J. Phys. Chem. B 2000, 104, 10727-10737
10727
ARTICLES Resonance Raman Intensity Analysis of the Carbazole/Tetracyanoethylene Charge-Transfer Complex: Mode-Specific Reorganization Energies for a Hole-Transport Molecule† Debra S. Egolf,‡ Mark R. Waterland,§ and Anne Myers Kelley*,§ Department of Chemistry and Center for Photoinduced Charge Transfer, UniVersity of Rochester, Rochester, New York, 14627-0219 ReceiVed: October 26, 1999
A resonance Raman intensity analysis is presented for the carbazole/tetracyanoethylene donor-acceptor chargetransfer complex in dichloromethane solution. The intent is to determine the nuclear reorganization contributions to the rates of charge hopping in carbazole polymers used as hole-transport agents in the xerographic (electrophotographic) process. Resonance Raman cross sections have been measured at seven excitation wavelengths spanning the broad visible charge-transfer absorption known to consist of two strongly overlapping charge-transfer electronic transitions. Interference between the Raman amplitudes from these two transitions manifests itself in the Raman excitation profiles for a number of resonantly enhanced modes. Simultaneous modeling of the absorption spectrum and the Raman cross sections shows that explicit consideration of the signs as well as the magnitudes of the normal mode displacements is required to reproduce the experimental data. The signs of the mode displacements obtained from the modeling are shown to be consistent with qualitative predictions based on the nodal patterns of the donor molecular orbitals and the forms of the resonantly enhanced normal modes. Mode-specific reorganization energies for the carbazole donor and tetracyanoethylene acceptor are obtained from the analysis along with parameters describing the magnitude and time scale of the solvent contributions to the reorganization energy. Approximately 60% of the total reorganization energy of 5100 cm-1 in each charge-transfer state is attributed to solvent and any other classically behaved low-frequency modes, with the remainder about equally divided between donor and acceptor modes. The partitioning of the reorganization energy among the carbazole modes is significantly different for the two electronic transitions.
I. Introduction The rate of electron transfer in the nonadiabatic limit is typically described by a Golden Rule expression1-3
ket ) (2π/p)|V|2 FC
(1)
that leads to the familiar behavior of the electron-transfer rate in the “Marcus inverted region”.4-6 V is the electronic coupling matrix element, and the Franck-Condon factor (FC) contains the dependence of the rate on the density of states and the total nuclear reorganization energy. The latter includes contributions from both low-frequency solvent and/or molecular modes (treated classically) and high-frequency molecular vibrations (treated quantum mechanically). The connection between the rate of electron transfer and the optical absorption line shape for a charge-transfer transition was recently illustrated.7-11 Both are Golden Rule expressions containing the linear response of †
Part of the special issue “Thomas Spiro Festschrift”. * To whom correspondence should be addressed. E-mail: amkelley@ ksu.edu. Address: Department of Chemistry, Kansas State University, 111 Willard Hall, Manhattan, KS 66506-3701. ‡ Permanent address: Department of Chemistry, Marietta College, Marietta, OH 45750-4014. § Current address: Department of Chemistry, Kansas State University, 111 Willard Hall, Manhattan, KS 66506-3701.
the material system. Employing a specific basis set leads to expressions similar to eq 1 (in the limit of static solvent dynamics). In particular, the Franck-Condon part of the Golden Rule expression for electron transfer is equivalent to that for photoemission with a hypothetical “zero-frequency” photon. Therefore, the system parameters, e.g., reorganization energies, that describe the material linear response and hence electron transfer are contained in the optical absorption line shape. Often, however, rapid electronic dephasing processes broaden the absorption line shape, obscuring the structure that contains such information and preventing these parameters from being reliably extracted. Resonance Raman spectroscopy has long been recognized as a powerful technique for determining changes in nuclear geometry following electronic excitation.12-17 The same material system parameters that determine the optical absorption line shape (related to a two-time correlation function) also determine the resonance Raman excitation profile (related to a four-time correlation function).18-20 The Raman line widths are determined by ground-state vibrational dephasing processes, while the electronic dephasing that contributes to the optical absorption and fluorescence line shapes affects only the shapes of the Raman excitation profiles and the overall intensity of the resonance Raman scattering. The resonance Raman intensities also depend on the geometry change for each normal mode that
10.1021/jp9938009 CCC: $19.00 © 2000 American Chemical Society Published on Web 02/03/2000
10728 J. Phys. Chem. B, Vol. 104, No. 46, 2000 is coupled to the resonant transition. Recently, the application of resonance Raman spectroscopy to systems in which electronic excitation corresponds to formal electron transfer from a neutral ground state, DA, to an ion-pair excited state, D•+/A•- , has provided a method for determining the mode-specific reorganization energies contained in the vibrational part of eq 1.11 Resonance Raman spectroscopy has been used to investigate a variety of charge-transfer transitions including metal-to-ligand (MLCT), ligand-to-metal (LMCT), ligand-to-ligand, and mixedvalence transitions of inorganic complexes, and charge-transfer transitions of organic covalently and noncovalently bound donor-acceptor (DA) complexes.21-33 Many resonance Raman intensity analyses for the determination of nuclear geometry displacements involve simplified expressions that are the result of several assumptions. One is that a single electronically excited state contributes to the scattering tensor. Often, however, absorption bands are composed of overlapping electronic transitions. This is particularly true for the LMCT and MLCT transitions of transition metal compounds.34 It has been shown previously that scattering from states quite distant in energy can contribute strongly to the scattering from the state of interest via preresonance enhancement, particularly when the resonant transition of interest is weak.35-37 Previous work by this group presented a general method for analyzing contributions from multiple electronic states to the resonant scattering process, irrespective of the relative magnitudes of the transition dipole moments or energy separation between the excited states.23,38 This work presents for the first time, to the best of our knowledge, an analysis of absolute Raman cross sections from strongly overlapping chargetransfer electronic states. Zink et al. have analyzed relatiVe Raman profiles from several charge-transfer and other systems in which interferences between two electronic states are important.39,40 This paper investigates the lowest energy charge-transfer band of the complex formed between carbazole as donor and tetracyanoethylene (TCNE) as acceptor. Simultaneous modeling of the resonance Raman excitation profiles and the absorption spectrum using a time-dependent description of the spectroscopic processes allows mode-specific reorganization energies to be determined. A new aspect of resonance Raman intensity analysis is investigated here. When only a single electronic state contributes to the Raman amplitude and the ground and excitedstate normal modes are assumed to be parallel (no Duschinsky mixing), the Raman cross section is insensitive to the sign of the displacement of a resonantly enhanced normal mode. However, as outlined below, the charge-transfer band of the carbazole/tetracyanoethylene complex is comprised of two strongly overlapping electronic transitions. In this case the Raman amplitude contains contributions from both excited states and the resulting Raman cross sections contain interference terms whose signs are determined by the products of the normal mode displacements for each state. Thus, the Raman cross sections are sensitive to the relative signs of the displacements in the two contributing electronic states. (This was pointed out in early work from Zink’s group,39 but they apparently did not explore the effect of varying the signs of the displacements in fitting their experimental data.) Additional information is still needed to establish one of the signs, but this is often obtainable from fairly qualitative representations of the molecular orbitals and the normal mode involved. Carbazole has properties that are of interest from practical as well as theoretical viewpoints. Following the discovery that poly(N-vinylcarbazole) (PVK) is an efficient organic photo-
Egolf et al. conductor41 and the discovery of the subsequent incorporation of PVK into a commercial process for electrophotography (xerography),42 a variety of studies of the charge-transfer complex between PVK and the electron acceptor trinitrofluorenone (TNF),43 as well as photogeneration and charge transport in films of this complex,44,45 were reported. Landman et al. demonstrated46 that the asymmetry observed in the lowest energy charge-transfer band of various carbazole/chloranil complexes is due to overlapping charge-transfer transitions originating from the HOMO and HOMO-1 molecular orbitals of the carbazole derivatives. Subsequent work focused on modeling transport under the influence of an external electric field, without regard to the specific molecular parameters that influence the mechanism.47-49 The mode-specific reorganization energies are important parameters in a complete theoretical description of this process. This paper presents the absolute resonance Raman scattering cross sections of the 1:1 carbazole/tetracyanoethylene complex measured at seven different excitation wavelengths spanning the lowest energy charge-transfer band. A total of 31 Raman active fundamental modes were observed. Mode-specific reorganization energies are determined via modeling with a timedependent description of the absorption and Raman scattering processes. The model includes explicit parameters for the two electronic states that contribute to the absorption and Raman scattering processes. The incorporation of two electronic states allows the relative signs of the normal mode displacements to be determined directly from the Raman cross sections. The solvent reorganization energy is also obtained. II. Experimental Methods Carbazole (Aldrich) was recrystallized twice from ethanol, and tetracyanoethylene (Aldrich) was purified through multiple sublimations. Solutions of carbazole, TCNE, and their donoracceptor complex were prepared in spectroscopic grade dichloromethane (Aldrich) with approximate concentrations of 30 mM carbazole and 60 mM TCNE. Absorption spectra, obtained on a Perkin-Elmer Lambda 19 UV-visible spectrophotometer, were used to determine the concentrations of the 1:1 complex, typically about 2 mM. The molar absorptivity at the absorption maximum of 601 nm is 3860 M-1 cm-1.50 Resonance Raman spectra were obtained with CW excitation at seven wavelengths using the general methods described previously.21,22 A Lexel 95-4 argon ion laser was used for excitation at 488 and 514 nm. An argon ion pumped dye laser (Spectra Physics 375B) provided excitation at 570, 625, 656, 687, and 723 nm. At 488 and 514 nm, spurious emission lines were removed with a SPEX 1450 prefilter while a Pellin-Broca prism was used for the redder wavelengths. The incident beam, at a power of 10-25 mW, was focused with a 15 cm focal length lens onto a spinning cell oriented in a backscattering geometry. Approximately 2 mL sample sizes were used. The scattered light was collected using reflective optics, dispersed using a SPEX 1877E 0.6 m triple spectrograph with a 1200 g/mm grating blazed at 500 nm, and detected with a liquid nitrogen cooled SPEX Spectrum One CCD detector. The signal was accumulated for 60 s, with a total of 15-120 individual accumulations comprising a single data set. No sample decomposition was observed during signal accumulation as monitored by absorption spectroscopy. Cosmic spikes were removed from each individual scan before averaging. Raman scattering from solvent was used for calibration of the frequency axis. All spectra were intensitycorrected using an Optronic Laboratories 245C tungsten-
Carbazole/Tetracyanoethylene Complex
J. Phys. Chem. B, Vol. 104, No. 46, 2000 10729
Rxx ) R1 + R2 cos2 θ
(2a)
Ryy ) R2 sin2 θ
(2b)
Rxy ) Ryx ) R2 cos θ sin θ
(2c)
Here, R1 and R2 are the contributions to the total Raman amplitude from the low-energy CT state and the high-energy CT state, respectively. In the carbazole/TCNE system the transition dipole moment vectors are assumed to be parallel (see the discussion section). Equations 2a-2c then reduce to
Riftot ) Rxx ) R1 + R2 Figure 1. Absorption spectrum of the carbazole/TCNE complex in CH2Cl2 (solid line) and a best fit to the spectrum using two log-normal functions to represent the two CT states plus a third to account for the next higher-energy state (long dashed line, CT1; short-long dashed line, CT2; solid line, higher state; their sum is the dotted curve almost superimposed on the experiment). The excitation wavelengths at which resonance Raman spectra were obtained are also indicated.
(3)
The Rk (k ) 1, 2) are calculated using the time-domain expression for the scattering amplitude:
Rk(ωL,δ) )
∫0∞ dt 〈χf|χi(k)(t)〉 exp[i(ωL - ωk - δ + ωi)t -
1 p
g(t)] (4)
halogen lamp.13 Spectra of the complex were further corrected for reabsorption using a two-peak Gaussian fit (Microcal Origin 5.0 Professional) of the absorption spectrum from 480 to 900 nm.13 Integrated peak intensities were derived by using the GRAMS/32 (Galactic Industries) curve-fitting algorithm to fit mixed Gaussian/Lorentzian peak profiles to regions of the spectra. Contributions to a Raman line from uncomplexed carbazole or TCNE, present at concentrations approximately 10 (carbazole) or 20 (TCNE) times greater than that of the complex, were subtracted to give the Raman cross section due solely to the carbazole/TCNE complex. These contributions were negligible for some Raman lines at some excitation wavelengths and large for others, in some cases accounting for more than half the measured intensity. Absolute Raman cross sections were calculated13 relative to the previously determined absolute Raman cross section of the 702/740 cm-1 doublet of dichloromethane,24 assuming F ) 0.33 for all carbazole/TCNE lines. Depolarization ratios at 612, 570, 514, and 488 nm excitation were measured in a similar manner with a polarization analyzer mounted immediately before the polarization scrambler at the entrance slit of the filter stage of the triple spectrograph. Linear polarization of the input beam was ensured by passing it through a Glan-Thomson prism immediately before the focusing lens.
where ωL is the incident laser frequency, ωk is the electronic zero-zero transition frequency to state k, pωi is the energy of the initial vibrational level of the ground electronic state (for these calculations all scattering was assumed to originate from the vibrational ground state, so ωi ) 0), 〈χf| ) 〈f|µk0 and |χi〉 ) µk0|i〉 are the multidimensional ground-state vibrational wave functions multiplied by the electronic transition moment, |χi(k)(t)〉 ) exp(-iHkt/p)|χi〉 is the initial vibrational wave function propagated for time t by the excited-state vibrational Hamiltonian Hk, δ is an electronic zero-zero frequency shift due to inhomogeneous broadening, and g(t) is a solvent broadening function modeled as an overdamped Brownian oscillator. The Raman amplitude is calculated within the Condon approximation; i.e., no coordinate dependence of the electronic transition moment. Previous analyses have incorporated non-Condon effects by using a Herzberg-Teller expansion to first order.21,23 Non-Condon effects were not included in the present analysis because the data showed no clear evidence for the importance of such contributions. The function exp[-g(t)] represents the dynamic contribution of the solvent environment treated as a frictionally overdamped oscillator53
III. Theory and Computational Methods
where, in the high-temperature limit (kT . ωBO where ωBO is the frequency of the underlying oscillator),
The absorption spectrum and resonance Raman cross sections were calculated using the time-domain expressions first introduced by Heller51,52 and recently summarized in refs 13, 14, and 23. Two electronic states are considered to contribute to the Raman amplitude for the carbazole/TCNE complex (see Figure 1). The effect of interferences between the Raman amplitudes for the two electronic states was considered directly by explicit modeling of the Raman cross sections obtained using a range of excitation wavelengths spanning the charge-transfer band. The interferences are influenced by the angle between the transition dipole moment vectors, µ(1) and µ(2), corresponding to the low-energy charge transfer (CT) and high-energy CT excitation, respectively. The molecule-fixed coordinate system is chosen so that µ(1) lies along the x axis and µ(2) lies in the xy plane. If θ is the angle between µ(1) and µ(2), the nonzero components of the scattering tensor are
g(t) ) gR(t) + igI(t)
(5a)
gR(t) ) (D2/Λ2)[exp(-Λt/p) - 1 + Λt/p]
(5b)
gI(t) ) [D2/(2kTΛ)][1 - exp(-Λt/p)]
(5c)
Here, D represents the strength of coupling between the electronic transition and the solvation coordinate and p/Λ gives the characteristic solvent time scale. The imaginary component of g(t) accounts for the solvation dynamics and is required to reproduce the Stokes shift.18 As the solvent time scale becomes long (κ ) Λ/D f 0), it assumes the form
gI(t) ) [D2/(2kT)]t/p ) λSt/p
(6)
where λS is the solvent contribution to the reorganization energy. The experimental observable, the differential Raman cross section, is calculated via
10730 J. Phys. Chem. B, Vol. 104, No. 46, 2000
() dσ
dΩ
)
|+⊥
∑i Bi∑f ∫ dωS
(
ωS3ωL c4
∫-∞ dδ G(δ) ∞
)
Rif (ωL,δ) Lif(ωL-ωS) (7) tot
where Bi is the Boltzmann population of initial state |i〉 (in the present case only the vibrational ground state was considered), Lif(ωL-ωS) is the normalized line shape of the Raman transition between the ground-state levels |i〉 and |f〉, ωS is the scattered frequency, and G(δ) is a normalized inhomogeneous distribution function taken to be Gaussian. Both CT states (i.e., 1 and 2) are assumed to have identical and totally correlated inhomogeneous distributions.23 At the same level of theory the absorption cross section at frequency ω is given by
σA(ω) ) Re
∑ k)1,2
∫0
∞
4π|µk0|2ω 3npc
∑i Bi∫-∞ dδ G(δ) ∞
dt〈χi|χik(t)〉 exp[i(ω - ωk - δ + ωi)t - g(t)] (8)
where n is the solvent refractive index, Re represents the real part, and the other symbols are defined below eq 4. The multidimensional time-dependent overlaps, 〈χi|χik(t)〉 and 〈χf|χik(t)〉, were calculated with algorithms given elsewhere.12 Separable harmonic potential surfaces of equal frequency for the ground and excited states, displaced relative to each other with no Duschinsky rotation, were used in the formulation of the excited-state Hamiltonian, Hk. Interference terms arise upon taking the modulus squared of the Raman amplitude. The resulting Raman intensity is dependent on not only the magnitude of the displacements of the resonantly enhanced normal modes but also the relative sign of the displacements. To determine the relative orientation of the transition dipole moment vectors for the excited CT states the dependence of the depolarization ratio upon excitation wavelength across the charge-transfer band was examined. The following expression describes the dependence of the depolarization ratio, F(θ), on the angle between µ(1) and µ(2) in terms of the corresponding Raman amplitudes,23
1 F(θ) ) 3
1 |R1|2 + |R2|2 + cos2 θ - sin2 θ (R/1R2 + R/2R1) 2 2 |R1|2 + |R2|2 + 1 - sin2 θ (R/1R2 + R/2R1) 3 (9)
(
)
(
)
At any angle θ different from zero, the frequency dependence of the Raman amplitudes determines the dependence of F(θ) on the excitation wavelength. Only in the case of parallel dipole moment vectors is the depolarization ratio independent of the excitation frequency. Under this condition, F is constant and equal to one-third. Visualization of the molecular orbitals of carbazole and TCNE was performed using PC Spartan Plus 1.5 (Wavefunction, Inc.). After the structure was input, molecular mechanics minimization was performed using the SYBYL force field (Tripos Inc.). The semiempirical AM1 method was used for further geometry optimization and generation of the molecular orbitals. The normal modes of carbazole were visualized by performing density function theory calculations (B3LYP functional, 6-31G* basis) with Gaussian 94 (Gaussian, Inc.) and then
Egolf et al. employing the XMol 1.3.1 utility (Network Computing Services, Inc.) to display the normal mode vectors. IV. Results Figure 1 presents the optical absorption spectrum of the carbazole/TCNE complex in dichloromethane solution and indicates the excitation wavelengths used to generate the resonance Raman scattering. The absorption spectrum is broad and somewhat asymmetric, with a maximum at 601 nm. A best fit using log-normal functions determined by the GRAMS/32 curve-fitting algorithm is also depicted in the figure. The fit obtained here is in good agreement with the analysis presented by Landman et al. on a variety of substituted and polymeric carbazole/chloranil donor-acceptor complexes.46 However, Okamoto et al., working on several substituted carbazole/TCNE complexes,54 and Klo¨pffer, for the complexes of n-isopropylcarbazole with both TCNE and chloranil,55 obtained their best fits with the higher-energy transition assigned as the stronger band (see Discussion). The resonance Raman spectra of the complex in dichloromethane obtained with 514.5 and 687 nm excitation are shown in Figure 2. The absolute resonance Raman intensities for all observed lines at all seven excitation wavelengths are presented in Table 1. The spectra are dominated by the TCNE ag CdC stretch mode at 1553 cm-1. The strongest carbazole mode is ν51 at 1629 cm-1. A number of qualitative differences are immediately obvious between the resonance Raman spectra obtained with red and blue excitation. The ν15 line at 654 cm-1 has relatively high intensity when excited at 687 nm but is barely discernible when obtained with 514.5 nm excitation. The modes assigned as ν41, ν42, ν43, ν45, and ν47 of carbazole at 1323, 1337, 1392, 1462, and 1495 cm-1, respectively, all show an increase in intensity, relative to other lines of the complex, as the excitation is tuned to the red edge of the absorption band. The underlying structure in the absorption spectrum produces these variations. As previously discussed by others,46,50 the CT1 and CT2 transitions originate from the HOMO and the HOMO-1 carbazole molecular orbitals, respectively. These orbitals have quite different nodal patterns (see Figure 3), so removal of an electron from each of these orbitals will enhance different normal modes of carbazole to different extents. This simple qualitative interpretation is complicated by interference effects and the relative magnitudes of the transition dipole moments. Lee and Boo56 have presented a normal mode assignment for carbazole based on earlier assignments57 and have refined it via ab initio and density functional theory calculations. Their assignments of the carbazole modes are used in this work. Assignments for the TCNE modes are taken from ref 21. The main focus of the present paper is the determination of modespecific reorganization energies. Normal mode descriptions are not required for this task; however, the assignments are required to exclude the possibility that an observed band is due to an overtone or combination band of lower frequency fundamentals. Furthermore, the normal mode assignments along with the molecular orbital descriptions can be used to make qualitative predictions of the sign of the normal mode displacement and eliminate a number of possible input parameter sets. Initial estimates for the resonantly enhanced normal mode displacements were made using the expression valid in the limit of “short-time” dynamics for Raman scattering.17,58 Under these conditions the relative intensity of a resonance Raman transition is directly proportional to the slope of the excited-state potential energy surface along the direction of the corresponding normal mode. Thus, the following expression was used to obtain an
Carbazole/Tetracyanoethylene Complex
J. Phys. Chem. B, Vol. 104, No. 46, 2000 10731
Figure 2. Resonance Raman spectra of the carbazole/TCNE complex in CH2Cl2 at 514.5 and 687 nm excitation. The complex concentration is approximately 2 mM for both spectra, and the spectra are scaled such that the solvent lines, marked with asterisks, have about the same intensity in both spectra. Lines assigned to carbazole modes (νi) or TCNE modes (tcne) are indicated; carbazole modes not labeled as a1 have b2 symmetry in uncomplexed carbazole. The slightly higher apparent intensity of the 1156 cm-1 solvent band in the 687 nm spectrum is due to the presence of an overlapping carbazole line that is slightly resonance-enhanced at this excitation wavelength. The contributions from uncomplexed carbazole and TCNE have not been subtracted from these spectra; daggers mark lines deriving more than half of their intensity from nonresonant scattering of the uncomplexed donor or acceptor.
TABLE 1: Experimental and Calculated Raman Cross Sections σR,solute/10-11 Å2 λ ) 488 nm
λ ) 514.5 nm
λ ) 570 nm
λ ) 625 nm
λ ) 656 nm
λ ) 687 nm
λ ) 723 nm
center/cm-1
expt
calc
expt
calc
expt
calc
expt
calc
expt
calc
expt
calc
expt
calc
241.8 428.4 528.9 550.9 595.1 615.2 653.8 1011.7 1108.1 1121.1 1204.4 1220.0 1237.8 1278.1 1286.3 1313.2 1323.1 1336.7 1392.1 1450.7 1461.8 1487.2 1494.7 1520.0 1530.1 1553.0 1577.1 1604.8 1629.0 2227.0 2236.4
b 2.60 0.05 0.84 0.93 0.14 b 0.99 1.04 b b 0.21 0.59 3.25 0.95 0.40 1.07 1.65 b 1.36 2.13 4.17 1.97 5.00 4.82 38.33 2.04 1.58 10.89 18.32 6.66
0.13 2.51 0.26 0.71 0.73 0.14 0.62 0.80 1.28 0.22 0.42 0.51 0.51 1.56 0.90 0.38 1.23 2.17 3.90 1.54 1.58 1.97 3.77 a a 55.20 2.97 1.50 7.33 19.10 1.21
b 3.58 0.29 1.19 0.93 0.18 0.50 2.35 1.68 b b 0.36 0.68 3.38 1.02 0.49 1.44 2.49 5.81 2.47 3.18 4.72 3.77 9.41 6.91 48.42 4.11 2.59 13.68 20.46 3.68
0.20 3.08 0.33 1.05 0.93 0.25 0.94 0.94 1.57 0.35 0.62 0.74 0.72 1.91 1.04 0.61 1.61 3.71 5.68 2.33 2.47 2.17 5.47 a a 66.30 4.95 2.09 12.10 22.10 1.39
1.03 5.88 0.53 2.71 1.24 0.45 1.18 0.33 3.32 2.02 0.10 0.81 1.58 2.92 1.12 0.65 2.90 7.18 8.91 4.53 5.71 5.88 6.10 12.36 7.10 54.10 8.37 4.41 20.20 c c
0.36 3.29 0.39 1.70 1.11 0.50 1.58 0.91 1.75 0.61 0.98 1.15 1.07 2.14 1.03 0.98 2.13 6.56 8.72 3.35 4.04 1.79 8.27 a a 72.40 8.38 2.98 20.10 22.40 1.41
b 2.99 b 2.10 0.73 0.57 1.33 b 1.38 1.32 0.99 1.25 1.19 1.05 0.59 0.73 2.81 6.27 9.81 3.48 4.58 3.00 7.95 11.59 7.16 59.88 9.06 3.75 19.07 c c
0.42 2.65 0.37 1.91 1.03 0.59 1.79 0.69 1.54 0.66 1.02 1.19 1.09 1.88 0.81 0.89 2.06 6.56 8.81 2.77 4.08 1.15 8.21 a a 62.30 8.07 2.88 18.70 17.90 1.13
0.25 1.84 0.30 1.97 0.70 0.38 1.57 0.13 0.97 0.95 1.98 1.30 1.22 1.14 0.82 1.01 2.29 4.99 9.64 2.56 3.77 1.45 8.96 12.42 8.86 57.85 8.67 4.60 12.75 35.61 7.41
0.41 2.38 0.34 1.82 0.96 0.55 1.70 0.61 1.40 0.60 0.93 1.09 1.00 1.71 0.72 0.75 1.87 5.72 7.95 2.23 3.63 0.98 7.35 a a 55.60 6.95 2.56 15.90 15.20 0.96
0.21 1.15 0.52 1.99 0.55 0.34 1.66 0.26 1.04 1.21 0.91 0.96 0.80 0.96 0.50 1.23 2.32 4.59 7.15 1.31 2.73 0.77 6.32 11.40 6.83 51.38 4.66 1.80 7.44 c c
0.35 2.00 0.29 1.50 0.80 0.43 1.38 0.49 1.12 0.45 0.71 0.83 0.76 1.33 0.57 0.49 1.43 3.96 5.88 1.36 2.61 0.74 5.36 a a 41.80 4.71 1.85 10.50 10.30 0.65
0.30 0.87 0.43 1.39 0.62 b 1.13 0.36 0.31 0.27 0.77 0.89 0.83 0.34 0.46 0.86 1.15 2.48 3.63 0.70 1.91 0.64 4.60 13.52 3.89 39.32 4.04 2.95 4.61 c c
0.27 1.59 0.22 1.11 0.62 0.30 1.00 0.36 0.80 0.30 0.48 0.56 0.51 0.93 0.40 0.29 0.97 2.46 3.86 0.77 1.67 0.52 3.47 a a 28.00 2.87 1.19 6.26 6.32 0.40
a
Combination bands not calculated (see text). b Band intensities too weak to be determined. c Bands outside spectral window.
initial set of displacements,
Ik I1553
)
ωk2 ∆k2 ω15532∆15532
(10)
where Ik and I1553 are the integrated intensities of mode k and
the reference mode (see below), respectively, ωk and ω1553 are the wavenumbers of the kth mode and reference modes, respectively, and ∆k and ∆1553 are the displacements of the same modes. Only relative displacements are provided with this method. Additional information is required to determine the absolute displacements. For this study it was assumed that the
10732 J. Phys. Chem. B, Vol. 104, No. 46, 2000
Egolf et al. TABLE 2: Spectral Modeling Parameters for the Carbazole/ TCNE Complex CT1 transition
Figure 3. Lowest-energy unoccupied molecular orbital of TCNE (LUMO) and the two highest-energy occupied molecular orbitals of carbazole (HOMO and HOMO-1), calculated using the AM1 method.
displacements of TCNE in the carbazole/TCNE system are very similar to those in hexamethylbenzene/TCNE. This is a reasonable assumption because in both systems the charge-transfer transition results in essentially a full electron being placed in the TCNE LUMO. Thus, ∆ ) 1.0 was used for the 1553 cm-1 mode.21,22 [An alternative assumption is to scale the displacements so that the absorption spectrum calculated with the parameter set determined by the Raman intensities has the same overall width as the experimental absorption spectrum. This assumes that the entire width of the absorption is due to vibronic activity in the high-frequency quantized modes; i.e., the solvent contribution to the absorption line width is ignored. This is unlikely to be an adequate assumption for charge-transfer transitions, where the solvent contribution to the line width (or reorganization energy) is considerable.] The remaining displacements were scaled through the use of eq 10. Although eq 10 assumes enhancement via a single electronic state, an estimate of the displacements for the CT1 state may be found by using the Raman cross sections obtained with the reddest (723 nm) excitation. Interference effects are expected to make only a minor contribution on the very red edge of the absorption band. Similarly, estimates for the CT2 state were obtained using the cross sections obtained with 488 nm excitation. The estimates for CT2 were expected to be poorer than for CT1 because the blue edge of the CT1 electronic transition still contributes significant intensity at 488 nm, and there may also be contributions from preresonance enhancement with higher-lying electronic excited states. Some restrictions were placed on the signs and magnitudes of the initial displacements. Because both charge-transfer transitions terminate in the LUMO of TCNE, the displacements for the TCNE modes were constrained to have the same values for both chargetransfer excitations. Initially, all the displacements obtained with the use of eq 10 were specified with the same sign (positive); under the assumptions that generate eq 10, the Raman cross sections are insensitive to the sign of the displacement (see Discussion). All parameters were then refined together to obtain the best simultaneous fit to the resonance Raman cross sections at all seven excitation wavelengths and the optical absorption spectrum. The final best-fit parameter set is summarized in Table 2. Table 1 and Figure 4 compare the calculated and experimental
CT2 transition
frequency/cm-1
assignmenta
∆b
λ/cm-1
∆b
λ/cm-1
2236 2227 1629 1605 1577 1553 1495 1487 1462 1451 1392 1337 1323 1313 1286 1278 1238 1220 1204 1121 1108 1012 654 615 595 551 529 428 242
TCNE TCNE ν51 (a1) ν49 (b2) ν48 (a1) TCNE ν47 (b2) ν46 (a1) ν45 (b2) ν44 (a1) ν43 (b2) ν42 (a1) ν41 (b2) ν40 (a1) ν39 (a1) TCNE ν38 (b2) ν37 (b2) ν36 (a1) ν33 (b2) ν32 (a1) ν30 (a1) ν15 (a1) ν14 (b2) TCNE ν11 (b2) TCNE ν8 (a1) TCNE
0.100 0.400 0.460 0.200 0.316 0.975 0.355 0.133 0.250 0.170 0.390 0.320 0.201 0.110 0.130 0.200 0.153 0.161 0.150 0.125 0.205 0.147 0.350 0.200 0.300 0.430 0.200 0.650 0.450
11 178 172 32 79 738 94 13 46 21 106 69 27 8 11 26 15 16 14 9 23 11 40 25 27 51 11 90 25
0.100 0.400 -0.550 0.030 -0.300 0.975 0.000 0.300 -0.080 -0.410 0.000 -0.350 0.100 -0.210 0.190 0.200 0.025 0.000 0.000 -0.050 0.220 0.246 0.000 -0.151 0.300 0.060 0.200 1.000 0.450
11 178 246 0.7 71 738 0 67 5 122 0 82 7 29 23 26 0.4 0 0 1 27 31 0 7 27 1 11 214 25
a Assignments (mode numbers and symmetries in uncomplexed carbazole) for carbazole modes from ref 56. b All displacements arbitrarily taken to be positive in CT1. Remaining modeling parameters: electronic zero-zero energy 10 525 cm-1 in CT1, 13 350 cm-1 in CT2; solvent broadening parameters D ) 2850 cm-1, Λ ) 285 cm-1, λs ) 3121 cm-1, zero inhomogeneous width in both states; transition dipole length 0.81 Å in CT1, 0.435 Å in CT2; T ) 298 K.
Figure 4. Experimental absorption spectrum of the carbazole/TCNE complex in CH2Cl2 (solid line, same as for Figure 1), the spectrum calculated from the best-fit modeling parameters of Table 2 (dotted line), and its decomposition into contributions from the two CT states (long dashed and short-long dashed lines).
resonance Raman intensities and absorption spectrum, respectively. In general, the fits are within the experimental error. The most glaring discrepancy is the complete lack of calculated intensity for the line at 1530 cm-1 and the very low calculated intensity for the line at 1520 cm-1. Both of these lines were previously assigned as combination bands of nontotally symmetric TCNE fundamentals.21 Within the assumptions employed in this analysis, combination bands can have intensity only if the
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fundamentals that comprise the combination have intensity. Here, the relevant fundamentals are missing (1530 cm-1 line) or extremely weak (1520 cm-1 line), and the combination band intensities presumably arise from anharmonic coupling between the combinations (which have overall ag symmetry) and the strong nearby 1553 cm-1 TCNE CdC stretch. The present model does not include anharmonicities. The TCNE modes are not the main focus of this work, however, and the carbazole mode parameters may be reliably extracted without requiring a satisfactory fit to the TCNE combination bands. Furthermore, because the absorption depends on the vibrational motions in a collective fashion, the absorption spectrum can be accurately fit without accounting for the combination band intensities. Thus, no attempt was made to fit these TCNE combination bands. We were able to obtain good quality resonance Raman spectra only down to about 200 cm-1. The HMB/TCNE complex has a line at 165 cm-1 that contributes ca. 250 cm-1 to the total reorganization energy. This line has long been assigned as the intermolecular donor-acceptor stretch, but Rubtsov and Yoshihara have recently provided rather compelling evidence, based on the excited-state frequency of the analogous vibration in a number of related complexes, that this strong low-frequency line is primarily an out-of-plane intramolecular vibration of the TCNE acceptor.59 Presumably, carbazole/TCNE also has a resonance-enhanced ground-state mode near 165 cm-1, but since we could not measure it in the Raman spectra, its reorganization energy will simply appear as an unresolved contribution to the low-frequency “solvent” reorganization. Raman depolarization ratios for the lines between 400 and 1700 cm-1 were measured at one excitation wavelength on resonance primarily with CT1 (612 nm) and three wavelengths in the region where the two absorptions overlap significantly (570, 514, and 488 nm). The uncertainty in these measurements is large because of the lower signal-to-noise ratio of the data, particularly for the perpendicular polarization, and the substantial contributions to many lines from uncomplexed carbazole or TCNE. The 1553 cm-1 TCNE mode and the a1 carbazole modes ν51 at 1629 cm-1 and ν48 at 1577 cm-1 show F ) 0.33 ( 0.05 at all four excitation wavelengths. Somewhat higher values of F are measured for the a1 mode ν42 at 1337 cm-1 (0.38-0.46) and the b2 mode ν11 at 550 cm-1 (0.33-0.41), while values slightly below 1/3 are measured for the TCNE mode at 595 cm-1 (0.23-0.36), the b2 carbazole mode ν43 at 1392 cm-1 (0.250.28), and the a1 carbazole modes ν32 at 1108 cm-1 (0.220.33) and ν8 at 428 cm-1 (0.20-0.29). No clear dependence on excitation wavelength is observed, and it is difficult to determine whether these values are truly different from 1/3 to within experimental uncertainty. We have assumed that they are not, which, referring to the discussion following eq 9, is consistent with parallel transition dipole moments for CT1 and CT2. V. Discussion It is well-known that both the absorption spectrum and the resonance Raman cross sections are insensitive to the sign of the displacement of a resonant normal mode. Attempts to extract the nuclear geometry changes upon electronic excitation are hindered by the existence of 2n possible geometry changes that are consistent with the observed intensity pattern.11,12 For two electronic states with parallel transition dipole moments, the differential Raman cross section is given by eq 7,
(dσ/dΩ) ∝ |R1 + R2|2 ) |R1|2 + |R2|2 + R1R2* + R1*R2 (11)
Figure 5. Experimental resonance Raman intensities of the indicated carbazole fundamentals (points), calculated excitation profiles using the best-fit parameters of Table 2 (solid), and calculated profiles using the same parameters but with the sign of ∆2 reversed (dashed). The very different profiles in the region of overlap between the two chargetransfer transitions arise from the change in sign of the interference term when the relative sign of the two displacements is changed.
The dependence of the Raman cross sections on the signs of the displacements in the two states may be illustrated most readily by specializing eqs 3 and 4 to the case of a single mode in the harmonic surface assumption with equal ground- and excited-state frequencies. In this case, the Raman polarizabilities are given by
Rk(ωL,δ) )
|µk0|2 p
∫0∞ dt 〈f|i(k)(t)〉 exp[i(ωL - ωk - δ + ωi)t - g(t)] (12)
which, when f ) 1 and i ) 0, becomes12
|µk0|2 ∆k Rk(ωL,δ) ) p x2
∫0∞ dt (e-iωt - 1) exp[-sk(1 -
e-iωt)] exp[i(ωL - ωk -δ)t - g(t)] (13)
where sk ) ∆k2/2 and ω is the vibrational frequency of the mode. Clearly, when only one electronic state (state 1) contributes, the cross section is proportional to |R1|2, which involves only even powers of ∆1, and the sign of the displacement is irrelevant. But when both states contribute, the terms R1R2* and R1*R2 depend on the products ∆1∆2, and the signs of these interference terms depend on the relative signs of the two displacements. (However, the overall sign still does not matter; changing the sign of both ∆1 and ∆2 leaves the cross section unchanged.) The interference effects are expected to be greatest when the two states have comparable transition dipole moments and where there is a large region of overlap between the two transitions. In the present case there is strong overlap between the two states, but CT1 is considerably stronger than CT2, making the interference terms less important than they might otherwise be. The effect of these interferences on the Raman cross sections is nevertheless quite significant for at least some of the carbazole modes as illustrated in Figure 5. The best fit for the excitation profile of the carbazole 1629 cm-1 mode (ν51) is obtained with
10734 J. Phys. Chem. B, Vol. 104, No. 46, 2000
Figure 6. Normal modes corresponding to the indicated carbazole vibrational frequencies and the numbering of the carbon and nitrogen atoms in the structure.
displacements that are similar in magnitude but of opposite sign in CT1 and CT2. For comparison, the figure also shows the profile calculated with displacements of the same sign. The signs of the displacements correlate in a qualitative fashion with predictions based on the form of the normal mode and the nodal patterns of the carbazole donor orbitals. The calculated MOs for the HOMO and HOMO-1 of carbazole (the donor orbitals for CT1 and CT2, respectively) are shown in Figure 3. Note the contrasting nodal patterns. A vectorial description of the 1629 cm-1 mode of carbazole is depicted in the upper portion of Figure 6. The relative displacements of the 1629 cm-1 mode in resonance with the charge-transfer transitions may be inferred by examining the changes to the bond orders caused by removing an electron from each of the carbazole orbitals as summarized in Table 3. The 1629 cm-1 mode involves mainly lengthening of the C4-C11 and C5-C12 bonds, which are antibonding in the carbazole HOMO and bonding in the HOMO1; shortening of the C1-C10 and C8-C13 bonds, which are bonding in the HOMO and antibonding in the HOMO-1; and shortening of the C11-C12 bond, which is bonding in the HOMO and antibonding in the HOMO-1. Removing an electron from the HOMO should shorten the C4-C11 and C5-C12 bonds and lengthen the C1-C10, C8-C13, and C11-C12 bonds, corresponding to a negative ∆ for the 1629 cm-1 mode (with the arbitrary overall phase of the normal mode as defined in Figure 6). On the other hand, removing an electron from the HOMO-1 should lengthen the C4-C11 and C5-C12 bonds and shorten the C1C10, C8-C13, and C11-C12 bonds, corresponding to a positive ∆ for this mode. Thus, the displacement of the 1629 cm-1 mode is predicted to be of opposite sign for the ground f CT1 and ground f CT2 transitions, as required for the best fit to the Raman cross sections. Note that we define ∆ as the value of the normal coordinate at the equilibrium geometry of the excited state minus its value in the ground state. Similar predictions can be made for the remainder of the enhanced modes, although the conclusions are less clear-cut in many cases. Figure 5 illustrates the effect of the relative signs of the displacements on the profiles for the 1577 and 1337 cm-1 lines, and Figure 6 illustrates the normal modes corresponding to these vibrations. Table 3 demonstrates that both of these modes should also have displacements of opposite sign in the
Egolf et al. two CT states, since they involve mainly motions of bonds that have opposite bonding characteristics in the HOMO and HOMO-1. Indeed, the calculated profiles show that the use of nearly equal and opposite displacements in CT1 and CT2 gives much better fits to the experimental data than displacements of the same sign. In fact, since most of the carbazole bonds that have bonding character in the HOMO are antibonding in the HOMO-1 and vice versa, one might reasonably expect that most of the carbazole stretching vibrations should have displacements of opposite sign in the two CT states. The exceptions should be modes dominated by motions of C1-C2/C7-C8 or C2-C3/C6C7 (nonbonding in HOMO, bonding in HOMO-1) or N-C10/ N-C13 (antibonding in HOMO, nonbonding in HOMO-1), in which case one might expect the displacements to have different magnitudes but not necessarily different signs in the two CT states. The nitrogen stretching appears mainly in the line at 1286 cm-1, for which we indeed find displacements of the same sign in both CT states (see Figure 5 and Table 3). This mode also has a large component of C1-C2/C7-C8 stretching. A large contribution from C1-C2/C7-C8 stretching is also found in the 1487 cm-1 line (along with significant C11-C12 stretching), and this mode has displacements of the same sign, though considerably different magnitudes, in the two states. The lower-frequency modes involve mainly bending motions, and their displacements are more difficult to estimate on the basis of simple electron density arguments. Figure 5 demonstrates that when the two resonant electronic states have displacements of opposite sign along a particular normal mode, the resonance Raman intensity in that mode’s fundamental is attenuated when excited at frequencies where the two states overlap strongly, above about 18 000 cm-1 according to Figures 1 and 4. A simple physical interpretation is that the two surfaces try to distort the molecule in opposite directions, so their effects partially cancel and the vibration ends up only weakly excited. More correctly, as indicated in Figure 7, a vibrational wave packet is launched on each of the two potential surfaces, where they propagate independently in opposite directions. The wave packet moving in one direction (|i1(t)〉) develops its overlap with the positive-going lobe of the final vibrational wave function |f〉 while the wave packet moving on the other surface (|i2(t)〉) overlaps the negative-going lobe, so the two time-dependent overlaps sum to nearly zero, the degree of cancellation being influenced by the energy mismatch between the actual energy of the initial wave packet, p(ωi + ωL), and the energy needed to reach potential surface k, pωk (refer to eq 4). The existence of two reasonably strong CT bands corresponding to the HOMO-1 f LUMO (CT2) and HOMO f LUMO (CT1) transitions, respectively, is supportive of an asymmetric geometry for the complex (lower than Cs). There exist two possible geometries having Cs symmetry, one in which the CdC bond of TCNE is along carbazole’s short axis and the other in which it lies parallel to the long axis. The latter would be expected to provide better π-electron interaction and be energetically favored. Consideration of the symmetries of the three orbitals shown in Figure 3, however, leads to the conclusion that for either Cs geometry only one of the two CT transitions is allowed for a transition dipole along the donoracceptor intermolecular axis (z): CT1 for the short-axis conformation and CT2 for the long-axis conformation. Thus, either there are two completely different complexes present, each giving rise to a different CT transition, or else the true structure does not have Cs symmetry, in which case both transitions are
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TABLE 3: Rationalization of Relative Sign of ∆ in CT1 and CT2 States for Four Normal Modes of Carbazole
mode and frequency (cm-1)
bonding (b), antibonding (a), or nonbonding (n) in HOMO HOMO-1
expected bond length change CT1 CT2
bond contribution to ∆ b CT1 CT2
bond
phase in normal modea
relative sign of ∆ c
ν51, 1629
C4-C11 C5-C12 C1-C10 C8-C13 C11-C12
+ + -
a a b b b
b b a a a
+ + +
+ + -
-
+ + + + +
opposite
ν48, 1577
C2-C3 C6-C7 C4-C11 C5-C12 C1-C10 C8-C13
+ + -
n n a a b b
b b b b a a
0 0 + +
+ + + + -
0 0 + + -
+ + + +
opposite
ν42, 1337
C2-C3 C6-C7 C1-C10 C8-C13 C4-C11 C5-C12 C11-C12
+
n n b b a a b
b b a a b b a
0 0 + + +
+ + + + -
0 0 + + +
+ + -
opposite
ν39, 1286
C10-N9 C13-N9 C1-C2 C7-C8
+ +
a a n n
n n b b
0 0
0 0 + +
+ + 0 0
0 0 + +
same
+ indicates an increase in the length of the indicated bond and - a decrease in bond length in the normal mode. Sign of product of phase of bond in normal mode and expected bond length change in electronic transition. c Relative sign of displacements in CT1 and CT2 from best-fit parameters of Table 2. a
Figure 7. Raman scattering in the fundamental of a normal mode along which two nearby resonant electronic states have displacements ∆ of similar magnitude and opposite sign. The wave packets launched on the two potential surfaces, |i1(t)〉 and |i2(t)〉, propagate in opposite directions, developing overlaps of opposite sign with the final state, |f〉.
allowed. Landman et al. concluded that the latter case holds for chloranil complexes with carbazoles, noting that BenesiHildebrand plots (admittedly not the most precise diagnostic) and other data give no evidence for more than one distinct complex.46 In the former situation there would be no Raman interferences between CT1 and CT2, since they would arise from different molecules. We have not attempted to determine whether the data could be modeled successfully under this
b
assumption because there are too many uncertainties (e.g., the need to know both the concentrations of the two complexes and their molar absorptivities.) The results of our direct modeling (parameters of Table 2) and our purely empirical curve fitting (Figure 1), which both indicate that CT1 is the stronger transition, also argue against two complexes of Cs symmetry, since it seems unlikely that the short-axis conformation would be more highly populated and/or have a much stronger CT absorption (although note that both Okamoto et al. and Klo¨pffer reached the opposite conclusion about the relative strengths of CT1 and CT2).54,55 We conclude that the most probable geometry for the complexes of carbazoles with TCNE as well as with chloranil is significantly asymmetric, with the TCNE translated to lie above one or the other of the phenyl rings rather than centered over the pyrrole ring. Further evidence for a complex geometry of low symmetry comes from the observation that some of the resonanceenhanced bands require assignments to nontotally symmetric (b2) carbazole vibrations as noted in Table 2. If either CT1 or CT2 were actually electronically forbidden, as in a complex of Cs symmetry, the vibronically induced component of the forbidden transition could induce resonance Raman activity in overtones and combination bands of these nontotally symmetric modes, but we do not observe such bands. For nontotally symmetric fundamentals to appear in the resonance Raman spectrum, the resonant electronic transition must have both electronically allowed and vibronically induced components (the “B-term” source of resonance Raman intensity).60,61 We have modeled the spectra under the assumption that the complex has C1 symmetry, so both CT transitions are electronically allowed, all vibrations of the complex are totally symmetric, and all observed Raman intensity comes from the purely FranckCondon mechanism (the “A-term”). There may, of course, be additional vibronically induced components to the CT transitions (i.e., both µ0 and ∂µ/∂q are nonzero), which would likely be
10736 J. Phys. Chem. B, Vol. 104, No. 46, 2000 largest for the modes that have b2 symmetry in uncomplexed carbazole, resulting in non-Condon contributions to the resonance Raman intensity. As mentioned in the theoretical methods section, we did not include non-Condon terms because they did not appear necessary to fit the experimental intensities, but it remains possible that a different choice of parameters with nonzero ∂µ/∂q would fit the data equally well. Comparing the reorganization energies obtained for the carbazole/TCNE system with those obtained for other donoracceptor complexes illustrates some interesting points. An earlier study of HMB/TCNE indicated the HMB contribution to the reorganization energy as 1040 cm-1.22 The carbazole contribution to the reorganization energy is a remarkably similar 933 cm-1. Carbazole/TCNE exhibits a greater number of enhanced modes in both CT transitions than HMB/TCNE, but each mode makes a smaller contribution to the reorganization energy. This is consistent with the hole being distributed over a larger area in the carbazole molecular orbitals (approximately 3 times the area compared to HMB). Consequently, each carbazole bond suffers a smaller perturbation upon electron transfer. The total internal reorganization energy of about 2000 cm-1 is in good qualitative agreement with that obtained from previous resonance Raman intensity analyses22,24 as well as simplified onemode fits to absorption and/or fluorescence spectra of a variety of related CT complexes.9,62 The dimensionless displacements obtained here for the TCNE modes are in good agreement with those obtained in other complexes of TCNE and related covalent charge-transfer molecules.63 Indeed, this was the criterion for scaling the initial set of displacements obtained with eq 10. These parameters required very little adjustment throughout the modeling procedure. This is again consistent with a simple molecular orbital interpretation. The electron is being transferred into the LUMO of TCNE in both systems; hence, the nuclear reorganization energy for the LUMO modes should be similar in both cases. Extraction of an actual excited-state geometry requires an accurate normal mode analysis and a complete determination of the sign combinations. However, an illustration of the magnitudes of bond length changes is appropriate. If the 1553 cm-1 mode of TCNE is taken to be an isolated CdC stretch, the dimensionless displacement for this mode of ∆ ) 0.975 corresponds to a bond length change of 0.06 Å. For comparison, the bond length difference between a carbon-carbon single bond and a double bond is ca. 0.20 Å. Perhaps the most remarkable result of the analysis is the very similar carbazole reorganization energy in both charge-transfer transitions. With the TCNE displacements constrained to be equal for each transition, the total intramolecular reorganization energies are 1985 and 1948 cm-1 for the low- and high-energy transitions, respectively. There are more modes resonantly enhanced by the ground f CT1 transition, but the contribution per mode is smaller than in the ground f CT2 transition. It is reasonable to suppose that the solvent contribution to the reorganization energy should be similar for both of the CT states. The HOMO and HOMO-1 electron densities are similar (although with differing nodal patterns), so the solvent should respond in a comparable fashion to the ground f CT1 and ground f CT2 transitions. As a first approximation, the coupling strength between the solvation coordinate and the electronic transition (D in eqs 5 and 6) was constrained to the same value for both electronic transitions. An adequate fit was obtained with this constraint, which was maintained in generating the calculated resonance Raman intensities in Table 2 and the absorption spectrum in Figure 4. A somewhat improved fit to
Egolf et al. the absorption spectrum could be obtained by decreasing the solvent coupling for the weaker transition and adjusting its zero-zero energy, but it is not clear whether the difference is significant. When the solvent contribution is allowed to vary, the best-fit solvent reorganization energies are 3121 cm-1 for the lower energy, more intense transition and 2033 cm-1 for the higher energy transition. Both are reasonably close to the values of 2000-3000 cm-1 (depending on assumptions about cavity radius) calculated from simple dielectric continuum theory64 and 2820 cm-1 obtained from a rather crude resonance Raman intensity analysis of the methoxybenzyltrimethylsilane/ TCNE complex in CH2Cl2,24 although a much larger value of 6200 cm-1 was obtained by McHale’s group for HMB/TCNE in the same solvent.64 An important parameter in determining the rate of charge hopping in actual applications of substituted carbazoles and carbazole polymers as hole-transport agents is the degree of energetic disorder, the extent to which the energy of a hole localized on different donors varies from one site to another.65-67 In the absence of disorder the reaction D + D•+ f D•+ + D is energetically neutral, while in its presence the reaction is uphill for some pairs of chromophores and downhill for others. Energetic disorder is analogous to the spectroscopic parameter of electronic inhomogeneous broadening but with some important differences. The spectroscopic experiment gives the variation in the energy of D + A f D•+ + A•- among different DA complexes, which depends not only on the environment of the different donors but also on any variations in D-A distance or relative orientation. Also, the purely environmental contribution for an ion pair (essentially a large dipole) solvated by a liquid at low concentration, as in the spectroscopic experiments, may be very different from that for a positive charge solvated at high concentration in a glassy or polycrystalline solid.68 The inhomogeneous line width is one of the parameters that must in principle be adjusted in modeling the absorption line shape and Raman cross sections, but when the absorption spectrum shows no vibronic structure, the fits tend to be relatively insensitive to the amount of inhomogeneous broadening used. (In the resonance Raman context, “inhomogeneous” broadening refers to those contributions to the solvent-induced spectral breadth that interconvert slowly relative to the inverse ground-state vibrational line width.69) Our best-fit value of zero for the inhomogeneous line width therefore should not be taken too literally or be used to draw conclusions about the extent of energetic disorder among carbazole donors present at high concentration in the solid state. It merely indicates that static inhomogeneous broadening does not make the major contribution to the electronic absorption line width of the carbazole/ TCNE complex in fluid solution. The Marcus theory of electron transfer, in its simplest form, predicts that the electron-transfer rate should be a maximum when the total reorganization energy equals the reaction exergonicity because this is where the reactant and product free energy surfaces cross with zero activation barrier. For an energetically neutral reaction such as charge hopping among nominally identical charge carriers, one should therefore attempt to minimize the total reorganization energy in order to maximize the rate, other factors being equal. The environmental reorganization is best minimized by choosing it to be relatively nonpolar and by choosing dopants that delocalize the charge over many bonds. Choosing a large, delocalized charge carrier is also expected to minimize the internal reorganization energy, although that is not experimentally clear from our work; we find only a slightly smaller reorganization energy for carbazole,
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