  Recently developed series representations of the Boltzmann operator are used
to obtain Quantum Mechanical results for the matrix elements, <x| exp(-{\beta}
H)|x>, of the imaginary time propagator. The calculations are done for two
different potential surfaces: one of them is an Eckart Barrier and the other
one is a double well potential surface. Numerical convergence of the series are
investigated. Although the zeroth order term is sufficient at high
temperatures, it does not lead to the correct saddle point structure at low
temperatures where the tunneling is important. Nevertheless the series
converges rapidly even at low temperatures. Some of the double well
calculations are also done with the bare potential (without Gaussian
averaging). Some equations of motion related with bare potentials are also
derived. The use of the bare potential results in faster integrations of
equations of motion. Although, it causes lower accuracy in the zeroth order
approximation, the series show similar convergence properties both for Gaussian
averaged calculations and the bare potential calculations. However, the series
may not converge for bare potential calculations at low temperatures because of
the low accuracy of zeroth order approximation. Interestingly, it is found that
the number of saddle points of <x| exp(-{\beta} H)|x> increases as the
temperature is lowered. An explanation of observed structures at low
temperatures remains as a challenge. Besides, it has implications for the
quantum instanton theory of reaction rates at very low temperatures.
