10484
J. Phys. Chem. 1995, 99, 10484-10491
Role of Vibronic Coupling and Correlation Effects on the Optical Properties of Mixed-Valent and Monovalent Dimer Compounds: the Creutz-Taube Ion and Its Monovalent Analogs Alessandro Ferretti,*>?Alessandro Lami: Mary Jo Ondrechen,' and Giovanni Villanit Istituto di Chimica Quantistica ed Energetica Molecolare del CNR, Via Risorgimento 35, I-56126 Pisa, Italy, and Department of Chemistry, Northeastem University, Boston, Massachusetts 02115 Received: December 13, 1994; In Final Form: March 27, 1995@
A three-site, three-harmonic oscillator vibronic model Hamiltonian that includes correlation effects is considered in order to study the optical absorption spectra of the Creutz-Taube ion at various values for the total charge. The model is made by a two band Hubbard Hamiltonian, plus a vibronic interaction that is linear in the nuclear coordinates. W e show that the main features of absorption spectra observed experimentally in the near-IR-visible are well reproduced by this model, as far as band position and shape are concerned. An approximate approach to the vibronic problem reveals that, in the present case, the vibronic interaction is strong and good results can be obtained only by solving the full vibronic problem.
1. Introduction A model for the optical absorption spectrum of bridged electron donor-acceptor systems with variable total charge is presented. We consider herein linear trinuclear clusters of the form, donor-bridge-acceptor, hereafter referred to as DBA. We illustrate our method with application to the extensively studied Creutz-Taube ion in three oxidation states: mixedvalent, fully reduced, and fully oxidized. Such complexes have numerous important applications in coordination chemistry, in molecular electronics, and in materials design. Electron donor-acceptor pairs joined together by a variable bridging ligand were first synthesized by Taube et a1.I for the purpose of studying inner-sphere electron transfer. A host of related bridged mixed-valent species that possess an odd charge and for which donor D and acceptor A are identical were also reported by the Taube group and These mixed-valent compounds run the gamut from strongly localized (slow electron transfer) to completely delocalized (averagedvalent). Probably the best known bridged mixed-valent dimer is the Creutz-Taube ion, a pyrazine-bridged mixed-valent dimer of ruthenium.6 This compound was the focus of intense controversy for more than a decade, but evidence later pointed toward a fully delocalized ground ~tate.~-'OMore recently, the Stark effect work of Oh and Boxer" and the resonance RamanI2 and asymmetric c~mplexation'~ work of Hupp and co-workers have contributed strong evidence in support of delocalization (valence averaging) in the ground state. Trinuclear systems of the DBA type were proposed as possible molecular-scale rectifiers in 1974.14 Recently reported evidence on this front is enco~raging.'~ Other molecules in this class recently were proposed as molecular switches with subpicosecond write times and at least microsecond retention times.'' Lanthanide sandwich compounds, in which a rare earth ion acts as a bridge between two macrocycles with an excess electron (a variation on the bridged mixed-valent theme), have applications in electrochromic devices." Extended-chain analogs of the above systems, in which multivalent metal ions M are joined together with bridging
@
Istituto di Chimica Quantistica ed Energetica Molecolare del CNR. Northeastem University. Abstract published in Advance ACS Abstracts, May 15, 1995.
0022-3654/95/2099- 10484$09.00/0
ligands L to form a linear chain of the form..-M-L-M-L.**, have been ~ y n t h e s i z e d . ' ~In . ' ~addition, one-, two-, and threedimensional assemblies of such systems have been suggested for particular applications in the areas of molecular electronics and nanoscale devices.20,*' In the systems discussed above, we have two chemical means for controlling their properties: (1) synthetic alteration of the bridging ligand and ( 2 ) variation of the total charge. Chemical changes in the bridging ligand can have profound effects on the spectroscopic and conductive properties of these bridged systems, and any reasonable model must incorporate the coupled bridge orbitals expli~itly.**-*~ Furthermore, whenever transition metals with variable total electron occupation are involved, as is the case in the systems of interest here, a simple one-electron model, such as the Huckel or tight binding models, will not be adequate to describe the system, and the effects of electronic correlation must be taken into a c c o ~ n t . ~ ~ - ~ ~ The Creutz-Taube ion is known to exhibit absorption bands in the near-IR and visible region that strongly depend upon the oxidation state of the two metal ions and hence on the total charge. The fully oxidized form of the complex, the one with both Ru ions in an oxidation state of f 3 , is characterized by the absence of absorption in the near-IR, while some bands appear above 3 eV. The complex with a total charge +5 still has some features above 4 eV but exhibits also two bands at 0.8 and 2.2 eV. Finally, the fully reduced form with a total charge of +4 has a spectrum very similar to the +5 species, except that at low frequency, the band at 0.8 eV is absent and the one at 2.2 eV is slightly blue shifted to 2.27 eV.' Early models for this complex considered only the two metal ions and neglected the role of the bridging Further investigation by Ondrechen et al.22-24revealed that a ligand orbital may play a fundamental role in the spectral features of the Creutz-Taube ion, and they fitted the experimental data for the +5 species with a Huckel Hamiltonian. Recent experimental evidence corroborates the conclusions of their three-site model Hamiltonian for the +5 species."-'3 The role of vibronic interaction with multicenter vibrations on the line shape has been investigated by Piepho for the +5 specie^,^',^^ using a M-L interaction term which is distance-dependent and a MO electronic basis. A recent improvement of the electronic three-site model involves the explicit introduction of the 0 1995 American Chemical Society
J. Phys. Chem., Vol. 99, No. 26, 1995 10485
Optical Properties of Mixed- and Monovalent Dimers electron-electron repulsion on the metal orbitals by a two-band Hubbard H a m i l t ~ n i a n .In ~ ~this way, a satisfactory description of the spectral behavior of Ru-pyrazine complexes with varying total charge is ~ b t a i n e d . ~ ~ - ~ O In the present work, we expand upon the earlier three-site models including vibronic c o ~ p l i n g ~to~incorporate -~~ electron correlation effects, along the lines explained in refs 28-30. This is achieved with a two-band Hubbard model with vibronic coupling. Previously, Prassides and S ~ h a t zutilized ~ ~ , ~a~single band, two-site Hubbard model to describe mixed-valent systems. The introduction of correlation effects via the on-site electronelectron repulsion term in the Hubbard Hamiltonian permits us to treat the multiple oxidation states of the systems of interest, a capability absent from the earlier three-site model^.^^-^^ The present work also has capabilities beyond the work of Prassides and Schatz in that systems with a strongly coupled ligand state, such as the Creutz-Taube ion, may be treated. We do not invoke the adiabatic approximation, as was done in refs 2224. The optical absorption line shape is obtained using the dipole autocorrelation function in the energy domain together with an iterative Lanczos technique. 2. The Model The vibronic model Hamiltonian which we utilize for studying the optical absorption spectra of trinuclear clusters with variable electron occupation is given by
Nsite
Nsite-1
Ru-atom
Nsite
H, =
w,(b:b, j
\
+);L1I
Nsite
TABLE 1: Parameters Utilized for the Hamiltonian of Equation 1 parameter (eV) ref parameter (eV) 11= A, = -0.071 337 10 t = -0.5 12 = -0.14 a A=5 O I= 03 = 0.061 977 6 10 u=4 02 = 0.075 488 7 10 This work.
ref 30 30
30
(I
Note that for present purposes, we assume that this coupling is linear in the nuclear coordinates, which physically corresponds to a displacement in the equilibrium position of an oscillator upon change in electron number at its site. The basic idea we are pursuing in this model is that, in the case of the Creutz-Taube ion, an essential role is played by the backbonding interaction of the d,, orbital of Ru atoms with the n* orbital of the pyrazine. This is supported by an electronic structure calculation: by line shape calculation^^^-^^ (which also pointed out, for the first time, that the ligand n* orbital must be considered explicitly in modeling these systems), by calculation of the components of the EPR g tensor,24 by recent resonance Raman experiments,I2 and by asymmetric complexation work.I3 Within our model of one orbital per site, an empty metal d, orbital corresponds to a Ru4+,thus the three-site system with two electrons models (R~-pz-Ru)~+ (dropping NH3 for simplicity), and so on. As far as the coupling with the nuclear degrees of freedom is concerned, in our approach we have only considered the most relevant vibrational mode on each fragment: the Ru-NH3 symmetric stretch of the octahedral coordination shell for each Ru site and the 6a mode for p y r a ~ i n e . ' ~ . ~ ~ The values of the parameters considered in this paper are given in Table 1. For Hel we have taken, for all the parameters, the values that in previous work28-30 appeared to give the best fit to experimental results. For the vibronic part of eq 1 we have taken the same values of the parameters considered in a previous paper,I0 except for the value of A 2 which in ref 10 appears to be overestimated. The dipole operator in second quantization form is given, for the present odd systems and taking the intrasite distance to be unity ((jlpv) = 1 and U) is the orbital on sitej), by
j,o
He]is a two-band Hubbard H a m i l t ~ n i a n , ~ where ~ - ~ ~u& > ~(a,,,,) ~ is the creation (annihilation) operator for one electron in the orbital of site j with spin u, n,,o = u l + , is the number operator for the electron at s i t e j with spin u, ~j = EL, ER" (A = EL - ER,,) is the site energy, and the parameters t and U, respectively, are the hopping energy of one electron between orbitals on adjacent sites (e.g., between Ru (d,,) and L (n*) in the case of [Ru(NH3)5-pz-Ru(NH3)5Im+) and the Coulomb repulsion for two electrons on the same Ru orbital. H , is the vibrational Hamiltonian for the harmonic oscillators, one for each site, in the second quantization form (b; is the creation operator for one-quantum excitation in the oscillator j and bj the corresponding annihilation operator; f i = 1). Hel-, is the vibronic interaction, which can be expressed as
with the substitution
where the summation is over sitesj and spins u and where Nsite is the total number of sites. For the present case of Nsite = 3, the dipole operator has then the form (dropping the spin index for simplicity)
p u n l -n3
(4)
Let us briefly illustrate the computational method used here. First of all, the basis set is obtained by direct product of electronic localized configurations and harmonic oscillator number states. For example, for the $6 case (two electrons), a typical member of the basis set is written as
We consider only electronic configurations having the minimal value of S, (S, = 0 for two- and four-electron cases and S, = '/2 for the three-electron case) so that their number is 9 in any case. For the harmonic oscillator states, the infinite sequence
10486 J. Phys. Chem., Vol. 99, No. 26, I995
Ferretti et al.
is truncated after the convergence of both ground state energy and absorption spectra. Typically about 40 000 vibronic basis states are sufficient in the present work. Once the ground state of the Hamiltonian (eq 1) is obtained using the Lanczos a l g ~ r i t h m , ~ ~the - ~ absorption ' spectra are computed from the dipole autocorrelation function in the energy domain
0.8-
i!
8 0.6-
i!
C
eQ
and utilizing the continued fraction expansion for the diagonal matrix element of G,38
8 n
< 0.40.2-
where the a's and p's, respectively, are the diagonal and off diagonal elements of the tridiagonal matrix obtained by propagating iteratively the doorway state pig) by the Lanczos algorithm. The tridiagonalization procedure converges quite rapidly (less than 200 steps). It should be noticed that, since k,fl = -iJ/t (where J is the current operator J = itC,,,(a~a,+, - a:,a,)), the spectra computed with eqs 5 and 6 are simply related to the optical conductivity ~pectra,~*-~O where the current-vector potential is used instead of the dipole-electric field form for the fieldmatter interaction. We wish to stress that the work presented here is the first reported numerical solution of the three-site, n-electron ( n = 2 , 3, 4) problem that takes into account exactly both electronic correlation and vibronic coupling. The system studied here represents a realistic model for the (Ru-pz-Ru)"+ and related complexes as far as backbonding features are concemed.
Figure 1. Measured absorption spectra of the Creutz-Taube ions (Rupz-Ru)"+ from ref 6. Below and above 2 eV spectra have been recorded in different solvents and with different concentrations.
3. Results In this section we report the results of our calculations of absorption spectra for the Hamiltonian of eq 1, which we shall show represents a realistic model for the Creutz-Taube complex (Ru-pz-Ru)"+ at various total charge m (4-6). In Figure 1 we report the experimental results taken from ref 6 for comparison. First, it is worthwhile to discuss some general points which depend on only the symmetry of the problem, as well as on the form of the dipole operator of eq 4. As is well known, the weight of transitions from the ground state to the various excited states is proportional to the square matrix element of the dipole. Expanding these states in our localized basis set, in which the dipole operator is diagonal (eq 4), one can easily verify that the only contributions come from localized configurations in which the populations of the two Ru atoms are different. However, it is clear from the form of the dipole operator that the optical excitation per se does not involve any charge transfer. In fact, the optical doorway state @IVg)) has the same charge Hence, upon excitation distribution as the ground state I+&. by a &pulse (which indeed corresponds to preparation of the doorway state), no charge transfer occurs at the beginning, but the charge distribution varies in the subsequent time evolution, with the only symmetry requirement of (nl) = (n3). The absorption spectrum, which reflects the decomposition of the doorway state on the eigenstates of the Hamiltonian, then exhibits charge transfer peaks.
2
0
4
6
Figure 2. Absorption spectra of the Creutz-Taube complexes at various total charge (m), modeled by the electronic model Hamiltonian He,of eq 1. Parameters are given in Table 1.
Recently, Simoni et al.39performed calculations of absorption spectra on a two-site model for mixed-valent compounds using a different transition operator, chosen ad hoc in order to mimic explicitly local state to local state charge transfer. Their transition operator, written as a dipole operator, is, in our notation, p = a,+a2 az+al. For the simple dimer, this operator does not possess the odd parity characteristic of the dipole operator. This choice of transition operator is not appropriate for the present case where we are dealing with absorption transitions polarized along the intemuclear a ~ i s . ~ ~ - ~ ~ First, we show and discuss the results obtained without considering vibronic coupling, that is utilizing the purely electronic part ( H e )of eq 1, which has been shown to be very useful also for studying longer chain analogs of the CreutzTaube ion.28-30 In Figure 2 the absorption spectra obtained by the two-band Hubbard model (Hei in eq 1) with the parameters of Table 1
+
Optical Properties of Mixed- and Monovalent Dimers
TABLE 2: Analysis of the Ground State ( 9 ) and of the Excited States (el, e,) involved in the Absorption Spectra for the Electronic Part of the Hamiltonian of Equation 1 (He,) with U = 4 and A = 5 (in eV), Modeling [Ru-pz-Ru]"+ CompleP
...,
tot m state (eV) %onM w
6
g
el e2 5
g
el e2 e3
e4 4
g
el e2 a
0.00 3.74 5.47 0.00 0.32 1.72 5.42 6.82 0.00 1.60 6.79
99.0 89.4 61.0 91.0 95.0 71.0 62.3 38.4 90.7 74.5 50.4
DM
WM
1
2
0.02 0.21 0.79 0.00
0.98 0.79 0.21 0.27 0.15 0.85 0.85 0.15 0.02 0.02 0.98
0.00 0.01 0.14 0.85 0.00 0.00 0.00
3
4
1
2
1
[0.02
p, MI =M3
L
0.99 0.89 0.61 1.36 1.42 1.06 0.93 0.58 1.81 1.49 1.01
0.02 0.21 0.79 0.28 0.15 0.87 1.13 1.85 0.37 1.02 1.98
0.00 0.79 0.21 0.73 0.73 0.85 0.85 0.14 0.14 0.01 0.86 0.00 0.14 0.32 0.66 0.32 0.66 0.98 0.00 1.00 0.00 0.02 0.00 1.00 0.00
The quantities M, WM, DM,and P,are defined in the text.
are shown. The relative positions of the lines as a function of the total number of electrons are in agreement with experiment, although the exact positions of the lines are not precisely those observed6 (see, e.g., Figure 1). The analysis of the ground state, and of the excited states connected to it by the dipole operator (eq 4),has been performed by computing the total population on metal sites M in percent (tot. % on M), as well as other quantities useful for characterizing the states as defined below. The total population of a site i is given the symbol P;, normalized to the total number of particles and defined as
Pi = (qpil?)); Pi = A;,
+ Ai,
(7)
where )1 is a generic eigenstate. Here i indicates either a metallic site (i = M I , M2) or a ligand site (i = L)in the complex M I-L-M2. Pi gives information regarding the local distribution of electrons on the sites. It is also useful to examine the total weight of states with k electrons on M (WM(~)),which can be defined as Naite
as well as the total weight of states with 1 doubly occupied M sites ( D M ( ~ ) ) Nsite
&(l) = (VlbM(1)IV);
bM(z)
= d(hM9z);
BM
=
Ai,fft,,J icM
(9)
The 6's appearing in eqs 8 and 9 are simple Kronecker 6's. Their effect on the state IV) is to destroy each component, in the configuration space, whose number of electrons on M is not the one required. The results of the analysis are reported in Table 2 for the three possible values of the total charge m. Notice that, for symmetry reasons, the ground state for the three forms is even with respect to reflection through a mirror plane bisecting the ligand, and since the dipole operator of eq 4 is odd with respect to this operation, the only allowed transitions are those to odd excited states. For the fully oxidized form (m = 6), the ground state has two electrons with one on each Ru site forming a singlet, with very little ligand mixing. From the ground state only two transitions are allowed, both to states with less metallic character, indicating a metal-to-ligand charge transfer (MLCT) but with
6
7
Figure 3. Absorption spectrum of the (R~-pz-Ru)~+ modeled by the vibronic Hamiltonian of eq 1. There is no absorption below 3 eV. Parameters are given in Table I.
different amounts of double occupation (Le., Ru2+) (see, e.g., Table 2 for m = 6). Upon reduction ( m = 5 and 4), since U and A are large compared to the hopping t but rather close to each other, the added electrons go onto ligand or metal sites (giving rise in this latter case to double occupation) with nearly the same probability. We have then strong charge fluctuations between the metal sites and the ligand site, responsible for the existence of low-frequency peaks; "strong fluctuations" is used here in a static sense, meaning that there is a large probability of finding electrons distributed differently on the sites from the average values reported in Table 2. This is a general property for the two-band Hubbard Hamiltonian with U a A and is large compared to the hopping t, as in the present case. When the lowest band is more than half filled, there are states close in energy to the ground state with slightly different charge d i s t r i b ~ t i o n ~and ~ - ~coupled ~ to it by the dipole operator. For m = 5 there are four allowed transitions from the ground state. From Table 2 for m = 5 one can see that the first (to el) corresponds to a ligand-to-metal charge transfer transition, the second (to e2) to a metal-to-ligand charge transfer from doubly occupied metal ions, the third (to e3) to metal-to-ligand charge transfer from singly occupied metal ions, and the fourth (to e4) to a metal-to-ligand double electron transfer transition. Finally, for m = 4 the two allowed transitions, respectively, are related to metal-to-ligand charge transfer transitions from doubly occupied metals (el) and metal-to-ligand double electron transfer (ez), as can be seen from Table 2 for m = 4. However, even if the two-band Hubbard Hamiltonian appears to reproduce the essential features observed experimentally, it fails to provide a complete understanding of the observed spectra. In particular, since it does not contain the coupling of electronic and nuclear motion, it cannot provide information on the line shape and cannot fit exactly the positions of the observed bands. The Hamiltonian of eq 1 contains the basic ingredients for studying the role of vibronic coupling in the Creutz-Taube and related ions. In Figures 3-5 the absorption spectra for the Hamiltonian of eq 1 with the parameters of Table 1 and varying numbers of electrons are shown in the frequency regions where vibronic transitions exist. Figure 3 is for two electrons, Figure 4 for
Ferretti et al.
10488 J. Phys. Chem., Vol. 99, No. 26, 1995 1.6-
I
-0.2 1.2
Lo
Io.,.
- 0.16
1.2-
h
.-E
-0.12
cn C
f
e
.-b
0.8 .-E cn c
a2
-C c
0.4
0
3 (0
0.12
Figure 5. Absorption spectrum of the ( R u - p z - R ~ ) ~modeled ~ by the vibronic Hamiltonian of eq 1. The only absorption occurs between 1 and 4 eV. Parameters are given in Table 1.
0.016
0.012
0.08
-r em
’g
.-c cn c
%
0 c
-
0.008
-C c
0.04
0.004
0
0 5
7
6
4
(W
(ev) Figure 4. Absorption spectrum of the ( R u - p ~ - R u ) ~ + modeled by the vibronic Hamiltonian of eq 1. The spectrum has been divided into
two parts, (a) and (b). to show clearly the various vibronic lines. Parameters are given in Table 1. three electrons, and Figure 5 for four electrons, corresponding, respectively, to the Creutz-Taube ion with m = 6, 5 , and 4. Comparing these results with those of Figure 2, one notices that the lower frequency bands are now displaced toward higher frequencies, and now the band positions observed experimentally (Figure 1) are well reproduced. Furthermore, the line shapes that result from the present computations agree well with those observed. The strongly asymmetric shape of the band observed at 0.8 eV for the +5 species is explained in the computation by a series of vibronic lines with decreasing intensity (see Figure 4a). The bands observed at 2.2 and 2.3 eV for m = 5 and 4, respectively, are less asymmetric, and this also is evident in the present numerical results (see Figures 4a and 5). Notice that the symmetry of the band profile is due to the dependence of Franck-Condon factors on the oscillators’ displacement: the more displaced the oscillators are, the more symmetric the band profile. The behavior at frequencies above 3 eV is quite different. In this region only the two bands for m = 5 appear clearly displaced to higher frequencies (Figure 4b), and there is almost no absorption for the m = 4 case (the spectrum is not
reported, but the spectral weight is very small). No noticeable variation is observed in the bands’ position for the m = 6 case (Figure 3). However, since the model is built taking into account only the d,, orbital of Ru and the z*orbital of pyrazine, it is unable to give complete predictions in this region of the spectrum, where the presence of transitions due to orbitals not included in the model is to be e ~ p e c t e d . ~ As far as the relative intensity of the various bands is concemed, in the experimental spectra6 the band at 0.8 eV, present form = 5, is almost twice the height of the band at 2.2 eV for the same species (see, e g . , Figure 1) and a little higher than the 2.3 eV band of the m = 4 complex; these two latter bands have almost the same width, which is larger than the 0.8 eV band of the +5 species. In our computations agreement is found with the experimental results, with the exception of the 2.2 eV band for the m = 5 case whose intensity appears underestimated by a factor of 2-3. This is to be expected, since the ab initio calculation9 shows that the band centered at 2.2 eV for the +5 species contains multiple components, and only one is included in the present model. The analysis of the ground state and of the single vibronic lines for the three values of the total charge m has been performed, computing the average number of quanta in the three oscillators as well as the quantities of eqs 7-9 by a sum over the vibrational part for each electronic basis state. In all spectra, only negligible variations in the quantities of eqs 7-9 are observed for the various lines in a band, compared to the corresponding pure electronic values of Table 2. The average number of quanta in each oscillator changes in the different lines of a band in an almost irregular manner, indicating that a strong mixing of the vibrational basis states occurs due to vibronic coupling. The only exceptions are the vibronic lines of the band centered at 3.8 eV for the m = 6 case, where the different lines are seen to correspond mainly to the vibrational progression in the two oscillators on the Ru sites and the oscillator on the ligand remains in its ground state. The above considerations are based on the results of an approximate diagonalization of the full vibronic problem. For the sake of simplifying the interpretation, as well as to facilitate the analysis of the results, we attempted to verify if, exploiting the full symmetry of the problem, some simplifications arise.
J. Phys. Chem., Vol. 99, No. 26, 1995 10489
Optical Properties of Mixed- and Monovalent Dimers
TABLE 3: Eigenvalues, Parity, and Average Population of the Sites (nj)for the Electronic Part (&I) of the Hamiltonian of Equation 1 with the Parameters of Table 1 at Various ma m 6
5
4
energy (ev) -12.10243 -8.37013 -8.36602 -6.72478 -6.63397 - 1.80267 - 14.56837 - 14.25126 -12.85183 - 12.62838 -9.14817 -9.14730 -7.74874 -7.65595 - 16.69629 -15.09808 -14.83857 - 13.56891 -9.90192 -9.89622
parity even even odd even odd even even odd odd even odd even odd even even odd even even odd even
( n +n3) ~
(n2)
1.9791 1.7773 1.7887 1.2048 1.2113 0.0388 2.7208 2.8490 2.1295 2.2409 1.8705 1.8790 1.1510 1.1593 3.6309 2.981 1 3.0036 2.3433 2.0189 2.0222
0.0209 0.2227 0.21 13 0.7952 0.7887 1.9612 0.2792 0.1510 0.8705 0.7591 1.1295 1.1210 1.8490 1.8407 0.3691 1.0189 0.9964 1.6567 1.9811 1.9778
Triplets have been omitted for m = 6 and m = 4, and the quartet has been omitted for m = 5. (I
First, we rewrite the Hamiltonian of eq 1 utilizing a symmetric (4,which can be obtained from the two oscillators on the Ru sites: (s) and an antisymmetric oscillator
d=
1
.Jz
1
- b3); 6'= -(bT .Jz
- b;)
(10)
of eq 1 has then the form
The vibronic Hamiltonian of eq 1 with the coupling term He1-v in the form of eq 11 exhibits clearly its invariance with respect to simultaneous permutation of indices 1 and 3 in all the operators. Each vibronic eigenstate is either odd or even with respect to this global (Le., involving both electrons and nuclei) symmetry operation. The parity of the vibronic ground state is even, while the excited states coupled by the dipole operator are odd since the electric dipole is an odd operator (eq 4). Even vibronic states consist of an electronic part and a nuclear part that are either both even or both odd; odd vibronic states have electronic and nuclear components of opposite parity. Examining the whole set of eigenstates of the electronic Hamiltonian at various total charge m for the parameters considered in this paper, given in Table 3, one notices that close in energy to every odd state is always an even state. The same, with the exception of the m = 5 case, does not hold for the ground state. Then, in order to get a simplified picture that can be helpful for discussing the effect of vibronic interaction, one may suppose that, if the pair of odd and even states close in energy is energetically isolated enough from other states (or pairs), in a first approximation only the vibronic mixing of the
3
5
4
6
7
w (eV) Figure 6. Absorption spectrum of the (R~-pz-Ru)~+modeled by the vibronic Hamiltonian of eq 11, obtained by the approximate solution described in the text. There is no absorption below 3 eV. Parameters are given in Table 1. states of the pair will give rise to the vibronic odd state responsible for the absorption band. The first and the second terms in the right-hand side of eq 11 mix electronic states of the same parity, while the third term mixes states with different parity. If the states of the pair have and on the same average population in the ligand site ((4) both metal sites ((nl n3) and notice that ( n ~ n3) (n2) = total number of electrons), the third term of eq 11 can be considered to mix pairs of basis states with the same number of quanta in both the symmetric oscillator (s) and the oscillator on ligand 2. For small vibronic coupling, the problem can then be managed by quasidegenerate perturbation theory by diagonalizing only the block concerning vibronic states of the two nearly degenerate electronic states. As discussed above, only the antisymmetric oscillator is involved in the vibronic coupling while the two others contribute to the spectrum simply by a Franck-Condon factor, being displaced with respect to the ground state. The results obtained with this approach are given in Figures 6-8, respectively, for m = 6 (Figure 6), m = 5 (Figure 7), and m = 4 (Figure 8). From Figure 6 we see that in this case the agreement between the spectrum of Figure 3 and the approximate spectrum is very good. In fact, the electronic ground state is, in this case, very well separated from the two pairs (even-odd) of excited states, and the average number (n2) (then n3)) does not change very much between the two also (nl states of the excited pairs (see Table 3). This does not hold for the m = 5 case where, from Table 3, one can see that the ground state is close in energy to an odd state, the one that gives rise to the very low-frequency peak of Figure 2, and the approximation fails. In fact, the vibronic ground state for m = 5 is then made by a mixture of these two states. Furthermore, the differences in (n2) are now larger than that in the previous case. For m = 4 the same problems hold; the ground state has a small mixing with the pair of electronic states which originate the 2.3 eV band, and neither the overall intensity nor the band position of the full vibronic calculations is obtained. From this approximate approach we may then get valuable predictions on the absorption spectra for only the m = 6 case, where the mixing of metal and ligand is very small. Once the
+
+
+ +
10490 J. Phys. Chem., Vol. 99, No. 26, 1995
Ferretti et al. 0.25
0.25
0.2
2-
0.2
1.5-
0.15
1.2
2
-0.15
9,
.-0 v)
C
-0.1
g -
-0.05
2,
.-I v)
e
.-cx.
C
$2C
-
-5 v)
C
1-
0.1
0.5-
0.05
-
0
01
3
2 0
Figure 8. Absorption spectrum of the ( R ~ - p z - R u ) ~ + modeled by the vibronic Hamiltonian of eq 11, obtained by the approximate solution described in the text. Parameters are given in Table 1.
0.004
0.003
2 Q a
v
-0.002
,g fn E
-p! C
- 0.001
-
t 0
5
6
7
the optical absorption spectra for the various combinations of Ru" and Ru"'. The computed absorption spectra in the nearIR-visible region show a remarkable agreement with the experimental ones and reproduce the characteristic features observed by progressive reduction of the f 6 ion. The bands observed experimentally at low frequency (below 3 eV) are very well reproduced, both in position and in shape, by our vibronic computations. We have also given some interpretation on the origin of the various bands on the basis of a pure electronic model, which holds for the vibronic spectrum, when summing over the vibrational quantum numbers. In the literature the very low-frequency band at about 0.8 eV, well known as the intervalence band (IT), is often assigned as a transition in which one electron goes from one metal to the other (Le., [3, 21 [ 2 , 31 for the present case of the Creutz-Taube ion with a total charge of +5). As we discuss extensively in the paper, this point of view is wrong since the dipole operator cannot transfer an electron from one site to another in this case. Our approach reveals that the IT transition corresponds to a dipole-allowed transition with a small variation in the charge distribution (actually it is a small ligand-to-metal charge transfer) due to charge fluctuations between doubly occupied metal orbital and ligand orbital, which, for the parameters U and A considered, are very close in energy. It is instructive to compare the predictions of the present correlated model with those of a simple Huckel model for the mixed-valent ( m = 5 ) species. The expressions [(a2 8 t Z ) ' I 2 - a]/2 and [(a2 8t2)"2 a]/2 represent the energies of the g-to-el and g-to-e2 transitions, respectively, in the Huckel model,I0 where a = A - U is the Huckel diagonal element for the bridge species. By use of the present parameters, Huckel theory predicts 0.37 and 1.37 eV for the g-to-el and g-to-e2 transitions, respectively, for the m = 5 species. The values obtained from the present model are 0.32 and 1.72 eV, respectively, for the same two transitions (see Table 3). Hence, when correlation is switched on, the el state is slightly stabilized relative to the ground state, while the e2 state is significantly destabilized relative to ground state. In Table 2, we find that the el state for the mixed-valent species has slightly more doubly occupied metal contribution than the ground state, while the e2 state has considerably less doubly occupied metal content. It is these doubly occupied metal configurations that are in
8
0 (eV) Figure 7. Absorption spectrum of the ( R u - p ~ - R u ) ~ +modeled by the vibronic Hamiltonian of eq 11. The spectrum has been divided into two parts, (a) and (b) by the approximate method. Parameters are given in Table 1.
onset of charge fluctuations occurs, as for m = 5 and m = 4, the presence of low-lying levels in the electronic spectrum makes such an approximate approach overly simplistic. 4. Conclusion
In this paper, we have presented and discussed a three-site model for studying trinuclear systems of the DBA type, a model that contains all the basic ingredients for understanding the properties associated with backbonding interactions. A ligand TC*orbital is explicitly taken into account, as well as electronic correlation on metal orbitals, by means of a two-band Hubbard Hamiltonian. The coupling of electronic motion to the nuclear degrees of freedom is included in the linear form, which corresponds to a variation of the equilibrium position of the oscillators upon electronic occupation, and with consideration of only one vibrational mode for each molecular fragment (D, B, and A). Our attention has been focused on the Creutz-Taube ion (D = A = Ru(NH3)5 and B = pyrazine) for which we have studied
+
+
+
J. Phys. Chem., Vol. 99, No. 26, I995 10491
Optical Properties of Mixed- and Monovalent Dimers resonance with the bridge orbital. Therefore, their presence (or absence) in a molecular state has a stabilizing (or destabilizing) effect on that state, relative to the simple Huckel predictions. This uncommon effect arises from the strong backbonding interaction. The present results indicate that both electronic correlation and vibronic interaction are needed to explain the spectral behavior of bridged mixed-valent compounds and are very encouraging for further investigation on these and other mixedvalent systems in which the bridge plays an active role. An attempt to simplify the problem by considering vibronic interaction only between selected electronic states succeeded only for the +6 ion, thereby showing that vibronic interaction is strong and in general can be accounted satisfactorily only by solving the full vibronic problem. Finally, the present results suggest that on-site repulsion (represented as U in the present model), together with vibronic coupling and the one-electron terms, is sufficient, at least in some cases, to describe the spectra of bridged dimers with variable total charge.
Acknowledgment. A.F., A.L., and G.V. acknowledge the financial support of the ‘‘Progetto Strategic0 Materiali Innovativi“ of Consiglio Nazionale delle Ricerche (C.N.R.). References and Notes (1) Isied, S.; Taube, H. J. Am. Chem. SOC. 1973, 95, 8198. Fischer, H.; Tom, G. M.; Taube, H. J. Am. Chem. SOC.1976, 98, 5512. (2) Tom, G. M.; Creutz, C.; Taube, H. J. Am. Chem. SOC. 1974, 96, 7827. (3) Creutz, C. f r o g . lnorg. Chem. 1983, 30, 1. (4) Tanner, M.; Ludi, A. lnorg. Chem. 1981, 20, 2348. ( 5 ) Mixed-Valency Systems: Applications to Chemistry, Physics and Biology; Prassides, K., Ed.; NATO Advanced Study Series C No. 343; Kluwer: Dordrecht, 1991. (6) Creutz, C.; Taube, H. J. Am. Chem. Soc 1969, 91, 3988. Creutz, C.; Taube, H. J. Am. Chem. SOC. 1973, 95, 1086. (7) Hush, N. S.; Edgar, A.; Beattie, J. K. Chem. Phys. Lett. 1980, 69, 128. (8) 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, 106, 121. (9) Zhang, L.-T.; KO,J.; Ondrechen, M. J. J. Am. Chem. SOC. 1987, 109, 1666.
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