Two-photon circular dichroism

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{array}{rcl} ^{n} R^{T P C D} (ω) & = & - b_{1} [B_{1} (ω)]_{0 n} - b_{2} [B_{2} (ω)]_{0 n} - b_{3} [B_{3} (ω)]_{0 n} \end{array}

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 $B_{1}$ , $B_{2}$ and $B_{3}$ take the form:

(50)#

\begin{array}{rcl} [B_{1} (ω)]_{0 n} & = & \frac{1}{ω^{3}} M_{ρ σ}^{p, 0 n} (ω) P_{ρ σ}^{p *, 0 n} (ω), \\ [B_{2} (ω)]_{0 n} & = & \frac{1}{2 ω^{3}} T_{ρ σ}^{+, 0 n} (ω) P_{ρ σ}^{p *, 0 n} (ω), \\ [B_{3} (ω)]_{0 n} & = & \frac{1}{ω^{3}} M_{ρ ρ}^{p, 0 n} (ω) P_{σ σ}^{p *, 0 n} (ω), \end{array}

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{array}{rcl} P_{α β}^{p, 0 n} (ω_{1}, ω_{2}) & = & \frac{1}{ℏ} \sum_{m} {\frac{({\hat{μ}}_{α}^{p})_{0 m} ({\hat{μ}}_{β}^{p})_{m n}}{ω_{m 0} - ω_{1}} + \frac{({\hat{μ}}_{β}^{p})_{0 m} ({\hat{μ}}_{α}^{p})_{m n}}{ω_{m 0} - ω_{2}}}, \\ M_{α β}^{p, 0 n} (ω_{1}, ω_{2}) & = & \frac{1}{ℏ} \sum_{m} {\frac{({\hat{μ}}_{α}^{p})_{0 m} ({\hat{m}}_{β})_{m n}}{ω_{m 0} - ω_{1}} + \frac{({\hat{m}}_{β})_{0 m} ({\hat{μ}}_{α}^{p})_{m n}}{ω_{m 0} - ω_{2}}}, \\ T_{α β}^{+, 0 n} (ω_{1}, ω_{2}) & = & \frac{1}{ℏ} ε_{β ρ σ} \sum_{m} {\frac{({\hat{T}}_{α ρ}^{+})_{0 m} ({\hat{μ}}_{σ}^{p})_{m n}}{ω_{m 0} - ω_{1}} + \frac{({\hat{μ}}_{σ}^{p})_{0 m} ({\hat{T}}_{α ρ}^{+})_{m n}}{ω_{m 0} - ω_{2}}}, \end{array}

where the velocity form of the dipole operator

{\hat{μ}}_{α}^{p} = - \frac{e}{m_{e}} \sum_{i} {\hat{p}}_{i α} .

and that of the quadrupole operator (mixed form)

{\hat{T}}_{α β}^{+} = - \frac{e}{m_{e}} \sum_{i} ({\hat{r}}_{i α} {\hat{p}}_{i β} + {\hat{p}}_{i α} {\hat{r}}_{i β}) .

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{array}{rcl} ⟨ ⟨ {\hat{μ}}_{α}^{p}; {\hat{μ}}_{β}^{p}, V^{ω_{n}} ⟩ ⟩_{ω_{1}, ω_{2}} & \Rightarrow & P_{α β}^{p, 0 n} (ω_{1}, ω_{2}), \\ ⟨ ⟨ {\hat{μ}}_{α}^{p}; {\hat{m}}_{β}, V^{ω_{n}} ⟩ ⟩_{ω_{1}, ω_{2}} & \Rightarrow & M_{α β}^{p, 0 n} (ω_{1}, ω_{2}), \\ ε_{β ρ σ} ⟨ ⟨ {\hat{T}}_{α ρ}^{+}; {\hat{μ}}_{σ}^{p}, V^{ω_{n}} ⟩ ⟩_{ω_{1}, ω_{2}} & \Rightarrow & T_{α β}^{+, 0 n} (ω_{1}, ω_{2}), \end{array}

where $V^{ω_{n}}$ 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.