerned only with complex molecules, which are always non-linear, we shall ignore the linear case. "utin, N., in Reference 2 uses a slightly different formalism in discussing the diagram in F i e 2(a).He plots potential energy versus atomic configuration since there will be the same number of atoms as nuclei in the system. However, he defines only N independent variables (which are equal to 3 N coordinates in the Marcus formulation) to get an N-dimensional surface in N + 1atomic configurational space (as apposed to 3 N + 1nuclear configurational space in the Marcus description). The N 1space arises from the combination of an N-dimensional surface plus potential energy. The potential energy of the products is represented by an N-dimensional surface (En) in (N + 1) space and the potential energy of the products is represented by a similar surface ( E d (see Figure 2(a) and substitute "atomic configuration" for "nuclear configuration").Again, the two surfaces intersect at S to give a surface of one less dimension (an (N - 1)dimensional surfacein this formalism or (3 N - 1)in the Marcus description). Sutin's method seems unnecessarily confusing and we will continue to use the original Marcus formalism, as have most other investigatorsin this field. Recently, there has been a tendency to use the term "reaction coordinate" instead of nuclear configuration in labelling the horizontal axis of a potential energy diagram. This labelling is misleading and should be avoided. The activated complex in any reaction lies at the point of minimum energy for all nuclear coordinates but one and for that one it is a maximum. The one vibrational mode for which the potential energy is a maximum is called the reaction coordinate. It is not proper to use this term interchangeably with the termnuclear configuration.
+
Volume 57, Number 7, July 1980 1 479
For the heteronuclear, or the unsymmetrical homonuclear cases (Fig. 2(a)), the energies of the donating and accepting orbitals must again he equal and again equality is accomplished by stretching and compresing the internuclear bonds of the reactants. However, it will no longer he true that the hond distances in the activated complex will he equal. Thus, the equal energy criterion (Franck-Condon restriction) in the activated complex isstiuvalid, hut the equality of internuclear bond lengths is not required. In both the diagrams of Figure 2, no electron transfer will occur, even though, a t the S surface, the Franck-Condon energy restrictions are met and the energies of the donating and accentine orbitals are eaual. The reactants. which are on the E R surface, on passing through the intersection point S , will continue to move on the E ~ s u r f a c eIf . electrdn transfer were to occur, the reactants would be transformed into products on the S surface and would then move down the E p surfare toward the B minimum. Thus, in order for electron transfer to occur in the activated complex, there must be electronic delocalization. In Figure 2, there is no electronic interaction and these diagrams therefore describe the "zero interaction" case. If there is electronic interaction, by coupling of electronic and vibrational motion. a t the intersection S. the two zeroorder states $R and $p produce two new stat& where7
.
-
~
~
~~
~
J.+=ah*ht~ (5) Assuming that the degree of interaction is very small, we may use first-order perturbation theory to solve this problem. A variation method can then he applied to give the energies E + and E- of the two new states formed. When ER = Ep, (the condition a t the intersection point), then
where S is the overlap integral and H R is~ the interaction enerw. A splitting now o c c w and the upper portion of the E ~ a n d E p zero-order surfaces form the first-order surface E- and the lower portions of the ERand E p zero-order surfaces form the first-order surface E + as is shown in Figure 3. The first-order surfaces do not c r m . i t the crossover point for the zero-interaction surfaces the enerw .. difference between the first order surfaces is AE where (7) AE=E--E+ Suhstituting the values of E + and E- fromeqn. (6) into eqn. (7) and simplifying gives
I t is usually assumed, for the purposes of doing calculations, that the amount of electronic interaction occurring is very small so that the overlap integral is vanishingly small (i.e. S N 0). Making the approximation in eqn. (8)gives AE = -2HRp (9) The intbradion energy Hxp is often called J (following the Heitler-London convention) in the electron transfer literature, hut we will continue to use the term HRP. We may distinguish three classes of electron transfer reactions according to their degree of electronic interactwn: (1) very weakly interacting (non-adiabatir). (2) weakly interacting (adiabatic) and (3) strongly interacting fstronelv - " adiahatic)." transmission coeffirient..w..is defined whose value ranges from zero to one. The term "adiahatic electron transfer reaction" is defined as beiw one in which the probability of electron transfer occurring when the system reacheb the activated complex, is unity (i.e. x = 1). The transmission coefficient appears on the right-hand side of the rate equation developed for the "Activated Complex Theory" (abbreviated ACT) (6). k = n -J- T. AF*lRT (10) h The transmission coefficient will be e a u h to one for the strongly interacting case. Strong coupling (and hence adiabatic transfer) occurs when the reacting centers are close together and a minimum of bond distortions are required to equalize the energ~esof the donating and accepting orbitalr. The transmission coefficient will also he eoual to one for the weakly interacting case since only a small amount of electronic imrra~tionis ~ q u i r e 10 h produce aiitabaric Aectron transfer. It is this fact which permits the theoretical development of the area. All of the eiectron transfer theories in common use treat only the weakly interacting case where adiahatic transfer ensues (see potential energy diagram in Fig. 4(b)). The amount of electronic interaction will he very small, on the other hand. if the orbital mismatch is such that considerable hond distortions are necessary to achieve the equal e n e" m.condition for reacting- orbitals. or if the reactine- centers are remote from each other with no low energy pathway for 7 A more detailed description of first-order surfaces may be found in Reference (3)from which this account has been adapted. 8In quantum theory, adiabaticity means that when a system changes, there is a smooth correlation of states. In order to have a smooth correlation of states, the system must remain on the lower first-order surface.
. 'wcLEm I.,
NUCLEAR CONFIGURATION Figure 3. A general potential energy diagram frw the h e t m u c l s a r reaction red, +ox*
?3OX,
R
+ red2 P
where electronic interaction Is allowed and a spliiing occus in me intersectian reaim to ~rodueeuooer . . iE-I . . and lower fEcl . .. first order surfaces. The ouantitv . . AE. is the overall free enerav chanoe in the reaction laiven bv the difierence on nalt-wave potent,ab oeween the reactants and products) and .&?, 0s me thermal enagy of an~vatmnexper mentally aetermlnea oy measurq A# am ,sag ma relat onsn p I F = 1 t P APV
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480 / Journal of Chemical Education
CowwRAIOW ,b,
'*ucLEm
-
C O N F ~ O ~ R ~ ~ "
,.I
Figure 4. (a) The very weakly interacting case. The arrows show lhe pathway taken by the reactants undergoing "on-adiabatic slectrtron bander. Very few products are fwmed since only accasio~llyin the m-adiabatic reaction dthe system remain on the lower surface. The pbints A and B have the same meaning as in Figure 5. (b) The weakly interacting case when me imerection energy-2hp is small. The arrows show the pathway taken by the reactants undergoing adiabatic electron hansfer. Each time the position of the activated complex is reached. electron transfer occurs and products are formed. The paint A indicates the mition of the r e a m in meir lowest vibatimal state and W e paint B indicates the position of the products in their lowest vibrational state. (c) The case of strong electronic interaction. There is no chance af Me system " W i n g " to the uppef surface.
electron transfer between them. In this case, there is some possibility of the molecule in the activated complex "jumping" to the upper surface (E-) rather than remaining on the lower (adiabatic) surface (E+). When, as a result of these non-adiabatic . iumps . in the interaction reeion. . the orobabilitv of a chemical reaction occurring is small per passage of the reactants through the intersection region, the reaction itself is described as "non-adiabatic" and 0 < K < 1. Thus, sometimes, on passing through the intersection reeion. the reaction will go i n to Grm priducts, but most of th;! time the system will remain on the reactants' surfaces, as is shown in Fieure
..
The amount of electronic interaction needed to ensure adiahatic electron transfer is very small. Calculations show that an interaction energy -HRP of 0.25 kcal (or a splitting -2 HRPof 0.5 kcal) is sufficient (7). For strong electronic coupling, the splitting AE can no longer be approximated by -2 HRP(since S,the overlap internal, is now much larger than zero). As the extent of coupkg increases, the thermalbarrier to electron transfer (AE*th) decreases until finally there is only one potential well (Fig. 4(c)). The probability of a system "hopping" from the lower first-order surface to the upper one in the strongly adiabatic case is vanishingly small (K= 1). Simple, electrostatic theory is incapable of treatine the strongly coupled case since thk assumption is made th; the amount of electronic overlap is small and the overlap intend. S , is set equal to zero. Of cburse, S cannot really bk equal ti zero or we would have the zero interaction case and no splitting of the zero-order surfaces would occur (Fig. 2). The theory of Marcus applies to the regime in which the amount of electronic overlap must be large enough to ensure adiahatic behavior but not so large that the overlap integral cannot be approximated to zero. The case where there is only slight electronic coupling (non-adiabatic electron transfer, 0 < K < 1)can be treated in the normal fashion by Marcus theory if the transmission coefficient, K,can be calculated. This matter is not trivial, however. The mechanism for electron transfer may always involve tunnelling, either by the electron or the nuclei (especiallysmall nuclei such as hydrogen atoms which are light). In fact, electron tunnelling is pr&ably involved in mosf electron transfer processes ( 8 ) .However, the tunnelling step may not be ratedetermining in most cases. Potential Energy Diagrams In Intervalence Electron Transfer Electron transfer reactions mav. in eeneral. oroceed bv IOU& In the fv; radiative (optical) or radiationless (il;e&) case, the reaction is initiated by bombarding the sample with light of the appropriate wavelength. The radiationless mechanism is the "dark"~outesince it does not require light. The activation parameters (AH*, AS*) are usually reported for thermal electron transfer reactions and are ohtained bv measuring the rates of reaction as a function of the tempera"Cure of the solution. The potential energy diagram for the thermal electron transfer process was discussed in detail in the previous section. Thermal electron transfer was shown to result from couoline of electronic and vibrational motion. . We will be concerned in this section with the optical electron transfer process and with its relationship to the thermal reaction. Optical intervalence transfer has been defined (9) as an optical transition which involves transfer of an electron from one nearly localized site to an adjacent one, the donor and acceptor being metal ions which possess more than one accessible oxidation state. I t was first ohserved for molecules such as Prussian blue. This complex results when solutions of Fe (11) salts are mixed with ferricyanide solutions.
Obviously, the visible spectra of the reactants are not simply additive. The oroduct exhibits an intense band which is aunique feature bf that product. It arises from electron transfer from Fe (11) to Fe (111) and hence is termed "intervalence" electron transfer. The problem with comdexes like Prussian blue is that discrete molecular units cannot be isolated. The complexes are formed from labile metal species which readily dissociate and reform in solution. This dissociation and reformation limits the amount of useful chemical information which mav he obtained from such a system. The systematic study of intervalence electron transfer reactions has been put on a solid experimental footing by the work of Taube and his associates and these ideas have been extended hv manv others. Taube decided to orenare discrete molecular "nits which would exhibit inteGaleke ele&& transfer. He selected the ruthenium atom which has two accessible oxidation states, +2 and +3, hoth of which produce inert complexes. This choice allowed ~roductionof discrete dimers ofthe type (NH3)6Ru11- L - RulU(NH&+ where L is some bifunctional ligand. As an illustrative example, we shall look a t the molecule in which L = 4,4'=hipyridine (1) (10).
(1)
A shorthand re~resentationfor a molecule of this tvne showing the oxidation states of the metal would he (2,3).he advantaee of working with molecules like (1) is that thev are very r o b k t and survive for long periods of time in solition under a variety of conditions. For example, complex (I) has a half-life of several days in aqueous solution. This life span makes these complexes ideal for studies which are amenable to theoretical int&pretation. The metal atoms in complex (I) will undergo a homonuclear one-electron intewalence transfer. Binuclear complexes (or dimers) such as these could be studied conveniently in solution as well as in the solid state.
-
-
4Fe(OHh2++ 3Fe(CN)e3- Fer[Fe(CN)&+ 24H20 (11) Both Fez+ and Fe(CN)e3- are very pale in color, hut the complex formed upon mixing these species is a deep hlue.
NUCLEAR CONFiGURATlON Figure 5. The potential energy diagram showing the orlgin of the intewalence transfer band whose enerov -.is oiven bv. A&. The diaaam shown is f a a hsteronuclear reaction where the donor metal atom is differem hamthe acceptw atom (complex (11)) or for an unsymmetrical homonuclear reaction where the donor a acceptor are the M m e kind of atom but have different iigands in Uwrir first wardination sphere (complex (Ill)). The point A is the position of minimum potential energy (vibrational ground state) of the reactant molecule, e.g., (NHp)sRu"(4.4'-blpyKch"'(NH3)Pt.In sharthand notation this would be (2.3). Point B is the position of minimum potential energy (vibrational ground state) d Uwr prodwl molecule, eg. (NH3)sRu"'(4,4'-bipytRh"(NH3)1~+.In shorthand notation this would be (3.2). Point C represents products in an excited vibrational state 13.2)'. The don4 line from A to D representsa non-verfical transition forbidden by the Franck-Condon restrictions.
-
-. -
~~
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Volume 57, Number 7, July 1980 1 481
.
The origin of the intervalenre band as an optical wansition for the heteronuclear (ex. complex 11) or unsymmetriral homonuclear (e.g., complex 111) case is shown in Figure 5.
ferring electron to two independent oscillators (which can be thought of as springs) of identical frequency. For the symmetrical, homonuclear, one-electron transfer, he arrived a t the conclusion that
Ah&
(13) i.e. the energy for optical transfer is predicted to be four times that of the thermal route. A h for this particular classof reactions, it can be shown that (9)
(rn)
..
The transition is a vertical one, again obeying the FranckCondon restriction that electronic motion is faster than nuclear motion. The reactants (at A) and products (at C) both have the same nuclear configurations but different potential enemies. If the transition were not vertical (e.g. A D), then nuclear motion would have had t o have occurred to give a different nuclear configuration for the products (on the upper first order surface a t D) than the reactants orieinallv had (at A). This nuclear motion would have occurredldurin~the act of electron transfer, but it is forbidden bv Franck-Condon restrictions. The intervalence transition in the ruthenium dimers corresponds to the transfer of an electron from one ruthenium t o the other. However, after the electron is transferred, i t is in a vibrationally excited state, i.e., the intervalence process may be represented as
-
inshorthand notation. This process is true for all intervalence electron transfers. The oroducts are alwavs formed in a vibrationally excited staG(position C on the upper first order surface. E - , in Fie. 5) since thev have the same nuclear configuration as the reactants in their ground vibrational state (at A) but the product (at C) has additional potential energy exhibited as vibrational energy. The vibrationally excited product (at C) must relax to the produds' vibrational ground state (position B on Fig. 5). Energy is evolved as thermal energy during this process, i.e., the solution heats up an infinitesimal amount during this vibrational relaxation process. The vibrational excited state should not be confused with an orthodox electronic excited state. The hinuclear (dimer) is usually in its electronic ground state at all times. This does not mean that electron transfer to an electronic excited state cannot occur. I t can, and does, but the intervalence transfer to an orthodox electronic excited state occurs a t much higher energy and the resulting spectral band often is found i'the ultraviolet region of the spectrum. In this region, it is usually obscured by other intense ligand-metal charge transfer transitions. The theory of intervalence transfer has been developed by Hush (9). He chose a simple model-coupling of the trans-
= 4AE*th
AE, = hum, - (16 In 2kT)-'AE1/22 (14) where k = Boltzman's constant, T = temperature in OA and hE112is the bandwidth a t half-height. At 300°C, this may he simphfied to
G., =
(%I
where i is in cm-I
(15)
Most intervalence hands are Gaussian in shape on an energy scale (recall that most spectrophotometers measure wavelength, which is inversely related to energy). For bands of this shape ( I I ) , the oscillator strength, f , is given by where t,. is the extinction roefficient of the hand maximum and AE1,2 is the band half-width at half.height in wavenumbers (cm-'). Thus, we can get a good deal of information about intervalence electron transfer for ruthenium binuclears by simply running a spectrum of the dimer in each of the (2,2), (2,3) and (3,3) oxidation states. Figure shows these spectra for the complexes
-
Because of the streneth of the rutheniumvibrations, the intervalence band is usually found in the near infrared spectrum (-700-1800 nm). This range is convenient since there are few other vibrations or electronic transitions associated with metal ions or their ligands, in this region. Thus, the (22) and (3,3) oxidation states of the dimers characteristically give no absorotion in the near-IR spectrum, but the (2,3) dimer gives a broad intervalence band with properties which m a y b e interoreted bv Hush's theorv. i s with-the Marcus iheory for outer-sphere electron transfer reactions (based on electrostatics), the theoryof Hush for intervalence electron transfer is only applicable to the case where there is slight electronic coupling . .(see Fig. 4(b)). The amount of electronic coupling determines the extent of electron delocalization over the dimer. When the coupling between the metal atoms (mediated by the ligand in M-L-M -
9Adapted from Figure 19, in the thesis of Creutz, C., Stanford University, California, U S A . , 1970.
I
650
850
1050 I250 WAVELENGTH (NMI
1450
Figure 6. Near infrared speetra of
A solution
MOI 6+
of 0.0 was timted wim o.oi MCr(ll1
482 1 Journal of Chemical ~ducaiion
1650
I
I NUCLEAR
CONFIGURATION
NUCLEAR CONFIGURATION
Figure 7. Potential energy diagrams showing possible optical transitions (intervalence transttions) from both potential energy wells. (a) Optical transitions possible In heteronuclear dimem with a small overall energy change AE. (b) Optical transition possible in heteronuciear d i m wllh a large averail enetgy change AE,.
usuallv) is small. the amount of delocalization will be small and the'oscillato~strengthofthe intervalence band (IT hand) will be small. Thus the oscillator strenath of the IT band is a measure of the amount of electron del&alization in the molecule. If there is strong coupling between the metals, the band will be narrower (AEI,~will be too small) than is predicted by Hush's theory (eqn. 15)and the extinction coefficient will be very high. The energy of the I T band (AE,) is a function of the distance between the metal centers. It is also sensitive to the nature of the metal atom (i.e. Rh versus Ru in complexes I and 11) and to the nature of non-brideine lieands mesent in the first coordination sphere (e.g. NHs versus C1- in complexes I and 111). It is further related to the energy of the FranckCondon barrier (AE*,,, in Fig. 5) by the relationship given in eqn. (13). There is one other prediction which arises from Hush's theory. Consider again a typical heteronuclear or unsymmetrical homonuclear electron transfer reaction (Fig. Val). There are really two possible transitions, one from each minimum on the notential enerw surface. Onlv one I T hand is usually observLd for binucle&l, even in the heteronuclear case for the followine reason. The ratio of the number of reactant molecules tn-the number of product molecules will be determined bv . AE,. Conseauentlv, there are many more product molecules than reactani molecules in Figure 7ia), and there will be many more transitions arising from the products' potential well (at B) than from the reactants' potential well (at A). These latter transitions will not he seen as an IT band because there will be very few of them,and they will not produce sufficient intensity. They may, however, contribute to a slieht broadening of the I'l'hand observed for the transitions from the potential well, in excess of what would be
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~redictedbv Hush's theorv- (em. . . 15). For the s&netrical homonucle~electrontransfer, it will not be wssible. of course. to sav from which well the transition originated sinde reactants and products are identical. If AE. is exceptionallylarge, it is possible for the potential energy surfaces of the reactants and products to intersect such that both potential minima are on the same side of the intersection point (Fig. 7(b)). As usual, two optical transitions are expected, one from each well. This time, however, one of them (hE0p,2)corresponds to an emission spectrum rather than an absorption, and one would expect to observe fluorescence. This has a ~ ~ a r e n tnot l v be& obsewed in a simole inoreanic binuclear alt'hough iiseems to be quite a commdn pbenomenon in organic charge transfer spectra (12).
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Acknowledgments
The author is grateful to Professors H. Taube and W. R. Fawcen for reading the various drafts of this manuscript and for many helpful suggestions and discussions.
.. (8) Taubo. H..and Could. E.
S..Accts. Chem. Rea.. Z. 321 (18691.
." (12) Faete.. R.. "Organic Charge Transfer Cmplara," Arsdrmie
h, London. 1w9,
Chapter 3B.
Volume 57, Number 7, July 1980 1 483
