J . Phys. Chem. 1989, 93, 3030-3034
3030
Nonadkbatk Quantum Mechanical Treatment of the Absorption Line Shape of Bridged Mixed-Valence Dimers Li-Tai Zhang,+ Jaeju KO,$and Mary Jo Ondrechen**$ Department of Chemistry, Northeastern University, Boston, Massachusetts 021 15, Department of Chemistry, Indiana University of Pennsylvania, Indiana, Pennsylvania 15705, and Kemisk Institut, Aarhus University, DK-8000 Aarhus C, Denmark (Received: August 2, 1988)
A nonadiabatic quantum mechanical treatment of the absorption line shape of bridged mixed-valence dimers is given. The method applies to delocalized and intermediate cases and is based on perturbation theory. A three-site, three-mode Hamiltonian is utilized. The zero-order basis set consists of products of molecular orbital electronic functions and harmonic oscillators. The vibronic coupling terms are treated perturbatively. It is shown that the formerly controversial Creutz-Taube ion, a pyrazine-bridged mixed-valence dimer of ruthenium, is strongly delocalized, consistent with experimental data and with our earlier work. Nonadiabatic mixing of upper and lower surfaces is negligible in this particular case. Expressions for calculating the optical absorption spectrum are given for the general class of compounds.
Introduction A nonadiabatic quantum mechanical treatment of vibronic coupling in bridged mixed-valence dimers is presented. Much attention has been given in recent years to bridged mixed-valence dimers.'-3 Studies of these compounds have raised important questions about spatial and temporal delocalization of electrons within molecules and about how these effects are manifested in common spectroscopic techniques. The general issue of the delocalization of molecular electronic states over space and over time is an important and recurring one in chemistry. These questions were raised in organic chemistry in the context of valence tautomerism4 and again in the context of classical versus nonclassical carbonium ions.5 Major advancements in inorganic synthesis in the past 20 years, pioneered by Taube,' have introduced new classes of mixed-valence and bimetallic complexes about which these same questions have arisen. The advent of bridged mixed-valence dimers, where two metal ions are joined together by some bridging ligand, has added a new dimension to the problem of intramolecular electron delocalization. Synthetic alteration of the bridging ligand can lead to substantial changes in the extent of metal-metal electron exchange coupling, in the rate of intramolecular electron transfer, and in the spectroscopic properties of the dimer.2,6 In addition, extended chains of metal ions and bridging ligands -M-B-M-B-Mare potentially very exciting, because of the possibility of conductivity along the M-B axis and because of the possibility of control of the conductive properties through synthetic alteration of the bridging ligand.' Of the dimeric compounds in this class, the Creutz-Taube ions probably has generated the most attention and c o n t r o ~ e r s y . ~ - ' ~ This cation is a pyrazine-bridged mixed-valence dimer of ruthenium with the structure
'0'
T+
~ H ~ ~ ~ R U - N ~ N - R Y I N H J ) ~
In the near-infrared region of the optical absorption spectrum there is a broad, strong transition (vmx = 6400 cm-I; AD,,, = 1200 cm-l; t = 5500 M-l cm-I) originally assigned to intervalence transfer (IT). The earlier papers about this ion were focused primarily on the question: Is the ground state localized (mixed-valent) or delocalized (average-valent)? Subsequent theoretical papers have been focused on development of a correct model for the ground state that also accounts for all of the experimentally observed properties 'Northeastern University. 'Alfred P. Sloan Foundation Fellow. On sabbatical leave at Aarhus University 1986-1 987. Permanent address: Northeastern University. * T o whom correspondence should be addressed. Indiana University of Pennsylvania.
0022-3654/89/2093-3030$01.50/0
of the complex. These observations include (1) the width and shape of the near-IR absorption band, (2) the solvent independence of this band's shape, (3) the absence of tunnelling transitions in the far infrared,19 and (4) the anisotropy of the electron paramagnetic resonance (EPR) g Piepho, Krausz, Schatz, and Wong were the first to investigate the complete shape of the near-IR absorption band for mixedvalence dimers.'2b'13b*21,22 Their approach is based on a two-site model with a polaron-type coupling. Their method solves the full vibronic coupling problem and incorporates nonadiabatic effects.
(1) Lippard, S. J., Ed. Prog. Inorg. Chem. 1983, 30. (2) Richardson, D. E.; Taube, H. Coord. Chem. Reu. 1984,60, 107-129. (3) Mikkelsen, K. V.; Ratner, M. A. Chem. Reu. 1987, 87, 113-153. (4) von E. Doering, W.; Roth, W. R. Angew. Chem., Int. Ed. Engl. 1963, 2, 115-122. (5) Grob, C. A. Acc. Chem. Res. 1983,16,426-431. Brown, H. C. Ibid. 432-440. Olah, G . A.; Prakash, G. K. S.; Saunders, M. Ibid. 440-448. Walling, C. Ibid. 448-454. (6) Creutz, C. Prog. Inorg. Chem. 1983, 30, 1-73. (7) Hanack, M. Isr. J . Chem. 1985, 25, 205-209. (8) (a) Creutz, C.; Taube, H. J. Am. Chem. SOC.1969,91,3988; (b) Ibid. 1973, 95, 1086-1094. (9) Furholz, U.; Burgi, H.-B.; Wagner, F. E.; Stebler, A.; Ammeter, J. H.; Krausz, E.; Clark, R. J. H.; Stead, M. J.; Ludi, A. J . Am. Chem. SOC.1984, 104, 121-123. (10) Dubicki, L.; Ferguson, J.; Krausz, E. R. J. Am. Chem. Soc. 1985,107, 179-182. (1 1) Neuenschwander, K.; Piepho, S. B.; Schatz, P. N. J. Am. Chem. SOC. 1985, 107, 7862-7869. (12) (a) Mixed Valence Compounds; Brown, D. B.; Ed.; Reidel: D. Dordrecht, Holland, 1980. (b) Schatz, P. N. Ibid. 115-150. (c) Hush, N. S. Ibid. 151-188. (13) (a) Mechanistic Aspects of Inorganic Reactions, Rorabacher, D. B., Endicott, J. F., Eds.; American Chemical Society: Washington, D.C., 1982. (b) Wong, K. Y.; Schatz, P. N. Ibid. 281-296. (c) Hush, N. S.Ibid. 301-329. (d) Schatz, P. N. Ibid., comment on p 330. (e) Hush, N. S.Ibid., comment on pp 330-331. (14) Bunker, B. C.; Drago, R. S.; Hendrickson, D. N.; Richman, R. M.; Kessel, S. L. J . Am. Chem. SOC.1978, 100, 3805. (15) Hush, N. S.; Edgar, A.; Beattie, J. K. Chem. Phys. Lett. 1980, 49, 128. (16) Stebler, A,; Ammeter, J. H., Furholz, U.; Ludi, A. Inorg. Chem. 1984, 23, 2764. (17) Hush, N. S. Chem. Phys. 1975, 10, 361. (18) Citrin, P. H.; Ginsberg, A. P. J . Am. Chem. SOC.1981, 103, 3673. (19) Krausz, E.; Burton, C.; Broomhead, J . Inorg. Chem. 1982, 20, 434-435. (20) Dubicki, L.; Ferguson, J.; Krausz, E. R.; Lay, P. A,; Maeder, M.; Magnuson, R. H.; Taube, H . J . Am. Chem. SOC.1985, 107, 2167. (21) Piepho, S. B.; Krausz, E. R.; Schatz, P. N. 1985, J . Am. Chem. SOC. 1978, 100, 2996-3005. (22) Wong, K. Y.; Schatz, P. N.; Piepho, S . 8. J . Am. Chem. SOC.1979, 101, 2793.
0 1989 American Chemical Society
Bridged Mixed-Valence Dimers
The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 3031
In recent papers we have argued that for systems like the Creutz-Taube ion, in which there is a strong back-bonding interaction between occupied 4d,, spin orbitals on the Ru ions and an empty s* orbital on the bridging ligand, the model must incorporate the strongly coupled electronic state on the In our application of our three-site to the Creutz-Taube ion, we introduced30 two new features: (1) the bridging ligand is explicitly included, and ( 2 ) the parameters in the purely electronic part of the Hamiltonian were obtained from an electronic structure calculation, rather than from a multiparameter fit. Some of the advantages of this approach are as follows: (1) The totally symmetric vibrational modes that are coupled to the electronic transition are properly incorporated into the line-shape problem. This resolved a long-standing d i s p ~ t e . ' ~ (2) - ' ~ The effects of the bridging ligand upon the observed properties of the complex became apparent. (3) Multiparameter fitting is obviated. (4) Nearly all of the observed properties of the Creutz-Taube ion have been accounted for by our treatment.30-31 In the present paper, our three-site description is extended to include nonadiabatic effects. Of the earlier treatments of electron transfer that incorporate nonadiabatic effects, probably the simplest (and most crude) is the calculation of probabilities of crossings to higher order potential surfaces by a Landau-Zener,, treatment. Nonadiabatic quantum mechanical descriptions of electron transfer have been given by L e ~ i c h , ~by, D ~ g o n a d z e , , ~ and by Kestner et al.35 Many electron-transfer systems have been observed to obey the adiabatic theory of Marcus.36 Hush has given expressions for the absorption line width for localized mixed-valence system^.^' The purposes of the present work are (1) to develop a nonadiabatic treatment of the absorption line shape for bridged dimers with strong metal-bridge electron exchange coupling, ( 2 ) to show that nonadiabatic effects are not important in the specific case of the Creutz-Taube ion, and (3) to calculate absorption profiles by a simple, approximate method that does not require heavy computation. We wish to emphasize that the simple method utilized here is applicable only to delocalized and intermediate cases and not to strongly localized systems. We show to what types of bridged mixed-valence dimers the present treatment applies. The Hamiltonian We begin with a three-site, multimode model Hamiltonian given by H = He + Hv + H e y (1) He = J(alta2+ a J a ,
+ hc) + aa2ta2
(2)
H, = 2m
(23) Root, L. J.; Ondrechen, M. J. Chem. Phys. Lett. 1982,93,421-424. (24) Ondrechen, M. J.; KO, J.; Root, L. J. J . Phys. Chem. 1984, 88, 5919-5923. (25) KO, J.; Ondrechen, M. J. Chem. Phys. Lett. 1984, 112, 507-512. (26) KO,J.; Ondrechen, M. J . J. Am. Chem. SOC.1985, 107, 6161-6167. (27) Ondrechen, M. J.; KO,J.; Zhang, L.-T. Int. J. Quantum Chem: Quantum Chem. Symp. 1986, 19, 393-401. (28) KO,J.; Zhang, L.-T.;Ondrechen, M. J. J . Am. Chem. SOC.1986,108, 171 2-1 71 3. (29) KO,J . Ph.D. Dissertation, Northeastern University, 1986. (30) Ondrechen, M. J.; KO,J.; Zhang, L.-T. J . Am. Chem. SOC.1987,109, 1672-1676. (31) Zhang, L.-T.;KO,J.;Ondrechen, M. J. J . Am. Chem. SOC.1987,109, 1666-167 1 . (32) Zener, C. Proc. R . SOC.1932, A137, 696. (33) Levich, V. G.Adu. Electrochem. Electrochem. Eng. 1966, 4 , 249. (34) Dogonadze, R. R.; Kuznetsov, A. M.; Vorotyntsev, M. A. Phys. Status Solidi E 1972, 54, 125. ( 3 5 ) Kestner, N . R.; Logan, J.; Jortner, J. J. Phys. Chem. 1974, 78, 2148. (36) Marcus, R. A. Annu. Rev. Phys. Chem. 1964, 15, 155. (37) Hush, N . S. Prog. Inorg. Chem. 1967, 8, 391-444.
where a,+and a, are the fermion creation and annihilation operators for the ith local electronic basis state and wherep, and qi are the momentum and coordinate for the ith local vibrational degree of freedom, assumed to be a harmonic oscillator with reduced mass m,and frequency w,. Local states 1 and 3 are the parent metal states (which are 4d, orbitals of ruthenium in the case of the Creutz-Taube ion) and are assumed to be degenerate. State 2 is the parent bridge state on the bridging ligand and is a vacant T * orbital of pyrazine in the case of the Creutz-Taube ion. The energy gap between the parent bridge state and the parent terminus states is given by a;J is the electron-exchange coupling between the bridge and each metal state. We consider here the case where the parent states 1 and 3 are identical, so that m = m l = m,, u 1 u1 = u,,and A = A I = A,. Only one vibrational mode on each metal site is assumed to be important, but multiple modes on the bridge may couple to the electronic manifold. These vibrational modes may be either intramolecular vibrational modes or solvent modes of motion. The vibronic coupling is assumed to be of the small-polaron type: linear in the oscillator displacement q, and linear in the electron number operator, with vibronic coupling constant A , . It is assumed that only one electronic state on each site is important, consistent with our electronic structure results in the case of the Creutz-Taube ion.,' Spin-orbit coupling, the inclusion of which is of course vital for the proper description of some important properties of the complex such as the anisotropic EPR g tensor, is neglected for present purposes. He is taken to be of the one-electron type: This means that for transition-metal systems, the values used for the parameters in He are valid only for fixed total charge and fixed spin. We consider systems with large electron-exchange coupling J and transform into the appropriate basis set, which is given by
3032 The Journal of Physical Chemistry, Vol. 93, No. 8, 1989
Zhang et al. numbers different by f 1 in one degree of freedom and the same in the other vibrational degrees of freedom. Equations 24 and 25, which we call the nonadiabatic secondorder perturbative (NASOP) method, then give the approximate energies and wave functions. This method is valid provided that A2E,[(E, - E,)(-E,)k]-' < 1 .
The purely vibrational term is given by LJU= 1 P+2 1 + P-2 - + -mw2q+2 - + -mw2q_2 + 2m 2m 2 2
(22) To treat the problem perturbatively, we adopt the B, N, A (for bonding, nonbonding, and antibonding) basis set as the zero-order basis set for the purely electronic manifold. The zero-order vibronic eigenkets are a product of one of these zero-order electronic functions and three harmonic oscillator functions and may be written as li)
E I@'rXm+X/-Xkb)
(23)
where I = B, N, or A and where m, I , and k are the vibrational quantum numbers in the +, -, and b vibrational modes, respectively. Note that eq 23 applies when only one vibrational mode on the bridge is important. Through second order, the energies are given by
and the eigenfunctions (before normalization) are given by
A Treatment in the Adiabatic Approximation Equations 1-4 may be solved in the adiabatic (Born-Oppenheimer) approximation, to obtain three exact adiabatic potential surfaces.24 These surfaces may be labeled UB,U,, and UAand are functions of the three vibrational coordinates q+, q-, and q2. Since these expressions are quite cumbersome, we prefer instead to work with the second-order approximate expressions to calculate the Franck-Condon overlap integral^.^^.^^ In the present case, the zero-order Hamiltonian is given by eq 1 1 plus the harmonic oscillator potential terms, and the perturbation is given by eq 15-18. We then apply the second-order harmonic oscillator (SOHO) method,26 where each surface is approximated as a harmonic oscillator along each of the three coordinates. The position of potential minimum is found for each surface along each coordinate, by using the second-order approximate adiabatic potentials. The second partial derivative of each surface along each coordinate then gives an effective harmonic force constant. The Franck-Condon factors are then simply the overlap integrals for displaced harmonic oscillators. We have shown previously that the antisymmetric coordinate q- does not contribute to the absorption line shape in the adiabatic a p p r ~ x i m a t i o n . ~In ~ - addition, ~~ the effective harmonic force constants along any given coordinate are nearly identical in the lower (B) and upper (N) surfaces. Thus the broadening of the optical absorption line shape arises from displacement of the minima between the upper and lower surfaces along the coordinates q+ and q2. With use of the present transformation of eq 5-7, the adiabatic potentials are calculated to second order as
where U,, is the original harmonic potential, given by 1
uh = Xmw2(q+' + q2)+ ~ Z/ m p I Z q 2 ?
(35)
To calculate the absorption line shape with the SOHO method, we make one further approximation. Since the effective harmonic force constants along a given coordinate are very nearly equal in the upper and lower surfaces, we take the force constant for both surfaces to be the average of the two, as K=
K2 =
I)$(+
i[(2)
:[(s)+ ( 31
(36)
(37)
The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 3033
Bridged Mixed-Valence Dimers For displaced two-dimensional harmonic oscillators of equal frequency, the Franck-Condon integrals are given by
for 1 > n and i > j , and where Aqx represents the displacement of the minimum along the x coordinate between the B and the N surfaces. LF is an associated Laguerre polynominal, given by
I
a
bl
and L, is a Laguerre polynomial, given by
+
The variable p x , where x = or 2, is proportional to the square of the displacement of the minima, and may be written as P x = (kxmx)’/2(Aqx)2/2h
cI
(41)
Transition intensities may then be calculated from Boltzmannweighted Franck-Condon factors, written as
where fi represents the dipole moment operator and where S represents the overlap integral of eq 38. The energies of the transitions between the vibronic levels are given by AE(B,/+N,tp)
= AEO
+ (n’- n)hv+ + (1’-
l ) h ~ 2 (43)
where AEO is the energy difference between the minima of the two-dimensional upper and lower surfaces and u+ and u2 are the effective (averaged) frequencies along the sum and bridge cobrdinates. For greater accuracy we calculate AEo using the energy difference between the minima of exact two-dimensional adiabatic potentials (for which we have analytic expressions).
Numerical Nonadiabatic Treatment For a numerical test of the perturbative scheme, we use the transformed Hamiltonian of eq 9-1 5 and express the total wave function as
where I = B, N, or A and where r denotes the vibronic level. The energies and eigenvectors for the vibronic levels are then found numerically by expansion on a large basis set. Transition intensities may then be calculated2’ as
where Z is given by Z = Cexp(-Er/kBT)
(46)
r
and wherefis given by
f = [ ~mc zl ( ak ! m . l , k c f ; ? , . l , k+ c f ; ! m , l , k a , h , k cf;!m,/,kCji?m,/,k
+
+ a ! m , / , k c f ; ? , , / , k ) ] ’ (47)
Here we have assumed that the matrix elements of the dipole moment operator obey the following relations: (fi)
MEN I WNB E WNA E MAN WAB
WBA
=0
(48) (49)
Figure 1. Transition intensity (arbitrary units) at 300 K as a function of frequency in wavenumbers for the Creutz-Taube ion, as calculated by (a) NASOP, (b) SOHO,and (c) numerical methods.
because of the symmetry of the system. Note that eq 45 may be used to calculate the transition intensities between the vibronic levels in both the nonadiabatic numerical scheme and the NASOP scheme.
Examples The Creutz-Taube Ion. For the Creutz-Taube ion, we utilize the Hamiltonian parameters obtained in ref 30: a = 0.93 eV, J = 4 . 8 5 eV, A = -1.52 eV/A, A2 = -2.4 eV/& k = 15.6 eV/A2, eV s2/A2), m2 = k2 = 7.0 eV/A2, m = 17 amu (=1.76 X 5.08 amu (=5.27 X eV s2/A2). In the case of the CreutzTaube ion, the coupled vibrational modes are all internal (intramolecular) modes of motion. The line shape of the IT band has been shown to be independent of the solvent.8b Predicted spectra, showing absorption intensity in arbitrary units a t 300 K as a function of excitation frequency in wavenumbers, are shown in Figure la-c as calculated by the NASOP, SOHO, and numerical methods, respectively. To compare the stick spectra with the condensed-phase spectrum, one can place a Gaussian function on each of the sticks and set the width of this Gaussian function equal to the minimum necessary to eliminate structure in the spectrum (about 420 c ~ - ’ ) . ~ When O this is done to the spectra in Figure 1, the resulting three smooth line-shape functions are nearly identical. Examination of the eigenvectors obtained numerically and by the NASOP method shows only minuscule mixing of the B, N, and A electronic components. For example, in the low-lying levels of the ground state, the components of the wave function have almost entirely B (bonding) electronic character, and the sum of the squares of the coefficients with N (nonbonding) electronic character is less than 0.01 by the NASOP method. Thus, nonadiabatic mixing is minimal in the Creutz-Taube ion. A System Closer to the Localization Threshold. We now consider a hypothetical bridged dimer with a larger vibronic coupling parameter A . This system is less strongly delocalized than the Creutz-Taube ion. The ground-state adiabatic potential surface has single-minimum form, but it is flat in the region near q- = 0. For this example we take cy = 0.9, J = -0.8,A = -3.5, A2 = -2.0,k = 15.0, k2 = 8.0, m = 1.8 X eV s2/A2, and
3034 The Journal of Physical Chemistry, Vol. 93, No. 8, 1989
4
b
L HAVENUMEERS
8500.00
Figure 2. Transition intensity (arbitrary units) at 300 K as a function of frequency in wavenumbers for a hypothetical system that is closer to the localization threshold than the Creutz-Taube ion, as calculated by (a) NASOP, (b) SOHO, and (c) numerical methods.
m2 = 5.3 X eV s2/A2. Figure 2a shows the spectrum predicted by the NASOP method, Figure 2b by the SOHO method, and Figure 2c numerically for this example at a temperature of 300 K. For the low-lying levels of the ground state, which have predominately B (bonding) electronic character, the sum of the squares of the coefficients for basis states with N (nonbonding) character is a few percent by both the NASOP and numerical methods.
Discussion For the Creutz-Taube ion, nonadiabatic mixing of the potential surfaces is negligible and the motion of the unpaired electron is well within the adiabatic regime. Although formerly a point of
Zhang et al. intense controversy, the delocalization of the unpaired electron is now clear. The potential surfaces are very nearly harmonic, and the SOHO approximation is good for this particular complex. If the vibronic coupling parameter is increased so that the system approaches the localization threshold, the simple SOHO treatment becomes less accurate. In the example shown in Figure 2, the S O H O method fails to predict a transition at about 6100 cm-'. This transition results from thermal population of the qvibration in the initial state. The other transitions that contribute to the optical absorption line shape result from the vibronic ground state, and the NASOP method predicts the transition energies to within about 1%. The energy of the q- hot band is predicted with about 2% accuracy. This method represents an alternative to our purely adiabatic S O H O approach to bridged dimers like the Creutz-Taube ion. As one increases the vibronic coupling parameter A (or decreases the electron-exchange coupling), one approaches the localization threshold and numerical solutions become more and more computationally intensive, because the basis set size equals 3n3, where n is the number of harmonic oscillator functions per vibrational degree of freedom. Thus for systems not as well delocalized as the Creutz-Taube ion, the present method is a fast and reasonable approach. The coupled vibrational modes may be either medium or internal degrees of freedom. While there is now a multitude of evidence that the unpaired electron of the Creutz-Taube ion is shared equally by the two metal ions and that the complex is average-valent, there is still one interesting unsolved problem. No one has yet accounted for the coincidence of infrared bands at 700, 1230, and 1585 cm-' with Raman bands at 699, 1232, and 1594 ~ m - ' .We ~ believe that this observation can be accounted for, using the present method but extending the model Hamiltonian to include other metal 4d states. This work is in progress. We hope that experimentalists will be inspired to investigate further other members of this intriguing class of compounds and their extended-chain analogues. Acknowledgment. We thank the National Science Foundation for support of this research under Grant CHE-8607693 and Grant CHE-8820340. We thank the Alfred P. Sloan Foundation for a Fellowship awarded to M.J.O. Part of the computational work was performed on the Northeastern University Chemistry Department computer facility, the purchase of which was partially funded by the National Science Foundation under Grant CHE8700787. M.J.O. acknowledges the support of the Danish Natural Science Research Council and the hospitality of Professor Jan Linderberg during her stay at Aarhus University. Registry No. [(NH3)5Ru(p-pyrazine)Ru(NH3)5]5+, 35599-57-6.
