30
J . Phys. Chem. 1991, 95, 30-38
The Co( NH3)62t’3+Exchange Reaction: Ground-State versus Thermally Excited Pathways Marshall D. Newton Chemistry Department, Brookhaven National Laboratory, Upton, New York 1 1973 (Received: April 3, 1990; In Final Form: July 2, 1990)
The electron-exchangereaction involving the Co(NH3)?+ and Co(NH3)63+has been analyzed on the basis of electronic structure calculations. Equilibrium and inner-shell transition-state geometries, activation energies, multiplet splittings, and spin-orbit co,upling coefficients have been evaluated by using the results of ab initio electronic structure calculations for the separate reactants (at both the SCF(UHF) and correlated (UMP2) levels) in conjunction with empirical values for atomic spin-orbit matrix elements and the known value of the multiplet splitting for the Cos+ reactant. Electron-transfer matrix elements H ~ ) ~ ~ + supermolecule complex have been obtained from INDO calculations for the C O ( N H ~ ) ~ ~ + I C O ( Ntransition-state corresponding to apex-to-apex,edge-to-edge,and face-to-face relative orientation of the reactants. Combining this information with previous estimates of solvent reorganization energy and coordination sphere breathing frequencies, we have estimated for the orientationally averaged electronic transmission factor associated with the ground-state spin-ora value of bit-enhanced pathway. The calculations suggest that an alternative thermally excited pathway will not be competitive at room temperature. The present results are compared with earlier analyses of kinetic data which suggest K~~ k
-
I. Introduction Elucidation of the mechanism of electron exchange in the aqueous C O ( N H ~ ) ~ ~redox + / ~ +system
and in related systems involving amine ligands has posed an ongoing challenge to experimental and theoretical chemists for more than two decades.’-I0 As indicated in Scheme I, the reaction pathway involving ground-state configurations of the reactants and products (dashed vertical arrows) is expected to have very low probability since it constitutes a “three-electron” process:1° Le., in addition to the transfer of an electron between reactants, additional one-electron excitations (“shakeup” processes) occur within each reactant. Since each reactant in the ground-state process changes its spin quantum number by 3/2, instead of the more common value of the process is sometimes referred to loosely as being “spin forbidden” or involving a ”spin barrier”. However, there is no change in the overall spin quantum number, S = 3 / 2 9 in proceeding from initial to final ground-state configurations, and thus the low probability for the process does not directly involve spin. To achieve appreciable reaction probability, one must then invoke mechanisms that involve conventional ( I ) Stranks, D. R . Discuss. Faraday SOC.1960, 29, 73. (2) (a) Buhks, E.; Bixon, M.; Jortner, J.; Navon, G. Inorg. Chem. 1979, 18, 2014. (b) The use of a rate equation in ref 2a to treat electron transfer involving multiple initial and final states, and the neglect of phase coherence implicit in this treatment, may be justified provided that the spacing of the initial states exceeds their reactive widths,2c a condition satisfied by the Co(NH3),2t/3C system. (c) Kestner, N . R.; Logan, J.; Jortner, J. J . Phys. Chem. 1974, 78, 2148. (3) (a) Dwyer, F.P.; Sargeson, A. M. J . Phys. Chem. 1961,65, 1892. (b) Neumann, H.M . Quoted in: Farina, R.: Wilkins, R. G . Inorg. Chem. 1968, 7, 514. (c) Szalda, D. J.; Creutz, C.; Mahajan, D.; Sutin, N . Inorg. Chem. 1983, 22, 2372. (d) Sargeson, A . M . Chem. Br. 1979, 15, 23. (4) Brunschwig, B. S.;Creutz, C.; Macartney, D. H.; Sham. T.-K.; Sutin, N . Faraday Discuss. Chem. SOC.1982, 74, 1 1 3. (5) (a) Sutin, N. frog. Inorg. Chem. 1983, 30, 441. (b) Marcus, R. A,; Sutin, N . Biochim. Biophys. A c f a 1985, 811. 265. (6) Geselowitz, D.; Taube, H. A h . Inorg. Bioinorg. Mech. 1982, I, 391. (7) Hammershoi, A.; Geselowitz, D.; Taube, H. Inorg. Chem. 1984, 23, 979. (8) Larsson, S.: Stahl, K.:Zerner, M. C. Inorg. Chem. 1986, 25, 3033. (9) (a) Siders, P.; Marcus, R. A. J . A m . Chem. SOC.1981, 103, 741. (b) Zheng, C.; McCammon, J . A.: Wolynes, P. G . Proc. Natl. Acad. Sci. U.S.A. 1989,86, 6441. (IO) (a) Newton, M. D. J . Phys. Chem. 1986, 90, 3734. (b) Newton, M . D. J. Phys. Chem. 1988,92,3049.(c) Newton, M. D. ACS Symp. Ser. 1989, No. 394,3783. (d) Newton, M. C. In Perspectiues in Photosynthesis: Jortner, J., Pullman. B., Eds.; Kluwer: Dordrecht, 1990; p 157.
0022-365419 112095-0030$02.50/0
one-electron pathways. The most likely ones for the Co( N H 3 ) 6 2 + / 3exchange + are those involving the lowest excited configurations, 3Tl,(Co3+) and 2E,(Co2+),as indicated by the four slanted solid arrows in Scheme I. These excited configurations may participate in ground-state thermal electron transfer by spin-orbit coupling with the dominant ground-state configurations or, instead, as the dominant configurations in thermally excited
pathway^.*^^^*^'^^ Of crucial importance in resolving these questions is a knowledge of the vertical energy separations of the high and low spin states for the oxidized and reduced species, denoted in the following as AE3+and AE2+, respectively (or generically as AE2+): AE3+ = E(3T,,) - E(’AI,)
(2)
E(2E,) - E(4T,,)
(3)
AE2+
The AEz+values serve as energy denominators in the evaluation of spin-orbit coupling coefficients, C3+ and via first-order perturbation theory:2x’0E
e+,
where 4 P and $2+ are the ground-state spin-orbit wave functions for Co(NH,):+ and CO(NH!)6’+, respectively.” The hE2+value is also used below in obtaining an estimate of the difference in activation energy for ground-state and thermally excited pathways. While an experimental value (- 13 700 cm-I) is available for hE3+ at the equilibrium ‘A,, equilibrium geometry,12the corresponding ( 1 I ) We use the term “ground state” in referring collectively to the set of 12 low-lying states arising from the 12-fold degenerate 4T,,state (see section IIC).
0 1991 American Chemical Society
The Co(NHJ)?+I3+ Exchange Reaction
The Journal of Physical Chemistry, Vol. 95, No. 1, 1991 31
value of M2+ is uncertain, although it is expected to be appreciably smaller than AE3+ (estimates ranging from -3000 to 9000 cm-' have been reported).2v8,'0f The multiplet separations vary with metal-ligand bond length ( r rCoN)as a result of changes in the balance between d-electron pairing energies and ligand-field strength.sJOc As emphasized previously,8*10c the multiplet splittings pertinent to the present mechanistic analysis are those associated with the inner-shell transition-state geometry, AE2+(r*)(Le., with Co-N bond length, r * , lying between the equilibrium values for the oxidized and reduced species). It is convenient to continue the discussion in the context of a standard transition-state m ~ d e l : ~ ~ ~ ~
single-determinantal) initial- and final-state wave functions whose orbital occupancies remain unchanged except for the transfer of an eBelectron. As shown previouslylob,dfrom analysis of ab initio and semiempirical results, the Hit')elements can to a good approximation be taken as matrix elements of an effective oneelectron Hamiltonian connecting donor (D) and acceptor (A) orbitals, denoted hDA('),even though Hitf)is, strictly speaking, the matrix element of a many-electron Hamiltonian with respect to many-electron initial- and final-state configurations. Small many-electron contributions to Hit') arise from electronic relaxation effects accompanying the one-electron transfer. Exploiting the near equivalence of the one-electron and many-electron pictures (Le., #')((ri)) z i h ( l ) ( r i ) )we , will in later sections analyze the Hifelements in the framework of a one-electron model, but then evaluate the resulting matrix elements of h(')using elements obtained from actual many-electron calculations for suitable CO(NH3)62+-CO(NH3)63+supermolecule complexes. The other factor in eq 12, yS'O',is the mean attenuation factor associated with spin-orbit mixing and with the multideterminantal nature of some of the pure spin states, averaging in the rootmean-square (rms) sense, having been carried out over the various spin-orbit states, as described in detail in sections 1I.D and 1I.E and Appendix B. An early experimental study of reaction 1 suggested an upper limit of 2 X s-' M-' for the rate constant at 65 "C and 1 M ionic strength.' This result, in conjunction with current estimates of the "nuclear" contributions in eq 2 (E* = 24.4 kcal/mol, I?,, 9.0, ueff = 347 cm-I) and the preequilibrium constant ( K , = 0.013 M-'),4*5implies K,, lo", a conclusion that at the time was perhaps not surprising, in light of the "electronic" bottleneck anticipated above. Subsequent theoretical analysis of the ground-state mechanism yielded an estimate of for the spin-orbital coefficient, yS'O',in eq 12, and hence a spin-orbit attenuation factor of -lo4 in K , ~(assuming the nonadiabatic limit of eqs 7 and 8).2 The analysis of more recent experimental data6,' has led to an estimated rate constant larger than the prior upper limit by several orders of magnitude (25X lo-' s-I M-I at 25 OC and 1 M ionic strength),ls thus leaving little apparent room for appreciable departures of K,' from unity, in contrast to expectations based on the mechanism involving the spin-orbit-enhanced ground-state process.2 Sutin and co-workers have analyzed the experimental data for the C O ( N H & ~ + / ~exchange + and related processes in conjunction with eq 6, concluding that K,' is most likely L 10-2.435 Ab initio and semiempirical electronic structure calculations were carried out by Larsson, Stahl, and Zernera and used to investigate the relative energies of all the low-lying states contained in Scheme I, with the conclusion that the activation energy for the thermally excited pathway involving the low-spin 2E state of C O ( N H ~ )might ~ ~ + well be lower than that for the ground-state process. Stimulated by these latter results, we have embarked on a detailed study of the energetics of the relevant states, employing ab initio electronic structure techniques. A preliminary study'& focused on evaluation of the strength of spin-orbit coupling at the transition state for the ground-state pathway and yielded an estimate of -IO-* for K~~ when the reactants are oriented in an "apex-to-apex" configuration (Le., along a common 4-fold axis of the octahedral frameworks). In the current study we present the results of additional calculations which yield a refined estimate of the extent of spin-orbit coupling and permit a quantitative estimate of the relative importance of the ground-state and thermally excited pathways.
ke, =
KeqVcffKclrn
exp(-PE*)
(6)
where K,, is the preequilibrium factor for the bimolecular encounter complex, where the other three prefactors denote, respectively, the effective frequency associated with motion along the reaction coordinate, the electronic transmission factor, and the nuclear tunneling factor, and where E* is the activation energy ( P I/kBT). Using the Landau-Zener model,14 we represent K , ~ ass,10913 Kel = PO/ ( 1 + PO) (7) where the probability Po for hopping from the initial state ($i) to the final state (fif) diabatic energy surface is given by Po = 1 - exp(-2ry) (8)
and where, for a harmonic oscillator model in the high-temperature limit we have 2 r y = IHirlZr3/2/hu,ff(kBT~~)'/2
(9) The quantity E , in eq 9 is the reorganization en erg^,^^'^ which for a thermoneutral exchange process is approximately equal to 4 times the activation energy, E * . The initial ($i) and final ($J states correspond, respectively, to the left- and right-hand sides of the chemical reaction of interest (eq I ) , and Hifis the electron-transfer matrix element (the matrix element of the Schradinger electronic Hamiltonian for the reacting system which couples the initial and final states).1° When Hifis sufficiently small in magnitude, as is the case on the ground-state pathway for reaction 1 * l o then the exponential of eq 8 can be expanded and truncated at the linear term, yielding the nonadiabatic limit in which K , ~is proportional to y. In this limit, quantum effects in the prefactor associated with inner-sphere nuclear motion may be incorporated through the use of a standard semiclassical mode15JJ in which eq 9 is replaced by 2 ~ =7 IHifIZ~S/2/(kBT[E,,, + E , d csch b)])'/2( I O ) where E,,, and Einare, respectively, the inner-sphere and outer-sphere contributions to E,, and y = hui,/2kBT,with uin being the inner-sphere breathing frequency. The same semiclassical modeI5J3yields the following expression for the nuclear tunneling factor: In rn = (Ein/4kBT)(I - (tanh b/2))/b/2)) (11) In view of the rather special electronic aspects of the Co(NH3)62+/3+exchange process, as described above, it is useful to express Hiffor the ground-state spin-orbit-enhanced process as Hif = (ys,o,)(Hip) (12) Here Hit1)is the common (mean) value assumed for the matrix elements controlling the primary one-electron processes (slanted solid arrows in Scheme I), each of which can be viewed as occurring between a superposition of single-configuration (Le., (12) (a) Wilson, R. B.; Solomon, E.1. J . Am. Chem. Soc. 1980,102,4085.
(b) The 13 700 cm-' is based on the peak value of spectrum given in Figure 14, ref I2a; values of 13400 and 13 000 cm-' were employed in refs 2 and 8, respectively. (13) Newton, M. D.; Sutin. N. Annu. Rev. Phys. Chem. 1984, 35, 437. (14) (a) Zener, C. Proc. R. SOC.London, A 1932, 137,696. (b) Zener, C. Proc. R. SOC.London. A 1933, 140,660. (c) Landau, L. Phys. Z . Sowjerunion 1932, I , 88. (d) Landau, L. Phys. 2.Sowjetunion 1932, 2, 46.
-
-
-
11. Theoretical Models and Computational Details A. Electronic Structure Calculations. Ab initio electronic
structure calculations have been carried out as a function of (15) (a) The estimated rate constant at 25 'C and 1 M ionic strength (?IO'S-' M-')Isb was obtained from the actual data (based on 40 ' C and 2.5 M ionic strength)' by using eqs 6, 10, and 11 and parameter estimates from refs 4 and 5, together with data on ionic strength dependence reported by: Ekstrom, A,: McLaren, A. B.; Smythe, L. E. Inorg. Chem. 1976, 15, 2853. (b) A similar estimate is given in ref 5.
32 The Journal of Physical Chemistry, Vol. 95, No. I, 1991
Newton
metal-ligand separation for the total energies of the four primary electronic configurations indicated in Scheme I (the ground and lowest excited configuration for each of the chemical species involved, CO(NH3)63+and CO(NH~),~+). These energies are then used to provide estimates of the activation energies and, via first-order perturbation theory, of the spin-orbit coupling coefficients (eqs 4 and 5). Self-consistent-field (SCF) calculations were carried out at the unrestricted Hartree-Fock (UHF) level, using a flexible basis of s, p, and d Gaussian-type (GTO) basis functions, as described previously.'& Further details are provided in Appendix A. For the most part, an octahedral CON, framework was maintained (0, point group), and energies were evaluated as a function of the common CON bond length, r. For the 2Egstate of Co(NH3)2+, the Jahn-Teller effect was included by employing distinct axial and equatorial Co-N bond lengths, r,, and rq (D4, framework geometry). Effects of electron correlation on the calculated energies were estimated by exploiting the results of second-order Moller-Plesset calculations based on UHF orbitals (UMP2)I6 obtained with inclusion of f orbitals and reported in the earlier work.'& Due to very large demands on computer time and memory, these UMPZ calculations were carried out for the followin limited number of geometries restricted to 0,symmetry: 2.05 for the 'A, and 3T,gstates, near the calculated equilibrium r value (re) for BAl,; 2.25 A for 4T1,and 2E,, near the calculated 4T,gre value; and 2.15 A for all four states, the initial estimate of the geometry ( r * )of the inner-shell transition state for the process involving exchange between the ]Al, and 4T,, states (see section IIB). In the preliminary study,Ik variations in UMPZ values of AE3+ and AE2+ with r in the vicinity of the r values indicated above were approximated by assuming a linear dependence, based on the slopes calculated from the available energies (2.05 and 2.15 A for AE3+ and 2.15 and 2.25 A for AE2+). This approximation is reliable to the extent that the two states associated with a given AE* have harmonic potential energy functions with respect to r of similar curvature.Ik In a final stage of refinement, the UMP2 AE* values were empirically corrected1&by exploiting the known value of AE3+ at the !Al, re value. Reasoning that the electronic rearrangements involved in the high-spin/low-spin interconversions for the two oxidation states are very similar (each involves breaking an electron pair in a (t2J6 shell and promoting a t2, electron to an empty eBorbital),'& we argued that the same empirical correction term (assumed independent of r ) could be used for both AEZ+values as a reasonable means of approximating correlation effects beyond the UMPZ level:
tional zero-order diabatic model which involves the energies of the isolated r e a c t a n t ~ : ~ > ~ J ~
1
-
AEbkp2
AE&Lp2+ c
(13)
AE&+Mp2
AE2v+Mp2- c
(14)
where c (estimated to be -6000 cm-' in the earlier work)'& is the constant that makes AEvMp2(r..)+ c (based on the calculated re value) equal to the experimental value, 13 700 cm-I.l2 A similar procedure was adopted by Larsson et a1.8 In the present work, the linear approximation defined above was refined by applying it only to the correlation energy, AE&RR AE$,,p2 - AE;kF, which is then added to the AE&, values calculated at all required r values. The AE6kp2values obtained in this manner are once again corrected by a constant term c, as indicated in sequences 13 and 14. In contrast to the full UMP2 energy for the individual states, the correlation component is expected to vary with r in a smooth, monotonic fashion in the r range of interest. Thus the linear approximation should be more reliable for the correlation energy terms than for the total UMP2 terms, whether applied to the r variation of AEz+ or the variation of individual state energies. B. Activation Energies. For the purpose of evaluating the "inner-shell" contribution to the activation energy, Ein*,for the Co(NH3)62+/3+ exchange reaction (eq I ) , we adopt the conven(16) Krishnan, R.; Frisch, M. J.; Pople, J. A. J . Chem. Phys. 1980, 72,
4244.
Ein* = E2+(r*)+ E3+(r*)- E2+(rz+)- E3*(ra+) (15) where the superscripts (2+ or 3+) refer, as usual, to the oxidation states of the individual reactants and r* is the minimum-energy value of rCoNwhen constrained to be the same for each reactant. E* is evaluated at both the U H F and the UMP2 level. As it stands, eq 15 can be applied directly to the activation energy associated with the ground-state configurations: C O ~ + ( ~ T , ~ ) / Co3+('AIg).When the thermally excited process is considered, C O ~ + ()/Co3+(IAIg). ~E each r in eq 15 is replaced by distinct axial ( T a x ) and equatorial (!q) values, as noted above. The procedure adopted here for obtaining E* values is defined in terms of pure spin states. However, the modest degree of spin-orbit coupling which we estimate below will lead to only minor corrections in r* and E* values. Lowering of the calculated barriers through avoided crossing will occur when the interaction between reactants is taken into account, and the magnitude of this effect will be considered below. C. Spin-Orbit Coupling. The spin-orbit coupling coefficients (C3+and c2+ in eqs 4 and 5, respectively) are evaluated as2Joc C3+(r)= ( H I A , ~ , ~ T ~ ~ ) / A E ~ + ( ~(16) ) q + ( r ) = (H4T,,~E,)r/AE2+(r), 1 = 0-3
(17)
In the octahedral double group," the 12-fold degeneracy of the 4T,, state is split into the following 2-fold (E) and 4-fold (G) states (listed in order of increasing energy): 4Tlg
---c
E1/2g9 G3/2g, G3/2g'
and
ES/2g
(18)
The 2E, representation maintains its 4-fold degeneracy: ---+
G3/2g
(19)
Evidently, the excited 2Egstate will mix with only two of the 4Tlg double-group states (those of G3/2 symmetry). Hence in eq 17, only Cy and ' : C are nonzero. In the case of Co3+,the 'Al, state mixes with the totally symmetric double-group component of the jTIgstate (Alg), Employing a standard one-electron model,2,10cwe evaluate the numerators on the right-hand side of eqs 16 and 17 as HiAle?Tlg= 61 / 2t 33+d (H4T1,,2E,)I
= (1 /51/2)[%
(H4TIg,2E,)2 = (3 / 51'2)t%
(20) (21) (22)
where ;6: and are the effective spin-orbit coupling elements for the 3d orbital in the Co3+ and Co2+complexes, respectively (as inferred from spectral data).I8 The numerators in eq 17 for 1 = 0 and I = 3 are zero. The two G3 states lie ,$: and 4[:: above the ground G I l 2state in the Co2I ion, and the E5 state is accidentally degenerate with the second G3 state.2 @e take values of 600 cm-' for and 5 15 cm-I for t24! J following Buhks et aL2 Although eqs 20-22 correspond nominally to a simple crystal field model, the empirical 4 parameters reflect the influence of ligand-field mixing in the Co(NH,), complexes. D. Evaluation of H y Elements. The evaluation of the electron-transfer elements Hi/ is based on a modelIk in which the initial- and final-state wave functions are taken as superpositions of antisymmetrized products of the spin-orbit wave functions for the two reacting cobalt complexes (denoted by subscripts 1 and 2) :
[::
$i(/)
=
(+?+)1($3+)2
(23a)
+t(,)
=
(+3+)I(+i+)2
(23b)
(17) Herzberg, G . Electronic Spectra of Electronic Structure of Polyatomic Molecules; van Nostrand: New York, NY, 1967; Appendix 1. ( 1 8 ) Dunn, T . M. Trans. Faraday SOC.1961, 57, 1441.
The C O ( N H ~ ) ~ ~Exchange + / ~ + Reaction
The Journal of Physical Chemistry, Vol. 95, No. 1, 1991 33
where 1 and m refer to the four different spin-orbit states arising from the high-spin ground-state configuration of the reduced species ( I and m range from 0 to 3, as in eq 17). For simplicity in the present analysis (see below), we assume that the molecular orbitals of qicn and \kr(,) are mutually orthogonal, and that (the notation we employ for Hi,(i)in the particular case of interesf here) can be taken as an effective one-electron matrix element, h!:ie,, (see discussion following eq 12), where the subscripts eg,and eg2indicate the pair of eg orbitals involved in the one-electron transfer of interest. These orbitals are dominated by the Co 3d orbitals ( z 2and x2 - y 2 ) but also exhibit significant contributions from the ligand orbitals as a result of ligand-field mixing.'& Note that the assumption of orthogonality of the orbitals of $i(l) and $f(m) introduced above is only adopted for the limited purpose of obtaining a simple expression for the spin-orbit factor yS.O.in eq 12. The actual nonorthogonality effects are fully accounted for in the evaluation of h!:!e82. We develop expressions for Hi(/)r(m) in terms of the factors defined in eq 12, which we modify for present purposes as follows:
TABLE I: Calculated Euuilibrium CON Bond Lengths ( r , A)"
@:le,
Hi(/)f(m)
= 7% hiit BI 82
(24)
In terms of the basis of pure spin states, Scheme I shows that in general there will be four leading contributions to a given matrix element, each first order in both C? and C3+: Le., the four possible combinations of the slanted solid arrows arising from the two possible "paths" at each In an earlier study,2 the matrix element Hi(,)r(m) was represented by the form H i ( / ) r ( m ) = (C3++ q + ) ( C 3 ++ G+)h!:,Lg2
(25)
in which the first two factors correspond to y f t in eq 24. The simplicity of the first two factors in eq 25 derives from the fact that the pure spin states were treated effectively as having single-determinant wave functions.2 We find that eq 25 must be modified when the multideterminant nature of the contributing spin states is taken into account. In the cases of interest here, 7 :; can be expressed as yj;
= ( a / )a,)(E3+/AE3+ (
+ (2+/AE2+)2
(26)
where cyo = a? = 0; a I = 1/5lt2; cy2 = 3/5II2. Thus only the two G3/ZKstates yield nonzero values (in contrast to eq 25). Further details are given in Appendix B, which also discusses the specification of hi:)e values. In the remainder of the paper (including Appendix 'EBj we suppress the subscripts 1 and 2. E. Evaluation of K,,, Since the spin-orbit-enhanced groundstate pathway involves two different initial (+i(l),+i(2)) and final (+f(i),+f(2)) states (see eq 23), Le., the two distinct G3 2g states discussed above, eq 6 must be generalized to the Boltzmann-weighted average:'&
[allowing
where E, is the first-order energy of the lth spin-orbit state arising from the 4T,, spin state, with degeneracy g/, where the electronic partition function Z is given by
-
and where kl, is the electron-transfer rate constant for the process $I(/)
J/f(m)
(28)
with exo- (or endo-) thermicity AElm = E, - E/
(29) In using eq 27, we assume (as in ref 2) that phase coherence between initial states ( I = 1 or 2) and between final states ( m = 1 or 2) can be neglected.2b The rate constants, k,,, may be evaluated using eqs 6-1 1 for 1 = m or suitable generalization for the nonthermoneutral cases ( I # m). In these latter cases numerical results may be obtained by iterative solution for the saddle-point time which arises in the
ab initio state
INDO*
exptl
2.04 2.12
2.02 2.06
1.97' 2.03f
Co(NH3):+ 2.26 2.18
2.14 2.09
2.19' [2.09]*
UMP2d
SCF'
CO(NH&'+ ,A,,
3Tlg
4TI,
2E,
2.07 [2.03] 2.14 [2.08] 2.30 [2.27] 2.24 [2.20]g
"Results from present work based on Oh CoN6 framework, except as noted. bReference 8. 'Quantities in brackets are ab initio SCF results from ref 8. dBased on second-order Moller-Plesset theory,16 as described in section IIA and Appendix A. 'Reference 4. 'Reference 12a; weighted mean of r,, and rcq values. g I n both the present and earlier* SCF calculations, Jahn-Teller distortion to an equilibrium D4h CON, framework causes changes of +0.12 and -0.06 A, respectively, in axial and equatorial CON bond lengths. In the earlier SCF study,* an estimate of 2000 cm-' was reported for the distortion energy at equilibrium ( r e ) ,but the distortion energy at the IA,,/2E, transition state was included only indirectly through an empirical estimate based on the use of the 4T, Co-N force constant, since no experimental value is available for the $E, force constants. Estimate based on calculated (UMP2) results for all four states and experimental results for the 'A,,, )TI,, and 4T,, states.
semiclassical f0rmu1a.l~It is convenient to define a mean effective electronic transmission factor, zel, as follows:
E,,
= iZeik:~
(30)
where Re, is defined in eq 27 and where k$ is the adiabatic rate constant obtained from eq 6 (for the thermoneutral case) by setting K , ~ = 1. Note that eq 30 implicitly defines the rms value of y " O . (introduced in eq 12), since in the weak-coupling limit pertinent to the ground-state process, k,,, is proportional to Hi(l)f(m)2 and hence also to (yfi.)2. 111. Results and Discussion A . Structure and Activation Energies. Since the inner-shell activation energy is crucially dependent on the variation of re value with oxidation state,4 we compare calculated re values with available experimental data in Table I, including previous S C F and INDO results as well as the S C F and UMP2 results from the present study. The a b initio re values are clearly too large, although inclusion of electron correlation yields significant improvement. At either level (SCF or UMPZ), however, the deviations from experiment are nearly constant, 0.10-0.14 A for the present S C F results and -0.07-0.09 A for the a p p r ~ x i m a t e ' ~ UMPZ results, thus underscoring the fact that the quantities important for the activation energies (Le., Are = r r are treated reliably. The earlier INDO resultss are in qualitative conformity with the other data, but the overall variation in re values (0.12 A) is only 55% of the observed range (0.22 A). The states involving a single eg electron can be expect to display appreciable Jahn-Teller distortion, and we calculated at the S C F level a disproportionation of +0.12 and -0.06 A, respectively, for the axial and equatorial CONbonds (relative to the Qhstructure). The same structural result was obtained in an earlier a b initio study,s although the energy lowering reported there (-2000 cm-l) is more than twice that obtained (840 cm-l) from the present U H F calculations. Table I1 contains the inner-shell activation energies calculated for the 1Al,/4Tl, and 'AI,/ZE, redox pairs (see eq 15), together with the relevant geometrical parameters, r* and Are. We focus on the IA,,/2E, excited-state pair since this is found (see below) to provide a lower energy pathway than the alternative 3Tl,/4T,, excited pathway (see Scheme I). For the redox process involving ground-state configurations (1Aig/4Tig), both the S C F and MP2
e)
(19) The term "approximate UMP2" refers to the combined use of SCF energies and linearly fitted UMPZ correlation energies (see section IIA and Appendix A).
Newton
34 The Journal of Physical Chemistry, Vol. 95, No. I , 1991 TABLE I V Sensitivity of A E (10’ cm-I)
TABLE 11: Calculated Inner-Shell Activation Energies redox
r*,b A
E’,” kcal/mol
Are:
A
level“ U M PZd cxpt‘ ’AI,l2E,
SC F
UMPZd
15.9 16.8 17.6 9.6 8.6
2.18 2.14 2.05
0.23 0.22 0.22
2.15g.h 2.1 OBI*
0.171 0.148
SCF (no 4f) SCF ( 4 0 UMP2b (no 4f) UMP2b (4f) UMP2b (4f, empirical
“ S C F and UMPZ values are based on eq 15. bCommon value of 9’ and r3+ in inner-shell transition state (mean value of rar and rq for the - r:’. dSee footnote d, Table I. iA18/2E, redox pair). rRefcrcnccs 4 and 5 , with E’ based on eq 31. ’Includes calculated (SCF level) diffcrcntial Jahn-Teller stabilization at the transition state (440 cm-I) and at equilibrium (840 cm-I). XBased on 0,geometry (SCF results from the present study). D4,, Jahn-Teller distortion at the transition-state results (at the S C F level) in changes of +0.06 A ( r a x )and -0.03 8, (r,) relative to the indicated mean value for 0, framework geometry.
ce
*
TABLE 111: Calculated Vertical Multiplet Splittings” ( lo3 c d )
Co3+ equilibrium ( r e ) transition state
r 2.04 2.14
AE3+(r) 13.7b 8.86
co2+ r 2.26 2.10
Values to Level of Calculation AE3+(2.05 A)
AE2+(2.25 A)
0.6 1.2
18.9 18.6 13.7 13.2 8.8
8.6 13.0
a Results are based on calculations with or without f-type basis functions on Co, as indicated. bSee footnote d , Table 1. cEmpirically corrected as described in the text (see eqs 13 and 14).
levels of calculation is indicated in Table IV. While the excited spin states for both the 3+ and 2+ species are calculated to be relatively low lying at the equilibrium geometries for the ground-state configurations, of particular significance for the kinetic question at hand is the fact that the AEz+ values are even smaller at the 1Al,/4TI, transition state ( r * ) . Accordingly, the spin-orbit mixing at r* is enhanced relative to that occurring at equilibrium. Using the calculated data, we estimate the slopes of AE3+ and AE2+ at r* to be:21
AE2+(r) 9.1bd 5.3b.d
d --E2+
“See footnote d of Table I. bAdjusted by an empirical correction term ( f 4 . 4 X IO’ cm-I for Co” and Co2+, respectively), as described in the text (see eqs 13 and 14). ‘Estimates in the range 3.3-4.7 cm-’ were obtained from INDO SCF/CI calculations reported in ref 8. An earlier estimate of 9.0 X IO’ cm-I has been cited in the literature,2 but without detailed documentation. earlier estimatesi@of 6.4 X IO’ cm-I ( r e ) and 2.2 X IO’ cm-’ ( r * ) were based on more limited SCF results, as discusscd in the text. rFor the 1Al,/4T,, redox pair (see Table 11).
The AE3+(r*)and AE2+(r*)estimates together with eqs 16, 17, and 20-22 yield the following spin-orbit coupling coefficients:
results are close to the experimental estimate of 17.6 kcal/mol obtained on the basis of eq 3 I ,4*5 Ei,* = ( 7 2 ) k ( A r e ) 2
(31)
where k is a mean force constant based on experimental force constants for the totally symmetric breathing mode of the CON bonds ( k = 2k2+k3+/(k2+ k3+)) and where Are is obtained from diffraction data.4 Equation I5 is somewhat more general than eq 31 in that the latter is based on the assumption of a harmonic energy surface. The E* values obtained for the 1A,,/2E, redox pair are smaller than those for the 1Al,/4Tlqpair, as expected in view of the smaller Are value. The calculated AI.J2E, values include the differential Jahn-Teller stabilization energy at r* (440 cm-I) and at re (840 cm-I). At the transition state we calculate Tax* and res* to be, respectively, 0.06 %, longer and 0.03 8, shorter than the r* value for Oh symmetry. Comparison with Table I reveals that for both redox pairs the calculated r * values are within 0.01 %, of the arithmetric mean of the corresponding r y and r:+ values. The inclusion of correlation leads to a decrease of E* for the 1Al,/2E, pair, in contrast to the situation for the 1A1,/4TI pair, a result attributable to the larger reduction in Are in the Former case
+
B. M u l t i p l e t S p l i t t i n g and Spin-Orbit Coupling Coefficients.
The calculated vertical multiplet splittings are presented in Table 111. As described in section II.A, all UMP2 AEz+ values have been corrected by an additive constant which brings AE3+(re)into exact agreement with the experimental value (Le., c = 4.4 X IO3 cm-’ in eqs 1 3 and 14).20 The sensitivity of AEz+values to various (20) The determination of the correction term c by adjusting the calculated AE’+ value at the calculured re value through the use of the observed AE3+ value, which pertains to the experimental re value, yields a consistent basis for evaluating A/?+ and AE2+ as a function of r , provided that the various potential energy curves with respect to rare uniformry displaced (horizontally) relative to the true curves. A com rison of calculated (UMP2) and observed re values in Table I and the )Atg/ga TI, E’ value in Table I1 provides evidence of such uniformity.
dr
= +36 X IO3 cm-’/A
(33)
c + ( r * )= 0.17
(34)
c+(r*) = 0.04
(35)
q ( r * ) = 0.13
(36)
Although the strength of spin-orbit coupling is thus found to be greater for the 3+ oxidation state (in spite of its larger energy denominator), we see, nevertheless, from eq 26 and the p+ and A F + ( r * ) values, that the 2+ oxidation state makes a greater contribution to the spin-orbit enhancement of H i p As discussed in detail above, the present AE2+ values are somewhat larger (by -3000 c d ) than the previous estimates1& based on more limited computational data, but they still imply a significant degree of spin-orbit coupling (eqs 34-36). The present AE2+(re)values are also -5000 cm-’ larger than the earlier estimates* based on empirically adjusted INDO SCF/CI calculations. As a result of its empirical parametrization, the INDO method has given a generally good account of various electronic excitation energies.22 However, with specific reference to the C O ( N H ~complexes, )~ it should be noted that the discrepancies in bevalues obtained by the INDO calculations9 (see Tables 1 and 11) seem likely to bias the procedure used to determine AE2+ (including the evaluation of AE2+ using INDO energies based on estimates of the experimenral r* values) in favor of the low-spin (2E,) state. Furthermore, we note that a higher doublet state, 2T,,, is likely to have a vertical excitation energy of -15 X lo3 cm-123(it has been observed for other amine complexes ( C ~ ( e n ) , ~and + Co(sep)2+), but not for the parent, CO(NH,),~+),whereas the (corrected) INDO estimate is 5 1 3 X lo3 Thus the INDO estimate for the 2E, energy should perhaps be considered a lower limit. (21) These slopes, based on approximate UMPZ results,I9 may be comy e d with earlier resultslW(-46.3 X lo3 cm-I/A (3+) and +29.6 X lo3cm-I (2+)) which employed linear fits of the full UMPZ AEz+ values. (22) Anderson, W. P.; Edwards, W. D.; Zerner, M. C. Inorg. Chem. 1986, 25, 2128. (23) (a) Yang, M. C.-L.; Palmer, R. A. J . Am. Chem. Soc. 1975,97,5390. (b),Creaser, I. I.; Harrowfield, J. MacB.; Herit, A. J.; Sargeson, A. M.; Springborg, J.; Geue, R. J.; Snow, M. R. J . Am. Chem. SOC.1977,99,3181. ( c ) Geselowitz, D. Unpublished work.
The C O ( N H ~ ) ~ ' + /Exchange ~+ Reaction
The Journal of Physical Chemistry, Vol. 95, No. 1, 1991 35
The UMP2 level of calculation is certainly not adequate for quantitative determination of multiplet splittings. However, the combination of a plausible empirical correction with the demonstrated systematic treatment of re values (Table I)z4suggests that the present approach can yield a useful estimate of AE2+. In the remainder of this paper we proceed to evaluate the likelihood of ground vs thermally excited pathways on the basis of the empirically adjusted UMP2 energetics. In the final section we consider how our conclusions may be affected by uncertainties in the AE2+ estimates. C.Kef for Ground-State Processes. Using the approach outlined in section 11 and Appendix B, we may evaluate K,, according to eqs 27 and 30. I n addition to the parameters that characterize the spin-orbit states and those that arise in eq 10 (taken from refs 4 and 5 ) , we re uire estimates of the mean "one-electron" transfer elements, h$ (see Appendix B). With the A F + ( r * ) values given in Table 111, we calculate
TABLE V: Differential Contributions to the Energies of the lAl#"le and lAle/*EeTransition States
AG* = G*(lA,,/2Eg) - G*(1Al,/4TI,)"
contribution zero order (0, diabatic states)b
+4.6 (IO3
nuclear tunnelingC Jahn-Teller distortion avoided crossing" electronic nonadiabaticityd,e net difference
+0.3
cm-I)
-0.4
-0.7 -1.4 +2.2
'These energy differences are labeled AG* since the contributions arising from the r, and K,, prefactors involve effective - T U * terms. bunlike the two E* quantities in Table I1 (where the zeroes of energy are the energies of the respective equilibrium configurations ('Al,/'T1, or 'Al,/2E,), this quantity is the difference of the energies (relative to a common zero of energy) of the two transition-state configurations. CTunnelingfactors (r,)of 9.0 and 2.4 were obtained, respectively, for the IA18/4TIB and lA1J2E, pathways, using the harmonic semiclassical expression given in eq 69 of ref 5. For the 'A,,/4TI, pathway, the empirical parameters of ref 5 were employed. The corresponding parameters for the lA,,/2E8 case were obtained by scaling, using the ratios of the calculated Ar and zero-order inner-shell E* values for the two pathways. The effective -TAS* terms are obtained as -RT In (r,,). "This first-order correction arises essentially entirely from the Hirelement for the IA,,/2E, pathway, estimated to be -700 cm-I for the apex-to-apex approach, based on INDO calculations'w (see section lllC and Appendix B). In cases where the barrier is dominated by solvent reorganization (in contrast to the present situation), it has recently been shown that the leading correction to the diabatic barrier should be second-order in the matrix element.40 e The contributions due to electronic nonadiabaticity are included as effective -TAS* terms [is., -RT In ( K e 1 ) ] .
where the bar refers to an rms average over e, components, as described in Appendix B.zs The coefficient of 0.0135, when squared, implies an attenuation factor in K ~ of , 1.8 X lo4 due to spin-orbit coupling (yS'O'in eq 12). A similar factor (1.3 X IO4) was obtained by Buhks et aL2 although the detailed contributions are quite different as a result of their use of larger estimates of AEZ+and AE3+ (based on estimates for re) and the neglect of multideterminantal effects (see Appendix B). The magnitude of h& and hence &, depends on the particular orientation of reactants in the transition state. We first consider the apex-to-apex approach, for which estimates of h& (where eg d22, with the z axis parallel to the Co-Co vector) range from Employing the results of INDO calculationsz8for hL:La,as reported 700 cm-I (INDO)"" to 940 cm-I (an ab initio result, obtained'& by scaling the matrix element for the C O ( N H ~ ) ~ + ( N H ~ ) C O ~in+ Appendix B, and the simple statistical analysis presented in ref 8, we estimate an overall (i.e., orientationally averaged) mean cluster)'0 to 1800 cm-' (extended Huckel).* As discussed in value A pendix B, we estimate that the Boltzmann-weighted rms value, hi L,. for the apex-to-apex orientation is within 10% of the h&1]d,2 %I 0.1 (ReJapex-tc-apex (41) value. It should also be noted that the intervalence charge transfer spectral intensit for the C O ~ + ( N H ~ ) ~ ~ + / R U ( C ionN pairZ6?'' )~' and accordingly, that the orientationally averaged value, t,,, is suggests4 an h$, value 2800 cm-' in magnitude. While the based on the present computational modeLZ9 charge-transfer process is not well characterized, it may actually Before confronting the available experimental kinetic data, we be rather analogous to the 'A,,/'E, one-electron pathway for first consider the alternative pathway involving the thermally C O ( N H ~ ) ~ ~ + / C O ( Nexchange H ~ ) ~ ~ +since the final co2+state in excited 'E, state of C O ( N H ~ ) ~ ~ + . ~ ~ the ion-pair charge transfer is most likely low spin (2Eg),"and since D. Comparison of Ground and Thermally Excited Pathways. the tzs electron being transferred from Ruz+may have the shape The contributions to the difference in free energy of activation3' of a d2zorbital, where in this case, the z axis is along a 3-fold axis for the 'Al,/4Tlg and 'Al,/2E, pathways based on an apex-to-apex of the Ru(CN),~- system (i.e., the "alsn orbital discussed in orientation are given in Table V. Defining Appendix A and also in ref 10). If we select the INDO value AG* = G*('A,,/'E,) - G*('Ai.J4T1,) (42) (700 cm-I) as an estimate for the apex-to-apex matrix element, h& we obtain we note first a reference energy of +4.6 X lo3 cm-I associated with the crossing points of the diabatic energy surfaces, assuming Air = 9.5 cm-l (39) 0,symmetry for both reactants. This value is increased slightly R,, = 1.2 x 10-3 (40)
-
-P
-
Thus we conclude that at least for the apex-to-apex orientation, which allows very effective C-type overlap of dzz orbitals,Iobthe spin-orbit-facilitated ground-state process suffers a relatively modest degree of attenuation due to electronic effects (in comparison with the very small probability expected for the three-electron process occurring in the absence of participation by excited electronic configurations (see Scheme I). The overall kinetic process must, of course, include contributions from other reactant orientations, in proportion to their statistical significance.8
(28) For a number of smaller, model clusters (including (FeH20)2s+, (CONH&~+, and (Fe(H20)3)2st), where comparisons with a b initio results are possible, we find a very systematic correlation between the values of INDO and ab initio matrix elements, with agreement generally to within -20% (Newton, M. D. To be published). (29) The orientational averaging of K~~ is defined in terms of the meansquare value of H"' and thus differs from ref 8, where the magnitude of hit) was averaged. Thci2ariation in Co.-Co separation (rcaco)associated with vaf? der Waals contact of the reactants in the different orientations (see Appendix B) will affect the preequilibrium constant, Kq. and the solvent contribution to the activation energy (see eq 6). In the present approximate treatment, the values of these latter quantities were based on the mean value rcoc0 = 6.6
A.5 (24) That is, the reasonably uniform shift of r values relative to experiment, which yields an inner-shell E* value for the fAl,/4TI, pathway in good agreement with the empirical estimate (Table 11). (25) Since for weak coupling the rate constant is proportional to Hit (see eq IO and preceding discussion), the mean value of Hifis most appropriately defined in terms of (Hio2. Hence our use of the rms quantitytk (cf. ref 8). (26) Vogler, A.; Kisslinger, J. Angew. Chem., In?. Ed. Engl. 1982, 21, 77. (27) Vogler, A . Unpublished work.
(30) The calculated energies for the ground and excited surfaces neglect small second-order energies due to the spin-orbit coupling (the small firstorder terms in the ground state were included (see eq 27)). Spin-orbit contributions are of minor importance for Hi, on the excited pathway, for which K,, 1. (31) The quasi-thermodynamic terminology is used merely as a convenient device for comparing the various factors in the rate constants for the groundand excited-state pathways.
-
Newton
36 The Journal of Physical Chemistry, Vol. 95, No. I , 1991 TABLE VI: Comparison of Theoretical and Experimental Estimates of k,," ke,, s-I M-' k)!'"'/k".:t theoreticalb ground state (1A18/4T18)c nonadiabaticd 4 x 10-10 (adiabatic)' (4 x IO") experimental8 L I 0-7
TABLE VII: Electronic Configurations Employed in the SCF Calculations" low spin (]Alg) high spin (3Tl,)b
CO(NHg)63+ (t2$ (3dXy)-'(3d,2,2)
low spin (2E8) high spin (4Tl,)b.c
CO(NH&~+ (t2 )6(3d,4 (33aJ-I (3dzz) (3dxz-y2)
54 x 10-3e (54 x l0')C
Rate constant for the aqueous CO(NH,),~+/)+exchange at 25 OC, 1
M ionic strength. bEquation 6. CThe rate constant for the excitedstate pathway is estimated to be several orders of magnitude less than that for the ground-state pathway (see section IIID). d K , ] (eq IO) evaluated on the basis of the present calculations (with estimated orientational averaging), together with quantities given in ref 5 (vi, = 409 cm-': Eint= 17.6 kcal/mol; E,,,,' = 6.8 kcal/mol); the other factors (see cq 6) are given in ref 5 . 'Inequalities based on the assumption that the experimental estimate' is a lower limit. /Based on eq 6 , but with K,, = I (see ref 5 ) . gReferences 7 and 15.
(+275 cm-I) by differential nuclear tunneling effects (i.e., the difference of the two -RT In (r,)terms), but decreased by 1140 cm-l due to the combined effects of Jahn-Teller distortion and avoided crossing, both of which occur on the 1A,,/2E, surface. An additional lowering of 1400 cm-I arises from the effective - T a t term due to the electronic nonadiabaticity associated with the 1A,,/4TI, pathway, yielding for the net difference an estimate of 2.2 X 1 O3 cm-' or 1OKT at room temperature. The avoided crossing and the nonadiabaticity terms arise from the interaction of the two reactants and depend on the magnitude of the transfer element Hip Recalling eq 41, we expect the rate constants of both the ground-state and the excited-state processes to be reduced by roughly an order of ma nitude relative to those based on the apex-to-apex value for when orientational averaging is taken into account.32 In short, the present calculations indicate that the thermally excited pathway is not likely to be competitive with the ground-state process at room temperature. In contrast, the INDO studys could not rule out the possibility of a lower activation energy for the 1Al,/2E, pathway. E . Comparison with Experiment and Concluding Remarks. In Table VI we compare the estimates for k,, based on theoretical calculation and on experiment. The present theoretical value is seen to be more than 2 orders of magnitude smaller than the most recent experimental estimate.' The latter quantity should perhaps be considered as a lower limit, although the analysis of the kinetic data is complicated because of the large number of chemical species and competing processes that are present. Because of these complications it was not possible to obtain activation parameters. A knowledge of these quantities would clearly help to pinpoint the source of discrepancies between theory and experiment. In the process closely analogous to the hexaamine reaction, eq 1, in which the bidentate ethylenediamine ligands replace the ammonias, there is less uncertainty regarding the kinetic parameters. Theoretical analysis of the kinetic data has suggested4 a value of iiel > to be compared to the present estimate of (the comparison is meaningful to the extent that the two reactions are governed by similar electronic states and electronic coupling). Although the present ab initio calculations and previous empirically based calculation^^*^ yield very similar estimates for the inner-shell portion of the lAl,/4T,, activation energy, both of these approaches employed the standard zero-order model based on separate reaction partners and may conceivably involve some systematic bias. This procedure has worked remarkably well in general4 but may be somewhat less satisfactory when dealing with the particularly large changes in bond lengths that arise in the cobalt ammine systems.33
-
&
(32) Based on an orientationally averaged rms estimate of -200 cm-l for The order-of-magnitude reduction in k, arises from the influence of h(') on K~~ and barrier suppression, respectively, for the IA1,/4T,, and 'Al,/2fi2 pathways. (33) For example, an uncertainty of iO.01 A in Arc causes an uncertainty of a factor of -IO*' in the calculated "nuclear factor" verrrn exp(-BE') (obtained using eq 1 I ) .
"The configurations listed are the nominal 3d-orbital configurations, based on the real orbitals: 3d,, 3d,,, 3d,,, 3d,z, and 3dx2-y2, where "3d$ stands for [3/2][2(3d,Z) - 3dx2 - 3dy2]. In the calculations, the CON bonds were parallel to the Cartesian axes. bFor the high-spin configurations only the hole in the tZgshell is indicated. CThe a,,-hole configuration, (3d,,J', where 3daI82 (1/31/2)(3dxy+ 3d,, + 3dy,), corresponds to a charge density that is totally symmetric in the D 3 d subgroup of the CON, framework. This D3d subgroup (with 3-fold axis corresponding to the symmetric sum of the Cartesian vectors) is the same as the Dld symmetry of the full Co(NH,), complex. The a,,-hole configuration yields lower U H F and UMP2 energies than the alternative hole state (3dX;') employed in an earlier study.'&
Aside from the activation energy, one must recognize the uncertainties in the calculation of the spin-orbit-enhanced transfer matrix element, Hif,including those due to the assumption that a single empirical correction term (independent of rCoN)can be applied to the calculated multiplet splittings for both charge states, as well as those associated with the evaluation of hL'J8. However, we emphasize that changes in the magnitude of & would have comparable effects on the rate constants for the ground and thermally excited pathways.34 We also note, for example, that a reduction of AE2+(r*)by as much as 2000 cm-I (Le., from 5400 to 3400 cm-') would lead only to a factor of 3 enhancement of K , ~ . If, in fact, the actual AE2+(r*)value is appreciably smaller than the present estimate (as, for example, suggested on the basis of the earlier INDO calculation^^),^^ then one may expect an interesting non-Arrhenius temperature dependence of the rate constant, reflecting the presence of two competitive processes at room temperature. It is hoped that the magnitude of AE2+ will soon become accessible to an experimental probe. In any case, the present calculations of multiplet splittings have focussed attention on the fact that spin-orbit coupling is considerably more effective at the transition state than at the equilibrium configurations of the reaction partners. Furthermore, we have seen that the spin-orbit ground-state mechanism can bring the kinetics of the Co(NH3)62+I3+exchange reaction reasonably close to "normal" behavior as defined by the adiabatic Marcus and as exemplified by the data of ref 4, in sharp contrast to the very slow kinetics expected in the absence of spin-orbit coupling. Acknowledgment. We acknowledge helpful correspondence with Prof. A. Vogler and also the receipt of material prior to publication. We are also grateful to Prof. D. Geselowitz for providing unpublished material. This research was carried out at Brookhaven National Laboratory under Contract No. DEAC02-76CH00016 with the US. Department of Energy and supported by its Division of Chemical Sciences, Office of Basic Energy Sciences. Appendix A. Details of the UHF Calculations The U H F calculations reported in the present study3' employ the same flexible spd basis of Gaussian-type orbitals discussed in detail in the earlier study:Ik ( 8 / 5 / 3 ) on Co, based on a contraction of the Wachters/Hay basis for C O , ~extended * with diffuse 2p-type GTOs, and a 4-31G basis39 for NH3. The U H F wave (34) For reasons analogous to those given in ref 32. (35) See, however, discussion in section IIIB, in connection with ref 23. (36) Marcus, R. A. Annu. Reo. Phys. Chem. 1964, 15, 155. (37) The calculations were executed with the GAUSSIAN 82 computer code: Binkley, S.; Whiteside, R. A,; Krishnan, R.; Schlegel, H. B.; Seeger, R.; De Frees, D.J.; Pople, J. A. Gaussian, Inc. ( 38) (a) Wachters, A. J. H. J . Chem. Phys. 1970, 52, 1033. (b) Hay, P. J. J . Chem. Phys. 1911, 66, 4377.
The C O ( N H ~ ) ~ ~Exchange + / ~ + Reaction
The Journal of Physical Chemistry, Vol. 95, No. I , 1991 37
functions employed in the UMPZ calculations a t r = 2.05, 2.15, and 2.25 A, also included a set of optimized 4f-type GTOs.'OC Hence in the procedure described in section IIA, in which UMPZ energies at other r values were obtained by combining UHF energies with linearly fitted estimates of the UMP2 correlation energy, E C O R R . the difference between U H F energies with and without the 4f functions, 6EUHF, was also included in the linear fit; i.e., EUMP2(,spdf)was obtained as EUHF(spd) 6EUHF EcoRR(spdf), with the latter two terms obtained from the linear fits. The C0(NH3), complexes were oriented so that the CON bond vectors were parallel to the Cartesian axes, and the N H 3 groups (with 3-fold axes coaxial with the respective CON vectors) were rotated so as to yield overall D3dsymmetry (with 3-fold axis having direction cosines of 1/3'12 with the Cartesian axes). The Djd perturbation of the Oh framework led to small splittings of the t26levels (S10-3au) in the closed-shell IA,, state (where each of the orbitals has the same occupancy), and for convenience we employ oh notation in referring to the various states and orbitals. The electronic configurations employed in the S C F calculations are given in Table VI1. The closed-shell !Al, state is totally symmetric in D3d symmetry. The three open-shell states are unrestricted with respect to both spin and (for the 3Tlgand 2E, states) the spatial symmetry (DJd) of the nuclei. The spin contamination is relatively minor: (S2) values of 0.75, 2.06, and 3.76, were obtained, respectively, for the ZE,, 3T1,, and 4T,gstates. In the previous study reported in ref IOc, the tig shell in both high-spin states (3TI,and 4TI,) was obtained by creating a 3dXyhole in the t;, shell. For the 3Tigstate, this procedure was required by the constraint of a single determinant. In the case of the 4T1,state it was subsequently determined that placing the hole in a t2, orbital of the type ( 1 (3)1/2)(d,y + d,, + d,,,) yields an S C F energy lower by -0.01 au. Relative to the filled !,t shell (which is essentially totally symmetric with respect to the Oh point group), this hole constitutes a D3d perturbation of the electronic density (the hole has a,, symmetry in D3d and is denoted below by "alg") which causes a slight mixing between the filled t2g orbitals and the half-filled e, orbitals (all of which transform as e, in the lower D3dsymmetry). This mixing is more significant than that caused by the minor D3d perturbation arising from the NH3 protons and accounts for the energy lower of 0.01 au relative to that of the 3dXyhole state. The d, hole constitutes a D4hperturbation (relative to oh),which does not cause any mixing between tzg and eg electrons. The present study employs the lower energy 4TIgS C F states based on the ai, hole. The UMPZ energies are then approximated by adding the previously obtained UMP2 correlation energies based on xy-hole states to the a,,-hole S C F energies. The additivity assumed in this procedure has been demonstrated to be reliable to -0.001 au in model UMPZ calculations carried out for a Co2+ion in the presence of a crystal field of Ohsymmetry. On the basis of the model studies, the 4TI, UMP2 energies obtained for C O ( N H & ~ +have been corrected by +0.001 au to compensate for the small effect of nonadditivity. Of the three spatially degenerate states, Jahn-Teller distortion was considered only for the 2E, state (a tetragonal distortion of the octahedral CON, framework in which the z axis is maintained as a 4-fold axis (ignoring the small perturbation due to the N H 3 protons)). The 3T1gstate also has an imbalance in the 3degorbital occupations and exhibits pronounced Jahn-Teller distortion.lZa However, in the present study, the 3Tlgstate arises only in connection with vertical energy splittings relative to the octahedral ] A l 6state. The 4TI, was assumed to have minor Jahn-Teller distortion (which we neglect) since the occupancies in the 3deg shell are balanced.
Indices I and m range over the four low-lying double-group levels 2G3/zg,and E5/2g). We assume that the Cartesian coordinate systems for the two interacting Co complexes (as defined in Appendix A) are related by translation, and we adopt the CY and spin-orbitals involving the 3d,z and 3d,tYz eg orbitals as the basis for the G3/2grepresentation at each Co center. We further assume that the interspecies matrix elements of h(I) are diagonal in this basis; i.e.
+
+
Appendix B. Evaluation of Matrix Elements Hi(r)f(m)
The evaluation of the elements involves the spin-orbit coefficients, $2 and the matrix elements of h(') (see section IID). (39) Ditchfield, R.; Hehre, W.J.; Pople, J . A. J . Chem. Phys. 1971, 54, 724. (40) Kim, H . J.; Hynes, J. T. J . Phys. Chem. 1990, 94, 2736.
h$i)e8b) = hLf)i)eg(i)6ij
(B1)
where i a n d j run over the four e, spin-orbitals (in the diagonal form we need only consider the spatial component, zz or x2 - yz). These assumptions are compatible with various limiting cases of relative orientation of reactants to which we confine our attention: Le., apex-to-apex, edge-to-edge, and face-to-face,8,i0where the Co-Co vector is taken, respectively, as the z axis, the bisector of the x and y axes, and the symmetric sum of the three Cartesian vectors. Then we find (cf. eq 25) that 7;: may be expressed as 7% = ((C3+)P,+ q+,((c3+)P, + G+)
where the coefficients
(B2)
PI are given by Po = P3 = 0
PI =
1/30'/2
P2 = 3/30i/2
(B3)
These coefficients (with magnitude C1, in contrast to eq 25, which implies all P = 1) arise from the multideterminantal nature of the wave functions for the various degenerate components of the 3T1,and 4Tl states. From eq B3 we see that nonzero values arise only when rand m refer to the 4-fold degenerate G3/2, representation; 7s; is diagonal with respect to the degenerate components of G3/2r(and independent of the particular component) if these components are taken as the e, spin-orbit basis defined above. Equations B2 and B3, together with eqs 16 and 17, and eqs 20-22, yield eq 26. It remains to specify the matrix elements h:1{,)e8(i)(see eq B1) which arise in the effective one-electron model for Hifintroduced in section IID. We evaluate the h(') elements using calculated values for the corresponding Mi)elements (see discussion following eq 12), emplo ing the notation Hi;') (see section IID). We recall that $Ag represents a mean of the values pertaining to the four distinct pathways indicated in Scheme I. In the following, we base values on calculations carried out for the pathway in which ah species are in low-spin states (IAIg and 2Eg). In previous (unpublished) ab initio studiesioaYbwe have ascertained that the values obtained for the different athways are roughly constant (Le., to within -20%). The @ values are based on a comprehensive set of IND022 calculations for suitable Co(NH3)62+/Co(NH3)63+ super-molecule clusters (in each orientation, the reactants are in van der Waals contact) employing delocalized wave functions as described in ref 10d (data given in ref 10d for metallocene/metallocinium redox couples indicate that similar matrix element magnitudes are obtained with localized and delocalized representations). We obtain the following values for apex-to-apex, 700 cm-I (dt), -0.0 (dA,,z); edge-to-edge, 120 cm-I (d,z-,,z), 40 cm-I (dzz); face-to-face, 75 cm-I (dzz and d,z-yz). Finally, we consider the relative contributions of the two e6 components ( d t and dA?) for each orientation of the reactants. While eq 27 constitutes a Boltzmann average based on the degeneracies of the isolated CO(NH&~+spin-orbit states (Le., the degeneracy), we recognize that, in the transition state, 4-fold G3/2g the crystal field from the Cos+ partner will in general cause some splitting of the Co2+ states (Le., a splitting of the d,z and d,z+ degeneracy). The simple form of eq 27 may be maintained provided that appropriate weighted values of are employed (denoted h$J. Using a simple a b initio S C F crystal field model, in which a Co2+ion interacts with a + 3 charge at the distance appropriate to each of the three reactant orientations (7.0, 6.9, and 5.8 A,
R
38
J. Phys. Chem. 1991, 95, 38-42
respectively, for the apex-to-apex, edge-to-edge, and face-to-face approaches), we estimate the following Boltzmann-weighted populations at room temperature (quantities given refer to % contributions from dZ2and dXLyl,respectively): 75, 25 (apex-toapex): 35, 65 (edge-to-edge); 50, 50 (face-to-face, where the eg
degeneracy is maintained). These populations are then used to average the hi:: values (in the rms sense). In section IIIC, we continue with the effective one-electron Hamiltonian notation, hi:!, but with the understanding that these matrix elements are evaluated in terms of the H(,:2g elements, as described above.
Heavy-Atom Effects on the Excited Singlet State Electron-Transfer Reaction Koichi Kikuchi,* Masato Hoshi,+Taeko Niwa, Yasutake Takahashi, and Tsutomu Miyashi Department of Chemistry, Faculty of Science, Tohoku University, Aramaki Aoba, Aoba- ku, Sendai 980, Japan (Received: April 5, 1990; In Final Form: July 2, 1990)
Heavy-atom effectson the free radical yield @R and the triplet yield GT of the fluorescence quenching were studied in acetonitrile by using 9,lO-dicyanoanthracene as the electron-accepting fluorescer and a series of para-halogenated anisole (I), aniline ( I I ) , and N,N-dimethylaniline (111) as the electron-donating quenchers. @T increases as the atomic number of the halogen substituent increases for all the systems, whereas @R decreases for the system I and does not change for the systems 11 and 111. These heavy-atom effects are interpreted in terms of the spin-orbit coupling between the singlet exciplex and the locally excited triplet state for the system 1, and in terms of the spin-orbit coupling between the geminate radical pair state with singlet spin and the locally excited triplet state for the systems I1 and 111.
Introduction
Heavy-atom effects on the photoinduced electron-transfer (ET) reaction have been extensively studied for the triplet-state ET reaction, where the free-radical yield extremely decreases on substituting a heavy atom into the triplet quencher.l** In a previous work3 we demonstrated that such heavy-atom effect is caused by the spin-orbit coupling between the geminate or encountered radical pair state with triplet spin and the singlet ground state, when the members of the radical pair contact with each other before the diffusive separation of the radical pair into free radicals. In this sense, an electronic interaction between the members of the radical pair might lead to the display of characteristics similar to those of a singlet exciplex. In the case of the excited singlet state ET reaction, the triplet molecules may be produced in addition to the radical In a highly polar solvent such as acetonitrile, it has been indicated that the ET fluorescence quenching occurs at a long distance (-7 A) of the flouorescer and the q ~ e n c h e r Thus . ~ it has been supposed that the ET fluorescence quenching in acetonitrile is caused by a weak electronic exchange interaction between the fluorescer and the quencher, and hence the primary quenching product is the geminate radical pair with singlet spin, the members of which are separated by solvent molecules. This type of ET may be called as the full ET or the direct ET. This supposition has recently been confirmed by the studies on the free enthalpy dependence of the free radical aRof the ET fluorescence quenching:,' because aR has satisfactorily been interpreted with a semiclassical ET theory8-Io which assumes a weak electronic exchange interaction between the electron donor and acceptor. If the energy of a radical pair is higher than the energy of a locally excited triplet state, and if there is a spin-orbit coupling between the geminate radical pair state and the locally excited triplet state when the members of the geminate radical pair contact with each other, the triplet formation is expected to be enhanced by substituting a heavy atom into the quencher. The fluorescence of an electron donor-acceptor system may also be quenched by the partial ET, Le., the exciplex formation. Therefore, the fluorescence may be competitively quenched by the full and partial ET according to the general reaction scheme given by Weller." The former is predominant in a nonpolar solvent whcreas the latter in a polar solvent. When the exciplex 'Present address: Biological Laboratory, Kao Co.,2606 Akabane Ichikawamachi, Haga-gun, Tochigi 321-34, Japan.
is formed in a highly polar solvent such as acetonitrile, it may rapidly dissociate into the geminate radical pair without giving the exciplex fluorescence. Even if the exciplex fluorescence cannot be observed, therefore, it has to be kept in mind that there are two possible pathways for the excited singlet state ET reaction in a polar solvent which are very hard to discriminate. In fact Iwa et al.'* have already suggested the competitive formation of the geminate radical pairs and the exciplex in the fluorescence quenching in methanol. It is noted that the difference between the exciplex state and the contact ion pair state has clearly been depicted by FO11 et aI.l3 In this work we study the fluorescence quenching of 9,lO-dicyanoanthracene (DCA) with para-halogenated anisole (X-AS), aniline (X-AL), and N,N-dimethylaniline (X-DMA) in acetonitrile. It is shown that the heavy-atom effects on the free-radical yield aR and the triplet yield aTof fluorescence quenching are useful to discriminate whether the quenching is due to the partial ET (the exciplex formation) or the full ET, and that aT is extremely increased by the spin-orbit coupling between the exciplex and the locally excited triplet state, but slightly by the spin-orbit coupling between the geminate radical pair state with singlet spin and the locally excited triplet state. Experimental Section
Materials. Aniline (H-AL), N,N-dimethylaniline (H-AL) (Nakarai), anisole (H-AS), 4-chloroanisole (CI-AS), and 4(1) Steiner, U.; Winter, G . Chem. Phys. Lett. 1978, 55, 364. (21 Ulrich. T.: Steiner. U. E.: Foll. R. E. J . Phvs. Chem. 1983. 87. 1873.
(3) Kikuchi, K.; Hoshi, M.; Abe, E.; Kokubun: H . J . Photochem. Photobioi., A: Chem. 1988, 45, 1. (4) Leonhardt, H.; Weller, A. 2.Phys. Chem. (Munich) 1961, 29, 277. (5) Nibbe, H.; Rehm, D.; Weller, A. Ber. Bunsen-Ges. Phys. Chem. 1968, 7-, 2 -7 5 7.
(6) Kikuchi, K.; Takahashi, Y.; Koike, K.; Wakamatsu, K.; Ikeda, H.; Miyashi, T. Z.Phys. Chem. (Munich), in press. (7) (a) Could, I. R.; Ege, D.;Mattes, S. L.; Farid, S. J . Am. Chem. SOC. 1987, 109, 3794. (b) Could, 1. R.; Moser, J. E.; Ege, D.;Farid, S. J . Am. Chem. SOC.1988, 110, 1991. (c) Could, I. R.; Ege, D.;Moser, J. E.; Farid, S. J . Am. Chem. SOC.1990, 112, 4290. (8) Ulstrup, J.; Jortner, J. J . Chem. Phys. 1975, 63, 4358. (9) Efrima, S.;Bixon, M. Chem. Phys. 1976, 13, 447. (10) Miller, J. R.; Beitz, J. V.; Huddleston, R. K. J . Am. Chem. Soc. 1984, 106, 5057. (11) Weller, A. Z . Phys. Chem. (Munich)1982, 130, 129. (12) Iwa, P.; Steiner, U. E.: Voneimann. E.: Kramer. H. E. A. J . Phvs. Chem. 1982, 86, 1277. (13) Foil, R. E.; Kramer, H. E. A.; Steiner, U . E. J . Phys. Chem. 1990, 94, 2476.
0022-365419 112095-0038$02.50/0 0 1991 American Chemical Society
