3734
J. Phys. Chem. 1986,90, 3734-3739
small variations with Em. Therefore, eq A.12 and A.13 can be expressed as follows. knr = ko ~xP(-AEoo) (A.14) A = y’/hum
ko
Ck2Wk(T/2hWmEk)1/2eXp(-S,)
(A.15)
S ( E ) = (B2r)-’j2 exp[-(E - C ’ X , ) ~ / B ~ ] (A.17) (A.18)
modes and into the dominant acceptor mode is constrained to be equal to the effective energy gap Ek. As mentioned previously, the maximum overlap for the low-frequency modes occurs when The rate for this optimal the amount of energy they accept is C ’ X ~ decay pathway is expressed as in eq A.19. The first-order cor-
k,, = ko(B2n)-’/2eXp[-A(Ek - C’XI)]
(A.19)
rection of decreasing the effective energy gap from Ek to EOkis seen to result. Contributions must also be included in all of the near-optimal decay pathways where the low-frequency modes absorb an amount of energy C’x, E and the dominant acceptor - E. The summation over the different pathways mode absorbs EOk can be expressed by the following integral.
+
(A.20)
This integral is readily solved by standard methods to give the result
k,, = ko eXp(-AEok
(A.16)
where k , and A can be taken as constant. The contributions of the low-frequency modes are now included. The overlap function S ( E ) for these is taken to be represented by a Gaussian bandshape centered at C’x,as given in eq A.17 and A.18. The sum of the energy that is dissipated into these B2 = C ’ 2 h ~ l ~ 1 ( 2 + 9 , 1)
k,, = ko(B2r)-’/z~exp[-A(Eok - E ) - E2/B2] d E
+ A2B2/4)
(A.21)
With the original definitions, the form of the extra term is A2B2/4 = ( y ’ / h ~ , ) ~ C ’ h ~ , x , ( 2 9+, 1)/2
(A.22)
This term is very similar to the fourth term on the rhs of eq 9, which is the second-order correction for the presence of the low-frequency modes. The only differences are in the substitution 1, which also involves Ek in place of EOkin the of y’ for y definition of y. A likely origin for these in the mathematics of the present derivation is in the assumption that the value for y is constant and not a function of Ekand in the approximation of the overlap function as a Gaussian. The net origin of the extra term remains the same. It arises from the incorporation of nonoptimal decay channels because of the breadth of the overlap function for the low-frequency modes. Registry No. 1, 89711-31-9; 2, 80502-69-8; 3, 102725-01-9; 4,
+
80502-67-6; 5, 81831-21-2; 6, 89689-79-2; 7, 75441-74-6; 8, 80502-63-2; 9, 80514-59-6; 10, 75441-72-4; 11, 80558-57-2; 12, 80502-57-4; 13, 80502-55-2; 14, 80502-59-6; 15, 80502-53-0; 16, 75441-70-2; 17, 102725-02-0; 18, 23648-06-8; 19, 47779-78-2; 20, 96964-80-6; 21, 102725-03-1; 22, 75446-26-3; 23, 80502-83-6; 24, 80502-85-8; 25, 75446-24-1; 26, 80502-81-4; 27, 80502-79-0; 28, 80558-59-4; 29, 80502-77-8; 30, 80502-75-6; 31, 3 1067-98-8; 32, 80502-73-4; 33, 80502-71-2.
Comparison of Electron-Transfer Matrix Elements for Transition-Metal Complexes: t,, vs. e, Transfer and NH, vs. H,O Ligands Marshall D. Newton Chemistry Department, Brookhaven National Laboratory, Upton, New York 1 I973 (Received: January 21, 1986; In Final Form: April 1 , 1986)
Electron-transfer matrix elements are calculated for several pairs of transition-metal complexes of the type ML62+-ML63 + 7 chosen so as to allow a comparison of t2,(r) vs. e,(o) transfer and NH3vs. H 2 0 ligands, primarily for the apex-to-apex orientation of reactants (M = Fe, Co, and Ru). The magnitude of the matrix element has a pronounced dependence on transfer type (e, > tZE)and ligand type (NH3 > H20), an effect which is correlated with the degree of mixing of the metal and ligand orbitals, based on a corresponding orbital analysis. Use of the calculated matrix element for the CO(NH~):+/~+electron exchange, in conjunction with a semiclassical kinetic formalism and a spin-orbit attenuation factor, yields a rate constant for the ground-state reaction path which is about 2 orders of magnitude less than the experimental rate constant. (Orientational averaging would lower the calculated rate constant even further.)
I. Introduction Mechanisms of electron-transfer reactions are strongly influenced by the electronic structure of the reacting species and the medium through which the transfer occurs.’ A quantity of central importance in this regard is the electron-transfer matrix element, H{f, which couples the initial ($J and final ($f) states, Le., reactants and products, in an electron-transfer process. In the = Sir# 0, we have general case where l+i*$f H’if = ( H ; f- HjjSi,)( 1 - Si?)-’ (1) where Hif = l$i*H$rand H i s the Hamiltonian operator for the system of The classification of an electron-transfer process with respect to adiabatic vs. nonadiabatic limiting cases (1) Newton, M. D.; Sutin, N. Annu. Rev. Phys. Chem. 1984, 35, 437. (2) Logan, J.; Newton, M. D. J . Chem. Phys. 1983, 78,4086.
0022-3654/86/2090-3734$01 .SO10
is strongly dependent on H{f: typically, IHtifl> K T (-200 cm-’ a t room temperature) corresponds to the criterion for adiabatic behavior.’ The dominant role of H{rin determining the electronic transmission factor, K ~ ’ ,in an electron-transfer process is evident in the following expression, valid in the nonadiabatic regime (K,,
