Absolute excited state molecular geometries revealed by resonance Raman signals

Photochemical and photophysical reactions are dictated by excited-state (ES) potential energy surfaces that displace relative to the ground state (GS), leading to changes in bond lengths, angles, and dihedrals that underlie photoisomerization, charge separation, and molecular motor function. Determining not only the magnitude but also the sensed direction (sign) of the ES displacements along GS normal modes is crucial, yet conventional absorption and spontaneous resonance Raman (RR) spectroscopies report only on the absolute values of Franck-Condon (FC) overlaps, making the sign inaccessible and often yielding ambiguous interpretations. Examples motivating this need include bond-length changes in stilbene isomerization, dihedral angle changes in retinal isomerization in vision, structural changes in diaryl thiophenes and push-pull systems, and asymmetric ES PESs in molecular motors. To overcome these limitations, the study introduces a two-color broadband impulsive stimulated Raman scattering (ISRS) scheme that enables direct measurement of complex (real and imaginary) Raman excitation profiles (REPs) and thus the sign of ES displacements referenced to GS eigenvectors.

The work situates itself within spontaneous RR spectroscopy, which provides vibrational information enhanced near electronic resonances but is fundamentally limited to the square modulus of FC-weighted transition moments, obscuring displacement sign. Prior developments in time-domain Raman methods, especially ISRS, have enabled phase and amplitude retrieval of vibrational coherences and suppression of fluorescence backgrounds. However, conventional implementations or frequency-domain approaches often encounter artifacts (e.g., cross-phase modulation) or still lack direct access to displacement sign. The authors build on diagrammatic treatments of nonlinear Raman responses and transform theory for REPs, and on applications of ISRS to track ultrafast structural dynamics in diverse systems (e.g., heme proteins, retinal, perovskites), to propose a chirp- and color-engineered ISRS protocol that linearly depends on FC overlaps and thus is sensitive to the sign of ES displacements.

• Concept and theory: The two-color broadband ISRS scheme employs an electronically off-resonant Raman pump (RP) to impulsively drive vibrational coherences in the GS via the molecular polarizability (H_RP = α·E_R·E_P), and a broadband resonant probe pulse (PP) tuned across an allowed electronic transition to stimulate RR responses. Using a semiclassical perturbative expansion, the third-order polarization is expressed via two classes of Feynman pathways (A/B) depending on whether RP acts on the ket or bra side. The measured spectrally dispersed ISRS signal S(ω,ΔT) ∝ Im{[I_RP on − I_P]/I_P} ∝ −3[P^(3)(ω,ΔT)/E_P(ω)], with P^(3) = P_A^(3)+P_B^(3). Because the RP is off-resonant, the preparation of coherences depends on ∂α/∂Q_j (GS polarizability derivative), while projection onto the ES manifold by the PP introduces sum-over-states terms proportional to products of ES–GS dipole matrix elements μ_eg^k μ_ge^k weighted by FC overlaps. This linear dependence on μ_eg^k μ_ge^k (odd in displacement) renders the ISRS signal antisymmetric with displacement and thus sensitive to its sign.
• Chirp engineering and REP reconstruction: The PP group-delay dispersion (chirp) C2 introduces a controllable phase offset between A and B pathways. By choosing specific C2 values (e.g., C2 = τ_vib and C2 = 2(m+1)τ_vib) and dechirped delays τ = nτ_vib or τ = (n+1)τ_vib, slices of the time-domain signal isolate linear combinations of the real/imaginary parts of Stokes and anti-Stokes complex REPs, enabling separate reconstruction of Re/Im components. The signal is analyzed prior to FFT to preserve phase information; FFT over ΔT is used to identify vibrational frequencies.
• Experimental setup: A Ti:sapphire laser (800 nm, 40 fs, 1 kHz, 3 mJ) provides both beams. The RP is generated by a non-collinear OPA centered at 700 nm, filtered (>600 nm) and compressed to 18 fs (chirped mirrors). The PP is a vertically polarized white-light continuum from a 3 mm sapphire crystal, compressed with chirped mirrors; fine chirp control via thin glass windows. RP and PP are focused non-collinearly into Rhodamine B (RhB) in methanol in a 500 μm quartz cuvette. A spectrometer/CCD detects the PP spectrum; a 500 Hz chopper alternates RP on/off to compute S(ω,ΔT). Probe chirps used include C2 = −4, 7, 30, 44, 71 fs^2.
• Measurements and analysis: Time-domain ISRS maps were recorded versus PP wavelength λ_p and RP–PP delay ΔT from −0.3 to 3 ps (10 fs steps), then fine-scanned around 0.35–0.85 ps (4 fs steps). FFT identified modes at ~490, 620, 735, 1280, 1360 cm⁻1; pulses (~18 fs) enable coherences up to ~1400 cm⁻1. For each mode i, signals S(λ,ΔT) were fit as Σ_i A_i(λ) sin(ω_i ΔT + φ_i(λ)), with ω_i fixed from FFT; A_i and φ_i versus λ and C2 were used to reconstruct time-domain responses and extract complex REPs using the chirp/delay protocol. A least-squares solution of an overdetermined linear system parameterized by ΔT and C2 yields four complex REP terms [R_g^g(ω)], [R_e^g(ω)], [R_g^e(ω)], [R_e^e(ω)]. Corrections were applied for spectral weighting by red/blue-shifted PP components contributing to S_A and S_B, probe spectral evolution in high optical density, and higher-order dispersion across the >50 nm bandwidth.
• Computational modeling: TD-DFT (CAM-B3LYP/6-311++G(2d,2p)) with PCM solvent was used to optimize GS and ES geometries, compute normal modes and transition dipoles, and evaluate polarizability derivatives ∂α/∂Q_j along GS eigenvectors. Transform theory was used to model REPs from the experimental absorption spectrum, including multimode participation with an adiabatic 1D projection along the mode of interest. Duschinsky mixing was evaluated and found negligible for modes considered. Displacements from TD-DFT were scaled (low-frequency <1000 cm⁻1 by 1.2, high-frequency by 1.4) to better match the absorption profile.

• Demonstration of a two-color broadband ISRS protocol that retrieves the complex (real and imaginary) Raman excitation profiles (REPs) and determines the sign of excited-state (ES) displacements along specific GS normal modes.
• Phase-sensitive time-domain analysis with an off-resonant pump and resonant probe makes the ISRS signal linear in Franck-Condon overlaps (μ_eg μ_ge), enabling sign sensitivity absent in spontaneous RR spectroscopy.
• In Rhodamine B (methanol), five vibrational modes were observed: 490, 620, 735, 1280, and 1360 cm⁻1. The method reconstructs REPs across 520–585 nm and separates Re/Im components by tuning probe chirp (e.g., C2 ≈ −4 fs^2 and 71 fs^2 ≈ τ_vib) and selecting appropriate dechirped delays.
• Extracted ES displacements (dimensionless d_EXP; signs indicate direction relative to GS eigenvectors):
  • 490 cm⁻1: d_EXP = −0.12 (Q_EXP = −0.049 a.u.)
  • 620 cm⁻1: d_EXP = −0.24 (Q_EXP = −0.058 a.u.)
  • 735 cm⁻1: d_EXP = −0.08 (Q_EXP = −0.020 a.u.)
  • 1280 cm⁻1: d_EXP = −0.17 (Q_EXP = −0.064 a.u.)
  • 1360 cm⁻1: d_EXP = +0.13 (Q_EXP = +0.038 a.u.)
• Experimental complex REPs agree in sign and largely in lineshape with profiles modeled from the absorption spectrum via transform theory using measured displacements, confirming correct determination of displacement directions.
• Structural interpretation: ES elongation of RhB’s central pyran ring due to increased O–distal carbon distance; oxygen motion induces out-of-plane rotation of the carboxylic acid; lateral benzenes undergo outward expansion.
• Additional observations: For C2 = 0, modes exhibit a π phase flip near the absorption maximum (~550 nm) where oscillation amplitudes vanish; heterodyne detection suppresses fluorescence, and the Stokes/anti-Stokes amplitudes do not follow Boltzmann factors due to impulsive preparation.

The presented two-color ISRS scheme addresses the long-standing limitation of spontaneous RR spectra, which only report absolute FC overlap magnitudes, by constructing a signal linearly sensitive to FC overlaps and thus to the sign of ES displacements. By controlling probe chirp and dechirped time delays, the method separates real and imaginary parts of Stokes/anti-Stokes REPs, whose odd symmetry with displacement allows unambiguous assignment of displacement direction in the GS eigenvector frame, independent of eigenvector sign convention. Application to Rhodamine B validates the approach: reconstructed REPs match transform-theory predictions and TD-DFT-guided models, and yield consistent displacement signs across five modes, enabling detailed ES structural insights (pyran ring elongation, functional group motions). Discrepancies in lineshape highlight practical considerations such as probe spectral variation in high optical density samples, higher-order dispersion, and subtle differences between TD-DFT displacements and experimental values, but do not alter the central conclusion that displacement sign and magnitude can be measured. The method’s time-domain, heterodyne nature also suppresses fluorescence and cross-phase modulation artifacts, enhancing robustness.

This study introduces and validates a two-color broadband ISRS methodology that directly measures complex Raman excitation profiles and reveals both magnitude and sign of excited-state displacements along specified normal modes. Benchmarking on Rhodamine B demonstrates accurate recovery of REPs and ES structural rearrangements, including pyran ring elongation and associated group motions, overcoming fundamental ambiguities of spontaneous RR spectroscopy. Because only two probe chirp settings and a few time delays are sufficient to determine complex REPs, the protocol enables fast, high-SNR acquisition. Future work can apply this approach to map reaction coordinates in diverse photoactive systems, incorporate comprehensive corrections for probe spectral evolution and dispersion, and refine ab initio modeling to further tighten experiment-theory agreement.

• Data analysis requires corrections for contributions of red/blue-shifted probe spectral components to different pathways (S_A/S_B), which can affect reconstructed REPs if neglected.
• High optical density leads to significant probe spectral evolution across the sample, altering effective interaction length and lineshapes.
• Higher-order dispersion across the broad PP bandwidth (>50 nm) must be accounted for to precisely time-order spectral components.
• TD-DFT-predicted displacements required empirical scaling to match the absorption profile, indicating potential inaccuracies in computed ES displacements or solvent/environment effects.
• Assumption of negligible Duschinsky mixing holds for the modes examined; strong mode mixing would complicate 1D adiabatic projections.
• Accurate chirp control and timing are critical; experimental imperfections can degrade phase-sensitive reconstruction.