This paper shows that an exact calculation for transition probability can make some systems deviate seriously from Fermi's golden rule. The paper also shows that the corresponding exact calculation of the phonon-induced hopping rate for a deuteron in the Pd?D system plus many-body electron screening, proposed by Ichimaru, can explain experimental factors observed in the Pd?D system. It also predicts that the perfect and low dimensional forms of the Pd lattice are very important factors for the phonon-induced hopping rate enhancement of deuterons in the Pd?D system.