## 1 Introduction  
It is nowadays well established that a laser pulse can break the electronic structure symmetry of atoms or molecules (as in Refs. [1; 2; 3; 4; 5; 6; 7]), a process that it often followed by charge migration (as in Refs. [8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18]). Recently, we showed that one can employ a well-designed second laser pulse that restores the symmetry of the electronic structure after application of a first symmetry-breaking pulse [19; 20; 21]. In the molecular case, by \"symmetry\" we mean the molecular point group, e.g., \(D_{6h}\) for benzene, or in general the group of all symmetry operations such as rotations, reflections, and inversions that map the initial electronic eigenfunction (or a set of degenerate eigenfunctions) on itself, with the characteristic phase factors that are called \"characters\". The characters determine the irreducible representations (IRREP) for the given symmetry, e.g., benzene has \(IRREP_{g}=A_{1g}\) and various other \(IRREP_{e}\) (e.g., \(E_{1u}\)) in the electronic ground and excited states, respectively. Symmetry breaking means that the first laser pulse distorts the electronic wave function such that it can no longer be assigned to the original symmetry group. For example, in the case of benzene, the distorted electronic wave function can no longer be assigned to \(D_{6h}\) symmetry. Likewise, symmetry restoration means that the second pulse re-shapes the wave function such that it can be re-assigned to the original symmetry group, irrespective of any possible changes of the IRREP. For example, in the case of benzene, symmetry restoration of electronic structure means that the electronic wave function can be re-assigned to \(D_{6h}\) symmetry. For reference, we use the next paragraph of this Introduction to explain our previous approaches to electronic structure symmetry breaking and symmetry restoration. This long paragraph also serves to introduce some important terminology that is employed in the rest of the paper. The following paragraph points to some related previous work. The remaining paragraphs define the goal of this paper, namely to develop a new strategy for breaking and restoring electronic structure symmetry. The previous approaches to breaking and restoring electronic structure symmetry by two laser pulses have been demonstrated for two model systems, the \({}^{87}Rb\) atom[19] and the oriented benzene molecule[19; 20; 21]. To achieve this goal, we designed two laser pulses centered at \(t_{b}<0\) and at \(t_{r}=-t_{b}>0\), respectively. The time delay \(t_{d}=t_{r}-t_{b}\) between the pulses was chosen sufficiently long, and their durations sufficiently short such that, at the \"central\" time \(t_{c}=(t_{b}+t_{r})/2=0\), the systems evolve in quasi-field-free environment. At the same time, \(t_{d}\) must be chosen short enough such that effects of decoherence (e.g., due to nuclear motions [13; 14; 22]) are negligible. Specifically, the coherence time of benzene exceeds ten femtosecond [13; 14]; we note in passing that this is significantly longer than for the benzene cation where it lasts for just a few femtoseconds [23]. This allows using the model of fixed nuclei to describe the laser induced electron dynamics. In our simulations, the nuclei of benzene were fixed at the global minimum structure of the electronic ground state with \(D_{6h}\) symmetry, adapted from Refs. [17; 24; 25; 26]. Initially, at time \(t_{i}\ll t_{b}\), the systems were in their highly symmetric electronic ground states, \(S_{0}(3)\) and \(D_{6h}\) for \({}^{87}Rb\) and benzene, respectively. The first laser pulse then broke symmetry by partial population transfer from the ground state with initial irreducible representation \(IRREP_{g}(=D^{(0)}\) and \(A_{1g}\), respectively) to an excited state with different \(IRREP_{e}\
eq IRREP_{g}\) (specifically, \(IRREP_{e}=D^{(2)}\) and \(E_{1u}\), respectively).