J. Phys. Chem. B 2000, 104, 10909-10914
10909
Raman Excitation Profiles with Self-Consistent Excited-State Displacements† M. H. Hennessy, Z. G. Soos,* D. F. Watson, and A. B. Bocarsly Department of Chemistry, Princeton UniVersity, Princeton, New Jersey 08544 ReceiVed: March 23, 2000; In Final Form: August 17, 2000
Raman excitation profiles (REPs) are computed in the time domain for harmonic modes with excited-state displacement g and measured at multiple excitation frequencies ωL in a bimetallic complex with a metal-tometal charge-transfer band and 10 coupled modes. Self-consistency is required because the time evolution depends on all displaced modes rather than the individual intensities used in the short-time approximation for g(0). Self-consistency ensures that g is independent of ωL for a single resonance. Exact REPs for displaced oscillators show the importance of vibronic lifetimes and marked deviations from the short-time approximation at molecular frequencies. Spectra at multiple ωL indicate scattering from another electronic excitation with displacement h that is also found self-consistently. Mode-specific Raman data at multiple ωL require additional parameters to yield reliable structural information. Harmonic oscillators with self-consistent displacements are a first approximation to systems with many coupled modes.
1. Introduction Tang and Albrecht1 related resonance Raman (RR) intensities to excited-state structure. Raman excitation profiles (REPs) have evolved into a general approach to excited states with dipoleallowed transitions, as reviewed by Myers.2-4 RR spectra yield contributions of individual vibrations that are typically unresolved in absorption. The analysis of multimode Raman intensities relies on time-correlator methods developed by Heller5,6 that bypass tedious sums over excited states. REPs in the time domain are associated with projections of ground-state vibrational functions on excited-state potentials and their subsequent time evolution. Additional information and approximations are required to convert REPs into changes of bond lengths and angles, as shown by Spiro et al.7 for metal-to-ligand charge transfer (MLCT) in bispyridium iron(II) heme. Current approaches to REPs share some basic assumptions: as sketched in Figure 1 for the jth normal mode, the ground and excited-state potentials are harmonic and have equal force constants along the same normal modes. Displaced harmonic oscillators and linear electron-vibration coupling are also the basis of solid-state models8,9 of polarons, excitons, or solitons. The coupling constant is the slope of the excited-state potential for a vertical transition in Figure 1. The observed frequencies ωj of totally symmetric (ag) modes and displacements gj completely specify the potential. RR intensities can be found for any potential, but this requires excited-state normal coordinates, frequencies, and anharmonicities. Displaced harmonic oscillators provide an exact first approximation10 to more complicated surfaces. The ground-state vibrational function |p〉 for R modes is a product of R harmonic oscillator functions |pj〉 with pj ) 0,1,2... quanta. Vertical (Condon) transitions lead to Franck-Condon factors that are known analytically and to the time evolution of |p(t)〉 in harmonic potentials. In this paper, we analyze Raman excitation profiles for two resonant states and present a self-consistent solution for displacements g ) (g1, ..., gR). The REP for mode j is the intensity Ij(ωL) at laser frequencies ωL, which peaks for ωL in †
Part of the special issue “Thomas Spiro Festschrift”.
Figure 1. Schematic potentials for resonance Raman scattering from the ground-state |0〉 of mode ωj, with displacement gj and hj in two excited states separated by ∆E. The three potentials are harmonic and have equal ωj; the mode-specific reorganization energy is λj.
the absorption band. Since the gj cannot vary with ωL, the excitation frequency provides a simple test for the internal consistency of g in systems with one resonant excitation. Scattering from a higher energy state, with displacement h, contributes as ωL increases. The intensity Ij(ωL) goes as (gj + hj)2 and the cross terms depend on the relative signs of the displacements. This possibility has been recognized11 and its implementation requires detailed information.12 Our selfconsistent approach to displacements g and h is general. Although based on measured Ij(ωL), the results are illustrative because the available data do not specify all parameters. The minimum of the excited-state potential in Figure 1 is lowered by λj ) pωjgj2 for displacement gj and frequency ωj, with gj ) -(∂V/∂zj)0 in these units. Doorn and Hupp13 exploited the connection between mode-specific λj and the inner-sphere reorganization energy in Marcus theory of electron transfer14,15
λj )
∑j pωjgj2
10.1021/jp0011084 CCC: $19.00 © 2000 American Chemical Society Published on Web 10/26/2000
(1)
10910 J. Phys. Chem. B, Vol. 104, No. 46, 2000
Hennessy et al. resonant excitation of mode j is
Rj(xL) )
Figure 2. Linear absorption of Fe(CN)6Pt(NH3)5 in aqueous solution. The Raman excitation frequencies ωL and MMCT, CT bands are indicated. The simulated spectrum is based on eq 7, Table 2, and broadening given in the text.
They considered13 MLCT in Ru(II) complexes or metal-to-metal charge transfer (MMCT) in bimetallic complexes, with the shorttime approximation, relative Ij(ωL) and an overall scaling of (1) to the absorption width. We also use relative Ij(ωL) and the linear absorption but assign some width to the solvent and fix λi in each excited state. The trimetallic complex (NC)5Fe(II)CN-Pt(IV)(NH3)4-NCFe(II)(CN)54- has been extensively characterized by Bocarsly and co-workers.16-18 The broad MMCT at 424 nm is Fe-Pt-Fe polarized. Preliminary single-crystal spectra indicate both parallel and perpendicular bands. Figure 2 shows the linear absorption in water of (NC)5Fe(II)CN-Pt(IV)(NH3)5. The MMCT is assigned as before and another CT band is assumed; the peak positions are not known accurately. The arrows indicate the laser frequencies used. Ten displaced modes18 ωj ranging from 300 to 2000 cm-1 are inputs for REP calculations. Section 2 summarizes the explicit REP expression for displaced harmonic oscillators with R coupled modes and presents a general method for determining self-consistent g at excitation ωL. Scattering from several electronic states requires additional spectroscopic information. The limitations of the short-time approximation leading to g(0) are illustrated in section 3 for molecular frequencies. Preliminary analysis of the Fe(II)Pt(IV) complex indicates two excitations, dependence on homogeneous broadening, and major deviations from g(0). The Discussion emphasizes that REPs give accurate information about vertical transitions, but not about the dynamics necessary for structure. 2. Self-Consistent REPs of Displaced Harmonic Oscillators We consider resonant Stokes scattering with excited-state displacement g for R modes. The laser frequency is ωL ) xL + E(0,0)/p, where E(0,0) is the 0-0 excitation energy of a dipoleallowed transition. Detailed derivations of REPs are given elsewhere.2,6,19 The ground vibrational function |p〉 has pk quanta in mode k. Resonant scattering involves excited-state vibrations that, as indicated in Figure 1, are displaced along the coupled modes. In units of pωj, the coupling constant gj is the derivative of the excited-state potential at the ground-state geometry and is characteristic of linear electron-vibration coupling. Within scale factors common to all modes, the polarizability tensor for
i π
∫0∞dt〈pb1|pb(t)〉eix te-Γt L
(2)
where the final state |p1〉 differs from |p〉 only in having pj + 1 quanta in mode j in the ground-state potential. The same homogeneous broadening Γ is assumed for each vibronic pω· p. This choice has significant consequences below. The time correlator vanishes at t ) 0, but the evolution of |p(t)〉 on the excited-state surface generates overlap for t > 0. The time evolution of displaced harmonic oscillators is completely coherent and, more importantly, is known. The assumption of displaced oscillators and equal Γ make the evaluation of eq 2 practical, in contrast to the corresponding R sums over the displaced oscillators. Since the relevant vibrational quanta are kT or larger, we focus on the ground state, |0〉, with pj ) 0 for all j, rather than the general case found in formal developments.2,6,19 The time autocorrelation function is
〈0 B|0 B(t)〉 ) exp[-
∑j gj2(1 - eix ω t)] L j
(3)
The matrix elements for mode j in eq 2 can be evaluated directly or related to partial derivatives with respect to gj for the vibrational ground state to give10
Rj(xL) ) gjF(ωj,xL)
(4)
The linear coupling gj appears explicitly. The integral is
F(ωj,xL) )
∫0∞dt(1 - eω t)〈0B1|0B(t)〉eix te-Γt
i π
j
L
(5)
We use eqs 4 and 5 for 1-0 scattering from the ground vibrational state throughout. Mode j and coupling gj appear explicitly and all displaced modes collectively through eq 3. Raman scattering Rj(xL) from general |p〉 in eq 2 is more complicated but can be worked out. The calculated RR intensity for mode j is gj2|F(ωj,xL)|2. We approximate inhomogeneous broadening in solution by convolving with a normalized Gaussian to obtain
Ij(xL,Γ,σ) ) gj2(2πσ2)-1/2
∫∞-∞dx|F(ωj,x)|2 exp[-(x - xL)2/2σ2] (6)
The dependence on ωL is specified by eq 5 in terms of the displacement g and broadening Γ. The intensity increases as gj2 and all components of g contribute through the time dependence of |p(t)〉. The sign of gj is not determined for displaced harmonic oscillators. The normalized line-shape function for linear absorption is
I(x,Γ,σ) ) -Re
i π
∫0∞dt〈pb|pb(t)〉eix te-Γt-σ t /2 L
22
(7)
where x ) ω - E(0,0)/p is again relative to 0-0 and the autocorrelation function is unity at t ) 0. The convolution integral (eq 6) for solvent broadening is elementary for linear processes and is the factor of exp(-σ2t2/2). The ωjt f 0 limit in eq 3 has linear terms -iωjgj2t that give the Franck-Condon maximum (eq 1) for the absorption. The steepest descent approximation to I(x) stops with the quadratic terms -gj2ωj2t2/2 of eq 3, which contribute to the width of the Gaussian profile.
Raman Excitation Profiles
J. Phys. Chem. B, Vol. 104, No. 46, 2000 10911
Hush20 developed this approximation for I(x) for electron transfer in solution. The low-frequency limit, ωjt f 0, of eq 5 is analogous to short time. The leading contribution is
F(ωj,xL) f ωjF(0,xL) ) (ωj/π)
∫0∞t dt〈0B|0B(t)〉eix te-Γt L
(8)
The time integral no longer depends on the mode j, and Ij(xL) is proportional to gj2ωj2. The ratio Ij/Ir immediately yields (gj(0)/ gr(0))2 in this, the short time approximation. An overall scale factor such as the reorganization energy (eq 1) converts ratios into absolute displacements. Hupp and co-workers13 took σ ) 0 and obtained upper bounds for g(0) that neglect solvent contributions to the width of I(x). Both σ and λi can be extracted from the asymmetry of the linear spectrum.10 The approximation (eq 8) is accurate below 100 cm-1, but is poor for molecular vibrations. The evaluation of g is a self-consistent problem, because the time evolution in eq 2 or 5 contains all displaced modes. With g(0) determined from eq 8, we evaluate eq 5 to obtain Fj(0) for j ) 1, 2, ..., R and find the zeroth-order scattering tensors, Rj(xL) in eq 4, whose magnitudes are Ij(0)(xL). The calculated and measured scattering differ due to the ωj dependence of F(ωj,xL). The iteration starts by finding gj(1) to give the correct Ij when the time evolution is governed by g(0). The g(1) are scaled to satisfy eq 1. The time evolution under g(1) is then used to obtain Fj(1) in eq 5 and intensities Ij(1). The calculated and measured Ij again differ because the time evolution depends on g and changes as successive displacements are readjusted. In the next iteration, we adjust gj(2) to match Ij when g(1) controls the time evolution. Self-consistency is achieved when the calculated and observed Ij are equal. The displacement g then produces a time dependence in eq 5 that is consistent with gjF(ωj,xL). Convergence typically takes five cycles. The self-consistent procedure specifies g at fixed ωL and is exact for displaced harmonic oscillators. The approximation eq 8 also fixes g(0) uniquely at ωL, since there are R equations in R unknowns. But the components of g(0) vary by factors of 2-3 on changing ωL in the same resonance. Any dependence on the excitation frequency is unphysical for a single potential. Selfconsistent g completely eliminate the ωL dependence for displaced harmonic oscillators, as typically assumed for REPs. Variations with ωL then point to contributions from other electronic states. Broad, overlapping transitions in solution make such scattering likely. The generalization of REPs to two allowed electronic transitions is straightforward. We again use the Condon approximation, now for transition dipoles µA and µB with fixed relative orientation in the molecule. We have ∆E ) EB(0,0) EA(0,0) > 0 in Figure 1 and displacements g, h. The Raman scattering tensor (4) contains the transition moments
Rj(xL) ) gjF(ωj,xL)µA2cos2 θ + hjF(ωj,xL - ∆E)µB2 sin2 θ (9) With the MMCT and CT bands of Figure 2 in mind, we take A to be z-polarized, B to be x,y polarized, and can easily treat other cases. Displaced harmonic oscillators with ground-state frequencies gives the same scattering, up to an energy offset, for all excited states; this underscores their first-order nature. We retain the ground vibrational state |0〉, the autocorrelation function (3), and identical Γ, σ, and λi for simplicity. The time evolution of F(ωj,xL,σ) is given by (5) with g for the first state and xL - ∆E, h for the second. The amplitude |Rj(xL)|2 gives
TABLE 1: Resonance Raman Intensities of the Fe(II)Pt(IV) Complex at Excitation ωL, Relative to the 981 cm-1 Sulfate Referencea pωL (cm-1) 300 365 422 473 505 567 590 626 2085 2129 24 588 24 207 21 839 21 155 20 491 19 932 19 436 18 835 17 599 15 453
0.28 0.33 0.81 0.73 0.95 1.37 1.05 0.89 0.67 0.27
0.72 0.95 2.39 1.78 1.69 2.65 2.63 1.85 1.86 1.44
0.09 0.20 0.39 0.15 0.20 0.61 0.41 0.23 0.04 0.18
0.51 0.72 1.08 0.88 0.89 1.46 1.12 0.77 0.54 0.44
0.37 0.55 0.67 0.75 0.61 1.16 0.76 0.43 0.45 0.30
1.44 1.92 2.74 3.02 3.30 4.67 4.30 2.81 3.49 0.87
0.70 0.73 1.62 1.51 1.72 1.72 1.97 1.61 1.30 0.85
1.56 1.97 3.43 3.74 4.69 5.54 5.54 3.71 4.34 1.83
3.04 10.1 3.33 11.0 2.16 5.55 1.86 4.36
a The displaced modes, with frequencies from 300 to 2129 cm-1, are identified in Table 2.
the intensity Ij at frequency ωL ) xL + EA(0,0)/p for an axial molecule with azimuthal angle θ relative to the oscillating field. The solution averages are cos4 θ ) 1/5, sin4 θ ) 8/15 and sin2θ cos2 θ ) 2/15. Scattering from A dominates when xL is on the low-energy side of EA in Figure 1. The self-consistent determination of g is carried out at such xL. As noted above, g cannot depend on xL. Scattering at higher frequency, when xL is between the two peaks, is from both A and B. With fixed A, the B contribution can be extracted from the observed Ij. The practical limitations are that additional displacements require accurate intensities and reasonably harmonic excited-state potentials. We suppose that g has been determined and consider intensities Ij at higher xL where A and B contribute according to eq 9. The solution average of |Rj(xL)|2 leads to
Ij(xL) ) |FA|2 + c(FA*FB + FAFB*) + d|FB|2
(10)
with c ) (4/3)(µB/µA)2 and d ) (8/3)(µB/µA).4 Since the scattering FA due to g is known, we seek the R components of h that satisfy ratios of eq 10 and the reorganization energy λi. An iterative scheme is again required because the time evolution of eq 5 depends on all modes. Any convenient choice of h(0) can be used, such as equal displacements hj(0) for all j. The Ij now give quadratic equations for hj(1) due to the cross terms in eq 10 and h(1) is fixed by λi. The time evolution under h(1) is used for the next iteration and the process is repeated until selfconsistent h are obtained. The second displacement usually requires more cycles, about 10. 3. Model Calculations for Fe(II)Pt(IV) RR intensities of the Fe(II)Pt(IV) complex were measured in aqueous solution containing K2SO4, as previously reported18 for the trimetallic complex. The concentrations are 40 mM for the complex and 0.5 M for K2SO4. The intensities are relative to a sulfate line far from resonance. The 981 cm-1 mode has been calibrated21 and its intensity increases by almost an order of magnitude between 15000 and 25000 cm-1. The corrected intensities Ij(xL) relative to sulfate are listed in Table 1 at 10 laser frequencies ωL that span the Fe(II)Pt(IV) absorption. Ten displaced modes ω yield almost 100 entries. The assignments in Table 2 follow18 the trimetallic and related complexes; the modes range from 300 cm-1 bends to 2129 cm-1 stretches. In principle, the 10 components of g should reproduce all intensities in Table 1 for scattering from one displaced harmonic oscillator, while the 20 components of g and h should suffice for two harmonic excited states. The reorganization energy (eq 1) is needed to obtain g and h from relative intensities. In addition to ω, g, and λi, parameters
10912 J. Phys. Chem. B, Vol. 104, No. 46, 2000
Hennessy et al.
TABLE 2: Frequencies, Assignments, and Displacements of Fe(CN)6Pt(NH3)5a displacement pωL(cm-1)
assignment
g
h
g(0)
2129 2085 626 590 567 505 473 422 365 300
ν(CN) bridge ν(CN) axial ν(Fe-CN) axial ν(Fe-CN) radial ν(Fe-CN) bridge ν(Pt-NH3) δ(Fe-CN) δ(Fe-CN) ν(Pt-NC) bridge δ(H3N-Pt-NH3)
0.42 0.23 0.93 0.52 0.85 0.28 0.29 0.08 0.52 0.31
0.80 0.43 0.13 0.11 0.17 0.12 0.14 0.08 0.07 0.04
0.45 0.25 0.65 0.42 0.71 0.43 0.52 0.31 0.77 0.56
a Excitation at 568.2 nm (17 599 cm-1) gives the self-consistent g and short-time g(0) discussed in the text; excitation at 413.1 nm (24 207 cm-1) is modeled by g + h and two resonant states.
Figure 4. Raman excitation profiles (REPs) of displaced harmonic oscillators and measured intensities of Fe(CN)6Pt(NH3)5. EA is the MMCT 0-0 line; the indicated multiplicative factors are relative to the first panel; the dashed and solid lines have displacement g and g + h in Table 2 and identical parameters given in the text. A single scale factor relates calculated and measured intensities. Figure 3. Scattering intensities computed as Ij(xL,Γ,σ)ωj-2gj-2, eq 6, for molecular modes ranging from ωj ) 100 to 2000 cm-1 with Γ ) 50 cm-1 (solid lines) and Γ ) 500 cm-1 (dashed lines) and otherwise identical parameters given in the text. There is no ωj dependence in the short-time approximation, eq 8.
such as Γ and σ depend on the state but are taken equal for simplicity. The integral (eq 6) then gives Fj(xL) and the calculated Ij(xL) ) gj2|F(ωj,xL)|2. The Raman scattering (eq 9) for two states also requires ∆E and µA, µB; its squared amplitude and solvent broadening leads to the intensity (eq 10). Different inputs can readily be used, but there are far too many possibilities to explore the full range, and we prefer estimates for actual complexes. We start with the REPs (eq 6) for a typical2 Γ ) 50 cm-1, one-electronic state with 0-0 at xL ) 0, equal displacements for the 10 modes in Table 2, λi ) 1800 cm-1, and σ ) 1500 cm-1. In the short-time approximation (eq 8), the spectrum of Ij(xL,Γ,σ)ωj-2gj-2 is identical for all modes. A logarithmic scale is needed for the solid curves in Figure 3 to include modes from 100 to 2000 cm-1. The intensity increase is far less than ωj2 when Γ < ωj. The dashed lines curves in Figure 3 are REPs with same inputs except for Γ ) 500 cm-1. They are 100-fold weaker and cross at xL > λi for these parameters. The integrand of eq 5 is small for Γt > 1, which accounts for a roughly Γ-2 intensity dependence. The integrand is also small for ωjt < 1, which is the basis of the approximation (8). As expected, the REPs in Figure 3 peak around xL ∼ 0 for small ωj and shift toward the Franck-Condon maximum at xL ) λi for large ωj. The Ij/Ik ratios clearly vary with xL, and g(0) consequently
changes. This unphysical result is not due to harmonic oscillators, which are assumed in either case, but to approximating F(ωj,xL) in eq 5 by its low-frequency limit (eq 8). Both Γ and σ broaden the REPs. Solvent broadening in eq 6 is a convolution of the σ ) 0 spectrum that conserves area. The Raman scattering tensor (eq 5) diverges as Γ f 0, as noted above, and large Γ improves the short-time approximation. Γ > 1000 cm-1 is sometimes used11 in REP simulations to represent all broadening. The magnitude matters when the REPs are computed directly. We note that equal Γ is used in the basic time-correlator expression (eq 2). This assumption resembles displaced harmonic oscillators in being extremely powerful and clearly approximate. The calculated linear absorption in Figure 2 is based on eq 7 and the following parameters, which are also used for the intensities in Table 1. The frequencies and displacements in Table 2 specify the time dependence of the autocorrelation function (eq 3). We take Γ ) 50 cm-1, λi ) 1800 cm-1 for both MMCT and CT, (µB/µA)2 ) 1.28 and orthogonal moments in eq 9, EA ) 19500 cm-1, ∆E ) 3500 cm-1, and σ ) 1900 cm-1. The decomposition in Figure 2 is not unique and another electronic state is partly resolved at higher energy.18 The intensities in Table 1 span two orders of magnitude. We used 568.2 nm data to obtain g and then 413.1 nm data to obtain h for the second state and scaled the total to the calculated intensities at this ωL. Comparison with experiment is shown in Figure 4 with EA at the MMCT 0-0. The solid and dashed lines represent, respectively, the total intensity g + h, which is forced by construction to pass through the 413.1 nm data, and the intensity due to g alone. The strongest scattering is from
Raman Excitation Profiles the 2129 cm-1 CN stretch in the first panel. To facilitate inspection, a factor is shown in parentheses to put other modes on the same scale. These are calculated, since a single scale relates all points and curves in Figure 4. The weakest mode (422 cm-1) shows considerable experimental scatter that is accentuated on the enlarged scale. The top panels are CN stretches measured at four xL; weak scattering 1000 cm-1 below EA is consistent with previous data.18 The other eight panels are REPs for the indicated modes and 10 measurements. The low-frequency modes follow the MMCT, as indicated by the dashed line. The calculated displacements g and h in Table 2 are out of phase. Negative cross terms are responsible, notably in the 626, 590, and 567 cm-1 panels, for total scattering above EA that barely exceeds the MMCT contribution. The present fits are preliminary for several reasons: The data have scatter and the CN intensities are limited to the blue; the calculations contain parameters that could be estimated from additional spectra; larger Γ or different overall scaling improves the fit slightly. The g(0) entries in Table 2 are the low-frequency approximation (eq 8) to the 568.2 nm data with the same parameters. They differ significantly from g because, as seen in Figure 3, the scattering of high-frequency modes is overestimated. This translates into reduced gj(0) for CN stretches and increased gj(0) for low-frequency modes at fixed reorganization energy λi. Published g(0) can readily be converted to self-consistent g using the same inputs. Such corrections are mandatory: displaced harmonic potentials are assumed in the short-time approximation and they unequivocally lead to g rather than g(0). Multiple excitation frequencies and the short-time approximation are in fact related, since “exact” fits based on g(0) are always possible and their inadequacy emerges with a second frequency. For harmonic potential potentials, g then gives the same displacements and nearby ωL in the absorption peak can be fit quantitatively. Williams et al.22 took three frequencies and iterated displacements for six coupled modes of an organic radical cation. Dong and Spiro23 also used three frequencies and avoided the short-time approximation. These recent examples illustrate the shift from this approximation. Our wide range of ωL cannot be fit with a single harmonic excited state and the fit in Figure 4 is limited for two states. RR spectra give ωj directly and absolute intensities yield mode-specific reorganization energies. There is far more information than in linear absorption, but additional inputs are required for displacements, even for harmonic excited states. Detailed analysis of the displacements in Table 2 is premature in our opinion. The assumed CT band is dominated by CN stretches; the contributions of other modes to the reorganization energy (eq 1) are negligible. The MMCT has large components of g for Fe-CN-Pt modes. Other choices of λi for either state, (µB/µA)2 or Γ and σ, yield fits comparable to Figure 4 for different g and h. The parameters tend to compensate and independent estimates are needed. Additional information could come from computations, spectroscopy, or physical reasoning. We emphasize self-consistent displacements and differences between g(0) and g for molecular frequencies. 4. Discussion We have presented a self-consistent method for excited-state displacement g of harmonic oscillators in the time domain. The general expression (eq 5) for F(ωj,xL,σ) holds for Raman scattering from the ground vibrational state for any number of displaced ag modes. Absolute RR intensities are directly related to gj2 while the more easily obtained relative intensities Ij(xL)/
J. Phys. Chem. B, Vol. 104, No. 46, 2000 10913 Ir(xL) also require the reorganization energy (eq 1). Assumptions are made in either case about homogeneous and inhomogeneous broadening. The extension to another resonant state also requires information about the magnitude and relative orientation of transition dipoles. The general expression eq 2 relates RR intensities to the evolution of ground-state vibrations |p(t)〉 on the excited-state surface. Its direct physical interpretation is a major strength. There are no limitations to similar ground and excited-state normal modes, to identical frequencies, or to harmonic potentials. The assumption of constant Γ deserves more attention in view of its impact on the relative intensities of molecular modes. A linear dependence Γ(E) on excess energy is a simple generalization that has been examined22 for Raman scattering. Raman data for vertical or nearly vertical transitions must be supplemented by excited-state dynamics to obtain structural information. Displaced harmonic oscillators and constant Γ are natural first approximations. They can be tested, as illustrated above, by combining multiple excitation frequencies with selfconsistent treatment of REPs and linear absorption. The shorttime approximation fails for molecular modes, but can readily be corrected with self-consistent g. The complexities of excitedstate potentials limit the use of RR intensities as a structural tool. Computations are making accessible more realistic potentials that may well change the role of RR spectra to testing rather than finding excited-state structure, since frequencies and intensities are the measured quantities. Difficulties can be expected in systems with anharmonic potentials, with large amplitude torsional modes, or Duchinsky rotation. They are beyond the present model. Displaced harmonic oscillators may be a starting point for small molecules, but they remain the preferred approach to large systems, where the approximation probably improves for many coupled modes. The gradient (∇V)0 for a vertical transition in Figure 1 is at most ∼10 eV/Å for exciting an electron from a bonding to antibonding orbital. Smaller gradients account directly for RR spectra of polyenes and conjugated polymers.23 Delocalization means small changes in many bonds that may correspond to one or many normal modes. The R ) 10 modes of the Fe(II)Pt(IV) complex have small individual displacements in Table 2, consistent with the absence of harmonics in the closely related trinuclear complex.18 Small displacements are expected in large systems whose RR spectra can be modeled as displaced harmonic oscillators with constant vibronic lifetimes. Acknowledgment. We thank the reviewer for pointing out that the sulfate line in the 2-3 eV range is calibrated in ref 21 rather than the higher energy calibration used originally. Z.G.S. thanks T. G. Spiro, K. K. Lehmann, and A. Painelli for stimulating discussions. We gratefully acknowledge support by the National Science Foundation through DMR-9530116 and CHE-963180. References and Notes (1) Tang, J.; Albrecht, A. C. J. Chem. Phys. 1968, 49, 1144. (2) Myers, A. B.; Mathies, R. A. In Biological Applications of Raman Spectroscopy; Spiro, T. G., Ed.; Wiley: New York, 1987; Vol. 2, pp 1-58. (3) Myers, A. B. Acc. Chem. Res. 1997, 30, 519. (4) Myers, A. B. Chem. ReV. 1996, 96, 911. (5) Lee, S. Y.; Heller, E. J. J. Chem. Phys. 1979, 71, 4777. (6) Tannor, D. J.; Heller, E. J. J. Chem. Phys. 1982, 77, 202. (7) Wright, P. G.; Stein, P.; Burke, J. M.; Spiro, T. G. J. Am. Chem. Soc. 1979, 101, 3531. (8) Painelli, A.; Girlando, A. J. Chem. Phys. 1986, 84, 5655. (9) Heeger, A. J.; Kivelson, S.; Schrieffer, J. R.; Su, W. P. ReV. Mod. Phys. 1988, 60, 81.
10914 J. Phys. Chem. B, Vol. 104, No. 46, 2000 (10) Hennessy, M.; Wu, Y.; Bocarsly, A. B.; Soos, Z. G. J. Phys. Chem. A 1998, 102, 8312. (11) Clark, R. J. H.; Dines, T. J. Mol. Phys. 1981, 42, 193. (12) Egolf, D. S.; Waterland, M. R.; Myers Kelley, A. J. Phys. Chem. In press. (13) Doorn, S. K.; Hupp, J. T. J. Am. Chem. Soc. 111, 1989, 1142, 4712. Doorn, S. K.; Blackbourn, R. L.; Johnson, C. S.; Hupp, J. T. Electrochim. Acta 1991, 36, 1775. (14) Marcus, R. A. ReV. Mod. Phys. 1993, 65, 599; Annu. ReV. Phys. Chem. 1964, 15, 155. (15) Barbara, P. F.; Meyer, T. J.; Ratner, M. A. J. Phys. Chem. 1996, 100, 13148. Ondrechen, M. J. Int. ReV. Phys. Chem. 1995, 14, 1. (16) Pfennig, B. W.; Bocarsly, A. B. J. Phys. Chem. 1992, 96, 226; Wu, Y.; Cohran, C.; Bocarsly, A. B. Inorg. Chim. Acta 1994, 226, 251. (17) Wu, Y. Ph.D. Thesis, Princeton University, 1996 (unpublished). (18) Pfennig, B. W.; Wu, Y.; Kumble, R.; Spiro, T. G.; Bocarsly, A. B. J. Phys. Chem. 1996, 100, 5745.
Hennessy et al. (19) Page, J. B.; Tonks, D. L. J. Chem. Phys. 1981, 75, 5694. Gussoni, M.; Castiglione, C.; Zerbi, G. In Spectroscopy of AdVanced Materials, AdVances in Spectroscopy; Clark, R. J. H., Hester, R. E., Eds.; Wiley: New York, 1991; Vol. 19, pp 251-353. (20) Hush, N. S.; Electrochim. Acta 1968, 13, 1005; Prog. Inorg. Chem. 1957, 8, 391. (21) Dudik, J. M.; Johnson, C. R.; Page, S. A. J. Chem. Phys. 1985, 82, 1732, Table 1. (22) Williams, R. D.; Hupp, J. T.; Ramm, M. T.; Nelson, S. F. J. Phys. Chem. 1999, 103, 11172. (23) Dong, S.; Spiro, T. G. J. Am. Chem. Soc. 1998, 120, 10434. (24) Li, P.; Champion, P. M. J. Chem. Phys. 1988, 88, 761. (25) Soos, Z. G.; Mukhopadhyay, D.; Painelli, A.; Girlando, A. In Handbook of Conducting Polymers, 2nd ed.; Skotheim, T. A., Elsenbaumer, R., Reynolds, J. R., Eds.; Marcel Dekker: New York, 1998; pp 168-196.
