## 2 Laser Control of Ring Currents in a Molecule Having a Single Aromatic Ring
### Beyond the Frozen-Nuclei Approximation: Nonadiabatically Coupled Vibronic Dynamics
#### 2.3.3 Angular Momentum Expectation Value and Vibrational Spectrum
The frequency spectrum of \(\psi_{X}(\mathbf{Q},t)\) after the nonadiabatic transition from \(|Y\rangle\) to \(|X\rangle\) can be obtained by the Fourier transform of its autocorrelation function [84]:  
\[\sigma_{X}(\omega)\equiv\text{Re}\!\int_{t_{i}}^{t_{i}+u}dte^{(i\omega-1/\tau) (t-t_{i})}\int d\mathbf{Q}\psi_{X}^{*}(\mathbf{Q},t_{i})\psi_{X}(\mathbf{Q},t). \tag{34}\]  
The parameter \(\tau\) was introduced to smooth the spectra and set to 39.6 fs, which is longer than the vibrational periods of the breathing and distortion modes (28.8 and 21.2 fs). The values of \(t_{i}\) for \(\mathbf{e}^{(+)}\)  
Figure 5: Expectation value of the electronic angular momentum \(l_{z}(t)\) in DCP irradiated by (**a**) linearly and (**b**) circularly polarized VUV laser pulses calculated from nuclear WP simulations. In panel (**a**), the expectation values for \(\mathbf{e}^{(+)}\) and \(\mathbf{e}^{(-)}\) are denoted by the solid and dotted lines, respectively; in panel (**b**), those for \(\mathbf{e}_{+1}\) and \(\mathbf{e}_{-1}\) are denoted by the solid and dotted lines, respectively. The laser pulses fully decay at \(t=7.26\) fs. Reprinted from [56] with permission of the American Chemical Society. \(\mathbf{e}^{(-)}\), \(\mathbf{e}_{+1}\), and \(\mathbf{e}_{-1}\) were 14.0, 10.0, 34.0, and 13.0 fs, respectively, and \(u=99.1\) fs for all of them. The zero of \(\omega\) was set to the minimum of the lower PES. The frequency spectra for the four types of excitations are depicted in Figure 7. For \(\mathbf{e}^{(+)}\), the maximum of \(\sigma_{X}(\omega)\) is identified at \(\widetilde{v}\sim\) 1400 cm\({}^{-1}\) and another peak emerges at \(\widetilde{v}\sim\) 2500 cm\({}^{-1}\) in Figure 7a; for \(\mathbf{e}^{(-)}\), the strongest peak of \(\sigma_{X}(\omega)\) is located at \(\widetilde{v}\sim\) 2500 cm\({}^{-1}\) and a couple of strong peaks are also found at \(\widetilde{v}>\) 3000 cm\({}^{-1}\). The wave numbers of 1400, 2500, and 3000 cm\({}^{-1}\) are almost equal to those of the lowest three vibrational states in \(|G\rangle\) owing to the analogy between \(|G\rangle\) and \(|X\rangle\) in the PES around its minimum. The frequency spectra for linear polarizations in Figure 7a corroborate that at \(t>t_{i}\), \(\psi_{X}(\mathbf{Q},t)\) mainly consists of low (high) vibrational states for \(\mathbf{e}^{(+)}\) [\(\mathbf{e}^{(-)}\)]. The spectral features for circular ones in Figure 7b are very similar to each other: The primary peaks of \(\sigma_{X}(\omega)\) for \(\mathbf{e}_{+1}\) and \(\mathbf{e}_{-1}\) are both at \(\widetilde{v}\sim\) 1400 cm\({}^{-1}\), while the intensities of the other peaks are a little stronger in the latter case. This indicates that \(\psi_{X}(\mathbf{Q},t)\) is composed of the same frequency components for \(\mathbf{e}_{+1}\) and \(\mathbf{e}_{-1}\) in the time domain after the nonadiabatic transition. Figure 6: Expectation value of the normal coordinates \(\mathbf{Q}(t)\) of the breathing and distortion modes in DCP irradiated by (**a**) linearly and (**b**) circularly polarized VUV laser pulses. In panel (**a**), the expectation values for \(\mathbf{e}^{(+)}\) and \(\mathbf{e}^{(-)}\) are denoted by the solid and dotted lines, respectively; in panel (**b**), those for \(\mathbf{e}_{+1}\) and \(\mathbf{e}_{-1}\) are denoted by the solid and dotted lines, respectively. The laser pulses fully decay at \(t=\) 7.26 fs. Reprinted from [56] with permission of the American Chemical Society. The remarkable difference in the vibrational amplitude between \(\mathbf{e}^{(+)}\) and \(\mathbf{e}^{(-)}\) suggests that ring currents in aromatic molecules such as DCP irradiated by linearly polarized laser pulses can be distinguished experimentally by vibrational spectroscopy in the frequency domain without time-resolved measurements. This suggestion may be useful for rapid identification of molecular chirality.