## 2 Laser Control of Ring Currents in a Molecule Having a Single Aromatic Ring
### Basic Ide of Laser-Driven Ring Currents: Molecular Symmetry and Angular Momentum Eigenstates  
First of all, we briefly summarize the mechanism of ring currents in benzene excited by a circularly polarized laser pulse [51] as an example. Benzene belongs to the \(D_{\rm 6h}\) point group at its equilibrium geometry and its \(C_{6}\) axis is chosen as the \(z\) axis. Its highest occupied and lowest unoccupied molecular orbitals (HOMO and LUMO) are \(\rm e_{1g}\) and \(\rm e_{2u}\) orbitals, respectively. As for multielectron states constructed from MOs, there exist doubly degenerate \({}^{1}\)E\({}_{\rm 1u}\) excited states whose major components are the HOMO-LUMO excitations. By approximating the hexagonal structure of benzene with a complete cylindrical ring, its symmetry becomes \(D_{\rm 6h}\) and the \(z\) component of electronic angular momentum is quantized in units of the Dirac constant \(\hbar\). The doubly degenerate \({}^{1}\)E\({}_{\rm 1u}\) excited states can be regarded as the eigenstates of the angular momentum operator \(\hat{l}_{z}\) with the quantum number \(m=\pm 1\). [55; 56; 57; 58] We denote the \({}^{1}\)E\({}_{\rm 1u}\) states with \(m=+1\) and \(-1\) by \(\left|1^{1}\rm E_{1u+}\right>\) and \(\left|1^{1}\rm E_{1u-}\right>\)respectively. The angular momentum eigenstates \(\left|1^{1}\mathrm{E}_{1u\pm}\right\rangle\) have a complex wave function that can be expressed as a linear combination of two real ones \(\left|1^{1}\mathrm{E}_{1u}\right\rangle\) and \(\left|1^{1}\mathrm{E}_{1 uy}\right\rangle\):  
\[\left|1^{1}\mathrm{E}_{1u\pm}\right\rangle=2^{-1/2}\Big{(}\left|1^{1}\mathrm{E }_{1u}\right\rangle\pm i\Big{|}1^{1}\mathrm{E}_{1 uy}\Big{\rangle}\Big{)}. \tag{1}\]  
Recalling that complex atomic orbitals (AOs) \(\left|2\mathrm{p}_{+1}\right\rangle\) and \(\left|2\mathrm{p}_{-1}\right\rangle\) are angular momentum eigenstates of an electron in a hydrogen atom, the relation in Equation (1) is similar to that between the complex AOs and real ones \(\left|2\mathrm{p}_{x}\right\rangle\) and \(\left|2\mathrm{p}_{y}\right\rangle\). The transition moments \(\left\langle 1^{1}\mathrm{A}_{1g}|\hat{\mathbf{\mu}}|1^{1}\mathrm{E}_{1 ux}\right\rangle\) and \(\left\langle 1^{1}\mathrm{A}_{1g}|\hat{\mathbf{\mu}}|1^{1}\mathrm{E}_{1u_{y}}\right\rangle\), where \(\left|1^{1}\mathrm{A}_{1g}\right\rangle\) is the ground state and \(\hat{\mathbf{\mu}}\) is the electric dipole moment operator, have the same magnitude and are perpendicular to each other in the \(xy\) plane. The \(x\) and \(y\) axes can thus be taken parallel to \(\left\langle 1^{1}\mathrm{A}_{1g}|\hat{\mathbf{\mu}}|1^{1}\mathrm{E}_{1 ux}\right\rangle\) and \(\left\langle 1^{1}\mathrm{A}_{1g}|\hat{\mathbf{\mu}}|1^{1}\mathrm{E}_{1 uy}\right\rangle\), respectively. When a circularly polarized laser pulse propagates along the \(z\) axis, the spin angular momentum of a photon selects either \(\left|1^{1}\mathrm{E}_{1u+}\right\rangle\) or \(\left|1^{1}\mathrm{E}_{1u-}\right\rangle\) and accordingly \(\pi\) electrons travel clockwise or anticlockwise around a ring depending on the sign of \(m\). This is the origin of the unique correspondence between the direction of ring currents and the helicity of circular polarization. One can immediately find it impossible to rotate \(\pi\) electrons in benzene by a linearly polarized laser pulse, which has no spin angular momentum. We next assume that the symmetry of benzene is lowered, e.g., by introducing functional groups and/or replacing some carbon atoms in its ring with heteroatoms. The degeneracy between \(\left|1^{1}\mathrm{E}_{1u+}\right\rangle\) and \(\left|1^{1}\mathrm{E}_{1u-}\right\rangle\) (\(\left|1^{1}\mathrm{E}_{1 ux}\right\rangle\) and \(\left|1^{1}\mathrm{E}_{1 uy}\right\rangle\)) is then broken and they split into a pair of quasi-degenerate states. We denote the real wave functions of the states closer to \(\left|1^{1}\mathrm{E}_{1 ux}\right\rangle\) and \(\left|1^{1}\mathrm{E}_{1 uy}\right\rangle\) by \(\left|X\right\rangle\) and \(\left|Y\right\rangle\), respectively.