## 1 Introduction  
Time-resolved photoelectron emission spectroscopy has been used widely as an efficient technique for investigating nuclear and electronic dynamics in molecular reactions [1; 2; 3; 4; 5; 6; 7; 8]. An ultrashort pump laser pulse initiates a coherent transition between ground and excited states in molecules and the time evolution is subsequently monitored after a variable time delay by a second probe pulse. Advances in synthesizing ultrashort intense pulses [9; 10; 11] allow one to visualize and control molecular reaction processes from femtosecond (1 fs = 10\({}^{-15}\) s) time scales for nuclear vibrations to attosecond (1 as = 10\({}^{-18}\) s) scales for electron motion. Currently, the shortest attosecond pulses with a duration of 43 as are available for such new photoelectron imaging techniques [12]. Electrons have a great potential for probing the time-resolved transient structure of matter via ultrashort photoelectron spectroscopy. One now can envisage the emergence of laser-induced electron interference and diffraction [13; 14; 15; 16; 17; 18; 19; 20] as efficient methods of measuring molecular electronic structure and electron motion for different fixed nuclear configurations. Recently, ultrafast charge migration arising from a coherent population excitation in multiple electronic states has attracted considerable attention in the field of photophysics and photochemistry [21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35]. It has been shown that quantum control of electron flux during intramolecular charge migration can be produced by designing ultrafast laser pulse that prepare the system in selective electronic state [36; 37; 38]. The electron currents are sensitive to the polarization and helicity of the driving pulse, i.e., the symmetry of the excited electronic state [39]. Recent work has shown how to control molecular electric symmetry via charge migration with ultrashort laser [40; 41]. By measuring high-order harmonic generation in ionized iodoacetylene via attosecond electron recollision, one can probe attosecond charge migration [42]. Angular electronic fluxes can also be used to reconstruct electron charge migration in excited benzene by preparing a coherent electronic state [37; 39], which can be monitored in photoelectron momentum spectra by a time-delayed high frequency attosecond pulses [43]. It is also found that molecular photoelectron angular distributions depend on the symmetry of molecular orbitals and molecular bondings, thus allowing to monitor electron coherent and charge migration [44; 45; 46; 47]. Most recently, nonresonant ultrafast X-ray scattering from a molecular wave packet has been used to observe an adiabatic electron transfer in molecules [48]. Time-resolved photoelectron imaging as a variant of time-resolved photoelectron spectroscopy has been used to study molecular dynamics [49; 50]. In the present work, we theoretically studied time resolved photoelectron emission with attosecond resolutions in molecular coherent excitation and charge migration processes. A coherent superposition of quantum states was created by a linearly polarized XUV pump pulse. Subsequently, a time delayed soft X-ray pulse was used to ionize the excited molecules, as illustrated in Figure 1. Simulations were performed on the benchmark molecular ion H\({}_{2}^{+}\) by numerically solving the corresponding time-dependent Schrodinger equation (TDSE). Results show that photoelectron momentum distributions exhibite asymmetric structure and evolve periodically with time, reflecting the electron coherence in molecular charge migration. It was found that the evolution of the photoelectron distributions is dependent on the polarization and helicity of the probe pulse. The paper is arranged as follows: In Section 2, we briefly describe the computation methods. The numerical results obtained by time-dependent quantum electron wave-packet calculations from the corresponding TDSE for a prealigned H\({}_{2}^{+}\) are presented and discussed in Section 3.