Two-photon circular dichroism

Two-photon circular dichroism Tinoco (1975) arises in chiral systems due to the differential absorption of two photons, of which at least one is circularly polarized De Boni et al. (2008). In this sense it can be seen as the nonlinear extension of ECD. The observable, the anisotropy of the two-photon transition strength, is proportional to the two-photon rotatory strength:

\begin{align} ^nR^{\rm TPCD}(\omega) & = & -b_1 [{\mathcal{B}}_1(\omega)]_{0n} - b_2 [{\mathcal{B}}_2(\omega)]_{0n} -b_3 [{\mathcal{B}}_3(\omega)]_{0n} \end{align}

(1)

where $b_1$ , $b_2$ and $b_3$ are numbers, combinations of the analogous polarization and set-up-related coefficients $F$ , $G$ and $H$ given for two-photon absorption. The molecule-related parameters ${\mathcal{B}}_1$ , ${\mathcal{B}}_2$ and ${\mathcal{B}}_3$ take the form:

\begin{align} [{\mathcal{B}}_1 (\omega)]_{0n} & = & \frac{1}{\omega ^3} {\mathcal{M}}_{\rho \sigma }^{{\rm {p}},0n} (\omega) {\mathcal{P}}_{\rho \sigma}^{{\rm p}*,0n}(\omega), \\ [{\mathcal{B}}_2 (\omega)] _{0n} & = & \frac{1}{2 \omega ^3} {{\mathcal{T}}_{\rho \sigma }^{ + ,0n}} (\omega) {\mathcal{P}}_{\rho \sigma }^{{\rm p}*,0n}(\omega), \\ [{\mathcal{B}}_3 (\omega)] _{0n} & = & \frac{1}{\omega ^3} {{\mathcal{M}}_{\rho \rho }^{{\rm p},0n}} (\omega) {\mathcal{P}}_{\sigma \sigma}^{{\rm p}*,0n}(\omega), \end{align}

(2)

and they are therefore appropriate contractions of generalized two-photon second-rank tensors. Indeed, these tensors are defined (for the general case of two photons of different frequency) as follows:

\begin{align} {\mathcal{P}}_{\alpha \beta}^{{\rm p},0n}(\omega_1,\omega_2) & = & \frac{1}{\hbar} \sum_{m} \Big \{ \frac{(\hat{\mu}^{\rm p}_{\alpha})_{0m} (\hat{\mu}^{\rm p}_{\beta})_{mn} } {\omega_{m0} - \omega_1} + \frac{(\hat{\mu}^{\rm p}_{\beta})_{0m} (\hat{\mu}^{\rm p}_{\alpha})_{mn} } {\omega_{m0} - \omega_2} \Big \}, \\ {\mathcal{M}}_{\alpha \beta }^{{\rm p},0n}(\omega_1,\omega_2) & = & \frac{1}{\hbar} \sum_{m} \Big \{ \frac{(\hat{\mu}^{\rm p}_{\alpha})_{0m} (\hat{m}_{\beta})_{mn} } {\omega_{m0} - \omega_1} + \frac{(\hat{m}_{\beta})_{0m} (\hat{\mu}^{\rm p}_{\alpha})_{mn} } {\omega_{m0} - \omega_2} \Big \}, \\ {\mathcal{T}}_{\alpha \beta }^{+ ,0n}(\omega_1,\omega_2) & = & \frac{1}{\hbar} \varepsilon_{\beta \rho \sigma} \sum_{m} \Big \{ %\frac{(\hat{T}_{\alpha \rho}^+)_{0m} (\hat{\mu}_{\sigma})_{mn} } {\omega_{m0} - %\omega_1} + \frac{(\hat{\mu}_{\sigma})_{0m} (\hat{T}_{\alpha \rho}^+)_{mn} } \frac{(\hat{T}_{\alpha \rho}^+)_{0m} (\hat{\mu}^{\rm p}_{\sigma})_{mn}} {\omega_{m0} - \omega_1} + \frac{(\hat{\mu}^{\rm p}_{\sigma})_{0m} (\hat{T}_{\alpha \rho}^+)_{mn} } {\omega_{m0} - \omega_2} \Big \}, \end{align}

(3)

where the velocity form of the dipole operator

\hat{\mu}^{\rm p}_\alpha = - \frac{e}{m_e} \sum_i {\hat p}_{i\alpha}.

(4)

and that of the quadrupole operator (mixed form)

\hat{T}_{\alpha \beta}^+ = - \frac{e}{m_e} \sum_i ({\hat r}_{i\alpha} {\hat p}_{i\beta} + {\hat p}_{i\alpha} {\hat r}_{i\beta}).

(5)

Within the formalism of response theory, the second-rank tensors of interest are obtained from the single residues of appropriate quadratic response functions. The quadratic response functions of relevance for two-photon circular dichroism are

\begin{align} \langle \langle \hat{\mu}^{\rm p}_\alpha;\hat{\mu}^{\rm p}_\beta,V^{\omega_n} \rangle \rangle_{\omega_1,\omega_2} & \Rightarrow & {\mathcal{P}}^{{\rm p},0n}_{\alpha \beta}(\omega_1,\omega_2), \\ \langle \langle \hat{\mu}^{\rm p}_\alpha;\hat{m}_\beta,V^{\omega_n} \rangle \rangle_{\omega_1,\omega_2} & \Rightarrow & {\mathcal{M}}^{{\rm p},0n}_{\alpha \beta}(\omega_1,\omega_2), \\ \varepsilon_{\beta \rho \sigma} \langle \langle %\hat{T}^+_{\alpha\rho};\hat{\mu}_\sigma,V^{\omega_n} \rangle \hat{T}^+_{\alpha\rho};\hat{\mu}^{\rm p}_\sigma,V^{\omega_n} \rangle \rangle_{\omega_1,\omega_2} & \Rightarrow & \mathcal{T}^{+,0n}_{\alpha \beta}(\omega_1,\omega_2), \end{align}

(6)

where $V^{\omega_n}$ is an arbitrary operator (corresponding to the excitation vector to the state $n$ ). Single residues of quadratic response functions are efficiently and accurately computed nowadays with a number of wave function models. Nevertheless, DFT has been used almost exclusively in the few theoretical studies of two-photon circular dichroism that were published.

References¶

Tinoco, I. (1975). Two-photon circular dichroism. J. Chem. Phys., 62, 1006. 10.1063/1.430566
De Boni, L., Toro, C., & Hernández, F. E. (2008). Synchronized double L-scan technique for the simultaneous measurement of polarization-dependent two-photon absorption in chiral molecules. Opt. Lett., 33, 2958. 10.1364/OL.33.002958