  The quantum dynamics of two distant H atoms excited by ultrashort and
spatially shaped laser pulses is studied by the numerical solution of the
non-Born-Oppenheimer time-dependent Schr\"odinger equation within a
three-dimensional (3D) model, including the internuclear distance R and the two
z coordinates of the electrons, z1 and z2. The two 1D hydrogen atoms, A and B,
are assumed to be initially in their ground states with a large (but otherwise
arbitrary) internuclear separation of R = 100 a.u. (5.29 nm). Two types of a
spatial envelope of a laser field linearly po- larized along the z-axis are
considered: (i) a broad Gaussian envelope, such that atom A is excited by the
laser field predominantly, and (ii) a narrow envelope, such that practically
only atom A is excited by the laser field. With the laser carrier frequency
{\omega} = 1.0 a.u. and the pulse duration tp = 5 fs, in both cases an
efficient energy transfer from atom A to atom B has been found. The ionization
of atom B achieved mostly after the end of the laser pulse is close to or even
higher than that of atom A. It is shown that with a narrow spatial envelope of
the laser field, the underlying mechanisms of the energy transfer from A to B
and the ionization of B are the Coulomb attraction of the laser driven electron
by the proton of atom B and a short-range Coulomb repulsion of the two
electrons when their wave functions significantly overlap in the domain of atom
B. In the case of a broad Gaussian spatial envelope of the laser field, the
opposite process also occurs, but with smaller probability: the energy is
transferred from the weakly excited atom B to atom A, and the ionization of
atom A is also induced by the electron-electron repulsion in the domain of atom
A due to a strong overlap of the electronic wave functions.
