With the development of the laser technology, the interaction of intense ultrashort laser pulses with material has been explored many times both in experiment and theory [1-3]. Compared to atoms in intense fields, molecules exhibit versatile phenomena due to their additional degrees of freedom [4-6]. Water has attracted much attention with a variety of experimental and theoretical techniques due to its various physical states [7-9]. Water cluster is always used as a vehicle for better understanding how bulk properties of water arise from the isolated water molecule as the cluster size is increased. Recently, the experimental study of the photo-induced real time dynamics of a water cluster in intense sub-50 fs vacuum ultraviolet laser pulses has been performed and three relevant time scales are distinguished [10]. The rates of proton transfer along a hydrogen bond in a water cluster cation have been investigated by means of a direct ab initio molecular dynamics method [11]. In the present work, applying the time-dependent local-density approximation coupled with molecular dynamics method (TDLDA-MD) we explore the irradiation dynamics of water tetramer in intense femtosecond laser pulses with different laser frequencies. TDLDA-MD has become a full-fledged method to explore the real-time dynamics of multi-electron system under intense laser field [12-18]. Our aim is to investigate the irradiation effect on a water cluster under ultrashort laser pulses. The water tetramer owning the ring structure with S4 symmetry is expected to behave rather differently from both the trimer and the pentamer [19]. The result presented here is in succession to our foregoing study [14, 18]. We expect that the present calculated results would help in better understanding of the process of photo-induced dynamics of water clusters in laser fields.

In the TDLDA-MD framework [12], the degrees of the molecule are the wave functions of valence electrons and the coordinates of the ionic cores. For a water tetramer, it is composed of 32 valence electrons and 12 ions. A norm-conserving pseudopotential including a local part and a non-local part is applied to describe the interaction between ions and electrons [20]. The time-dependent Kohn-Sham (TDKS) equation [21] for single-particle orbitals ϕ_j (r, t) is applied to describe the motion of valence electrons. An average-density self-interaction correction (ADSIC) [22] is used to put the single-particle energies at their correct values. Ions are treated as classical particles and the ionic motion of the water tetramer is described within standard molecular dynamics. The ground state wavefunctions are achieved through the damped gradient method [12] and the TDLDA equations are solved numerically on a grid 3D coordinate space. A constant time step Δt = 0.000605 fs is used for the evolution of electrons and ions. The absorbing boundary condition is applied to avoid periodic reflecting electrons [23]. The laser field is defined as E_las(t) = E₀f_las(t) cos(ωt), where E₀ and ω denote the maximum amplitude of electric field and the laser frequency. f_las(t) is the pulse profile taking the form of cos² and the Full Width Half Maximum (FWHM) is 8 fs. The number of escaped electrons is defined as N_esc(t) = N(t=0) − ∫_vd³rρ(r, t), where ρ(r, t) denotes the electronic density within the finite volume V centered around ions. The ionization out of state j is defined as n_esc,j(t) = ∫_vd³r|ϕ_j (r, t)|² = ∫_vd³rρ_j(r, t) and ρ_j(r,t) is the single-particle Kohn-Sham density. The real-time evolution of the dipole moment is given by D_i(t) = ∫d³rr_iρ(r,t), with i = x, y, z. The optical absorption strength is the imaginary part of which is obtained by Fourier transform the dipole momentum D(t) [24].

In the present study, the water tetramer is placed in a cuboid box with the size of 96 × 96 × 96 spatial grid points with each mesh spacing of 0.412a₀ to ensure a convergent calculation. The left panel in Fig. 1 shows the optimal configuration of the water tetramer owning the ring structure with S₄ symmetry, which is nearly planar structure.

In the water tetramer four water molecules act as both proton donors and acceptors. Thus, there are four hydrogen bonds in water tetramer which are O₁H₄, O₁₀H₉, O₅ H₁₂ and O₇H₂. Moreover, there are eight OH bonds which could be classified into two categories, i.e. free OH bonds (O₁H₃, O₅H₆, O₇H₈ and O₁₀H₁₁) and bonded OH bonds (O₁H₂, O₅H₄, O₇H₉ and O₁₀H₁₂). Due to the degeneracy of spin up and spin down states, there are 16 occupied states. The right panel in Fig. 1 displays the calculated electronic density of the respective orbitals from the HOMO level and to the most deep occupied electronic level (the occupied single-particle state owning the lowest energy) in the xy plane. The red and gray circles represent the oxygen and hydrogen ions respectively. One can see that the electronic density distributes intensively around the water tetramer in the numerical box indicating that the computation space is large enough to launch the dynamic simulation.

We calculate the ionization potential I_p of the water tetramer. It corresponds to the energy of the last occupied electron state of the water tetramer. Table 1 compiles the ionization potential of the water tetramer obtained by different methods, together with experimental value. It is shown that our calculated ionization potential accords well with the experimental value, indicating the accuracy of our simulation.

It is known that the irradiation dynamics can be implemented by adjusting pulse parameters such as frequency, pulse number, intensity and phase [15, 28]. We explore the non-adiabatic dynamics of the water tetramer in different laser fields by varying the laser frequencies, which are ω = 1.55 eV, 5.44 eV and 13.0 eV, respectively. It should be noted that in the former two cases the laser frequencies are far below all eigenmodes in the optical absorption spectra of the water tetramer, while ω = 13.0 eV falls into the continuum region and is above the ionization potential. In all cases the laser polarization is along the x axis. The Keldysh parameter [29] takes the form of , where I_p and U_p are the ionization potential and the ponderomotive energy respectively. In these three cases γ = 0.54, 1.89 and 4.52 respectively. It is known that the Keldysh parameter separates the multiphotonic domain (γ >1) and the tunnelling regime (γ <1). Therefore, the first case belongs to the tunnelling ionization and the latter two cases pertain to the multiphotonic ionization. The laser pulse is turned off at 16 fs. In order to observe the dynamics of the water tetramer during the relaxation time, we track the calculations until 35 fs in the case of ω = 1.55 eV and until 55 fs in the cases of ω = 5.44 eV and 13.0 eV.

Figure 2 shows the time evolutions of the ionization N_esc of the water tetramer in three different laser fields by varying laser frequencies. Figure 3 gives the snapshots of the evolving process of the water tetramer in three cases as shown in Fig. 2. In the first case of ω = 1.55 eV, one can see that the ionization increases quickly from about 4 fs to 12 fs. There are about two electrons emitted when the laser pulse is switched off at 16 fs and no more electrons are emitted in the remaining time. Thus, the water tetramer keeps on the gentle stretching vibration coupled with the rotation mode as shown in Fig. 3(a). In addition, the calculated average stretching frequency of OH bonds in the present case is about 3243 cm^-1. Compared with the experimental values of 3714 cm^-1 and 3416 cm^-1 corresponding to the free OH stretching frequency and the bonded OH stretching frequency [30], the present average stretching frequency is red shifted due to the fact that the OH bond lengths are enlarged as a result of the ionization.

In the second situation of ω= 5.44 eV as shown in Fig. 2, there are about 3.7 electrons emitted rapidly from about 4 fs to 16 fs, making the water tetramer unstable. Thus, one can see from Fig. 3(b) the breaks of free OH bonds O₅H₆ and O₇H₈ indicating the stronger response of free OH bonds than that of bonded OH bonds. Although the ionization is almost saturated after the laser pulse is turned off, there are about 0.4 electrons emitted additionally from 40 fs to 48 fs (see the red dashed line in Fig. 2). This is because the hydrogen ions (H₆ and H₈) carrying the electrons move toward the boundary of the numerical box (as shown in Fig. 3(b)). Furthermore, it can be seen in Fig. 3(b) that after the breaks of two free OH bonds, the remaining parts keep on vibrating and rotating.

For the third case of ω = 13.0 eV, it can be seen from Fig. 2 that the ionization increases rapidly from about 2 fs and almost five electrons are emitted directly until t = 16 fs leaving a highly excited water tetramer which then experiences Coulomb explosion. The snapshots in Fig. 3(c) show that the Coulomb explosion results in fast breaking of all OH bonds. Furthermore, each escaping hydrogen ion carries away the surrounding electron cloud leading to the continuous electron emission during the relaxation time.

In general, it can be seen in Fig. 2 that in the respective cases the ionization takes place predominantly in a time slot of about 8 fs around the peak of the laser pulse. The ionization is enhanced and the corresponding fragmentation effect is found more notably when increasing the laser frequency especially for the frequency falling in the resonant region of the absorption spectra. This pattern is similar to that found in the metal cluster and water molecule subjected to laser pulses [12, 17]. While it is different from the reaction of fullerene C₆₀ in the laser pulse [31]. For C₆₀, it is found that the fragmentation effect is greater at low laser frequency rather than at the resonant frequency.

Figure 4 exhibits the time evolutions of the dipole moment of the water tetramer in the x direction in three cases. In the first case as shown in Fig. 4(a), the associated dipole signal in the x direction closely follows the laser profile. It attains an appreciable amplitude but fades away as soon as the laser pulse is turned off and keeps on slight oscillation in the relaxation time. This pattern is similar to the response of the water monomer and the metal cluster irradiated in the laser field with the frequency far below the resonant region [17, 23].

For the case of ω = 5.44 eV, Fig. 4(b) exhibits clearly that from 0 fs to 8 fs the profile of the dipole moment is similar to that of the laser pulse, while then the dipole response shows the damping on account of the quick ionization shown in the red dashed line in Fig. 3. After switching off the laser pulse, because of the higher excitation than in the first case the dipole signal holds on to the stronger vibration than that in the first case.

In the case of ω = 13.0 eV, the behavior of the dipole signal shown in Fig. 4(c) is quite different. It follows the laser amplitude from 0 fs to 6 fs, while then undergoes a much stronger damping in the course of time. This is mainly due to the high energetic excitation resulting in the rapid electron emission. Even when the pulse is turned off, because of the continuous electron emission and the ionic motion relating to the breaks of the OH bonds in the relaxation time, the amplitude of dipole signal fails and the dipole signal keeps on the much stronger oscillation than that in the former two cases.

In short, Fig. 4 exhibits visually how the electronic features of the water tetramer are dynamically influenced by the irradiation of laser pulses.

Figure 5 shows the stationary single-electron spectrum with the amount of depletion of each occupied single-particle state. The positions at the y-axis stand for the 16 Kohn-Sham energies of the occupied single-particle states visualized in Fig. 1. The lowest position of -28.07 eV and the highest position of -10.37 eV represent the most deep occupied level and the HOMO level respectively. The length of the horizontal bar denotes the final depletion from each occupied single-particle state, i.e. the ionization from the respective single-particle state. As there are 16 single-particle states and some of states are degenerate with different values of m (three 1d states and two 1p states, and so on), the lines in Fig. 5 are plotted in different colors and types. It can be seen in Fig. 5(a) that in the case of ω = 1.55 eV, i.e. in the tunnelling ionization case, all levels except the deepest occupied electronic levels contribute the emission equivalently. For the multiphotonic ionization cases as shown in Figs. 5(b) and 5(c), emission comes from all levels even the deepest occupied levels and the HOMO level contributes the most. Furthermore, one can also see in Fig. 5(c) that the emission from the deepest occupied levels are comparable with that from other states lower than the HOMO level. This can be traced back to the fact that the laser frequency above I_p is high enough to remove electrons from all states.

In summary, applying the TDLDA-MD approach, we simulate the non-adiabatic dynamics of the water tetramer induced by intense laser pulses. We especially focus on the influence of the laser frequency on the electronic and ionic reactions of the water tetramer subject to the laser fields. Three laser frequencies are considered, i.e. ω = 1.55 eV (far below the resonant region), ω= 5.44 eV (below the resonant region) and ω = 13.0 eV (in the resonant region and above the I_p). Three typical reaction channels, which are normal vibration with enlarged OH bonds, free OH bonds breaking and the pure Coulomb explosion are visually presented respectively. The study shows that the ionization is enhanced and the corresponding fragmentation effect as well as the damping of the dipole moment are found more notably when increasing the laser frequency especially for the frequency falling in the resonant region of the absorption spectra. The study of the level depletion reveals that in the tunnelling ionization case all levels except the deepest occupied electronic levels contribute the emission equivalently while in the multiphotonic ionization case the emission from the HOMO level is more than that from other states. In addition, the emission ratio from the deepest occupied levels is enhanced when the laser frequency is higher than the ionization potential. Our study illustrates that the various irradiation dynamics of the water tetramer can be realized by controlling the laser frequency. We hope the results will help the further experimental study.