## 2 Laser Control of Ring Currents in a Molecule Having a Single Aromatic Ring
### Laser-Polarization Effects on Ring Currents within the Frozen-Nuclei Approximation
#### 2.2.5 Numerical Demonstration for 2,5-Dichloropyrazine  
The model system with a single aromatic ring adopted for numerical simulations is 2,5-dichloropyrazine (hereafter abbreviated as DCP), whose molecular formula is illustrated in Figure 2a. The molecule is assumed to be pre-oriented, e.g., by the nonadiabatic optical alignment technique [68; 69]. We executed ab initio electronic structure calculations for DCP with the 6-31G* Gaussian basis set [70] by using the quantum chemistry program MOLPRO [71]. The geometry of DCP was optimized in the ground state at the level of the second-order Moller-Plesset perturbation theory (MP2) [72] and the optimized geometry was of \(C_{\mathrm{2h}}\) symmetry. The excited-state properties at this geometry were obtained by the single-point calculation at the state-averaged complete-active-space self-consistent field (CASSCF) [73; 74] level of theory. The active space was composed of ten electrons distributed among eight (four \(\mathrm{a_{u}}\) and four \(\mathrm{b_{g}}\)) orbitals [abbreviated as CASSCF(10,8)]. At the optimized geometry in the ground state \(|G\rangle=\left|1^{1}\mathrm{A_{g}}\right\rangle\), there exists a pair of optically allowed quasi-degenerate excited states \(|X\rangle=\left|3^{1}\mathrm{B_{u}}\right\rangle\) and \(|Y\rangle=\left|4^{1}\mathrm{B_{u}}\right\rangle\) with the energy gap \(2\hbar\Delta\omega=0.44\) eV. The difference between their average energy and the ground-state energy is 9.62 eV. The magnitudes of the transition moments \(\mathbf{\mu}_{X}\) and \(\mathbf{\mu}_{Y}\) are \(1.66 ea_{0}\) and \(1.58 ea_{0}\), respectively, where \(e\) is the elementary charge and \(a_{0}\) is the Bohr radius. The directions of \(\mathbf{\mu}_{X}\) and \(\mathbf{\mu}_{Y}\) are illustrated in Figure 2b and the angle between them is \(\eta_{Y}\)\(-\)\(\eta_{X}=0.35\pi\). The linear combinations of \(|X\rangle\) and \(|Y\rangle\) defined by Equation (2) are the approximate angular momentum eigenstates \(|+\rangle\) and \(|-\rangle\) with \(\lambda\) = 0.98\(\hbar\). Positive (negative) angular momentum designates \(\pi\) electrons traveling anticlockwise (clockwise) around the ring in Figure 2a. We performed electronic WP simulations for linear (\(\beta=\pi/4\)) and circular (\(\beta=0\) and \(\pi/2\)) polarizations. The directions of the linear polarization vectors \(\mathbf{e}^{(+)}\) and \(\mathbf{e}^{(-)}\) evaluated from Equation (31) are depicted in Figure 2b. For circular polarizations (\(\mathbf{e}_{+1}\) and \(\mathbf{e}_{-1}\)), we set \(\delta=0\). Instead of a rectangular envelope, we assumed a \(\sin^{2}\) function: \(f(t)=\sin^{2}\left(\pi t/t_{f}\right)\) for \([0,t_{f}]\) and otherwise zero. The common laser parameters for the above four types of excitations were \(t_{f}=7.26\) fs, \(\omega=9.62\) eV/\(\hbar\) [vacuum UV (VUV) light with a wavelength of 129 nm], and \(\varphi=0\). In analogy to the full-excitation pulse in the rectangular-envelope case, the peak field strength \(E^{0}\) was chosen so that the pulse area \(\Omega F(t_{f})\) (equal to \(\Omega t_{f}/2\) for a \(\sin^{2}\) envelope) was \(2\arccos\left[-\left(2\Delta\omega/\overline{\Omega}\right)^{2}\right]\) under the constraint \(2|\Delta\omega|\leq\overline{\Omega}\). The values of \(E^{0}\) so determined for \(\mathbf{e}^{(+)}\), \(\mathbf{e}^{(-)}\), \(\mathbf{e}_{+1}\), and \(\mathbf{e}_{-1}\) were 5.99, 9.80, 7.22, and 7.22 G\(\mathrm{V}\mathrm{m}^{-1}\), respectively. Since \(|\mid\mu_{X}\mid\simeq|\mid\mu_{Y}\mid\) for DCP, the condition \(|\Omega_{X}|\simeq|\Omega_{Y}|\) approximately holds for circular polarizations. For the four types of excitations, we numerically integrated Equation (9) without the RWA by the fourth-order Runge-Kutta method [75]. The time increment for the numerical integration was 0.024 fs (2.4 as). Figure 3 displays the temporal behavior of \(l_{z}(t)\) for the four types of excitations.