  The electron-screening acceleration of laboratory fusion reactions at
astrophysical energies is an unsolved problem of great importance to
astrophysics. That effect is modeled here by considering the fusion of
hydrogen-like atoms whose electron probability density is used in Poisson's
equation in order to derive the corresponding screened Coulomb potential
energy. That way atomic excitations and deformations of the fusing atoms can be
taken into account. Those potentials are then treated semiclassically in order
to obtain the screening (accelerating) factor of the reaction. By means of the
proposed model the effect of a superstrong magnetic field on laboratory
Hydrogen fusion reactions is investigated here for the first time showing that,
despite the considerable increase in the cross section of the $% dd$ reaction,
the $pp$ reaction is still too slow to justify experimentation. The proposed
model is finally applied on the $H^{2}(d,p) H^{3}$ fusion reaction describing
satisfactorily the experimental data although some ambiguity remains regarding
the molecular nature of the deuteron target. Notably, the present method gives
a sufficiently high screening energy for Hydrogen fusion reactions so that the
take-away energy of the spectator nucleus can also be taken into account.
