Two-photon circular dichroism

Two-photon circular dichroism [Tinoco75] arises in chiral systems due to the differential absorption of two photons, of which at least one is circularly polarized [DBTHernandez08]. 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:

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

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:

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

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:

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

where the velocity form of the dipole operator

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

and that of the quadrupole operator (mixed form)

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

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

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

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.