Novel coherent two-dimensional optical spectroscopy probes of chirality exchange and fluctuations in molecules

Abstract

We demonstrate how stochastic transitions between molecular configurations with opposite senses of chirality may be probed by 2D optical signals with specific pulse polarization configurations. The third-order optical response of molecular dimers (such as biphenyls) with dynamical axial chirality is calculated to order of k(2) in the wavevector of light. Spectroscopic signatures of equilibrium chirality fluctuations are predicted for three dynamical models (Ornstein-Uhlenbeck, two-state jump, and diffusion in double well) of the dihedral angle that controls the chirality.

Figure 1

Dimer geometry. Top: For φ = 0, the dimer is in xz plane, and is symmetric under the reflection through yz plane. Left chromophore corresponds to ${\vec{\mu }}_1{|}_{\varphi =0}=({\tilde{\mu }}_x,0,{\tilde{\mu }}_z)$, ${\vec{M}}_1$, . . . , right chromophore corresponds to ${\vec{\mu }}_2{|}_{\varphi =0}=(-{\tilde{\mu }}_x,0,{\tilde{\mu }}_z)$, ${\vec{M}}_2$, . . . . Bottom: Axial chirality induced by twisting the groups by angle 2φ along x axis. The variation of multipoles with φ are given by Eq. 1 and in Appendix B.

Figure 2

Free energy profiles for the three models of diffusive (stochastic) dynamics of φ. Details are given in Appendix A.

Figure 3

Top: Angular density of the CD signal of the symmetric exciton $B_{+}^{†}$ (Eq. 16) ${\mathcal{I}}_{yz}(\omega ,\varphi )$ for model (i) with equilibrium width σ = π/6. Left: Real part. Right : Imaginary part. Bottom: the same for model (iii) (with potential minima at φ₀ = ±π/3, and width σ = π/12). Other parameters: J_Δ/Γ = 20, $\frac{k}{6}R{\left|{\tilde{\mu }}_z\right|}^2=1$.

Figure 4

Integrated absolute values of the signals from Fig. 3. Green dashed line shows integrated real part Re Y(ω), red solid line shows the integrated imaginary part Im Y(ω). Absorption line shape I_xx(ω) (blue dotted) is given for comparison.

Figure 5

Chiral photon echo signal ${\mathcal{R}}_{c,\omega }$ (Eq. 30) of the symmetric exciton level of a dimer undergoing a two-state jump (model (ii)) between configurations with opposite chirality (Eq. A7). Shown are the frequency-frequency correlation plots ${\mathcal{R}}_{c,\omega }({\omega }_3,t_2,{\omega }_1)$ at two delay times as indicated. Other parameters: φ₀ = π/3.0, J_Δ/Γ = 20, $\frac{k^2}{3}{R}^2{\left|{\tilde{\mu }}_z\right|}^4=1$.

Figure 6

Same as Fig. 5 but for model (i) (Eq. A5) at delay times Dt₂ = 0 (top panel), Dt₂ = 1 (central panel), and Dt₂ = 10 (bottom panel). Parameters: σ = π/6, J_Δ/Γ = 20, $\frac{k^2}{3}{R}^2{\left|{\tilde{\mu }}_z\right|}^4=1$.

Figure 7

Same as Fig. 5 but for model (iii) (Eq. A8) at delay times (from top to bottom) Dt₂ = 0, 1, 100, 1000. Parameters: Λ/D = 0.01, φ₀ = π/3.0, σ = π/12.0, J_Δ/Γ = 20, $\frac{k^2}{3}{R}^2{\left|{\tilde{\mu }}_z\right|}^4=1$.

Figure 8

Decay of the peak volume ${\mathcal{R}}_C\equiv \int d{\omega }_{3}d{\omega }_1\mathcal{R}({\omega }_3,t_2,{\omega }_1)$ with delay time t₂ for model (i) (Eq. G2, blue dashed-dotted line), model (ii) (Eq. G3, red solid line), and model (iii) (Eq. G4, magenta dotted line). Straight lines indicate exponential decay. Parameters: Λ/D = 0.1, φ₀ = π/3.0, σ = π/6.0.

Figure 9

Variation of the signal ${\mathcal{R}}_D\left(t_2\right)$ with delay time t₂ for model (i) (Eq. G5, blue dashed-dotted line), model (ii) (Eq. G6, red solid line), and model (iii) (Eq. G7, magenta dotted line). Straight lines indicate exponential decay. Parameters: Λ/D = 0.1, φ₀ = π/3.0, σ = π/6.0.