Resonance stimulated Raman signal and line shape are evaluated analytically under common electronic/vibrational dephasing and exponential Raman/probe pulse, exp(−|t|/τ). Generally, the signal from a particular state includes contributions from higher and lower electronic states. Thus, with S0 → S1 actinic excitation, the Raman signal consists of 15 Feynman diagrams entering with different signs. The negative sign indicates vibrational coherences in S1 or higher Sn, whereas the positive sign reveals coherences in S0 or Sn via S1 → Sn → Sm (n < m) coupling. The signal complexity is in contrast to spontaneous Raman with its single diagram only. The results are applied to femtosecond stimulated Raman spectra of trans–trans, cis–trans (ct), and cis–cis (cc) 1,4-diphenyl-1,3-butadiene, the ct and cc being reported for the first time. Upon actinic excitation, the Stokes spectra show negative bands from S1 or Sn. When approaching higher resonances Sn → Sm, some Raman bands switch their sign from negative to positive, thus, indicating new coherences in Sn. The results are discussed, and the measured Raman spectra are compared to the computed quantum-chemical spectra.
Femtosecond stimulated Raman (FSR) spectroscopy is a powerful tool for probing photoinduced molecular rearrangements. Proposed by Yoshizawa in 1994,1 the technique has been largely developed by the group of Mathies2–9 and his collaborators10–21 as well as by other researchers.22–36 The technique is unique, as it combines high temporal (femtosecond) and high spectral (several cm−1) resolutions. This is in contrast to conventional spontaneous Raman37 where the temporal resolution is limited by the picosecond Raman pulse. The trade-off is the complexity of the FSR signal that generally contains both positive and negative contributions,30 a result not recognized in previous theoretical reports.4,7,10–13,23,34
In early FSR experiments,6,7,16,22 researchers noticed a variety of Raman line shapes that may change in the vicinity of electronic resonances from symmetric to dispersive shape or even switch the sign of the signal. The features were explained by inverse Raman and hot luminescence contributions,7,10,11,17 by resonance-dependent Franck–Condon activity,22 or by the interplay between pulse parameters and intramolecular dynamics.23,24 The theoretical analysis assumed vibrational coherences in actinically excited state S1 and in lower S0 or higher state Sn.10–13,16,17 However, it was not considered that these coherences may contribute to different signs. In particular, without actinic excitation, resonance stimulated Raman spectra contain negative contributions from S0 and positive and negative contributions from S1.30,31
In this paper, we focus on the above issues. We provide a detailed procedure for calculating the excited-state FSR signal in the resonance case. For simplicity, we consider here only the Stokes spectra, as the generalization to the anti-Stokes case is straightforward and detailed in the supplementary material. We also pay attention to the similarity between transient absorption (TA) and stimulated Raman in connection with their spontaneous counterparts, fluorescence, and spontaneous Raman.
The theoretical analysis is illustrated by experimental FSR spectra of trans–trans (tt), cis–trans (ct), and cis–cis (cc) 1,4-diphenyl-1,3-butadiene (DPB). The first Raman spectra of this compound were calculated38 and measured39–46 in 1980–1990, but the excited-state spectra were recorded only in a high frequency range >600 cm−1 and only for the tt-isomer.
II. TRANSIENT ABSORPTION VS STIMULATED RAMAN
It is necessary to stress a deep similarity between TA and FSR spectroscopy.28–31 They both deliver a stimulated signal of differential absorption ΔA, which is usually measured in mOD units, with 1 mOD = 0.001 ln 10. The difference between the techniques is only that the TA results from the pump and probe, while the FSR comes from an actinic pump and simultaneously acting Raman and probe. It is, therefore, advantageous to measure the FSR signal in the mOD units as well.28–31 This is different from a common approach for measuring the Raman signal, the conversion to conventional Raman gain/loss units being given by
Importantly, the sign of ΔA may be positive or negative. We follow the convention from TA spectroscopy, where ΔA is positive for absorption and negative for emission and bleach. Since the non-resonant Stokes FSR signal appears in emission, it should be considered negative, while the corresponding anti-Stokes absorptive signal is positive. Recall that this definition is opposite to that of common Raman literature, where the Raman gain is taken to be positive.
It is also instructive to compare TA and FSR with regard to their spontaneous counterparts. For TA, this is time-resolved fluorescence. The latter appears in TA as stimulated emission (SE), but there are also two other contributions, bleach and excited-state absorption (ESA). Similarly, for FSR, there is a term corresponding to spontaneous Raman from S1, but there are many others in the resonance case. These additional contributions result from lower S0 and higher Sn electronic states since the Raman and probe fields are capable of creating vibrational coherences there.
III. CALCULATING THE FSR SIGNAL
Nonlinear spectroscopic signals can be calculated with the response function formalism developed by Mukamel and co-workers.47–50 The basic concepts and key definitions from Refs. 47–54 are adapted here.
The femtosecond stimulated Raman signal is treated within the fifth-order perturbation formalism. A useful approximation assumes well separated in time pump and probe pulses with delay t between them. The actinic pump prepares the system with population p in S1. Delayed Raman ER and probe E fields induce the optical changes in the third-order approximation. The sequential contribution (ER, ER, E) generates a Raman background that should be subtracted. We focus on the coherent contribution arising from Raman and probe fields and resulting in a background-free Raman signal. The quantity of interest is the transient signal induced by the actinic pulse and probed upon delay t by the Raman and probe pulses as functions of probe frequency ω.
The FSR signal in the probe direction k + kR − kR is then given by47,48,50,52
where population p(t), created by the actinic pump, evolves in time, while Raman polarization is given by triple integration and summation over all possible paths,
where is the nonlinear response, FT is the Fourier transform, and the nonlinear field depends on the ordering of Raman ER and probe E,
with [s] = α or β. Assuming a common electronic/vibrational dephasing rate Γ, the response function is given by
where , , and Γab is the dephasing rate between states and . The path is specified by subscripts , and is the product of the transition dipole moments along the path. The Fourier transform gives
The triple integration in (3) can be done analytically with exponential pulses and . Upon integration with τ ≪ τR and omitting p(t) and the nonlinearity index, the FSR signal is given by
The explicit expressions for ΔA(ω) are summarized in the Appendix.
Each term in ΔA(ω) can be represented by a double-sided Feynman diagram or by an energy-level diagram as shown in Fig. 1. Here, each electronic state S0, S1, Sn, Sm (m > n) is associated with a pair of vibrational levels: S0 with 1 and 2, S1 with 3 and 4, Sn with 5 and 6, and Sm with 7. Raman ER and probe E fields are marked by the blue and orange arrows, respectively. In the energy level diagrams, -ket/bra- is represented by a solid/dashed arrow, while in the Feynman diagrams, -ket is on the left and bra- is on the right.
Actinic excitation S0 → S1 transfers population to level 3, which serves as the initial state for the stimulated Raman from S1. Three successive interactions (arrows) result in the signal ΔA in Eq. (7a) given by a sum over products . The first interaction leads to electronic coherence, the second gkl gives vibrational coherence, and the third fmn again leads to electronic coherence. enters as when the Raman field ER is the first or as when the probe E is the first. The factor is the product of the transition dipole moments along the path [s]. For example, for diagram 1, , where μmn include the vibrational overlap integral in the Condon approximation. Below, when studying the signal dependence on the Raman detuning from vibronic resonance, one can take .
Note that the diagrams contribute to the signal with a different sign σ = ±1, unlike in spontaneous Raman. In the spontaneous Raman, the probe E is zero and the only contributing diagram is 1 (top left), where the Raman field is absorbed to emit the Stokes probe field. To be consistent with TA spectroscopy, such a diagram must enter with a negative sign, similar to bleach or stimulated emission. This finally determines the sign σ of a diagram,47
As shown, the sign is negative for odd Nket (the number of solid arrows) and positive for even Nket.
Consider upper diagrams 1, 2a, 2b, 3a, and 3b in Fig. 1 reflecting S1 → Sn → Sm transitions from S1 to higher states Sn and Sm (m > n). Only diagram 1 prepares coherences in S1, while the four others deal with coherences in Sn. When the Raman field ER is fully off-resonant, these four become negligible, and the only 1 delivers the S1 Raman spectrum of interest. Note that the non-resonant FSR spectrum is identical to that from spontaneous Raman, as both are given by the same diagram 1.
Furthermore, when ER becomes resonant with S1–Sn but still non-resonant with Sn–Sm, diagrams 3a and 3b start to contribute, and the resulting Raman spectrum acquires negative bands from Sn, in addition to those from S1. Note that these additional Sn bands can be distinguished by comparison with the non-resonant FSR spectrum (that delivers the S1 contributions only).
When ER is in resonance with both S1–Sn and Sn–Sm, diagrams 2a and 2b contribute, and some Raman bands may switch their sign from negative to positive. This happens if μnm > μ1n, where μ1n and μnm are the corresponding transition dipole moments. In that case, the positive bands can be clearly ascribed to Sn.
Next, the actinic pump produces not only a population in S1 but also a bleach (lack of population) in S0. These bleach terms can be described by the same diagrams 1, 2a, 2b, 3a, and 3b, but starting from S0 and with the opposite sign as the bleach population is negative. In the non-resonant case, the bleach term reproduces the S0 spectra, taken with the opposite sign and scaled by the actinic population.
In the absence of actinic excitation, one starts directly from S0. Again, diagrams 1, 2a, 2b, 3a, and 3b apply with minor modifications: S1, Sn, and Sm are substituted with S0, S1, and Sn, and levels 3, 4, 5, 6, and 7 change accordingly to 1, 2, 3, 4, and 5 (see the supplementary material for details).
Now consider diagrams 4, 5, 7, 8a, and 8b in Fig. 1 reflecting the transitions to a lower electronic state S0. Here, diagram 6 is not shown to match the numbering of Ref. 30. Far from S0–S1 and S1–Sn resonances, only diagrams 4, 5, and 7 survive. The 5 and 7 are close in magnitude but opposite in sign and, therefore, cancel each other. The Raman signal results from 4 only, which is positive and associated with S1.
To summarize the non-resonant case, the FSR signal comes from diagrams 1 and 4. Both deal with coherency in S1 but contribute with opposite signs. The final sign of ΔA depends on the transition dipoles μ01 and μ1n and on how far ER is detuned from S0–S1 and S1–Sn.
When ER is in resonance with S0–S1, diagrams 8a and 8b contribute additional positive bands associated with S0, as was experimentally observed by us in trans-azobenzene.30
Now we discuss in more detail how different diagrams behave near electronic resonances S1–Sn, Sn–Sm, and S0–S1. We simulate the Raman signal with Eq. (7a) assuming ν12 = 1100 cm−1, ν34 = ν56 = 900 cm−1, Γv = Γ12 = Γ34 = Γ56 = 0.2 ps−1, and Γe = Γ13 = Γ35 = Γ57 = 200 ps−1. The ER detuning from the resonances was taken in units of νij, from −1 to +1, being within the electronic bandwidth (∼1000 cm−1). The resulting line shapes are displayed under each diagram, always for five fixed detuning −νij, −νij/2, 0, νij/2, and νij. As shown, when the Raman field acts first (diagrams 1, 2b, 3b, 7, and 8b), no substantial distortion of the line shape occurs. On the contrary, when the probe is the first, the distortion is large and results in a dispersive line shape. This happens when the Raman detuning is approximately equal to the vibrational frequency. Another reason for the dispersive line shape is the simultaneous contributions with opposite signs, from higher and lower electronic states.
Previously, the dispersive line shapes were experimentally observed in stimulated Raman measurements.6,7,16,22 Contributions from S0 were considered by Lee and co-workers,10–13 while the Sn contributions and the dependence on resonance conditions were discussed by Oscar et al.,17 although without specifying the sign of the contributing terms.
The above consideration implies that only level 3 is initially populated by an actinic pump. However, for low-frequency modes, level 4 can be populated as well, which doubles the number of contributing diagrams (see Figs. S4, S5, S8, and S9 for hot Stokes and anti-Stokes diagrams). For example, diagram 4 of Fig. 1 would have a counterpart given by diagram 4 of Fig. S8. It follows that the latter contributes with the same positive sign, leaving the total signal (the sum from levels 3 and 4) to be the same as the full population would be at level 3.
IV. QUANTUM CHEMICAL COMPUTATIONS
The ground and excited state Raman spectra are calculated at the density functional theory (DFT) and the linear response time-dependent DFT (TDDFT), respectively, with the use of the Firefly software.55 The PBE0 hybrid exchange–correlation functional and the Def2-TZVPP basis set were used. The spectra are calculated in the standard non-resonant approximation. To our experience with related stilbene compounds, the inclusion of the resonant effects via the pre-resonance methodology rarely delivers a decisive improvement; furthermore, such an approach is hardly feasible for the excited states. The calculated harmonic frequencies were scaled by a factor of 0.962 to achieve better correspondence with the experiment, as is common practice.60
Our TA and FSR setups are described in great detail elsewhere.28–31,56–59
For TA,56–59 30 fs pulses at 800 nm, 500 Hz are converted to 400 nm and then pass through a 1 mm CaF2 to generate a broadband continuum probe in the range 275–700 nm. The probe is split into signal and reference and registered with two polychromators. Fifty-fs pump pulses are derived from an optical parametric amplifier in the range 270–900 nm. The sample flows through a measurement cell of 0.3 mm thickness with fused silica windows of 0.2 mm. The pump and probe are focused to a ∼0.15 mm spot in the cell and intersect at ∼15°. The pump is chopped to block every second pulse to record the baseline. The TA spectra ΔA(λ, t) are measured with parallel, perpendicular, and magic angle polarization. Multiple (10–40) back and forth pump–probe scans are applied to improve signal-to-noise. The spectra at negative delays are subtracted to eliminate pump scattering and sample fluorescence.
For FSR,28–31 the general scheme is basically similar to TA, with a ∼2-ps Raman pump added in boxcar geometry. Thirty-fs actinic pump and probe are derived from noncollinear optical parametric amplifiers (NOPAs).28 The probe pulse of 1000 cm−1 width is used directly without continuum generation. The Raman pump is chopped, and thus, the actinic pump is included in the baseline. The Raman spectra at negative delays are subtracted to eliminate the solvent and ground-state contributions.28
VI. S0 RAMAN SPECTRA
Ground-state Stokes Raman spectra are shown in Fig. 2. These are recorded without actinic excitation, with the Raman pump at 645 or 621 nm, far from the stationary absorption band at ∼330 nm. In this strongly non-resonant case, only diagram 1 contributes to the stimulated Raman, and as discussed above, the FSR spectra reproduce those from the stationary spontaneous Raman.
The low-frequency region <800 cm−1 is dominated by bending and deformation modes with low Raman activity, very similar to stilbene. The activity in the low-frequency range strongly increases from tt- to ct- and further to cc-isomer, reflecting the lower rigidity of the cis-conjugation. The two most intense lines are located around 1600 cm−1 and correspond to C–C– and C=C– stretching modes.
VII. EXCITED-STATE SPECTRA
Stokes Raman spectra from S1, under actinic excitation of tt, ct, and cc, are presented in Fig. 3 (right). Here, the computed spectra are also shown by vertical bars. The experimental spectra reveal mainly negative bands from vibrational coherences in S1 or Sn. A band at 580 cm−1 is positive, which may indicate a resonance between ER and Sn → Sm (diagrams 2a and 2b). We shall discuss this point in Sec. VIII. Here, we briefly discuss the TA spectra at the left of Fig. 3.59
For tt (top), the TA spectra reveal three well-separated regions: negative bleach about 300 nm, negative stimulated emission (SE) at 400 nm, and positive excited-state absorption (ESA) at 650 nm associated with an S1 → Sn transition. The TA decay with ∼500 ps is mainly due to photoisomerization tt → ct.59 For ct (middle), the general evolution looks similar. The ESA is nearly the same, while the SE band is slightly red-shifted, and the ct decay occurs with ∼20 ps, reflecting faster photoisomerization ct → tt. Finally for cc (bottom), the ESA band shifts a bit more to the red, and the SE is too weak to be observed, while instead the P-band (of perpendicular molecular configuration)59 at 400 nm appears at early time t = 0.3 ps. The cc → ct photoisomerization proceeds even faster, with 6 ps.59
The optimized geometry of the DPB molecule and the electronic structure of the S0 and S1 states have been reported in Ref. 59. The solution-phase S1 is due to a single-electron excitation from the HOMO to the LUMO and is, therefore, correctly describable by TDDFT. Contrariwise, the gas-phase S1 includes a considerable contribution of the double excitation, and in a non-polar solution, the two states will remain rather close to each other. Therefore, one cannot confidently rule out some involvement of the doubly excited state via, e.g., its partial thermal population. A quantitatively reliable description of those subtle effects of solvent-dependent state reordering in DPB, perhaps, needs more accurate experimental data to calibrate the computational results against. One more complication may arise from the very short lifetime of the excited cc-isomer. Its Raman signal may correspond to some ensemble of configurations along its photoisomerization pathway, which is not entirely accessible by TDDFT due to diminishing the S0–S1 gap upon the cc → ct twisting.
As one can see that, while the ground-state calculations reproduce the major spectral bands (see Fig. 2) rather well, the excited state data in Fig. 3 show less satisfactory agreement. Possibly, the discrepancies in the cc and ct isomers are rather quantitative than qualitative, and the major calculated bands are just unevenly displaced from the experimental ones. Perhaps, those discrepancies could be refined by a costly higher-level calculation of vibrational frequencies. Surprisingly, the tt-isomer whose excited-state lifetime is, conveniently, the longest and seems to pose a more important problem: it lacks highly intense bands around 1600 cm−1. Those bands are due to the symmetric stretching vibrations in the conjugated system, and our calculations predict them, rather expectedly, to dominate in all isomers both in the ground and in the excited state. One can hypothesize that here we indeed observe some resonant effects that result from non-trivial couplings with the respective Sn manifold.
VIII. EFFECT OF Sn → Sm RESONANCE
The Raman pump ER at 690–700 nm hits the ESA band in resonance with the S1–Sn transition and is far from the S1 → S0 resonance. Hence, the lower diagrams 4, 5, 7, 8a, and 8b are strongly suppressed and can be neglected. Among the rest, diagram 1 brings a contribution from vibrational coherences in S1, while 2a, 2b, 3a, and 3b deliver the signals from coherences in Sn. The positive signal may appear from 2a and 2b when ER is in resonance with Sn → Sm.
The position of the Sn → Sm band can be established in a double-pump experiment, S0 → S1 → Sn, when the S0 → S1 excitation is followed by a delayed S1 → Sn excitation. Here, the first pulse populates S1, while the second, after a 1 ps delay, transfers the population further to Sn. The first pump pulse is not chopped and is included in the baseline, the pump–probe delay t being countered from the second pulse. The resulting TA spectra are presented in Fig. 4.
The Sn → Sm absorption band can be restored by subtracting the early bleach at t = 0.08 ps from the top spectra. The bleach shape is given by the ESA band in Fig. 3 (top left), and its amplitude is adjusted to get approximately zero signal at t = 0.08 ps at the band peak (a nearly zero signal is expected upon subtraction as the bleach dominates in that range). The resulting spectra are displayed at the bottom of Fig. 4. It is shown that the Sn–Sm band is peaked at 560 nm, and the Raman pump at 700 nm hits its absorption edge only. One hardly expects a large positive Raman signal at such conditions; however, it should increase when the Raman pump is tuned closer to the Sn–Sm peak. We, therefore, perform FSR measurements with tunable Raman pump ER in the range λR = 645–700 nm.
The top frame in Fig. 5 shows the FSR spectra of tt in n-hexane recorded under these conditions. One sees that a band at 225 cm−1 switches its sign from negative to positive when ER shifts to the blue, closer to the Sn → Sm resonance. Other bands at 142, 471, and 581 cm−1 behave similarly although without switching their sign. Such behavior qualitatively agrees with an increasing contribution from diagrams 2a and 2b, and therefore, the bands can be ascribed to Sn. As already mentioned, the Sn bands should disappear when tuning the Raman pump far to the red. Unfortunately, the non-resonant case cannot be realized in our setup.
The bottom frame of Fig. 5 displays our simulations, using Eq. (7a) or Eqs. (A1)–(A3) of the Appendix, with the Sn lifetime τ = 0.1 ps, and Γ35 = Γ57 = 200 ps−1 and Γ56 = 0.2 ps−1. Note that τ = 0.1 ps corresponds to the spectral width (FWHM) of 56 cm−1. The simulated spectra generally well reproduce the experimental features but differ by long red tails of the bands, not seen in the experiment. These tails originate mainly from broadening due to shortly lived Sn and partly from the exponential pulses. In the experimental spectra, the tails may be hidden under a complicated Raman background (see Fig. S10, top), which has to be subtracted to get background-free Raman spectra.
We have analytically calculated the FSR signal and its line shape upon S0 → S1 actinic excitation at different detuning of the Raman pump from higher (S1 → Sn → Sm) and lower (S1 → S0) electronic resonances. Generally, the signal includes 15 Feynman diagrams. Unlike spontaneous Raman, the FSR signal contains contributions of different signs, resulting in a net positive or negative signal or producing a dispersive line shape that strongly depends on the resonance conditions.
In the case of the S1 → Sn resonance only, the Stokes Raman spectra are negative and originate both from S1 and Sn vibrational coherencies. The Sn coherences will disappear in the non-resonance case, and, thus, they can be distinguished from the coherences in S1. When the Sn → Sm resonance starts to contribute, some Raman bands may switch their sign from negative to positive.
If S1 → S0 is also in resonance, the spectra reveal positive bands from S0. The spectra, recorded with different Raman detuning from the electronic resonances, agree qualitatively with analytically simulated spectra.
We pay attention to the deep connection between transient absorption and stimulated Raman spectroscopy. When higher electronic resonances S1 → Sn → Sm contribute to the resonance Raman signal, the relevant information on the excited states can be obtained from the TA measurements.
In addition, the Raman spectra of tt-, ct-, and cc-isomers of diphenylbutadiene are measured for the first time, and the S0 spectra are in reasonable agreement with theoretically computed spectra.
The supplementary material includes a detailed theoretical description of the stimulated Raman signal; the Stokes and anti-Stokes diagrams from ground S0 and excited S1 and Sn electronic states, for cold and hot vibrational states; and the computed Raman frequencies for three isomers of diphenylbutadiene.
O.A.K. acknowledges the National Science Foundation for support under Grant No. CHE 1900226. The computations were carried out using the facilities of HPC computing resources at Lomonosov Moscow State University.
Conflict of Interest
The authors have no conflicts to disclose.
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
APPENDIX: CALCULATING STIMULATED RAMAN SIGNAL
A compact form of the Raman signal is obtained by taking the imaginary part in Eqs. (7a) and (7b). The Raman signal can be presented as the product (A3) of three Lorentzians multiplied by a second-order polynomial . Lorentzians , , and are determined by the electronic resonance conditions; the vibrational coherence gives rise to the Lorentzian , where ων is the vibrational frequency. Depending on the roots of the polynomial, the Raman signal may change sign. If the polynomial passes through zero at the resonance frequency ων, the band is dispersive with negative and positive wings. The FSR signal can be expressed as
Here, ωs is the detuning from ΩR, ωv is the vibrational frequency, and Γij is the half width at half maximum. The FRS signal is given by the sum over all pathways [s] through the vibrational manifolds in S0, S1, and Sn.
The potential energy for the electronic states is assumed to be separable into normal harmonic modes with frequencies νj along intramolecular coordinates qj. The relative geometry of Sn/Sm is taken to be identical to S0/S1 for simplicity. All vibronic transitions are organized in Franck–Condon progressions. Within the Born–Oppenheimer approximation, , where is the electronic transition dipole operator and is the vibrational overlap integral, with its square equals the Franck–Condon factors .