In this Letter we show an elegant application of the path integral based method developed recently [J. Chem. Phys. 124 (2006) 204111] to calculate the lower bound to the survival probability of a particle diffusing in a harmonic well in presence of an exponential sink. Our lower bound is always finite and well behaved and does not blow up like the approximate survival probability when calculated using second-order cumulant expansion for some decorrelation time scales [J. Phys. Chem. B 110 (2006) 19061]. The problem can be mapped onto the problem of electron transfer dynamics in single protein molecule where the electron transfer rate shows an exponential dependence on the edge to edge distance between the donor and the acceptor. (C) 2010 Elsevier B.V. All rights reserved.