  We investigate the effect of inertia on barrierless electronic reactions in
solution by suggesting and examining different probability distribution
functions (PDF) of relevant Brownian functionals associated with the lifetime
and reactivity of the process. Activationless electronic reaction in solution
can be modeled as a free Brownian motion with inertial term in the underdamped
regime. In this context we suggest several important distribution functions
that can characterize the reaction kinetics. Most of the studies on Brownian
functional which has vast potential application in diverse fields, are confined
in the overdamped regime. To the best of our knowledge, we are attempting first
time to incorporate the much important inertial effects on the study of
different PDFs related with Brownian functionals of an underdamped Brownian
motion with time dependent drift and diffusion coefficients using celebrated
backward Fokker-Planck and path decomposition methods. We have explored
nontrivial scaling behaviour of different PDFs and calculated explicitly the
critical exponents related with the asymptotic limits in time.
