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In this paper we derive a general expression for the transmission coefficientusing the method of reactive flux for a particle coupled to a harmonic bathsurmounting a one dimensional inverted parabolic barrier. Unlike Kohen andTannor [J. Chem. Phys. 103, 6013 (1995)] we use a normal mode analysis wherethe unstable and the other modes have a complete physical meaning. Importantlyour approach results a very general expression for the time dependenttransmission coefficient not restricted to overdamped limit. Once the spectraldensity for the problem is know one can use our formula to evaluate the timedependent transmission coefficient. We have done the calculations with timedependent friction used by Xie [Phys. Rev. Lett 93, 180603 (2004)] and also theone used by Kohen and Tannor [J. Chem. Phys. 103, 6013 (1995)]. Like theformula of Kohen and Tannor our formula also reproduces the results oftransition state theory as well as the Kramers theory in the limits t->0 andt->infinity respectively

Exact analytical evaluation of time dependent transmission coefficient from the method of reactive flux for an inverted parabolic barrier