  We investigate the deexcitation of the $^{229}$Th nucleus via the excitation
of an electron. Detailed calculations are performed for the enhancement of the
nuclear decay width due to this so called electron bridge (EB) compared to the
direct photoemission from the nucleus. The results are obtianed for triply
ionized thorium by using a B-spline pseudo basis approach to solve the Dirac
equation for a local $x_\alpha$ potential. This approach allows for an
approximation of the full electron propagator including the positive and
negative continuum. We show that the contribution of continua slightly
increases the enhancement compared to a propagator calculated by a direct
summation over bound states. Moreover we put special emphasis on the
interference between the direct and exchange Feynman diagrams that can have a
strong influence on the enhancement.
