  We study numerically, in the framework of the Cooper approach from 1956,
mechanisms of pair formation in a model of La-based cuprate superconductors
with longer-ranged hopping parameters reported in the literature at different
values of center of mass momentum. An efficient numerical method allows to
study lattices with more than a million sites. We consider the cases of
attractive Hubbard and d-wave type interactions and a repulsive Coulomb
interaction. The approach based on a frozen Fermi sea leads to a complex
structure of accessible relative momentum states which is very sensitive to the
total pair momentum of static or mobile pairs. It is found that interactions
with attraction of approximately half of an electronvolt give a satisfactory
agreement with experimentally reported results for the critical superconducting
temperature and its dependence on hole doping. Ground states exhibit d-wave
symmetries for both attractive Hubbard and d-wave interactions which is
essentially due to the particular Fermi surface structure and not entirely to
an eventual d-wave symmetry of the interaction. We also find pair states
created by Coulomb repulsion at excited energies above the Fermi energy and
determine the different mechanisms of their formation. In particular, we
identify such pairs in a region of negative mass at rather modest excitation
energies which is due to a particular band structure.
