## 2 Laser Control of Ring Currents in a Molecule Having a Single Aromatic Ring
### Beyond the Frozen-Nuclei Approximation: Nonadiabatically Coupled Vibronic Dynamics
#### 2.3.1 Selection of Effective Vibrational Modes and Method of Nuclear Wave Packet Propagation  
To pick out effective vibrational degrees of freedom, excited-state geometry optimization was executed for \(|X\rangle\) and \(|Y\rangle\) at the CASSCF(10,8) level of theory. The optimized geometries of \(|X\rangle\) and \(|Y\rangle\) are of \(C_{\rm 2h}\) symmetry as that in \(|G\rangle\). Hence, the displacements from the optimized geometry of \(|G\rangle\)  
Figure 3: Expectation value of the electronic angular momentum \(l_{z}(t)\) in DCP irradiated by **(a)** linearly and **(b)** circularly polarized vacuum UV (VUV) laser pulses calculated from electronic WP simulations at the optimized geometry of \(|G\rangle\). In panel **(a)**, the expectation values for \(\mathbf{e}^{(+)}\) and \(\mathbf{e}^{(-)}\) are denoted by the solid and dotted lines, respectively. The peaks at \(t=5.4\) fs are indicated by the vertical broken line. In panel **(b)**, the expectation values for \(\mathbf{e}_{+1}\) and \(\mathbf{e}_{-1}\) are denoted by the solid and dotted lines, respectively. The peaks at \(t=4.5\) and \(7.0\) fs are indicated by the vertical broken lines. The laser pulses fully decay at \(t=7.26\) fs. Modified from [58] with permission of the Physical Society of the Republic of China. to that of \(|X\rangle\) and \(|Y\rangle\) are totally symmetric. What is more, vibrational modes that nonadiabatically couple two \({}^{1}\)B\({}_{\text{u}}\) states \(|X\rangle\) and \(|Y\rangle\) are totally symmetric A\({}_{\text{g}}\) modes as well. We therefore selected two A\({}_{\text{g}}\) normal modes with large potential displacements and calculated the relevant nonadiabatic coupling matrix element between \(|X\rangle\) and \(|Y\rangle\); they are the breathing and distortion modes, whose vibrational vectors are illustrated in Figure 2c,d, respectively. The harmonic wave numbers of the breathing and distortion modes in \(|G\rangle\) are 1160 and 1570 cm\({}^{-1}\), respectively. We omitted nonadiabatic couplings between the ground and two excited states because \(|X\rangle\) and \(|Y\rangle\) lie far above \(|G\rangle\) and there exists no potential crossing between them near the Franck-Condon region. The adiabatic PESs of \(|X\rangle\) and \(|Y\rangle\) with respect to the normal coordinates \(\mathbf{Q}\) of the breathing and distortion modes were evaluated at the CASSCF(10,8) level of theory. As will be shown in Section 2.3.4, there exists an avoided crossing (not a conical intersection) between the PESs, although whether it is an avoided crossing or a conical intersection does not affect the qualitative discussion below. At the crossing point, the gap between the two PESs is about 190 cm\({}^{-1}\). We also calculated the PESs at the level of the second-order CAS perturbation theory (CASPT2) [76; 77] and confirmed that the geometry at the crossing point remains unchanged when dynamical electron correlation is included, while the PESs are lowered by about 3 eV. The initial nuclear WP is the vibrational ground-state wave function of \(|G\rangle\). The system is then electronically excited by a laser pulse \(\mathbf{E}(t)\) defined by Equation (4). The subsequent propagation of the WP was carried out in the diabatic basis. Rigorous adiabatic-to-diabatic transformation requires the line integral of the derivative coupling matrix, which is in general path dependent [78; 79]. We instead used the quasi-diabatization scheme implemented in MOLPRO, which is based on an analysis of configuration interaction vectors [80]. The total wave function of DCP was then expanded in terms of the three diabatic states, each of which is a linear combination of the adiabatic states \(|G\rangle\), \(|X\rangle\) and \(|Y\rangle\).