2 chain, a 2D lattice of 12 × 10 sites, and a finite quantum wire with parabolic lateral confinement. Conclusions and discussions are presented in Section 4. body description of quantum transport in an open sys- tem and established a connection between the GME and non-equilibrium Greeen’s functions. They studied simple systems in the steady state regime: a resonant level cou- pled to a a single vibration mode, an interacting dot with two spins, and a two-level bridge. Another recent work by Darau et al. 21 implemented the GME for a benzene single-electron transistor and used exact MES to compute steady state currents within the Markov approximation. 21 The stability diagram and the conductance peaks were obtained and a current blocking due to interferences be- tween degenerated orbitals was noticed. II. GME METHOD AND COULOMB INTERACTION In this section we summarize the main lines of our method. The equations apply both to the lattice and continuous models. The time-dependent transport prob- lem is considered within the partitioning approach which is known both from the pioneering work of Caroli 22 and from the derivation of the GME. Prior to an initial time t$_{0}$ the left lead (L) having a “source” role, and the right lead (R) having a “drain” role, are not connected to the sample and therefore can be characterized by equilibrium states with chemical potentials µ$_{L}$ and µ$_{R}$ respectively. Our aim is to compute the time dependent currents flow- ing through the sample and leads starting at moment t$_{0}$ , when the three subsystems are connected, until a station- ary state is reached. In our previous papers 7,8 we considered the GME method for the RDO of independent electrons in the Fock space. We discussed the transient transport through quantum dots and quantum wires. The contact between the leads and the sample was switched on at a certain ini- tial moment t$_{0}$ . We discussed extensively the occupation of the states within the bias window and the geometrical effects on the transient currents. We described the cou- pling between the sample and the leads via a tunneling Hamiltonian in which we took into account the spatial extension of the wave functions of both subsystems in the contact region. The generic Hamiltonian of the total system consisting of the sample plus the leads is: H$_{l}$ with l = L, R are the Hamiltonians of the leads. We denote by ε$_{ql}$ and ψ$_{ql}$ the single-particle energies and wave functions respectively, for each lead. Using the cre- ation and annihilation operators associated to the single- particle states, c † ql and c$_{ql}$ , we can write In spite of earlier or more recent attempts a complete description of the Coulomb effects in the time-dependent transport is still missing, especially in sample models larger than a few sites. In the present work we com- bine the GME method with the Coulomb interaction in the sample and we analyze the dynamics of the electrons starting with the moment when the leads are coupled to the sample until a steady state is reached. The Coulomb interaction is included in the Hamiltonian of the isolated sample and the interacting MES are calculated with the exact diagonalization method. This means the Coulomb interaction is fully included with no mean field assump- tion or density-functional model. The number of single- electron states (SES) used to define the matrix elements of the Hamiltonian of interacting electrons is sufficiently large such that the MES of interest are convergent. Due to the finite bias window only a limited number of MES participate to the charge transport through the sample, i. e. only those energetically compatible with the elec- trons in the leads. Hence the MES of interest are selected by the chemical potentials in the leads. We calculate the RDO matrix elements in the subspace of these MES using the GME. The electron-electron interaction in the leads is neglected. H$_{S}$ is the Hamiltonian of the sample. In the absence of the interaction the SES have discrete energies denoted as E$_{n}$ and corresponding one-body wave functions φ$_{n}$ ( r ). Using now the creation and annihilation operators for the sample SES, d † n and d$_{n}$ , we can write The second term in Eq. (3) is the Coulomb interaction. In the SES basis the two-body matrix elements are given by: It is well known that the Fock space increases expo- nentially with the number of SES. In addition the time dependent numerical solution of the GME is also com- putational expensive. So at this stage we are limited to describe only few electrons in the system: up to five in a small system, but only up to three in a larger one. The paper is organized as follows. In Section 2 we briefly describe the GME, the inclusion of the Coulomb interaction, and the selection of the MES. Next, in Sec- tion 3, we show results for three models: a short 1D where u ( r − r $^{′}$) is the Coulomb potential. The third term of Eq. (1) is the so-called tunneling Hamiltonian describing the transfer of particles between the leads and the sample: H$_{S}$ = ∑ n E$_{n}$d † $_{n}$d$_{n}$ + 1 2 ∑ nm n $^{′}$m′ V$_{nm,n}$ $^{′}$m′ d † $_{n}$d † $_{m}$d$_{m}$ ′ d$_{n}$ ′ . (3) V$_{nm,n}$ $^{′}$m′ = ∫ d r d r $^{′}$φ∗ $_{n}$( r ) φ ∗ $_{m}$( r $^{′}$) u ( r − r $^{′}$) φ$_{n}$ ′ ( r ) φ$_{m}$ ′ ( r $^{′}$) , (4) H$_{T}$ ( t ) = ∑ l = L,R ∑ n ∫ dqχ$_{l}$ ( t )( T l $_{qn}$c † $_{ql}$d$_{n}$ + h.c. ) . (5) H ( t ) = H$_{L}$ + H$_{R}$ + H$_{S}$ + H$_{T}$ ( t ) . (1) H$_{l}$ = ∫ dq ε$_{ql}$ c † $_{ql}$c$_{ql}$ . (2)