Two-Dimensional Resonance Raman Signatures of Vibronic Coherence Transfer in Chemical Reactions

Abstract

Two-dimensional resonance Raman (2DRR) spectroscopy has been developed for studies of photochemical reaction mechanisms and structural heterogeneity in condensed phase systems. 2DRR spectroscopy is motivated by knowledge of non-equilibrium effects that cannot be detected with traditional resonance Raman spectroscopy. For example, 2DRR spectra may reveal correlated distributions of reactant and product geometries in systems that undergo chemical reactions on the femtosecond time scale. Structural heterogeneity in an ensemble may also be reflected in the 2D spectroscopic line shapes of both reactive and non-reactive systems. In this chapter, these capabilities of 2DRR spectroscopy are discussed in the context of recent applications to the photodissociation reactions of triiodide. We show that signatures of “vibronic coherence transfer” in the photodissociation process can be targeted with particular 2DRR pulse sequences. Key differences between the signal generation mechanisms for 2DRR and off-resonant 2D Raman spectroscopy techniques are also addressed. Overall, recent experimental developments and applications of the 2DRR method suggest that it will be a valuable tool for elucidating ultrafast chemical reaction mechanisms.

Appendices

Appendix A: Vibrational Hamiltonians

The present model assumes that both triiodide and diiodide possess two electronic levels and one nuclear coordinate with displaced ground and excited state potential energy minima. The anharmonic vibrational wave functions for the Franck–Condon active bond stretching mode of diiodide and the symmetric stretching coordinate of triiodide are generated using a Hamiltonian with the following form [73]

$$ H_{\alpha } = \frac{{\hbar \omega_{{\alpha ,{\text{vib}}}} }}{2}\left( {2a^{\dag } a + 1} \right) + U_{3,\alpha } \left[ {a^{\dag } a^{\dag } a^{\dag } + 3a^{\dag } a^{\dag } a + 3a^{\dag } aa + aaa + 3a^{\dag } + 3a} \right], $$

(7)

where

$$ U_{3,\alpha } = \frac{1}{{3!\sqrt {2^{3} m^{3} \omega^{3} \hbar^{ - 3} } }}\left( {\frac{{d^{3} V}}{{dq^{3} }}} \right)_{0} . $$

(8)

The wavefunctions are obtained by diagonalizing this Hamiltonian in a basis set of harmonic oscillators that includes states with up to the 40 vibrational quanta. Parameters of the vibrational Hamiltonian are given in Table 1. We use a notation in which α represents the molecule (r for triiodide or p for diiodide) and an asterisk indicates an electronically excited state.

Table 1 Parameters of Model Used to Compute 2DRR Spectra

The vibrational overlap integrals used to evaluate the response functions of diiodide are obtained using

$$ \left\langle {n} \mathrel{\left | {\vphantom {n m}} \right. \kern-0pt} {m} \right\rangle = \sum\limits_{jk} {\varphi_{nk} \varphi_{mj} \left\langle {k} \mathrel{\left | {\vphantom {k j}} \right. \kern-0pt} {j} \right\rangle } , $$

(9)

where φ _nk is the expansion coefficient for harmonic basis vector, k, and the anharmonic excited state vibrational wave function, n. Vibrational overlap integrals of triiodide are given by a different formula,

$$ \left\langle {n} \mathrel{\left | {\vphantom {n m}} \right. \kern-0pt} {m} \right\rangle = \sum\limits_{k} {\varphi_{nk} \left\langle {k} \mathrel{\left | {\vphantom {k m}} \right. \kern-0pt} {m} \right\rangle } , $$

(10)

because the ground and excited states are taken to be harmonic and anharmonic, respectively (see discussion in Sect. 2). In order to evaluate the overlap integrals, we assume a dimensionless displacement of 7.0 based on spontaneous Raman measurements for triiodide [45] and our earlier 2DRR study [22]. A displacement of 7.0 produces an excited state potential energy gradient of 225 eV/pm in diiodide which is identical to that associated with a previously employed exponential surface [42, 43].

Appendix B: Two-Dimensional Resonance Raman Signal Components

The Feynman diagrams presented in Fig. 3 include dummy indices for vibrational levels (m, n, j, k, l, u, v, w) associated with the ground and excited electronic states (r and r* for triiodide, p and p* for diiodide). Transition dipoles are written as product of integrals over electronic and nuclear degrees of freedom based on the Condon approximation. For example, an interaction that couples vibrational level m in the ground electronic state of the reactant and vibrational level n in the excited electronic state of the reactant contributes the product, μ _r*r $\left\langle {n} \mathrel{\left | {\vphantom {n m}} \right. \kern-0pt} {m} \right\rangle$, to the response function, where μ _r*r is the electronic transition dipole and $\left\langle {n} \mathrel{\left | {\vphantom {n m}} \right. \kern-0pt} {m} \right\rangle$ is a vibrational overlap integral. We use a notation in which the excited state vibrational energy level is always written in the bra [48].

The first polarization component is given in Eq. (3). The remaining 11 polarization components are given by

$$ \begin{aligned} P_{2}^{\left( 5 \right)} (\omega_{1} ,\omega_{2} ) & = - \frac{{N\xi_{\text{UV}}^{5} \left| {\mu_{r*r} } \right|^{6} }}{{\hbar^{5} }}\sum\limits_{mnjklu} {B_{m} } \left\langle {n} \mathrel{\left | {\vphantom {n m}} \right. \kern-0pt} {m} \right\rangle \left\langle {n} \mathrel{\left | {\vphantom {n j}} \right. \kern-0pt} {j} \right\rangle \left\langle {k} \mathrel{\left | {\vphantom {k m}} \right. \kern-0pt} {m} \right\rangle \left\langle {k} \mathrel{\left | {\vphantom {k l}} \right. \kern-0pt} {l} \right\rangle \left\langle {u} \mathrel{\left | {\vphantom {u j}} \right. \kern-0pt} {j} \right\rangle \left\langle {u} \mathrel{\left | {\vphantom {u l}} \right. \kern-0pt} {l} \right\rangle , \\ & \quad \times L_{r*n,rm} \left( {\omega_{\text{UV}} } \right)D_{rj,rm} \left( {\omega_{1} } \right)L_{rj,r*k} \left( { - \omega_{\text{UV}} } \right)D_{rj,rl} \left( {\omega_{2} } \right)L_{r*u,rl} \left( {\omega_{t} } \right) \\ \end{aligned} $$

(11)

$$ \begin{aligned} P_{3}^{\left( 5 \right)} (\omega_{1} ,\omega_{2} ) & = - \frac{{N\xi_{\text{UV}}^{5} \left| {\mu_{r*r} } \right|^{6} }}{{\hbar^{5} }}\sum\limits_{mnjklu} {B_{m} } \left\langle {n} \mathrel{\left | {\vphantom {n m}} \right. \kern-0pt} {m} \right\rangle \left\langle {n} \mathrel{\left | {\vphantom {n j}} \right. \kern-0pt} {j} \right\rangle \left\langle {k} \mathrel{\left | {\vphantom {k j}} \right. \kern-0pt} {j} \right\rangle \left\langle {k} \mathrel{\left | {\vphantom {k l}} \right. \kern-0pt} {l} \right\rangle \left\langle {u} \mathrel{\left | {\vphantom {u m}} \right. \kern-0pt} {m} \right\rangle \left\langle {u} \mathrel{\left | {\vphantom {u l}} \right. \kern-0pt} {l} \right\rangle , \\ & \quad \times L_{rm,r*n} \left( { - \omega_{\text{UV}} } \right)D_{rm,rj} \left( {\omega_{1} } \right)L_{rm,r*k} \left( { - \omega_{\text{UV}} } \right)D_{rm,rl} \left( {\omega_{2} } \right)L_{r*u,rl} \left( {\omega_{t} } \right) \\ \end{aligned} $$

(12)

$$ \begin{aligned} P_{4}^{\left( 5 \right)} (\omega_{1} ,\omega_{2} ) & = - \frac{{N\xi_{\text{UV}}^{5} \left| {\mu_{r*r} } \right|^{6} }}{{\hbar^{5} }}\sum\limits_{mnjklu} {B_{m} } \left\langle {n} \mathrel{\left | {\vphantom {n m}} \right. \kern-0pt} {m} \right\rangle \left\langle {n} \mathrel{\left | {\vphantom {n j}} \right. \kern-0pt} {j} \right\rangle \left\langle {k} \mathrel{\left | {\vphantom {k m}} \right. \kern-0pt} {m} \right\rangle \left\langle {k} \mathrel{\left | {\vphantom {k l}} \right. \kern-0pt} {l} \right\rangle \left\langle {u} \mathrel{\left | {\vphantom {u l}} \right. \kern-0pt} {l} \right\rangle \left\langle {u} \mathrel{\left | {\vphantom {u j}} \right. \kern-0pt} {j} \right\rangle , \\ & \quad \times L_{rm,r*n} \left( { - \omega_{\text{UV}} } \right)D_{rm,rj} \left( {\omega_{1} } \right)L_{r*k,rj} \left( {\omega_{\text{UV}} } \right)D_{rl,rj} \left( {\omega_{2} } \right)L_{r*u,rj} \left( {\omega_{t} } \right) \\ \end{aligned} $$

(13)

$$ \begin{aligned} P_{5}^{\left( 5 \right)} (\omega_{1} ,\omega_{2} ) & = - \frac{{N\xi_{\text{UV}}^{2} \xi_{VIS}^{3} \left| {\mu_{r*r} } \right|^{2} \left| {\mu_{p*p} } \right|^{4} }}{{\hbar^{5} }}\sum\limits_{mnjkluvw} {B_{m} } \left\langle {n} \mathrel{\left | {\vphantom {n m}} \right. \kern-0pt} {m} \right\rangle \left\langle {j} \mathrel{\left | {\vphantom {j m}} \right. \kern-0pt} {m} \right\rangle \left\langle {u} \mathrel{\left | {\vphantom {u k}} \right. \kern-0pt} {k} \right\rangle \left\langle {u} \mathrel{\left | {\vphantom {u v}} \right. \kern-0pt} {v} \right\rangle \left\langle {w} \mathrel{\left | {\vphantom {w v}} \right. \kern-0pt} {v} \right\rangle \left\langle {w} \mathrel{\left | {\vphantom {w l}} \right. \kern-0pt} {l} \right\rangle , \\ & \quad \times L_{r*n,rm} \left( {\omega_{\text{UV}} } \right)D_{pk,pl} \left( {\omega_{1} } \right)L_{p*u,pl} \left( {\omega_{\text{VIS}} } \right)D_{pv,pl} \left( {\omega_{2} } \right)L_{p*w,pl} \left( {\omega_{t} } \right) \\ \end{aligned} $$

(14)

$$ \begin{aligned} P_{6}^{\left( 5 \right)} (\omega_{1} ,\omega_{2} ) & = - \frac{{N\xi_{\text{UV}}^{2} \xi_{\text{VIS}}^{3} \left| {\mu_{r*r} } \right|^{2} \left| {\mu_{p*p} } \right|^{4} }}{{\hbar^{5} }}\sum\limits_{mnjkluvw} {B_{m} } \left\langle {n} \mathrel{\left | {\vphantom {n m}} \right. \kern-0pt} {m} \right\rangle \left\langle {j} \mathrel{\left | {\vphantom {j m}} \right. \kern-0pt} {m} \right\rangle \left\langle {u} \mathrel{\left | {\vphantom {u k}} \right. \kern-0pt} {k} \right\rangle \left\langle {u} \mathrel{\left | {\vphantom {u v}} \right. \kern-0pt} {v} \right\rangle \left\langle {w} \mathrel{\left | {\vphantom {w v}} \right. \kern-0pt} {v} \right\rangle \left\langle {w} \mathrel{\left | {\vphantom {w l}} \right. \kern-0pt} {l} \right\rangle , \\ & \quad \times L_{rm,r*n} \left( { - \omega_{\text{UV}} } \right)D_{pk,pl} \left( {\omega_{1} } \right)L_{p*u,pl} \left( {\omega_{\text{VIS}} } \right)D_{pv,pl} \left( {\omega_{2} } \right)L_{p*w,pl} \left( {\omega_{t} } \right) \\ \end{aligned} $$

(15)

$$ \begin{aligned} P_{7}^{\left( 5 \right)} (\omega_{1} ,\omega_{2} ) & = - \frac{{N\xi_{\text{UV}}^{2} \xi_{\text{VIS}}^{3} \left| {\mu_{r*r} } \right|^{2} \left| {\mu_{p*p} } \right|^{4} }}{{\hbar^{5} }}\sum\limits_{mnjkluvw} {B_{m} } \left\langle {n} \mathrel{\left | {\vphantom {n m}} \right. \kern-0pt} {m} \right\rangle \left\langle {j} \mathrel{\left | {\vphantom {j m}} \right. \kern-0pt} {m} \right\rangle \left\langle {u} \mathrel{\left | {\vphantom {u l}} \right. \kern-0pt} {l} \right\rangle \left\langle {u} \mathrel{\left | {\vphantom {u v}} \right. \kern-0pt} {v} \right\rangle \left\langle {w} \mathrel{\left | {\vphantom {w k}} \right. \kern-0pt} {k} \right\rangle \left\langle {w} \mathrel{\left | {\vphantom {w v}} \right. \kern-0pt} {v} \right\rangle , \\ & \quad \times L_{r*n,rm} \left( {\omega_{\text{UV}} } \right)D_{pk,pl} \left( {\omega_{1} } \right)L_{pk,p*u} \left( { - \omega_{\text{VIS}} } \right)D_{pk,pv} \left( {\omega_{2} } \right)L_{p*w,pv} \left( {\omega_{t} } \right) \\ \end{aligned} $$

(16)

$$ \begin{aligned} P_{8}^{\left( 5 \right)} (\omega_{1} ,\omega_{2} ) & = - \frac{{N\xi_{\text{UV}}^{2} \xi_{\text{VIS}}^{3} \left| {\mu_{r*r} } \right|^{2} \left| {\mu_{p*p} } \right|^{4} }}{{\hbar^{5} }}\sum\limits_{mnjkluvw} {B_{m} } \left\langle {n} \mathrel{\left | {\vphantom {n m}} \right. \kern-0pt} {m} \right\rangle \left\langle {j} \mathrel{\left | {\vphantom {j m}} \right. \kern-0pt} {m} \right\rangle \left\langle {u} \mathrel{\left | {\vphantom {u l}} \right. \kern-0pt} {l} \right\rangle \left\langle {u} \mathrel{\left | {\vphantom {u v}} \right. \kern-0pt} {v} \right\rangle \left\langle {w} \mathrel{\left | {\vphantom {w k}} \right. \kern-0pt} {k} \right\rangle \left\langle {w} \mathrel{\left | {\vphantom {w v}} \right. \kern-0pt} {v} \right\rangle , \\ & \quad \times L_{rm,r*n} \left( { - \omega_{\text{UV}} } \right)D_{pk,pl} \left( {\omega_{1} } \right)L_{pk,p*u} \left( { - \omega_{\text{VIS}} } \right)D_{pk,pv} \left( {\omega_{2} } \right)L_{p*w,pv} \left( {\omega_{t} } \right) \\ \end{aligned} $$

(17)

$$ \begin{aligned} P_{9}^{\left( 5 \right)} (\omega_{1} ,\omega_{2} ) & = - \frac{{N\xi_{\text{UV}}^{4} \xi_{\text{VIS}} \left| {\mu_{r*r} } \right|^{4} \left| {\mu_{p*p} } \right|^{2} }}{{\hbar^{5} }}\sum\limits_{mnjkluvw} {B_{m} } \left\langle {n} \mathrel{\left | {\vphantom {n m}} \right. \kern-0pt} {m} \right\rangle \left\langle {n} \mathrel{\left | {\vphantom {n j}} \right. \kern-0pt} {j} \right\rangle \left\langle {k} \mathrel{\left | {\vphantom {k j}} \right. \kern-0pt} {j} \right\rangle \left\langle {l} \mathrel{\left | {\vphantom {l m}} \right. \kern-0pt} {m} \right\rangle \left\langle {w} \mathrel{\left | {\vphantom {w u}} \right. \kern-0pt} {u} \right\rangle \left\langle {w} \mathrel{\left | {\vphantom {w v}} \right. \kern-0pt} {v} \right\rangle , \\ & \quad \times L_{r*n,rm} \left( {\omega_{UV} } \right)D_{rj,rm} \left( {\omega_{1} } \right)L_{r*k,rm} \left( {\omega_{UV} } \right)D_{pu,pv} \left( {\omega_{2} } \right)L_{p*w,pv} \left( {\omega_{t} } \right) \\ \end{aligned} $$

(18)

$$ \begin{aligned} P_{10}^{\left( 5 \right)} (\omega_{1} ,\omega_{2} ) & = - \frac{{N\xi_{\text{UV}}^{4} \xi_{\text{VIS}}^{{}} \left| {\mu_{r*r} } \right|^{4} \left| {\mu_{p*p} } \right|^{2} }}{{\hbar^{5} }}\sum\limits_{mnjkluvw} {B_{m} } \left\langle {n} \mathrel{\left | {\vphantom {n m}} \right. \kern-0pt} {m} \right\rangle \left\langle {n} \mathrel{\left | {\vphantom {n j}} \right. \kern-0pt} {j} \right\rangle \left\langle {k} \mathrel{\left | {\vphantom {k m}} \right. \kern-0pt} {m} \right\rangle \left\langle {l} \mathrel{\left | {\vphantom {l j}} \right. \kern-0pt} {j} \right\rangle \left\langle {w} \mathrel{\left | {\vphantom {w u}} \right. \kern-0pt} {u} \right\rangle \left\langle {w} \mathrel{\left | {\vphantom {w v}} \right. \kern-0pt} {v} \right\rangle , \\ & \quad \times L_{rm,r*n} \left( { - \omega_{\text{UV}} } \right)D_{rm,rj} \left( {\omega_{1} } \right)L_{r*k,rj} \left( {\omega_{\text{UV}} } \right)D_{pu,pv} \left( {\omega_{2} } \right)L_{p*w,pv} \left( {\omega_{t} } \right) \\ \end{aligned} $$

(19)

$$ \begin{aligned} P_{11}^{\left( 5 \right)} (\omega_{1} ,\omega_{2} ) & = - \frac{{N\xi_{\text{UV}}^{4} \xi_{\text{VIS}}^{{}} \left| {\mu_{r*r} } \right|^{4} \left| {\mu_{p*p} } \right|^{2} }}{{\hbar^{5} }}\sum\limits_{mnjkluvw} {B_{m} } \left\langle {n} \mathrel{\left | {\vphantom {n m}} \right. \kern-0pt} {m} \right\rangle \left\langle {n} \mathrel{\left | {\vphantom {n j}} \right. \kern-0pt} {j} \right\rangle \left\langle {k} \mathrel{\left | {\vphantom {k m}} \right. \kern-0pt} {m} \right\rangle \left\langle {l} \mathrel{\left | {\vphantom {l j}} \right. \kern-0pt} {j} \right\rangle \left\langle {w} \mathrel{\left | {\vphantom {w u}} \right. \kern-0pt} {u} \right\rangle \left\langle {w} \mathrel{\left | {\vphantom {w v}} \right. \kern-0pt} {v} \right\rangle , \\ & \quad \times L_{r*n,rm} \left( {\omega_{\text{UV}} } \right)D_{rj,rm} \left( {\omega_{1} } \right)L_{rj,r*k} \left( { - \omega_{UV} } \right)D_{pu,pv} \left( {\omega_{2} } \right)L_{p*w,pv} \left( {\omega_{t} } \right) \\ \end{aligned} $$

(20)

$$ \begin{aligned} P_{12}^{\left( 5 \right)} (\omega_{1} ,\omega_{2} ) & = - \frac{{N\xi_{\text{UV}}^{4} \xi_{\text{VIS}}^{{}} \left| {\mu_{r*r} } \right|^{4} \left| {\mu_{p*p} } \right|^{2} }}{{\hbar^{5} }}\sum\limits_{mnjkluvw} {B_{m} } \left\langle {n} \mathrel{\left | {\vphantom {n m}} \right. \kern-0pt} {m} \right\rangle \left\langle {n} \mathrel{\left | {\vphantom {n j}} \right. \kern-0pt} {j} \right\rangle \left\langle {k} \mathrel{\left | {\vphantom {k j}} \right. \kern-0pt} {j} \right\rangle \left\langle {l} \mathrel{\left | {\vphantom {l m}} \right. \kern-0pt} {m} \right\rangle \left\langle {w} \mathrel{\left | {\vphantom {w u}} \right. \kern-0pt} {u} \right\rangle \left\langle {w} \mathrel{\left | {\vphantom {w v}} \right. \kern-0pt} {v} \right\rangle . \\ & \quad \times L_{rm,r*n} \left( { - \omega_{\text{UV}} } \right)D_{rm,rj} \left( {\omega_{1} } \right)L_{rm,r*k} \left( { - \omega_{\text{UV}} } \right)D_{pu,pv} \left( {\omega_{2} } \right)L_{p*w,pv} \left( {\omega_{t} } \right) \\ \end{aligned} $$

(21)

In the above polarization components, laser pulses with the subscripts UV and VIS are taken to interact with triiodide and diiodide, respectively.

For convenience, we further group the terms into three classes of signal fields under the assumption of perfect phase-matching conditions

$$ E_{r,r}^{\left( 5 \right)} (\omega_{1} ,\omega_{2} ) = \left( {\frac{{i\omega_{t} l}}{{2\varepsilon_{0} n\left( {\omega_{t} } \right)c}}} \right)\sum\limits_{m = 1}^{4} {P_{m}^{\left( 5 \right)} (\omega_{1} ,\omega_{2} )} , $$

(22)

$$ E_{p,p}^{\left( 5 \right)} (\omega_{1} ,\omega_{2} ) = \left( {\frac{{i\omega_{t} l}}{{2\varepsilon_{0} n\left( {\omega_{t} } \right)c}}} \right)\sum\limits_{m = 5}^{8} {P_{m}^{\left( 5 \right)} (\omega_{1} ,\omega_{2} )\,} , $$

(23)

and

$$ E_{r,p}^{\left( 5 \right)} (\omega_{1} ,\omega_{2} )\, = \left( {\frac{{i\omega_{t} l}}{{2\varepsilon_{0} n\left( {\omega_{t} } \right)c}}} \right)\sum\limits_{m = 9}^{12} {P_{m}^{\left( 5 \right)} (\omega_{1} ,\omega_{2} )\,\,} . $$

(24)

Here, the two subscripts of the signal fields represent sensitivity to the triiodide reactant (subscript r) and diiodide product (subscript p) in the two frequency dimensions, ω ₁ and ω ₂.