For laser acceleration of electrons (and other charged particles) by lasers to the TeV energy range in vacuum, the scheme of trapping electrons in spatially moving and accelerated intensity gradients or minima of laser fields, the single electron motion in standing wave fields is evaluated in details numerically. Acceleration of the minima results in the acceleration of the electrons as expected from global results of the nonlinear forces. If half-wave length laser pulses propagating in vacuum are used the relativistic exact solutions are derived and evaluated. A disadvantage is the lateral motion requiring a large laser focus. For TeV electron energy, MJ KrF-laser pulses are necessary and the acceleration length is about 10 cm.