## 1 Introduction  
Transition-metal dichalcogenides (TMDs) are layered compounds that are a testbed to study electron correlations and electron-lattice interactions in two dimensions [1; 2; 3]. In particular, 1T-TaS\({}_{2}\) shows a variety of phases including charge density waves (CDWs) accompanied by periodic lattice distortions (PLDs) at low temperatures [4; 5; 6; 7; 8]. In the lowest-temperature phase below 180 K, electrons experience a lock-in to the commensurate CDW (CCDW) that forms a \(\sqrt{13}\times\sqrt{13}\) hexagram structure (see Figure 1), which has been also referred to as the Star of David. The CCDW phase is insulating as revealed by early resistivity and susceptibility measurements [1], and this is confirmed by recent experiments of angle-resolved photo-emission spectroscopy (ARPES) [9]. This is believed to be a Mott insulator [10] (see also [11; 12] for alternative scenarios), and the possibility of spin liquid is under debate [13; 14].  
1T-TaS\({}_{2}\) has also attracted renewed interest for its nonequilibrium phenomena driven by photoexcitation, and the electron-lattice cooperative dynamics have been extensively studied [15; 16; 17; 18]. Photoinduced phase transitions have been observed between the different CDWs in the phase diagram [19]. In addition, photoexcitation [20; 21; 22; 23; 24] and a gate-voltage pulse [25; 26; 27; 28] have realized transitions to metastable states, which are often referred to as hidden states [29; 30; 31]. Since these states are quite stable, the phase transitions have potential application to nonvolatile ultrafast memory devices [26; 28; 32].  
Theoretical studies for these intriguing nonequilibrium phenomena are quite limited except phenomenological Ginzburg-Landau theories [33; 34; 35]. This is partly because modeling the electronic structure has not been settled yet for the CCDW phase in 1T-TaS\({}_{2}\). Constructing an empirical tight-binding model dates back to [PERSON] [36], who proposed a three-orbital Hamiltonian for describing the normal state without PLD. [PERSON] et al. [37] studied the case in the presence of the PLD and discussed the band structure well below the Fermi energy. [PERSON] and [PERSON] [38] included the spin-orbit interaction and showed that this isolates one narrow band near the Fermi energy. They conjectured the Mott insulating phase of this band, but it remains an open problem to examine its validity.  
In this paper, we extend the empirical tight-binding model of [PERSON] and [PERSON] [38] by adding the on-site Coulomb interaction, and obtain a band gap at the Fermi level within a mean-field approach. The gap formation is in line with their conjecture [38] and consistent with the electronic property of the CCDW phase of 1T-TaS\({}_{2}\). We then treat the PLD of the hexagram shape as a dynamical variable and formulate the coupled equations of motion for the electrons and the PLD. On the basis of these equations, we study the electron-lattice cooperative dynamics induced by an ultrashort laser pulse within a time-dependent mean-field approximation. On a short time scale, the lattice distortion is almost frozen, but the laser input induces coherent electron dynamics and the charge density wave decreases and even disappears. On a longer time scale, the lattice motion sets in and causes a further decrease in the charge-density-wave order even after the laser irradiation.