Theory for the ultrafast structural response of optically excited small clusters: Time dependence of the ionization potential.

Combining an electronic theory with molecular dynamics simulations we present results for the ultrafast structural changes in small clusters. We determine the time scale for the change from the linear to a triangular structure after the photodetachment process Ag−3 → Ag3. We show that the timedependent change of the ionization potential reflects in detail the internal degrees of freedom, in particular coherent and incoherent motion, and that it is sensitive to the initial temperature. We compare with experiment and point out the general significance of our results. Typeset using REVTEX 1 The excitation of a cluster by a laser pulse induces time-dependent changes of its electronic and atomic structure. These changes involve bond formation and bond breaking. The understanding of the relaxation mechanisms induced by the excitation is of general interest. In particular, it is of fundamental importance to study how the system approaches equilibrium in order to determine how the time-scales of the relaxation processes can be controlled by varying the experimental conditions. Recently, a pump&probe experiment has been performed on mass selected Ag−3 clusters, which serves as an example for the investigation of the structural relaxation times [1]. The initially negatively charged clusters were neutralized through photodetachment by the pump pulse and after a delay time ∆t ionized by the probe pulse in order to be detected. Due to the remarkable differences in the equilibrium geometries of the ground states of Ag−3 , which is linear [2], and Ag3, which consists of an obtuse isosceles triangle [3], the ultrashort photodetachment process puts the neutralized trimer in an extreme nonequilibrium situation. As a consequence of that, a structural relaxation process occurs. The experimental signal, consisting in the yield of Ag+3 was measured as a function of ∆t and the frequency of the probe laser pulse. For a frequency slightly above the ionization potential (IP) of Ag3 a sharp rise of the signal is observed at ∆t ≃ 750fs. After a maximum is reached, there is a saturation of the signal, which then remains constant for at least 100ps, which is the longest time delay used in the experiment. New features appear for higher frequencies. Again the signal increases sharply, but after reaching a maximum it decreases to a constant value. A preliminary interpretation of these results uses the Franck-Condon-principle [1]. The first laser pulse creates a neutral linear silver trimer which bends and comes to a turning point near the equilateral equilibrium geometry of the positive ion [3]. After rebounding, the neutral trimer starts pseudo-rotating through its three equivalent obtuse isosceles equilibrium geometries. This would explain the saturation behaviour of the signal. However, this would mean that the pseudo-rotations have an extremely long mean life, which seems improbable. Furthermore, this model does not explain why the signal changes as a function of the frequency of the laser pulse. 2 In this paper we perform a theoretical analysis of the physics underlying the ultrafast dynamics of Ag3 clusters produced by photodetachment. In particular, we analyze the time evolution of the ionization potential and the dependence of the dynamics on the initial temperature of the clusters. We show that the experimental results can be explained using a physical picture which can be generally applied to other ultrashort-time processes. In our calculations, we combine molecular dynamics (MD) simulations in the Born-Oppenheimer approximation with a microscopic theory to describe the time-dependent electronic structure of the clusters. In order to determine the potential energy surface (PES) needed for the MD simulations, we start from a Hamiltonian of the form H = HTB+1/2 ∑ i6=j φ(rij), where the tight-binding part HTB is given by HTB = ∑ i,α,σ εiαc + iασciασ + ∑ i6=j,σ α,β Viαjβc + iαciβ. (1) Here, the operator c+iασ (ciασ) creates (annihilates) an electron with spin σ at the site i and orbital α (α = 5s, 5px, 5py, 5pz). εiα stands for the on-site energy, and Viαjβ for the hopping matrix elements. For simplicity, and since the 5s electrons are expected to be rather delocalized, we neglect the intraatomic Coulomb matrix elements. φ(rij) refers to the repulsive potential between the atomic cores i and j. For the distance dependence of the hopping elements and the repulsive potential we use the functional form proposed in Ref. [4]. By diagonalizing HTB, (taking into account the angular dependence of the hopping elements [5]), and summing over the occupied states, we calculate as a function of the atomic coordinates the attractive parts of the electronic ground-state energies E attr and E 0 attr of Ag − 3 and Ag03, respectively. Then, by adding the repulsive part of H we obtain the PES, which we need to perform the MD simulations. In order to determine the forces acting on the atoms we make use of the Hellman-Feynman theorem. Thus, the α-component of the force acting on atom i Fiα = −∂E/∂riα is given by Fiα = − ∑ k occ. 〈k | ∂HTB ∂riα | k〉 − 1 2 ∑ i6=j ∂φ(rij) ∂riα . (2)