## 2 Model, Concept, Theory and Methods
### Conceptual Background for the New Symmetry Restoration Strategy
(**Bottom**) One-electron density of the oriented benzene molecule in the ground state labeled \(A_{1g}\) (symmetry \(D_{6h}\), irreducible representation \(IRREP_{g}=A_{1g}\)). The first circularly-polarized laser pulse is centered at time \(t_{b}=-4.5T\). It breaks symmetry by exciting the ground state to the superposition labeled “\(A_{1g}-iE_{1u}\)” of the ground state and an excited state with \(IRREP_{e}=E_{1u}\). This laser excitation is symbolized by the first red arrow. The superposition state has symmetry \(C_{s}\). (**Middle**) Periodic charge migration from “\(A_{1g}-iE_{1u}\)” via “\(A_{1g}+iE_{1u}\)” back to “\(A_{1g}-iE_{1u}\)”, with period \(T=504\) as. This is symbolized by the two curved arrows, with snapshots of the one-electron densities for state “\(A_{1g}-iE_{1u}\)” (left) at central time \(t=t_{c}=0\) (and also at \(t=T,2T\), etc.) and for state “\(A_{1g}+iE_{1u}\)” (right) at time \(t=T/2\) (and also at \(3T/2,5T/2\), etc.) The second laser pulse centered at \(t_{T}=4.5T\) restores \(D_{6h}\) symmetry by transferring the superposition state to the excited state with \(IRREP_{e}=E_{1u}\). This laser excitation is symbolized by the second red arrow. (**Top**) One-electron density of the excited target state labeled \(E_{1u}\). The time delay \(t_{d}=t_{r}-t_{b}=NT\) between the centers of the laser pulses must be equal to an integer number \(N\) of periods \(T\) of charge migration, with precision of few attoseconds. Here, \(N=9\). Any attempts to restore electronic structure symmetry at delay times that correspond to incomplete cycles of charge migration are useless—this is indicated by the crossed-out arrows. The Gaussian shape functions (dashed lines) and the \(x\)- and \(y\)-components of the electric field (red and green continuous lines) of the circularly-polarized laser pulses are also sketched. All densities were created using detCl@ORBKIT [43; 44; 45] and plotted using Matplotlib [46]. but different field strength \(\epsilon_{b}\
eq\epsilon_{r}\). More specifically, we show that, to satisfy the condition in Equation (29), the field strength of the second pulse must be larger than for the first pulse, \(\epsilon_{b}>\epsilon_{r}\), as illustrated in Figure 1.0 Figure 1 also shows that the \(y\)-components are equal to zero at the times \(t_{b}\) and \(t_{r}\) of maximum intensities, i.e., at these instants, the laser pulses point along the same direction \(\vec{e}_{x}\). We close this subsection by adding information about the consequences of the different polarizations of the first laser pulse for cases where the excited state in the superposition (Equation (5)) is degenerate. Here, this is explained for the example of benzene's doubly degenerate excited state with \(IRREP_{e}=E_{1u}\). As shown in Ref. [17], the laser polarization may be used as a control knob for selective excitation of specific target states \(|\Psi_{ex}\rangle\equiv|\Psi_{E_{1 ux}}\rangle,|\Psi_{cy}\rangle\equiv|\Psi_{E_ {1 uy}}\rangle\) or \(|\Psi_{e+}\rangle\equiv|\Psi_{E_{1u+}}\rangle,|\Psi_{e-}\rangle\equiv|\Psi_{E_ {1u-}}\rangle\) out of benzene's lowest set of degenerate eigenstates with \(IRREP_{e}=E_{1u}\). Specifically, linearly \(x\)- and \(y\)-polarized pulses excite \(|\Psi_{g}\rangle\) to \(|\Psi_{e}\rangle=|\Psi_{ex}\rangle\) and \(|\Psi_{cy}\rangle\), respectively. This selectivity is due to the symmetry rules for the \(x\)- and \(y\)-components of the transition dipole matrix elements [17; 24; 25; 26]  
\[ \begin{array}{c}d\equiv d_{x,gx}=\langle\Psi_{g}|d_{x}|\Psi_{ex}\rangle=d_{y,gy}=\langle\Psi_{g}|d_{y}|\Psi_{ey}\rangle\
eq 0,\\ d_{x,gy}=\langle\Psi_{g}|d_{x}|\Psi_{ey}\rangle=d_{y,gx}=\langle\Psi_{g}|d_{y} |\Psi_{ex}\rangle=0.\end{array}  \tag{38}\]  
Circularly right (+) or left (-) polarized laser pulses excite \(|\Psi_{g}\rangle\) to  
\[|\Psi_{e+}\rangle=\frac{1}{\sqrt{2}}\left(|\Psi_{ex}\rangle+i|\Psi_{ey}\rangle \right), \tag{39}\]  
and  
\[|\Psi_{e-}\rangle=\frac{1}{\sqrt{2}}\left(|\Psi_{ex}\rangle-i|\Psi_{ey}\rangle \right), \tag{40}\]  
respectively.