A THEORETICAL STUDY OF THE DYNAMICS OF VIBRATIONAL WAVE PACKETS IN THE lg STATE OF Xez

We present a theoretical description of the dynamics of vibrational wave packets in the lg state of Xe2. As an illustration, a simulation of a picosecond pump excitation-probe ionization laser experiment is carried out. The initial wavepacket is calculated using an explicit modelling of the short-pulse excitation process and propagated for up to 160picoseconds. Evidence of fractional and full revivals of the wavepacket has been found and analyzed. The time delayed ionization signal is simulated using first order perturbation theory and shows clear oscillations corresponding to the temporal development of the wavepacket.


INTRODUCTION
In recent years there has been considerable interest in time-resolved experiments by means of ultrashort laser pulses (for a review, see Ref. [1]). For molecular systems, with bound potential energy surfaces, a pump laser pulse creates a coherent superposition of rovibrational states known as a wavepacket. This wavepacket initially oscillates between the classical inner and outer turning points of the potential with a periodicity determined by the local energy spacing between vibrational states. Because of the anharmonicity of the potential the *Present address: Physical and Theoretical Chemistry Laboratory, South Parks Road, Oxford OXI 3QZ, U.K.
Corresponding author.
wavepacket spreads quickly. However, at specific times interesting partial localization of the wavepacket can take place causing what are known as fractional revivals. Experimentally the evolution of the wavepacket can be monitored using a time-delayed probe laser pulse which ionizes the molecule. The ionization signal will show oscillations that reflect the fact that the vibrational overlap between the wavepacket and vibrational levels of the molecular ion depends on time. If the internuclear equilibrium distance of the excited state is larger than that of the ionic state the probability of ionization will be greater when the wavepacket is near to inner turning point of the potential. To date, vibrational wave packets have been experimentally observed for I2 [2-3,], Na2 [4], CS 2 [5], NaI [6], NaK [7], Li2 [8], Br2 [9],  [10][11][12]. We assume that the molecule is in its ground electronic and vibrational state, e.g., following supersonic expansion. The theoretical description of the pump pulse excitation and subsequent propagation of the wavepacket is accomplished using the FFT split operator method [13][14]. The nuclear wave functions for the ground state g({Rm}, t) and the excited state u({Rm}, t) are evaluated at the spatial grid points {Rm}. The wavefunctions at + 6t are calculated using the short time propagator: where the time evolution operator is split into two parts corresponding to the kinetic and potential energy operators: 9(R, t) is the potential matrix that contains the electronic diagonal potentials and the non-diagonal radiative coupling: We assume that the laser pulse has a Gaussian profile so the lasermolecule interaction is taken to be: where the temporal full width at half maximum is At 2-p(ln 2)1/2, a; is the mean carrier frequency, E0 is the peak amplitude of the electric field and #(R) is the transition dipole moment. The kinetic energy operator is defined in the momentum representation, and the potential operator in coordinate space, using the diagonal representation of the potential matrix 9(R, t). The transformation of the wavefunction back and forth from coordinate to momentum space is performed using a Fast Fourier Transform (FFT) Danielson-Lanczos algorithm. The wavefunction on the ionic potential created by the probe pulse is a function of two different times: the time delay td between the pump and probe pulses and the total time. In the case of low laser intensities a perturbative methodology can be used without loss of accuracy. Following the theoretical approach described by Engel et al. [15]

-i(lq/h)[t-(t'-t)]lu(R, td)
where: where ek is the kinetic energy of the photoelectron. We assume that #zu is constant over the interval [0, e] and zero otherwise. The corresponding probability of photoionization at time delay td between the two pulses is: Pz(e, td) -olim f dRII(R, t, ek)l 2 (7) and the total ionization probability: Pl(td) is proportional to the ionization signal S (td), that is, the yield of photoelectrons or molecular ions detected. In practice the integral over the electron kinetic energies is performed over an equidistant grid of discrete energies ek.

RESULTS AND DISCUSSION
In order to simulate the pump laser pulse we have chosen a central wavelength A 293.3nm with a pulse FWHM of At-3ps (Aco 10cm-1) and intensity I 10W/cm 2. At this wavelength the absorption of two photons is required to reach the g state [12]. The probe pulse has identical parameters but with A 363.4 nm. The transition dipole moments #,g and ]3,ui are not known so we have set them to be independent of the internuclear distance (Condon approximation). In order to evaluate the integral (7)   Morse-like anharmonic systems has been given by Vetchinkin et al. [16]. They showed that the revival times hold the relation p/q Trey with p and q are integers. In this study we have observed revivals at times 1/3 Trev, 1/2 Trey, 2/3 Trev and =Trev.
The quantity that could be measured experimentally is the Xeor photoelectron signal as a function of the time delay between the pulses td. The ionization signal S(ta) is plotted in Figure 2a.

CONCLUSIONS
We have carried out wavepacket calculations to model a picosecond pump-probe experiment on the Xe dimer. A fully quantum mechanical approach has been used to describe the pump pulse excitation, propagation of the wavepacket and the probe pulse ionization processes. In this study we have observed fractional revivals where the wavepacket splits into several sub-wavepackets, each oscillating at a classical frequency but shifted against each other. The predicted ionization signal shows clearly that this system is a prototyical example for vibrational wavepacket observations.