## 3 Coherent \(\pi\)-Electron Angular Momentum and Current of a Molecule Having Two Aromatic Rings
### Two Ring Components of \(\pi\)-Electron Angular Momentum and Current
#### 3.1.3 Coherent Ring Current  
The ring current passing through a surface \(S\) at time \(t\) is defined as follows:  
\[\left\langle J(t)\right\rangle\equiv\int_{S}d^{2}r_{\perp}\overrightarrow{n} _{\perp}\cdot\left\langle\hat{I}(\overrightarrow{r},t)\right\rangle \tag{39}\]  
Here, \(\overrightarrow{n}_{\perp}\) is a unit vector perpendicular to surface \(S\), and \(\left\langle\hat{I}(\overrightarrow{r},t)\right\rangle\equiv\left\langle \Psi(t)\middle|\hat{I}(\overrightarrow{r})\middle|\Psi(t)\right\rangle\), which can be expressed as:  
\[ \begin{split}\left\langle\hat{J}(\overrightarrow{r},t)\right\rangle& =2 in\sum_{\alpha<\beta}\text{Im}\rho_{\beta\alpha}(t)\sum_{ij}\left(\delta_{ ab}c_{a^{\prime}i}^{*}c_{b^{\prime}j}+\delta_{a^{\prime}b^{\prime}}c_{ai}^{*}c_{bj} \right)\chi_{i}^{*}\hat{J}(\overrightarrow{r})\chi_{j}\\ &=\frac{2n\epsilon}{m_{e}}\sum_{\alpha<\beta}\text{Im}\rho_{\beta \alpha}(t)\sum_{ij}\left(\delta_{ab}c_{a^{\prime}i}^{*}c_{b^{\prime}j}+\delta_{a ^{\prime}b^{\prime}}c_{ai}^{*}c_{bj}\right)\chi_{i}^{*}\overrightarrow{\
abla} \chi_{j}\end{split} , \tag{40}\]  
where \(\hat{J}(\overrightarrow{r})=(eh/2m_{e}i)\left(\overrightarrow{\
abla}- \overleftarrow{\
abla}\right)\) is the current density operator. Here, \(\overrightarrow{\
abla}\) (\(\overleftarrow{\
abla}\)) denotes the nabra operating on the AO on the right-hand (left-hand) side. Surface integration in Equation (39) is carried out over a planar plane \(S\) perpendicular to a C-C bond of an aromatic ring. Since the \(L\) and \(R\) rings are not round but consist of nonequivalent C-C bonds, we introduce the bond current \(\left\langle J(t)\right\rangle_{ij}\) from the nearest neighbor atoms at sites \(j\) to \(i\), which is given in terms of inter-atomic bond current (IABC) \(J_{ij}^{IABC}\) as  
\[\left\langle J(t)\right\rangle_{ij}=\frac{2 neh}{m_{e}}\sum_{\alpha<\beta}\text {Im}\rho_{\beta\alpha}(t)\left[\delta_{ab}\left(c_{a^{\prime}i}^{*}c_{b^{ \prime}j}-c_{b^{\prime}i}^{*}c_{a^{\prime}j}\right)+\delta_{a^{\prime}b^{ \prime}}\left(c_{ai}^{*}c_{bj}-c_{bi}^{*}c_{aj}\right)\right]J_{ij}^{IABC}, \tag{41}\]  
where  
\[J_{ij}^{IABC}=\int_{S}d^{2}r_{\perp}x_{i}^{*}\overrightarrow{n}_{\perp}\cdot \overrightarrow{\
abla}\chi_{j}\left[\equiv J_{ij}^{IABC(S\ at\ center)}\right] \tag{42}\]  
Here, surface \(S\) is set to be perpendicular to the C\({}_{i}\) and C\({}_{j}\) bond at the center. Equation (42) is given in the 2p carbon AO basis set, \(\left\{\chi_{i}\right\}\), as [87]:  
\[J_{ij}^{IABC(S\ at\ center)}=\frac{r_{ij}\cos\Theta}{2a_{\beta}^{6}}\int_{0} ^{\infty}dr\frac{r^{3}\exp\left(-2\sqrt{r_{ij}^{2}/4+r^{2}}/a_{2}\right)}{ \sqrt{r_{ij}^{2}/4+r^{2}}}. \tag{43}\]  
Here, \(\Theta\) = \(\pi\) for the bond current of the chemical bond belonging to one of the two aromatic rings, \(L\) or \(R\), while \(\Theta\) = \(\Theta_{d}\) for the bridge bond current of bond C\({}_{i}\) - C\({}_{j}\) with a dihedral angle between the two rings \(\Theta_{d}\). C\({}_{i}\) (C\({}_{j}\)) refers to the bridge carbon belonging to the \(R\) (\(L\)) ring. We now define an effective ring current along ring \(K\), \(\left\langle J(t)\right\rangle_{K}\), by taking the average over all of the bond currents as [87]:  
\[\left\langle J(t)\right\rangle_{K}\approx\frac{1}{N_{K}}\sum_{(ij)\subset K} ^{N_{K}}\left\langle J(t)\right\rangle_{ij}, \tag{44}\]  
where \(N_{K}\) is the number of bonds of ring \(K\).