Absolute excited state molecular geometries revealed by resonance Raman signals

Abstract

Ultrafast reactions activated by light absorption are governed by multidimensional excited-state (ES) potential energy surfaces (PESs), which describe how the molecular potential varies with the nuclear coordinates. ES PESs ad-hoc displaced with respect to the ground state can drive subtle structural rearrangements, accompanying molecular biological activity and regulating physical/chemical properties. Such displacements are encoded in the Franck-Condon overlap integrals, which in turn determine the resonant Raman response. Conventional spectroscopic approaches only access their absolute value, and hence cannot determine the sense of ES displacements. Here, we introduce a two-color broadband impulsive Raman experimental scheme, to directly measure complex Raman excitation profiles along desired normal modes. The key to achieve this task is in the signal linear dependence on the Frank-Condon overlaps, brought about by non-degenerate resonant probe and off-resonant pump pulses, which ultimately enables time-domain sensitivity to the phase of the stimulated vibrational coherences. Our results provide the tool to determine the magnitude and the sensed direction of ES displacements, unambiguously relating them to the ground state eigenvectors reference frame.

Fig. 1. Concept of the proposed Impulsive Stimulated Raman Scattering (ISRS) scheme for a diatomic molecule.

A pair of ultrashort pulses is exploited to measure the complex Raman excitation profile (REP) and access the excited state (ES) displacement, determining whether the atomic distance is increased or decreased upon the photoexcitation a, b. Since the ISRS response is proportional to the product between the ground state (GS) polarizability derivative and the Franck-Condon (FC) overlaps, the measured signals S(λ) uniquely determines the ES geometry and the sense of the geometrical rearrangement c, d, being antisymmetric with respect to the nuclear distance modification. Notably, while the sense of the GS eigenvector is arbitrary, as both outgoing (Q) or in-going arrows (${\mathit{Q}}^{\prime }=-\mathit{Q}$) can be selected e, reversing such a sense changes both the sign of the polarizability derivative as well as the sign of the FC overlaps f, g, making the experimental signals invariant on the selection of the reference frame. For example, if the complex REP reconstructed from the ISRS signal is the one reported in c, the consequences of the two possible eigenmode choices e are depicted in the corresponding f: the eigenmode direction Q is associated to a positive $\frac{\partial \alpha }{\partial Q}$ and a positive displacement d, while the direction ${\mathbf{Q}}^{\prime }=-\mathbf{Q}$ corresponds to a negative polarizability derivative and a negative d. Crucially, the two formal descriptions yield to the same physical observable, i.e., an increase of the bond length in the excited state (highlighted in red). On the other hand, the measured REP in c excludes an ES bond-length shortening (blue), as the corresponding descriptions shown in g are both associated to the signal in d, which has a reversed sign with respect to the measured one.

Fig. 2. Schematic of the ISRS third-order nonlinear processes.

Energy level diagrams accounting the for impulsive stimulated Raman processes are reported in a, b. Two consecutive interactions with the RP, which exerts a driving force on the vibrational normal mode Q_j with a coupling ruled by the electronic polarizability derivative $\frac{\partial \alpha }{\partial Q_j}$, prepare a vibrational coherence on the ket a or on the bra side b of the density matrix. Interactions with the ket/bra side of the density matrix and the (classical) electromagnetic fields are presented by continuous/dashed vertical arrows, respectively. The sample density matrix is then promoted on the electronic excited state (ES) manifold e_k via an interaction, proportional to FC overlaps ${\mu }_{g{e}_k}{\mu }_{e_k{g}^{\prime }}$, with the broadband resonant probe pulse, enabling to record upon a free induction decay the ground state (GS) Raman response in the time-domain. One-dimensional projections of the ES PES along a specific normal mode are reported in c. Since the a/b pathways responsible for the measured signal depend on the product between two dipole moments ${\mu }_{g{e}_k}{\mu }_{e_k{g}^{\prime }}$ (reported in d for a single mode) and not on their absolute squared value, the experimental response, reported in arbitrary units in e, is sensitive to the linear combination of complex Stokes/anti-Stokes REPs (R_S and R_AS, respectively), which reveal the sign of the molecular displacement.

Fig. 3. Rhodamine B spectral response.

Time-domain broadband impulsive stimulated Raman scattering spectra of Rhodamine B dissolved in methanol as a function of the probe wavelength λ_P recorded with a vanishing probe chirp a; a magnification of the oscillating ISRS trace around the 450–650 fs temporal range is reported in b. The frequency-domain ISRS spectra obtained upon Fourier transforming over the RP-PP time delay are reported (in arbitrary units) in d as a function of λ_P, while the integrated ISRS spectrum, computed averaging the frequency-domain map over the probe wavelengths from 520 to 585 nm is shown in c. A strong mode at 620 cm⁻¹ dominates the signal, with respect to lower intensity peaks at 490, 735, 1280, and 1360 cm⁻¹. Small contributions from the solvent and from the glass cuvette may be expected at 1040 and 490 cm⁻¹, respectively, and can be suppressed for a vanishing PP chirp. Panel e shows the PP spectrum (yellow area) and the sample’s absorbance profile (blue area). This latter is compared with the theoretical curves computed via DFT calculations, as discussed in the text, with and without the displacement scaling factor (blue and red dashed lines, respectively).

Fig. 4. Amplitude and the phase of the experimental ISRS signals.

The amplitude and the phase of the different RhB normal mode components in the ISRS map, recorded for different probe chirp values (reported in the legend), are evaluated across the sample absorption profile as the sum of sinusoidal terms. As expected, for a vanishing PP chirp all the modes exhibit a π phase shift around the absorption maximum at 550 nm, where the oscillation amplitudes vanish.

Fig. 5. Chirp dependence of the ISRS response and REP reconstruction.

Contributions of the 735 cm⁻¹ mode to the ISRS experimental map, measured with an approximately vanishing chirp (C₂ = − 4 fs²) and with a $C_2=71f{s}^2\approx \frac{1}{8\pi }{T}_{vib}^2$ are reported in a, b, respectively. The dashed purple lines indicate the slope of the chirp and a magnification of the ISRS maps around 500–600 fs is reported in c, d as a function of time delay upon dechirp ($\tilde{T}$ bottom axis) and $\tilde{T}/T_{vib}$ (top axis). Slices of such maps at $\tilde{T}/T_{vib}=n$ and $\tilde{T}/T_{vib}=\left(n-1/4\right)$, which allows the complete reconstruction of the real and imaginary parts of the Raman excitation profiles (see the text and Table 1), are reported (as continuous lines) in e, f and compared with the REPs modeled by TD-DFT (dashed lines), normalized to the area of the measured traces at $\tilde{T}=$ 12 T_vib. Notably, the experimental and the theoretical signals share the same sign, validating hence the sign of the reconstructed normal mode ES displacement, while discrepancies between their spectral profiles suggest a role of the sample absorption, third-order dispersion and probe spectrum, which have not been included in the data analysis reported in this figure.

Fig. 6. Complex REPs and direction of the ES displacements.

Real and Imaginary part of the Stokes/anti-Stokes (red and blue lines) Raman excitation profiles measured for the different vibrational modes of Rhodamine B dissolved in methanol are reported in a. The REPs have been directly extracted from the time-domain traces, from 400 to 700 fs, considering multiple PP chirp values (C₂ = −4, 7, 30, 44, 71 fs²). The anti-Stokes Raman excitation profiles have been horizontally shifted by one vibrational quantum for a direct comparison with the Stokes counterparts. Shaded areas indicate the 90% confidence intervals. The experimentally extracted complex REPs are compared with the ones (black dashed lines) modeled from the absorption spectrum by using the transform method^,, while the directions of the corresponding ES displacements along the normal coordinates are depicted in b.