## 3 Coherent \(\pi\)-Electron Angular Momentum and Current of a Molecule Having Two Aromatic Rings
### Quantum Localization of Coherent \(\pi\)-Electron Angular Momentum
#### 3.3.3 Simulation Results of Maintaining Angular Momentum Localization  
Figure 16 shows the quantum control results of the second step for the localization, i.e., maintaining the unidirectional (anticlockwise or clockwise) rotation of \(\pi\) electrons around the \(L\)- or \(R\)-axis. After the localization has been completed, the localized angular momentum undergoes free propagation if there is no control field. Figure 16a shows the time dependence of the expectation values of the \(\pi\)-electron angular momentum under the field-free condition after the [AO] localization is achieved. Here, the initial time \(t=0\) was set to be the time when the first step of localization has been achieved. The expectation value of the \(\pi\)-electron angular momentum around the \(L\)-axis, \(l_{L}(t)\), starts to oscillate, and that around the \(R\)-axis, \(l_{R}(t)\), deviates from zero and oscillates in the opposite way. Figure 16b shows the results of the second step control to the initial localization demonstrated in Figure 15a. The control time was again set at \(t_{f}=30\) fs. Figure 16b indicates that \(l_{L}(t)\) keeps positive values under the optimal field. It is possible to keep the localization at any longer times \(t_{f}>30\) fs. Although there exists small deviation, \(l_{R}(t)\) is kept at around zero. The time-averaged expectation value of the target states defined as \(J_{L}=\int_{0}^{t_{f}}dt\langle\Psi(t)\big{|}\big{|}\Psi(t)\rangle/t_{f}\) was calculated to be 0.85, while that of the free propagation was 0.22. Figure 16c shows the absolute values of the optimal electric fields, \(E_{X}\), \(E_{Y}\), and \(E_{Z}\) in the frequency domain. The main peaks of all components appear at around 6.9 eV just above the highest eigenstates \(b_{2}\). The optimal field \(\mathbf{E}(t)\) in the time domain is also shown in the inserted figure to indicate the phase relations. It can be seen that compared to \(E_{X}(t)\), \(E_{Z}(t)\) is shifted by \(\pi\), and \(E_{Y}(t)\) is shifted by \(\pi/2\), and these relations are almost kept during the control time. This means that the resultant optimal field has an elliptically polarized character. We now briefly mention how to observe the localization of \(\pi\)-electron rotations around one of the benzene rings. This may be carried out by separately measuring the ring currents and magnetic fields generated on the two phenol groups since the two benzene rings are not planar but tilted with the dihedral angle of \(108.8^{\circ}\).  
Figure 16: (**a**) Expectation values of \(\pi\)-electron angular momentum for the free rotations after the [AO] localization. (**b**) Results of the optimal control for maintaining the localization. The control time is \(t_{f}=30\) fs. (**c**) Absolute values of the optimal electric field. The inset figure is the field in the restricted time domain. Reprinted from [91] with permission of the American Chemical Society.  
The simulations in this section were performed using a simplified model omitting both the effects of vibrational motions and those of heat bath modes. In real aromatic ring molecules these effects influence on \(\pi\)-electron rotations. [PERSON] et al. recently found that intramolecular vibrational effects make significant contributions to dephasing of \(\pi\)-electron rotations under finite temperature conditions [95] as well as bath mode effects in condensed phases beyond the Markov approximation [96]. It has been also shown in Section 2.3 that nonadiabatic transitions strongly influence \(\pi\)-electron rotations [97]. Application of optimal control theory to such open systems is one of the important subsequent issues.