  The rate constants for recombination and exchange processes are studied in
terms of two different flux correlation approaches: one is the Yamamoto
approach, which is based on the linear response theory, and the other is the
Miller one. Using those approaches we consider two exactly solvable cases,
i.e., the free particle and the parabolic potential models. Since the rate
constants for recombination and exchange processes are calculated by Laplace
transforms of the flux correlation functions, the two approaches give different
results. In the present calculation, we find that the rate constant in the
Yamamoto approach is larger than that in the Miller approach by about 40% at
low temperature ($\sim 100$ K) and high pressure ($\sim 1$ GPa). The difference
is considerable in the region where quantum effects dominate.
